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Quantum Phase Transitions G. H. Lai May 5, 2006 Abstract A quantum phase transition (QPT) is a zero-temperature, generically continuous transition tuned by a parameter in the Hamiltonian at which quantum fluctuations of diverging size and duration (and vanishing energy) take the system between two distinct ground states [4]. This short review discusses the characteristic aspects of QPT and illustrates possible applications in physics and biology. Contents 1 What are quantum phase transitions? 1 2 Features of a quantum phase transition 3 3 Relevance in experimental Physics and Biology 5 4 Further work 9 1 What are quantum phase transitions? A phase transition is a fundamental change in the state of a system when one of the parameters of the system (the order parameter) passes through its critical point. The states on opposite sides of the critical point are characterized by different types of ordering, typically from a symmetric or disordered state, which incorporates some symmetry of the Hamiltonian, to a brokensymmetry or ordered state, which does not have that symmetry, although the Hamiltonian still possesses it. 1 As we approach the phase transition, the correlations of the order parameter become long-ranged. Fluctuations of diverging size and duration (and vanishing energy) take the system between two distinct ground states across the critical point. When are quantum effects significant? Surprisingly, all non-zero temperature transitions are considered “classical”, even in highly quantum-mechanical systems like superfluid helium or superconductors. It turns out that while quantum mechanics is needed for the existence of an order parameter in such systems, it is classical thermal fluctuations that govern the behaviour at long wave-lengths. The fact that the critical behaviour is independent of the microscopic details of the actual Hamiltonian is due to the diverging correlation length and correlation time: close to the critical point, the system performs an average over all length scales that are smaller than the very large correlation length. As a result, to correctly describe universal critical behaviour, it should suffice to work with an effective theory that keeps explicitly only the asymptotic long-wavelength behaviour of the original Hamiltonian (for instance, the phenomenological Landau-Ginsburg free-energy functional). So far, we have shown that classical theory suffices. However, consider what happens when the temperature around the critical point is below some characteristic energy of the system under consideration [2]. For example, the characteristic energy of an atom would be the Ryberg energy. We see a characteristic frequency ωc and it follows that quantum mechanics should be important when kb T < ~ωc . In the same spirit, if kb T >> ~ωc close to the transition, the critical fluctuations should behave classically. This argument also shows that zero-temperature phase transitions, where Tc = 0, are qualitatively different and their critical fluctuations have to be treated quantum mechanically. In such zero-temperature or quantum phase transitions (QPT), instead of varying the temperature through a critical point, we tune a dimensionless coupling constant J in the controlling Hamiltonian H(J). Generically [1], the ground state energy of H(J) will be a smooth, analytic function of g for finite lattices. However, suppose we have the Hamiltonian H(J) = H0 +JH1 , where H0 and H1 commute. This means that H0 and H1 can be simultaneously diagonalized and the eigenstates of the system will remain the same even though the eigen-energies will vary with J. A level crossing is possible where an excited level crosses the ground energy level at some coupling constant Jc , creating a point of non-analyticity in the ground state energy as a function of J, which manifests as a quantum phase transition. 2 11 Quantum Phase Transitions 10 D.BELITZ AND T.R.KIRKPATRICK critical region T T PM Tc (J) FM classical PM FM QM Jc QM Jc J (A) J (B) Figure Schematic phase diagram as in Fig. 1 showing the vicinity of the quantum Figure 1. Schematic phase diagram showing a paramagnetic (PM) and2.a ferromagnetic critical apoint (J = phase Jc , T = 0). Indicated are the critical region, and the regions dominated (FM) phase. The path indicated by the dashed line path represents classical by the classical and quantum mechanical critical behavior, respectively. A measurement transition, and the one indicated by the solid line a quantum phase transition. along the path shown will observe the crossover from quantum critical behavior away from the