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Eigenstate Phase Transitions Bo Zhao A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy Recommended for Acceptance by the Department of Physics Adviser: Professor David A. Huse September 2015 c Copyright by Bo Zhao, 2015. All rights reserved. Abstract Phase transitions are one of the most exciting physical phenomena ever discovered. The understanding of phase transitions has long been of interest. Recently eigenstate phase transitions have been discovered and studied; they are drastically different from traditional thermal phase transitions. In eigenstate phase transitions, a sharp change is exhibited in properties of the many-body eigenstates of the Hamiltonian of a quantum system, but not the thermal equilibrium properties of the same system. In this thesis, we study two different types of eigenstate phase transitions. The first is the eigenstate phase transition within the ferromagnetic phase of an infinite-range spin model. By studying the interplay of the eigenstate thermalization hypothesis and Ising symmetry breaking, we find two eigenstate phase transitions within the ferromagnetic phase: In the lowest-temperature phase the magnetization can macroscopically oscillate by quantum tunneling between up and down. The relaxation of the magnetization is always overdamped in the remainder of the ferromagnetic phase, which is further divided into phases where the system thermally activates itself over the barrier between the up and down states, and where it quantum tunnels. The second is the many-body localization phase transition. The eigenstates on one side of the transition obey the eigenstate thermalization hypothesis; the eigenstates on the other side are many-body localized, and thus thermal equilibrium need not be achieved for an initial state even after evolving for an arbitrary long time. We study this many-body localization phase transition in the strong disorder renormalization group framework. After setting up a set of coarse-graining rules for a general one dimensional chain, we get a simple “toy model” and obtain an almost purely analytical solution to the infinite-randomness critical fixed point renormalization group equation. We also get an estimate of the correlation length critical exponent ν ≈ 2.5. iii Acknowledgements No one can achieve a PhD without others’ help. I am very thankful to have this opportunity of pursuing the PhD in physics department in Princeton. First of all, the honor should be given to my advisor, Professor David Huse. It is he who led me to the palace of physics. Before entering graduate school in Princeton, I was just a student who had some ability in learning in courses and doing homework. One of the most important lessons David has shown me is how to tackle a research topic where there might be no preexisting theory. By doing research with David during these PhD years , I have experienced both exciting and painful moments. But David has taught me, through his own attitude, that when doing research, one should keep cautious, positive and energetic. What typically happens during research is that one keeps trying and failing until stages where a new method, model or direction are found. I am very grateful to learn from David, in terms of both his strong physical intuition and his encouraging personality. Secondly, I would like to thank my colleagues who have helped me in my academic life in my PhD years. In David’s group, I have had many meaningful discussions about physics with Hyungwon Kim and Liangsheng Zhang. Both of them have helped me a lot in digesting the background physics in our research projects. I would also like to thank my colleagues, Aris Alexandradinata and Vedika Khemani who have also given me lots of advice about many academic aspects. Thirdly, PhD’s life is not all about research. I enjoyed these years in Jadwin with all of my friends, including, but not limited to, Chaney Lin, Akshay Kumar, Bin Xu, Yu Shen and many more graduate students who often appeared on the 4th floor in Jadwin. Without them, my life in Jadwin would be much more lonely and much less interesting. There are countless many friends who I enjoyed staying with, so I also would like to thank all of them. iv Last, but definitely not the least, I must give special thanks to my family. The most selfless love and support were consistently given by my parents. Without them, this thesis would have never been completed. v To my parents. vi Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi 1 Introduction 1 1.1 Quantum Thermalization . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Eigenstate Thermalization Hypothesis . . . . . . . . . . . . . . . . . 5 1.3 Many-body Localization . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Eigenstate Phase Transition . . . . . . . . . . . . . . . . . . . . . . . 10 1.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Three ‘Species’ of Schrödinger Cat States in an Infinite-range Spin Model 12 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Infinite Range Transverse Field Ising Model . . . . . . . . . . . . . . 16 2.2.1 Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.2 Energy Level Degeneracy . . . . . . . . . . . . . . . . . . . . . 17 2.2.3 Discrete WKB Method . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 Eigenstate Thermalization . . . . . . . . . . . . . . . . . . . . . . . . 24 vii 2.5 2.6 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.5.1 Thermal Activation and Quantum Tunneling . . . . . . . . . . 26 2.5.2 Paired States and Unpaired States . . . . . . . . . . . . . . . 31 Analytical Calculation of the Thermal Activation and Quantum Tunneling Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 32 Asymptotic Behaviors of Both Transitions Near Both QCP and u = 0 , Γ = 0 Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.7.1 Near QCP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.7.2 Near u = 0 , Γ = 0 Point . . . . . . . . . . . . . . . . . . . . . 43 2.8 Numerical Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3 Strong Disorder Renormalization Group Approach to Many-body Localization Transition 54 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.2 RG Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.2.1 RG Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.2.2 RG Flow of Probability Distributions . . . . . . . . . . . . . . 67 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.3.1 Fixed Point Distribution . . . . . . . . . . . . . . . . . . . . . 68 3.3.2 Critical Exponent . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.3.3 Numerical Evidence . . . . . . . . . . . . . . . . . . . . . . . . 91 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.3 3.4 A Mathematical Proof of the Degeneracy Formula B Estimation of the Magnitude of Disorder 97 103 B.1 Lower Bound of Disorder . . . . . . . . . . . . . . . . . . . . . . . . . 104 B.2 Upper Bound of Disorder . . . . . . . . . . . . . . . . . . . . . . . . . 107 viii C Divergence of First Order Derivative of γ(s̄) at s̄ = s̄min 109 D Discussion on Convexity of Function γ(s̄) 112 Bibliography 116 ix List of Tables 1.1 A list of some properties of the many-body localized phase contrasted with properties of the thermal phases. Table from [19]. . . . . . . . . x 9 List of Figures 2.1 The phase diagram of our model. u is the energy per spin and Γ is the transverse field. The ground state (zero temperature) is indicated by the black (solid) line, with the quantum critical point (QCP) indicated. The green (dot-dashed) line is the thermodynamic phase transition (PT) between the paramagnetic and ferromagnetic (F) phases. There are two dynamical (eigenstate) phase transitions within the ferromagnetic phases, indicated by the blue (dashed) lines. See the text for discussions of the sharp distinctions between these three phases F1, F2 and F3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Two “potential-energy curves” in our model. . . . . . . . . . . . . . . 20 2.3 Constant energy (E) line described by equation (2.25) by setting s̄y = 0 or just by equation (2.22). It determines the WKB turning point xt when the total spin density s̄ is fixed. . . . . . . . . . . . . . . . . . . 27 2.4 A sketch of the entropy Σ(s̄) and the tunneling rate γ(s̄). 29 2.5 A sketch of the difference between thermal activation and quantum . . . . . . tunneling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi 30 2.6 The level-spacing statistics using 100 realizations of H at N = 15 in phase F1 within the even sector. δ< /δ> is the ratio between the smaller level spacing δ< to the larger level spacing δ> for three consecutive eigenenergies in the even sector. f is the relative frequency for each bin in this histogram. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 46 Averages of log (D) in phases F1 and F3, respectively, where D is the ‘eigenstate distance’ defined in the text.. The energy density range we used in F1 is from the first excited state in each sector up to uc − 0.02 where uc is the energy density at the phase boundary between F1 and F2, whereas in F3 we used the phase’s full energy density range. N is the total number of spins varying from 8 to 15. The exponential decrease of D with increasing N indicates thermalization. The error bars come from averaging over 100 realizations. . . . . . . . . . . . . 2.8 48 The mean ᾱn and the standard deviation ∆αn of the quantity αn defined in Eq. (2.81). The number of realizations is 1600 for N = 11 (blue dash-dotted lines) and 100 for N = 15 (red dashed lines). The green (solid) line gives the theoretical quantity α(u, Γ) defined in Eq. (2.32) for the system size N → ∞. . . . . . . . . . . . . . . . . . . . 3.1 49 A sketch of typical RG moves. For example, (a) is to fuse two adjacent thermal (T) blocks into one thermal block, called “TT move”; all others are similar. 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Q∗ (η) up to η = 20 using composite Trapezoidal rule. (a) shows the curve and (b) demonstrates it in a semi-log plot with a linear regression fit. (c) plots the cumulative area under the curve. . . . . . . . . . . . 3.3 80 The eigenfunction f− (η) corresponding to eigenvalue 0.3995: (a) Linear scale (b) Absolute value on log scale. . . . . . . . . . . . . . . . . . . xii 90 3.4 The eigenfunction f− (η) using either the numerical integration directly or the diagonalization. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 91 The cumulative distribution function (Cdf) for both the T-blocks and I-blocks as the total number of blocks N decreases from 107 to 103 and the cutoff Λ grows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 93 The difference of cumulative distribution functions (Cdf) between the T-blocks and I-blocks as the total number of blocks N decreases from 107 to 103 and the cutoff Λ grows. . . . . . . . . . . . . . . . . . . . . 94 D.1 Comparison between numerical and analytical result of the thermal activation and quantum tunneling transition line; the dots are numerical points and the line underneath the dots is the analytical result given by equation (2.47). They fit perfectly well. . . . . . . . . . . . . . . . 113 D.2 Numerical results of ∂γ ∂s̄ as a function of s̄ by sowing different pairs of Γ and u randomly within the entire ferromagnetic phase region. . . . 114 xiii Chapter 1 Introduction Our understanding of the laws of nature has taken giant leaps in the past several centuries, beginning from the time of Galileo. For mechanics, Newton first postulated the classical equations of motion that every macroscopic object should obey, these equations, we now call “Newton’s Laws”. These laws solve nearly every mechanical phenomenon we observe in daily life! However, when we would like to further understand the microscopic structures of thermal behaviors, we must consider many interacting degrees of freedom, too many to explicitly solve for the dynamics, say from Newton’s Laws or from Schrödinger equation for a quantum system. This gave birth to the area of statistical mechanics. Classical statistical mechanics is very successful at explaining equilibrium thermal behavior in daily life, for example the concept of temperature. Thus, at least phenomenologically speaking, we know the solution based on the essence of classical equilibrium statistical mechanics: the ensemble theory. But the foundation of the ensemble theory is to assume a probabilistic nature of so-called microstates. If we want to get the equilibrium probability distributions in ensemble theory purely from Newton’s Laws, either we assume that the initial state is itself an ensemble, or we get the ensemble by including all long time states of the system (ergodicity). Fortunately, 1 people have successfully discovered a more fundamental theory, quantum mechanics, which is itself a probabilistic theory. Quantum statistical mechanics is the product of combining quantum mechanics and statistical mechanics. In quantum statistical mechanics, ensemble theory naturally arises from the probabilistic nature of quantum mechanics, so the equilibrium ensemble can emerge at any specific long time even if the initial state is a pure state. In quantum statistical mechanics, one of the most fundamental questions is whether a closed quantum system equilibrates. This question is by no means as easy as it appears. One of the difficulties is the apparent time reversal symmetry breaking which arises from low entropy initial states. However, the quantum mechanical formalism obeys the time reversal symmetry, through the unitary time evolution of a quantum state. Thus, at least, we need to formulate a more rigorous definition of equilibration called quantum thermalization. In the remaining sections of this chapter, I will briefly review the concepts of quantum statistical mechanics, including, but not limited to, quantum thermalization, eigenstate thermalization hypothesis and many-body localization. 1.1 Quantum Thermalization Consider a closed quantum system with a time independent Hamiltonian H. In quantum mechanics, the system is described by a ket vector |ψi in the Hilbert space. But in quantum statistical mechanics, |ψi is not enough. It only covers the pure states, a subset of the quantum states described by the probability operator (also known as density matrix) ρ. As a probability operator, ρ needs to satisfy the following constraints: ρ† = ρ , tr ρ = 1 , 2 (1.1) and also, the spectrum of ρ is in the interval [0, 1]. After describing the quantum state of the system, what we need next is the law of time evolution. If we choose the Schrödinger picture, the time evolution of ρ(t) as a function of time t is given by the unitary operator e−iHt/~ : ρ(t) = e−iHt/~ ρ(0)eiHt/~ , (1.2) or the differential equation form: i~ d ρ(t) = [H, ρ(t)] . dt (1.3) By establishing both the quantum state and the rule of time evolution, we are able to formulate a definition for quantum thermalization. Roughly speaking, as mentioned earlier, thermalization means that the system goes to equilibrium in the limit of a long time evolution. But if we consider the full probability operator of an entire closed quantum system, the time evolution is unitary; it will never reach equilibrium, because such a state is, by definition, fully determined by initial conditions. In other words, the unitary time evolution “remembers” all information about the initial state. Instead, we consider thermalization in terms of subsystems. And the idea is to use the rest of the full system to serve as a heat bath to thermalize the selected subsystem. When we are talking about thermalization, we need to send our system to the thermodynamic limit where the total degrees of freedom N goes to infinity. In general, the full system can have several extensive conserved quantities, for example, total energy, particle number, total spin, etc. But to keep the discussion simple, we assume the system only has one extensive conserved quantity, the total energy. So if the system goes to thermal equilibrium, the equilibrium state has only one thermodynamic parameter, the temperature T . In order to take the thermodynamic limit, we need 3 a sequence of both the initial states ρN (t = 0) and the Hamiltonian HN labeled by N . In addition, we need to restrict to initial states ρN (0) where the uncertainty in total energy is subextensive, so T has zero uncertainty for N → ∞. Now we consider a subsystem S. The degrees of freedom in the full system F can be decomposed into the degrees of freedom in the subsystem S and the remaining degrees of freedom —called the “bath”— B: F =S⊕B, (1.4) for the sets of degrees of freedom. (It is a tensor product in Hilbert space.) Then the thermodynamic limit corresponds to taking both F and B to infinity but keeping S finite. We consider the probability operator ρS (t) on S (known as the “reduced” density matrix) ρS (t) = trB ρN (t) . (1.5) Thermalization on the subsystem S means that by sending the system to the thermodynamic limit, the probability operator of the subsystem S goes to its thermal equilibrium value indicated by the temperature T : lim ρS (t) = ρeq S (T ) , N →∞ t→∞ (1.6) where eq ρeq S (T ) = lim trB ρN (T ) N →∞ 4 (1.7) and ρeq N (T ) = 1 −βH e . Z (1.8) The last expression is the standard quantum statistical mechanical formula describing the equilibrium state in the canonical ensemble where Z is the partition function Z = trF e−βH and β is the temperature parameter β = 1/kB T . In other words, quantum thermalization of a subsystem S means that the subsystem behaves as if the full system is exactly at the thermal equilibrium state. And the rest of the full system serves as a heat bath. And if for all subsystems S, at long time t the above thermalization condition is true for the same temperature T , and for all initial states corresponding to that T , we say the system thermalizes for this temperature. [10, 26, 27, 22, 19] 1.2 Eigenstate Thermalization Hypothesis By defining quantum thermalization, we can further study the destiny of a given closed quantum system H. From everyday thermal phenomena, one would expect that if a system can equilibrate, then the system must thermalize from any initial conditions. Then, it follows that all the exact many-body eigenstates of the full system must be thermal because the probability operators, ρeigen = |ψihψ|, do not change with time, where |ψi satisfies H|ψi = E|ψi. This motivates the following hypothesis called the Eigenstate Thermalization Hypothesis (ETH):[10, 26, 27, 22] Every single eigenstate thermalizes. By saying eigenstate, we are making no approximations. We mean the exact manybody eigenstate of the full system. By saying thermalize, we mean all subsystems thermalize for the temperature determined by the eigenenergy of the eigenstate. 