Hard-core Model Research Articles

Uniqueness and nonuniqueness conditions for weakly periodic Gibbs measures for the hard-core model

We study a “hard-core” model on a Cayley tree. In the case of a normal divisor of index 4, we show the uniqueness of weakly periodic Gibbs measures under certain conditions on the parameters. Moreover, we prove that there exist weakly periodic (nonperiodic) Gibbs measures different from those previously known.

Coloring triangle-free graphs with local list sizes.

We prove two distinct and natural refinements of a recent breakthrough result of Molloy (and a follow‐up work of Bernshteyn) on the (list) chromatic number of triangle‐free graphs. In both our results, we permit the amount of color made available to vertices of lower degree to be accordingly lower. One result concerns list coloring and correspondence coloring, while the other concerns fractional coloring. Our proof of the second illustrates the use of the hard‐core model to prove a Johansson‐type result, which may be of independent interest.

Fine Grained Tensor Network Methods.

We develop a strategy for tensor network algorithms that allows to deal very efficiently with lattices of high connectivity. The basic idea is to fine grain the physical degrees of freedom, i.e., decompose them into more fundamental units which, after a suitable coarse graining, provide the original ones. Thanks to this procedure, the original lattice with high connectivity is transformed by an isometry into a simpler structure, which is easier to simulate via usual tensor network methods. In particular this enables the use of standard schemes to contract infinite 2D tensor networks-such as corner transfer matrix renormalization schemes-which are more involved on complex lattice structures. We prove the validity of our approach by numerically computing the ground-state properties of the ferromagnetic spin-1 transverse-field Ising model on the 2D triangular and 3D stacked triangular lattice, as well as of the hardcore and softcore Bose-Hubbard models on the triangular lattice. Our results are benchmarked against those obtained with other techniques, such as perturbative continuous unitary transformations and graph projected entangled pair states, showing excellent agreement and also improved performance in several regimes.

Permutation cycles of hardcore Bose-Hubbard models on square and kagome lattices

In this paper, we study the statistics of permutation cycles of ground-state hardcore lattice bosons described by various two-dimensional Bose-Hubbard-type models on both square and Kagome lattices. We find that it is possible to differentiate quantum phases by the statistics of permutations cycles. Indeed, features in the permutation cycles statistics can be used to uniquely identify certain insulating phases, and are consistent with local resonances of occupation numbers in the ground-state expansion of the phase. We also confirm that suitable quantities derived from the probability distribution of the length of permutation cycles can be used to detect superfluid to insulator phase transitions.

Finitary codings for spatial mixing Markov random fields

It has been shown by van den Berg and Steif (Ann. Probab. 27 (1999) 1501–1522) that the subcritical and critical Ising model on $\mathbb{Z}^{d}$ is a finitary factor of an i.i.d. process (ffiid), whereas the super-critical model is not. In fact, they showed that the latter is a general phenomenon in that a phase transition presents an obstruction for being ffiid. The question remained whether this is the only such obstruction. We make progress on this, showing that certain spatial mixing conditions (notions of weak dependence on boundary conditions, not to be confused with other notions of mixing in ergodic theory) imply ffiid. Our main result is that weak spatial mixing implies ffiid with power-law tails for the coding radius, and that strong spatial mixing implies ffiid with exponential tails for the coding radius. The weak spatial mixing condition can be relaxed to a condition which is satisfied by some critical two-dimensional models. Using a result of the author (Spinka (2018)), we deduce that strong spatial mixing also implies ffiid with stretched-exponential tails from a finite-valued i.i.d. process. We give several applications to models such as the Potts model, proper colorings, the hard-core model, the Widom–Rowlinson model and the beach model. For instance, for the ferromagnetic $q$-state Potts model on $\mathbb{Z}^{d}$ at inverse temperature $\beta $, we show that it is ffiid with exponential tails if $\beta $ is sufficiently small, it is ffiid if $\beta <\beta _{c}(q,d)$, it is not ffiid if $\beta >\beta_{c}(q,d)$ and, when $d=2$ and $\beta =\beta _{c}(q,d)$, it is ffiid if and only if $q\le 4$.

