Characteristics of 2D lattice models from fermionic realization: Ising and XYZ models
aa r X i v : . [ c ond - m a t . s t r- e l ] N ov Characteristics of 2D lattice models from fermionic realization: Ising and
XY Z models
Sh. Khachatryan and A. Sedrakyan Yerevan Physics Institute, Br. Alikhanian 2, Yerevan 36, Armenia
We develop a field theoretical approach to the classical two-dimensional models, particularly to2D Ising model (2DIM) and
XY Z model, which is simple to apply for calculation of various corre-lation functions. We calculate the partition function of 2DIM and XY model within the developedframework. Determinant representation of spin-spin correlation functions is derived using fermionicrealization for the Boltzmann weights. The approach also allows formulation of the partition func-tion of 2DIM in the presence of an external magnetic field. PACS numbers: 71.10.Fd,71.10.Pm
I. INTRODUCTION
Two-dimensional Ising model (2DIM) is one of themost attractive models in physics of low dimensions thatdescribe physical properties of real materials and ad-mit exact solution . Originally, 2DIM wassolved by Onsager in 1944, and, subsequently, had at-tracted a steady interest of field theorists and mathemat-ical physicists. Many effective and interesting approacheswere developed to calculate the free energy, magnetiza-tion, and correlation functions of the model at large dis-tances and all temperatures. Behavior of the model atthe critical point is governed by the conformal symme-try, and thus, can be well described by the conformalfield theory,, which was developed in the seminal articleby Belavin, Polyakov and Zamolodchikov . All the criti-cal indices of 2DIM were calculated within the conformalfield theory approach, in full agreement with the originallattice calculations .Although various physical characteristics of 2DIM havebeen derived using different approaches, still there areopen questions that need to be answered. Some ofthe most important characteristics of 2DIM include lat-tice correlation functions and form factors . Thesequantities attract considerable interest in connectionwith the condensed-matter problems , as well as withthe problems in string theory . Importance of formfactors becomes especially visible when one switcheson the magnetic field . Then the system exhibitsthe phenomenon, known in particle physics as quarkconfinement , observed also in spin-1/2 Heisenbergchain with frustration and dimerization .One of the effective approaches to 2DIM is based onits equivalence to the theory of two-dimensional freefermions (see Ref. and references therein) due to thepresence of Kac-Word sign-factor in the path integralrepresentation of the partition function. Though manyworks have been dedicated to the investigation of the2DIM problem by means of the fermionic (Grassmann)variables, none of them had linked fermionic representa-tion with vertex R matrix formulation and possible ex-tensions to other integrable models.One of the motivations of the present work is to fillthis gap and present a systematically developed field theoretical approach (action formulation of the parti-tion function) to the 2D Ising and XY Z models on asquare lattice, which is based on the Grassmann fields.The developed theory utilizes the graded R operatorformalism and allows the generalization toother integrable models, which is demonstrated in thiswork 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 squarelattice and demonstrate that the R matrices, constructedvia Boltzmann weights, satisfy Yang-Baxter equations.Then in Sec. III the description of fermionic realizationfor the R matrices is followed, with particular cases ofthe eight vertex model, which is equivalent to the one-dimensional (1D) quantum XY Z model and 2DIM: thecase of finite magnetic fields is also considered. In Sec.IV the partition function is written in the coherent-statebasis in terms of scalar fermions. It is represented as acontinual integral over the fermionic fields with quadraticaction for the 2DIM, when magnetic field vanishes, andfor the free-fermionic limit of the eight-vertex model ( XY model). The non-local fermionic action is obtainedin Sec. IV A for the case with non-zero magnetic field.Continuum limit of the action is derived in Sec. IV C. InSec. IV D the classical results for the free energy and thethermal capacity are re-obtained within the developedtheory.In Sec. V we present the technique for fermionic repre-sentation of correlation functions (with details includedin the Appendix). In particular, the two-point correla-tion functions in 2DIM are considered on the lattice andtheir expressions are written in the Fourier coordinate ba-sis. In the limit of infinite lattice, large distance spin-spincorrelation functions can be presented as a determinant(Sec. V A), which coincides with the Toeplitz determi-nant, studied in Ref. . Sec. VI is devoted to the in-vestigation of the spectrum of one-dimensional quantumchain problem, which is equivalent to the classical 2DIM.The work is supplemented with an appendix with ratherdetailed description of the Jordan-Wigner spin-fermiontransformation on 2D lattice, which we have used in thecourse of the calculations. ❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅(cid:0)(cid:0) ❅❅(cid:0)(cid:0)❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅(cid:0)(cid:0) ❅❅(cid:0)(cid:0)❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅(cid:0)(cid:0) ❅❅(cid:0)(cid:0) ❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅(cid:0)(cid:0) ❅❅(cid:0)(cid:0)❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅(cid:0)(cid:0) ❅❅(cid:0)(cid:0)❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅(cid:0)(cid:0) ❅❅(cid:0)(cid:0) ❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅(cid:0)(cid:0) ❅❅(cid:0)(cid:0)❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅(cid:0)(cid:0) ❅❅(cid:0)(cid:0)❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅(cid:0)(cid:0) ❅❅(cid:0)(cid:0) ❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅(cid:0)(cid:0) ❅❅(cid:0)(cid:0)❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅(cid:0)(cid:0) ❅❅(cid:0)(cid:0)❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅(cid:0)(cid:0) ❅❅(cid:0)(cid:0) W α ′ β ′ αβ α β β ′ α ′ ttt ttt ttt ttt ttt❞❞❞❞ ❞❞❞❞ ❞❞❞❞ ❞❞❞❞rrrr rrrr rrrr rrrr ✲✻ xy FIG. 1: A fragment of the lattice of 2DIM: spin variablescorrespond to vertices, local Boltzmann weights correspondto dashed squares.