transition to classical critical behavior asymptotically close to it. Figure 1: (A) Schematic phase diagram showing a paramagnetic (PM) and ferromagnetic (FM) phase. The dotted path represents a classical phase 1.3.a AN EXAMPLE: THE PARAMAGNET-TO-FERROMAGNET TRANSITION transition while the solid line indicates a quantum phase transition. (B) speaking, related to the classical one in a different spatial dimensionality, Let us illustrate the concepts introduced above by means of a concrete and since we usually Vicinity of the we quantum critical point (Jknow = that Jc , changing T = Tthe Indicated aremeans thechangc ).dimensionality example. For definiteness, consider a metallic or itinerant ferromagnet. ing the universality class, we expect the critical behavior at this quantum 5 critical region, as well as the regions by the clasical andatquantum criticaldominated point to be different from the one observed any other point on the coexistence curve. mechanical critical behaviour(QM) [2]. 1.3.1. The phase diagram This brings to the question of how continuity is guaranteed when one Figure 1 shows a schematic phase diagram in the T -J plane, with Justhe along the coexistence curve. The answer, which was found by Suzuki strength of the exchance coupling that is responsible moves for ferromagnetism. [9], also explains why the behavior at the T = 0 critical point is relevant The coexistence curve separates the paramagnetic phase at large T and for observations atJ, small but non-zero temperatures. Consider Fig. 2, which small J from the ferromagnetic one at small T and large J. For a given an enlarged there is a critical temperature, the Curie temperature shows Tc , where the phasesection of the phase diagram near the quantum critical point. The critical transition occurs. This is the usual situation: A particular material has a region, i.e. the region in parameter space where the critical laws given value of J, and the classical transition is triggered by power lowering thecan be observed, is bounded by the two dashed lines. It then out that the critical region is divided into two subregions, temperature through Tc . Alternatively, however, we can imageturns changing J by ‘QM’ and ‘classical’ in Fig. 2, in which the observed critical at zero temperature (e.g. by alloying the magnet withdenoted some non-magnetic behavior is predominantly quantum mechanical and classical, respectively. material). Then we will encounter the paramagnet-to-ferromagnet transiThetransition division between tion at the critical value Jc . This is the quantum phase we are these two regions (shown as a dotted line in Fig. 2) is not sharp neither is the boundary of the critical region), but rather interested in. Since we have seen that the quantum transition is,(and loosely 2 Features of a quantum phase transition To illustrate the concepts, we consider the example of a metallic or itinerant ferromagnet [2]. Figure 1A shows a schematic phase diagram in the T − J plane, with T the temperature and J the strength of the exchange coupling that is responsible for ferromagnetism. The coexistence curve separates the paramagnetic phasewe(large Tconsider , small J)spins, from the ferromagnetic phase (small For the purposes of this subsection might as well localized but the theory discussed in Sec. 2 below applies to itinerant magnets only. T , large J). For any given J, there is an associated Curie temperature Tc and classical phase transition occurs if we vary the temperature through Tc . On the other hand, imagine changing J at zero temperature, for instance, by alloying/doping the magnet with some non-magnetic material. Here, we encounter a quantum phase transition between the paramagnet and the ferromagnet at the critical value Jc . The behaviour at the T = 0, J = Jc is very different from T 6= 0. In order to understand what happens in the vicinity of a QPT, a special mathematical trick which allows us to connect QPT with classical statistical mechanics is needed [3]. Consider the partition function of a d-dimension classical system governed by a Hamiltonian H, 5 3 Z(β) = Tr e−β Ĥ (1) which can be written, via Feynman’s path integral formulation, as a function integral of the form (generalized for quantum many-body systems) Z Z = D[ψ̄, ψ] eS[ψ̄,ψ] . (2) Here, we let Ĥ be the Hamiltonian operator and S is the action of the system, Z 1 kB T ∂ dτ ψ̄(x, τ )[− +µ]ψ(x, τ )− dτ H(ψ̄(x, τ ), ψ(x, τ )). S[ψ̄, ψ] = dx ∂τ 0 0 (3) ψ̄ and ψ are the conjugate fields isomorphic to the creation and annihilation operators in the second quantized formulation of the Hamiltonian. We have sneaked in the trick of Wick rotation: analytically