Logarithmic Sobolev inequalities for finite spin systems and applications

We derive sufficient conditions for a probability measure on a finite product space (a spin system) to satisfy a (modified) logarithmic Sobolev inequality. We establish these conditions for various examples, such as the (vertex-weighted) exponential random graph model, the random coloring and the hard-core model with fugacity. This leads to two separate branches of applications. The first branch is given by mixing time estimates of the Glauber dynamics. The proofs do not rely on coupling arguments, but instead use functional inequalities. As a byproduct, this also yields exponential decay of the relative entropy along the Glauber semigroup. Secondly, we investigate the concentration of measure phenomenon (particularly of higher order) for these spin systems. We show the effect of better concentration properties by centering not around the mean, but around a stochastic term in the exponential random graph model. From there, one can deduce a central limit theorem for the number of triangles from the CLT of the edge count. In the Erdős–Rényi model the first-order approximation leads to a quantification and a proof of a central limit theorem for subgraph counts.

Guide to Exact Diagonalization Study of Quantum Thermalization

Exact diagonalization is a powerful numerical method to study isolated quantum many-body systems. This paper provides a review of numerical algorithms to diagonalize the Hamiltonian matrix. Symmetry and the conservation law help us perform the numerical study efficiently. We explain the method to block-diagonalize the Hamiltonian matrix by using particle number conservation, translational symmetry, particle-hole symmetry, and spatial reflection symmetry in the context of the spin-1/2 XXZ model or the hard-core boson model in a one-dimensional lattice. We also explain the method to study the unitary time evolution governed by the Schr\"odinger equation and to calculate thermodynamic quantities such as the entanglement entropy. As an application, we demonstrate numerical results that support that the eigenstate thermalization hypothesis holds in the XXZ model.

Application of hard-core Exponential-6 intermolecular potential function to determine the second osmotic virial coefficients of polymer solutions

The osmotic pressures of polymer solutions are related to their corresponding osmotic virial coefficients. There are several experimental methods to determine the osmotic virial coefficients. However, these experimental virial coefficients are restricted to the experimental conditions. An alternative way of determining the osmotic virial coefficient is with the help of statistical thermodynamics, and unlike experimental methods, because of its theoretical basis, it is not restricted to limited ranges of experimental conditions. The osmotic virial coefficients can be used to calculate the different properties of polymers and polymer solutions, such as the average molecular weight of the polymer (M) and the theta temperature. In this work, for the first time, the second osmotic virial coefficients of a number of polymeric solutions are calculated using the Kihara and Exponential-6 intermolecular potential functions. Furthermore, the hard-core Exponential-6 model, that we have proposed, is also investigated. Additionally, the molecular weights and theta temperatures of these solutions are determined using the calculated second virial coefficients and are compared to literature data. The results show that the recent hard-core Exponential-6 potential model is more accurate than the Kihara and Exponential-6 models and is applicable for calculating other properties of the polymers and polymer solutions.

Biexciton Condensation in Electron-Hole-Doped Hubbard Bilayers: A Sign-Problem-Free Quantum MonteCarlo Study.

The bilayer Hubbard model with electron-hole doping is an ideal platform to study excitonic orders due to suppressed recombination via spatial separation of electrons and holes. However, suffering from the sign problem, previous quantum MonteCarlo studies could not arrive at an unequivocal conclusion regarding the presence of phases with clear signatures of excitonic condensation in bilayer Hubbard models. Here, we develop a determinant quantum MonteCarlo algorithm for the bilayer Hubbard model that is sign-problem-free for equal and opposite doping in the two layers and study excitonic order and charge and spin density modulations as a function of chemical potential difference between the two layers, on-site Coulomb repulsion, and interlayer interaction. In the intermediate coupling regime and in proximity to the SU(4)-symmetric point, we find a biexcitonic condensate phase at finite electron-hole doping, as well as a competing (π,π) charge density wave state. We extract the Berezinskii-Kosterlitz-Thouless transition temperature from superfluid density and a finite-size scaling analysis of the correlation functions and explain our results in terms of an effective biexcitonic hard-core boson model.

Persistent oscillations versus thermalization in the quench dynamics of quantum gases with long-range interactions

Searching for nonthermalized dynamics in interacting quantum systems is not only of fundamental theoretical interest in nonequilibrium quantum physics, but also of immense practical significance in quantum information processing. In this paper, we study quantum quench dynamics in an hard-core bosonic model with infinite-range interactions, which have been realized in recent high-finesse cavity experiments. We show the long-time dynamics of this model can exhibit either undamped oscillations or thermalization depending on the choice of initial states. The long-range nature of the interactions rather than conserved quantities are responsible for such nonergodic dynamical behaviors.

