Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. B, Solid state

Ising Transfer Matrix and a d -Dimensional Fermion Model

William J. Camp

Published Nov 1, 1973• 3 citations
Publisher

Abstract

The $d$-dimensional Ising transfer matrix is expressed exactly in terms of Fermi-type operators. In two dimensions this leads to the standard solution of the model. In three or more dimensions the transfer matrix is no longer a quadratic form in the Fermi variables, but rather contains quartic and higher-order terms. However, it can be expressed as a well-defined expansion about a quadratic form in Fermi operators. The zeroth-order transfer-matrix-called herein the "fermion model"---can be exactly diagonalized, and is well approximated in the critical region by ${V}^{(0)}\ensuremath{\approx}{[2sinh(2K)]}^{\frac{N}{2}}\mathrm{exp}\left[\ensuremath{-}N\ensuremath{\int}d\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}{\left({\ensuremath{\kappa}}^{2}+\ensuremath{\Sigma}{n=1}^{d\ensuremath{-}1}\ensuremath{\Sigma}{m=1}^{d\ensuremath{-}1}{q}_{n}{q}_{m}\right)}^{\frac{1}{2}}t(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})\right]$where $N$ is the number of sites per layer, $\ensuremath{\mu}$ is the inverse correlation length, $t(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$ takes values \ifmmode\pm\else\textpm\fi{}1/2, $K$ is the reduced inverse temperature, and the integral extends over a Brillouin zone of the ($d\ensuremath{-}1$)-dimensional layer. The critical point of the model is located by the equation ${x}_{c}^{d}+{x}_{c}^{d\ensuremath{-}1}+{x}_{c}=1$, where ${x}_{c}={e}^{\ensuremath{-}2{K}_{c}}$. The critical exponent $\ensuremath{\nu}$ describing the divergence of the correlation length at ${K}_{c}$ is found to take on its two-dimensional value, $\ensuremath{\nu}=1$, in all dimensions $d$. The possibility of using the fermion model as the basis for an expansion of the Ising critical exponents as a function of $\ensuremath{\delta}(=d\ensuremath{-}2)$ is pointed out.

DOI
10.1103/physrevb.8.4307

Related papers

Study of One-Dimensional Fermion Model by the Transfer-Matrix Method

Terufumi Yokota, Hiroshi Betsuyaku

On transforme le modele de fermion a une dimension en le modele d'Ising a 2 dimensions en utilisant la formule de Suzuki-Trotter. Par la methode de la matrice de transfert numeriquement exacte, l'energie et les fonctions de correlation du modele equivalent. On etudie la dependance des fonctions de correlation par rapport a la force de l'interaction entre fermions

Progress of Theoretical Physics•Jan 1, 1986•12 citations

Application of many-fermion techniques to lattice statistical problems

R. W. Gibberd

The partition functions of many lattice statistical problems, such as the Ising models, the dimer model, and the ferroelectric model, have been written in the following general form:[Formula: see text]For models that can be solved by the pfaffian technique, H(j) is a quadratic function of fermion operators, but otherwise H(j) also contains quartic functions of fermion operators, making the above expression intractable to exact evaluation. General field theoretical perturbation methods, such as the Green's function techniques and the diagonalization (by a Fourier transformation) of the Hamiltonian, Ho(j), are developed in relation to the above expression. It is shown that the partition function for unsolved problems can be expressed in terms of an "irreducible vertex part" for which low-order approximations can be obtained. This general formalism is applied to the next-nearest neighbor two-dimensional Ising lattice. A first-order approximation is obtained for the irreducible vertex part, and the critical properties are obtained. For the equivalent-neighbor model, the critical temperature obtained is in excellent agreement with the series expansion results, being only 0.4 % higher.

