Showing papers on "Ising model published in 2012"
TL;DR: In this article, the constraints of crossing symmetry and unitarity in general 3D conformal field theories were studied, and it was shown that the 3D Ising model lies at a corner point on the boundary of the allowed parameter space.
Abstract: We study the constraints of crossing symmetry and unitarity in general 3D conformal field theories. In doing so we derive new results for conformal blocks appearing in four-point functions of scalars and present an efficient method for their computation in arbitrary space-time dimension. Comparing the resulting bounds on operator dimensions and product-expansion coefficients in 3D to known results, we find that the 3D Ising model lies at a corner point on the boundary of the allowed parameter space. We also derive general upper bounds on the dimensions of higher spin operators, relevant in the context of theories with weakly broken higher spin symmetries.
862 citations
TL;DR: In this article, a two-dimensional (2D) quantum spin model was constructed that realizes an Ising paramagnet with gapless edge modes protected by Ising symmetry.
Abstract: We construct a two-dimensional (2D) quantum spin model that realizes an Ising paramagnet with gapless edge modes protected by Ising symmetry. This model provides an example of a ``symmetry-protected topological phase.'' We describe a simple physical construction that distinguishes this system from a conventional paramagnet: We couple the system to a ${\mathbb{Z}}_{2}$ gauge field and then show that the $\ensuremath{\pi}$-flux excitations have different braiding statistics from that of a usual paramagnet. In addition, we show that these braiding statistics directly imply the existence of protected edge modes. Finally, we analyze a particular microscopic model for the edge and derive a field theoretic description of the low energy excitations. We believe that the braiding statistics approach outlined in this paper can be generalized to a large class of symmetry-protected topological phases.
355 citations
TL;DR: In this paper, the authors investigated the dimensions of unitary higher-dimensional conformal field theories (CFTs) via the crossing equations in the light-cone limit and found that CFTs become free at large spin and 1/s is a weak coupling parameter.
Abstract: We consider several aspects of unitary higher-dimensional conformal field theories (CFTs). We first study massive deformations that trigger a flow to a gapped phase. Deep inelastic scattering in the gapped phase leads to a convexity property of dimensions of spinning operators of the original CFT. We further investigate the dimensions of spinning operators via the crossing equations in the light-cone limit. We find that, in a sense, CFTs become free at large spin and 1/s is a weak coupling parameter. The spectrum of CFTs enjoys additivity: if two twists tau_1, tau_2 appear in the spectrum, there are operators whose twists are arbitrarily close to tau_1+tau_2. We characterize how tau_1+tau_2 is approached at large spin by solving the crossing equations analytically. We find the precise form of the leading correction, including the prefactor. We compare with examples where these observables were computed in perturbation theory, or via gauge-gravity duality, and find complete agreement. The crossing equations show that certain operators have a convex spectrum in twist space. We also observe a connection between convexity and the ratio of dimension to charge. Applications include the 3d Ising model, theories with a gravity dual, SCFTs, and patterns of higher spin symmetry breaking.
353 citations
TL;DR: In this article, a coarse-graining tensor renormalization group method based on higher-order singular value decomposition was proposed for studying both classical and quantum lattice models in two or three dimensions.
Abstract: We propose a novel coarse-graining tensor renormalization group method based on the higher-order singular value decomposition. This method provides an accurate but low computational cost technique for studying both classical and quantum lattice models in two or three dimensions. We have demonstrated this method using the Ising model on the square and cubic lattices. By keeping up to 16 bond basis states, we obtain by far the most accurate numerical renormalization group results for the three-dimensional Ising model. We have also applied the method to study the ground state as well as finite temperature properties for the two-dimensional quantum transverse Ising model and obtain the results which are consistent with published data.
288 citations
TL;DR: In this paper, discrete holomorphic fermions for the 2D Ising model at criticality on a large family of planar graphs were introduced, and they were shown to have universal and conformally invariant scaling limits on bounded domains with appropriate boundary conditions.
