arXiv subjects

Kazuhiko Minami

Publications and source records attributed to Kazuhiko Minami.

At least 19 recordsLinked to original sources

arXiv2026

The free energy of the square lattice Ising model with interactions alternating in horizontal and vertical directions

The free energy of the Ising model on the square lattice with alternating interactions in both horizontal and vertical directions is exactly derived. This model is distinct from the checkerboard Ising model. The result includes Onsager's free energy as a special case, and also includes Lee-Yang's free energy with an imaginary field, and relates these two solutions via continuous parameters. It is also derived that each imaginary magnetic field $i\pi/2$ applied to a lattice site corresponds to a single frustrated square in its dual lattice.

cond-mat.stat-mech↗

arXiv2025

Conserved Charges of Series of Solvable Lattice Models

An infinite number of solvable Hamiltonians, including the transverse Ising chain, the XY chain with an external field, the cluster model with next-nearest-neighbor x-x interactions, or with next-nearest-neighbor z-z interactions, and other solvable models that can be mapped to the free fermion system are considered. All the conserved charges of these models written by the string-type products of the interactions are obtained. In the case of the transverse Ising chain, all the known charges are rederived, and in the case of the other models, new conserved charges are obtained.

cond-mat.stat-mech↗

arXiv2023

The exact susceptibility of the spin-S transverse Ising chain with next-nearest-neighbor interactions

The zero-field susceptibility of the spin-S transverse Ising chain with next-nearest-neighbor interactions is obtained exactly. The susceptibility is given in an explicit form for S=1/2, and expressed in terms of the eigenvectors of the transfer matrix for general spin S. It is found that the low-temperature limit is independent of spin S, and is divergent at the transition point.

cond-mat.stat-mech↗

arXiv2021

Onsager algebra and algebraic generalization of Jordan-Wigner transformation

Recently, an algebraic generalization of the Jordan-Wigner transformation was introduced and applied to one- and two-dimensional systems. This transformation is composed of the interactions $η_{i}$ that appear in the Hamiltonian ${\cal H}$ as ${\cal H}=\sum_{i=1}^{N}J_{i}η_{i}$, where $J_{i}$ are coupling constants. In this short note, it is derived that operators that are composed of $η_{i}$, or its $n$-state clock generalizations, generate the Onsager algebra, which was introduced in the original solution of the rectangular Ising model, and appears in some integrable models.

cond-mat.stat-mech↗

arXiv2021

Exact solution of a cluster model with next-nearest-neighbor interaction

A one-dimensional cluster model with next-nearest-neighbor interactions and two additional composite interactions is solved; the free energy is obtained and a correlation function is derived exactly. The model is diagonalized by a transformation obtained automatically from its interactions, which is an algebraic generalization of the Jordan-Wigner transformation. The gapless condition is expressed as a condition on the roots of a cubic equation, and the phase diagram is obtained exactly. We find that the distribution of roots for this algebraic equation determines the existence of long-range order, and we again obtain the ground-state phase diagram. We also derive the central charges of the corresponding conformal field theory. Finally, we note that our results are universally valid for an infinite number of solvable spin chains whose interactions obey the same algebraic relations.

cond-mat.stat-mech↗

arXiv2020

Geometric Criterion for Solvability of Lattice Spin Systems

We present a simple criterion for solvability of lattice spin systems on the basis of the graph theory and the simplicial homology. The lattice systems satisfy algebras with graphical representations. It is shown that the null spaces of adjacency matrices of the graphs provide conserved quantities of the systems. Furthermore, when the graphs belong to a class of simplicial complexes, the Hamiltonians are found to be mapped to bilinear forms of Majorana fermions, from which the full spectra of the systems are obtained. In the latter situation, we find a relation between conserved quantities and the first homology group of the graph, and the relation enables us to interpret the conserved quantities as flux excitations of the systems. The validity of our theory is confirmed in several known solvable spin systems including the 1d transverse-field Ising chain, the 2d Kitaev honeycomb model and the 3d diamond lattice model. We also present new solvable models on a 1d tri-junction, 2d and 3d fractal lattices, and the 3d cubic lattice.

cond-mat.stat-mech↗

arXiv2019

Honeycomb lattice Kitaev model with Wen-Toric-code interactions, and anyon excitations