Canadian Journal of Physics•Apr 1, 1969•2 citations

Emergent relativistic symmetry from interacting fermions on the honeycomb bilayer

Zi Hong Liu, Lukas Janssen

We study the phase diagram of interacting spinless fermions on the honeycomb bilayer at charge neutrality using large-scale quantum Monte Carlo simulations. In the noninteracting limit, the low-energy spectrum features quadratically dispersing bands that touch at the corners of the hexagonal Brillouin zone. Weak to intermediate interactions induce a splitting of each of the quadratic band touching points into four Dirac points, located along high-symmetry directions of the reciprocal lattice. Strong interactions lead to the formation of a layer-polarized charge density wave, which spontaneously breaks the $\mathbb Z_2$ layer inversion symmetry and opens an insulating gap in the spectrum. We show that the semimetal-to-insulator quantum phase transition as a function of interaction is continuous and characterized by emergent relativistic symmetry. Our results for the values of the correlation-length exponent $ν$, the order-parameter anomalous dimension $η_ϕ$, and the fermion anomalous dimension $η_ψ$ agree with those of the theoretically predicted 2+1D Gross-Neveu-Ising universality class with eight two-component Dirac fermions within less than 5\%\ deviation. We also determine the crossover scale as a function of interaction strength between the nonrelativistic semimetal state at high temperatures, characterized by dynamical critical exponent $z = 2$, and the Dirac semimetal state at intermediates temperatures, characterized by $z=1$. Further reducing the temperature below the crossover scale at a fixed value of the interaction strength above the quantum critical point results in a classical ordering transition in the 2D Ising universality class.

arXiv:2603.22259•Physical review. B./Physical review. B•Sep 22, 2026•0 citations

Transfer matrix studies of d≥3 Ising models

M. A. Novotny

A novel numerical transfer matrix approach to obtain the critical behavior of Ising models in dimensions higher than two is described. The approach augments the physical transfer matrices with other matrices which do not necessarily have a straightforward physical interpretation. These additional matrices have approximately the same form as the physical transfer matrices, and for high dimensions may provide the only way of performing finite size scaling using transfer matrix calculations. Results are presented for the Ising model in three, four, and five dimensions.

Journal of Applied Physics•May 1, 1990•27 citations

Decay of Order in Classical Many-Body Systems. II. Ising Model at High Temperatures

William J. Camp

We employ the transfer-matrix formalism expounded in Paper I to study the decay of pair correlation functions at high temperatures in the $d$-dimensional Ising model in an arbitrary magnetic field $H$. A general correlation function decays according to $〈\ensuremath{\delta}L(\stackrel{\ensuremath{\rightarrow}}{0})\ensuremath{\delta}Q(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})〉\ensuremath{\approx}{R}^{\frac{(d\ensuremath{-}1)}{2}} {e}^{\ensuremath{-}\ensuremath{\kappa}R}({A}_{0}+{A}_{1}{R}^{\ensuremath{-}1}+\ensuremath{\cdots})+{R}^{\ensuremath{-}x}{e}^{\ensuremath{-}{\ensuremath{\kappa}}^{\ensuremath{'}}R}({B}_{0}+{B}_{1}{R}^{\ensuremath{-}1}+\ensuremath{\cdots})+\ensuremath{\cdots}$ as $R\ensuremath{\rightarrow}\ensuremath{\infty}$. For sufficiently small $H$ and high temperature $T$, the exponent $x$ is equal to the dimension of the lattice and ${\ensuremath{\kappa}}^{\ensuremath{'}}=2\ensuremath{\kappa}$. The coefficients ${A}_{n}$ and ${B}_{n}$ factor as ${A}_{n}(Q){A}_{n}(L)$ and ${B}_{n}(Q){B}_{n}(L)$, respectively. If $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$ is an operator involving an odd number of closely spaced spins, ${A}_{n}(Q)$ tends to a finite limit and ${B}_{n}(Q)$ tends to zero as $H$ tends to zero. In contrast, if $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$ involves an even number of closely spaced spins, ${A}_{n}(Q)$ tends to zero and ${B}_{n}(Q)$ to a finite limit as $H$ tends to zero. Thus, for finite $H$ an arbitrary pair correlation function verifies the Ornstein-Zernike (OZ) prediction; whereas in the zero field, (i) if both $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$ and $\stackrel{\ensuremath{\rightarrow}}{\mathrm{L}}$ are products of odd numbers of spin operators, ${G}_{\mathrm{LQ}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})$ will verify the OZ prediction in zero field; (ii) if both $\stackrel{\ensuremath{\rightarrow}}{\mathrm{L}}$ and $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$ involve even numbers of spins, ${G}_{\mathrm{LQ}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})$ fails to verify the OZ prediction in zero field; and (iii) if $\stackrel{\ensuremath{\rightarrow}}{\mathrm{L}}$ involves an odd number of spins while $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$ involves an even number of spins, ${G}_{\mathrm{LQ}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})$ is identically zero in zero field for all $\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}$. The failure of the zero-field Ising model to completely verify the OZ prediction at high temperatures is due to the symmetry of this model about the $H=0$ line.

