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Recent progresses on diagrammatic determinant QMC Lei Wang, ETH Zürich Trento 2015.10 better scaling entanglement & fidelity Iazzi and Troyer, PRB 2015 LW and Troyer, PRL 2014 LW, Iazzi, Corboz and Troyer, PRB 2015 LW, Liu, Imriška, Ma and Troyer, PRX 2015 sign problem Huffman and Chandrasekharan, PRB 2014 Li, Jiang and Yao, PRB 2015 LW, Liu, Iazzi, Troyer and Harcos, 1506.05349 Recent progresses on diagrammatic determinant QMC Lei Wang, ETH Zürich Trento 2015.10 better scaling entanglement & fidelity Iazzi and Troyer, PRB 2015 LW and Troyer, PRL 2014 LW, Iazzi, Corboz and Troyer, PRB 2015 LW, Liu, Imriška, Ma and Troyer, PRX 2015 sign problem Huffman and Chandrasekharan, PRB 2014 Li, Jiang and Yao, PRB 2015 LW, Liu, Iazzi, Troyer and Harcos, 1506.05349 About me Background: condensed matter physics Interested in quantum many-body systems, quantum phase transitions, etc Temperature Please forgive my ignorance! Quantum Critical Phase 1 Phase 2 c Hubbard model of fermions in this talk Solid materials Optical lattices Quantum Monte Carlo achieved. We shall continue our discussion of Markov-chain Monte Carlo methods in Subsection 1.1.4, but want to first take a brief look at the history of stochastic computing. The first recorded Monte Carlo simulation Historical origins The idea of direct sampling was introduced into modern science in the late 1940s by the mathematician Ulam, not without pride, as one can find out from his autobiography Adventures of a Mathematician (Ulam (1991)). Much earlier, in 1777, the French naturalist Buffon (1707–1788) imagined a legendary needle-throwing experiment, and analyzed it completely. All through thehits eighteenth and nineteenth centuries, royal courts and learned circles were intrigued by this game, and the theory was developed further. After a basic treatment of the Buffon needle problem, we shall describe the particularly brilliant idea of Barbier (1860), which foreshadows modern techniques of variance reduction. The Count is shown in Fig. 1.6 randomly throwing needles of length a onto a wooden floor with cracks a distance b apart. We introduce hN 2` i= ⇡d rcenter φ 0 xcenter b 2b 3b 4b Fig. 1.7 Variables xcenter and φ in Buffon’s needle experiment. The needles are of length a. d{ ` { coordinates rcenter and φ as in Fig. 1.7, and assume that the needles’ centers rcenter are uniformly distributed on an infinite floor. The needles do not roll into cracks, as they do in real life, nor do they interact with each other. Furthermore, the angle φ is uniformly distributed between 0 and 2 . This is the mathematical model for Buffon’s experiment. All the cracks in the floor are equivalent, and there are symmetries xcenter ↔ b − xcenter and φ ↔ −φ. The variable y is irrelevant to the Buffon 1777 Fig. 1.6 Georges Louis Leclerc, Co of Buffon (1707–1788), performing first recorded Monte Carlo simulati Statistical Mechanics: in 1777. (Published with permission Algorithms and Computations Le Monde.) Fig. 1.10 Buffon’s experiment with 2000 needles (a = b). 1.1.3 Werner Krauth THE JOURNAL OF CHEMICAL PHYSICS VOLUME 21, NUMBER 6 JUNE, 1953 Equation of State Calculations by Fast Computing Machines NICHOLAS METROPOLIS, ARIANNA W. ROSENBLUTH, MARSHALL N. ROSENBLUTH, AND AUGUSTA H. TELLER, Los Alamos Scientific Laboratory, Los Alamos, New Mexico AND EDWARD TELLER, * Department of Physics, University of Chicago, Chicago, Illinois (Received March 6, 1953) A general method, suitable for fast computing machines, for investigatiflg such properties as equations of state for substances consisting of interacting individual molecules is described. The method consists of a modified Monte Carlo integration over configuration space. Results for the two-dimensional rigid-sphere system have been obtained on the Los Alamos MANIAC and are presented here. These results are compared to the free volume equation of state and to a four-term virial coefficient expansion. I. INTRODUCTION T HE purpose of this paper is to describe a general method, suitable for fast electronic computing disks and spheres 102 Hard disks and spheres machines, of calculating the properties of any substance which may be considered as composed of interacting individual molecules. Classical statistics is assumed, only two-body forces are considered, and the potential field of a molecule is assumed spherically symmetric. These are the usual assumptions made in theories of liquids. Subject to the above assumptions, the method is not restricted to any range of temperature or density. This paper will also present results of a preliminary twodimensional calculation for the rigid-sphere system. Work on the two-dimensional case with a Lennardbe reported Jones potential is in progress and will η = 0.48 in a η = 0.48 II. THE GENERAL METHOD FOR AN ARBITRARY POTENTIAL BETWEEN THE PARTICLES In order to reduce the problem to a feasible size for numerical work, we can, of course, consider only a finite number of particles. This number N may be as high as several hundred. Our system consists of a squaret containing N particles. In order to minimize the surface effects we suppose the complete substance to be periodic, consisting of many such squares, each square containing N particles in the same configuration. Thus we define dAB, the minimum distance between particles A and B, as the shortest distance between A and any of the particles B, of which there is one in each of the squares which comprise the complete substance. If we have a potential which falls off rapidly with distance, there ηwill be at η =most 0.72 one of the distances AB which = 0.72 THE JOURNAL OF CHEMICAL PHYSICS VOLUME 21, NUMBER 6 JUNE, 1953 Equation of State Calculations by Fast Computing Machines NICHOLAS METROPOLIS, ARIANNA W. ROSENBLUTH, MARSHALL N. ROSENBLUTH, AND AUGUSTA H. TELLER, Los Alamos Scientific Laboratory, Los Alamos, New Mexico AND EDWARD TELLER, * Department of Physics, University of Chicago, Chicago, Illinois (Received March 6, 1953) A general method, suitable for fast computing machines, for investigatiflg such properties as equations of state for substances consisting of interacting individual molecules is described. The method consists of a modified Monte Carlo integration over configuration space. Results