Abstract: It is widely believed that the celebrated 2D Ising model at criticality has a universal and conformally invariant scaling limit, which is used in deriving many of its properties. However, no mathematical proof has ever been given, and even physics arguments support (a priori weaker) Mobius invariance. We introduce discrete holomorphic fermions for the 2D Ising model at criticality on a large family of planar graphs. We show that on bounded domains with appropriate boundary conditions, those have universal and conformally invariant scaling limits, thus proving the universality and conformal invariance conjectures.
284 citations
TL;DR: In this paper, the authors studied the constraints of crossing symmetry and unitarity for conformal field theories in the presence of a boundary, with a focus on the Ising model in various dimensions.
Abstract: We study the constraints of crossing symmetry and unitarity for conformal field theories in the presence of a boundary, with a focus on the Ising model in various dimensions. We show that an analytic approach to the bootstrap is feasible for free-field theory and at one loop in the epsilon expansion, but more generally one has to resort to numerical methods. Using the recently developed linear programming techniques we find several interesting bounds for operator dimensions and OPE coefficients and comment on their physical relevance. We also show that the "boundary bootstrap" can be easily applied to correlation functions of tensorial operators and study the stress tensor as an example. In the appendices we present conformal block decompositions of a variety of physically interesting correlation functions.
279 citations
TL;DR: It is found that disorder can decrease the reliability of an analog quantum simulator of this model, although large errors in local observables are introduced only for strong levels of disorder.
Abstract: Various fundamental phenomena of strongly correlated quantum systems such as high-Tc superconductivity, the fractional quantum-Hall effect and quark confinement are still awaiting a universally accepted explanation. The main obstacle is the computational complexity of solving even the most simplified theoretical models which are designed to capture the relevant quantum correlations of the many-body system of interest. In his seminal 1982 paper (Feynman 1982 Int. J. Theor. Phys. 21 467), Richard Feynman suggested that such models might be solved by ‘simulation’ with a new type of computer whose constituent parts are effectively governed by a desired quantum many-body dynamics. Measurements on this engineered machine, now known as a ‘quantum simulator,’ would reveal some unknown or difficult to compute properties of a model of interest. We argue that a useful quantum simulator must satisfy four conditions: relevance, controllability, reliability and efficiency. We review the current state of the art of digital and analog quantum simulators. Whereas so far the majority of the focus, both theoretically and experimentally, has been on controllability of relevant models, we emphasize here the need for a careful analysis of reliability and efficiency in the presence of imperfections. We discuss how disorder and noise can impact these conditions, and illustrate our concerns with novel numerical simulations of a paradigmatic example: a disordered quantum spin chain governed by the Ising model in a transverse magnetic field. We find that disorder can decrease the reliability of an analog quantum simulator of this model, although large errors in local observables are introduced only for strong levels of disorder. We conclude that the answer to the question ‘Can we trust quantum simulators?’ is … to some extent.
250 citations
TL;DR: In this article, the existence of edge zero modes in the Z2-invariant Ising/Majorana chain with Zn symmetry has been studied, and it has been shown that for appropriate couplings they are exact.
Abstract: A sign of topological order in a gapped one-dimensional quantum chain is the existence of edge zero modes. These occur in the Z2-invariant Ising/Majorana chain, where they can be understood using free-fermion techniques. Here I discuss their presence in spin chains with Zn symmetry, and prove that for appropriate couplings they are exact, even in this strongly interacting system. These modes are naturally expressed in terms of parafermions, generalizations of fermions to the Zn case. I show that parafermionic edge zero modes do not occur in the usual ferromagnetic and antiferromagnetic cases, but rather only when the interactions are chiral, so that spatial-parity and time-reversal symmetries are broken.
239 citations
TL;DR: In this article, the existence of edge zero modes in spin chains with Z-n symmetry has been studied in terms of parafermions, generalizations of fermions to the Z n case.