II. BOLTZMANN WEIGHTS ANDYANG-BAXTER EQUATION
1. Boltzmann weights.
Classical two-dimensional Ising model on the square lattice can be de-fined via its local Boltzmann weights W α ′ β ′ αβ = e J (¯ σ α ¯ σ α ′ +¯ σ β ¯ σ β ′ )+ J (¯ σ α ¯ σ β +¯ σ α ′ ¯ σ β ′ ) , (2.1) α, β, α ′ , β ′ = 0 , , where the two state spin variables ¯ σ α = {± } are as-signed to the vertices of the lattice. The partition func-tion Z ( J , J ) = X { α,β } Y W α ′ β ′ αβ (2.2)is a sum over spin configurations of products of the Boltz-mann weights W α ′ β ′ αβ , each associated with the elemen-tary square plaquette with vertices α, β, α ′ , β ′ andarranged in a checkerboard pattern (dashed squares inFig.1). There are imposed periodic boundary conditionson the spin variables.Boltzmann weight W α ′ β ′ αβ in Eq. (2.1) can be regarded as a matrix, W = e J + J ) e − J + J ) e J − J ) e − J + J ) e − J + J ) e J − J ) e − J + J ) e J + J ) , (2.3)acting as a linear operator on the direct product of twotwo-dimensional linear vector spaces, | α i | β i , α, β = 0 , h β ′ | h α ′ | W | α i | β i = W α ′ β ′ αβ , (2.4)where | i , | i are orthonormalized vectors (cid:0) (cid:1) , (cid:0) (cid:1) .The matrix (2.3) can be represented as a tensor productof spin operators, W = e J J (ˆ1 ⊗ ˆ1 + σ z ⊗ σ z ) (2.5)+ e J − J (ˆ1 ⊗ ˆ1 − σ z ⊗ σ z ) + e − J J ( σ ⊗ σ − σ ⊗ σ )+ e − J J ( σ ⊗ σ + σ ⊗ σ ) + (ˆ1 ⊗ σ + σ ⊗ ˆ1) , where σ α ( α = 1 , , z ) are Pauli matrices and ˆ1 is thetwo-dimensional identity operator.By use of a unitary transformation, one can representthe matrix W in the form of the R matrix, correspond-ing to the eight vertex model. Let us define the unitarymatrix U = √ (cid:0) − (cid:1) , such that U σ z U − = σ , U σ U − = σ z , U σ U − = − σ . Then the action of these unitary transformations on thelinear spaces, which are assigned to every site of the two-dimensional square lattice (Fig.1), yields R = (cid:0) U − ⊗ U − (cid:1) W ( U ⊗ U ) = e J J (ˆ1 ⊗ ˆ1 + σ ⊗ σ )+ e J − J (ˆ1 ⊗ ˆ1 − σ ⊗ σ ) + e J − J ( σ z ⊗ σ z − σ ⊗ σ )+ e − J J ( σ z ⊗ σ z + σ ⊗ σ ) + (ˆ1 ⊗ σ z + σ z ⊗ ˆ1) . (2.6)Partition function (2.2) of the model can be expressedvia new weights (2.6), as Z = X { α,β } Y R α ′ β ′ αβ , (2.7)where R operator has the following matrix form R = 2 cosh [2 J ] cosh [2 J ] + 1 0 0 cosh [2 J ] sinh [2 J ]0 sinh [2 J ] cosh [2 J ] sinh [2 J ] sinh [2 J ] 00 sinh [2 J ] sinh [2 J ] sinh [2 J ] cosh [2 J ] 0cosh [2 J ] sinh [2 J ] 0 0 cosh [2 J ] cosh [2 J ] − . (2.8)¿From (2.8) it is apparent that R α ′ β ′ αβ has the form of R matrix corresponding to the XY model. It fulfills the”free-fermionic” condition of the XY model: R R − R R = R R − R R . (2.9)