for the two-dimensional rigid-sphere system have been obtained on the Los Alamos MANIAC and are presented here. These results are compared to the free volume equation of state and to a four-term virial coefficient expansion. I. INTRODUCTION T HE purpose of this paper is to describe a general method, suitable for fast electronic computing disks and spheres 102 Hard disks and spheres machines, of calculating the properties of any substance which may be considered as composed of interacting individual molecules. Classical statistics is assumed, only two-body forces are considered, and the potential field of a molecule is assumed spherically symmetric. These are the usual assumptions made in theories of liquids. Subject to the above assumptions, the method is not restricted to any range of temperature or density. This paper will also present results of a preliminary twodimensional calculation for the rigid-sphere system. Work on the two-dimensional case with a Lennardbe reported Jones potential is in progress and will η = 0.48 in a η = 0.48 II. THE GENERAL METHOD FOR AN ARBITRARY POTENTIAL BETWEEN THE PARTICLES In order to reduce the problem to a feasible size for numerical work, we can, of course, consider only a finite number of particles. This number N may be as high as several hundred. Our system consists of a squaret containing N particles. In order to minimize the surface effects we suppose the complete substance to be periodic, consisting of many such squares, each square containing N particles in the same configuration. Thus we define dAB, the minimum distance between particles A and B, as the shortest distance between A and any of the particles B, of which there is one in each of the squares which comprise the complete substance. If we have a potential which falls off rapidly with distance, there ηwill be at η =most 0.72 one of the distances AB which = 0.72 Quantum to classical mapping ⇣ Z = Tr e Ĥ ⌘ Trotterization ⇣ imaginary-time Z = Tr e M Ĥ ...e Ĥ = Ĥ0 + Ĥ1 Diagrammatic approach M Ĥ ⌘ .. . Z= 1 X k=0 k Z h d⌧1 . . . 0 Tr ( 1)k e ( Z ⌧k 1 d⌧k ⇥ ⌧k )Ĥ0 Ĥ1 . . . Ĥ1 e /M 0 space ⌧1 Ĥ0 0 imaginary-time axis Beard and Wiese, 1996 Prokof ’ev, Svistunov, Tupitsyn,1996 i Time 11 10 9 8 7 6 5 4 3 2 1 0 p World-line Approach Stochastic Series Expansion Prokof ’ev et al, JETP, 87, 310 (1998) Sandvik et al, PRB, 43, 5950 (1991) 44 18 40 36 31 32 14 28 19 24 20 16 46 12 34 8 4 38 0 35 l=0 45 30 41 37 7 33 15 29 6 25 21 17 47 13 2 9 5 39 1 3 l=1 46 16 42 38 4 34 12 30 45 26 22 18 44 14 32 10 6 29 2 13 l=2 47 17 43 39 5 35 0 31 36 27 23 19 28 15 33 11 7 37 3 1 l=3 v X(v) v X(v) v X(v) v X(v) bosons quantum spins 1 0 0 2 1 3 Determinantal Methods 0 2 1 3 fermions Gull et al, RMP, 83, 349 (2011) 0 2 1 3 RE 58. Allowed vertices for the isotropic S = 1/2 Heisenberg model. The numbering l = 0, 1, 2, 3 vertex legs corresponds to the position v = 4p + l, in linked-list storage illustrated in Fig. 57. 3 2 RE 57. Linked vertex storage of the configuration in Fig. 55. In the graphical representation to ft, constant spin states between operators have been replaced by lines (links) connecting the spins efore and after the operator acts. The links can be stored in a list X(v), where the four elements p + l, l = 0, 1, 2, 3, correspond to the legs (with the numbering convention shown in Fig. 58) of the x at position p in the sequence SL . For two linked legs v and v′ , X(v) = v′ and X(v′ ) = v. Space Diagrammatic approaches Diagrammatic approaches bosons quantum spins fermions World-line Approach Stochastic Series Expansion Determinantal Methods Prokof ’ev et al, JETP, 87, 310 (1998) Sandvik et al, PRB, 43, 5950 (1991) Gull et al, RMP, 83, 349 (2011) 0 Diagrammatic determinant QMC Z= 1 X k=0 = 1 X k=0 k Z d⌧1 . . . 0 k X Ck Z ⌧k w(Ck ) 1 h d⌧k Tr ( 1)k e ( ⌧k )Ĥ0 Ĥ1 . . . Ĥ1 e ⌧1 Ĥ0 i Diagrammatic determinant QMC Z= 1 X k=0 = 1 X k=0 k Z d⌧1 . . . 0 k X Z ⌧k w(Ck ) Ck 0 1 h d⌧k Tr ( 1)k e ( ⌧k )Ĥ0 Ĥ1 . . . Ĥ1 e ⌧1 Ĥ0 i Diagrammatic determinant QMC Z= 1 X k=0 = 1 X k=0 k Z d⌧1 . . . 0 k X Z ⌧k w(Ck ) Ck 0 1 h d⌧k Tr ( 1)k e ( ⌧k )Ĥ0 Ĥ1 . . . Ĥ1 e ⌧1 Ĥ0 i Diagrammatic determinant QMC Z= 1 X k=0 = 1 X k=0 k Z d⌧1 . . . 0 k X Z ⌧k 1 h d⌧k Tr ( 1)k e ( ⌧k )Ĥ0 ⌧1 Ĥ0 Ĥ1 . . . Ĥ1 e i Rubtsov et al, PRB 2005 Gull et al, RMP 2011 w(Ck ) Ck det ✓ hki ⇠ 0 Noninteracting Green0 s functions N , scales as O( ◆ k⇥k 3 3 3 N ) Diagrammatic determinant QMC Z= 1 X k=0 = 1 X k=0 k Z d⌧1 . . . 0 k X Z ⌧k 1 h d⌧k Tr ( 1)k e ( ⌧k )Ĥ0 ⌧1 Ĥ0 Ĥ1 . . . Ĥ1 e i Rubtsov et al, PRB 2005 Gull et al, RMP 2011 w(Ck ) Ck det ✓ hki ⇠ Noninteracting Green0 s functions N , scales as O( ◆ k⇥k 3 3 3 N ) 0 Rombouts, Heyde and Jachowicz, PRL 1999 Iazzi and Troyer, PRB 2015 LW, Iazzi, Corboz and Troyer, PRB 2015 LCT-QMC Methods ⇣ det I + T e R 0 thus achieving O( d⌧ HCk (⌧ ) ⌘ N ⇥N N 3 ) scaling! Fidelity susceptibility made simple! LW, Liu, Imriška, Ma and Troyer, PRX 2015 F = hkL kR i 2 hkL i hkR i 2 kR = 4 0 kL = 2 Signifies quantum phase transitions without need of the local order parameter kL Time = kR Worldline Algorithms Stochastic Series Expansion Determinantal Methods (bosons) 44 18 40 36 31 32 14 28 19 24 20 16 46 12 34 8 4 38 0 35 l=0 45 30 41 37 7 33 15 29 6 25 21 17 47 13 2 9 5 39 1 3 l=1 46 16 42 38 4 34 12 30 45 26 22 18 44 14 32 10 6 29 2 13 l=2 47 17 43 39 5 35 0 31 36 27 23 19 28 15 33 11 7 37 3 1 l=3 v X(v) v X(v) v X(v) v X(v) 2 11 10 9 8 7 6 5 4 3 2 1 0 F hkL kR i p LW, Liu, Imriška, Ma and Troyer, PRX 2015 hkL i hkR i 2 kL Signifies quantum phase transitions without need of the local order parameter (quantum spins) kR 1 0 3 kL 0 2 kR 1 (fermions) 0 2 1 3 0 2 1 3 FIGURE 58. Allowed vertices for the isotropic S = 1/2 Heisenberg model. The numbering l = 0, 1, 2, 3 of the vertex legs corresponds to the position v = 4p + l, in linked-list storage illustrated in Fig. 57. 