Abstract: A sign of topological order in a gapped one-dimensional quantum chain is the existence of edge zero modes. These occur in the Z_2-invariant Ising/Majorana chain, where they can be understood using free-fermion techniques. Here I discuss their presence in spin chains with Z_n symmetry, and prove that for appropriate coupling they are exact, even in this strongly interacting system. These modes are naturally expressed in terms of parafermions, generalizations of fermions to the Z_n case. I show that parafermionic edge zero modes do not occur in the usual ferromagnetic and antiferromagnetic cases, but rather only when the interactions are chiral, so that spatial-parity and time-reversal symmetries are broken.
239 citations
TL;DR: The Ising model is considered in a transverse field with long-range antiferromagnetic interactions that decay as a power law with their distance, and the phase diagram and the entanglement properties are studied as a function of the exponent of the interaction.
Abstract: We consider the Ising model in a transverse field with long-range antiferromagnetic interactions that decay as a power law with their distance. We study both the phase diagram and the entanglement properties as a function of the exponent of the interaction. The phase diagram can be used as a guide for future experiments with trapped ions. We find two gapped phases, one dominated by the transverse field, exhibiting quasi-long-range order, and one dominated by the long-range interaction, with long-range N\'eel ordered ground states. We determine the location of the quantum critical points separating those two phases. We determine their critical exponents and central charges. In the phase with quasi-long-range order the ground states exhibit exotic corrections to the area law for the entanglement entropy coexisting with gapped entanglement spectra.
199 citations
TL;DR: In this article, the magnetic structure and magnetization dynamics of systems of plane frustrated Ising chain lattices are reviewed for three groups of compounds:,,, and. The available experimental data are analyzed and compared in detail.
Abstract: The magnetic structure and magnetization dynamics of systems of plane frustrated Ising chain lattices are reviewed for three groups of compounds: , , and . The available experimental data are analyzed and compared in detail. It is shown that a high-temperature magnetic phase on a triangle lattice is normally and universally a partially disordered antiferromagnetic (PDA) structure. The diversity of low-temperature phases results from weak interactions that lift the degeneracy of a 2D antiferromagnetic Ising model on the triangle lattice. Mean-field models, Monte Carlo simulation results on the static magnetization curve, and results on slow magnetization dynamics obtained with Glauber's theory are discussed in detail.
TL;DR: It is shown that a method based on logistic regression, using all the data, solves the inverse Ising problem far better than mean-field calculations relying only on sample pairwise correlation functions, while still computationally feasible for hundreds of nodes.
Abstract: We show that a method based on logistic regression, using all the data, solves the inverse Ising problem far better than mean-field calculations relying only on sample pairwise correlation functions, while still computationally feasible for hundreds of nodes. The largest improvement in reconstruction occurs for strong interactions. Using two examples, a diluted Sherrington-Kirkpatrick model and a two-dimensional lattice, we also show that interaction topologies can be recovered from few samples with good accuracy and that the use of l(1) regularization is beneficial in this process, pushing inference abilities further into low-temperature regimes.
TL;DR: In this paper, a one-dimensional Ising model in a transverse field can be mapped onto a system of spinless fermions with $p$-wave superconductivity.
Abstract: A one-dimensional Ising model in a transverse field can be mapped onto a system of spinless fermions with $p$-wave superconductivity. In the weak-coupling BCS regime, it exhibits a zero-energy Majorana mode at each end of the chain. Here, we consider a variation of the model, which represents a superconductor with longer-ranged kinetic energy and pairing amplitudes, as is likely to occur in more realistic systems. It possesses a richer zero-temperature phase diagram and has several quantum phase transitions. From an exact solution of the model, we find that these phases can be classified according to the number of Majorana zero modes of an open chain: zero, one, or two at each end. The model possesses a multicritical point where phases with zero, one, and two Majorana end modes meet. The number of Majorana modes at each end of the chain is identical to the topological winding number of the Anderson pseudospin vector that describes the BCS Hamiltonian. The topological classification of the phases requires a unitary time-reversal symmetry to be present. When this symmetry is broken, only the number of Majorana end modes modulo 2 can be used to distinguish two phases. In one of the regimes, the wave functions of the two phase-shifted Majorana zero modes decay exponentially in space but in an oscillatory manner. The wavelength of oscillation is identical to that in the asymptotic connected spin-spin correlation of the $XY$ model in a transverse field, to which our model is dual.
