Characteristics of two-dimensional lattice models from a fermionic realization: Ising and XYZ models Sh. Khachatryan and A. Sedrakyan Yerevan Physics Institute, Br. Alikhanian 2, Yerevan 36, Armenia Received 23 May 2009; revised manuscript received 18 July 2009; published 28 September 2009 We develop a field theoretical approach to the classical two-dimensional 2D models, particularly to 2D Ising model 2DIM and XYZ model, which is simple to apply for calculation of various correlation functions. We calculate the partition function of 2DIM and XY model within the developed framework. Determinant representation of spin-spin correlation functions is derived using fermionic realization for the Boltzmann weights. The approach also allows formulation of the partition function of 2DIM in the presence of an external magnetic field. DOI: 10.1103/PhysRevB.80.125128 PACS numbers: 71.10.Fd, 71.10.Pm I. INTRODUCTION Two-dimensional Ising model 1 2DIM is one of the most attractive models in physics of low dimensions that describe physical properties of real materials and admit exact solution. 2–11 Originally, 2DIM was solved by Onsager 3 in 1944, and, subsequently, had attracted a steady interest of field theorists and mathematical physicists. Many effective and interesting approaches were developed to calculate the free energy, magnetization, and correlation functions of the model at large distances and all temperatures. Behavior of the model at the critical point is governed by the conformal symmetry, and thus, can be well described by the conformal field theory, which was developed in the seminal article by Belavin et al. 12 All the critical indices of 2DIM were calcu- lated within the conformal field theory approach, in full agreement with the original lattice calculations. 3,8 Although various physical characteristics of 2DIM have been derived using different approaches, still there are open questions that need to be answered. Some of the most impor- tant characteristics of 2DIM include lattice correlation func- tions and form factors. 13–15 These quantities attract consider- able interest in connection with the condensed-matter problems, 16–20 as well as with the problems in string theory. 21 Importance of form factors becomes especially vis- ible when one switches on the magnetic field. 15,22 Then the system exhibits the phenomenon, known in particle physics as quark confinement, 22 observed also in spin-1/2 Heisenberg chain with frustration and dimerization. 18,23,24 One of the effective approaches to 2DIM is based on its equivalence to the theory of two-dimensional free fermions see Ref. 11 and references therein due to the presence of Kac-Word sign-factor 6 in the path-integral representation of the partition function. Though many works have been dedi- cated to the investigation of the 2DIM problem by means of the fermionic Grassmann variables, none of them had linked fermionic representation with vertex R-matrix formu- lation and possible extensions to other integrable models. One of the motivations of the present work is to fill this gap and present a systematically developed field theoretical approach action formulation of the partition function to the 2D Ising and XYZ models on a square lattice, which is based on the Grassmann fields. The developed theory utilizes the graded R operator formalism 25–29 and allows the generaliza- tion to other integrable models, which is demonstrated in this work by operating with rather general R operator. The paper is organized as follows. In Sec. II first we in- troduce the partition function of the 2DIM on the square lattice and demonstrate that the R matrices, constructed via Boltzmann weights, satisfy Yang-Baxter equations. Then in Sec. III the description of fermionic realization for the R matrices is followed, with particular cases of the eight-vertex model, which is equivalent to the one-dimensional 1D quantum XYZ model and 2DIM: the case of finite magnetic fields is also considered. In Sec. IV the partition function is written in the coherent-state basis in terms of scalar fermi- ons. It is represented as a continual integral over the fermi- onic fields with quadratic action for the 2DIM, when mag- netic field vanishes, and for the free-fermionic limit of the eight-vertex model XY model. The nonlocal fermionic ac- tion is obtained in Sec. IV A for the case with nonzero mag- netic field. Continuum limit of the action is derived in Sec. IV C. In Sec. IV D the classical results for the free energy and the thermal capacity are re-obtained within the devel- oped theory. In Sec. V we present the technique for fermionic repre- sentation of correlation functions with details included in the Appendix. In particular, the two-point correlation func- tions in 2DIM are considered on the lattice and their expres- sions are written in the Fourier coordinate basis. In the limit of infinite lattice, large distance spin-spin correlation func- tions can be presented as a determinant Sec. V, which co- incides with the Toeplitz determinant, studied in Ref. 8. Sec- tion VI is devoted to the investigation of the spectrum of one-dimensional quantum chain problem, which is equiva- lent to the classical 2DIM. The work is supplemented with an appendix with rather detailed description of the Jordan- Wigner spin-fermion transformation on 2D lattice, which we have used in the course of the calculations. II. BOLTZMANN WEIGHTS AND YANG-BAXTER EQUATION 1 Boltzmann weights. Classical two-dimensional Ising model on the square lattice can be defined via its local Bolt- zmann weights PHYSICAL REVIEW B 80, 125128 2009 1098-0121/2009/8012/12512819 ©2009 The American Physical Society 125128-1