3 2 FIGURE 57. Linked vertex storage of the configuration in Fig. 55. In the graphical representation to the left, constant spin states between operators have been replaced by lines (links) connecting the spins just before and after the operator acts. The links can be stored in a list X(v), where the four elements v = 4p + l, l = 0, 1, 2, 3, correspond to the legs (with the numbering convention shown in Fig. 58) of the vertex at position p in the sequence SL . For two linked legs v and v′ , X(v) = v′ and X(v′ ) = v. Space Fidelity susceptibility made simple! More advantages 1 X hÔi = Z k @hÔi hÔki = @ hÔihki w(Ck )O(Ck ) Ck Histogram reweighing Lee-Yang zeros = hÔi(k) Observable derivatives k X x x x x x x x x x x k Directly sample derivatives of any observable x Can obtain observables in a continuous range of coupling strengths x x x x x < Partition function zeros in the complex coupling strength plane Ferrenberg et al, 1988 Wang and Landau, 2001 Troyer et al, 2003 What about the sign problem ? Sign problem free time-reversal symmetry M" = w(Ck ) = det M" ⇥ det M# = | det M" | 2 0 ⇤ M# Lang et al, Phys. Rev. C, 1993 Koonin et al, Phys. Rep, 1997 Hands et al, EPJC, 2000 Wu et al, PRB, 2005 uantum Monte Carlo simulations. The quantum spin liquid of the k spin-orbit interaction, and is not adiabatically connected to the ritical value of U > Uc both states are unstable toward magnetic ate we study the spin, charge, and single-particle dynamics of the the Hubbard interaction only on a ribbon edge. The Hubbard currents along the edge and promotes edge magnetism but leaves elical liquid intact. What about the sign problem ? PACS numbers: 75.10.!b, 03.65.Vf, 71.10.Pm, 71.30.+h del on the þ HU , cj ; Sign problem free time-reversal symmetry Bulk phase diagram.—For bulk simulations we use the projective auxiliary field QMC approach. The ground state j!0 i is filtered out of a trial wave function j!T i with h!T j!0 i ! 0; a very good choice is the ground state of the KM model. For an arbitrary observable, h!0 jOj!0 i ¼ lim"!1 h!T je!"H=2 Oe!"H=2 j!T i=h!T je!"H j!T i. The absence of the negative sign problem at half filling follows from the fact that, after a discrete Hubbard-Stratonovich transformation of HU and subsequent integration over the fermionic degrees of freedom, the fermionic determinants in the up and down spin sectors are linked via complex conjugation such that their product is positive. We employ an SUð2Þ invariant Hubbard-Stratonovich transformation and an imaginary time discretization of #"t ¼ 0:1. Projection parameters "t ¼ 40 prove sufficient for converged (within statistical errors) ground-state results. For details of the algorithm, see [19]. The SO coupling reduces the SUð2Þ symmetry to a Uð1Þ symmetry corresponding to spin rotations around the z axis. The Hubbard interaction promotes transverse, x-y magnetic ordering [15] which can be tracked by M" = w(Ck ) = det M" ⇥ det M# = | det M" | 2 0 Lang et al, Phys. Rev. C, 1993 Koonin et al, Phys. Rep, 1997 Hands et al, EPJC, 2000 B C Wu et al, PRB, 2005 A K K Attractive interaction at any filling on any lattice Repulsive interaction at half-filling on bipartite lattices And more … PRL 111, 066401 (2013) PHYSICAL REVIEW LETTERS week ending 9 AUGUST 2013 exactly N=2 particles on each lattice site, H would reduce to an exact SUðNÞ-symmetric Heisenberg(b) model HH ¼ P λ U U J=2N hi;ji Si $ Sj , with Si being the vector of the SUðNÞ λ P y a a t spin operators Si ¼ !;" ci! T!" ci" , expressed in terms of t a the generators T a ,2 a ¼ 1; . . . ; N 2 % 1; of SUðNÞ in the δ 1 fundamental representation, with TrðT a T b Þ ¼ #ab =2 [e.g., T a ¼ $a =2 in terms of the Pauli matrices $a for SU(2)]. 1 the large U limit of a model that in This would result, e.g.,ain δ3 to H also includes a local Hubbard-U interaction, addition δ2the local particle fluctuations around the which reduces mean value of N=2. In the large-N limit, and considering only the paramagnetic saddle point, these fluctuations become irrelevant and the Hubbard-U interaction merely fixes the average particle number to N=2. The FIG. 1 (color online). Periodic latticethe structure the 1 (color online). Ground state phase diagram of fermions Hamiltonian(a) in Eq. (1) indeed equals U ¼ 0 limit ofof FIG. the Hubbard-Heisenberg Hamiltonian of the seminal with SUðNÞ-symmetric flavor exchange on the honeycomb latKM-Hubbard model with nearest-neighbor hopping t, spin-orbit works in Refs. [2–4], and represents an unrestricted tice. Crosses denote parameters at which QMC simulations have coupling !, and SUðNÞ-symmetric Coulomb repulsion U.It has Arrows indicate been the carried out. For all considered (even) N the system undert % J model. been considered goes a quantum phase transition from a semimetal (SM) to an using with QMC simulations on the square lattice, current direction previously associated the spin-orbit term for one insulator. For N ( 6 the insulating state is a columnar valence where an exotic gapless spin liquid was obtained for N ¼ 4 spin species and sublattice. (b) Effective model on a semi-infinite bond solid (cVBS), while at N ¼ 4 it is a valence bond solid with flavors [27]. In the following, we consider the case of the Fig. 1. (A) Fermi surface of the Fermi liquid phase of a single band model on the square unit lattice spacing. The “hot spots” are denoted by the filled circles. (B) The reconstru surface in the metal with SDW order. The dashed lines show the Fermi surface in the metal w order, and its translation by K. Gaps have opened at the hot spots, leading to small “poc surfaces. (C) A deformed Fermi surface of the metal without SDW order, in which the vicin hot spots are unchanged from (A). The horizontal and vertical Fermi surfaces now belong t electronic bands. 1 (a) a Wannier ree nearest '"3 g, see ext-nearest =j"n ( "m j article-hole ted with a Hohenadler, Lang and Assaad, PRL 2011 ing the in- Topological insulators SU(2N) models Lang, Meng, Muramatsu, Wessel and Assaad, PRL 2013 resonating valence bond plaquettes (pVBS); both are depicted in 1 A 0 K −0.5 −1 −1 B 0.5 0.5 ky /π (1) ⇤ M# ky /π arises from time reverdge states, s spin and ven by the ces to two gns of the me reversal ential scat[4,5], and vious work wo routes: or (as here) b repulsion rlo (QMC) transition se of [18] t magnetic the helical that the renormalization group and Feyn expansions of the field theory flow to s pling in two spatial dimensions, mak analytical progress difficult. Here, we show that the universa field theory can be realized in lattice m are free of the sign problem and so is to large-scale QMC studies. Our claim contradict the no-go theorem of (5), b do not provide a general recipe for e the sign problem. However, we will it for the specific case