TL;DR: The nonergodicity of the correlations is shown analytically to imply the equivalence with the generalized Gibbs ensemble for quantum Ising and XX spin chains as well as for the Luttinger model the thermodynamic limit.
Abstract: The generalized Gibbs ensemble introduced for describing few-body correlations in exactly solvable systems following a quantum quench is related to the nonergodic way in which operators sample, in the limit of infinite time after the quench, the quantum correlations present in the initial state. The nonergodicity of the correlations is thus shown analyticallyto imply the equivalence with the generalized Gibbs ensemble for quantum Ising and $XX$ spin chains as well as for the Luttinger model the thermodynamic limit, and for a broad class of initial states and correlation functions of both local and nonlocal operators.
20 Oct 2012
TL;DR: In this paper, it was shown that for both the hard core (independent set) model and the anti-ferromagnetic Ising model with arbitrary external field, it is NP-hard to approximate the partition function or approximately sample from the model on regular graphs when the model has non-uniqueness on the corresponding regular tree.
Abstract: The class of two-spin systems contains several important models, including random independent sets and the Ising model of statistical physics. We show that for both the hard-core (independent set) model and the anti-ferromagnetic Ising model with arbitrary external field, it is NP-hard to approximate the partition function or approximately sample from the model on regular graphs when the model has non-uniqueness on the corresponding regular tree. Together with results of Jerrum -- Sinclair, Weitz, and Sinclair -- Srivastava -- Thurley giving FPRAS's for all other two-spin systems except at the uniqueness threshold, this gives an almost complete classification of the computational complexity of two-spin systems on bounded-degree graphs. Our proof establishes that the normalized log-partition function of any two-spin system on bipartite locally tree-like graphs converges to a limiting ``free energy density'' which coincides with the (non-rigorous) Be the prediction of statistical physics. We use this result to characterize the local structure of two-spin systems on locally tree-like bipartite expander graphs, which then become the basic gadgets in a randomized reduction to approximate MAX-CUT. Our approach is novel in that it makes no use of the second moment method employed in previous works on these questions.
TL;DR: In this article, the authors employ the concept of a dynamical, activity order parameter to study the Ising model in a transverse magnetic field coupled to a Markovian bath, and demonstrate that dynamical phase coexistence becomes manifest in an intermittent behavior of the bath quanta emission.
Abstract: We employ the concept of a dynamical, activity order parameter to study the Ising model in a transverse magnetic field coupled to a Markovian bath. For a certain range of values of the spin-spin coupling, magnetic field, and dissipation rate, we identify a first-order dynamical phase transition between active and inactive dynamical phases. We demonstrate that dynamical phase coexistence becomes manifest in an intermittent behavior of the bath quanta emission. Moreover, we establish the connection between the dynamical order parameter that quantifies the activity and the longitudinal magnetization that serves as static order parameter. The system that we consider can be implemented in current experiments with Rydberg atoms and trapped ions.
TL;DR: In this paper, the authors studied the real-time dynamics of the order parameter σ(t) in the Ising field theory after a quench in the fermion mass, which corresponds to a quen in the transverse field of the corresponding transverse-field Ising chain.
