Partition Functions Of Models Research Articles

Partition functions from orthogonal and symplectic group invariants

We introduce a new class of graph parameters coming from invariants of the orthosymplectic group, which we call mixed partition functions. This class strictly contains partition functions of edge-coloring models and we give a characterization of this class by means of rank growth of associated connection matrices.

Correspondence between spanning trees and the Ising model on a square lattice.

An important problem in statistical physics concerns the fascinating connections between partition functions of lattice models studied in equilibrium statistical mechanics on the one hand and graph theoretical enumeration problems on the other hand. We investigate the nature of the relationship between the number of spanning trees and the partition function of the Ising model on the square lattice. The spanning tree generating function T(z) gives the spanning tree constant when evaluated at z=1, while giving the lattice green function when differentiated. It is known that for the infinite square lattice the partition function Z(K) of the Ising model evaluated at the critical temperature K=K_{c} is related to T(1). Here we show that this idea in fact generalizes to all real temperatures. We prove that [Z(K)sech2K]^{2}=kexp[T(k)], where k=2tanh(2K)sech(2K). The identical Mahler measure connects the two seemingly disparate quantities T(z) and Z(K). In turn, the Mahler measure is determined by the random walk structure function. Finally, we show that the the above correspondence does not generalize in a straightforward manner to nonplanar lattices.

Transport properties of density wave phases in three-dimensional metals and semimetals under high magnetic field

Three-dimensional (3D) metals/semimetals under magnetic field have an instability toward a density wave (DW) ordering which breaks a translational symmetry along the field direction. Effective boson models for the DW phases take forms of XY models with/without Potts terms. Longitudinal conductivity along the field direction is calculated in the DW phases with inclusion of effects of low-energy charge fluctuation (phason) and disorder. A single-particle imaginary-time Green function is identified with a partition function of 3D XY models in the presence of pairs of magnetic monopoles. In terms of the celebrated electromagnetic duality, electronic spectral function is calculated near the DW phase transition. The result shows that the single-particle spectral function acquires an additional low-energy feature due to the strong phason fluctuation. Relevance to an in-plane conductance due to surface chiral Fermi arc states are also discussed.

Bilayer linearized tensor renormalization group approach for thermal tensor networks

In this paper, we perform a comprehensive study of the renormalization group (RG) method on thermal tensor networks (TTN). By Trotter-Suzuki decomposition, one obtains the 1+1D TTN representing the partition function of 1D quantum lattice models, and then employs efficient RG contractions to obtain the thermodynamic properties with high precision. The linearized tensor renormalization group (LTRG) method, which can be used to contract TTN in an efficient and accurate way, is briefly reviewed. In addition, the single-layer LTRG can be generalized to a bilayer form, dubbed as LTRG++, in both finite- and infinite-size systems, with accuracies significantly improved. We provide the details of LTRG++ in finite-size system, comparing its accuracy with single-layer algorithm, and elaborate the infinite-size LTRG++ in the context of fermion chain model. We show that the LTRG++ algorithm in infinite-size system is in essence equivalent to transfer-matrix renormalization group method, while expressed in a tensor network language. LTRG++ is then applied to study an extended Hubbard model, where the phase separation phenomenon, groundstate phase diagram, as well as quantum criticality-enhanced magnetocaloric effects under external fields, are investigated.

Commuting quantum circuits and complexity of Ising partition functions

Instantaneous quantum polynomial-time (IQP) computation is a class of quantum computation consisting only of commuting two-qubit gates and is not universal. Nevertheless, it has been shown that if there is a classical algorithm that can simulate IQP efficiently, the polynomial hierarchy collapses to the third level, which is highly implausible. However, the origin of the classical intractability is still less understood. Here we establish a relationship between IQP and computational complexity of calculating the imaginary-valued partition functions of Ising models. We apply the established relationship in two opposite directions. One direction is to find subclasses of IQP that are classically efficiently simulatable by using exact solvability of certain types of Ising models. Another direction is applying quantum computational complexity of IQP to investigate (im)possibility of efficient classical approximations of Ising partition functions with imaginary coupling constants. Specifically, we show that a multiplicative approximation of Ising partition functions is #P-hard for almost all imaginary coupling constants even on planar lattices of a bounded degree.

Deterministic Polynomial-Time Approximation Algorithms for Partition Functions and Graph Polynomials

In this paper we show a new way of constructing deterministic polynomial-time approximation algorithms for computing complex-valued evaluations of a large class of graph polynomials on bounded degree graphs. In particular, our approach works for the Tutte polynomial and independence polynomial, as well as partition functions of complex-valued spin and edge-coloring models. More specifically, we define a large class of graph polynomials $\mathcal C$ and show that if $p\in \cal C$ and there is a disk $D$ centered at zero in the complex plane such that $p(G)$ does not vanish on $D$ for all bounded degree graphs $G$, then for each $z$ in the interior of $D$ there exists a deterministic polynomial-time approximation algorithm for evaluating $p(G)$ at $z$. This gives an explicit connection between absence of zeros of graph polynomials and the existence of efficient approximation algorithms, allowing us to show new relationships between well-known conjectures. Our work builds on a recent line of work initiated by. Barvinok, which provides a new algorithmic approach besides the existing Markov chain Monte Carlo method and the correlation decay method for these types of problems.