continuing the inverse temperature β into imaginary time τ via τ = −i~β, so that the operator density matrix of Z, eβ Ĥ , looks like the time-evolution operator e−iĤτ /~ . The end result is seen in the action, which looks like that of a d + 1 Euclidean space-time integral, except that the extra temporal dimension is finite in extent (from 0 to β). As T → 0, we get the same (infinite) limits for a d + 1 effective classical system. This equivalent mapping between a d-dimension quantum system and a d + 1-dimensional classical system allows for great simplifications in our understanding of QPT. Since we know that the quantum transition is related to a classical analog in a different spatial dimensionality, and since changing the dimensionality usually means changing the universality class, the critical behaviour at the quantum critical point Jc should be different from that observed at any other point along the coexistence curve in Figure 1. Also, another interesting aspect of QPT is that dynamics and thermodynamics(i.e. statics, as thermodynamics is very much a misnomer) cannot be independently analyzed, unlike the case for classical statistical mechanics. This loss of freedom is due to the non-commutability of coordinates and momenta in the quantum problem. As a result, both the form of the Hamiltonian as well as the equations of motion are required, meaning that one cannot solve the thermodynamics without also solving the dynamics – Z Z 1 kB T 4 a feature that makes quantum statistical mechanics much harder to solve. Hence, even though it was mentioned in the earlier paragraph that a quantum transition can, in some fashion, be mapped into a classical thermodynamic transition, information about the correlations in excited states which drive the system out of the ground state cannot be extracted from this map. 3 Relevance in experimental Physics and Biology Quantum phase transitions attract intense attention because they are relevant to a host of experimental issues, despite the fact that T = 0 cannot be achieved experimentally. This is because associated with the quantum critical point, there is a sizable region where quantum critical behaviour is observable (labeled QM in Fig. 1B). Some examples are: • Anderson-Mott models and metal-insulator transitions [4], • superconductor-insulator (SI) transition in granular superconductors [5], • transitions between quantum hall states [3], • the physics of vortices in the presence of columnar disorder [6]. Besides the above, knowledge of dissipative effects [7] on quantum coherence and QPT is essential to assess the reliability of mesoscopic quantum devices in performing tasks that strongly depend on their ability to maintain entanglement (for instance, in quantum computation). Among quantum devices widely used, many are based on a collection of regularly arranged or single small Josephson junctions (JJ). A Josephson junction comprises two superconductors linked by a very thin insulating oxide barrier and the current that tunnels through the barrier is the Josephson current. Josephson junction arrays also constitute a particularly attractive testing ground for the superconducting-insulating (SI) transition, because all parameters are well under control and are widely tunable. Cooper pairs of electrons are able to tunnel back and forth between grains and hence communicate about the quantum state on each grain. If the Cooper pairs are able to move freely from grain to grain throughout the array, the system behaves like a superconductor. If the grains are very small, a large charging energy is incurred to 5 Sondhi et al.: Continuous quantum phase317 transitions Sondhi 318 et al.: Continuous quantum phase transitions rameter on the metallic elements connected by the junctions7 and their conjugate variables, the charges (excess Cooper pairs, or equivalently the voltages) on each grain. TheSondhi intermediate state ! m ! , at time " j #j $ " , that FIG. 1. Schematic representation of a 1D Josephson-junction 317 et al.: Continuous jquantum phase transitions enters the quantum partition function of Eq. (5) can array. The crosses represent the tunnel junctions between superconducting segments, and . i are the phases of the thus be superdefined by specifying the set of phase angles conducting order parameter in the latter. % & ( " j ) ' # ( & 1 ( " j ), & 2 ( " j ), . . . , & L ( " j ) ) , where & i ( " j ) is the phase angle of the ith grain at time " j . Two typical paths A. Partition functions and path integrals or time histories on the interval ( 0,* + ) are illustrated in FIG. 1. Schematic representation 1D Josephson-junction Figs. of 2 aand 