arXiv subjects

De-Zhang Li

Publications and source records attributed to De-Zhang Li.

10 recordsLinked to original sources

arXiv2026

Tensor network investigation of the monomer-dimer model on the honeycomb lattice

The monomer-dimer model is one of the most well-known unsolved lattice models. In this paper we study the monomer-dimer model on the honeycomb lattice using the tensor network method, in the case that the dimer and monomer activities are 1. The monomer-dimer configurations are exactly mapped into the ground states of the antiferromagnetic Ising model on the Kagomé lattice in the critical field $H_{\rm{ex}}=4J$, and the tensor network is constructed based on the local ground states of each Ising triangle. The VUMPS approach is employed to contract the tensor network, providing a high-precision result of the monomer-dimer problem. We also revisit the edge coloring problem on the honeycomb lattice and discuss its relationship to the monomer-dimer model. Finally we formulate the monomer-dimer problem in the language of the sixteen-vertex model, and discuss the non-integrability of the general monomer-dimer model and the integrability of the pure dimer model.

cond-mat.stat-mech↗

arXiv2026

Solution of the Ising model with Brascamp-Kunz boundary conditions by the transfer matrix method

The square lattice Ising model under the Brascamp-Kunz boundary conditions is a well-known exactly solvable lattice model. The exact solution of this system has been derived within the framework of Pfaffian-type method. In this paper we provide a derivation for the solution by the Schultz-Mattis-Lieb method in the transfer matrix formalism. We set special interactions on the boundaries and take certain limit of these interactions, so that the system under the Brascamp-Kunz boundary conditions is transformed into another system under the toroidal boundary conditions. The Schultz-Mattis-Lieb method is applied to the mapping system and the partition function is exactly solved in the fermionic representation. The Fisher zeros are analytically calculated and the physical critical point is identified. We also discuss the difference between the transfer matrix approaches to the Brascamp-Kunz and to the toroidal boundary conditions. Our work introduces a member to the family of transfer-matrix-based studies for Ising model under various boundary conditions.

cond-mat.stat-mech↗

arXiv2025

Low-temperature series expansion of square lattice Ising model: A study based on Fisher zeros

Low-temperature expansion of Ising model has long been a topic of significant interest in condensed matter and statistical physics. In this paper we present new results of the coefficients in the low-temperature series of the Ising partition function on the square lattice, in the cases of a zero field and of an imaginary field $i(π/2)k_BT$. The coefficients in the low-temperature series of the free energy in the thermodynamic limit are represented using the explicit expression of the density function of the Fisher zeros. The asymptotic behaviour of the sequence of the coefficients when the order goes to infinity is determined exactly, for both the series of the free energy and of the partition function. Our analytic and numerical results demonstrate that, the convergence radius of the sequence is dependent on the accumulation points of the Fisher zeros which have the smallest modulus. In the zero field case this accumulation point is the physical critical point, while in the imaginary field case it corresponds to a non-physical singularity. We further discuss the relation between the series coefficients and the energy state degeneracies, using the combinatorial expression of the coefficients and the subgraph expansion.

cond-mat.stat-mech↗

arXiv2025

Free-fermion models and the two-dimensional Ising models under the zero field and imaginary field $i(π/2){k_B}T$

Ising model is famous in condensed matter and statistical physics. In this work we present a free-fermion formulation of the two-dimensional classical Ising models on the honeycomb, triangular and Kagomé lattices. Each Ising model is studied in the cases of a zero field and of an imaginary field $i(π/2){k_B}T$. We employ the decorated lattice technique, star-triangle transformation and weak-graph expansion method to exactly map each Ising model in both cases into an eight-vertex model on the square lattice. The resulting vertex weights are shown to satisfy the free-fermion condition. In the zero field case, each Ising model is an even free-fermion model. In the case of the imaginary field, the Ising model on the honeycomb lattice is an even free-fermion model while the models on the triangular and Kagomé lattices are odd free-fermion models. We obtain the exact solution of the Kagomé lattice Ising model under the imaginary field $i(π/2){k_B}T$, a result not previously reported in the literature. We also show that the frustrated Ising models on the triangular and Kagomé lattices in the imaginary field still exhibit a non-zero residual entropy.

cond-mat.stat-mech↗

arXiv2025

Free-fermion approach to the partition function zeros : Special boundary conditions and product form of solution

Partition function zeros are powerful tools in understanding critical behavior. In this paper we present new results of the Fisher zeros of two-dimensional Ising models, in the framework of free-fermion eight-vertex model. First we succeed in finding special boundary conditions for the free-fermion model, under which the partition function of a finite lattice can be expressed in a double product form. Using appropriate mappings, these boundary conditions are transformed into the corresponding versions of the square, triangular and honeycomb lattice Ising models. Each Ising model is studied in the cases of a zero field and of an imaginary field $i(π/2)k_BT$. For the square lattice model we rediscover the famous Brascamp-Kunz (B-K) boundary conditions. For the triangular and honeycomb lattice models we obtain the B-K type boundary conditions, and the Fisher zeros are conveniently solved from the product form of partition function. The advantage of B-K type boundary conditions is that the Fisher zeros of any finite lattice exactly lie on certain loci, and the accumulation points of zeros can be easily determined in the thermodynamic limit. Our finding and method would be very helpful in studying the partition function zeros of vertex and Ising models.