Physical review. B, Solid state•Aug 1, 1972•23 citations

Critical properties of a two-dimensional planar model

Alan H. Luther, Douglas J. Scalapino

Recently, critical exponents in the continuum Baxter model have been calculated using an equivalence of this model to the Luttinger model of a one-dimensional spinless fermion gas. We discuss here the extension of this method to the two-dimensional planar model, using the Stoeckly-Scalapino representation of the transfer matrix. This representation is shown to be equivalent to the spin-1/2 fermion gas in one dimension, which further separates into a Luttinger model and the quantum-mechanical sine-Gordon equation. We emphasize the role that soliton bound states play in determining the critical properties. We find that a critical temperature ${T}_{c}$ exists below which the susceptibility is infinite but without long-range order. At ${T}_{c}$, the correlation-function exponent $\ensuremath{\eta}$ takes the values $1\sqrt{8}$, and decreases as the temperature is lowered, consistent with low-temperature results. As in the Baxter-model calculations, we argue that these results are exact for the continuum limit of the transfer matrix, and therefore provide a solution for the asymptotic properties near ${T}_{c}$. We discuss the correlation length and susceptibility, and sugget that nonuniversality is responsible for the disagreement between different numerical calculations.

Physical review. B, Solid state•Aug 1, 1977•103 citations
The ddd-dimensional Ising transfer matrix is expressed exactly in terms of Fermi-type operators. In two dimensions this leads to the standard solution of the model. In three or more dimensions the transfer matrix is no longer a quadratic form in the Fermi variables, but rather contains quartic and higher-order terms. However, it can be expressed as a well-defined expansion about a quadratic form in Fermi operators. The zeroth-order transfer-matrix-called herein the "fermion model"---can be exactly diagonalized, and is well approximated in the critical region by V(0)\ensuremath≈[2sinh(2K)]N2exp[\ensuremath−N\ensuremath∫dq\ensuremath→(\ensuremathκ2+\ensuremathΣn=1d\ensuremath−1\ensuremathΣm=1d\ensuremath−1qnqm)12t(q\ensuremath→)]{V}^{(0)}\ensuremath{\approx}{[2sinh(2K)]}^{\frac{N}{2}}\mathrm{exp}\left[\ensuremath{-}N\ensuremath{\int}d\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}{\left({\ensuremath{\kappa}}^{2}+\ensuremath{\Sigma}{n=1}^{d\ensuremath{-}1}\ensuremath{\Sigma}{m=1}^{d\ensuremath{-}1}{q}_{n}{q}_{m}\right)}^{\frac{1}{2}}t(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})\right]V(0)\ensuremath≈[2sinh(2K)]2N​exp[\ensuremath−N\ensuremath∫dq\ensuremath→​(\ensuremathκ2+\ensuremathΣn=1d\ensuremath−1\ensuremathΣm=1d\ensuremath−1qn​qm​)21​t(q\ensuremath→​)]where NNN is the number of sites per layer, \ensuremathμ\ensuremath{\mu}\ensuremathμ is the inverse correlation length, t(q\ensuremath→)t(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})t(q\ensuremath→​) takes values \ifmmode\pm\else\textpm\fi{}1/2, KKK is the reduced inverse temperature, and the integral extends over a Brillouin zone of the (d\ensuremath−1d\ensuremath{-}1d\ensuremath−1)-dimensional layer. The critical point of the model is located by the equation xcd+xcd\ensuremath−1+xc=1{x}_{c}^{d}+{x}_{c}^{d\ensuremath{-}1}+{x}_{c}=1xcd​+xcd\ensuremath−1​+xc​=1, where xc=e\ensuremath−2Kc{x}_{c}={e}^{\ensuremath{-}2{K}_{c}}xc​=e\ensuremath−2Kc​. The critical exponent \ensuremathν\ensuremath{\nu}\ensuremathν describing the divergence of the correlation length at Kc{K}_{c}Kc​ is found to take on its two-dimensional value, \ensuremathν=1\ensuremath{\nu}=1\ensuremathν=1, in all dimensions ddd. The possibility of using the fermion model as the basis for an expansion of the Ising critical exponents as a function of \ensuremathδ(=d\ensuremath−2)\ensuremath{\delta}(=d\ensuremath{-}2)\ensuremathδ(=d\ensuremath−2) is pointed out.