The honeycomb lattice Kitaev model H_{K} with two kinds of Wen-Toric-code four-body interactions H_{WT} is investigated exactly using a new fermionization method, and the ground state phase diagram is obtained. Six kinds of three-body interactions are also considered. A Hamiltonian equivalent to the honeycomb lattice Kitaev model is also introduced. The fermionization method is generalized to two-dimensional systems, and the two-dimensional Jordan-Wigner transformation is obtained as a special case of this formula. The model H_{K}+H_{WT} is symmetric in four-dimensional space of coupling constants, and the anyon type excitations appear in each phase.

cond-mat.stat-mech↗

arXiv2017

Infinite number of solvable generalizations of XY-chain, with cluster state, and with central charge c=m/2

An infinite number of spin chains are solved and it is derived that the ground-state phase transitions belong to the universality classes with central charge c=m/2, where m is an integer. The models are diagonalized by automatically obtained transformations, many of which are different from the Jordan-Wigner transformation. The free energies, correlation functions, string order parameters, exponents, central charges, and the phase diagram are obtained. Most of the examples consist of the stabilizers of the cluster state. A unified structure of the one-dimensional XY and cluster-type spin chains is revealed, and other series of solvable models can be obtained through this formula.

cond-mat.stat-mech↗

arXiv2012

Equivalence between two-dimensional alternating/random Ising model and the ground state of one-dimensional alternating/random XY chain

It is derived that the two-dimensional Ising model with alternating/random interactions and with periodic/free boundary conditions is equivalent to the ground state of the one-dimensional alternating/random XY model with the corresponding periodic/free boundary conditions. This provides an exact equivalence between a random rectangular Ising model, in which the Griffiths-McCoy phase appears, and a random XY chain.

cond-mat.stat-mech↗

arXiv2011

Generalized boundary condition, free energies of six-vertex models and fractal dimension

The structures of the configuration space of the six-vertex models with various boundaries and boundary conditions are investigated, and it is derived that the free energies depend on the boundary conditions, and that they are classified by the fractal structures. The "n-equivalences" of the boundary conditions are defined with a property that the models with n-equivalent boundary conditions result in the identical free energy. The configurations which satisfy the six-vertex restriction are classified, through the n-equivalences, into sets of configurations called islands. It is derived that each island shows a fractal structure when it is mapped to the real axis. Each free energy is expressed by the weighted fractal dimension (multi-fractal dimension) of the island. The fractal dimension of the island has a strict relation with the maximum eigenvalue of the corresponding block element of the transfer matrix. It is also found that the fractal dimensions of islands take the absolute maximum when the boundary is "alternative"; in this case, the free energy equals to the solution obtained by Lieb and Sutherland.

cond-mat.stat-mech↗

arXiv2011

Equivalence between two-dimensional cell-sorting and one-dimensional generalized random walk -- spin representations of generating operators

The two-dimensional cell-sorting problem is found to be mathematically equivalent to the one-dimensional random walk problem with pair creations and annihilations, i.e. the adhesion probabilities in the cell-sorting model relate analytically to the expectation values in the random walk problem. This is an example demonstrating that two completely different biological systems are governed by a common mathematical structure. This result is obtained through the equivalences of these systems with lattice spin models. It is also shown that arbitrary generation operators can be written by the spin operators, and hence all biological stochastic problems can in principle be analyzed utilizing the techniques and knowledge previously obtained in the study of lattice spin systems.

q-bio.CB↗

arXiv2009

Weak-universal critical behavior and quantum critical point of the exactly soluble spin-1/2 Ising-Heisenberg model with the pair XYZ Heisenberg and quartic Ising interactions

Spin-1/2 Ising-Heisenberg model with XYZ Heisenberg pair interaction and two different Ising quartic interactions is exactly solved with the help of the generalized star-square transformation, which establishes a precise mapping equivalence with the corresponding eight-vertex model on a square lattice generally satisfying Baxter's zero-field (symmetric) condition. The investigated model exhibits a remarkable weak-universal critical behavior with two marked wings of critical lines along which critical exponents vary continuously with the interaction parameters. Both wings of critical lines merge together at a very special quantum critical point of the infinite order, which can be characterized through diverging critical exponents. The possibility of observing reentrant phase transitions in a close vicinity of the quantum critical point is related to a relative strength of the exchange anisotropy in the XYZ Heisenberg pair interaction.