5 ETH is a hypothesis. It is not true for systems that are many-body localized (MBL) as I will discuss in the next section. Also, it is not true for integrable systems which contain infinite number of local conservation laws. By saying local, we do not necessarily mean local in real space, it can be local in for example momentum space. But it should not be global because it is trivial that for any given full system eigenstate |ni, the projection operator |nihn| is a conserved operator. And so is any weighted sum over them. Thus there are infinite number of “global” conservation laws for any given system. But these projection operators are global and presently inaccessible to measurement. ETH is a hypothesis in another sense, that even for those systems where ETH appears to be true, for example everyday thermal phenomena, it is extremely hard to prove, even numerically, because one needs to test the exact many-body eigenstates which requires the exact diagonalization of the full system. In practice, this approach is limited to N only up to about 20, due to the exponential growth of the dimension of H. But that has not deferred people from finding numerical evidence for ETH. Indeed, there has been plenty of research tackling the numerical test of ETH and there is strong support that ETH is true for a large number of systems, for example Rigol, Dunjko and Olshanii [22]; Pal and Huse [20]; Kim and Huse [16]; Beugeling, Moessner, and Haque [6] and many more. Thus it is worthwhile to mention some consequences based on ETH. One of them is for the diagonal ensemble. We write the probability operator in the basis of the exact many-body eigenstates of the full Hamiltonian: ρ= X |miρmn hn| , (1.9) m,n then based on the time evolution of the probability operator, we have that the diagonal matrix element ρnn stays constant and the off-diagonal matrix element ρmn , m 6= n, 6 evolves by multiplying a “simple” phase factor: ρmn (t) = e−i(Em −En )t/~ ρmn (0) , m 6= n , (1.10) where Em and En are the eigenenergies of the corresponding eigenstates |mi and |ni: H|ni = En |ni . (1.11) When t → ∞, the phase factors in the off-diagonal terms become essentially random and when they contribute to the probability operator of any local subsystem, they effectively go to their mean value 0. This procedure is called dephasing which causes the full system to thermalize when ETH is true. After dephasing, the probability operator becomes diagonal: ρD = X |niρnn hn| , (1.12) n which is called the diagonal ensemble. It neglects all the off-diagonal terms, and the diagonal terms are set by the initial condition. As being two special cases of the diagonal ensemble, we also have the canonical ensemble as appears in the standard equilibrium statistical mechanics ρeq = 1 −βH 1X e = |nie−βEn hn| Z Z n (1.13) and the single many-body eigenstate “ensemble” ρ(n) = |nihn| (1.14) as a limit of the standard microcanonical ensemble by sending the available energy window ∆E → 0. When ETH is true, all different forms of the diagonal ensembles 7 including the two special cases above are equivalent for small subsystems, provided the uncertainty in the total energy remains less than extensive. 1.3 Many-body Localization As mentioned in the previous section, there exists a large class of systems, localized systems, which do not obey ETH. The idea of localization first came from Anderson [1] in 1958. In this section, I will focus on one subset of the localized systems: interacting many-body localized (MBL) systems. To briefly demonstrate the idea of many-body localization, I use the model of Pal and Huse [20]. It is a one dimensional spin chain with spin-1/2 with Hamiltonian: H= X hi σiz + i X J~σi · ~σi+1 , (1.15) i where σi are the Pauli operators for the spin-1/2 at ‘site’ i. The onsite magnetic fields hi are static random variables with a probability distribution that is uniform in [−h, h]. At J = 0, the many-body eigenstates are just the product states which are the basis of the σ z representation and the system is fully localized. For nonzero J, if we assume J h, we can apply perturbation theory with zeroth order J = 0 to construct the many-body eigenstates [5]. Because the local level spacing produced by the J = 0 product states are comparable to h, it is in general much larger than the interaction J. It implies that the eigenstates are very weakly hybridized. This argument further implies that there is no DC spin transport or energy transport and so the quantum thermalization is violated [5]. In addition to this perturbative argument, Pal and Huse [20] demonstrated numerical evidence for the generic nonperturbative case by doing the exact diagonalization up to 16 spins. They showed 8 that when h/J > hc /J ≈ 3.5, the system fails to behave thermal and goes into the localized phase. There are several properties that differ between thermalization and localization. Some are summarized in Table 1.1 cited from [19]. Note that these differences are all in dynamical properties or properties of the exact eigenstates. In fact, the two phases do not differ at all in their static thermal equilibrium properties. Thermal Phase Many-body Localized Phase Memory of initial conditions ‘hid- Some memory of local initial conden’ in global operators at long ditions preserved in local observtimes ables at long times ETH true ETH false May have non-zero DC conduc- Zero DC conductivity tivity Continuous local spectrum Discrete local spectrum Eigenstates with volume-law entanglement Eigenstates with area-law entanglement Power-law in time spreading of Logarithmic in time spreading of entanglement from non-entangled entanglement from non-entangled initial condition initial condition Dephasing and dissipation Dephasing but no dissipation Table 1.1: A list of some properties of the many-body localized phase contrasted with properties of the thermal phases. Table from [19]. I close this section by mentioning the definition of temperature in the MBL phase. As we know in the standard statistical mechanics, temperature is a crucial concept in the equilibrium state. But in the MBL phase, the system fails to obey the ETH hence the system cannot reach the equilibrium state. So temperature is ill-defined in the MBL phase. One way to define temperature is to say that it is the corresponding temperature if the system could thermalize. This implies a brand new type of phase transition called the eigenstate phase transition and it will be discussed in the next section. 9 1.4 Eigenstate Phase Transition The eigenstate phase transition, as formulated e.g. in [13], is a brand new type of phase transition compared with traditional thermal phase transition. In thermal phase transition, we see a sharp change in terms of the thermal equilibrium when we go from one phase to another. For example, from water to ice, the sharp change can be observed directly by solving the minimal value of the free energy in the standard statistical mechanical way and the minimal point of the free energy just corresponds to the thermal equilibrium. However, in eigenstate phase transitions, the sharp change is in properties of the many-body eigenstates but not the thermal equilibrium because thermal equilibrium is an average over lots of eigenstates and this average washes out the sharp change in terms of single many-body eigenstates. Thus, eigenstate phase transition is invisible to the equilibrium statistical mechanics. For example, as I mentioned in the previous section, the phase transition between the thermal phase and the many-body localized phase is an eigenstate phase transition. In the MBL phase, the system cannot even evolve to its thermal equilibrium value and some information is still stored locally for a long time. In fact, if one would use the standard equilibrium statistical mechanics to solve the MBL system, one could still get some result and there would be no sharp change between the two phases in this sense. The traditional equilibrium statistical mechanics however, does not correctly give the long-time behavior in the MBL phase. Because the eigenstate phase transition is a brand new type of phase transition, it by itself has become a topic of research. Also, the usual equilibrium statistical mechanics may break down when applied to the eigenstate phase transition, so the properties of the critical point in the eigenstate phase transition, for example the MBL transition, need to be reestablished and many questions about it are still open [19]. In addition, eigenstate phase transition is not limited to the MBL transition. Even in the regime where ETH is true on both sides, there remain eigenstate phase 10 transitions, at least in the limit of long-ranged interaction [32]. These are one of the main topics in this thesis. 1.5 Thesis Outline In this thesis, I will discuss two examples of eigenstate phase transitions as mentioned in the previous section. In Chapter 2, we explore a transverse-field Ising model that exhibits both spontaneous symmetry-breaking and eigenstate thermalization. Within its ferromagnetic phase, the exact eigenstates of the Hamiltonian of any large but finite-sized system are all Schrödinger cat states: superpositions of states with ‘up’ and ‘down’ spontaneous magnetization. This model exhibits two eigenstate phase transitions within its ferromagnetic phase: In the lowest-temperature phase the magnetization can macroscopically oscillate by quantum tunneling between up and down. The relaxation of the magnetization is always overdamped in the remainder of the ferromagnetic phase, which is further divided into phases where the system thermally activates itself over the barrier between the up and down states, and where it quantum tunnels. In Chapter 3, we study the many-body localization transition in one dimensional systems via the strong disorder renormalization group approach. In this framework, we impose a set of rules for coarse-graining the system. The result from this set of rules turns out to be a beautifully simple “toy” renormalization group. We can almost solve for the critical fixed point distribution analytically. In addition, we also get an estimate of the correlation length critical exponent ν ≈ 2.5 by both solving the analytical equations numerically and directly simulating the coarse-graining procedure. 11 Chapter 2 Three ‘Species’ of Schrödinger Cat States in an Infinite-range Spin Model 2.1 Introduction The dynamical properties of isolated many-body quantum systems have long been of interest, due to their role in the fundamentals of quantum statistical mechanics. More recently, experiments approximating this ideal of isolated many-body quantum systems have become feasible in systems of trapped atoms [17, 8] and ions [7], and as a consequence this topic has received renewed attention. It appears that a broad class of such systems obey the Eigenstate Thermalization Hypothesis (ETH) [10, 26, 27, 22, 19]. The ETH asserts that each exact many-body eigenstate of a system’s Hamiltonian is, all by itself, a proper microcanonical ensemble in the thermodynamic limit, in which any small subsystem is thermally equilibrated, with the remainder of the system acting as a reservoir. In the present chapter we present some interesting 12 results for an infinite-range transverse-field Ising model that obeys the ETH and also has spontaneous symmetry-breaking. Quantum many-body systems with static randomness may fail to obey the ETH due to many-body Anderson localization stopping thermalization [1, 5, 20]. The interesting interplay of many-body localization and discrete symmetry-breaking was recently explored in Refs. [13, 21, 29, 18]; the present chapter instead explores an example of the interplay of the ETH and Ising symmetry breaking. We start with the infinite-range transverse-field Ising model: N N X 1 X z z H0 = − si sj − Γ sxi , N 1=i<j i=1 (2.1) where ~si = (sxi , syi , szi ) = ~σi /2, and ~σi are the Pauli operators for the spin-1/2 at ‘site’ i. We choose to set ~ = kB = 1. This model has been extensively studied recently, particularly as an example for exploring quantum information issues where it is known as the ‘Lipkin-Meshkov-Glick model’; see, e.g., [31] and references therein. One can determine and use many properties of the exact eigenstates of this Hamiltonian, but there are extensive degeneracies in its spectrum due to its symmetry under any permutation of the N spins, which give it eigenstates that do not obey the ETH. Thus we add static random Ising interactions to the Hamiltonian to break the permutation symmetry and lift all the degeneracies (with probability one), so the full Hamiltonian of the system we consider is H = H0 + H1 = H0 + N λ X εij szi szj , N p 1=i<j (2.2) where 1 ≥ λ > 0, the εij are independent Gaussian random numbers of mean zero and variance one, and the power p satisfies 1/2 < p < 1. As we argue below, the eigenstates of this Hamiltonian should obey the ETH, but the randomness is weak enough so that many of the properties of H0 , such as the thermodynamics, are 13 unchanged and can be used in our analysis. This randomness is too weak to produce any localization. We have chosen to put the randomness on the interactions, since we have found in exact diagonalizations that this produces much better thermalization at numerically accessible system sizes as compared to, e.g., only making the local transverse fields random. 0 Para −0.05 −0.1 Ground State Thermodynamic PT Eigenstate PT QCP F1 u−0.15 F3 F2 −0.2 −0.25 −0.3 −0.35 0 0.1 0.2 0.3 Γ 0.4 0.5 0.6 0.7 Figure 2.1: The phase diagram of our model. u is the energy per spin and Γ is the transverse field. The ground state (zero temperature) is indicated by the black (solid) line, with the quantum critical point (QCP) indicated. The green (dot-dashed) line is the thermodynamic phase transition (PT) between the paramagnetic and ferromagnetic (F) phases. There are two dynamical (eigenstate) phase transitions within the ferromagnetic phases, indicated by the blue (dashed) lines. See the text for discussions of the sharp distinctions between these three phases F1, F2 and F3. We now briefly summarize our results, before deriving and discussing them in more detail below. The phase diagram of this spin model as a function of the energy 14 per spin u = hHi/N and the transverse field Γ is shown in Fig 2.1. There are the usual two thermodynamic phases of a ferromagnet: the paramagnetic phase (Para) at high energy and/or high |Γ|, and the ferromagnetic phase (F) when both |Γ| and the energy are low enough. In the ferromagnetic phase, for any finite N , we can ask about the dynamics of the system’s order parameter. There are three regimes of behavior that are sharply distinguished from one another in the thermodynamic limit N → ∞: At the highest energies within region F3 of the ferromagnetic phase the system is a sufficiently large thermal reservoir for itself so that the most probable path by which it flips from ‘up’ to ‘down’ magnetization under unitary time evolution is by thermally activating itself over the free energy barrier between the two ordered states. At lower energies (F1 and F2) the barrier is higher and wider and as a result the reservoir is inadequate, so the system quantum tunnels through the barrier when it flips the Ising order parameter. At the lowest energies in region F1 one can in principle prepare a state that is a linear combination of two Schrödinger cat eigenstates of H that will coherently oscillate via macroscopic quantum tunneling between up and down magnetizations. In the intermediate energy regime (F2) the magnetization dynamics due to quantum tunneling is always overdamped. Throughout the ferromagnetic phase, the exact eigenstates of H for any finite N are Schrödinger cat states that are superpositions of up and down magnetized states, and the properties of these cats differ in the three regimes of the ferromagnetic phase that are indicated in Fig 2.1. Thus the two phase transitions between these three dynamically distinct ferromagnetic phases are not only dynamical phase transitions but also ‘eigenstate phase transitions’ [13], while the equilibrium thermodynamic properties are perfectly analytic through these two phase transitions. We consider the infinite-range model not only because this allows a controlled calculation of this novel physics within the ferromagnetic phases, but also because finite-range, finite-dimensional models obeying ETH do not show these features. In 15 the latter models the free energy needed to flip the magnetization by making a domain wall and sweeping it across the system is sub-extensive, while at any nonzero temperature the system is a reservoir of extensive size, so a large system will always flip via the thermal process without macroscopic quantum tunneling through the energy barrier; phases F1 and F2 thus do not exist for such models. One can also consider intermediate cases of transverse-field Ising models with interactions that fall off as a power of the distance between spins. When this power is small enough, the resulting free energy barrier to flip the magnetization in the ferromagnetic phase is extensive, and we thus expect phases F1 and F2 to also occur in those models, although we do not see a way to simply calculate the locations of the phase boundaries as we can for the infinite-range model. 2.2 2.2.1 Infinite Range Transverse Field Ising Model Integrability First we examine the the unperturbed Hamiltonian H0 , which has the same thermodynamics as our full model H. The Hamiltonian H0 commutes with all permutations of the spins, as does the total spin operator: ~≡ S N X s~i . (2.3) i=1 The magnitude S 2 of the total spin squared also commutes with H0 , so we can choose a set of eigenstates of H0 that are also eigenstates of S 2 . This unperturbed Hamiltonian 16 only depends on the total spin and thus can be written as N N X 1 X z z H0 = − s s −Γ sxi N 1=i<j i j i=1 =− N N X 1 X z z si sj − Γ sxi 2N i6=j=1 i=1 N N N X 1 X z z X z2 si sj − si − Γ sxi =− 2N i,j=1 i=1 i=1 N N X 1 X z 2 1 =− si + − Γ sxi 2N i=1 8 i=1 =− 1 1 2 Sz − ΓSx + . 2N 8 (2.4) The magnitude S of the total spin is of order N and ranges from zero up to S = N/2. 2.2.2 Energy Level Degeneracy ~ 2 = 0 and also H0 , S ~ 2 = H0 , S ~ 2 = H0 , S ~ 2 = · · · = 0, Manifestly, H0 , S 1 12 123 ~123···n ≡ Pn ~si . It means if we couple the angular momentum one by one where S i=1 from the very beginning site, all “partial summations” are good quantum numbers. So the Hamiltonian is block diagonal with each specific angular momentum coupling. ~tot , it means all Moreover, since H0 only contains the information of total spin S different ways of the angular momentum coupling have exactly the same spectrum if the total spin is unchanged. It implies a large amount of degeneracy of each energy level. The degeneracy is then determined by counting how many different ways of constructing an angular momentum coupling to have a specific value of total spin S. Let us denote the degeneracy by fN (S) where N is the total number of sites and S 17 is the total spin value, then it is shown1 that N fN (S) = CN2 −S N − CN2 −S−1 . (2.5) Then the entropy per spin Σ(S/N ) defined by S 1 )≡ ln fN (S) N N (2.6) 1 S 1 S 1 S 1 S S ) = − ( − ) ln( − ) + ( + ) ln( + ) , N 2 N 2 N 2 N 2 N (2.7) Σ( is given by2 Σ( due to all the different ways one can add together N spin-1/2’s to get total spin S depends only on the ratio S/N . For each value of S the spectrum of H0 has (2S + 1) eigenenergies. These eigenenergies and the corresponding eigenstates can be approximated for large S using a discrete version of the WKB method [9], as we discuss below. 2.2.3 Discrete WKB Method After solving the degeneracy of the energy level, the remaining task we need to deal with is one copy of the block diagonal Hamiltonian with the total spin S. The subspace is 2S + 1 dimensional and the Hamiltonian in the subspace is 1 2 S − ΓSx 2N z 1 2 Γ =− S − (S+ + S− ) , 2N z 2 H0 = − 1 2 Please see Appendix A for the mathematical proof. Please see the end of Appendix A for the proof. 18 (2.8) where S+ , S− are raising and lowering operators. In (S 2 , Sz ) representation, we have Sz2 |S, mi = m2 |S, mi p S+ |S, mi = (S − m)(S + m + 1)|S, m + 1i p S− |S, mi = (S + m)(S − m + 1)|S, m − 1i . (2.9) If we set the eigenstate wave function as |ψi = S X Cm |S, mi , (2.10) m=−S we arrive at the discrete version of the Schrödinger equation: − 1 2 Γp Γp m Cm − (S + m)(S − m + 1)Cm−1 − (S − m)(S + m + 1)Cm+1 = E · Cm . 2N 2 2 (2.11) By applying the discrete WKB method reviewed by P.A.Braun[9], we define 1 2 m 2N Γp pm = (S + m)(S − m + 1) , 2 wm = (2.12) then the Schrödinger equation reduces to the standard form in [9]:3 pm Cm−1 + (wm + E)Cm + pm+1 Cm+1 = 0 . (2.13) ∂ By following the steps mentioned in [9], we introduce “momentum” ϕ = −i ∂m , then H0 = −(wm + pm e−iϕ + pm+1 eiϕ ) ' −(wm + 2pm+ 1 cos ϕ) . 