The partition functions of many lattice statistical problems, such as the Ising models, the dimer model, and the ferroelectric model, have been written in the following general form:[Formula: see text]For models that can be solved by the pfaffian technique, H(j) is a quadratic function of fermion operators, but otherwise H(j) also contains quartic functions of fermion operators, making the above expression intractable to exact evaluation. General field theoretical perturbation methods, such as the Green's function techniques and the diagonalization (by a Fourier transformation) of the Hamiltonian, Ho(j), are developed in relation to the above expression. It is shown that the partition function for unsolved problems can be expressed in terms of an "irreducible vertex part" for which low-order approximations can be obtained. This general formalism is applied to the next-nearest neighbor two-dimensional Ising lattice. A first-order approximation is obtained for the irreducible vertex part, and the critical properties are obtained. For the equivalent-neighbor model, the critical temperature obtained is in excellent agreement with the series expansion results, being only 0.4 % higher.
We study the phase diagram of interacting spinless fermions on the honeycomb bilayer at charge neutrality using large-scale quantum Monte Carlo simulations. In the noninteracting limit, the low-energy spectrum features quadratically dispersing bands that touch at the corners of the hexagonal Brillouin zone. Weak to intermediate interactions induce a splitting of each of the quadratic band touching points into four Dirac points, located along high-symmetry directions of the reciprocal lattice. Strong interactions lead to the formation of a layer-polarized charge density wave, which spontaneously breaks the Z2\mathbb Z_2Z2​ layer inversion symmetry and opens an insulating gap in the spectrum. We show that the semimetal-to-insulator quantum phase transition as a function of interaction is continuous and characterized by emergent relativistic symmetry. Our results for the values of the correlation-length exponent ννν, the order-parameter anomalous dimension ηϕη_ϕηϕ​, and the fermion anomalous dimension ηψη_ψηψ​ agree with those of the theoretically predicted 2+1D Gross-Neveu-Ising universality class with eight two-component Dirac fermions within less than 5\%\ deviation. We also determine the crossover scale as a function of interaction strength between the nonrelativistic semimetal state at high temperatures, characterized by dynamical critical exponent z=2z = 2z=2, and the Dirac semimetal state at intermediates temperatures, characterized by z=1z=1z=1. Further reducing the temperature below the crossover scale at a fixed value of the interaction strength above the quantum critical point results in a classical ordering transition in the 2D Ising universality class.
A novel numerical transfer matrix approach to obtain the critical behavior of Ising models in dimensions higher than two is described. The approach augments the physical transfer matrices with other matrices which do not necessarily have a straightforward physical interpretation. These additional matrices have approximately the same form as the physical transfer matrices, and for high dimensions may provide the only way of performing finite size scaling using transfer matrix calculations. Results are presented for the Ising model in three, four, and five dimensions.