of the onset romagnetic order in a two-dimensio provided the perturbative arguments portance of the hot spots to the qua theory (7, 8, 10, 11) apply. Our modi model has at least two bands. Therefor in which there is only a single active b transition, such as in the electron-doped our method requires modifying the face far away from the hot spots; we this can be done while preserving the low-energy properties of the antiferr critical point. On the other hand, o applies to multiband situations (suc sign problem. Such algorithms express the partition function as a sum over Feynman histories, and the sign problem arises when the weights assigned to the trajectories are not all positive because of quantum interference effects. A general solution to the fermion sign problem has been proved to be in the computational complexity class of nondeterministic polynomial (NP) hard (5), and so there has been little hope that the antiferromagnetic quantum critical point could be elucidated by computational studies. Application of the methods of quantum field theory and the renormalization group to the onset of antiferromagnetism in a metal (6) has identified (7, 8) a universal quantum field theory that captures all the singular low-energy quantum fluctuations that control the quantum critical point and deviations from the Fermi liquid physics of traditional metals. The field theory is expressed in terms of fermionic excitations in the vicinity of a finite number of “hot spots” on the Fermi surface, and is thus independent of the details of the fermionic band structure, except for the number of hot spots and Fermi velocities at the hot spots (9). Recent work (10, 11) has shown 0 −0.5 0 kx /π 1 −1 −1 0 kx Fig. 2. (A) Fermi surfaces (solid lines) of LF for free yx,y fermions with the parameters li text. The dashed lines show the portion of the Fermi surface in Fig. 1C that was shifte obtain the yy Fermi surface. The hot spots are now at the intersections of the Fermi su → Mean-field yx,y Fermi surfaces with SDW order j〈f〉j = 0:25. Two-orbital model Berg, Metliski and Sachdev, Science SCIENCE 2012 www.sciencemag.org VOL 338 What about the sign problem ? Sign problem free time-reversal symmetry M" = w(Ck ) = det M" ⇥ det M# = | det M" | 2 ⇤ M# Lang et al, Phys. Rev. C, 1993 Koonin et al, Phys. Rep, 1997 Hands et al, EPJC, 2000 Wu et al, PRB, 2005 0 But, how about this ? spinless fermions Ĥ = t X⇣ ĉ†i ĉj hi,ji + ĉ†j ĉi ⌘ +V X n̂i n̂j hi,ji w(Ck ) = det M Scalapino et al, PRB 1984 Gubernatis et al, PRB 1985 Meron cluster approach, Chandrasekharan and Wiese, PRL 1999 up to 8*8 square lattice and T≥0.3t solves sign problem for V ≥ 2t Solutions ! RAPID COMMUNICATIONS PHYSICAL REVIEW B 89, 111101(R) (2014) Solution to sign problems in half-filled spin-polarized electronic systems Emilie Fulton Huffman and Shailesh Chandrasekharan Department of Physics, Duke University, Durham, North Carolina 27708, USA (Received 19 December 2013; revised manuscript received 14 February 2014; published 12 March 2014) We solve the sign problem in a particle-hole symmetric spin-polarized fermion model on bipartite lattices using the idea of fermion bags. The solution can be extended to a class of models at half filling but without particle-hole symmetry. Attractive Hubbard models with an odd number of fermion species can also be solved. Our solutions should allow us to study quantum phase transitions that have remained unexplored so far due to sign problems. DOI: 10.1103/PhysRevB.89.111101 Quantum Monte Carlo methods for many-body fermionic systems in thermal equilibrium usually require one to be able to rewrite quantum partition functions as a sum over classical configurations with positive Boltzmann weights that are computable in polynomial time. Unfortunately, due to the underlying quantum nature of the problem, the Boltzmann weights can be negative or even complex in general. Such expansions are said to suffer from a sign problem [1]. The discovery of an expansion with positive Boltzmann weights is referred to as a solution to the sign problem. Solutions to sign problems in many quantum systems are considered to be outstanding problems in computational complexity [2]. Traditionally, solutions are based on rewriting the interacting problem as a free fermion problem where fermions only interact with background auxiliary fields [3–6]. The Boltzmann weight then depends on the determinant of the free fermion matrix, which can still be negative or complex. However, in electronic systems, a symmetric treatment of both spin components of the electron can sometimes make the Boltzmann weight positive since it can be written as the product of two real determinants that come with the same sign [7]. Sign problems PACS number(s): 71.10.Fd, 71.27.+a, 71.30.+h neighbor Hubbard-type interaction. The model is well known as the tV model and was considered on square lattices a long time ago [9,10]. Although the model has a particle-hole symmetry, as far as we know its sign problem has not been solved by traditional methods for any value of V . Thus, it seems like the tV model at half filling in spin-polarized systems is more difficult to solve than the traditional Hubbard model with an on-site U interaction between the two spins. Unlike the traditional Hubbard model, here the V < 0 model cannot be mapped into the V > 0 model through a unitary transformation. In the repulsive case for V ! 2t the sign problem could indeed be solved using a nontraditional method called the meron-cluster approach [11]. Unfortunately, that solution could not be extended to smaller values of V . The spin-polarized t-V model (1) is of interest from a fundamental quantum field theory perspective since it describes a minimally doubled lattice fermion system [12]. A similar minimally doubled fermion system can be obtained with Hamiltonian staggered fermions on a square lattice [13,14]. These models contain an interesting quantum phase transition between a semimetal phase (containing massless Dirac Solutions ! RAPID COMMUNICATIONS PHYSICAL REVIEW B 89, 111101(R) (2014) Solution to sign problems in half-filled spin-polarized electronic systems Emilie Fulton Huffman and Shailesh Chandrasekharan Department of Physics, Duke University, Durham, North Carolina 27708, USA (Received 19 December 2013; revised manuscript received 14 February 2014; published 12 March 2014) RAPID COMMUNICATIONS We solve the sign problem in a particle-hole symmetric spin-polarized fermion model on bipartite lattices REVIEW 241117(R) (2015) using the idea of fermion bags. The solution can bePHYSICAL extended to a class Bof91, models at half filling but without particle-hole symmetry. Attractive Hubbard models with an odd number of fermion species can also be solved. Solving theallow fermion signquantum problem intransitions quantum Carlo unexplored simulations bydue Majorana Our solutions should us to study phase thatMonte have remained so far to sign problems. Zi-Xiang Li,1 Yi-Fan Jiang,1,2 and Hong Yao1,3,* DOI: 10.1103/PhysRevB.89.111101 1 Institute for Advanced Study, PACSTsinghua number(s): 71.10.Fd, 71.27.