2 3 (2.14) In the original paper, the coefficient before Cm is “wm − E”, what we only need to do is to replace all E’s into −E. 19 From now on, we will simplify the notation by denoting w ≡ wm and p ≡ pm+ 1 = 2 q Γ (S + 12 )2 − m2 . Then we introduce the “potential-energy curve”:4 2 U + (m) ≡ w + 2|p| (2.15) U − (m) ≡ w − 2|p| . The result given by [9] about U + and U − is that the classical accessible energy region is confined by U + and U − : −U + (m) ≤ E ≤ − U − (m) . (2.16) In our model, the relation among U + , U − and E is demonstrated by Fig. 2.2. From U -U - m_t m E -U + Figure 2.2: Two “potential-energy curves” in our model. Fig. 2.2, the classical accessible region is between the two curves, so if the energy is specified as shown in Fig. 2.2, it would generically have the quantum tunneling effect between the left and right well. In the original paper, the definition was U ± ≡ w ± 2p by assuming p > 0. Actually when p < 0, the situation is the same as p > 0 if we define U ± ≡ w ± 2|p|. 4 20 Based on the connection relation associated by U + turning point mt , we have the WKB wave function[9]: Z m A π C = cos arccos B dm − √ m>m t vm 4 mt Z mt A p C = arccosh B dm , exp − m<mt 2 |vm | m p where A is the normalization factor; vm ≡ ity; B ≡ −E−w 2|p| (2.17) (U + + E)(U − + E) is the classical veloc- has the property when m is reaching the edge of the classical region or the turning point mt , B(m = mt ) = 1. Thus, the leading effect of the level splitting due to double-well quantum tunneling is given by: Z mt (2.18) arccosh B dm (2.19) arccosh B dm ≡ e−γN , ∆E ∼ exp − −mt where 1 γ= N Z mt −mt is the scaled logarithm of the tunneling rate and mt satisfies the turning point equation B(mt ) = 1 which is r 1 1 2 mt + |Γ| (S + )2 − m2t + E = 0 . 2N 2 (2.20) When we send the number of sites N → ∞, we would notice that the total energy E, total spin S and the z-component of the total spin m are all proportional to N . Therefore it is better to define the number density of all the above quantities in order to explicitly demonstrate the N dependence: u≡ E , N s̄ ≡ S , N 21 x≡ m . N (2.21) Then the turning point equation (2.20) reduces to: p 1 2 xt + |Γ| s̄2 − x2t + u = 0 ; 2 ⇒ xt = √ q p 2 · −u − Γ2 − Γ2 (2u + s̄2 + Γ2 ) . (2.22) (2.23) Meanwhile, the scaled logarithm of the tunneling rate γ reduces to: 1 −E − 2N m2 arccosh √ dm |Γ| S 2 − m2 −mt xt −u − 21 x2 arccosh √ dx . |Γ| s̄2 − x2 0 1 γ= N Z =2 2.3 Z mt (2.24) Thermodynamics For the thermodynamics (but not the dynamics) of this system in the limit of large ~ classically and ignore their nonzero commutators N we can treat the components of S when obtaining the extensive thermodynamic properties (energies, entropies, magnetizations). Then we obtain the energy density u= E H0 1 = = − s̄2z − Γs̄x , N N 2 (2.25) where q s̄x = ± s̄2 − s̄2y − s̄2z . (2.26) Note that this equation is the same as the turning point equation (2.22) by setting s̄y = 0 and s̄z = xt and choosing the sign of s̄x as the same as the sign of Γ. It means that we can also use the semiclassical point of view to determine the WKB 22 turning point xt and further obtain the tunneling rate γ(s̄). The ground state is given by minimizing the energy density u as a function of s̄, s̄y and s̄z . Here we choose the sign of s̄x as the same as the sign of Γ so that the term −Γs̄x gives a negative contribution to u. Since s̄ ≤ 1/2, it is easy to see that when s̄ = 1/2 and s̄y = 0, u will be minimized in terms of them. By plugging these two conditions into u, we have r 1 2 1 u = − s̄z − |Γ| − s̄2z . 2 4 (2.27) By setting the first order derivative of s̄z equals zero, we have s̄z = 0 or s̄z = q 1 4 − Γ2 . After evaluating the second order derivative, it becomes clear that when 41 − Γ2 < 0, q 1 2 s̄z = 0 is the minimal point; when 4 − Γ > 0, s̄z = 14 − Γ2 minimizes u: q when |Γ| < 1 , s̄z = 1 − Γ2 , umin = − 1 − 1 Γ2 2 4 8 2 when |Γ| > 1 , s̄z = 0, 2 . (2.28) umin = − 21 |Γ| Thus, the ground state of H0 always has the maximum value of S = N/2. For |Γ| ≥ 1/2, the ground state is paramagnetic with the spins polarized along the xdirection: s̄y = s̄z = 0, s̄x = 21 sign(Γ) and u = hH0 i/N = −|Γ|/2. For |Γ| < 1/2, the √ two nearly-degenerate ground states are ferromagnetic, with s̄z = ±(1/2) 1 − 4Γ2 , s̄x = Γ, s̄y = 0 and u = −(1 + 4Γ2 )/8. So the system has ground state quantum phase transitions at Γ = ±1/2 between ferromagnetic phase (s̄z 6= 0, |Γ| < 1/2) and paramagnetic phase (s̄z = 0, |Γ| > 1/2). The corresponding quantum critical points are at Γc = ±1/2 , uc = −1/4. For the excited states, the phase transition between ferromagnetic and paramagnetic phase also exists. Since we are interested here in eigenstates, which are at a given energy E = N u, we will do the statistical mechanics in the microcanonical ensemble. For a given transverse field Γ and energy E, the equilibrium (most probable) 23 state of the system is the one that maximizes the entropy, which means minimizing the total spin S. To minimize S for a given E, clearly we set Sy = 0, since Sy does not appear in the Hamiltonian. In the paramagnetic phase the total spin points along the x-direction and the equilibrium value of the total spin is thus s̄eq = |u/Γ|. In the ferromagnetic phase the system can go to higher entropy (lower total spin) for a given u by making s̄z 6= 0. Some algebra shows that in the ferromagnetic phase, which is |Γ| < 1/2 and −(1 + 4Γ2 )/8 ≤ u < −Γ2 , the equilibrium is at s̄x = Γ, p √ s̄z = ± 2(−u − Γ2 ) and s̄eq = −2u − Γ2 . The line of critical points separating the para- and ferromagnetic phases is u = −Γ2 , for |Γ| < 1/2, as indicated in Fig 2.1; this critical line ends at the quantum critical points at |Γ| = 1/2, u = −1/4. Note that in this whole ferromagnetic phase regime, the sign of s̄x is the same as the sign of Γ. Since all the quantities related to Γ, except s̄x , are even functions of Γ (or functions of |Γ|), from now on we assume Γ > 0 and use Γ and |Γ| interchangeably. So it also implies that s̄x ≥ 0. 2.4 Eigenstate Thermalization Due to its full symmetry under all permutations of the N spins, the Hamiltonian H0 is integrable, with all the good quantum numbers associated with this permutation symmetry, including the magnitude S of the total spin. We want to study a more generic system, so we add to the Hamiltonian the small term H1 (see Eq. (2.2)) to break the permutation symmetry, lift all degeneracies, and make the eigenstates thermal. The only symmetry that remains in our full H is the Ising (Z2 ) symmetry under a global rotation of all spins by angle π about their x axes. For a given E and Γ, the eigenstates of H0 have total spin ranging from the minimum and equilibrium value Seq up to the maximum value of S = N/2. To make these in to thermal eigenstates we need H1 to perturb the system enough so that 24 the eigenstates of the full H are linear combinations of all these total spin values, weighted as at thermal equilibrium. At first order, the perturbation we are adding, H1 , flips at most two spins, so it can change the total spin by at most ±2. The spectrum of H0 at each value of total spin S contains (2S + 1) eigenenergies spread over a range of energy that is of order S. Thus the level-spacing in the spectrum of H0 at a given S remains of order one in the limit of large N . This is reflected in the dynamics under H0 , which is spin precession about the mean field, and the mean field is of order one, so the rate of precession is also of order one. For the eigenstates of H to strongly and thermally mix the different values of S we thus need the matrix elements of H1 between states at different S to be large compared to the (order-one) level spacing of H0 in the large N limit. This is why we require that the exponent p in the definition of H1 satisfies p < 1, since this is the condition for these matrix elements to diverge in the large N limit.5 This should be sufficient to make all the eigenstates of H thermal in that limit. We have not yet found a way to actually prove that this is sufficient to make all the eigenstates of H satisfy the ETH, but below we provide some numerical evidence for this from exact diagonalization of finite-size systems. 2.5 Dynamics In addition to making sure that H1 is strong enough to thermalize the system, we also want it to be weak enough so that we can use the well-understood dynamics and thermodynamics of H0 in our analysis. By restricting the exponent p to be greater than 1/2, in the large N limit the effective field that each spin is precessing about is the mean field from H0 , with only a small correction from H1 that vanishes as N → ∞.6 This small correction is enough to thermalize the system for 1/2 < p < 1, 5 6 For the details of the order analysis of the lower bound of disorder, see Appendix B.1. For the details of the order analysis of the upper bound of disorder, see Appendix B.2. 25 which is the range where the perturbation due to H1 on a single spin’s dynamics vanishes for N → ∞, while the perturbation to the dynamics of the full many-body system diverges. In this regime, the system’s primary dynamics is the S-conserving dynamics due to H0 , which is spin precession at a rate of order one, and, assuming we are in the ferromagnetic phase, ‘attempts’ at rate of order one to tunnel through the energy barrier between total Sz up and down. At a rate that is slower by a power of N , the dynamics due to H1 allow ‘hopping’ between different values of the total spin S, and thus thermalization to the equilibrium probability distribution of S dictated by the entropy Σ. And at a rate that is slower still, exponentially slow in N , the system succeeds in crossing the free energy barrier between up and down magnetizations. It is the separation between these three time scales that allows us to systematically understand the dynamics of this system for large N . Since our full Hamiltonian H has Ising symmetry under a global spin flip, and the randomness in H1 means there are no exact degeneracies in the spectrum of H, for finite N any eigenstate of H is either even or odd under this Ising symmetry (with probability one). In the ferromagnetic phase, this means the exact eigenstates of H are all Schrödinger cat states that are either even or odd linear combinations of states with total Sz up and down. These two equal (in magnitude) and opposite (in sign) values of Sz are extensive, thus ‘macroscopically’ different, which is why it is appropriate to call these eigenstates ‘Schrödinger cats’. 2.5.1 Thermal Activation and Quantum Tunneling Next we examine the rate at which this system, in its ferromagnetic phase, will flip from the up state with positive total Sz to the down state with negative Sz under the unitary time evolution due to its Hamiltonian H. In Section 2.2.3, we noticed that the system has an energy level splitting due to double-well quantum tunneling which is given by ∆E ∼ e−γ(s̄)N where γ(s̄) is the scaled logarithm of the tunneling rate 26 as a function of the total spin density s̄ = S/N . By drawing the classically constant energy line shown in Fig. 2.3 described by equation (2.25) by setting s̄y = 0 or (2.22) s E s_0 s_min x_t s_z Figure 2.3: Constant energy (E) line described by equation (2.25) by setting s̄y = 0 or just by equation (2.22). It determines the WKB turning point xt when the total spin density s̄ is fixed. in the ferromagnetic phase region, we obtain a double-well shaped curve which has a local maximum at s̄z = 0 where s̄ = s̄0 = −u/Γ and two local minima at |s̄z | 6= 0 where s̄ = s̄min = Seq /N . There are two steps to this process: First the system gets ‘excited’ from its usual (high entropy) total spin Seq ‘up’ to a larger total spin S with lower entropy, with a probability ∼ exp {−N (Σ(Seq /N ) − Σ(S/N ))} given by the resulting decrease of the entropy. As S is increased, the energy barrier, whose top is at energy E = −ΓS, decreases. For E ≥ −ΓN/2, one way the system can flip is to increase S enough so that E ≥ −ΓS and then it will simply cross over the top of the barrier without quantum tunneling. In the higher-energy part (F3) of the ferromagnetic phase, in the limit of large N this is the dominant process that flips the magnetization: the system ‘thermally activates’ itself (via its unitary time-evolution) to a low-entropy, hightotal-spin state where the energy barrier can be crossed without quantum tunneling. 27 The ‘height’ of the entropy barrier it must cross to do this is extensive: √ N ∆Σ = N (Σ( −2u − Γ2 ) − Σ(−u/Γ)) , (2.29) where ∆Σ is the reduction in entropy per site needed to go over the barrier. If the system does not or can not go over the energy barrier by increasing the total spin S, then in order to flip the magnetization it must quantum tunnel through the barrier. For large N , this tunneling probability can be estimated using a version of the WKB method [9] I have explained in Section 2.2.3. As a small summary of Section 2.2.3, the total Sz serves as the ‘position’, while the operator −ΓSx serves as the ‘kinetic energy’. What we need to calculate is the probability of the system tunneling between positive and negative total Sz for a given total spin S = N s̄ satisfying Seq ≤ S < −E/Γ. This probability behaves as ∼ exp (−N γ), with γ(u, s̄, Γ) being an intensive quantity. If we define a scaled ‘position’ x = Sz /N , the WKB ‘turning points’ adjacent to the barrier are at x = ±xt with xt = √ q p 2 · −u − Γ2 − Γ2 (2u + s̄2 + Γ2 ) . (2.30) Then the intensive factor in the exponent of the tunneling probability is given by the WKB tunneling integral Z γ=2 0 xt −u − 1 x2 arcosh √ 2 dx . Γ s̄2 − x2 (2.31) Since the probabilities of being ‘excited’ to total spin S and of quantum tunneling through the barrier with total spin S are both exponentially small in N , in the limit of large N the dominant process by which the magnetization flips is given by a standard ‘saddle point’ condition. The total spin S = N s̄ at which the system tunnels is the value that maximizes the product of these two probabilities and thus minimizes the 28 quantity α(u, Γ) = mins̄ {Σ(s̄min = √ −2u − Γ2 ) − Σ(s̄) + γ(u, s̄, Γ)} . (2.32) This means in order to tunnel through the barrier, the system will finally choose one particular value of the total spin s̄ to maximize {Σ(s̄) − Σ(s̄min ) − γ(u, s̄, Γ)}. Then the distinction between thermal and quantum tunneling becomes clear: if the total spin is chosen to be in the interval (s̄min , s̄0 ) which is below the top of the barrier, it is quantum tunneling; if the total spin is chosen to be bigger than or equal to s̄0 = −u/Γ which is at the top of the barrier, it is thermal activation over the barrier. The sketch of Σ(s̄) and γ(s̄) is shown in Fig. 2.4, where dγ(s̄) ds̄ diverges7 at s̄ = s̄min ; S, Γ S(s) Γ(s) 0 s_min s_0 1/2 s Figure 2.4: A sketch of the entropy Σ(s̄) and the tunneling rate γ(s̄). γ(s̄) becomes zero at s̄ = s̄0 since it is at the top of the barrier. From Fig. 2.4, if we assume γ(s̄) is a convex8 function of s̄, then the distinction between the thermal activation and quantum tunneling would be shown straightforward in Fig. 2.5 where in thermal activation, Σ − γ reaches its maximum at s̄ = s̄0 ; in quantum tunneling, Σ − γ reaches its maximum in the middle between s̄min and s̄0 . Then, the condition 7 8 Please see Appendix C for proof of the divergence. For detailed discussion of this issue, see Appendix D. 29 S-Γ S-Γ s_min s_0 s_min s (a) Thermal activation s_0 s (b) Quantum tunneling Figure 2.5: A sketch of the difference between thermal activation and quantum tunneling. reduces to: if − γ) > 0 ⇒ Thermal Activation; s̄=s̄0 −0 if ∂ (Σ − γ) < 0 ⇒ Quantum Tunneling. ∂s̄ ∂ (Σ ∂s̄ (2.33) s̄=s̄0 −0 We have located the saddle point numerically at many points within the ferromagnetic phase and it appears to always be unique, without any discontinuities as the parameters u and Γ are varied. Some straightforward analysis as I will discuss in Section 2.6 in details shows that in the higher-energy part (F3) of the ferromagnetic phase where √ πΓ Γ − 2u ≥ ln Γ + 2u −u − Γ2 (2.34) the saddle point is ‘thermal’: the ‘entropy cost’ of going to higher S is less than the ‘tunneling cost’, and the system goes over the barrier without any quantum tunneling. We call the eigenstates in this regime ‘thermal cats’, since these Schrödinger cat states flip by thermally activating themselves over the barrier. In regions F1 and F2 this inequality is instead false, and the system quantum tunnels through the barrier at a value of S satisfying Seq ≤ S < −E/Γ, so the eigenstates are instead ‘quantum cats’. 30 The location of the dynamical phase transition between phases F2 and F3 is given by converting the above inequality (2.88) to an equality. The asymptotic behavior of this transition curve will be discussed in details in Section 2.7 and the result is as follows. Near the quantum critical points (∆Γ = 1/2 − |Γ| 1), this transition line becomes exponentially adjacent to (and above) √ the straight line u = −|Γ|/2: u = −1/4 + (∆Γ/2) + O(exp(−1/ ∆Γ)); near Γ = 0, p it behaves as a power law: −u ≈ (|Γ| π/4)4/3 . 2.5.2 Paired States and Unpaired States Within the lower-energy phases (F1 and F2) of ‘quantum cats’, there is a second dynamical phase transition within the ferromagnetic phase. This is also a ‘spectral phase transition’ [13] in the level-spacing statistics of the eigenenergies. If one starts with a state (not an eigenstate) that is magnetized up, the rate at which the system crosses the barrier to down is ∼ exp (−N α(u, Γ)), and as a result the uncertainty of the energy of this initial ‘up’ state must be at least this large, by the time-energy uncertainty relation. Compare this minimum energy uncertainty to the typical many√ body level spacing ∼ exp (−N Σ( −2u − Γ2 )) of the eigenstates of H at that energy. There is clearly a sharp change at the phase transition line, which is the line where √ α(u, Γ) = Σ( −2u − Γ2 ). In another word, if max(Σ − γ) < 0 ⇒ Paired States; s̄ (2.35) if max(Σ − γ) > 0 ⇒ Unpaired States. s̄ This transition line between phases F1 and F2 is shown in Fig 2.1. The location of this transition was obtained numerically, since we do not have a simple closed-form expression for α(u, Γ). The asymptotic behavior of this transition curve will also be discussed in details in Section 2.7 and the result is as follows. Near the quantum critical points (∆Γ = 31 1/2 − |Γ| 1), this transition line also becomes exponentially adjacent to (but now √ below) the straight line u = −|Γ|/2: u = −1/4 + (∆Γ/2) − O(exp(−1/ ∆Γ)); near Γ = 0, it is logarithmically tangent to the u-axis: −u ∼ 1/(ln |Γ|)2 . 2.6 Analytical Calculation of the Thermal Activation and Quantum Tunneling Transition The transition between thermal activation and quantum tunneling is illustrated by condition (2.33). Thus, in order to find the transition, we need to calculate the derivatives ∂Σ ∂s̄ and ∂γ ∂s̄ at s̄ = s̄0 − 0. The first one is straightforward from equation (2.7): ∂Σ 1 + 2s̄ = − ln . ∂s̄ 1 − 2s̄ Therefore we need to focus on solving ∂γ ∂s̄ (2.36) at s̄ = s̄0 − 0. When s̄ = s̄0 , by definition we have the turning point xt = 0 (It implies γ(s̄0 ) = 0.). Plugging xt = 0 into the turning point equation (2.22) gives us: s̄0 = −u . Γ Now we perturb xt a little away from 0 by setting xt = ε 1. Then we have: √ 1 2 ε + Γ s̄2 − ε2 + u = 0 2 √ 1 Γ s̄2 − ε2 = −u − ε2 2 1 1 s̄2 − ε2 = 2 (−u − ε2 )2 Γ 2 2 2 u uε s̄2 = 2 + ε2 + 2 + O(ε4 ) . Γ Γ 32 (2.37) Then, r u2 uε2 2+ + ε + O(ε4 ) 2 Γr Γ2 −u u + Γ2 2 1+ ε + O(ε4 ) = Γ u2 i u + Γ2 2 −u h 4 1+ = ε + O(ε ) Γ 2u2 u u + Γ2 2 =− − ε + O(ε4 ) . Γ 2uΓ ⇒ s̄ = Thus the infinitesimal change of s̄ is: ∆s̄ = − u + Γ2 2 ε + O(ε4 ) < 0 . 