Abstract: We study the real-time dynamics of the order parameter σ(t) in the Ising field theory after a quench in the fermion mass, which corresponds to a quench in the transverse field of the corresponding transverse field Ising chain. We focus on quenches within the ordered phase. The long-time behaviour is obtained analytically by a resummation of the leading divergent terms in a form-factor expansion for σ(t). Our main result is the development of a method for treating divergences associated with working directly in the field theory limit. We recover the scaling limit of the corresponding result by Calabrese et al (2011 Phys. Rev. Lett. 106 227203), which was obtained for the lattice model. Our formalism generalizes to integrable quantum quenches in other integrable models.
TL;DR: The gas-liquid phase transition of the three-dimensional Lennard-Jones particles system is studied by molecular dynamics simulations and the obtained values of β and ν are consistent with those of the Ising universality class.
Abstract: The gas-liquid phase transition of the three-dimensional Lennard-Jones particles system is studied by molecular dynamics simulations. The gas and liquid densities in the coexisting state are determined with high accuracy. The critical point is determined by the block density analysis of the Binder parameter with the aid of the law of rectilinear diameter. From the critical behavior of the gas-liquid coexisting density, the critical exponent of the order parameter is estimated to be β = 0.3285(7). Surface tension is estimated from interface broadening behavior due to capillary waves. From the critical behavior of the surface tension, the critical exponent of the correlation length is estimated to be ν = 0.63(4). The obtained values of β and ν are consistent with those of the Ising universality class.
Posted Content•
TL;DR: In this article, the existence and the conformal invariance of scaling limits of the magnetization and multi-point spin correlations in the critical Ising model on arbitrary simply connected planar domains were proved.
Abstract: We rigorously prove the existence and the conformal invariance of scaling limits of the magnetization and multi-point spin correlations in the critical Ising model on arbitrary simply connected planar domains. This solves a number of conjectures coming from the physical and the mathematical literature. The proof relies on convergence results for discrete holomorphic spinor observables and probabilistic techniques.
TL;DR: It is observed that the interface-free-energy estimates based on CNT are generally in error, and extensive calculations for the Ising model show that corrections due to a nonsharp and thermally fluctuating interface account for the barrier shape with excellent accuracy.
Abstract: We reconsider the applicability of classical nucleation theory (CNT) to the calculation of the free energy of solid cluster formation in a liquid and its use to the evaluation of interface free energies from nucleation barriers. Using two different freezing transitions (hard spheres and NaCl) as test cases, we first observe that the interface-free-energy estimates based on CNT are generally in error. As successive refinements of nucleation-barrier theory, we consider corrections due to a nonsharp solid-liquid interface and to a nonspherical cluster shape. Extensive calculations for the Ising model show that corrections due to a nonsharp and thermally fluctuating interface account for the barrier shape with excellent accuracy. The experimental solid nucleation rates that are measured in colloids are better accounted for by these non-CNT terms, whose effect appears to be crucial in the interpretation of data and in the extraction of the interface tension from them.
17 Jan 2012
TL;DR: The results of this paper indicate a tight relationship between complexity theory and phase transition phenomena in two-state anti-ferromagnetic spin systems on graphs of maximum degree $$d$$d for parameters outside the uniqueness region.
Abstract: In a seminal paper [12], Weitz gave a deterministic fully polynomial approximation scheme for counting exponentially weighted independent sets (equivalently, approximating the partition function of the hard-core model from statistical physics) on graphs of degree at most d, up to the critical activity for the uniqueness of the Gibbs measure on the infinite d-regular tree. More recently Sly [10] showed that this is optimal in the sense that if there is an FPRAS for the hard-core partition function on graphs of maximum degree d for activities larger than the critical activity on the infinite d-regular tree then NP = RP. In this paper, we extend Weitz's approach to derive a deterministic fully polynomial approximation scheme for the partition function of the anti-ferromagnetic Ising model with arbitrary field on graphs of maximum degree d, up to the corresponding critical point on the d-regular tree. The main ingredient of our result is a proof that for two-state anti-ferromagnetic spin systems on the d-regular tree, weak spatial mixing implies strong spatial mixing. This in turn uses a message-decay argument which extends a similar approach proposed recently for the hard-core model by Restrepo et al [9] to the case of the anti-ferromagnetic Ising model with arbitrary field. By a standard correspondence, these results translate to arbitrary two-state anti-ferromagnetic spin systems with soft constraints.