3, where the orientation of the arrows array. represent for the Z. tunnel junctions between suLet us focus for nowThe oncrosses the expression Notice (‘‘spins’’) indicates the local phase angle of the order segments, .same phases of the superthat the operatorperconducting density matrix e ! ! H isand the as the i are the parameter. The statistical weight of a given path, in the conducting order, parameter in latter.the time-evolution operator e !iHT/" providedsum wethe assign in Eq. (5), is given by the product of the matrix imaginary value T"!i" ! to the time elements interval over A. Partition and path when integrals which the system evolves. functions More precisely, the 3. Typical path or time history of a 1D JosephsonFIG. 2. Typical path or time history of a FIG. 1D JosephsonH(B) (C) e " . 1/* / $ "array. !%&. " j /'! , (6) trace is written in terms of (A) a complete set of states, - % & . " j!1 / ' !junction junction array one of the con-in the insulating phase, where correlations fall Let us focus for now on the j expression for Z. Notice Note that this is equivalent to off exponentially figurations of a 1+1D classical XY model. The long-range cor- in both space and time. This corresponds to !!H that! !the operator density where matrix e is the same as the the disordered relations shown here are typical of the superconducting phasephase in the classical model. Z# ! $" ! e H! n ' , (3) & ntime-evolution operator e !iHT/" , provided we assign the C n of the 1D array ˆ " ˆ or, equivalently, of the H# V 2j "E (7)ordered phase of the imaginary value T"!i" ! to the2 time interval J cos. &over j & j!1 / classical model. j easily identified, finding the classical Hamiltonian for Z takes the formwhich of a sum imaginary-time transition the of system evolves. More precisely, when the FIG. 2. Typical path or time history of a is1D Josephsonthis X-Y model somewhat more work and requires an isstate quantum Josephson-junction amplitudes for the system to start some ! n ' andsetHamiltonian icontrace is written in in terms of athe complete of states, of the junction array. Note that this is equivalent to oneofofthe the matrix explicit evaluation elements, which interreturn to the same state after an imaginaryarray. time Here interval &ˆ j is the the phase willoperator observerepresenting thatfigurations the expression forof the quantum partiof a 1+1D classical XY model. The long-range cor8 readers can find in the Appendix. !i" ! . Thus we see that calculating the thermodynamics parameter on jth grain, tionorder function (5) the has the here formare oftypical a ested classical partirelations shown of the superconducting phase Z# ! $" (3) in Eq. & n ! e ! ! H ! nthe ' , superconducting 9 It expressed is shown in the Appendix that, in an approximation of a quantum system is the same as calculating transition 1 / 1tion V j #"i(2e/C)( & j ) isfunction, conjugate the phase andequivalently, is i.e., sum over n oftoa the 1D array configurations or, of the ordered phase of the preserves the universality class of the problem,10 the amplitudes for its evolution in imaginary the time, with the voltage on the in jthterms junction, E J is matrix, the Josephson of a and transfer if we thinkthat of imaginary classical model. Z takes the form of a sum of imaginary-time transition product of matrix elements in Eq. (6) can be rewritten in total time interval fixed by the temperature of interest. coupling energy. The two in the Hamiltonian then In time as terms an additional spatial dimension. particular, if for happens the system to start in some state ! n ' and thethe form e "H XY , where the Hamiltonian of the equivaThe fact that theamplitudes time interval torepresent be imaginary the charging energy ofsystem each grain Joour quantum lives and in dthe dimensions, expresreturn to the same state after an imaginary time interval will observe that the expression for the quantum lent classical X-Y model partiis is not central. The key idea we hope to transmit the sephson to coupling of the across function the junction sion forphase its partition looksbelike a classical parti. Thusevoke we seeanthat calculating the thermodynamics function Eq. (5) has the form of a classical partireader is that Eq.!i" (3) !should image of quantum tween grains. tion function fortion a system withind#1 dimensions, except 1 of a quantum system is the same as calculating transition tion the function, i.e.,in aextent—" sum overH dynamics and temporal propagation. As indicated previously, candimension map quantum sta(8) that the we extra is finite ! configurations in # units cos. &expressed XY i" & j / amplitudes for its evolution ina