2uΓ (2.38) Now it is time to see ∆γ: ε Z ∆γ = γ(xt = ε) = 2 0 −u − 12 x2 arccosh √ dx . Γ s̄2 − x2 (2.39) Since in this integral x runs from 0 to ε, we may do the rescaling to let x ≡ εb where b runs from 0 to 1. After that, we have: −u − 21 ε2 b2 B= √ Γ s̄2 − ε2 b2 −u − 12 ε2 b2 = q 2 Γ Γu2 + (1 + Γu2 )ε2 − ε2 b2 + O(ε4 ) −u − 21 ε2 b2 =p u2 + (u + Γ2 − b2 Γ2 )ε2 + O(ε4 ) −u − 21 ε2 b2 q = 2 2 Γ2 (−u) 1 + u+Γ u−b ε2 + O(ε4 ) 2 = b2 2 ε 2u u+Γ2 −b2 Γ2 2 + ε + 2u2 2 2 1+ O(ε4 ) b u + Γ − b2 Γ 2 2 =1+ − ε + O(ε4 ) . 2u 2u2 1 33 (2.40) Since arccosh B = ln(B + √ B 2 − 1), then if B = 1 + ∆ where ∆ 1, we have √ 2∆ + ∆2 ) √ √ = ln[1 + ∆ + 2∆ + O(∆ ∆)] √ √ √ √ √ 1 = ∆ + 2∆ + O(∆ ∆) − [∆ + 2∆ + O(∆ ∆)]2 + O(∆ ∆) 2 √ √ 1 = ∆ + 2∆ − · 2∆ + O(∆ ∆) 2 √ √ = 2∆ + O(∆ ∆) . (2.41) arccosh(1 + ∆) = ln(1 + ∆ + So if ∆ = b2 2u − u+Γ2 −b2 Γ2 2u2 ε2 + O(ε4 ), then r arccosh(1 + ∆) = 1 r Z 1 Z ⇒ γ(xt = ε) = 2 0 = 2ε 2 0 = 2ε 2 Z 0 1 b2 u + Γ2 − b2 Γ2 − ε + O(ε3 ) . 2 u u (2.42) b2 u + Γ2 − b2 Γ2 − ε · ε db + O(ε4 ) u u2 1 p 2 ub − (Γ2 + u) + b2 Γ2 db + O(ε4 ) −u √ −u − Γ2 √ 1 − b2 db + O(ε4 ) . −u If we change the variable b = sin θ, then R1√ 0 1 − b2 db = Rπ 2 0 cos θ ·cos θ dθ = (2.43) π 4 . Thus we arrive at: √ −u − Γ2 π 2 γ(xt = ε) = 2 · · · ε + O(ε4 ) , −u 4 34 (2.44) ∂γ γ(xt = ε) − 0 = lim ∂s̄ s̄=s̄0 −0 ε→0√ ∆s̄ −u−Γ2 π 2 · −u · 4 = 2 − u+Γ 2uΓ πΓ . = −√ −u − Γ2 (2.45) In addition, ∂Σ 1 + 2s̄0 = − ln ∂s̄ s̄=s̄0 1 − 2s̄0 Γ − 2u . = − ln Γ + 2u (2.46) Therefore the boundary between thermal activation and quantum tunneling is given by: ⇒ ∂γ ∂Σ = ∂s̄ s̄=s̄0 −0 ∂s̄ s̄=s̄0 πΓ Γ − 2u √ . = ln Γ + 2u −u − Γ2 (2.47) To be more specific, we arrive at the result: if √ πΓ −u−Γ2 ⇒ > ln Γ−2u Γ+2u if √ πΓ −u−Γ2 < ln Γ−2u ⇒ Γ+2u − γ) >0 ⇒ s̄=s̄0 −0 ∂ (Σ − γ) <0 ⇒ ∂s̄ ∂ (Σ ∂s̄ s̄=s̄0 −0 Thermal Activation; Quantum Tunneling. (2.48) 2.7 Asymptotic Behaviors of Both Transitions Near Both QCP and u = 0 , Γ = 0 Point Up to now, we have already obtained the mathematical condition of both transition lines. To further explore the properties of both lines, it is better to also have the asymptotic behaviors near the edges of the ferromagnetic regime demonstrated 35 by Fig. 2.1. One edge is the Quantum Critical Point(QCP); the other is the origin of the phase diagram: u = 0 , Γ = 0 point. We shall study the asymptotic behaviors one by one. 2.7.1 Near QCP The QCP is at Γc = 1/2 , uc = −1/4. We perturb the system a little away from the QCP into the ferromagnetic region to let Γ = 1 2 − ∆Γ where ∆Γ 1. Then the goal is to see what asymptotic behavior the energy density u should have in terms of ∆Γ. Let us discuss these two transitions one by one. Thermal Activation and Quantum Tunneling Transition In the thermal activation and quantum tunneling transition case, the transition line must be placed above the straight line u = −Γ/2. If not, the top of the barrier s̄0 = −u Γ in Fig. 2.3 would be greater than 1/2 which is the upper bound of the available total spin density s̄. Then there would be always a quantum tunneling effect and no thermal activation can be achieved. So the transition line must be placed in the regime where s̄0 = −u Γ ≤ 1 2 or u ≥ −Γ/2. Also we know that u ≤ −Γ2 since it is in ferromagnetic region. Note that u = −Γ2 has a first order derivative −1 at Γc = 1/2. Therefore we set9 1 1 u = − + ( + δ) · ∆Γ , 4 2 (2.49) where 0 ≤ δ ≤ 1/2. Plugging this definition into the transition line equation: Γ − 2u πΓ √ = ln , 2 Γ + 2u −u − Γ 9 (2.50) This is only a redefinition of a variable but not a Taylor expansion since the essential singularity occurs in general. And the same concern applies to all cases afterwards. 36 we then arrive at: π( 21 − ∆Γ) 1 L.H.S. = q ; &O √ ∆Γ ( 12 − δ)∆Γ − ∆Γ2 1 − 2(1 + δ)∆Γ R.H.S. = ln ∼ O − ln(δ∆Γ) . 2δ∆Γ (2.51) Thus, if δ ∼ O(1), then the right hand side would be O − ln(∆Γ) which is much 1 smaller than O √∆Γ . So in order to let the right hand side big enough, we need 1 . By setting the δ 1. Under this condition, the left hand side is exactly O √∆Γ left hand side to be equal to the right hand side, we finally get: 1 ∼ − ln(δ∆Γ) ∆Γ 1 − √1 ⇒ δ∼ e ∆Γ . ∆Γ √ Note that the form e − √1 ∆Γ (2.52) behaves essentially singular at ∆Γ = 0 point and does not have a proper Taylor expansion in terms of ∆Γ; in fact, it is much smaller than any finite power of ∆Γ, so all the Taylor coefficients would be zero. It implies the transition line is extremely near (and above) the straight line u = −Γ/2. So finally, the asymptotic behavior of the thermal activation and quantum tunneling transition is: 1 Γ = − ∆Γ 2 1 u = − 1 + 1 ∆Γ + O e− √∆Γ , 4 2 where ∆Γ 1. 37 (2.53) Paired and Unpaired States Transition In the paired and unpaired states transition case, on the contrary, the transition line must be placed below the straight line u = −Γ/2. If not, then s̄0 = −u Γ would be an available candidate for taking the maximal value of Σ(s̄) − γ(s̄). By noting that γ(s̄0 ) = 0, we would have max(Σ − γ) ≥ Σ(s̄0 ) − γ(s̄0 ) s̄ = Σ(s̄0 ) > 0 , (2.54) then it would be always in unpaired states region. So the transition line must be placed in the regime where u ≤ − Γ/2. Also we know that u ≥ − 81 − 12 Γ2 since the right hand side is the ground state energy. Note that at Γc = 1/2, u = − 81 − 12 Γ2 has a first order derivative −1/2 which is the same as that of the straight line u = −Γ/2. Since the transition line must be placed between these two bounds, we obtain the first order derivative of the transition line must also be −1/2. Therefore we set 1 1 u = − + ∆Γ − λ1 ∆Γ2 , 4 2 (2.55) where 0 ≤ λ1 ≤ 1/2. In order to solve max(Σ − γ), we need to first solve the upper and lower bound s̄ of the available s̄. The upper bound is 1/2 since in this case s̄0 ≥ 1/2. The lower bound is exactly s̄min indicated in Fig. 2.3. By solving s̄ in the turning point equation (2.22), we arrive at: s̄2 = u 1 u2 + (1 + 2 )x2t + 2 x4t , 2 Γ Γ 4Γ 38 (2.56) then the minimal available s̄2 is exactly s̄2min = −2u − Γ2 (2.57) at (xt )2min = −2u − 2Γ2 . By plugging the definition of u and Γ in terms of ∆Γ and λ1 into s̄2min , we have: s̄2min = −2u − Γ2 = 1 − (1 − 2λ1 )∆Γ2 . 4 (2.58) Then we define s̄2 ≡ 1 − (1 − 2λ1 ) · α1 · ∆Γ2 , 4 (2.59) where 0 ≤ α1 ≤ 1. Then the task converts into finding the maximal value of Σ − γ in terms of α1 . Since ∆Γ 1 and λ1 , α1 . O(1), we have s̄ = ⇒ 2 1 − (1 − 2λ1 )α1 ∆Γ2 + O (1 − 2λ1 )α1 ∆Γ2 2 (2.60) 1 1 1 1 Σ(s̄) = − ( − s̄) ln( − s̄) + ( − s̄) ln( − s̄) 2 2 2 2 h i 2 2 2 = − (1 − 2λ1 )α1 ∆Γ ln (1 − 2λ1 )α1 ∆Γ + O (1 − 2λ1 )α1 ∆Γ . − ∆Γ2 ln(∆Γ2 ) . (2.61) Then for −u − 21 x2 γ(s̄) = 2 arccosh √ dx Γ s̄2 − x2 0 Z 1 −u − 1 x2 b2 =2 arccosh p 2 t xt db , Γ s̄2 − x2t b2 0 Z xt 39 (2.62) we have: x2t =2· √ 2 2 − u − Γ − Γ 2u + s̄ + Γ 2 1 1 − ∆Γ + λ1 ∆Γ2 − ( − ∆Γ)2 4 2 2 r i 1 1 1 2 2 − ( − ∆Γ) − (1 − 2λ1 )α1 ∆Γ − + (1 − 2λ1 )∆Γ 2 4 4 h 1 1 p i p =2· − (1 − 2λ1 )(1 − α1 ) ∆Γ + λ1 − 1 + (1 − 2λ1 )(1 − α1 ) ∆Γ2 2 2 =2· h1 ≡ 2 · (a1 ∆Γ + a2 ∆Γ2 ) , where a1 ≡ 1 2 − 1 2 (2.63) p p (1 − 2λ1 )(1 − α1 ) , a2 ≡ λ1 − 1 + (1 − 2λ1 )(1 − α1 ) . Then, we have: −u − 12 x2t b2 p Γ s̄2 − x2t b2 1 − 21 ∆Γ + λ1 ∆Γ2 − (a1 ∆Γ + a2 ∆Γ2 )b2 4 q = 1 ( 2 − ∆Γ) 14 − (1 − 2λ1 )α1 ∆Γ2 − 2(a1 ∆Γ + a2 ∆Γ2 )b2 i h 1 λ ∆Γ2 − a b2 ∆Γ − a b2 ∆Γ2 i h 1 1 2 2 2 2 2 2 + · 2 1 + 2(1 − 2λ )α ∆Γ + 4a b ∆Γ + 4a b ∆Γ ≈ 2 1 1 1 1 2 − ∆Γ 2 2 2(λ1 ∆Γ − a1 b2 ∆Γ − a2 b2 ∆Γ2 ) + 2(1 − 2λ1 )α1 ∆Γ2 + 4a1 b2 ∆Γ + 4a2 b2 ∆Γ2 =1+ 1 − ∆Γ 2 h 1 =1+ 1 2λ1 ∆Γ2 − 2a1 b2 ∆Γ − 2a2 b2 ∆Γ2 + (1 − 2∆Γ)(1 − 2λ1 )α1 ∆Γ2 − ∆Γ 2 i + 2a1 b2 ∆Γ − 4a1 b2 ∆Γ2 + 2a2 b2 ∆Γ2 − 4a2 b2 ∆Γ3 h i ≈ 1 + 2∆Γ2 2λ1 + (1 − 2λ1 )α1 − 4a1 b2 − 4a2 b2 ∆Γ , (2.64) where all the terms I have omitted are definitely much smaller than at least one of the present terms in the last line given ∆Γ 1 , λ1 . O(1) , α1 . O(1) . Next comes the key point: in order to have Σ and γ in same order at some value of α1 , both α1 and λ1 have to be much smaller than 1. Let me prove it by contradiction. If the conclusion is not true, then at least one of α1 or λ1 would be 40 √ O(1). Then, we would have both a1 ∼ O(1) and a2 ∼ O(1). So xt ∼ ∆Γ and √ √ −u− 12 x2t b2 √ 2 2 2 ∼ 1 + O(∆Γ2 ). Since we know arccosh(1 + ∆) = 2∆ + O(∆ ∆) when Γ s̄ −xt b ∆ 1, it means that: −u − 21 x2t b2 arccosh p γ(s̄) = 2 xt db Γ s̄2 − x2t b2 0 √ √ ∼ ∆Γ2 · ∆Γ Z 1 3 = ∆Γ 2 . (2.65) However, from equation (2.61), we have Σ . ∆Γ2 ln(∆Γ2 ). So we would arrive at Σ(s̄) γ(s̄) which is not the condition of the transition line. Thus, by contradiction, we conclude both α1 1 and λ1 1 have to be true in order to satisfy the transition condition. By knowing α1 1 and λ1 1, all quantities are further simplified. First, we have: 1 1 a1 ≈ λ1 + α1 , 2 4 1 a2 = λ1 − 2a1 ≈ − α1 , 2 (2.66) which are both linear in λ1 and α1 . Then, we get: r p xt ≈ 2a1 ∆Γ ≈ 1 (λ1 + α1 )∆Γ , 2 (2.67) and −u − 21 x2t b2 p ≈ 1 + 2(2λ1 + α1 − 4a1 b2 )∆Γ2 2 2 2 Γ s̄ − xt b ≈ 1 + 2(2λ1 + α1 )(1 − b2 )∆Γ2 . 41 (2.68) r p 1 γ(s̄) ≈ 2 2 · 2(2λ1 + α1 )(1 − b2 )∆Γ2 (λ1 + α1 )∆Γ db 2 0 Z 1√ √ = 2 2(2λ1 + α1 )∆Γ3/2 1 − b2 db 0 √ 2 = π(2λ1 + α1 )∆Γ3/2 . 2 Z ⇒ 1 (2.69) In addition, from equation (2.61) we have: Σ(s̄) ≈ − (1 − 2λ1 )α1 ∆Γ2 ln (1 − 2λ1 )α1 ∆Γ2 ≈ α1 ∆Γ2 ln α1 ∆Γ2 . (2.70) In order to let Σ and γ be in same order, we finally arrive at: (2λ1 + α1 )∆Γ3/2 ∼ α1 ∆Γ2 ln α1 ∆Γ2 . (2.71) Then it is better to let O(λ1 ) ≤ O(α1 ), since if not, both λ1 and α1 would have an extra factor λ1 /α1 on the exponent which would make the value far smaller than the case O(λ1 ) ≤ O(α1 ). Therefore we finally get the result: α1 ∼ 1 − √1 e ∆Γ , ∆Γ2 λ1 ∼ 1 − √1 e ∆Γ . ∆Γ2 (2.72) They also behave essentially singular at ∆Γ = 0 point so the transition line is also extremely near (but this time below) the straight line u = −Γ/2. Finally, the asymptotic behavior of the paired and unpaired transition is also: 1 Γ = − ∆Γ 2 1 u = − 1 + 1 ∆Γ + O e− √∆Γ , 4 2 where ∆Γ 1. 42 (2.73) 2.7.2 Near u = 0 , Γ = 0 Point The other interested point is the origin of the phase diagram: u = 0 , Γ = 0 point. Also, we will perturb the system a little away from Γ = 0 or equivalently let Γ 1. Then the goal is to see what u behaves in terms of Γ asymptotically when Γ → 0. Thermal Activation and Quantum Tunneling Transition In this case, we know that Γ2 ≤ − u ≤ Γ/2. First consider two extreme cases: √ πΓ Γ−2u & O(1). So we have 1. If −u ∼ O(Γ), then √−u−Γ ∼ O( Γ) , ln 2 Γ+2u √ πΓ ln Γ−2u . It is in thermal activation region. Γ+2u −u−Γ2 πΓ Γ−2u 2. If −u ∼ O(Γ2 ), then √−u−Γ ∼ O(1) , ln ∼ O(Γ). Then we have 2 Γ+2u √ πΓ . It is in quantum tunneling region. ln Γ−2u Γ+2u −u−Γ2 Thus, the transition line must be placed in between where Γ2 − u Γ. When this is true, πΓ ≈ −u − Γ2 Γ − 2u ln ≈ Γ + 2u √ ⇒ πΓ √ , −u −4u . Γ πΓ −4u √ = , Γ −u 3 π 2 Γ = (−u) 2 , 4 r π 34 4 −u = Γ ∼ Γ3 . 4 (2.74) (2.75) This is a power law behavior when Γ → 0. Paired and Unpaired States Transition In this case, we have Γ/2 ≤ − u ≤ 1/8. Also consider the two extreme cases first: 43 1. If −u ∼ O(1), then xt = −u− 21 x2t b2 √ √ p √ √ 2 −u − Γ2 − Γ 2u + s̄2 + Γ2 ≈ −2u ∼ O(1). ) → ∞. ∼ O( −u Γ In addition, arccosh B = ln(B + Z 1 √ −u − 1 x2 b2 2 B − 1) → ln 2B when B → ∞. So γ = 2 arccosh p 2 t xt db ∼ Γ s̄2 − x2t b2 0 − ln Γ → ∞. But Σ . O(1), so it means max(Σ − γ) < 0. It is in paired states But B = Γ s̄2 −x2t b2 s̄ region. Thus the transition line tends to reach u = 0 point when Γ → 0. 2. If −u ∼ O(Γ), then we define −u = α2 Γ, where α2 ∼ O(1) and α2 ≥ 1/2 since we have −u ≥ Γ/2. Under this assumption, we have: q √ √ xt = 2 −u − Γ2 − Γ 2u + s̄2 + Γ2 p √ q = 2 α2 Γ − Γ2 − Γ s̄2 − 2α2 Γ + Γ2 p √ q ≈ 2 α2 Γ − Γ s̄2 − 2α2 Γ p √ √ q = 2 · Γ · α2 − s̄2 − 2α2 Γ √ . O( Γ) , (2.76) whereas, −u − 1 x2 b2 B= p 2 t Γ s̄2 − x2t b2 √ α2 Γ − b2 Γ(α2 − s̄2 − 2α2 Γ) p ≈ Γ s̄2 − x2t b2 √ α2 − b2 (α2 − s̄2 − 2α2 Γ) p . = s̄2 − x2t b2 (2.77) Z 1 So for a given finite s̄, we have B ∼ O(1). Thus, γ = 2 (arccosh B)xt db . 0 √ O( Γ) → 0, but Σ ∼ O(1), so it means max(Σ − γ) > 0. It is in unpaired s̄ states region. Thus, the transition line must also be placed in between where Γ − u 1. √ p √ √ −u− 1 x2 b2 Then we get xt = 2 −u − Γ2 − Γ 2u + s̄2 + Γ2 ∼ O( −u) and B = √ 22 t2 2 ∼ Γ 44 s̄ −xt b O( −u ) 1. Γ Z ⇒ γ=2 0 1 −u √ i (arccosh B)xt db ∼ O ln · −u . Γ h (2.78) In order to let γ comparable to Σ which is typically O(1), we should have: ⇒ h −u √ i O ln · −u ∼ O(1) Γ √ ln(−u) − ln Γ −u ∼ 1 ⇒ √ − ln Γ −u ∼ 1 ⇒ This is equivalent to Γ ∼ e 1 − √−u −u∼ 1 . (ln Γ)2 (2.79) which is also an essential singularity at u = 0 , Γ = 0 point. It means that the transition line is extremely near the u-axis but not the u = −Γ/2 line. 2.8 Numerical Evidence We now present some numerical results about the thermalization properties and the distinction between phases F1 and F3 (phase F2 is too narrow to clearly see in the size systems we can diagonalize). We exactly diagonalize Hamiltonians with up to N = 15 spins. We first put the Hamiltonian into a block diagonal form using basis states that are even and odd with respect to the system’s Z2 Ising symmetry, and then diagonalize each sector numerically to obtain all of the eigenstates within that sector. For finite N , the coefficient λ in the disorder term H1 in Eq. (2.2) matters. If λ is too large the system will become a spin glass rather than a ferromagnet. Thus λ needs to be carefully chosen to be large enough for our finite systems to show thermalization, 45 but small enough to avoid the spin glass regime. After some exploration, we chose to use the parameters λ = 0.7, p = 3/4 and Γ = 1/8 for our exact diagonalizations. 0.05 0.04 0.03 f 0.02 0.01 0 0 0.2 0.4 0.6 0.8 1 δ< /δ> Figure 2.6: The level-spacing statistics using 100 realizations of H at N = 15 in phase F1 within the even sector. δ< /δ> is the ratio between the smaller level spacing δ< to the larger level spacing δ> for three consecutive eigenenergies in the even sector. f is the relative frequency for each bin in this histogram. First, we show the level-spacing statistics, which should be GOE if the eigenstates are thermal. This must be done within one Z2 symmetry sector, since there is no levelrepulsion between states in different sectors. We look at each set of three consecutive levels in one sector and denote δ< as the smaller level spacing and δ> as the larger level spacing. Then the histogram of the ratio δ< /δ> can be compared to GOE level statistics [2]. The even sector results in the F1 phase for 100 realizations at N = 15 are shown in Fig. 2.6. We see the expected strong level repulsion, consistent with the 46 thermalization. All other phases and symmetry sectors were also examined and the results are also thermal, since phase F1 is the lowest-energy phase and thus the most difficult to thermalize. Next, we examine a ‘distance’ between two eigenstates that are adjacent in the energy spectrum by comparing their probability distributions for the total spin Sz . We define this distance between eigenstate 1 and eigenstate 2 as D12 = N/2 X P1 (Sz ) − P2 (Sz ) , (2.80) Sz =−N/2 where P1 (Sz ) and P2 (Sz ) are the probability distributions of Sz in eigenstates 1 and 2, respectively. We tested three different distances: Deo the distance between an even parity state (eigenstate 1) and the nearest-energy odd parity state (eigenstate 2), and similarly for Dee and Doo . If a system thermalizes, each eigenstate is equivalent to a microcanonical ensemble characterized by its energy. For two eigenstates that are adjacent in energy, the energy difference ∆E ∼ 2−N , therefore we expect the eigenstate distances Deo , Dee and Doo should decrease exponentially with N . In phase F1, since the spectrum consists of nearly-degenerate pairs of states, the energy differences satisfy ∆Eeo ∆Eee , ∆Eoo . Thus we expect in phase F1, Deo Dee , Doo . In addition, if we choose the upper bound of the energy window we average over to be well within the F1 phase, we would also expect that Eq. (2.84) holds, so the exponential decay rate of Deo would be greater than those of Dee and Doo . Meanwhile, in phases other than F1, we expect all three D’s are well coincident. The numerical results are shown in Fig. 2.7. As we expected, all these eigenstate distances decay exponentially with N . In addition, Deo in phase F1 is much smaller and decreasing much faster than the other two distances, and in phase F3 all three D’s are well coincident. This demonstrates the clear distinction between phases F1 and F3, and further tests the thermalization in phase F3. 47 −1 log(D) −2 −3 e−o e−e o−o −4 −5 −6 8 9 10 11 12 13 14 15 N (a) Eigenstate distance in phase F1 −1.5 e−o e−e o−o log(D) −2 −2.5 −3 −3.5 8 9 10 11 12 13 14 15 N (b) Eigenstate distance in phase F3 Figure 2.7: Averages of log (D) in phases F1 and F3, respectively, where D is the ‘eigenstate distance’ defined in the text.. The energy density range we used in F1 is from the first excited state in each sector up to uc − 0.02 where uc is the energy density at the phase boundary between F1 and F2, whereas in F3 we used the phase’s full energy density range. N is the total number of spins varying from 8 to 15. The exponential decrease of D with increasing N indicates thermalization. The error bars come from averaging over 100 realizations. 48 1.2 ᾱn at N = 11 ᾱn at N = 15 ∆αn at N = 11 ∆αn at N = 15 N →∞ 1 F1 0.8 αn 0.6 F2 0.4 F3 0.2 0 −0.14 −0.12 −0.1 −0.08 −0.06 u −0.04 −0.02 0 Figure 2.8: The mean ᾱn and the standard deviation ∆αn of the quantity αn defined in Eq. (2.81). The number of realizations is 1600 for N = 11 (blue dash-dotted lines) and 100 for N = 15 (red dashed lines). The green (solid) line gives the theoretical quantity α(u, Γ) defined in Eq. (2.32) for the system size N → ∞. In the ferromagnetic phases, the system spontaneously flips between magnetization up and down at a rate that behaves as ∼ exp(−N α(u, Γ)), where the quantity α(u, Γ) is defined in Eq. (2.32). The probability of the system having total magnetization zero (or 1/2 for systems with odd N ) also behaves as ∼ exp(−N α(u, Γ)). Thus from a single many-body eigenstate |ni we can obtain an estimate of the quantity α as αn = 1 maxSz {Pn (Sz )} ln , N Pn (Sz = 0 or 1/2) 49 (2.81) where Pn (Sz ) is the probability distribution of the total magnetization Sz in this eigenstate. If the ETH is true, the magnetization thermalizes and these estimates αn will converge to α(u, Γ) in the limit of large N . In Fig. 2.8 we show the mean (ᾱn ) and the standard deviation (∆αn ) of αn within energy bins for systems of size N = 11 and N = 15. The standard deviation decreases with increasing N , as expected for these Schrödinger cat states that obey the ETH. In the limit of large N every eigenstate at a given energy density will have the same probability distribution of the total magnetization. That distribution is a thermal distribution for the magnetizations that are accessed by thermal excitation and ‘tails’ due to quantum tunneling in the remainder of the distribution. The model we are studying here is a ferromagnet (H0 ) with a small added spinglass term (H1 ) in its Hamiltonian. In terms of its affect on the system’s thermodynamics, the relative strength of the spin-glass term scales as ∼ N −1/4 for the case p = 3/4 that we have here. Thus it is perhaps reasonable to expect the mean value of αn to exhibit a finite-size correction that vanishes for large N as ∼ N −1/4 . The spin-glass term weakly frustrates the ferromagnetism, so will cause a reduction in the apparent α. Given the very modest range of N for which we can do exact diagonalizations, this ∼ N −1/4 is a very slow convergence towards the thermodynamic limit. In Fig. 2.8 we also show the expected value of α in the limit of large N . As N is increased from 11 to 15, ᾱn does indeed increase slowly toward α(u, Γ), as expected. Thus the results of these exact diagonalizations provide some numerical support for the theoretical results derived above assuming this system obeys the ETH. Of course, given the very small size systems that can be diagonalized, there are strong finite-size effects that make a detailed demonstration of thermalization in the large system limit not possible. This is the situation with all numerical tests of thermalization. 