We employ the transfer-matrix formalism expounded in Paper I to study the decay of pair correlation functions at high temperatures in the ddd-dimensional Ising model in an arbitrary magnetic field HHH. A general correlation function decays according to $〈\ensuremath{\delta}L(\stackrel{\ensuremath{\rightarrow}}{0})\ensuremath{\delta}Q(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})〉\ensuremath{\approx}{R}^{\frac{(d\ensuremath{-}1)}{2}} {e}^{\ensuremath{-}\ensuremath{\kappa}R}({A}_{0}+{A}_{1}{R}^{\ensuremath{-}1}+\ensuremath{\cdots})+{R}^{\ensuremath{-}x}{e}^{\ensuremath{-}{\ensuremath{\kappa}}^{\ensuremath{'}}R}({B}_{0}+{B}_{1}{R}^{\ensuremath{-}1}+\ensuremath{\cdots})+\ensuremath{\cdots}$ as $R\ensuremath{\rightarrow}\ensuremath{\infty}$. For sufficiently small HHH and high temperature TTT, the exponent xxx is equal to the dimension of the lattice and ${\ensuremath{\kappa}}^{\ensuremath{'}}=2\ensuremath{\kappa}$. The coefficients An{A}_{n}An​ and Bn{B}_{n}Bn​ factor as An(Q)An(L){A}_{n}(Q){A}_{n}(L)An​(Q)An​(L) and Bn(Q)Bn(L){B}_{n}(Q){B}_{n}(L)Bn​(Q)Bn​(L), respectively. If $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$ is an operator involving an odd number of closely spaced spins, An(Q){A}_{n}(Q)An​(Q) tends to a finite limit and Bn(Q){B}_{n}(Q)Bn​(Q) tends to zero as HHH tends to zero. In contrast, if $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$ involves an even number of closely spaced spins, An(Q){A}_{n}(Q)An​(Q) tends to zero and Bn(Q){B}_{n}(Q)Bn​(Q) to a finite limit as HHH tends to zero. Thus, for finite HHH an arbitrary pair correlation function verifies the Ornstein-Zernike (OZ) prediction; whereas in the zero field, (i) if both $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$ and $\stackrel{\ensuremath{\rightarrow}}{\mathrm{L}}$ are products of odd numbers of spin operators, ${G}_{\mathrm{LQ}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})$ will verify the OZ prediction in zero field; (ii) if both $\stackrel{\ensuremath{\rightarrow}}{\mathrm{L}}$ and $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$ involve even numbers of spins, ${G}_{\mathrm{LQ}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})$ fails to verify the OZ prediction in zero field; and (iii) if $\stackrel{\ensuremath{\rightarrow}}{\mathrm{L}}$ involves an odd number of spins while $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$ involves an even number of spins, ${G}_{\mathrm{LQ}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})$ is identically zero in zero field for all $\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}$. The failure of the zero-field Ising model to completely verify the OZ prediction at high temperatures is due to the symmetry of this model about the H=0H=0H=0 line.
Recently, critical exponents in the continuum Baxter model have been calculated using an equivalence of this model to the Luttinger model of a one-dimensional spinless fermion gas. We discuss here the extension of this method to the two-dimensional planar model, using the Stoeckly-Scalapino representation of the transfer matrix. This representation is shown to be equivalent to the spin-1/2 fermion gas in one dimension, which further separates into a Luttinger model and the quantum-mechanical sine-Gordon equation. We emphasize the role that soliton bound states play in determining the critical properties. We find that a critical temperature Tc{T}_{c}Tc​ exists below which the susceptibility is infinite but without long-range order. At Tc{T}_{c}Tc​, the correlation-function exponent $\ensuremath{\eta}$ takes the values 181\sqrt{8}18​, and decreases as the temperature is lowered, consistent with low-temperature results. As in the Baxter-model calculations, we argue that these results are exact for the continuum limit of the transfer matrix, and therefore provide a solution for the asymptotic properties near Tc{T}_{c}Tc​. We discuss the correlation length and susceptibility, and sugget that nonuniversality is responsible for the disagreement between different numerical calculations.