cond-mat.stat-mech↗

arXiv2009

Spin-1/2 Ising-Heisenberg model with the pair XYZ Heisenberg interaction and quartic Ising interactions as the exactly soluble zero-field eight-vertex model

The spin-1/2 Ising-Heisenberg model with the pair XYZ Heisenberg interaction and quartic Ising interactions is exactly solved by establishing a precise mapping relationship with the corresponding zero-field (symmetric) eight-vertex model. It is shown that the Ising-Heisenberg model with the ferromagnetic Heisenberg interaction exhibits a striking critical behavior, which manifests itself through re-entrant phase transitions as well as continuously varying critical exponents. The changes of critical exponents are in accordance with the weak universality hypothesis in spite of a peculiar singular behavior to emerge at a quantum critical point of the infinite order, which occurs at the isotropic limit of the Heisenberg interaction. On the other hand, the Ising-Heisenberg model with the antiferromagnetic Heisenberg interaction surprisingly exhibits less significant changes of both critical temperatures as well as critical exponents upon varying a strength of the exchange anisotropy in the Heisenberg interaction.

cond-mat.stat-mech↗

arXiv2009

Fractal structure of a solvable lattice model

Fractal structure of the six-vertex model is introduced with the use of the IFS (Iterated Function Systems). The fractal dimension satisfies an equation written by the free energy of the six-vertex model. It is pointed out that the transfer matrix method and the $n$-equivalence relation introduced in lattice theories have also been introduced in the area of fractal geometry. All the results can be generalized for the models suitable to the transfer matrix treatment, and hence this gives general relation between solvable lattice models and fractal geometry.

cond-mat.stat-mech↗

arXiv2006

An equivalence relation of boundary/initial conditions, and the infinite limit properties

The 'n-equivalences' of boundary conditions of lattice models are introduced and it is derived that the models with n-equivalent boundary conditions result in the identical free energy. It is shown that the free energy of the six-vertex model is classified through the density of left/down arrows on the boundary. The free energy becomes identical to that obtained by Lieb and Sutherland with the periodic boundary condition, if the density of the arrows is equal to 1/2. The relation to the structure of the transfer matrix and a relation to stochastic processes are noted.

cond-mat.stat-mech↗

arXiv2006

The free energies of six-vertex models and the n-equivalence relation

The free energies of six-vertex models on general domain D with various boundary conditions are investigated with the use of the n-equivalence relation which classifies the thermodynamic limit properties. It is derived that the free energy of the six-vertex model on the rectangle is unique in the limit in which both the height and the width goes to infinity. It is derived that the free energies of the model on D are classified through the densities of left/down arrows on the boundary. Specifically the free energy is identical to that obtained by Lieb and Sutherland with the cyclic boundary condition when the densities are both equal to 1/2. This fact explains several results already obtained through the transfer matrix calculations. The relation to the domino tiling (or dimer, or matching) problems is also noted.

cond-mat.stat-mech↗

arXiv2005

Thermodynamic properties of a tetramer ferro-ferro-antiferro-antiferromagnetic Ising-Heisenberg bond alternating chain as a model system for Cu(3-Clpy)$_2$(N$_3$)$_2$

Thermodynamic properties of a tetramer ferro-ferro-antiferro-antiferromagnetic Ising-Heisenberg bond alternating chain are investigated by the use of an exact mapping transformation technique. Exact results for the magnetization, susceptibility and specific heat in the zero as well as nonzero magnetic field are presented and discussed in detail. The results obtained from the mapping are compared with the relevant experimental data of Cu(3-Clpy)$_2$(N$_3$)$_2$ (3-Clpy=3-Chloropyridine).

cond-mat.mtrl-sci↗

arXiv2004

Non-Universal Critical Behaviour of Two-Dimensional Ising Systems

Two conditions are derived for Ising models to show non-universal critical behaviour, namely conditions concerning 1) logarithmic singularity of the specific heat and 2) degeneracy of the ground state. These conditions are satisfied with the eight-vertex model, the Ashkin-Teller model, some Ising models with short- or long-range interactions and even Ising systems without the translational or the rotational invariance.

cond-mat.stat-mech↗