+a, University, Beijing 100084,71.30.+h China representation 2 Department of Physics, Stanford University, Stanford, California 94305, USA Collaborative Innovation Center of Quantum Matter, Beijing 100084, China for (Received many-body fermionic neighbor Hubbard-type interaction. The model is well known 27 August 2014; revised manuscript received 13 October 2014; published 30 June 2015) 3 Quantum Monte Carlo methods systems in thermal equilibrium usually require one to be as the tV model and was considered on square lattices a discover as a quantum (QMC) method to solve the fermion sign problem in interacting fermion able to rewrite quantum partition We functions a sumMonte over Carlolong time agoPHYSICAL [9,10]. Although model has a particle-hole REVIEWthe B 91, 235151 (2015) models by employing a Majorana of complex We call the “Majorana (MQMC). classical configurations with positive Boltzmann weights that representation symmetry, as far fermions. as we know itsit sign problemQMC” has not been MQMC simulations can be performed efficiently both at finite and zero temperatures. Especially, MQMC are computable in polynomial time. Unfortunately, due to the solved by traditional methods for any value of V . Thus, isit Efficient continuous-time Carlo ground fermion sign free in Boltzmann simulating a class seems of quantum spinless fermion models on bipartite lattices atfor half filling and with anstate of correlated fermions underlying quantum nature of the problem, the like theMonte tV model at method half filling inthe spin-polarized range of (unfrustrated) interactions. Moreover, we find a class of SU (N ) fermionic models with odd weights can be negative or evenarbitrary complex in general. Such systems is1more difficult1to solve than the traditional Hubbard 1 2 N , which are sign free in MQMC but whose sign problem cannot be Philippe in solved in other QMC methods, such as Lei Wang, Mauro Iazzi, Corboz, and Matthias Troyer expansions are said to suffer from a sign problem [1]. The model1 with an on-site U interaction between the two spins. continuous-time QMC. To the best of our knowledge, MQMC is theETH first Zurich, auxiliary8093 fieldZurich, QMC method to solve Theoretische Physik, Switzerland discovery of an expansion with positive Boltzmann weights Unlike the traditional Hubbard model, here the V < 0 model 2 the fermion sign problem in spinless (more generally, number ofScience species)Park fermion conjecture Institute for Theoretical Physics, Universityanofodd Amsterdam, 904models. PostbusWe 94485, 1090 GL Amsterdam, The Netherlands is referred to as a solution to the that signMQMC problem. tosolve cannot be sign mapped into the V > 0 fermionic model through a unitary couldSolutions be applied to the fermion problem in more generic models. January 2015; revised manuscript received 13 March 2015; published 30 June 2015) sign problems in many quantum systems are considered(Received to be 12transformation. In the repulsive case for V ! 2t the sign DOI: 10.1103/PhysRevB.91.241117 PACS number(s): 05.30.Fk, 02.70.Ss, 71.27.+a outstanding problems in computational complexity [2]. problem could indeedof be using a nontraditional method We present the ground state extension thesolved efficient continuous-time quantum Monte Carlo algorithm for Traditionally, solutions are based on rewriting lattice the interactcalled meron-cluster [11]. Unfortunately, fermions of M. Iazzithe and M. Troyer, Phys. approach Rev. B 91, 241118 (2015). Based on that continuous-time expansion ing problem as a free fermion problem where fermions only insolution could not be extended to smaller values of V . of an imaginary-time projection the algorithm is free of fermions. systematicConsequently, error and scales with Introduction. Interacting fermionic quantum systems with operator, represented as two Majorana welinearly can teract with background auxiliary fields and/or [3–6].topological Theprojection Boltzmann The spin-polarized t-V model (1) is of interest from funtime andhave interaction strength. Compared to the conventional quantum Carlo methods strong correlations properties attracted express spinless fermion Hamiltonians inMonte aaMajorana represen-for lattice weight then depends increasing on the determinant of theNonetheless, free fermion quantum field theory perspective sincepowerful it describes fermions, thisinapproach hashigher greater flexibility and is easier to combine with machinery such as histogram attention [1,2]. twodamental and tation and then perform Hubbard-Stratonovich (HS) transformatrix, which can still be negative or complex. However, inand extended spatial dimensions, strongly interacting are doubled mations to decouple by introducing a systems minimally lattice fermion systemthe [12]. A similar ofbackground reweightingquantum ensemble simulation techniques. Weinteractions discuss implementation the continuous-time the reach ofprojection analytical in thethe sense auxiliary fields. Under certain conditions, as particle-hole electronic systems, a generically symmetricbeyond treatment of both spin cominmethods detailminimally using spinless t-V fermion model as an example compare thesuch numerical results with exact doubled system canandbe obtained with of solving those quantum in an unbiased way.matrix As anrenormalization symmetry, we can find symmetric treatment species ponents of the electron can sometimes make models thediagonalization, Boltzmann density group, and infinite projected