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Physical review. B, Solid state

Critical properties of a two-dimensional planar model

Alan H. Luther, Douglas J. Scalapino

Published Aug 1, 1977• 103 citations

Abstract

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.

DOI: 10.1103/physrevb.16.1153

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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.

We construct a relationship between the Baxter model in two dimensions and the Luttinger model in one, and use it to calculate critical exponents for the Baxter model from appropriate Luttinger-model correlation functions. An important part of this work involves the generalization of the Jordan-Wigner transformation to provide a representation for continuum spin operators. With this generalization, we are also able to calculate spin correlation functions for a continuum generalization of the spin-\textonehalf{} Heisenberg-Ising chain. We discuss the difference between the continuum and discrete lattice models, and with the help of a new scaling law, use previous results for the Baxter model to calculate new exponents for the Baxter and Heisenberg-Ising model on a lattice.

We present an exact derivation for the asymptotic large distance behavior of the spin two-point correlation function in the XY-model. This allows for the exact obtainment of the critical exponent η = 1/4 at the Kosterlitz-Thouless transition that occurs in this model and in the 2D neutral Coulomb gas and which has been previously obtained by scaling arguments. In order to do that, we use the language of sine-Gordon theory to obtain a Coulomb Gas description of the XY-model spin correlation function, which becomes identified with the soliton correlator of that theory. Using a representation in terms of bipolar coordinates we obtain an exact expression for the asymptotic large distance behavior of the relevant correlator at β 2 = 8π, which corresponds to the Kosterlitz-Thouless transition. The result is obtained by approaching this point from the plasma (high-temperature) phase of the gas. The vortex correlator of the XY-model is also obtained using the same procedure. KEY WORDS: soliton correlation functions, sine-Gordon theory, XY-model, 2D Coulomb gas, Kosterlitz-Thouless transition

Transfer matrix formalism has been used to study the phase transition in a two-dimensional isotropic planar model where one dimension is taken to be spatial and the second dimension is temporal. Character expansion has been used to calculate the eigenvalues of the transfer matrix operator. This has ensured very rapid convergence around the critical point. Fluxes have been generated at each lattice site of the spatial dimension by Monte Carlo simulation. Mass gap and free energy have been found in both theoretical calculation and computer simulation separately for different values of temperature. From the results I infer an algebraic divergence of correlation length rather than a Kosterlitz-Thouless type. The value of critical temperature is found to be

{\mathrm{k}}_{\mathrm{B}}

{\mathrm{T}}_{\mathrm{c}}

/J=0.899.

The equivalence of the one-dimensional two-fermion model, on one hand, with the two-dimensional Coulomb gas and sine-Gordon models on the other is worked out by employing the Jordan boson representation and the point-splitting regularization. The distinction is made throughout between the bandwidth cut-off and momentum-transfer cut-off. It is shown that the coupling constants, both in the weak-coupling regime and near the Luther-Emery exact solution, scale with the momentum-transfer cut-off. The relationship between the bosonized form of the two-fermion model and the corresponding Fermi gas model is briefly discussed.

New universality classes in two-dimensional critical phenomena have recently been discovered using methods of quantum field theory and conformal invariance. These can equally well describe classical two-dimensional systems at their critical temperature or one-dimensional gapless quantum systems at (or near) zero temperature. (The two types of systems are related by the transfer matrix or Feynman path integral.) It is argued that isotropic one-dimensional antiferromagnets are realizations of these new universality classes. Exact predictions for the low temperature specific heat, susceptibility, correlation length, and other critical properties are made. These agree exactly with exactly solveable models and well with experimental results on CuCl2⋅2(NC5H5).

A Trotter-Suzuki mapping is used to calculate the finite-temperature properties of the one-dimensional supersymmetric t − J model. This approach allows for the exact calculation of various thermodynamical properties by means of the quantum transfer matrix (QTM). The free energy and other interesting quantities are obtained such as the specific heat and compressibility. For the largest eigenvalue of the QTM leading to the free energy a set of just two non-linear integral equations is presented. These equations are studied analytically and numerically for different particle densities and temperatures. The structure of the specific heat is discussed in terms of the elementary charge as well as spin excitations. Special emphasis is placed on the study of the low-temperature behavior confirming scaling predictions by conformal field theory and Luttinger liquid theory. To our knowledge this is the first complete investigation of a strongly correlated electron system on a lattice at finite temperature.