cond-mat.stat-mech↗

arXiv2024

Residual Entropy of Ice: A Study Based on Transfer Matrices

Residual entropy of ice systems has long been a significant and intriguing issue in condensed matter physics and statistical mechanics. The exact solutions for the residual entropy of realistic three-dimensional ice systems remain unknown. In this study, we focus on two typical realistic ice systems, namely the hexagonal ice (ice Ih) and cubic ice (ice Ic). We present a transfer matrix description of the number of ice-ruled configurations for these two systems. First, a transfer matrix $M$ is constructed for ice Ic, where each element is the number of ice-ruled configurations of a hexagonal monolayer under certain condition. The product of $M$ and its transpose corresponds to a bilayer unit in ice Ih lattice, therefore is exactly a transfer matrix for ice Ih. Making use of this, we simply show that the residual entropy of ice Ih is not less than that of ice Ic in the thermodynamic limit, which was first proved by Onsager in 1960s. Furthermore, we find an alternative transfer matrix $M'$ for ice Ih, which is based on a monolayer periodic unit. Some interesting properties of $M$, $MM^T$ and $M'$ are illustrated, specifically the summation of all elements, the element in the first row and first column, and the trace. Each property is equivalent with the residual entropy of a two-dimensional ice model. Our work rediscovers the relationship between the residual entropies of ice Ih and ice Ic, and provides an effective description for various two-dimensional ice models.

cond-mat.stat-mech↗

arXiv2022

Analysis of Transition Path Ensemble in the Exactly Solvable Models via Overdamped Langevin Equation

Transition of a system between two states is an important but difficult problem in natural science. In this article we study the transition problem in the framework of transition path ensemble. Using the overdamped Langevin method, we introduce the path integral formulation of the transition probability and obtain the equation for the minimum action path in the transition path space. For the effective sampling in the transition path ensemble, we derive a conditional overdamped Langevin equation. In two exactly solvable models, the free particle system and the harmonic system, we present the expression of the conditional probability density and the explicit solutions for the conditional Langevin equation and the minimum action path. The analytic results demonstrate the consistence of the conditional Langevin equation with the desired probability distribution in the transition. It is confirmed that the conditional Langevin equation is an effective tool to sample the transition path ensemble, and the minimum action principle actually leads to the most probable path.

cond-mat.stat-mech↗

arXiv2022

Residual Entropy of a Two-dimensional Ising Model with Crossing and Four-spin Interactions

We study the residual entropy of a two-dimensional Ising model with crossing and four-spin interactions, both for the case that in zero magnetic field and that in an imaginary magnetic field i(π/2)kT. The spin configurations of this Ising model can be mapped into the hydrogen configurations of square ice with the defined standard direction of the hydrogen bonds. Making use of the equivalence of this Ising system with the exactly solved eight-vertex model and taking the low temperature limit, we obtain the residual entropy. Two soluble cases in zero field and one soluble case in imaginary field are examined. In the case that the free-fermion condition holds in zero field, we find the ground states in low temperature limit include the configurations disobeying the ice rules. In another case in zero field that the four-spin interactions are -{\infty}, and the case in imaginary field that the four-spin interactions are 0, the residual entropy exactly agrees with the result of square ice determined by Lieb in 1967. In the solutions of the latter two cases, we have shown alternative approaches to the residual entropy problem of square ice.

cond-mat.stat-mech↗

arXiv2022

Exact Results for the Residual Entropy of Ice Hexagonal Monolayer

Since the problem of the residual entropy of square ice was exactly solved, exact solutions for two-dimensional realistic ice models have been of interest. In this paper, we study the exact residual entropy of ice hexagonal monolayer in two cases. In the case that the external electric field along the z-axis exists, we map the hydrogen configurations into the spin configurations of the Ising model on the Kagomé lattice. By taking the low temperature limit of the Ising model, we derive the exact residual entropy, which agrees with the result determined previously from the dimer model on the honeycomb lattice. In another case that the ice hexagonal monolayer is under the periodic boundary conditions in the cubic ice lattice, we employ the six-vertex model on the square lattice to represent the hydrogen configurations obeying the ice rules. The exact residual entropy in this case is obtained from the solution of the equivalent six-vertex model. Our work provides more examples of the exactly soluble two-dimensional models.

cond-mat.stat-mech↗

arXiv2022

On the Numerical Stationary Distribution of Overdamped Langevin Equation in Harmonic System

Efficient numerical algorithm for stochastic differential equation has been an important object in the research of statistical physics and mathematics for a long time. In this paper we study the highly accurate numerical algorithm of the overdamped Langevin equation. In particular, our interest is the behaviour of the numerical schemes for solving the overdamped Langevin equation in the harmonic system. Three algorithms are obtained for overdamped Langevin equation, from the large friction limit of the schemes for underdamped Langevin dynamics. We derive the explicit expression of the stationary distribution of each algorithm by analysing the discrete time trajectory, for both one-dimensional and multi-dimensional cases. The accuracy of the stationary distribution of each algorithm is illustrated by comparing to the exact Boltzmann distribution. Our results demonstrate that, the "BAOA-limit" algorithm generates the exact distribution for the harmonic system in the canonical ensemble, within the stable regime of the time interval. The other algorithms do not produce the exact distribution of the harmonic system.

cond-mat.stat-mech↗