Magnetization Plateaus and Thermal Entanglement of Spin Systems

The geometrically magnetic frustrations and quantum thermal entanglement of antiferromagnetic metal-containing compounds are considered on a diamond chain. We researched the magnetic and thermal properties of the symmetric Hubbard dimers with delocalized interstitial spins and the quantum entanglement states. It is presented magnetization plateaus and negativity in spin-1 Ising-Heisenberg model using transfer matrix technique. Applying the dynamic system approach we study the magnetic curves, Lyapunov exponents and superstable point in the two-dimensional mapping for the partition function of spin-1 classical and Ising-Heisenberg models at T → 0 on a diamond chain.

Ising model from intertwiners

Spin networks appear in a number of areas, for instance in lattice gauge theories and in quantum gravity. They describe the contraction of intertwiners according to the underlying network. We show how a certain generating function of intertwiner contractions for arbitrary networks, when restricted to a square lattice is exactly related to the high temperature expansion of the 2d Ising model partition function. This implies that the intertwiner model possesses a second order phase transition, thus leading to a continuum limit with propagating degrees of freedom.

Graph parameters from symplectic group invariants

In this paper we introduce, and characterize, a class of graph parameters obtained from tensor invariants of the symplectic group. These parameters are similar to partition functions of vertex models, as introduced by de la Harpe and Jones (1993) [5]. Yet they give a completely different class of graph invariants. We moreover show that certain evaluations of the cycle partition polynomial, as defined by Martin (1977) [15], give examples of graph parameters that can be obtained this way.

Efficient algorithm for computing exact partition functions of lattice polymer models

Polymers are important macromolecules in many physical, chemical, biological and industrial problems. Studies on simple lattice polymer models are very helpful for understanding behaviors of polymers. We develop an efficient algorithm for computing exact partition functions of lattice polymer models, and we use this algorithm and personal computers to obtain exact partition functions of the interacting self-avoiding walks with N monomers on the simple cubic lattice up to N=28 and on the square lattice up to N=40. Our algorithm can be extended to study other lattice polymer models, such as the HP model for protein folding and the charged HP model for protein aggregation. It also provides references for checking accuracy of numerical partition functions obtained by simulations.

Partition function of the elliptic solid-on-solid model as a single determinant.

In this Rapid Communication we express the partition function of the integrable elliptic solid-on-solid model with domain-wall boundary conditions as a single determinant. This representation appears naturally as the solution of a system of functional equations governing the model's partition function.

Refined Cauchy/Littlewood identities and six-vertex model partition functions: III. Deformed bosons

We study Hall–Littlewood polynomials using an integrable lattice model of t-deformed bosons. Working with row-to-row transfer matrices, we review the construction of Hall–Littlewood polynomials (of the An root system) within the framework of this model. Introducing appropriate double-row transfer matrices, we extend this formalism to Hall–Littlewood polynomials based on the BCn root system, and obtain a new combinatorial formula for them. We then apply our methods to prove a series of refined Cauchy and Littlewood identities involving Hall–Littlewood polynomials. The last two of these identities are new, and relate infinite sums over hyperoctahedrally symmetric Hall–Littlewood polynomials with partition functions of the six-vertex model on finite domains.

Tutte Polynomial of Scale-Free Networks

The Tutte polynomial of a graph, or equivalently the q-state Potts model partition function, is a two-variable polynomial graph invariant of considerable importance in both statistical physics and combinatorics. The computation of this invariant for a graph is NP-hard in general. In this paper, we focus on two iteratively growing scale-free networks, which are ubiquitous in real-life systems. Based on their self-similar structures, we mainly obtain recursive formulas for the Tutte polynomials of two scale-free networks (lattices), one is fractal and “large world”, while the other is non-fractal but possess the small-world property. Furthermore, we give some exact analytical expressions of the Tutte polynomial for several special points at (x, y)-plane, such as, the number of spanning trees, the number of acyclic orientations, etc.

Quiver gauge theories and integrable lattice models

We discuss connections between certain classes of supersymmetric quiver gauge theories and integrable lattice models from the point of view of topological quantum field theories (TQFTs). The relevant classes include 4d $\mathcal{N} = 1$ theories known as brane box and brane tilling models, 3d $\mathcal{N} = 2$ and 2d $\mathcal{N} = (2,2)$ theories obtained from them by compactification, and 2d $\mathcal{N} = (0,2)$ theories closely related to these theories. We argue that their supersymmetric indices carry structures of TQFTs equipped with line operators, and as a consequence, are equal to the partition functions of lattice models. The integrability of these models follows from the existence of extra dimension in the TQFTs, which emerges after the theories are embedded in M-theory. The Yang-Baxter equation expresses the invariance of supersymmetric indices under Seiberg duality and its lower-dimensional analogs.