Posted Content•
TL;DR: The proof establishes that the normalized log-partition function of any two-spin system on bipartite locally tree-like graphs converges to a limiting ``free energy density'' which coincides with the (non-rigorous) Be the prediction of statistical physics.
Abstract: The class of two-spin systems contains several important models, including random independent sets and the Ising model of statistical physics. We show that for both the hard-core (independent set) model and the anti-ferromagnetic Ising model with arbitrary external field, it is NP-hard to approximate the partition function or approximately sample from the model on d-regular graphs when the model has non-uniqueness on the d-regular tree. Together with results of Jerrum--Sinclair, Weitz, and Sinclair--Srivastava--Thurley giving FPRAS's for all other two-spin systems except at the uniqueness threshold, this gives an almost complete classification of the computational complexity of two-spin systems on bounded-degree graphs.
Our proof establishes that the normalized log-partition function of any two-spin system on bipartite locally tree-like graphs converges to a limiting "free energy density" which coincides with the (non-rigorous) Bethe prediction of statistical physics. We use this result to characterize the local structure of two-spin systems on locally tree-like bipartite expander graphs, which then become the basic gadgets in a randomized reduction to approximate MAX-CUT. Our approach is novel in that it makes no use of the second moment method employed in previous works on these questions.
TL;DR: In this article, the coexistence of distinct normal and relaxor ferroelectric phases within the framework of percolation theory has been studied, and it has been shown that a strong frequency dispersion of the dielectric response and an apparent lack of macroscopic symmetry breaking in the low temperature phase are related to mesoscopic RF-driven phase transitions, which give rise to irregularly shaped quasi-stable polar nanoregions below the characteristic temperature T*, but above the global transition temperature Tc.
Abstract: Substitutional charge disorder as in PbMg1/3Nb2/3O3, structural cation vacancies as in SrxBa1-xNb2O6 and isovalent substitution of off-centered cations as in BaTi1-xSnxO3 and BaTi1-xZrxO3 give rise to quenched electric random-fields (RFs), which we proposed to be at the origin of the peculiar behavior of relaxor ferroelectrics 20 years ago. These are, e.g. a strong frequency dispersion of the dielectric response and an apparent lack of macroscopic symmetry breaking in the low temperature phase. Both are related to mesoscopic RF-driven phase transitions, which give rise to irregularly shaped quasi-stable polar nanoregions below the characteristic temperature T*, but above the global transition temperature Tc. Their co-existence with the paraelectric parent phase can be modeled by time-dependent field equations under the control of quenched RFs and stress-free strain (in the case of order parameter dimension n ≥ 2). Transitions into global polar order at Tc may occur in uniaxial relaxors as observed on the uniaxial relaxor ferroelectric Sr0.8Ba0.2Nb2O6 and come close to RF Ising model criticality. Re-entrant relaxor transitions as observed in solid solutions of Ba2Pr0.6Nd0.4(FeNb4)O15 are proposed to evidence the coexistence of distinct normal and relaxor ferroelectric phases within the framework of percolation theory.
TL;DR: In this paper, the authors evaluate the partition function of three dimensional theories of gravity in the quantum regime, where the AdS radius is Planck scale and the central charge is of order one.