50 In addition to the model discussed above, we tested two other ways of adding disorder to H0 . The other Hamiltonians that we diagonalized are N 1 X x H = H0 + p0 εi s i , N i=1 (2.82) N N X 1 X z z H =− (ηi + 1)(ηj + 1)si sj − Γ sxi , N 1=i<j i=1 (2.83) 0 and 00 where in H 0 , the εi are also independent Gaussian random variables of mean zero and variance one, and the power p0 here satisfies 0 < p0 < 1/2; specifically we looked at p0 = 1/4. In H 00 , the ηi are independent random variables uniformly distributed in the interval [−1, 1]. These latter two models only have N random parameters, one per spin, unlike Eq. (2.2) which has one per pair of spins: N (N − 1)/2 random couplings. This difference appears to be quantitatively important, as neither of these latter two models showed good evidence of thermalization under the tests illustrated above for the sizes that can be exactly diagonalized. But we expect that in the limit of large N this difference should go away, with all of these models thermalizing. 2.9 Conclusion In this chapter, we confirm that there are indeed three different eigenstate phases (F1, F2, F3) in the ferromagnetic phase of our infinite-range spin model by demonstrating both analytical and numerical evidence. Let me conclude this chapter by indicating the physical differences among these phases. In the lowest-energy phase (F1) where √ α(u, Γ) > Σ( −2u − Γ2 ) , 51 (2.84) the many-body eigenstates come in almost-degenerate pairs that are well separated in energy from other pairs of eigenstates. These eigenstates are still thermal in their fluctuations near equilibrium, so the inter-pair eigenenergy spacings have the level statistics of the Gaussian Orthogonal Ensemble (GOE). Each pair of eigenstates within phase F1 is well-approximated by the two states 1 |n, ±i = √ (| ↑in ± | ↓in ) , 2 (2.85) where | ↑in is a thermal state that is magnetized up, while | ↓in is its opposite under the global spin flip symmetry; here the label n refers to such a pair of eigenstates. In this regime (F1), one can make an up-magnetized initial state using a simple linear combination of only these two eigenstates, and this linear combination will then oscillate with a frequency ∼ exp (−N α(u, Γ)). Thus in this phase the Schrödinger cats can in principle be made to oscillate between ‘alive’ (up) and ‘dead’ (down). In the higher-energy parts of the ferromagnetic phase (F2 and F3 in Fig 2.1) where √ α(u, Γ) < Σ( −2u − Γ2 ) , (2.86) the energy associated with the tunneling is large compared to the many-body level spacing, so there are no closely degenerate pairs of states from which we can make a coherently oscillating cat. Instead, the eigenstates are of the form 1 |ni = √ (| ↑in + (−1)zn | ↓in ) , 2 (2.87) with each eigenstate being either even (zn = 0) or odd (zn = 1) under the global spin flip; here the label n refers to just one eigenstate. States of opposite symmetry (zn 6= zm ) that are nearly degenerate are not made out of the same thermal states in each well, so n h↑ | ↑im → δnm for N → ∞. To make a state that is initially magnetized 52 up requires coherently adding together exponentially many eigenstates in order to destructively cancel all the amplitudes for down magnetizations. Under the unitary time evolution this special initial linear combination will dephase and the probability of down magnetization will increase, but presumably in an overdamped fashion as the state relaxes to equal probability of up and down magnetization. Thus we expect the dynamics of the average magnetization to always be an overdamped relaxation in phases F2 and F3. In phase F1, on the other hand, one can in principle prepare an initial state whose macroscopic magnetization will oscillate, as discussed above. However, even in phase F1, if one starts in a generic state that has a given energy density and is magnetized up, this will also be a linear combination of exponentially many eigenstates and also presumable show an overdamped relaxation of the average magnetization. In the highest-energy phase (F3) where √ Γ − 2u πΓ ≥ ln Γ + 2u −u − Γ2 (2.88) the saddle point is ‘thermal’: the ‘entropy cost’ of going to higher S is less than the ‘tunneling cost’, and the system goes over the barrier without any quantum tunneling. We call the eigenstates in this regime ‘thermal cats’, since these Schrödinger cat states flip by thermally activating themselves over the barrier. In regions F1 and F2 this inequality is instead false, and the system quantum tunnels through the barrier at a value of S satisfying Seq ≤ S < −E/Γ, so the eigenstates are instead ‘quantum cats’. 53 Chapter 3 Strong Disorder Renormalization Group Approach to Many-body Localization Transition 3.1 Introduction As mentioned in Chapter 1, the many-body localization transition is an eigenstate phase transition. But it is not as standard as the eigenstate phase transition in Chapter 2 where the eigenstates on both sides of the transition and at the transition are all satisfying the ETH. Instead, the MBL phase is the only known generic exception to thermalization [19]. We know that there are many properties of the MBL phase which are not shared by the thermal phase, which does obey the ETH. For example, the area law of the eigenstate entanglement entropy for the MBL phase versus the volume law of the entanglement entropy for the thermal phase. These distinctions between the two phases drive us to study this special phase transition. Although there has been much recent progress in understanding the MBL phase by itself, for example [28, 25, 14, 12, 15, 3, 23, 33, 4, 24], very little is known about the eigenstate 54 phase transition between the MBL phase and thermal phases. In 2014, Vosk, Huse and Altman [30] formulated a new theoretical approach, a strong disorder renormalization group (RG) framework, to explore the physics of this critical point. However, in their framework, parts of the RG rules are ad hoc and, in addition, their approach seems to be more complicated than is necessary. This motivates us to think about variations of the rules. Ultimately, the set of RG rules we want is different from that found in [30], simpler and less ad hoc. But it is not easy, and may not even be possible, to simultaneously achieve all of these goals. I will not explain the RG scheme in detail until the next section, but in order to get a rough idea of how we arrived at a certain set of RG rules, here I would like to briefly mention the rules we initially tried. The RG scheme is for a one dimensional chain containing many blocks. Each block is a section of the system that all by itself looks either thermal or insulating and has a dimensionless quantity g to measure how thermal or how insulating the block is. The RG rules are just the rules for coarse-graining: we fuse several adjacent blocks into a new longer block and determine the parameters of the new block. As the RG flows, the number of blocks reduces steadily. The first two simple rules are the case of combining two adjacent thermal blocks and the case of combining two adjacent insulating blocks as shown in Fig. 3.1(a) and 3.1(b). As expected, after the combination, the new block becomes more thermal, or more insulating, respectively. But these two rules are of course not sufficient since they do not address the situation of a thermal block next to an insulating block. Thus, we need to construct rules for the cases where a thermal block meets with an insulating block. But not all such cases are simple, and it is for such cases that Ref. [30] introduced somewhat arbitrary rules, and in the end, so do we here. Also, it is in these cases where the competition between thermalization and localization lies. A configuration for which there exists a natural expectation for the RG flow is when two blocks of the same type surround a block of the opposite 55 type: that is, either a sequence of thermal, insulating and thermal, or a sequence of insulating, thermal and insulating blocks as shown in Fig. 3.1(c) and 3.1(d). If the Thermal Block Thermal Block Insulating Insulating Block Block Thermal Block Insulating Block (a) TT move Thermal Block Insulating Block (b) II move Thermal Block Insulating Block Thermal Block Thermal Block Insulating Block Insulating Block (c) TIT move (d) ITI move Figure 3.1: A sketch of typical RG moves. For example, (a) is to fuse two adjacent thermal (T) blocks into one thermal block, called “TT move”; all others are similar. blocks are consecutively thermal, insulating and thermal, we are able to set a simple rule that combines them into a new effective block that ensures, as expected, that this new block is thermal. This RG rule applies when the two thermal blocks are large enough to thermalize the insulating block in between them. However, difficulties arise in the other scenario, a sequence of insulating, thermal and insulating blocks. Our first trial at setting this rule in this case was an attempt to stay close to the microscopic physics and avoid an ad hoc RG rule. But the natural formula for such an RG step does not ensure the new block is insulating. In this first 56 trial, we simply forbid this RG step from happening if the new block turns out not to be insulating. But the numerical results show that this version of our RG gets stuck when the system is near the phase transition. This means that the system flows under this RG to configurations from which no moves are possible. This means that our basic goal of studying the RG fixed point governing the phase transition is not possible since no such fixed point exists in this version of the RG. In order to resolve this, we apparently need to modify the rules so the system never gets stuck. This motivates our second trial: allowing the move (combining insulating (I), thermal (T) and insulating (I) blocks, called “ITI move”) any time. And in this scenario, when the new block happens to not be insulating, we set it as a thermal block, and at the same time combine it with the two other thermal blocks which are adjacent to this new thermal block. After this treatment, the final new block is thermal. This successfully solves the previous issue of the RG getting stuck, however, this introduces new difficulties. When we tried this set of RG rules, we found that the RG becomes very unstable and oscillates, such that we cannot even see an RG fixed point and the RG flow appears to go towards a limit cycle at the phase transition, again preventing our goal of studying the fixed point. After these two failed attempts at developing a new RG, we decided to be less ambitious and to give up the goal of avoiding ad hoc rules. The hope is to develop a simple “toy” RG for this phase transition that can then be improved and built upon. One solution to the difficulties with the first two RG schemes is to impose a constraint that the new block of the basic ITI move must be an insulating block. We include a rule for our ITI move that satisfies this constraint. This gives us our “final” version of the RG rules. Although this version is still ad hoc, the RG equation is largely simplified and we can nearly get an analytical solution to the fixed point distribution and also the critical exponent. Thus, this version becomes a “toy model” and we can explore this model in great detail. It also gives us hope for studying the 57 real physical models by perturbing this toy model to be less ad hoc and to use this toy model as a reference point. Here in this chapter, I will discuss this toy model in detail after setting the RG scheme. 3.2 RG Scheme Let us consider a generic one dimensional chain of “blocks” with different length. The microscopic details of the chain are coarse-grained by doing the renormalization group flow and only a minimal set of parameters remains in the coarse-grained system. This suggests that this approach can describe several microscopically distinct systems, including, but not limited to, spin chains and lattice particles. For now, let us assume for specificity that it is a spin-1/2 chain. Assume that there are, in total, N blocks numbered from 1 to N . We impose periodic boundary conditions such that the N th block is adjacent to the 1st block. This is convenient because the final goal is to take the thermodynamic limit N → ∞ and the periodic boundary conditions eliminate boundary effects when we coarsegrain the system with finite N . We characterize each block i by only two parameters: Γi and ∆i . Γi is the end-to-end entanglement rate. The physical meaning of Γi can be obtained in the following setup: One connects the block i at one end to a much longer thermalizing block j where j can serve as a heat bath to thermalize i, and one isolates them from the environment. Then the two block system starts with a pure product state (no entanglement between blocks i and j) and one evolves the state under the unitary time evolution due to the Hamiltonian of these two blocks. The entanglement entropy will grow from 0 and saturate at the time scale τi = 1/Γi . ∆i is the typical many-body level spacing of block i which is set by the length of the block. If we denote the length of block i as li , then for a spin-1/2 chain the dimension of the Hilbert space is 2li . So we have the typical level spacing ∆i ∼ 2−li , up to some 58 prefactors that are a power of li . Note that we are considering the strong disorder RG and assuming the critical fixed point is at infinite randomness, which means that at the fixed point we study, the logarithm of the rates Γi and ∆i have a broad probability distribution [11]. Thus, these power law prefactors can be neglected in this strong disorder RG scheme. Next, we divide these N blocks into two groups, depending on whether a block locally behaves more like an insulator (called “I-block”) or more like a thermalizing system (called “T-block”). If a block only exhibits short range entanglement in its eigenstates, it is considered an insulator. On the other hand, in a thermal block, the entanglement is spreading fast and the whole block is entangled. In terms of the two parameters Γi and ∆i , block i is considered as an I-block when Γi ∆i and it is a T-block when Γi ∆i . If we define a dimensionless parameter gi as follows: gi ≡ Γi , ∆i (3.1) we compare whether gi is much smaller or larger than 1 to decide whether block i is insulating or thermal, respectively. Still, note that we are considering the strong disorder RG, so when we are approaching the critical point, the probability of having g ∼ 1 vanishes. In practice, we treat any g > 1 as g 1, and any g < 1 as g 1. Before proceeding, let me mention several issues about Γ and ∆. The first issue is the dimension. Although both Γ and ∆ are dimensionful quantities with the same dimensions (energy), we can redefine them as dimensionless quantities, relative to a typical microscopic energy scale in the system: Γ0i ≡ Γi /E , ∆0i ≡ ∆i /E where E is a typical energy scale at the microscopic scale. The second issue is the regime. We are interested in the low energy regime where both Γi 1 and ∆i 1 for any block i. For ∆i , because ∆i ∼ 2−li for block i where li is the length of the block, it is reasonable to assume ∆i 1, since we are considering long, coarse-grained blocks. For Γi , we 59 split the discussion into two cases: one for insulating blocks, one for thermal blocks. First, if the block is insulating, the many-body level spacing for that single isolated block is large and the eigenstates are localized. This means that the entanglement spreading is basically due to quantum tunneling and is exponentially slow in the length. In the second case, for a thermal block, if the block were deeply thermal, the entanglement would spread fast. But the system is now near the critical point and the Griffiths effect takes place [30]. The Griffiths effect says that even within a thermal block there will be shorter segments that are localized and act as a bottleneck for the entanglement spreading across the thermal block. The entanglement needs to tunnel through these bottlenecks. Hence the entanglement spreading rate Γi is also slow even in thermal blocks, when we are near the phase transition. Thus, we have Γi 1 for all blocks. After parametrizing each block, we can discuss the coarse-graining procedure, the essence of the RG flow. By coarse-graining the chain, we are combining several blocks into one single “new” block with only two new parameters Γnew and ∆new remaining as a function of the previous parameters. In this way, we reduce the amount of microscopically detailed information. In this framework, we focus more and more on the physics on longer length scales, which correspond to lower energy scales. In the theory of phase transition, it is this low-energy, long-length-scale limit where the behavior is universal, and the RG is set up to capture that universal behavior by focusing on these limits. Here we are assuming that this general framework will also apply to this new type of eigenstate phase transition between MBL and thermalization. 3.2.1 RG Rules After briefly describing the RG framework, we are now ready to discuss the RG rules, which govern how the new parameters relate to the old ones. 60 Building Up from Simple Cases First comes the simple rules for combining two adjacent I-blocks or two adjacent T-blocks. Consider the two adjacent I-blocks case, and call the blocks block 1 and block 2. Let the parameters for block i be Γi and ∆i as before, where i = 1, 2. For the new Γ, we multiply the two old Γ’s. This is because in insulators, the entanglement spreads basically by quantum tunneling and the spreading time is exponentially slow. For the new ∆, we similarly multiply the two old ∆’s. This is because ∆ is roughly the inverse of the dimension of the Hilbert space of the corresponding block and when we combine the blocks, the dimension of the Hilbert space multiplies. And this is true in general, so for all the rules for ∆, we just get the new ∆ by multiplying all the original ∆’s involved in the combination. So for the two I-blocks case, we get the following RG rule: Γnew = Γ1 Γ2 , (3.2) ∆new = ∆1 ∆2 . Consider now two adjacent T-blocks. For the new Γ, because both blocks are thermal blocks, they are both well entangled and the entanglement is spreading fast and the spreading time basically adds. This means that the inverse of the rates add: Γnew = 1 . 1 1 + Γ1 Γ2 (3.3) But we are dealing with the strong disorder RG and we will assume that we are working near an infinite-randomness fixed point, which means that Γ1 will be either much larger than or much smaller than Γ2 . In both cases, the right hand side of the above formula is approximately equal to the minimum of Γ1 and Γ2 (the slower of the 61 two rates is the “bottleneck”): Γnew ≈ min{Γ1 , Γ2 } . (3.4) And the formula for ∆new is the same as in the I-blocks case, for the same reason. So for the two T-blocks case, we arrive at the following rules: Γnew = min{Γ1 , Γ2 } , (3.5) ∆new = ∆1 ∆2 . By combining neighboring blocks of the same type, one would naturally expect that the resulting new block must share the same type. This can be seen by comparing the new Γ and new ∆. For the insulating case, we have Γ1 ∆1 and Γ2 ∆2 by definition. So we have Γnew = Γ1 Γ2 ∆1 ∆2 = ∆new . (3.6) Thus, the new block is an insulating block, as expected. And in fact, the new block is more insulating than the original ones in the sense that the ratio g ≡ Γ/∆ in the new block is smaller than the original g’s. Note that g 1 is the condition for a block being insulating, and the smaller value a block has for g, the more insulating it is. For the thermal case, we have Γ1 ∆1 and Γ2 ∆2 by definition. As mentioned before, we are interested in the regime where Γi , ∆i 1 for any block i. So we have Γ1 ∆1 ∆2 Γ2 ∆1 ∆2 . (3.7) Γnew = min{Γ1 , Γ2 } ∆1 ∆2 = ∆new . (3.8) and Thus, 62 Therefore, the new block is a thermal block, as expected. Similarly, we can also show as follows that the new ratio gnew ≡ Γnew /∆new is greater than both g1 and g2 which indicates that the new block is more thermal. By the exchange symmetry between subscripts 1 and 2, we only need to show gnew > g1 which is gnew = min{Γ1 , Γ2 } Γ1 > g1 = . ∆1 ∆2 ∆1 (3.9) We know that Γ1 Γ1 > ∆1 ∆2 ∆1 (3.10) Γ2 1 Γ1 > > ∆1 ∆2 ∆1 ∆1 (3.11) because ∆2 < 1; because g2 > 1 and Γ1 < 1. Combining the above two expressions, we get the desired statement gnew > g1 . In these two simple cases, we showed that any two adjacent blocks which are both insulating or both thermal can combine into one block which is even more insulating or more thermal, respectively. This means that as a first step in the RG, we can combine sequences of adjacent I-blocks or T-blocks together, which then result in a sequence of the blocks alternating between I-blocks and T-blocks: ··· I T I T I T ··· . Next we must set up a series of RG rules to do the coarse-graining starting from this new pattern. 