Zeros of Jones Polynomials of Graphs

In this paper, we introduce the Jones polynomial of a graph $G=(V,E)$ with $k$ components as the following specialization of the Tutte polynomial:$$J_G(t)=(-1)^{|V|-k}t^{|E|-|V|+k}T_G(-t,-t^{-1}).$$We first study its basic properties and determine certain extreme coefficients. Then we prove that $(-\infty, 0]$ is a zero-free interval of Jones polynomials of connected bridgeless graphs while for any small $\epsilon>0$ or large $M>0$, there is a zero of the Jones polynomial of a plane graph in $(0,\epsilon)$, $(1-\epsilon,1)$, $(1,1+\epsilon)$ or $(M,+\infty)$. Let $r(G)$ be the maximum moduli of zeros of $J_G(t)$. By applying Sokal's result on zeros of Potts model partition functions and Lucas's theorem, we prove that\begin{eqnarray*}{q_s-|V|+1\over |E|}\leq r(G)<1+6.907652\Delta_G\end{eqnarray*}for any connected bridgeless and loopless graph $G=(V,E)$ of maximum degree $\Delta_G$ with $q_s$ parallel classes. As a consequence of the upper bound, X.-S. Lin's conjecture holds if the positive checkerboard graph of a connected alternating link has a fixed maximum degree and a sufficiently large number of edges.

Temperature-reflection symmetry

We point out the presence of a $T\ensuremath{\rightarrow}\ensuremath{-}T$ temperature-reflection ($T$-reflection) symmetry for the partition functions of many physical systems. Without knowledge of the origin of the symmetry, we have only been able to test the existence of $T$-reflection symmetry in systems with exactly calculable partition functions. We show that $T$-reflection symmetry is present in a variety of conformal and nonconformal field theories and statistical mechanics models with known partition functions. For example, all minimal model partition functions in two-dimensional conformal field theories are invariant under $T$-reflections. An interesting property of the $T$-reflection symmetry is that it can be broken by shifts of the vacuum energy.

Refined Cauchy/Littlewood identities and six-vertex model partition functions: II. Proofs and new conjectures

We prove two identities of Hall---Littlewood polynomials, which appeared recently in Betea and Wheeler (2014). We also conjecture, and in some cases prove, new identities which relate infinite sums of symmetric polynomials and partition functions associated with symmetry classes of alternating sign matrices. These identities generalize those already found in Betea and Wheeler (2014), via the introduction of additional parameters. The left-hand side of each of our identities is a simple refinement of a relevant Cauchy or Littlewood identity. The right-hand side of each identity is (one of the two factors present in) the partition function of the six-vertex model on a relevant domain.

Entanglement of an alternating bipartition in spin chains: relation with classical integrable models

We study the entanglement properties of a class of ground states defined by matrix product states, which are generalizations of the valence bond solid (VBS) state in one dimension. It is shown that the transfer matrix of these states can be related to representations of the Temperley–Lieb algebra, allowing an exact computation of Renyi entropy. For an alternating bipartition, we find that the Renyi entropy can be mapped to an eight vertex model partition function on a rotated lattice. We also show that for the VBS state, the Renyi entropy of the alternating partition is described by a critical field theory with central charge c = 1. The generalization to SU(n) VBS and its connection with a dimerization transition in the entanglement Hamiltonian is discussed.

Note on a duality of topological branes

We show a duality of branes in the topological B-model by inserting two kinds of the non-compact branes simultaneously. We explicitly derive the integral formula for the matrix model partition function describing this situation, which correspondingly includes both of the characteristic polynomial and the external source. We show that these two descriptions are dual to each other through the Fourier transformation, and the brane partition function satisfies integrable equations in one and two dimensions.

Tutte polynomial of the Apollonian network

The Tutte polynomial of a graph, or equivalently the q-state Potts model partition function, is a two-variable polynomial graph invariant of considerable importance in both combinatorics and statistical physics. The computation of this invariant for a graph is, in general, NP-hard. The aim of this paper is to compute the Tutte polynomial of the Apollonian network. Based on the well-known duality property of the Tutte polynomial, we extend the subgraph-decomposition method. In particular, we do not calculate the Tutte polynomial of the Apollonian network directly, instead we calculate the Tutte polynomial of the Apollonian dual graph. By using the close relation between the Apollonian dual graph and the Hanoi graph, we express the Tutte polynomial of the Apollonian dual graph in terms of that of the Hanoi graph. As an application, we also give the number of spanning trees of the Apollonian network.