Abstract: We evaluate the partition function of three dimensional theories of gravity in the quantum regime, where the AdS radius is Planck scale and the central charge is of order one. The contribution from the AdS vacuum sector can – with certain assumptions – be computed and equals the vacuum character of a minimal model CFT. The torus partition function is given by a sum over geometries which is finite and computable. For generic values of Newton’s constant G and the AdS radius l the result has no Hilbert space interpretation, but in certain cases it agrees with the partition function of a known CFT. For example, the partition function of pure Einstein gravity with G = 3l equals that of the Ising model, providing evidence that these theories are dual. We also present somewhat weaker evidence that the 3-state and tricritical Potts models are dual to pure higher spin theories of gravity based on SL(3) and E6, respectively.
TL;DR: In this paper, the authors compute the stability of the toric code to depolarization by mapping the quantum problem onto a classical disordered eight-vertex Ising model, and demonstrate an increased error threshold of 18.9(3)% when noise correlations are taken into account.
Abstract: The inevitable presence of decoherence effects in systems suitable for quantum computation necessitates effective error-correction schemes to protect information from noise. We compute the stability of the toric code to depolarization by mapping the quantum problem onto a classical disordered eight-vertex Ising model. By studying the stability of the related ferromagnetic phase via both large-scale Monte Carlo simulations and the duality method, we are able to demonstrate an increased error threshold of 18.9(3)% when noise correlations are taken into account. Remarkably, this result agrees within error bars with the result for a different class of codes—topological color codes—where the mapping yields interesting new types of interacting eight-vertex models.
TL;DR: In this article, the authors compared the performance of different mean-field approximations on several models (diluted ferromagnets and spin glasses) defined on random graphs and regular lattices, showing which one is in general more effective.
Abstract: The inverse Ising problem consists in inferring the coupling constants of an Ising model given the correlation matrix. The fastest methods for solving this problem are based on mean-field approximations, but which one performs better in the general case is still not completely clear. In the first part of this work, I summarize the formulas for several mean-field approximations and I derive new analytical expressions for the Bethe approximation, which allow one to solve the inverse Ising problem without running the susceptibility propagation algorithm (thus avoiding the lack of convergence). In the second part, I compare the accuracy of different mean-field approximations on several models (diluted ferromagnets and spin glasses) defined on random graphs and regular lattices, showing which one is in general more effective. A simple improvement over these approximations is proposed. Also a fundamental limitation is found in using methods based on TAP and Bethe approximations in the presence of an external field.
TL;DR: In this article, a variational approach for matrix product states (MPSs) was proposed to avoid the errors inherent in the multi-state targeting approach of DMRG.
Abstract: Theoretical understanding of strongly correlated systems in one spatial dimension (1D) has been greatly advanced by the density-matrix renormalization group (DMRG) algorithm, which is a variational approach using a class of entanglement-restricted states called matrix product states (MPSs). However, DMRG suffers from inherent accuracy restrictions when multiple states are involved due to multi-state targeting and also the approximate representation of the Hamiltonian and other operators. By formulating the variational approach of DMRG explicitly for MPSs one can avoid errors inherent in the multi-state targeting approach. Furthermore, by using the matrix product operator (MPO) formalism, one can exactly represent the Hamiltonian and other operators relevant for the calculation. The MPO approach allows 1D Hamiltonians to be templated using a small set of finite state automaton (FSA) rules without reference to the particular microscopic degrees of freedom. We present two algorithms which take advantage of these properties: eMPS to find excited states of 1D Hamiltonians and tMPS for the time evolution of a generic time-dependent 1D Hamiltonian. We properly account for time-ordering of the propagator such that the error does not depend on the rate of change of the Hamiltonian. Our algorithms use only the MPO form of the Hamiltonian, and so are applicable to microscopic degrees of freedom of any variety, and do not require Hamiltonian-specialized implementation. We benchmark our algorithms with a case study of the Ising model, where the critical point is located using entanglement measures. We then study the dynamics of this model under a time-dependent quench of the transverse field through the critical point. Finally, we present studies of a dipolar, or long-range Ising model, again using entanglement measures to find the critical point and study the dynamics of a time-dependent quench through the critical point.