Dynamical Gibbs\u2013Non-Gibbs Transitions in Lattice Widom\u2013Rowlinson Models with Hard-Core and Soft-Core Interactions

We consider the Widom–Rowlinson model on the lattice {mathbb {Z}}^d in two versions, comparing the cases of a hard-core repulsion and of a soft-core repulsion between particles carrying opposite signs. For both versions we investigate their dynamical Gibbs–non-Gibbs transitions under an independent stochastic symmetric spin-flip dynamics. While both models have a similar phase transition in the high-intensity regime in equilibrium, we show that they behave differently under time-evolution: the time-evolved soft-core model is Gibbs for small times and loses the Gibbs property for large enough times. By contrast, the time-evolved hard-core model loses the Gibbs property immediately, and for asymmetric intensities, shows a transition back to the Gibbsian regime at a sharp transition time.

Algorithms for #BIS-Hard Problems on Expander Graphs

We give a fully polynomial-time approximation scheme (FPTAS) and an efficient sampling algorithm for the high-fugacity hard-core model on bounded-degree bipartite expander graphs and the low-temper...

Spontaneous Symmetry Breaking in Coupled Bose–Einstein Condensates

We study a system of two hardcore bosonic Hubbard models weakly coupled with each other by tunneling. Assuming that the single uncoupled model exhibits off-diagonal long-range order, we prove that the coupled system exhibits spontaneous symmetry breaking (SSB) in the infinite volume limit, in the sense that the two subsystems maintain a definite relative U(1) phase when the tunneling is turned off. Although SSB of the U(1) phase is never observable in a single system, SSB of the relative U(1) phase is physically meaningful and observable by interference experiments. The present theorem is made possible by the rigorous theory of low-lying states and SSB in quantum antiferromagnets developed over the years.

Pair hopping in systems of strongly interacting hard-core bosons

We have used the Stochastic Series Expansion quantum Monte Carlo method to study interacting hard-core bosons on the square lattice, with pair-hopping processes supplementing the standard single-particle hopping. Such pair hopping arises in effective models for frustrated quantum magnets. Our goal is to investigate the effects of the pair hopping process on the commonly observed superfluid, insulating (Mott), and super-solid ground-state phases in the standard hard-core boson model with various interaction terms. The model is specifically motivated by the observation of finite dispersion of 2-magnon bound states in neutron diffraction experiments SrCu$_2$(BO$_3$)$_2$. Our results show that the pair hopping has different effects on Mott phases at different filling fractions, "melting" them at different critical pair-hopping amplitudes. Thus, it appears that pair hopping may have an important role in determining which out of a potentially large number of Mott phases (stabilized by details of the charge-diagonal interaction terms) actually survive the totality of quantum fluctuations present.

Uniqueness of Gibbs measures for continuous hardcore models

We formulate a continuous version of the well-known discrete hardcore (or independent set) model on a locally finite graph, parameterized by the so-called activity parameter $\lambda >0$. In this version the state or “spin value” $x_{u}$ of any node $u$ of the graph lies in the interval $[0,1]$, the hardcore constraint $x_{u}+x_{v}\leq 1$ is satisfied for every edge $(u,v)$ of the graph, and the space of feasible configurations is given by a convex polytope. When the graph is a regular tree, we show that there is a unique Gibbs measure associated to each activity parameter $\lambda >0$. Our result shows that, in contrast to the standard discrete hardcore model, the continuous hardcore model does not exhibit a phase transition on the infinite regular tree. We also consider a family of continuous models that interpolate between the discrete and continuous hardcore models on a regular tree when $\lambda =1$ and show that each member of the family has a unique Gibbs measure, even when the discrete model does not. In each case the proof entails the analysis of an associated Hamiltonian dynamical system that describes a certain limit of the marginal distribution at a node. Furthermore, given any sequence of regular graphs with fixed degree and girth diverging to infinity, we apply our results to compute the asymptotic limit of suitably normalized volumes of the corresponding sequence of convex polytopes of feasible configurations. In particular this yields an approximation for the partition function of the continuous hard core model on a regular graph with large girth in the case $\lambda =1$.

Algorithmic Pirogov–Sinai theory

We develop an efficient algorithmic approach for approximate counting and sampling in the low-temperature regime of a broad class of statistical physics models on finite subsets of the lattice $\mathbb Z^d$ and on the torus $(\mathbb Z/n \mathbb Z)^d$. Our approach is based on combining contour representations from Pirogov-Sinai theory with Barvinok's approach to approximate counting using truncated Taylor series. Some consequences of our main results include an FPTAS for approximating the partition function of the hard-core model at sufficiently high fugacity on subsets of $\mathbb Z^d$ with appropriate boundary conditions and an efficient sampling algorithm for the ferromagnetic Potts model on the discrete torus $(\mathbb Z/n \mathbb Z)^d$ at sufficiently low temperature.