63 Order of Flow: the Cutoff In our “toy” model, we have two RG rules. The first rule is to combine 3 adjacent T-I-T blocks into a new T-block (called “TIT rule”) and the second rule is to combine 3 adjacent I-T-I blocks into a new I-block (called “ITI rule”). Then the RG will run by “integrating out” the fastest block. To determine how fast a block is, we define the rate Ωi for each block i as follows. For thermal blocks we use ∆ and for insulating blocks we use Γ: ∆T , Ωi = Γ , I if the block is thermal; (3.12) if the block is insulating. In this way, we can make a list ordered by Ωi . Then the maximal value in this list is defined as the cutoff Ω: Ω = max{Ωi } . i (3.13) At every iteration of the RG, we select the move in which the central block i has the Ωi value equal to the cutoff Ω. In this way, we first ensure that the flow will never get stuck because the maximal value of Ωi always exists. Second, by selecting the maximal energy Ω every time, we lower the cutoff steadily, and focus more and more on the lower energy behavior, which is the spirit of the RG framework. After setting the cutoff Ω, let me discuss both rules one by one in detail. TIT Rule In order to address both rules, we need to understand what happens physically when we connect a T-block with parameters ΓT , ∆T to an I-block with parameters ΓI , ∆I . If we treat the combined two blocks as an isolated quantum system, it is natural to expect the final result depends on whether the thermal block can serve 64 as a heat bath that can thermalize the I-block. We argue that the condition for a T-block to thermalize an I-block is ΓI ∆T . Note that, by definition, ΓI is the end-to-end entanglement rate of the I-block when it combines with a large enough thermal block. So the rate will remain ΓI as long as the T-block considered here has the level spacing ∆T ΓI . In this way, the T-block has a continuum spectrum relative to ΓI . So ΓI ∆T is the condition that the T-block serves as a heat bath and thermalizes the I-block. After getting the condition for a T-block thermalizing an adjacent I-block, we then set our TIT rule. We denote these three blocks as 1, 2 and 3 with the type T, I and T, respectively. In order for these three blocks to combine to one new thermal block, we need at least one of the T-blocks to be able to thermalize the middle I-block. The RG only does this move if the I-block is faster than both T-blocks, since it “integrates out” the fastest blocks first. So we have Γ2 ∆1 and Γ2 ∆3 . (3.14) Under this condition, we can derive the parameters of the new block Γnew and ∆new . For Γnew , the argument is similar as the two T-blocks (TT) simple case. Since the middle block is successfully thermalized, it is well entangled with the thermal blocks and the entanglement spreading is fast. Thus the entanglement time still adds as in the TT case. So Γnew becomes the minimal value of all 3 original Γ’s as it is in the TT case: Γnew = min{Γ1 , Γ2 , Γ3 } . (3.15) And the rule for ∆new is still as simple as before: ∆new = ∆1 ∆2 ∆3 . 65 (3.16) Note that these two formulae ensure that the new block is indeed a thermal block. We can check it by evaluating the parameter gnew = Γnew /∆new . Because Γ1 Γ1 1 ∆1 ∆2 ∆3 ∆1 (3.17) and similarly for Γ2 or Γ3 as being the numerator. Then we have gnew = Γnew min{Γ1 , Γ2 , Γ3 } = 1 ∆new ∆1 ∆2 ∆3 (3.18) which indicates the new block is automatically thermal. ITI Rule Next comes the ITI rule, which is the RG rule that is difficult to determine in any controlled way, and was the source of the difficulties in our attempts to avoid making ad hoc assumptions. Here we also denote the 3 consecutive blocks as 1, 2 and 3; this time block 2 is the middle T-block, which is the fastest of these three blocks. Similar to the TIT rule, this time we need the middle T-block to not be able to by itself thermalize either block 1 or block 3. So one condition for the ITI rule is that ∆2 Γ1 and ∆2 Γ3 . (3.19) The ∆new is still the product of the original ∆’s; this follows simply from the dimension of the product Hilbert space: ∆new = ∆1 ∆2 ∆3 . (3.20) For Γnew in this case, things become a little arbitrary, as mentioned before, and I will explain this later on. Because neither block 1 nor block 3 is thermalized by the middle T-block, both of them are insulating. The entanglement spreads very slowly 66 in both block 1 and block 3. As in the two I-blocks case, Γnew must at least contain the factor Γ1 Γ3 . But if we just set Γ1 Γ3 as the new Γ, it would not guarantee that the new block is insulating or gnew 1 as we would expect. In order to fix this issue, we add a multiplier which is smaller than or equal to ∆2 to ensure that the new block is insulating. To minimize the modification, we set Γnew = Γ1 ∆2 Γ3 . (3.21) Although this ITI rule for Γnew is a bit arbitrary, it turns out to give us a beautifully simple solution to the RG fixed point. Moreover, we can almost tackle it analytically a la Fisher [11]! I will present the results in the next section. 3.2.2 RG Flow of Probability Distributions We can run this RG flow from any given chain of alternating T and I blocks. For example, we can start from drawing Γ’s and ∆’s from a given probability distribution and form the chain. Then our RG rules give us an evolution of the probability distribution. In general, the distribution is 2n dimensional if the total number of blocks is n: P ({ln Γi }, {ln ∆i }) , (3.22) where we take the natural logarithm for convenience. Here we assume the parameters in different blocks are independent and one can show that this remains true under the RG. So the distribution simplifies to the following two joint probability distributions: PT (ln ΓT , ln ∆T ) and 67 PI (ln ΓI , ln ∆I ) , (3.23) where the subscripts T and I indicate the T-block distribution and I-block distribution because these two distributions should in general differ from each other. One further simplification can be made based on the following observation. If we focus on the parameter ΓT , by checking all the formulae, we see that the ΓT ’s do not influence any of the other quantities, only themselves. So we can ignore the parameter ΓT in the distribution PT . The distributions then become PT (ln ∆T ) and PI (ln ΓI , ln ∆I ) . (3.24) Our aim is to find out the scaled distributions for Γ’s and ∆’s at the critical point which, by assumption, should not change because the critical point typically behaves as a fixed point in the RG flow. And note that the parameters of the distributions should also be rescaled by the cutoff, so in the end, we are interested in the evolution of the following distribution functions: PT 3.3 ln ∆ T ln Ω and PI ln ∆I . ln Ω ln Ω ln Γ I , (3.25) Results As mentioned previously, the solution to the fixed point distribution can be found nearly analytically. In this section, I will present both analytical and numerical results regarding the fixed point distribution and the critical exponent. 3.3.1 Fixed Point Distribution First, let us try to simplify the form of the fixed point distribution. We have 3 variables: ln ∆T , ln ΓI and ln ∆I . All of them are proportional (or extensive) to ln Ω when we send the cutoff ln Ω → −∞. One nice feature is that when the RG flows, ln gI becomes sub-extensive to ln Ω because when we apply the ITI rule to three 68 consecutive blocks 1,2 and 3, ln gInew (ln Γ1 + ln ∆2 + ln Γ3 ) − (ln ∆1 + ln ∆2 + ln ∆3 ) = new ln ΓI ln Γ1 + ln ∆2 + ln Γ3 ln g1 + ln g3 = . ln Γ1 + ln Γ3 + ln ∆2 (3.26) If ln ∆2 were zero, then the ratio ln gI / ln ΓI would become fixed when the distribution goes towards the fixed point. But here ln ∆2 is extensive, which means that when all the other quantities approach the fixed point, ln gI / ln ΓI are still steadily decaying. Thus, ln gI ln gI ∼ → 0. ln Ω ln ΓI (3.27) This is a very nice feature because we can then treat the following two intensive quantities as equal at the fixed point: ln ΓI ln ∆I = . ln Ω ln Ω (3.28) Note that this does not affect the strong disorder of the fixed point where ln gT,I are far away from 0, because we have ln gI = ln g1 + ln g3 , (3.29) the absolute value of ln gI becomes larger and larger. The numerical simulation of the RG flow shows that | ln gI | ∼ | ln Ω|p where the power p ≈ 0.8. It verifies while ln gI is growing by itself, ln gI / ln Ω is still an irrelevant operator. Then we only have two variables: ∆T and ∆I . Because the rules for ∆T and ∆I are symmetric between T and I, their distributions will be equal at the fixed point. This further reduces the number of variables to one when we consider only the fixed point; we denote this 69 variable by ln ∆. And the RG rule for it is simply ln ∆new = ln ∆1 + ln ∆2 + ln ∆3 (3.30) when we merge blocks 1, 2 and 3, which we do only when ∆2 = Ω, so ∆2 ∆1 and ∆2 ∆3 . After reducing the problem to a single variable problem, we may then construct the analytical equation for the probability distribution, following Fisher [11]. We define two quantities as Fisher did: Λ ≡ − ln Ω (3.31) and ζ ≡ ln Ω ∆ . (3.32) Then we have the constraint ζ ≥ 0 and large ζ corresponds to small ∆. In terms of Λ and ζ, the RG rule Eq. (3.30) becomes ζnew = ζ1 + ζ2 + ζ3 + 2Λ = ζ1 + ζ3 + 2Λ , (3.33) where ζ2 = 0 because the second block is at the cutoff: ∆2 = Ω. Let the distribution density of ζ at cutoff Λ be ρ(ζ, Λ), such that the probability of ζ in the interval [ζ, ζ + dζ] is ρ(ζ, Λ)dζ. As the cutoff Λ gets large, the distribution of ζ becomes broad. This leads us to consider the following transformation: let η≡ ζ , Λ ρ(ζ, Λ) ≡ 1 1 ζ Q(η, Λ) = Q( , Λ) . Λ Λ Λ 70 (3.34) Thus, Q(η, Λ) is just the probability distribution for η when the cutoff is Λ, and the RG rule for η is ηnew = η1 + η3 + 2 , (3.35) which follows from Eq. (3.33). What we want to find out is the evolution of the distribution Q(η, Λ) when the cutoff Λ is increasing, namely distribution ρ(ζ, Λ), namely ∂Q(η,Λ) . ∂Λ ∂ρ(ζ,Λ) . ∂Λ But it is easier to find out the evolution of the Thus we analyze the process for the change of ρ when the cutoff Λ changes by dΛ. In order to successfully change the cutoff, we need to first eliminate all the blocks which have the ln ∆ in the interval [ln Ω + d ln Ω, ln Ω], or [ln Ω − dΛ, ln Ω] in terms of dΛ. In terms of ζ, the points (blocks) we need to remove have the value ζ in the interval [0, dΛ] which has a fraction of points equal to ρ(0, Λ)dΛ. After removing all the points in this interval, we then re-normalize the distribution so that it integrates to 1. Up to now, the available ζ is in the interval [dΛ, +∞). Thus, the next step after re-normalization is to shift the lower edge of the the distribution horizontally by dΛ so that the domain of ζ returns to be [0, +∞). By finishing these steps we get the evolution of the distribution ρ(ζ, Λ), and the final step is to rescale the variable ζ into η to get the evolution of the distribution Q(η, Λ). Let me analyze these steps one by one. The first step is to remove all ζ in the interval [0, dΛ]. These are the fast blocks that are being integrated out. Each time we remove one ζ in this interval, we randomly pick up two other points ζa and ζb with probabilities ρ(ζa , Λ)dζa and ρ(ζb , Λ)dζb separately. These are the two neighboring blocks. These three points form the three original blocks in the TIT or ITI move. Based on the RG rule Eq. (3.33), the new block we generate has ζc = ζa + ζb + 2Λ. But the increase at ζc is not only from the pair ζa and ζb , but also from all other pairs, so long as the sum of the pair is equal to ζa + ζb , or 71 ζc − 2Λ. Thus, the total probability of selecting ζc in dζc is ∞ Z dζc ∞ Z dζb ρ(ζa , Λ)ρ(ζb , Λ)δ(ζc − ζa − ζb − 2Λ) . dζa 0 (3.36) 0 If we consider the change of ρ at ζ, we have three cases: ζ = ζa , ζ = ζb or ζ = ζc . In other words, for the selection of ζa (or ζb , ζc ), it only affects the distribution at point ζa (or ζb , ζc ). And in the first step, we in total need to do a fraction ρ(0, Λ)dΛ of RG moves. Therefore, in this step the total change in the probability density function at ζ is h dρ(ζ, Λ) = ρ(0, Λ)dΛ − ρ(ζ, Λ) − ρ(ζ, Λ) Z ∞ Z ∞ i dζa dζb ρ(ζa , Λ)ρ(ζb , Λ)δ(ζ − ζa − ζb − 2Λ) . + 0 0 (3.37) The second step is to re-normalize ρ such that the integral of ζ is 1. In the first step, the integral of ρ(ζ, Λ) is decreased: for the selection of ζa , we lose ∞ Z ρ(0, Λ)dΛ dζa ρ(ζa , Λ) = ρ(0, Λ)dΛ ; (3.38) dζb ρ(ζb , Λ) = ρ(0, Λ)dΛ ; (3.39) 0 for the selection of ζb , we lose Z ρ(0, Λ)dΛ ∞ 0 for the selection of ζc , we gain Z ρ(0, Λ)dΛ ∞ Z dζc 0 ∞ Z 0 ∞ dζb ρ(ζa , Λ)ρ(ζb , Λ)δ(ζc − ζa − ζb − 2Λ) = ρ(0, Λ)dΛ ; dζa 0 (3.40) 72 for the selection at the cutoff, ζ = 0, we lose ρ(0, Λ)dΛ. Thus, we in total lose 2ρ(0, Λ)dΛ. In order to re-normalize ρ, we need to amplify it everywhere by a factor of 1 + 2ρ(0, Λ)dΛ. The resulting ρ after re-normalization is n h ρ(ζ, Λ) + ρ(0, Λ)dΛ − 2ρ(ζ, Λ) Z ∞ Z ∞ io + dζa dζb ρ(ζa , Λ)ρ(ζb , Λ)δ(ζ − ζa − ζb − 2Λ) · (1 + 2ρ(0, Λ)dΛ) 0 0 h = ρ(ζ, Λ) + ρ(0, Λ)dΛ − 2ρ(ζ, Λ) Z ∞ Z ∞ i dζa dζb ρ(ζa , Λ)ρ(ζb , Λ)δ(ζ − ζa − ζb − 2Λ) + ρ(ζ, Λ) · 2ρ(0, Λ)dΛ + 0 0 Z ∞ Z ∞ = ρ(ζ, Λ) + ρ(0, Λ)dΛ dζa dζb ρ(ζa , Λ)ρ(ζb , Λ)δ(ζ − ζa − ζb − 2Λ) . (3.41) 0 0 The third step is to shift the whole function horizontally to start from 0. The amount of this shift is just dΛ. So the change ρ in this step is ∂ρ dΛ. ∂ζ After this, we complete one infinitesimal process of changing the distribution. The total amount of change is Z ∞ Z ∞ dζb ρ(ζa , Λ)ρ(ζb , Λ)δ(ζ − ζa − ζb − 2Λ) + dζa dρ = ρ(0, Λ)dΛ 0 0 ∂ρ dΛ ∂ζ ∂ρ ≡ dΛ . ∂Λ (3.42) Thus, we obtain the equation for the RG flow of the probability distribution ρ(ζ, Λ): ∂ρ ∂ρ = + ρ(0, Λ) ∂Λ ∂ζ Z ∞ Z 0 ∞ dζb ρ(ζa , Λ)ρ(ζb , Λ)δ(ζ − ζa − ζb − 2Λ) . dζa (3.43) 0 The last step is to rescale ζ into η, and we have i ∂ρ ∂ h1 ζ = Q( , Λ) ∂Λ ∂Λ Λ Λ 1 η ∂Q 1 ∂Q + ; = − 2Q − 2 Λ Λ ∂η Λ ∂Λ 73 (3.44) and ∂ρ 1 ∂Q = 2 . ∂ζ Λ ∂η (3.45) The integral becomes Z ∞ Z ∞ dζb ρ(ζa , Λ)ρ(ζb , Λ)δ(ζ − ζa − ζb − 2Λ) dζa 0 0 ζ−2Λ Z ρ(ζa , Λ)ρ(ζ − ζa − 2Λ, Λ)dζa =Θ(ζ − 2Λ) 0 1 = Θ(η − 2) Λ Z η−2 Q(ηa , Λ)Q(η − ηa − 2, Λ)dηa , (3.46) 0 where Θ(x) is the step function. By plugging these relations into the equation of ρ, we arrive at Z η−2 i ∂Q ∂ h = (1 + η)Q + Q(0, Λ)Θ(η − 2) Q(ηa , Λ)Q(η − ηa − 2, Λ)dηa . Λ ∂Λ ∂η 0 (3.47) Given the “toy” RG rules we are using, this (3.47) is the RG equation for the probability distribution Q flowing as a function of the cutoff Λ at the critical point, where the T and I blocks have the same distributions of ∆. We will return to the RG flow away from this fixed point later. When the left hand side of this equation is equal to zero, the solution, denoted by Q∗ (η), becomes the critical fixed point distribution which is independent of Λ. The fixed point distribution Q∗ (η) satisfies i dh (1 + η)Q∗ + Q∗ (0)Θ(η − 2) dη Z η−2 Q∗ (ηa )Q∗ (η − ηa − 2)dηa = 0 . 0 74 (3.48) After getting the equation Eq. (3.48) for the fixed point distribution Q∗ (η), we can then try to solve it. First is the value at zero, Q∗ (0). We can show that Q∗ (0) = 1 2 (3.49) based on the following argument: Consider the sum of ln ∆i over all the blocks indicated by i when the system is at the fixed point. By the rule of ln ∆, Eq. (3.30) conserves this summation, so we know that as the RG flows, this quantity stays constant. And we have X ln ∆i = −Λ X (1 + ηi ) = −ΛN (Λ)hηif.p. i (3.50) i where N (Λ) is the total number of blocks when the cutoff is Λ and hηif.p. is the average value of η at the fixed point which is independent of Λ. This means the product ΛN is a constant, so ΛdN + N dΛ = 0 . (3.51) In addition, based on the analysis of obtaining ρ(ζ, Λ), when the cutoff is moving by an infinitesimal dΛ, we lose a fraction of 2ρ(0, Λ)dΛ points (blocks). This means dN = −2ρ(0, Λ)dΛ . N (3.52) Combining the two equations we get ρ(0, Λ) = − 1 dN 1 = , 2 N dΛ 2Λ 75 (3.53) thus Q∗ (0) ≡ Λρ(0, Λ) = 1 . 2 (3.54) It is very helpful to obtain this Q∗ (0) value because it serves as an initial condition for determining the rest of the Q∗ (η), and in fact, we can integrate Eq. (3.48) iteratively as follows. First consider the interval 0 < η < 2. In this interval, the step function Θ(η − 2) kills the contribution from the integral in Eq. (3.48). And the equation reduces to i dh (1 + η)Q∗ = 0 , dη for 0 < η < 2 . (3.55) Then the solution in this interval is simply given Q∗ (0) = 1/2: Q∗ (η) = 1 , 2(1 + η) for 0 < η < 2 . (3.56) Second is for the interval 2 < η < 4. When η is in this interval, the integral in Eq. (3.48) starts contributing. But it contributes in a simple way: the integrand only depends on values of Q∗ (η) up to η = 2, which I have just presented. So we have Z i 1 η−2 1 1 dh ∗ (1 + η)Q = − · dηa dη 2 0 2(1 + ηa ) 2[1 + (η − ηa − 2)] ln(η − 1) =− , for 2 < η < 4 . 4η (3.57) Thus, we obtain a closed form for Q∗ (η) in 2 < η < 4: 1 h1 Q (η) = − 1+η 2 ∗ Z 2 η ln(η 0 − 1) 0 i dη , 4η 0 76 for 2 < η < 4 , (3.58) where [(1 + η)Q∗ (η)]|η=2 = 1/2 serves as the initial condition. In principle, we can get the analytical form of Q∗ (η) for any η ≥ 0 by using exactly the same iterating method and treating Q∗ (η) piecewise. This shows that the physical solution to Eq. (3.48) is unique. In practice, we obtain Q∗ (η) to numerical precision by doing this iteration numerically, as is discussed below. After getting the solution Q∗ (η), let me study the behavior of this function to get an idea of what this function looks like. First, one can show that although Q∗ (η) is piecewise, it is continuous up to (2l − 1)th order derivative at the connecting point η = 2l, where l is a positive integer. This can be proven using mathematical induction as follows. At η = 2, from the analytical solution from both sides, Eq. (3.56) and (3.58), we see that Q∗ is continuous in its first order derivative but the second order derivative has a finite jump: ∆Q∗(2) (2) = Q∗(2) (2+ ) − Q∗(2) (2− ) 6= 0. Then by taking nth order derivatives on Eq. (3.48) at the argument η > 2, we have n−2 ∗(n) (1 + η)Q ∗(n−1) + nQ 1 X ∗(k) Q (η − 2)Q∗(n−2−k) (0+ ) + 2 k=0 Z 1 η−2 ∗ + Q (ηa )Q∗(n−1) (η − ηa − 2)dηa = 0 , 2 0 (3.59) where we have already plugged in Q∗ (0) = 1/2. Then, we evaluate this equation at η = 2l+ and η = 2l− and take the difference between them to get h i h i (1 + 2l) Q∗(n) (2l+ ) − Q∗(n) (2l− ) + n Q∗(n−1) (2l+ ) − Q∗(n−1) (2l− ) n−2 i 1 X ∗(n−2−k) + h ∗(k) + Q (0 ) Q (2(l − 1)+ ) − Q∗(k) (2(l − 1)− ) 2 k=0 Z + 1 2(l−1) ∗ + Q (ηa )Q∗(n−1) (2l − ηa − 2)dηa = 0 . 