TL;DR: In this article, the authors consider the problem of high-dimensional Ising (graphical) model selection and propose a simple algorithm for structure estimation based on the thresholding of the empirical conditional variation distances.
Abstract: We consider the problem of high-dimensional Ising (graphical) model selection. We propose a simple algorithm for structure estimation based on the thresholding of the empirical conditional variation distances. We introduce a novel criterion for tractable graph families, where this method is efficient, based on the presence of sparse local separators between node pairs in the underlying graph. For such graphs, the proposed algorithm has a sample complexity of $n=\Omega(J_{\min}^{-2}\log p)$, where $p$ is the number of variables, and $J_{\min}$ is the minimum (absolute) edge potential in the model. We also establish nonasymptotic necessary and sufficient conditions for structure estimation.
TL;DR: Li et al. as mentioned in this paper showed that for the antiferrogmanetic Ising model without external field, unless RP=NP, there is no FPRAS for approximating the partition function on graphs of maximum degree when the inverse temperature lies in the non-uniqueness regime of the infinite tree.
Abstract: Recent inapproximability results of Sly (2010), together with an approximation algorithm presented by Weitz (2006) establish a beautiful picture for the computational complexity of approximating the partition function of the hard-core model. Let $\lambda_c(T_\Delta)$ denote the critical activity for the hard-model on the infinite $\Delta$-regular tree. Weitz presented an FPTAS for the partition function when $\lambda 0$ such that (unless RP=NP) there is no FPRAS for approximating the partition function on graphs of maximum degree $\Delta$ for activities $\lambda$ satisfying $\lambda_c(T_\Delta)<\lambda<\lambda_c(T_\Delta)+\epsilon_\Delta$.
We prove that a similar phenomenon holds for the antiferromagnetic Ising model. Recent results of Li et al. and Sinclair et al. extend Weitz's approach to any 2-spin model, which includes the antiferromagnetic Ising model, to yield an FPTAS for the partition function for all graphs of constant maximum degree $\Delta$ when the parameters of the model lie in the uniqueness regime of the infinite tree $T_\Delta$. We prove the complementary result that for the antiferrogmanetic Ising model without external field that, unless RP=NP, for all $\Delta\geq 3$, there is no FPRAS for approximating the partition function on graphs of maximum degree $\Delta$ when the inverse temperature lies in the non-uniqueness regime of the infinite tree $T_\Delta$. Our results extend to a region of the parameter space for general 2-spin models. Our proof works by relating certain second moment calculations for random $\Delta$-regular bipartite graphs to the tree recursions used to establish the critical points on the infinite tree.
University of Tokyo1, Nagoya University2, Osaka University3, University of California, Santa Cruz4, Japan Atomic Energy Agency5, University of the Ryukyus6, Bandung Institute of Technology7, National Institute of Standards and Technology8, University of Maryland, College Park9, Johns Hopkins University10
TL;DR: Ba3CuSb2O9, which is magnetically anisotropic at the atomic scale but curiously isotropic on mesoscopic length and time scales, is found to have a broad spectrum of spin-dimer–like excitations and low-energy spin degrees of freedom that retain overall hexagonal symmetry.
Abstract: Frustrated magnetic materials, in which local conditions for energy minimization are incompatible because of the lattice structure, can remain disordered to the lowest temperatures. Such is the case for Ba(3)CuSb(2)O(9), which is magnetically anisotropic at the atomic scale but curiously isotropic on mesoscopic length and time scales. We find that the frustration of Wannier's Ising model on the triangular lattice is imprinted in a nanostructured honeycomb lattice of Cu(2+) ions that resists a coherent static Jahn-Teller distortion. The resulting two-dimensional random-bond spin-1/2 system on the honeycomb lattice has a broad spectrum of spin-dimer-like excitations and low-energy spin degrees of freedom that retain overall hexagonal symmetry.