imaginary time, with the in the terms of statistical a transfer matrix, we Kthink - ij ! of imaginary This way of looking at things can be given particutistical mechanics of of time. the array onto classical As T→0 system size in this extraif ‘‘time’’ total time interval fixed by the temperature of interest. time as an additional spatial dimension. In particular, if larly beautiful and practical implementation in the lanmechanics by identifying statistical weight of a getand athe sum direction thediverges, and we trulyruns over nearest-neighbor points in the The fact that the time interval of happens to be imaginary quantum system livestwo-dimensional in d dimensions,(space-time) the expres- lattice.11 The nearestguage of Feynman’s path-integral formulation quanspace-time path in d#1-dimensional, Eq. (6) with the our Boltzmann weight ofsystem. effective classical is not central. The key idea we hope to transmit to the sion for its partition function looks like a classical tum mechanics (Feynman, 1972). a two-dimensional Feynman’s spatial of abetween classical asysneighborquancharacter of theparticouplings identifies the classiSinceconfiguration this equivalence d-dimensional reader is that Eq. (3)amplitude should evoke an image of quantum function a system with d#1asdimensions, except prescription is that the net transition between tem. In this case the thereforefor a twocal model model, extensively studied in tumclassical systemsystem and tion a is d#1-dimensional classical system isthe 2D X-Y andbe temporal propagation. that the extra dimension is finite in extent—" ! in units two states of thedynamics system can calculated by summing dimensional X-Y model, its degreeselse of freedom crucial i.e., to everything we haveare to say the and context since Eq.of superfluid and superconducting films way of looking at things canThe be given a particuof time. As T→0 the system size in this extra ‘‘time’’ amplitudes for allThis possible paths between them. planar spins, specified by angles & , that live on a two(Goldenfeld, 1992; Chaikin and Lubensky, 1995). We (5) is probably not i very illuminating for readers not used larlysystem beautiful and practical implementation in the landirection diverges, and wevery getwhilea thetruly path taken by the is defined by dimensional specifying the squaretolattice. that temperatures that, straightforward identification a daily(Recall regimen ofattransfer matrices, itemphasize will be guage of Feynman’s path-integral formulation of quaneffective classical system. state of the system at a sequence of finelyabove spacedzero interthe lattice a finited#1-dimensional, width * + / $example " in the of the of freedom of the classical model in this usefulhas to consider a specific in order to degrees be able tumFormally mechanics 1972). Feynman’s Since equivalence between quan- of the resulting classimediate time steps. we write(Feynman, temporal direction.) While thewhat degrees of this freedom are exampleaisd-dimensional robust, this simplicity to visualize Eq. (5) means. prescription is that the net transition amplitude between tum system and a d#1-dimensional classical system isof a minor miracle. cal Hamiltonian is something !!H ! # 1/" $ ) * H N " ( e two states (4) e + , of the system can be calculated by summing crucial to everything else we to say to and since It have is essential note thatEq. the dimensionless coupling Example: Josephson-junction arrays readers not used amplitudes allispossible betweenB.them. The One-dimensional (5) is not very interval6for that small7It onpaths time where ) * is a time constant K for in H XY , which plays the role of the temperais the believed that neglecting fluctuations in probably the magnitude of illuminating taken where by the,system is ultraviolet defined by specifying the to a daily regimenand of transfer matrices, it will be very scales of interestpath ( ) * ""/,, is some the order parameter is a good approximation (see Bradley ture in the classical model, depends on the ratio of the Consider a one-dimensional array comprising a large the system at achosen sequence finely spaced useful to consider a specific example in orderenergy to be able Doniach, 1984; Wallin et al.,inter1994). cutoff) and N state is aof large integer so of that capacitive charging E C #(2e) 2 /C to the Josephnumber L of identical Josephson junctions as illustrated mediate time steps. Formally we write to visualize what Eq. (5) son means. 