entangled-pair states calculations. Hamiltonian staggered fermions on a asquare lattice [13,14].of two unbiased the the quantum of fermionic Majorana fermions, namely, freetransiMajorana weight positive since itintrinsically can be written as thenumerical productFinally ofmethod, twowereal use methodMonte to study contain the quantum critical point the of spinless fermions fermion on a honeycomb These models an interesting quantum phase Carlo (QMC) simulation plays a key role in understanding Hamiltonian obtained after HS transformations is a sum of two determinants that come with the same sign [7]. Sign problems lattice and confirm tion previous resultsaconcerning its phase critical (containing exponents. massless Dirac between semimetal Solutions ! RAPID COMMUNICATIONS PHYSICAL REVIEW B 89, 111101(R) (2014) Solution to sign problems in half-filled spin-polarized electronic systems Emilie Fulton Huffman and Shailesh Chandrasekharan Department of Physics, Duke University, Durham, North Carolina 27708, USA (Received 19 December 2013; revised manuscript received 14 February 2014; published 12 March 2014) RAPID COMMUNICATIONS We solve the sign problem in a particle-hole symmetric spin-polarized fermion model on bipartite lattices REVIEW 241117(R) (2015) using the idea of fermion bags. The solution can bePHYSICAL extended to a class Bof91, models at half filling but without particle-hole symmetry. Attractive Hubbard models with an odd number of fermion species can also be solved. Solving theallow fermion signquantum problem intransitions quantum Carlo unexplored simulations bydue Majorana Our solutions should us to study phase thatMonte have remained so far to sign problems. Zi-Xiang Li,1 Yi-Fan Jiang,1,2 and Hong Yao1,3,* DOI: 10.1103/PhysRevB.89.111101 1 Institute for Advanced Study, PACSTsinghua number(s): 71.10.Fd, 71.27.+a, University, Beijing 100084,71.30.+h China representation 2 Department of Physics, Stanford University, Stanford, California 94305, USA Collaborative Innovation Center of Quantum Matter, Beijing 100084, China for (Received many-body fermionic neighbor Hubbard-type interaction. The model is well known 27 August 2014; revised manuscript received 13 October 2014; published 30 June 2015) 3 Quantum Monte Carlo methods systems in thermal equilibrium usually require one to be as the tV model and was considered on square lattices a discover as a quantum (QMC) method to solve the fermion sign problem in interacting fermion able to rewrite quantum partition We functions a sumMonte over Carlolong time agoPHYSICAL [9,10]. Although model has a particle-hole REVIEWthe B 91, 235151 (2015) models by employing a Majorana of complex We call the “Majorana (MQMC). classical configurations with positive Boltzmann weights that representation symmetry, as far fermions. as we know itsit sign problemQMC” has not been MQMC simulations can be performed efficiently both at finite and zero temperatures. Especially, MQMC are computable in polynomial time. Unfortunately, due to the solved by traditional methods for any value of V . Thus, isit Efficient continuous-time Carlo ground fermion sign free in Boltzmann simulating a class seems of quantum spinless fermion models on bipartite lattices atfor half filling and with anstate of correlated fermions underlying quantum nature of the problem, the like theMonte tV model at method half filling inthe spin-polarized range of (unfrustrated) interactions. Moreover, we find a class of SU (N ) fermionic models with odd weights can be negative or evenarbitrary complex in general. Such systems is1more difficult1to solve than the traditional Hubbard 1 2 N , which are sign free in MQMC but whose sign problem cannot be Philippe in solved in other QMC methods, such as Lei Wang, Mauro Iazzi, Corboz, and Matthias Troyer expansions are said to suffer from a sign problem [1]. The model1 with an on-site U interaction between the two spins. continuous-time QMC. To the best of our knowledge, MQMC is theETH first Zurich, auxiliary8093 fieldZurich, QMC method to solve Theoretische Physik, Switzerland discovery of an expansion with positive Boltzmann weights Unlike the traditional Hubbard model, here the V < 0 model 2 the fermion sign problem in spinless (more generally, number ofScience species)Park fermion conjecture Institute for Theoretical Physics, Universityanofodd Amsterdam, 904models. PostbusWe 94485, 1090 GL Amsterdam, The Netherlands is referred to as a solution to the that signMQMC problem. tosolve cannot be sign mapped into the V > 0 fermionic model through a unitary couldSolutions be applied to the fermion problem in more generic models. January 2015; revised manuscript received 13 March 2015; published 30 June 2015) sign problems in many quantum systems are considered(Received to be 12transformation. In the repulsive case for V ! 2t the sign DOI: 10.1103/PhysRevB.91.241117 PACS number(s): 05.30.Fk, 02.70.Ss, 71.27.+a outstanding problems in computational complexity [2]. problem could indeedof be using a nontraditional method We present the ground state extension thesolved efficient continuous-time quantum Monte Carlo algorithm for Traditionally, solutions are based on rewriting lattice the interactcalled meron-cluster [11]. Unfortunately, fermions of M. Iazzithe and M. Troyer, Phys. approach Rev. B 91, 241118 (2015). Based on that continuous-time expansion 1506.05349 ing problem as a free fermion problem where fermions only insolution could not be extended to smaller values of V . of an imaginary-time projection the algorithm is free of fermions. systematicConsequently, error and scales with Introduction. Interacting fermionic quantum with operator, represented as two Majorana welinearly can Splitsystems orthogonal group: teract with background auxiliary [3–6].topological The Boltzmann The spin-polarized t-V model (1) is of interest from funprojection time andhave interaction strength. Compared to the conventional quantum Carlo methods strong correlations and/or properties attracted express spinless fermion Hamiltonians inMonte aaMajorana represen-for lattice Afields guiding principle for sign-problem-free fermionic simulations weight then depends increasing on the determinant of theNonetheless, free fermion quantum field theory perspective sincepowerful it describes