2 2(l−1)− (3.60) Assuming Q∗ (η) is continuous at 2l − 2 up to (2l − 3)th order derivative and with a finite jump at (2l − 2)th order derivative, and since by definition Q∗ (η) is everywhere 77 continuous, we can easily see that Q∗ (η) is continuous up to (2l−1)th order derivative. In addition, we have ∆Q∗(2l) (2l) = 1 ∆Q∗(2l−2) (2l − 2) , 2(1 + 2l) (3.61) so the jump size decreases exponentially fast. Second, one can also study the asymptotic behavior as η → ∞. In Fisher’s paper [11], he got the simple exponential distribution for the fixed point distribution from his RG equation. So we expect that in our Q∗ (η), it will also have an exponential decay when η → ∞. I can verify1 here that the Q∗ (η) in our case also decays exponentially. Assume Q∗ (η) ∼ CQ exp(−λQ η) (3.62) when η 1. Then we have Z η−2 ∗ ∗ Z η−2−A Q (ηa )Q (η − ηa − 2)dηa ≈ 0 CQ2 e−λQ ηa e−λQ (η−ηa −2) dηa A ≈ ηCQ2 e−λQ η+2λQ , (3.63) where A is a large constant. Thus, Eq. (3.48) becomes 1 CQ e−λQ η − λQ CQ (1 + η)e−λQ η + ηCQ2 e−λQ η+2λQ ≈ 0 . 2 (3.64) So the leading order is O(ηe−λQ η ) when η 1. By setting this leading order equal to zero, we get 1 −λQ CQ ηe−λQ η + ηCQ2 e−λQ η+2λQ = 0 . 2 1 (3.65) Although it is not a proof, the numerical integration result I will present later shows that it is indeed the case. 78 Then we have an estimation of the exponent λQ which satisfies 1 λQ = CQ e2λQ . 2 (3.66) This verifies the exponential decaying behavior when η 1. After getting the analytical solution of Q∗ (η) for small η and some other analytical properties, I would like to move on to the numerical solution of Q∗ (η) by doing the numerical integral iteratively. In particular, 20000 points are evenly sampled in each interval [2l, 2l + 2) for l = 1, 2, 3, · · · and the double integration for solving (1 + η)Q∗ (η) is numerically performed using composite trapezoidal rule. The integration is performed up to η = 20. The results are shown in Fig. 3.2. Fig. 3.2(a) gives the numerical curve and Fig. 3.2(b) shows the curve in a semi-log plot as well as a straight line from a linear regression for η > 5. It can be seen that the plot already looks exponential for small η ∼ 2 and the linear fit for relative large η is almost perfect. The fitting generates λQ ≈ 0.3725 and a normalization CQ ≈ 0.3537. Since 2λQ exp(−2λQ ) ≈ 0.3537, we see 2λQ exp(−2λQ ) = CQ is satisfied. Fig. 3.2(c) reports the cumulative area under the curve Q∗ (η) up to η = 20, and we see it seems to converge to 1, confirming the normalization of Q∗ (η) to 1. In particular, until η = 20 the total area under the curve is 0.9995, and if we assume Q∗ (η) = CQ exp(−λQ η) after η > 20, we can get an approximate asymptotic area 1.0000, which again confirms the normalization. 3.3.2 Critical Exponent After getting the theoretical solution for the critical fixed point distribution Q∗ (η) for this “toy” RG, we may perturb the distribution a bit away from Q∗ (η) to study the critical exponent given by the RG flow. In order to move off the critical point, we need to allow T and I blocks to have different distributions of η when the cutoff is Λ: 79 100 0.5 10-1 0.3 Q ∗ ( η) Q ∗ ( η) 0.4 0.2 10-2 10-3 0.1 0.00 RG solution Fitted Curve 5 10 η 15 10-4 0 20 (a) Q∗ (η) 5 10 η 15 20 (b) Q∗ (η) in log scale 1.0 Cumulative Q ∗ 0.8 0.6 0.4 0.2 0.00 5 10 η 15 20 (c) Cumulative Q∗ (η) Figure 3.2: Q∗ (η) up to η = 20 using composite Trapezoidal rule. (a) shows the curve and (b) demonstrates it in a semi-log plot with a linear regression fit. (c) plots the cumulative area under the curve. QT (η, Λ) and QI (η, Λ). In this section, we focus on perturbations that moves away from the critical point, so QT (η, Λ) 6= QI (η, Λ). In general, this difference between the distributions of ∆T and ∆I will drive the system away from the critical point and move to either a thermal phase or a localized phase as the RG flows. But for the same reason in the critical fixed point case, here we still set ln gI as sub-extensive, i.e. ln gI / ln Ω → 0. This gives us ΓI = ∆I . So what we obtain is a set of two-variable RG equations, and the cutoff Ω becomes Ω = max{∆T , ∆I } . 80 (3.67) Here we still use the parameters Λ, ζ and η defined in Eq. (3.31), (3.32) and (3.34). We set ρT (ζ, Λ) and ρI (ζ, Λ) as the distributions of ζ for thermal blocks and insulating blocks when the cutoff is Λ. Thus the relations between Q’s and ρ’s are still similar: QT (η, Λ) ≡ ΛρT (ζ, Λ) and QI (η, Λ) ≡ ΛρI (ζ, Λ) . (3.68) Similar to the derivation of the critical fixed point distribution, here we also consider the infinitesimal process when the cutoff is moving by dΛ. But now the contributions from TIT move and ITI move are different. For TIT move, we use points in the regime ρI (0, Λ)dΛ, because here the middle I-block will be at the cutoff. Then we randomly select ζa , ζb from the distribution ρT (ζ, Λ) and merge these three points into a new point with ζnew = ζa + ζb + 2Λ that adds to the T distribution. And similarly for ITI move because of the exchange symmetry T ↔ I. Only after we use all the points in ρT (0, Λ)dΛ and ρI (0, Λ)dΛ can we move the cutoff by dΛ. Let us focus on the change in ρT before re-normalization. The change contains two parts: first is to get rid of all points in ρT (0, Λ)dΛ and second is to get rid of ζa , ζb and add a new point at ζa + ζb + 2Λ. For the second part, the change of ρT is (similar to the analysis for the fixed point distribution) Z ρI (0, Λ)dΛ ∞ Z dζa 0 ∞ dζb ρT (ζa , Λ)ρT (ζb , Λ)· 0 h i · − δ(ζ − ζa ) − δ(ζ − ζb ) + δ(ζ − ζa − ζb − 2Λ) . (3.69) The prefactor ρI (0, Λ)dΛ is because this whole second part is from TIT move where the middle I-block is at the cutoff. And the next step is the re-normalization. The area decreases by ρT (0, Λ)dΛ + ρI (0, Λ)dΛ , 81 (3.70) where the first term comes from ITI move and the second term comes from TIT move. Thus, we need to multiply the whole distribution by a factor (1 + ρT (0, Λ)dΛ + ρI (0, Λ)dΛ). In this way, ρT becomes Z n ρT (ζ, Λ) + ρI (0, Λ)dΛ ∞ Z ∞ h dζb ρT (ζa , Λ)ρT (ζb , Λ) · − δ(ζ − ζa ) dζa 0 0 io − δ(ζ − ζb ) + δ(ζ − ζa − ζb − 2Λ) (1 + ρT (0, Λ)dΛ + ρI (0, Λ)dΛ) h i = ρT (ζ, Λ) + ρT (ζ, Λ) ρT (0, Λ)dΛ − ρI (0, Λ)dΛ Z ∞ Z ∞ + ρI (0, Λ)dΛ dζa dζb ρT (ζa , Λ)ρT (ζb , Λ)δ(ζ − ζa − ζb − 2Λ) 0 0 (3.71) The remaining procedure of shifting the function horizontally is exactly the same as that in the fixed point distribution. Thus, we get the equation for ρT (ζ, Λ): h i ∂ρT ∂ρT = + ρT (ζ, Λ) ρT (0, Λ) − ρI (0, Λ) ∂Λ ∂ζ Z ∞ Z ∞ + ρI (0, Λ) dζa dζb ρT (ζa , Λ)ρT (ζb , Λ)δ(ζ − ζa − ζb − 2Λ) . (3.72) 0 0 For ρI , all things are the same except for change in the subscript T ↔ I: h i ∂ρI ∂ρI = + ρI (ζ, Λ) ρI (0, Λ) − ρT (0, Λ) ∂Λ ∂ζ Z ∞ Z ∞ + ρT (0, Λ) dζa dζb ρI (ζa , Λ)ρI (ζb , Λ)δ(ζ − ζa − ζb − 2Λ) . (3.73) 0 0 These two equations together form the set of RG equations for ρT (ζ, Λ) and ρI (ζ, Λ). Then we still would like to transform the argument from ζ to η ≡ ζ/Λ. Similar to 82 the single variable case, after change of variables, we arrive at h i ∂QT ∂QT Λ = QT + (1 + η) + QT (η, Λ) QT (0, Λ) − QI (0, Λ) ∂Λ ∂η Z η−2 dηa QT (ηa , Λ)QT (η − ηa − 2, Λ) ; + QI (0, Λ)Θ(η − 2) 0 h i ∂QI ∂QI Λ = QI + (1 + η) + QI (η, Λ) QI (0, Λ) − QT (0, Λ) ∂Λ ∂η Z η−2 dηa QI (ηa , Λ)QI (η − ηa − 2, Λ) . + QT (0, Λ)Θ(η − 2) (3.74) 0 After getting the general set of RG equations for QT (η, Λ) and QI (η, Λ), we set these two functions close to the fixed point distribution Q∗ (η): QT (η, Λ) ≡ Q∗ (η) + δT (η, Λ) , (3.75) QI (η, Λ) ≡ Q∗ (η) + δI (η, Λ) , where δT,I are small compared to Q∗ so that we can omit the second order terms in δT,I . And also, δT,I satisfy Z ∞ δT,I (η, Λ)dη = 0 (3.76) 0 so that the QT,I are normalized. By plugging these into the equations for QT and QI and omitting the second order terms in δT,I , we obtain h i ∂δT ∂δT ∗ Λ = δT + (1 + η) + Q (η) δT (0, Λ) − δI (0, Λ) ∂Λ ∂η Z η−2 + δI (0, Λ)Θ(η − 2) dηa Q∗ (ηa )Q∗ (η − ηa − 2) Z η−2 0 + Θ(η − 2) dηa δT (ηa , Λ)Q∗ (η − ηa − 2) 0 83 (3.77) and h i ∂δI ∂δI ∗ Λ = δI + (1 + η) + Q (η) δI (0, Λ) − δT (0, Λ) ∂Λ ∂η Z η−2 dηa Q∗ (ηa )Q∗ (η − ηa − 2) + δT (0, Λ)Θ(η − 2) 0 Z η−2 dηa δI (ηa , Λ)Q∗ (η − ηa − 2) . + Θ(η − 2) (3.78) 0 In order to decouple δT and δI , we set their sum and difference as new variables: δ+ (η, Λ) ≡ δT (η, Λ) + δI (η, Λ) , (3.79) δ− (η, Λ) ≡ δT (η, Λ) − δI (η, Λ) . By adding and subtracting Eq. (3.77) and (3.78) separately, we get the decoupled equations for δ+ and δ− : Z η−2 ∂δ+ ∂δ+ Λ = δ+ + (1 + η) + δ+ (0, Λ)Θ(η − 2) dηa Q∗ (ηa )Q∗ (η − ηa − 2) ∂Λ ∂η 0 Z η−2 + Θ(η − 2) dηa δ+ (ηa , Λ)Q∗ (η − ηa − 2) (3.80) 0 and Λ ∂δ− ∂δ− = δ− + (1 + η) + 2Q∗ (η)δ− (0, Λ) ∂Λ ∂η Z η−2 − δ− (0, Λ)Θ(η − 2) dηa Q∗ (ηa )Q∗ (η − ηa − 2) 0 Z η−2 + Θ(η − 2) dηa δ− (ηa , Λ)Q∗ (η − ηa − 2) . 0 84 (3.81) In general, they are both linear integro-differential equations, so the method of separating of variables works here. If we set δ+ (η, Λ) ≡ f+ (η)g+ (Λ) , (3.82) δ− (η, Λ) ≡ f− (η)g− (Λ) , the equation for δ+ reduces to Z η−2 d ln g+ 1 + η df+ f+ (0) dηa Q∗ (ηa )Q∗ (η − ηa − 2) =1+ + Θ(η − 2) d ln Λ f+ (η) dη f+ (η) 0 Z η−2 f+ (ηa ) ∗ dηa + Θ(η − 2) Q (η − ηa − 2) . (3.83) f+ (η) 0 The left hand side of the above equation is a function of Λ while the right hand side is a function of η. So they must be constant. We set this constant to 1/ν+ and we get 1 d ln g+ = , d ln Λ ν+ (3.84) g+ ∝ Λ1/ν+ . (3.85) so This gives us a general power law relation to the cutoff Λ which is the typical behavior near the critical point of a phase transition. Then the remaining equation of f+ (η) becomes an eigenvalue equation: Z η−2 1 df+ f+ (η) = f+ (η) + (1 + η) + f+ (0)Θ(η − 2) dηa Q∗ (ηa )Q∗ (η − ηa − 2) ν+ dη 0 Z η−2 + Θ(η − 2) dηa f+ (ηa )Q∗ (η − ηa − 2) . (3.86) 0 85 Similar analysis applies to the equation for δ− , by setting the form g− ∝ Λ1/ν− , (3.87) 1 df− f− (η) = f− (η) + (1 + η) + 2Q∗ (η)f− (0) ν− dη Z η−2 dηa Q∗ (ηa )Q∗ (η − ηa − 2) − f− (0)Θ(η − 2) 0 Z η−2 + Θ(η − 2) dηa f− (ηa )Q∗ (η − ηa − 2) . (3.88) we arrive at 0 These two eigenvalue equations can be solved numerically by discretizing them into a matrix form and numerically diagonalizing the matrix. Before directly doing that, let me discuss one property of the eigenfunctions f± (η), the constraint Z ∞ f± (η)dη = 0 (3.89) 0 which comes from the similar constraint of δT,I , Eq. (3.76), and hence δ± . By integrating η from 0 to ∞ in both sides of Eq. (3.86), we have Z 0 ∞ 1 dη f+ (η) = ν+ Z ∞ i dh dη (1 + η)f+ dη 0 Z ∞ Z η−2 + f+ (0) dη dηa Q∗ (ηa )Q∗ (η − ηa − 2) Z ∞ 2Z η−2 0 + dη dηa f+ (ηa )Q∗ (η − ηa − 2) . 2 (3.90) 0 The first term in the right hand side of the above equation gives us −f+ (0) if we R∞ assume that f+ (η) decays faster than 1/η when η → ∞ or the integral 0 f+ (η)dη is finite. For the second and the third terms, we may apply the change of variables to 86 get the result: ∞ Z Z dη 2 η−2 dηa Q∗ (ηa )Q∗ (η − ηa − 2) = 1 (3.91) 0 and Z ∞ Z because R∞ 0 ∗ Z dηa f+ (ηa )Q (η − ηa − 2) = dη 2 η−2 0 ∞ f+ (η)dη Q∗ (η)dη = 1. These together imply 1 Z ∞ −1 f+ (η)dη = 0 ν+ 0 if R∞ 0 (3.92) 0 f+ (η)dη is finite. This gives us a trivial eigenvalue (3.93) 1 ν+ = 1, but we need to give up this solution because the corresponding eigenfunction does not satisfy the R∞ constraint 0 f+ (η)dη = 0, and hence it is not physical. But this also means that if we find an eigenvalue which is not 1 and also the eigenfunction is integrable, the integral must be zero. So when we are doing the numerical diagonalization, we can just rule out the solution with eigenvalue 1 if the integral of the eigenfunction is not zero. The same happens for f− (η). By doing the same with Eq. (3.88), we also arrive at 1 Z ∞ −1 f− (η)dη = 0 ν− 0 (3.94) The same argument applies: we can just rule out the solution with eigenvalue 1 if the integral of the eigenfunction is not zero. Next comes the numerical diagonalization. We evenly discretize the η axis from 0 up to ηu and set the minimal length δη so that ηu is a large multiple of δη. In this 87 way, we can have a matrix form of the eigenvalue equation. M± f± = 1 f± , ν± (3.95) where M± is the matrix we need to diagonalize, f± is the discretized version of f± (η) and 1/ν± is still the eigenvalue. When constructing the matrix M± , we make the following considerations. The first consideration is the derivative: Because f± (η) + (1 + η) i df± dh = (1 + η)f± dη dη (3.96) and this whole term helps us to determine the constraint equation Eq. (3.93) and Eq. (3.94), we would like to discretize the full derivative as a whole: i i dh 1h (1 + η)f± (η) → 1 + (i + 1)δη f± (i + 1)δη − (1 + iδη)f± (iδη) dη δη (3.97) for the place η = iδη where i is a non negative integer. Note that here we also set the derivative as the right derivative, because in the last step of the derivation of ρ(ζ, Λ), when we shift the whole function horizontally by an amount −dΛ, this actually corresponds in the discrete case to the right derivative. The second consideration is the boundary condition at η = ηu . In order to match the exponential decaying behavior of Q∗ (η) at large η, it turns out that the f± (η) must also be exponentially decaying with the same decay rate λQ . In other words, f± (η) ∼ exp(−λQ η) up to a prefactor for large η. This gives us the boundary condition, which is to match the first order derivative at the point ηu . Roughly speaking, f± (ηu + δη) ≈ f± (ηu ) exp(−λQ δη) 88 (3.98) so we have the discretized version of the derivative at the boundary: i i dh 1h (1 + η)f± (η) (1 + ηu + δη)f± (ηu + δη) − (1 + ηu )f± (ηu ) → dη δη η=ηu i 1h ≈ (1 + ηu + δη)e−λQ δη − (1 + ηu ) f± (ηu ) . (3.99) δη The third consideration is the integral, where we just use the standard trapezoidal rule to discretize the integral. After constructing the matrices, we can numerically diagonalize them. For δη = 0.01 and ηu = 39.99, we have two 4000 × 4000 matrices to diagonalize. Note that previously we only have the Q∗ (η) values up to η = 20; here for η > 20, we use the approximation Q∗ (η) ≈ CQ exp(−λQ η) directly and the values CQ and λQ from the linear fit in Fig. 3.2(b). The result shows that for M+ , if we sort the eigenvalues based on their real parts, then the largest eigenvalue is equal to 0.9993 which corresponds to the trivial 1 ν+ = 1 solution and the eigenfunction always has the same sign so the integral is nonzero. This confirms our analysis for the constraint about the integral for f± (η). The second largest has real part −3.4. It means that this is an irrelevant solution because it decays as Λ−3.4 when the RG is flowing. And all the other eigenvalues have real parts even smaller than or equal to −3.4, which means that for M+ or f+ (η), no relevant solution is found. This is as expected because this means that if we perturb the QT and QI in the same direction, this perturbation would in general decay back to zero as the cutoff Λ grows. Thus, in this direction, the fixed point distribution Q∗ (η) is stable because we actually do not put the system away from the critical point by setting QT and QI equal. Next comes the result for M− , which gives us the critical exponent. If we as before sort the eigenvalues based on their real parts, we again find the largest eigenvalue equal to 1.0000 corresponding to the trivial 1 ν− = 1 solution, and the integral of the eigenfunction is also nonzero. So we just rule out this solution as before. The next 89 largest eigenvalue is the most important one. It is real and equal to 0.3995. The next eigenvalue in the sorted list has real part −1.8, which is negative and hence irrelevant. This means that the only relevant solution is the one with eigenvalue 0.3995. We will focus on this solution from now on. 1.0 100 0.8 10-1 0.6 10-2 |f−(η)| f−(η) First we plot the eigenfunction as shown in Fig. 3.3. Fig. 3.3(a) gives us the eigen- 10-3 0.4 0.2 10-4 0.0 10-5 0.20 5 10 15 20 η 25 30 35 10-6 0 40 5 10 15 20 η 25 30 35 40 (b) |f− (η)| in log scale (a) f− (η) Figure 3.3: The eigenfunction f− (η) corresponding to eigenvalue 0.3995: (a) Linear scale (b) Absolute value on log scale. function on a linear scale. We choose f− (0) = 1 as a normalization factor because the eigenfunction has an arbitrary multiplier, meaning if we multiply an arbitrary constant everywhere in an eigenfunction f− (η), the new function still solves the eigenvalue equation with the same eigenvalue. One nice thing about this eigenfunction is that it has both positive and negative values so that the integral can be close to 0. Actually, R∞ in this case the area is −0.005 which confirms the constraint 0 f− (η) = 0. Fig. 3.3(b) gives us the absolute value of the eigenfunction on a log scale. The tip in the regime 0 < η < 5 is due to the changing sign of f− (η) at that point. Fig. 3.3(b) shows clearly that the eigenfunction f− (η) also decays exponentially at large η as expected. We can also do the numerical integration to directly obtain the eigenfunction by setting 1/ν− = 0.4 and f− (0) = 1. The result is shown in Fig. 3.4. Fig. 3.4 shows the two resulting curves of f− (η) coincide perfectly. This result further confirms that 90 1.0 Direct Integration Diagonalization 0.8 f−(η) 0.6 0.4 0.2 0.0 0.20 5 10 15 η 20 Figure 3.4: The eigenfunction f− (η) using either the numerical integration directly or the diagonalization. f− (η) is the correct solution to the eigenvalue equation. After solving this eigenvalue equation, we get an estimate of the critical exponent: 1 ≈ 0.4 or ν− ≈ 2.5 . ν− 3.3.3 (3.100) Numerical Evidence After deducing the analytical RG equations for the distributions and discussing the solutions to them, we now move on the numerical test by initializing a long one dimensional chain and running the RG numerically. The initial number of blocks is 107 and the final number of blocks we choose is 103 . The first step is to test the fixed point distribution Q∗ (η). We set the initial cutoff Λ equal to 1 and use the theoretical Q∗ (η) distribution, shown in Fig. 3.2, as the 91 initial distribution for both ηT ≡ ln ∆T ln Ω −1 and ηI ≡ ln ∆I ln Ω −1. And we set ln gI = −10−9 as a number close to zero. The resulting cumulative distribution function (Cdf) for the RG distribution flow is shown in Fig. 3.5. Fig. 3.5 clearly shows that for both the distributions in T-block and I-block, they do not change when the total number of blocks is decreasing from 107 all the way to 103 and simultaneously when the cutoff Λ is increasing. This confirms that our Q∗ (η) is indeed a fixed point distribution. In addition, we can also see the pairs of N and Λ in the legend in Fig. 3.5, and their product also keeps the same as expected in Eq. (3.51). Note that Eq. (3.51) is only true when the system is at the fixed point. This is further evidence that the system is at the fixed point. After verifying the fixed point distribution Q∗ (η), we can then perturb the initial distribution away from Q∗ (η) by setting QT (η) = Q∗ (η) + a · f− (η) , (3.101) QI (η) = Q∗ (η) − a · f− (η) , where f− (η) is the eigenfunction with eigenvalue ≈ 0.4 and a is a parameter we can tune. Note that we have an arbitrary multiplier in the eigenfunction f− (η) and it actually can be absorbed in the factor a in our numerical test. So we do not need to normalize the f− (0) value to 1 and we just use the numerical value directly outputted by diagonalization: f− (0) = 0.1119. In addition, we need to keep the a value neither too large nor too small. If a were too large, the perturbation approximation by omitting second order terms in δT,I would no longer be true. If a were too small, it would amplify the error coming from the random sampling and we can hardly tell the difference between QT (η) and QI (η). Based on these concerns, we choose a = 0.1 together with the normalization factor f− (0) = 0.1119. The result of the difference between the Cdfs is shown in Fig. 3.6. Fig. 3.6(a) shows that when the cutoff Λ is increasing, the difference is indeed increasing. In order to see whether this increasing 92 1 N = 10000000; Λ = 1 N = 2154434; Λ = 4.644 N = 464158; Λ = 21.5743 N = 99998; Λ = 99.9655 N = 21544; Λ = 465.511 N = 4640; Λ = 2152.89 N = 1000; Λ = 10001 0.9 0.8 0.7 Cdf(ηT) 0.6 0.5 0.4 0.3 0.2 0.1 0 0 5 10 15 20 η 25 30 35 40 T (a) Cdf flow of T-blocks 1 N = 10000000; Λ = 1 N = 2154434; Λ = 4.644 N = 464158; Λ = 21.5743 N = 99998; Λ = 99.9655 N = 21544; Λ = 465.511 N = 4640; Λ = 2152.89 N = 1000; Λ = 10001 0.9 0.8 0.7 Cdf(ηI) 0.6 0.5 0.4 0.3 0.2 0.1 0 0 5 10 15 20 ηI 25 30 35 40 (b) Cdf flow of I-blocks Figure 3.5: The cumulative distribution function (Cdf) for both the T-blocks and I-blocks as the total number of blocks N decreases from 107 to 103 and the cutoff Λ grows. 