8 N ) * "" ! . We then insert a sequence of sums over com- here is that % & ( " ) ' refers couplingbyE J in the array, Our notation to thehave configuration in Fig. 1. Such arrays recently been studied $ ) * Hthe plete sets of intermediate into expression (4) e ! ! H "states ( e ! # 1/" + N , of the entirefor set of Haviland angle variables at time (1996). slice " . A The &ˆ ’s and Delsing Josephson junction a J, !EisC /E K2 (9) Z( ! ): appearing Hamiltonian in Eq. (7) angulartwo coordinate B.areExample: One-dimensional Josephson-junction arrays 6 tunnel connecting superconducting metallic that in is the small on thejunction time where ) * is a time interval operators, and j is a site label. The state at time slice " is an and has nothing to do with the physical temperature (see grains. Cooper pairs of electrons are able to tunnel back scales of interest! #(1/" ) *$ )""/,, where , is some ultraviolet *H array comprising a large of and these operators: cos(grains &ˆ j"&ˆ a )and !%&(")hence the appendix). The physics here is that a large JosephZ# ! $" n!e ! meigenfunction '! j!1one-dimensional 1' forth betweenConsider the communicate and& N is a large integer chosen so that n mcutoff) 1 ,m 2 , . . . ,m N #cos(&j(")"&j!1("))!%&(")'!. number L of identical Josephson junctions as illustrated son coupling produces a small value of K, which favors information N ) * "" ! . We then insert a9 sequence of sums over com- about the quantum state on each grain. If ! # 1/" $ ) * H in Fig. 1. Such arrays have recently been studied by It is useful to think of this as a quantum rotor model. The -& m 1plete ! e sets of ! mintermediate ' the Cooper states into the expression forpairs are able to move freely from grain to 2 '& m 2 ! ••• ! m N state with wave function e im j & j has m j units angular and Delsing (1996). A Josephson junction is a 10 grain throughoutHaviland theofarray, themomensystem is a superconducZ( !!): That is, the approximation is such that the universal aspects # 1/" $ ) * H tum representing pairs on grain j. The -& m N ! e !n'. (5) mtor. junction superconducting metallic j excess Cooper tunnel If the grains are very small, connecting however, it two costs a large of the critical behavior, such as the exponents and scaling funcCooper-pair number operator in this representation isof electrons grains. Cooper pairs are able to tunnel back 1/" $ ) * H charging energy to excess Cooper pairwill onto tions, be agiven without error. However, nonuniversal This rather messy Z expression actually nhas ! mWallin # ! $" & n ! ae !1#rather / 1 & j ) (see et al., 1994).and Themove cosinean term in Eq. 1' j #"i( forth between thethe grains andpairs hence communicate quantities, such as the critical coupling, will differ from an exn Formally m 1 ,m 2 , . . . inclined ,m N grain. If transfers this energy enough, Cooper simple physical interpretation. (7) is a readers ‘‘torque’’ term which unitsisoflarge conserved anc termsJare irrelevant information about the quantum state on each grain. If act evaluation. Technically, the neglected fail to propagate and become stuck on individual grains, (Cooper pairs) from site to site. Note that H 2 -& m 1 ! e ! # 1/" $ ) *gular ! m 2momentum m ! ••• ! m '& 2 ' the Cooper pairs are ableatto from grain themove fixed freely point underlying thetotransition. Nwhich system to be an insulator. the potential energy of thecauses bosonsthe is represented, somewhat c grain throughout the array,11the system is a superconduc6 ! # 1/" * H so that the The essential degrees of freedom system arecrucial Notice this change in notation from Eq. (7), where paradoxically, by the kinetic energy the quantum rotors andin this For convenience we have chosen ) *! eto be$ )real, -& m N !n'. (5) of tor. If the grains are very jsmall, however, a large vice is versa. refers to a papoint it incosts 1D space, not 1+1D space-time. the phases of the complex order small interval of imaginary time that it represents !i ) * . J superconducting charging energy to move an excess Cooper pair onto a This rather messy expression actually has a rather grain. If this energy is large enough, the Cooper pairs simple physical interpretation. Formally inclined readers Rev. Mod. Phys., Vol. 69, No. 1, January 1997 Rev. Mod. Phys., Vol. 69, No. 1, January 1997 fail to propagate and become stuck on individual grains, which causes the system to be an insulator. 6 The essential degrees of freedom in this system are For convenience we have chosen ) * to be real, so that the the phases of the complex superconducting order pasmall interval of imaginary time that it represents is !i ) * . , % Figure 2: (A) Representation of a 1D Josephson Junction array. Crosses 0 represent junctions between superconducting segments, and θ are the phases of the% superconducting order parameter. (B) Typical path or time history of the 1D JJ array. Notice it is equivalent to the configuration of a 1+1D classical XY model. The long-ranged correlations are typical of the ordered (superconducting for 1D JJ) phase. (C) Typical path or time history of the 0 for 1+1D XY) [3]. 