fermions, thisinapproach hashigher greater flexibility and is easier to combine with machinery such as histogram attention [1,2]. twodamental and tation and then perform Hubbard-Stratonovich (HS) transformatrix, which can still be negative or complex. However, in spatial dimensions, strongly interacting mations to decouple by introducing a systems minimally lattice fermion systemthe [12]. A2 similar ofbackground reweightingquantum ensemble simulation techniques. Weinteractions discuss implementation the continuous-time 1 1and extended 1are doubled 1 Lei Wang , Ye-Hua Liu , Mauro Iazzi , Matthias Troyer and Gergely Harcos the reach ofprojection analytical in thethe sense auxiliary fields. Under certain conditions, as particle-hole electronic systems, a generically symmetricbeyond treatment of spin cominmethods detail using spinless t-V fermion model as an example andbe compare thesuch numerical results with exact doubled system canand obtained with 1 both Theoretische Physik, minimally ETH Zurich, 8093 Zurich, Switzerland of solving models in an unbiased way.matrix As anrenormalization symmetry, we can find symmetric treatment species ponents of the electron can sometimes makeInstitute thediagonalization, Boltzmann 2those quantum density group, and infinite projected entangled-pair states calculations. Hamiltonian staggered fermions on a asquare lattice [13,14].of two Alfréd Rényi of Mathematics, Reáltanoda utca 13-15., Budapest H-1053, Hungary unbiased the the quantum of fermionic Majorana fermions, namely, freetransiMajorana weight positive since itintrinsically can be written as thenumerical productFinally ofmethod, twowereal use methodMonte to study contain the quantum critical point the of spinless fermions fermion on a honeycomb These models an interesting quantum phase Carlo (QMC) simulation plays a key role in understanding Hamiltonian obtained after HS transformations is a sum of two determinants that come with theWe same sign a[7]. Sign problems lattice and confirm previous results its phase criticaland exponents. present guiding principle for designing fermionic Hamiltonians quantummassless Monte Carlo tion between aconcerning semimetal (containing Dirac A tale of open science ⇣ w(Ck ) ⇠ det I + T e R 0 d⌧ HCk (⌧ ) ⌘ Free fermions with an effective imaginary-time dependent A tale of open science ⇣ w(Ck ) ⇠ det I + T e R 0 Let real matrices Ai = then det I + e A1 A2 e d⌧ HCk (⌧ ) ✓ 0 BiT AN ...e ⌘ Free fermions with an Bi 0 effective imaginary-time dependent ◆ 0 http://mathoverflow.net/questions/204460/ how-to-prove-this-determinant-is-positive A tale of open science ⇣ w(Ck ) ⇠ det I + T e R 0 Let real matrices Ai = then det I + e A1 A2 e d⌧ HCk (⌧ ) ✓ 0 BiT ⌘ Free fermions with an Bi 0 AN ...e The conjecture was proved by Gergely Harcos and Terence Tao, with inputs from others https://terrytao.wordpress.com/2015/05/03/ the-standard-branch-of-the-matrix-logarithm/ Tao and Paul Erdős in 1985 effective imaginary-time dependent ◆ 0 http://mathoverflow.net/questions/204460/ how-to-prove-this-determinant-is-positive A tale of open science ⇣ w(Ck ) ⇠ det I + T e R 0 Let real matrices Ai = then det I + e A1 A2 e d⌧ HCk (⌧ ) ✓ 0 BiT ⌘ Free fermions with an Bi 0 AN ...e The conjecture was proved by Gergely Harcos and Terence Tao, with inputs from others https://terrytao.wordpress.com/2015/05/03/ the-standard-branch-of-the-matrix-logarithm/ Tao and Paul Erdős in 1985 effective imaginary-time dependent ◆ 0 http://mathoverflow.net/questions/204460/ how-to-prove-this-determinant-is-positive A tale of open science ⇣ w(Ck ) ⇠ det I + T e R 0 Let real matrices Ai = then det I + e A1 A2 e d⌧ HCk (⌧ ) ✓ 0 BiT ⌘ Free fermions with an Bi 0 AN ...e The conjecture was proved by Gergely Harcos and Terence Tao, with inputs from others https://terrytao.wordpress.com/2015/05/03/ the-standard-branch-of-the-matrix-logarithm/ Tao and Paul Erdős in 1985 effective imaginary-time dependent ◆ 0 http://mathoverflow.net/questions/204460/ how-to-prove-this-determinant-is-positive A tale of open science ⇣ w(Ck ) ⇠ det I + T e R 0 Let real matrices Ai = then det I + e A1 A2 e d⌧ HCk (⌧ ) ✓ 0 BiT ⌘ Free fermions with an Bi 0 AN ...e The conjecture was proved by Gergely Harcos and Terence Tao, with inputs from others https://terrytao.wordpress.com/2015/05/03/ the-standard-branch-of-the-matrix-logarithm/ Tao and Paul Erdős in 1985 effective imaginary-time dependent ◆ 0 http://mathoverflow.net/questions/204460/ how-to-prove-this-determinant-is-positive A new de-sign principle LW, Liu, Iazzi, Troyer and Harcos 1506.05349 M T ⌘M = ⌘ ⌘ = diag(I, I) A new de-sign principle LW, Liu, Iazzi, Troyer and Harcos 1506.05349 M T ⌘M = ⌘ ⌘ = diag(I, I) O+ (n, n) O++ (n, n) O O M 2 O(n, n) split orthogonal group (n, n) + (n, n) A new de-sign principle LW, Liu, Iazzi, Troyer and Harcos 1506.05349 M T ⌘M = ⌘ ⌘ = diag(I, I) O+ (n, n) 0 ⌘0 det (I + M ) has a definite sign for each component ! O++ (n, n) O (n, n) O 0 + (n, n) ⌘0 A new de-sign principle LW, Liu, Iazzi, Troyer and Harcos 1506.05349 M T ⌘M = ⌘ Te R 0 d⌧ HCk (⌧ ) ⌘ = diag(I, I) O+ (n, n) 0 ⌘0 det (I + M ) has a definite sign for each component ! O++ (n, n) O (n, n) O 0 + (n, n) ⌘0 A new de-sign principle LW, Liu, Iazzi, Troyer and Harcos 1506.05349 M T ⌘M = ⌘ Te R 0 d⌧ HCk (⌧ ) ⌘ = diag(I, I) O+ (n, n) 0 ⌘0 det (I + M ) has a definite sign for each component ! O++ (n, n) O (n, n) O + (n, n) 0 ⌘0 ky π spinless fermions split Dirac cone LW, Troyer, PRL 2014 LW, Corboz, Troyer, NJP 2014 LW, Iazzi, Corboz, Troyer, PRB, 2015 Liu and LW, 1510.00715 0 -π -π 0 π spin nematicity k x SU(3) A new de-sign principle LW, Liu, Iazzi, Troyer and Harcos 1506.05349 M T ⌘M = ⌘ Te R 0 d⌧ HCk (⌧ ) ⌘ = diag(I, I) O+ (n, n) 0 ⌘0 det (I + M ) has a definite sign for each component ! O++ (n, n) O (n, n) O + (n, n) 0 ⌘0 ky π spinless fermions split Dirac cone LW, Troyer, PRL 2014 LW, Corboz, Troyer, NJP 2014 LW, Iazzi, Corboz, Troyer, PRB, 2015 Liu and LW, 1510.00715 0 -π -π 0 π spin nematicity k x SU(3) Asymmetric Hubbard model t" 6= t# U Realization: mixture of ultracold fermions (e.g. Jotzu et al, PRL 2015 Now, continuously tunable by t# /t" 2 ( 1, 1) PRL 99, 220403 (2007) PHYSICAL REVIEW LETTERS al, PRL 2007 and many others FIG. 2. Lignier Dynamicaletsuppression of tunneling in an optical lattice. Shown here is jJeff =Jj as a function of the shaking parameter K0 for V0 =Erec ! 