Universal tensor-network algorithm for any infinite lattice

We present a general graph-based Projected Entangled-Pair State (gPEPS) algorithm to approximate ground states of nearest-neighbor local Hamiltonians on any lattice or graph of infinite size. By introducing the structural-matrix which codifies the details of tensor networks on any graphs in any dimension $d$, we are able to produce a code that can be essentially launched to simulate any lattice. We further introduce an optimized algorithm to compute simple tensor updates as well as expectation values and correlators with a mean-field-like effective environments. Though not being variational, this strategy allows to cope with PEPS of very large bond dimension (e.g., $D=100$), and produces remarkably accurate results in the thermodynamic limit in many situations, and specially when the correlation length is small and the connectivity of the lattice is large. We prove the validity of our approach by benchmarking the algorithm against known results for several models, i.e., the antiferromagnetic Heisenberg model on a chain, star and cubic lattices, the hardcore Bose-Hubbard model on square lattice, the ferromagnetic Heisenberg model in a field on the pyrochlore lattice, as well as the $3$-state quantum Potts model in field on the kagome lattice and the spin-$1$ bilinear-biquadratic Heisenberg model on the triangular lattice. We further demonstrate the performance of gPEPS by studying the quantum phase transition of the $2d$ quantum Ising model in transverse magnetic field on the square lattice, and the phase diagram of the Kitaev-Heisenberg model on the hyperhoneycomb lattice. Our results are in excellent agreement with previous studies.

Weighted counting of solutions to sparse systems of equations

AbstractGiven complex numbers w1,…,wn, we define the weight w(X) of a set X of 0–1 vectors as the sum of $w_1^{x_1} \cdots w_n^{x_n}$ over all vectors (x1,…,xn) in X. We present an algorithm which, for a set X defined by a system of homogeneous linear equations with at most r variables per equation and at most c equations per variable, computes w(X) within relative error ∊ &gt; 0 in (rc)O(lnn-ln∊) time provided $|w_j| \leq \beta (r \sqrt{c})^{-1}$ for an absolute constant β &gt; 0 and all j = 1,…,n. A similar algorithm is constructed for computing the weight of a linear code over ${\mathbb F}_p$. Applications include counting weighted perfect matchings in hypergraphs, counting weighted graph homomorphisms, computing weight enumerators of linear codes with sparse code generating matrices, and computing the partition functions of the ferromagnetic Potts model at low temperatures and of the hard-core model at high fugacity on biregular bipartite graphs.

Diagnosing Potts criticality and two-stage melting in one-dimensional hard-core boson models

We investigate a model of hard-core bosons with infinitely repulsive nearest- and next-nearest-neighbor interactions in one dimension, introduced by Fendley, Sengupta and Sachdev in Phys. Rev. B 69, 075106 (2004). Using a combination of exact diagonalization, tensor network, and quantum Monte Carlo simulations, we show how an intermediate incommensurate phase separates a crystalline and a disordered phase. We base our analysis on a variety of diagnostics, including entanglement measures, fidelity susceptibility, correlation functions, and spectral properties. According to theoretical expectations, the disordered-to-incommensurate-phase transition point is compatible with Berezinskii-Kosterlitz-Thouless universal behaviour. The second transition is instead non-relativistic, with dynamical critical exponent $z > 1$. For the sake of comparison, we illustrate how some of the techniques applied here work at the Potts critical point present in the phase diagram of the model for finite next-nearest-neighbor repulsion. This latter application also allows to quantitatively estimate which system sizes are needed to match the conformal field theory spectra with experiments performing level spectroscopy.

The number of independent sets in an irregular graph

Settling Kahn's conjecture (2001), we prove the following upper bound on the number i(G) of independent sets in a graph G without isolated vertices:i(G)≤∏uv∈E(G)i(Kdu,dv)1/(dudv), where du is the degree of vertex u in G. Equality occurs when G is a disjoint union of complete bipartite graphs. The inequality was previously proved for regular graphs by Kahn and Zhao.We also prove an analogous tight lower bound:i(G)≥∏v∈V(G)i(Kdv+1)1/(dv+1), where equality occurs for G a disjoint union of cliques. More generally, we prove bounds on the weighted versions of these quantities, i.e., the independent set polynomial, or equivalently the partition function of the hard-core model with a given fugacity on a graph.