1D JJ array in the insulating phase (or disordered phase move an excess Cooper pair onto a grain. When this energy is large enough, Cooper pairs cannot propagate and becomes confined on individual grains, leading to the quenching of the collective superconducting phase. As an example, we consider a one-dimensional array of identical JJs (Fig. 2). The essential degrees of freedom are the phases of the complex superconducting order parameter on the metallic segments connected by the junctions and their conjugate variables, the charges (excess Cooper pairs, or equivalently the voltages) on each grain. Even without doing further math, from % Fig. % 2B and 2C, one can draw an analogy between our system and a 2-D classical XY model. It turns out that, in an approximation that preserves the universality class of the system, we can indeed map our 1D JJ array p into a 2D % % X-Y system [3], with our dimensionless coupling constant K ∼ E /E playing the role of the temperature in the classical analog, where E = (2e) /C is the capacitive charging energy and E the Josephson coupling in the array. This equivalence generalizes to d-dimensional arrays and d + 1-dimensional classical X-Y models. Among the systems studied, two-dimensional ones are most interesting as no true long-range order is possible at finite temperatures while occurs at T = 0. Moreover, 2D samples can be easily Rev. Mod. Phys.,a Vol.genuine 69, No. 1, JanuaryQPT 1997 fabricated and experimentally characterized. 6 al symmetry and that at the transition the polyelectrolyte bundle adopts a universal response ar. numbers: 61.20.Qg, 87.15.-v, 74.40.+k ged DNA molecules in aqueous soludense hexagonal bundles (or tori) in w concentrations of positively charged or “counter-ions” [1, 2, 3]. This as interesting applications in biology bited as well by other highly charged attracted significant theoretical interclear disagreement with the classical n mean-field theory of aqueous electrol [4] and analytical [5, 6] studies of mitive model”–where the biopolymer is , charged rod and the counter-ions as cate that the attraction between two fact a fundamental feature of aqueous interaction, which has a short range, ut-of-phase” correlations between orarrays on the two rods. The purpose show that the problem of the finite er-ion freezing, or “Wigner crystallizaxagonal polyelectrolyte bundle can be = 0 quantum phase transition, and furping leads to surprising predictions for olyelectrolyte bundles. this claim, we begin with a single rod ed charge per unit length, −eρ0 , plus ribution of mobile polyvalent ions of ensed onto the rod. The rod is placed saline aqueous solution with Debye pae absence of thermal fluctuations, the -ions form a one-dimensional “lattice” = Z/ρ0 . The modulated part of the he rod can be expressed as a sum over ce vectors of the Wigner crystal: # $ % !" |ρn |ein Gz+φ(z) + cc. . (1) placement of the of the counter-ion lattice with respect to the rod. The phase variable obeys an effective Hamiltonian, & C L ' ∂φ (2 , (2) dz Frod [φ] = 2 0 ∂z where CG2 is the one-dimensional, T = 0 compression modulus of the lattice–roughly proportional to |ρ1 |2 e−κd – and L is the length of the rods. Due to phase fluctuations at finite temperatures, the contribution from the nth term in eq. (1) to the thermal expectation value "δρ(z)# van2 ishes as e−n L/βC and the distribution has only liquidlike correlations, "δρ(0)δρ(z)# ∼ e−|z|/ξz cos Gz, with ξz = 2βC. Below, we include only the lowest order terms, ẑ ŷ x̂ (a) (b) FIG. 1: The two degenerate, chiral ground states of the anti- Figure 3: Correspondence between and ferromagnetic XY model onpolyelectrolyte a triangular latticebundle are shown in T = 0 2D JJ Thedegenerate, helical ordering around a triangular plaquette the array. (A) The(a).two chiral ground states of theofantiferromagnetic s the reciprocal lattice vector, |ρn | an lattice corresponding to one of these ground states is XY model on bundle a triangular lattice. The helical ordering around a triangular shown in (b). The dark black lines depict the polymer backned by the charge distribution of a sinplaquette of the bundle lattice shown where the dark black lines bone while the spheres is depict peaksinin(B), the condensed charge nd φ(z) a phase variable restricted to density.backbone while the spheres depict peaks in the condensed depict Specifically, φ(z)G−1 is the local the