6, !=2! ! 1 kHz (squares), V0 =Erec ! 6, !=2! ! 0:5 kHz (circles), and V0 =Erec ! 4, !=2! ! 1 kHz week ending 30 NOVEMBER 2007 " # Dirac fermions with unequal Fermi velocities Asymmetric Hubbard model t" 6= t# U Realization: mixture of ultracold fermions (e.g. Jotzu et al, PRL 2015 Now, continuously tunable by t# /t" 2 ( 1, 1) PRL 99, 220403 (2007) PHYSICAL REVIEW LETTERS week ending 30 NOVEMBER 2007 " # U + al, PRL 2007 and many others FIG. 2. Lignier Dynamicaletsuppression of tunneling in an optical lattice. Shown here is jJeff =Jj as a function of the shaking parameter K0 for V0 =Erec ! 6, !=2! ! 1 kHz (squares), V0 =Erec ! 6, !=2! ! 0:5 kHz (circles), and V0 =Erec ! 4, !=2! ! 1 kHz ? Dirac fermions with unequal Fermi velocities Two limiting cases VOLUME 22, NUMBER 19 t" ✘ PHYSICAL REVIEW LETTERS t# U 12 MA+ 1969 t" K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsitz, and P. L. Kelley, Phys. Rev. 177, 306 (1969). Y. R. Shen and N. Bloembergen, Phys. Rev. 137, 1787 {1965). The calculation on the saturation effect in this reference has recently been verified by M. Maier and W. Kaiser, to be published. ~P. L. Kelley, Phys. Rev. Letters 15, 1005 (1965). ~ We have been informed that T. K. Gustafson and J. P. Taran have obtained similar results. M. Naier, W. Kaiser, and J. A. Giordmaine, Phys. Rev. Letters 17, 1275 (1966). ~2J. H. Marburger and E. L. Dawes, Phys. Rev. Letters 21, 556 (1968), and Phys. Rev. (to be published). i Teor. Fiz. —Pis'ma Redakt. 7, 153 (1968) [translation: JETP Letters 7, 117 (1968)]. 8M. T. Loy and Y. R. Shen, to be published. The essence of the technique is to extract the pulse signal s (t) from the convolution integral R(t) = f ~ s (v)g(t-v) x «, where R{t) and g(t) are response functions of the detection system to the pulse signal aad to a pulse of 6 function, respectively. Allowing the possibility of different pulse shapes, we were able to measure the pulse width with an accuracy of &0 psec for a 200psec pulse and of &0 psec for a 100-psec pulse. The YT. Falicov-Kamball Limit time constant of our detection system was about 400 psec. t# ⌧ U Strong Coupling Limit SIMPLE MODEL FOR SEMICONDUCTOR-METAL SmB~ AND TRANSITION-METAL TRANSITIONS: OXIDES L. M. Falicov* Department of Physics, University of California, Berkeley, California 94720 and J. C. Kimballg Department of Physics, and The James Franck Institute, University of Chicago, Chicago, Illinois 60637 (Received 12 March 1969) We propose a simple model for a semiconductor-metal transition, based on the existence of both localized (ionic) and band (Bloch) states. It differs from other theories in that we assume the one-electron states to be essentially unchanged by the transition. The electron-hole interaction is responsible for the anomalous temperature dependence of the number of conduction electrons. For interactions larger than a critical value, a first-order semiconductor-metal phase transition takes place. SmBS' and a numtions and (b) a set of localized states centered at ber of transition-metal oxides, ' exhibit semiconthe sites of the metallic ions in the crystal. As ductor-metal transitions. ' The transitions are T —0 the localized states are lower in energy in many cases first-order phase transitions (e.g. , than the band states and are fully occupied by in V, O, ); however, they can also result from a electrons. Therefore the quasiparticle excitagradual but anomalously large increase in cantions are either localized holes or itinerant elecductivity over a range of temperatures (e.g. , in trons. In the language of second quantization and SmB, and Ti, O, ). In addition, measurements of in the spirit of the Landau theory of Fermi liqlarge magnetic susceptibilities with anomalous uids, we write the one-particle terms as temperature dependences suggest that in many of (k)a-~aH +gab ig tb ig' these materials localized magnetic moments ex0 v vkg vkg . vkg Sg ist and that they are intimately connected with Kennedy and Lieb the transition. As an example, it has been hywhere avko4 creates an electron in state k, band pothesized' that in SmB, the conduction electrons v, with spin o, and b, o~ creates a hole with spin and the localized moments are produced simulo at site i. The energies ev(k) and F. are positive taneously by the promotion of a single localized definite and such that electron from the spherically symmetric Sm++ ion (4= 0) into a conduction band The Sm+. ++ ion n =— min[E + e (k) j & 0 ') acts as a localized moment. left behind (J= —, We present here a simple theory of the semiis the energy for the formation of an elecFreericks and RMP, 2003 gapZlatić, conductor-metal transition based on a model havtron-hole pair. We further assume that the quaing both localized and itinerant interacting quasisiparticle interaction is screened, and its range particle states. The relevant single-electron short enough so that only intra-atomic terms states consist of (a) bands of extended Bloch funcneed be considered. In this case Many substances, including Long-range spin order on bipartite lattices with infinitesimal repulsion =Pe “Fruit fly” of DMFT Jxy ⇣ x x Ŝi Ŝj + y y Ŝi Ŝj ⌘ + z z Jz Ŝi Ŝj 2(t"2 + t#2 ) 4t" t# U U XXZ model with t# /t" Phase diagram 0 U/t" t# /t" Phase diagram Dirac Fermions 0 U/t" t# /t" Phase diagram Dirac Fermions AF-Ising 0 Falicov-Kimball limit U/t" t# /t" Phase diagram Dirac Fermions XXZ limit AF-Ising 0 Falicov-Kimball limit U/t" t# /t" Phase diagram 1 AF-Heisenberg Dirac Fermions XXZ limit AF-Ising 0 Falicov-Kimball limit U/t" t# /t" Phase diagram 1 Meng et al 2010 Sorella et al 2012 Assaad et al 2013 ⇡ 3.8 AF-Heisenberg Dirac Fermions XXZ limit AF-Ising 0 Falicov-Kimball limit U/t" t# /t" Phase diagram 1 Meng et al 2010 Sorella et al 2012 Assaad et al 2013 ⇡ 3.8 AF-Heisenberg Dirac Fermions XXZ limit AF-Ising 0 Falicov-Kimball limit U/t" How to connect the phase boundary ? What is the Binder ratio M2 = * 1 X iQ·r n̂r" n̂r# e N r 2 !2 + M4 = * t# /t" = 0.15 1 X iQ·r n̂r" n̂r# e N r 2 !4 + Scaling analysis ⌫ = 0.84(4) z + ⌘ = 1.395(7) Summary Exciting time for QMC simulation of lattice fermions Thanks to my collaborators! Mauro Iazzi Philippe Corboz Ye-Hua Liu Jakub Imriška Ping Nang Ma Gergely Harcos Matthias Troyer