93 0.6 N = 10000000; Λ = 1 N = 2154434; Λ = 4.6415 N = 464158; Λ = 21.5246 N = 99998; Λ = 98.7657 N = 21544; Λ = 453.086 N = 4640; Λ = 1966.02 N = 1000; Λ = 7493.21 0.5 ∆ Cdf(η) 0.4 0.3 0.2 0.1 0 −0.1 0 5 10 15 20 η 25 30 35 40 (a) Difference of Cdf flow between T-blocks and I-blocks −3 20 x 10 N = 10000000; Λ = 1 N = 2154434; Λ = 4.6415 N = 464158; Λ = 21.5246 N = 99998; Λ = 98.7657 N = 21544; Λ = 453.086 N = 4640; Λ = 1966.02 N = 1000; Λ = 7493.21 15 ∆ Cdf(η) 10 5 0 −5 0 5 10 15 20 η 25 30 35 40 (b) Collapsing of difference of Cdf by multiplying Λ−0.4 Figure 3.6: The difference of cumulative distribution functions (Cdf) between the T-blocks and I-blocks as the total number of blocks N decreases from 107 to 103 and the cutoff Λ grows. 94 is just due to the power law formula δ− (η, Λ) ∝ Λ1/ν− f− (η) , (3.102) we multiply every single function by a factor Λ−0.4 . The result is in Fig. 3.6(b) and all the resulting functions collapse pretty well. This confirms our estimation of the critical exponent 1 ≈ 0.4 . ν− (3.103) Also note that products of the pairs N and Λ here are not constant. This also indicates that the RG in this case is indeed flowing and is not at the fixed point. 3.4 Conclusion and Outlook In this chapter, we studied a “toy model” in the strong disorder renormalization group framework to explore the eigenstate phase transition between MBL and thermalization. This model turns out to be quite simple. We solved the critical fixed point distribution Q∗ (η) almost analytically from the fixed point RG equation. It is an infinite-randomness fixed point with the energy scale Ω related to the length scale L by ln Ω ∼ −L, as in [30]. We also analytically derived the RG equations when the distributions in both T-blocks and I-blocks deviate from the fixed point distribution, and solved for the correlation length critical exponent ν = ν− ≈ 2.5 by both solving the eigenvalue equation and directly simulating the RG flow. 95 (3.104) One apparent issue of this model is that we do not avoid ad hoc rules. In the ITI move, we imposed that the move always flows to an insulating block to avoid the RG getting stuck and unstable as in the case of our initial trials. The solution to the issue of this “toy” model is to modify the RG rules such that the RG flow becomes closer to the microscopic physics and still shares the features of not getting stuck and unstable. The new model may not be as simple as our “toy model”, but we hope that our “toy model” can serve as a useful reference point for studying the more physical model, and thus for understanding the physics of the eigenstate phase transition between MBL and thermalization. 96 Appendix A Mathematical Proof of the Degeneracy Formula In Section 2.2.2, we stated that the energy level degeneracy denoted by fN (S) has the form: N fN (S) = CN2 −S N − CN2 −S−1 , (A.1) where N is the total number of sites and S is the total spin. Let me briefly prove this formula here. The method is basically induction by counting N from 1 to ∞. Let me list the first few cases: 1. N = 1: S = 1/2 is single degenerate. f1 ( 21 ) = 1. 2. N = 2: S = 0 or 1. Both are single degenerate. f2 (0) = f2 (1) = 1. 3. N = 3: Based on angular momentum coupling procedure, if the first two spins are coupled to have S12 = 0, then the total spin S would be 1/2; if the first two are coupled to have S12 = 1, then the total spin S would be 1/2 or 3/2. 97 So S = 1/2 is double degenerate; S = 3/2 is single degenerate. f3 ( 12 ) = 2, f3 ( 32 ) = 1. 4. N = 4: If S123 = 1/2, we would have either S = 0 or S = 1; if S123 = 3/2, we would have either S = 1 or S = 2. So f4 (0) = 2, f4 (1) = 3, f4 (2) = 1. As a double check, every single S expands a 2S + 1 dimensional Hilbert space, so the total dimension is (2 × 0 + 1) · 2 + (2 × 1 + 1) · 3 + (2 × 2 + 1) · 1 = 16 = 24 as expected. 5. N = 5: If S1234 = 0, then S = 1/2; if S1234 = 1, then S = 1/2 or S = 3/2; if S1234 = 2, then S = 3/2 or S = 5/2. So f5 ( 21 ) = 2 + 3 = 5, f5 ( 32 ) = 3 + 1 = 4, f5 ( 52 ) = 1. Also, as a double check, the total dimension of the Hilbert space is (2 × 12 + 1) · 5 + (2 × 23 + 1) · 4 + (2 × 52 + 1) · 1 = 32 = 25 as expected. From these examples, we conclude a general recursion relation of fN (S): 1 f2N +1 ( ) = f2N (0) + f2N (1) 2 3 f2N +1 ( ) = f2N (1) + f2N (2) 2 .. . (A.2) 1 f2N +1 (N − ) = f2N (N − 1) + f2N (N ) 2 f2N +1 (N + 1 ) = f2N (N ) 2 and 1 f2N +2 (0) = f2N +1 ( ) 2 1 3 f2N +2 (1) = f2N +1 ( ) + f2N +1 ( ) 2 2 .. . 1 1 f2N +2 (N ) = f2N +1 (N − ) + f2N +1 (N + ) 2 2 1 f2N +2 (N + 1) = f2N +1 (N + ) . 2 98 (A.3) First, I will show that the formula (A.1) is true for any positive even N . By induction, when N = 2, the conclusion is true from the above example. We assume for 2N case, the formula is true: N −S N −S−1 f2N (S) = C2N − C2N . (A.4) Then, for 2N + 2 case, by recursion relation, we have: f2N +2 (0) = f2N (0) + f2N (1) f2N +2 (S) = f2N (S − 1) + 2f2N (S) + f2N (S + 1) , (A.5) ∀S ≥ 1. n = 0 if n < 0. We do not need to worry about the case S ≥ N . Since by definition Cm So the above equation is also valid for S ≥ N by setting f2N (S ≥ N + 1) = 0 ≡ N −S N −S−1 C2N − C2N where N − S < 0 and N − S − 1 < 0. So for S = 0 case, we have: f2N +2 (0) = f2N (0) + f2N (1) N −1 N −1 N −2 N = (C2N − C2N ) + (C2N − C2N ) N −1 N −1 N −2 N = (C2N + C2N ) − (C2N + C2N ) N −1 N = C2N +1 − C2N +1 N +1 N −1 N N = (C2N +1 + C2N +1 ) − (C2N +1 + C2N +1 ) N +1 N = C2N +2 − C2N +2 , which is the desired equation. 99 (A.6) For S ≥ 1 case, we have: f2N +2 (S) = f2N (S − 1) + 2f2N (S) + f2N (S + 1) = [f2N (S − 1) + f2N (S)] + [f2N (S) + f2N (S + 1)] h i N −S+1 N −S N −S N −S−1 = (C2N − C2N ) + (C2N − C2N ) h i N −S N −S−1 N −S−1 N −S−2 + (C2N − C2N ) + (C2N − C2N ) h i N −S+1 N −S N −S N −S−1 = (C2N + C2N ) − (C2N + C2N ) h i N −S N −S−1 N −S−1 N −S−2 + (C2N + C2N ) − (C2N + C2N ) N −S+1 N −S N −S N −S−1 = C2N − C2N +1 +1 + C2N +1 − C2N +1 N −S+1 N −S N −S N −S−1 = (C2N + C2N +1 +1 ) − (C2N +1 + C2N +1 ) N −S+1 N −S = C2N − C2N +2 +2 (N +1)−S = C2N +2 (N +1)−S−1 − C2N +2 , (A.7) which is also the desired equation. Thus, by induction, we conclude that for any positive even N , the formula is valid. The next step is to prove the formula is also valid for positive odd N . Since now N −S N −S−1 f2N (S) = C2N − C2N is true, then for 2N + 1 case, we have: 1 1 f2N +1 (S) = f2N (S − ) + f2N (S + ) 2 2 N −S+ 21 = C2N N −S− 21 − C2N N −S+ 12 = (C2N N −S+ 12 = C2N +1 2N +1 −S = C2N2+1 N −S− 21 + C2N N −S− 12 + C2N N −S− 12 ) − (C2N N −S− 32 − C2N N −S− 32 + C2N ) N −S− 21 − C2N +1 2N +1 −S−1 − C2N2+1 which is the desired equation for the odd case. 100 , (A.8) After these two steps, we conclude that the degeneracy formula: N fN (S) = CN2 −S N − CN2 −S−1 (A.9) is valid for ∀N ∈ Z+ . Also, the total dimension of the Hilbert space is in general given by: N/2 X (2S + 1)fN (S) = CN0 + CN1 + · · · + CNN = 2N (A.10) S=0 or 1/2 as expected. Then, by using Stirling’s formula ln N ! = N ln N − N + O(ln N ), we may have the expression of the entropy defined by Σ(S) ≡ ln fN (S): N fN (S) = CN2 −S N − CN2 −S−1 N! N! − N N − S)!( 2 + S)! ( 2 − S − 1)!( N2 + S + 1)! h N i N! N · ( + S + 1) − ( − S) = N 2 2 ( 2 − S)!( N2 + S + 1)! N! = (2S + 1) · N ; ( 2 − S)!( N2 + S + 1)! = ( N2 101 (A.11) then N N − S)! − ln( + S + 1)! 2 2 N N N = N ln N − N − ( − S) ln( − S) + ( − S) 2 2 2 N N N − ( + S + 1) ln( + S + 1) + ( + S + 1) + O(ln N ) 2 2 2 N N N N = N ln N − ( − S) ln( − S) − ( + S + 1) ln( + S + 1) + O(ln N ) 2 2 2 2 N N = N ln N − ( − S) ln N − ( + S + 1) ln N 2 2 N 1 S N 1 S+1 − ( − S) ln( − ) − ( + S + 1) ln( + ) + O(ln N ) 2 2 N 2 2 N h 1 S 1 S 1 S+1 1 S+1 i = −N ( − ) ln( − ) + ( + ) ln( + ) + O(ln N ) 2 N 2 N 2 N 2 N h 1 S 1 S 1 S 1 S i = −N ( − ) ln( − ) + ( + ) ln( + ) + O(ln N ) , (A.12) 2 N 2 N 2 N 2 N ln fN (S) = ln(2S + 1) + ln N ! − ln( which is exactly the entropy equation (2.7) in Sec 2.2.2. 102 Appendix B Estimation of the Magnitude of Disorder In Section 2.4 and 2.5, we added a disorder term in the original Hamiltonian to get: H=− N N N X 1 X z z λ X si sj − Γ sxi + p εij szi szj , N 1=i<j N i=1 1=i<j (B.1) where 1 ≥ λ > 0, εij are independent and identically distributed random variables for different site i with expectation value zero and standard deviation one. This task is to estimate the upper and lower bound for the power p. For further convenience, we denote N N X 1 X z z si sj − Γ sxi , H0 ≡ − N 1=i<j i=1 N λ X Hr ≡ p εij szi szj . N 1=i<j (B.2) So H = H0 + Hr . Then the constrains acting on the magnitude of the disorder term Hr are: 1. Lower Bound: Hr should be big enough for the system to look thermal. 103 2. Upper Bound: Hr should be small enough not to vary macroscopic thermal property such as free energy. Let us discuss them one by one. B.1 Lower Bound of Disorder First is for the lower bound of disorder. When adding Hr , all degenerated energy levels are split due to Hr . By letting the system look thermal, we mean the typical value of the level splitting due to Hr should be big enough to compare with the typical value of the original level splitting of H0 which is typically of order 1. Using degenerate perturbation theory treating Hr perturbatively, we need to diagonalize Hr in the degenerated H0 subspace: Hr11 Hr12 · · · Hr1n Hr Hr22 · · · Hr2n 21 . .. .. ... .. . . Hrn1 Hrn2 · · · Hrnn 0 ··· 0 ∆E1 ∆E2 · · · 0 diagonalize 0 −−−−−−→ . .. .. ... .. . . 0 0 · · · ∆En , (B.3) where Hrmn ≡ hm|Hr |ni is the matrix element of Hr in H0 representation; n = eΣ(S) is the degeneracy. Since the diagonalization process keeps the trace invariant, we would have an estimation by taking the trace: X hn|Hr |ni = n n X ∆Ei . (B.4) i=1 Then, the left hand side is equal to: X n hn|Hr |ni = X n N N X λ X λ X z z hn| p εij si sj |ni = p εij hn|szi szj |ni . N 1=i<j N 1=i<j n 104 (B.5) Since the Hamiltonian H0 is invariant under permutation Pjk for any two sites j and k, we have [H0 , Pjk ] = 0, so H0 Pjk |ni = En Pjk |ni. It means Pjk is a block diagonalized form in terms of the same energy subspace. In another word, Pjk is also an operator in the same energy subspace. So we can deal with the operator totally in the subspace: X hn|szi szj |ni = trsubspace (szi szj ) n −1 = trsubspace (Pjk szi szj Pjk ) = trsubspace (szi szk ) X = hn|szi szk |ni . (B.6) n It implies that x x n hn|si sj |ni P is independent of site indices i and j as long as i 6= j. Because X X X hn|Sz · Sz |ni = (N 2 − N ) hn|szi szj |ni + N hn|sz2 i |ni n n = (N 2 − N ) X n n 1 hn|szi szj |ni + N · n , 4 (B.7) we have: X hn|szi szj |ni = n X 1 1 2 hn|Sz |ni − N · n . N2 − N n 4 (B.8) Every |ni in the subspace has the same quantum number of total S and m. Therefore, hn|Sz2 |ni ≡ hS, m|Sz2 |S, mi = m2 . (B.9) X m2 − 41 N z z hn|si sj |ni = n · 2 . N −N n (B.10) Then we arrive at: 105 Plugging this back we have: λ X m2 − 14 N tr(Hr ) = n p εij · 2 N N −N i6=j = n X ∆Ei , (B.11) i=1 n 1X λ X m2 − 14 N ∆Ei = p . εij · 2 n i=1 N N −N i6=j (B.12) We need the typical value of the energy shift ∆Ei to be O(1) in order to fully mix the energy spectrum up. Since ∆Ei s come from random variables εij s, they themselves can also be treated as random variables. From the law of large number in probability theory, n 1X ∆Ei = E(∆Ei ) & O(1) , n i=1 (B.13) where E(X) is the expectation value of a random variable X. Since εij has a variance 1 as mentioned at the beginning, we apply to the central limit theorem and have: X εij ∼ √ N 2 − N = O(N ) . (B.14) i6=j In addition, m2 = O(N 2 ), so m2 − 41 N N 2 −N ∼ O(1). Thus the requirement finally reduces to: N O(1) Np ⇒ 106 p < 1. (B.15) B.2 Upper Bound of Disorder For the upper bound of disorder, we require free energy should not change much by adding disorder. Let f0 be the free energy of the system H0 and f˜ be the free energy of the full system H0 + Hr . The requirement becomes: f˜ − f0 = 0. N →∞ f0 lim (B.16) Obviously, f0 ∼ O(N ), so the only thing we need to solve is the difference: f˜ − f0 . By definition, h i −β f˜ = ln Z̃ = ln tr e−β(H0 +Hr ) . (B.17) To take the trace, it is equivalent to use the saddle point approximation: to take the minimal eigenvalue of the following expression1 Σ(S) − β(H0 + Hr ) . (B.18) We know that Σ(S) ∼ O(N ) and βH0 ∼ O(N ) . (B.19) Then the task is to estimate the ground state energy of Hr . If the system only contains Hr , the system is a typical “spin glass”. And the ground state of this spin glass can be argued this way: for one spin, there are N adjacent spins exerting “random” fields and each one of the fields has a magnitude O( N1p ). So by the central limit theorem, √ the total effective field for one spin is O( N · N1p ). If we choose every spin to be The approximation exp Σ(S) exp − β(H0 + Hr ) ≈ exp Σ(S) − β(H0 + Hr ) is used since the commutator has a subleading order of N . 1 107 aligned with this effective field, the energy of Hr can be as low as O(N · √ N· 1 ) Np which is the order of magnitude of the ground state energy of a spin glass system. √ √ Thus if N · N · N1p O(N ) or N · N · N1p ∼ O(N ), the disorder term would have a non-eligible or even a leading effect on free energy. Therefore in order to maintain the same free energy as N → ∞, we need Hr ∼ N · √ N· 1 O(N ) Np ⇒ p> 1 . 2 (B.20) As a conclusion of the discussion in both Section B.1 and Section B.2, we finally get that the magnitude of the disorder should be within the following interval: 1 < p < 1. 2 108 (B.21) Appendix C Divergence of First Order Derivative of γ(s̄) at s̄ = s̄min In Section 2.5.1, we stated that dγ(s̄) ds̄ diverges at s̄ = s̄min . Let me show this divergence briefly here. The original expression of γ(s̄) is given by: Z γ(s̄) = 2 0 xt −u − 1 x2 arccosh √ 2 dx , Γ s̄2 − x2 (C.1) −u− 1 x2 2 where xt is the turning point. We denote B(x) ≡ arccosh Γ√s̄2 −x 2 , then xt satisfies √ p √ the turning point equation B(xt ) = 1 or xt = 2 −u − Γ2 − Γ 2u + s̄2 + Γ2 . Since the upper bound of the integral in γ(s̄) is also a function of s̄, then the derivative contains generically two terms: −u − 1 x2 ∂x ∂γ t = 2 arccosh p 2 t · +2 2 2 ∂s̄ ∂s̄ Γ s̄ − xt Z 0 xt ∂ arccosh B dx , ∂s̄ (C.2) 1 2 −u− x since B(xt ) = √ 22 t2 = 1, arccosh B(xt ) = 0, the first term would vanish. But there Γ are points where s̄ −xt ∂xt ∂s̄ = ∞ and it happens when xt = 0 and s̄ = s̄min . However, we do not need to take a special care even at these points since what we really care is 109 the asymptotic behavior near these points and every individual point does not matter that much. So the derivative reduces to: Z ∂γ =2 ∂s̄ xt Z0 xt =2 0 ∂ arccosh B dx ∂s̄ B −s̄ √ · 2 dx . B 2 − 1 s̄ − x2 (C.3) If we study the above equation at the point xt = 0 or s̄ = s̄0 , we would recover: πΓ ∂γ , = −√ ∂s̄ s̄=s̄0 −0 −u − Γ2 (C.4) but here the meaning is slightly different from that in Section 2.6. In Section 2.6, we calculated the derivative exactly at s̄ = s̄0 ; here, on the other hand, the procedure is that we first take the derivative at some other point and then send the point to s̄0 . The consistency of the two result shows the left continuity of the function ∂γ ∂s̄ at s̄0 . Now it is time to see the asymptotic behavior of ∂γ when s̄ → s̄min . When ∂s̄ √ √ s̄ → s̄min = −2u − Γ2 , we have xt → −2u − 2Γ2 ≡ xt0 . Thus, ∂γ =2 ∂s̄ s̄=s̄min Z 0 xt0 B(s̄min ) p B(s̄min )2 − 1 · −s̄min dx , − x2 s̄2min (C.5) when x → xt0 , the integrand behaves as: B(s̄min ) → 1, s̄2min − x2 → Γ2 , but √ 1 B(s̄min )2 −1 → ∞. We need to study how fast it goes to ∞. 110 If we perturb x a little away from xt0 by setting x = xt0 − ∆ and expand B(s̄min ) in terms of ∆, we have: B(s̄min ) −u − 12 x2 = √ Γ −2u − Γ2 − x2 −u − 12 (xt0 − ∆)2 = p Γ −2u − Γ2 − (xt0 − ∆)2 −u − 12 x2t0 + xt0 ∆ − 12 ∆2 √ = Γ Γ2 + 2xt0 ∆ − ∆2 Γ2 + xt0 ∆ − 12 ∆2 = q Γ2 1 + 2xΓ2t0 ∆ − Γ12 ∆2 2x xt0 1 2 h 1 2 i− 12 t0 = 1 + 2 ∆ − 2∆ 1 + ∆ − 2∆ Γ 2Γ Γ2 Γ h i xt0 1 2 1 2xt0 1 2 3 2xt0 1 2 2 3 = 1 + 2 ∆ − 2∆ 1 − ∆ − 2∆ + ∆ − 2∆ + O(∆ ) Γ 2Γ 2 Γ2 Γ 8 Γ2 Γ xt0 1 1 2xt0 xt0 1 2xt0 1 3 2xt0 2 = 1 + 2 ∆ − 2 ∆2 − · 2 ∆ − 2 ∆ · 2 ∆ + 2 ∆2 + ∆ + O(∆3 ) Γ 2Γ 2 Γ Γ 2 Γ 2Γ 8 Γ2 x2 = 1 + t04 ∆2 + O(∆3 ) , (C.6) 2Γ then, r xt0 x2t0 2 ∆ + O(∆3 ) = ∆[1 + O(∆)] , 4 Γ Γ (C.7) 1 Γ Γ p = [1 + O(∆)] = + O(1) . 2 xt0 ∆ xt0 ∆ B(s̄min ) − 1 (C.8) p B(s̄min )2 − 1 = ⇒ Hence, ∂γ ∼ − ∂s̄ s̄=s̄min Z xt0 0 which diverges logarithmically. 111 1 d∆ → − ∞ , ∆ (C.9) Appendix D Discussion on Convexity of Function γ(s̄) In Section 2.5.1, we assumed that γ(s̄) is a convex function of s̄ in order to get the analytical result of the thermal activation and quantum tunneling transition line. By saying convexity, we mean ∂ 2γ > 0. ∂s̄2 (D.1) In fact, we can do the numerics following directly from the thermal activation and quantum tunneling transition criteria which is to minimize the quantity α(u, Γ) in Eq. 2.32 without assuming the convexity of γ(s̄). The result is shown in Fig. D.1. The dots in Fig. D.1 are the numerical results given by actually doing the numerical integral for γ(s̄) without assuming the convexity of function γ(s̄). But this numerical result still fits the analytical one which is the line underneath the dots in Fig. D.1. This convinced us that the analytical result must be the right answer. While we already have the numerical evidence, we would also like to discuss the convexity of function γ(s̄) directly and see how much we could get. 112 0.00 -0.05 -0.10 u -0.15 -0.20 -0.25 G -0.30 0.0 0.1 0.2 0.3 0.4 0.5 Figure D.1: Comparison between numerical and analytical result of the thermal activation and quantum tunneling transition line; the dots are numerical points and the line underneath the dots is the analytical result given by equation (2.47). They fit perfectly well. One attempt is to directly calculate the integral: ∂γ =2 ∂s̄ Z 0 xt B −s̄ √ dx · 2 B 2 − 1 s̄ − x2 (D.2) numerically by sowing different pairs of Γ and u randomly. Some of the results are shown in Fig. D.2. In Fig. D.2, it is clear that for all pairs of Γ and u which have been randomly chosen, ∂γ ∂s̄ is a monotonic increasing function of s̄. This adds our belief that γ(s̄) is a convex function although we do not have a mathematical proof. 113 dΓ dΓ dΓ ds 0.356 0.357 0.358 0.359 0.360 ds ds s 0.361 0.5 1.0 1.5 2.0 s 2.5 -1 0.358 0.360 0.362 s 0.366 0.364 -1 -0.2 -2 -2 -0.4 -3 -3 -0.6 -4 -4 -5 -0.8 -6 -1.0 -5 -6 (a) Γ = −0.106658 0.295502 , u = (b) Γ = −0.0937125 dΓ 0.031665 , u = (c) Γ = −0.104641 dΓ ds 0.3856 0.3858 0.3860 0.3862 0.3864 0.3866 ds s 0.45 -2 -0.5 -4 -1.0 = dΓ ds 0.3854 0.285948 , u 0.50 0.55 0.60 0.65 0.70 s 0.49865 0.49870 s 0.49880 0.49875 -5 -10 -1.5 -6 -15 -2.0 -8 -2.5 -20 -10 (d) Γ = −0.136793 0.353770 , u = (e) Γ = −0.106585 0.145284 , u = (f) Γ = −0.242333 0.485834 , u = as a function of s̄ by sowing different pairs of Γ Figure D.2: Numerical results of ∂γ ∂s̄ and u randomly within the entire ferromagnetic phase region. Another attempt is based on trying to calculate it analytically by doing another order derivative from the expression of Z ∂γ =2 ∂s̄ xt 0 Z =2 0 1 ∂γ . ∂s̄ Since −s̄ B √ · 2 dx B 2 − 1 s̄ − x2 B −s̄ √ · 2 xt db , B 2 − 1 s̄ − x2t b2 (D.3) we have: ∂ 2γ = −2 ∂s̄2 Z 0 1 ∂ B −s̄ √ · 2 x t db . ∂s̄ B 2 − 1 s̄ − x2t b2 (D.4) −s̄ · s̄2 −x 2 b2 xt where in general both B and xt are functions of s̄: B = t √ p √ ≡ B(s̄, xt ) = B(s̄, xt (s̄)) and xt = 2 −u − Γ2 − Γ 2u + s̄2 + Γ2 ≡ √ B B 2 −1 −u− 12 x2t b2 Let g(s̄, b) ≡ arccosh √ s̄2 −x2t b2 Thus if ∂g ∂s̄ Γ xt (s̄). < 0 for all 0 < b < 1, we would have do is to test whether ∂g ∂s̄ ∂2γ ∂s̄2 > 0. Then all we need to < 0 or not. After a messy calculation, we successfully turn 114 this condition into whether a big polynomial containing about 50 terms is smaller than zero. Then the method of sowing points randomly also applies. 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