dis- polymer charge density [9]. =0 7 The electromagnetic properties of a Josephson array, such as conductivity and diamagnetic susceptibility, are determined by the response of the array to an applied vector potential. Specifically, the conductivity can be expressed by the Kubo relation as σ(ω) = ImCxx (q = TABLE I: Correspondence between the d = 3, classical polyelectrolyte system (left) and the d = 2+1, quantum frustrated Josephson array system (right). free-energy functional, βF [φ] height along polymer, z longitudinal phonon (or phase) stiffness, βC electrostatic, inter-rod coupling, βEij shear strain, 2G$⊥z elastic response, Gxzxz (qz ) imaginary-time action, h̄−1 S[φ] imaginary time, τ = it inverse grain charging energy, h̄Ec−1 Josephson inter-grain coupling, h̄−1 EJ vector potential, (2π/Φ0 )a⊥ current response, −Cxx (−iω) sponse to uniform sliding deformation Gxzxz (q⊥ , qz ) ∝ (kB T G2 )qz2 ξ(α). H columnar liquid crystal with a bendin the bundle which diverges at the crit correlation length, ξ(α). Turning next ducting phase (α < αc ) of the Josephso zero-frequency conductivity is infinite t current-current correlation function is p superfluid density, so that Cxx (q⊥ = 0 [14]. Therefore, the shear modulus of t bundle, Cxzxz = Gxzxz (q = 0) ∝ is finite in the low-temperature, char and vanishes at the critical point. T ulus corresponding to the uniform ela pictured in Fig. 2. Finally, right at (α = αc ) the Josephson is predicted to i.e. to have a finite zero-frequency c Cxx (q⊥ , −iω) = σ ∗ |ω|. Moreover, th tance is expected to be a universal q e2 /2h̄ [10]. The corresponding strain tion, Gxzxz (q⊥ , qz ) ∝ (kB T G2 )|qz |, wo Figure 4: Correspondence between d = 3 classical polyelectrolyte system (left) and d = 2 + 1 quantum frustrated JJ array system (right) [9]. In biology, nature has a small range of temperatures to play with, due to the protein nature of life chemistry, and most phase transitions occur with the variation of a parameter other than temperature. I hypothesize that it may be possible to draw analogies with QPTs, where the temperature is fixed (at T = 0) and some other parameter is varied, and hence tap into the immerse theoretical and experimental work that has been done on JJ arrays. Although i do not have anything solid to back up my statement, the field of polyelectrolyte condensation may show some promise. Polyelectrolyte chains naturally repel each other, but will nevertheless form condensed bundles in the presence of oppositely charged counterions beyond a certain valency (depending on the nature of the polyelectrolyte). Examples include DNA and F-actin. In fact, polyelectrolytes condense above a certain counterion concentration and dissociate above another counterion concentration. It has become clear that this condensation results from some form of organization of the counterions, either dynamical (correlated charge density fluctuations) or essentially static, in the form of a counterion lattice (positional correlations between condensed counterions). Some theoretical work on the counterion ‘melting’ transition suggest that the melting transition (i.e. the bundle-to-individual polyelectrolyte transition) is continuous and can be shown to be in the universality class of the three-dimensional XY model [8]. In another work [9], researchers managed to establish a mapping between a model for hexagonal polyelectrolyte bundles and a two-dimensional, frustrated Josephson-junction array (Fig.3). They found that the T = 0 SI transition of the quantum system corresponds to a continuous liquid-to-solid transition of the condensed charge in the finite 8 temperature classical polyelectrolyte system. Moreover, the role of the vector potential in the JJ system is played by elastic strain in the classical system (Fig. 4). The general conclusion of this work relates the elastic constants of polyelectrolyte bundles to the phase behaviour of the counterions, which is significant as elastic constants are easier to measure than the scattering amplitudes of counterion charge modulation. 4 Further work Due to constraints, much experimental and theoretical details have been left out. There are interesting instances of quantum phase transitions in two dimensional quantum magnets which cannot be predicted with an order parameter using the GLW (Ginzburg-Landau-Wilson) formalism that we have adopted in this study [10]. Also, studies on the dynamics of QPTs are still active and a quantum counterpart for the classical Kibble-Zurek mechanism of second order thermodynamic phase transitions was recently proposed [11, 12, 13]. 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