2. Yang-Baxter equations.
In this section weexamine whether matrix (2.8) is a solution of Yang-Baxter equations. We shall verify this by using Baxter’stransformation , e ± J = cn[ i u, k ] ∓ i sn[ i u, k ] ,e ± J = i (dn[ i u, k ] ± / ( k sn[ i u, k ]) . (2.10) It has been proven in Ref. , that for fixed k parameter,two transfer matrices with different parameters u com-mute. The case of k = 1 corresponds to the point ofphase transition.Now we rewrite the R matrix (2.8) in terms of functions(2.10), R ( u, k ) = i cn[ i u, k ] dn[ i u, k ] k sn[ i u, k ] dn[ i u, k ] k i cn[ i u, k ] k sn[ i u, k ] 1 k k i cn[ i u, k ] k sn[ i u, k ] dn[ i u, k ] k − i cn[ i u, k ] dn[ i u, k ] k sn[ i u, k ] . (2.11)Let us multiply matrix (2.11) by − i k sn( i u, k ) anddefine the matrix r ( u, k ) as r ( u, k ) = − i k sn[ i u, k ] R ( u, k ) . (2.12)It is straightforward to verify that r (0 , k ) = I , where I = ˆ1 ⊗ ˆ1 is the identity matrix. Importantly, it takesplace the relation r ( − u, k ) = r − ( u, k ) . (2.13)Using the properties of the Jacobi elliptic functions,one can verify that r ( u, k ) satisfies the Yang-Baxter equa- tion X β , β , β ′ r β β α α ( u − v, k ) r β ′ γ β α ( u, k ) r γ γ β β ′ ( v, k ) = X β , β ′ , β r β β α α ( v, k ) r γ β ′ α β ( u, k ) r γ γ β ′ β ( u − v, k ) . (2.14)Note that there is also another R matrix correspond-ing to 2DIM. It is known, that the classical 2DIM is aspecial case of the eight-vertex model . In general, the R matrix of the eight-vertex (or XY Z ) model can beparameterized by two model parameters, k and λ , as r xyz = sn [ i λ − u , k ] sn [ iλ, k ] − k sn[ i λ + u , k ]sn[ i λ − u , k ]0 1 sn [ i λ + u , k ] sn [ i λ, k ]
00 sn [ i λ + u , k ] sn [ i λ, k ] − k sn[ i λ + u , k ]sn[ i λ − u , k ] 0 0 sn [ i λ − u , k ] sn [ i λ, k ] . (2.15)The ”Ising” limit corresponds to the choice of λ = I ′ , where I ′ is one of two half-periods of the ellipticfunctions . In this case the eight-vertex model, definedon a rectangular lattice, splits into two independent Isingmodels defined on the two sublattices.
3. The transfer matrix and the Hamiltonian.
For convenience we denote the coordinates of the latticesites by even-even (2 i, j ) (black circles on Figs. 1 and 2),and odd-odd (2 i + 1 , j + 1) (white circles on Figs. 1 and2) numbers, and assign two-dimensional linear spaces of quantum states of spins | α i, j i and | α i +1 , j +1 i to eachof these spaces. Periodic boundary conditions imply | α , j i = | α N, j i and | α i +1 , i = | α i +1 , N +1 i . (2.16)Local R ( i, j ) operators (2.8) are acting linearly on theproduct of spaces | α i, j i| α i +1 , j − i at the sites (2 i, j )and (2 i + 1 , j − i +1 , j + 1) and (2 i + 2 , j ), R ( i, j ) : | α i, j i | α i +1 , j − i ⇒ | α i +1 , j +1 i | α i +2 , j i . (2.17) ❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅(cid:0)(cid:0) ❅❅(cid:0)(cid:0)❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅(cid:0)(cid:0) ❅❅(cid:0)(cid:0)❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅(cid:0)(cid:0) ❅❅(cid:0)(cid:0) ❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅(cid:0)(cid:0) ❅❅(cid:0)(cid:0)❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅(cid:0)(cid:0) ❅❅(cid:0)(cid:0)❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅(cid:0)(cid:0) ❅❅(cid:0)(cid:0) ❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅(cid:0)(cid:0) ❅❅(cid:0)(cid:0)❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅(cid:0)(cid:0) ❅❅(cid:0)(cid:0)❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅(cid:0)(cid:0) ❅❅(cid:0)(cid:0) ❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅(cid:0)(cid:0) ❅❅(cid:0)(cid:0)❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅(cid:0)(cid:0) ❅❅(cid:0)(cid:0)❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅(cid:0)(cid:0) ❅❅(cid:0)(cid:0) ttt ttt ttt ttt ttt❞❞❞❞ ❞❞❞❞ ❞❞❞❞ ❞❞❞❞rrrr rrrr rrrr rrrr ❄ ❄ ❄✲✲ ✲ ❤❤ a ✲✻ xy ij ❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅(cid:0)(cid:0) ❅❅(cid:0)(cid:0) t t❞❞rr (2 i, j )(2 i + 1 , j − i + 2 , j )(2 i + 1 , j + 1) = R ( i, j )b FIG. 2: (a) Bold lines and circled vertices represent a frag-ment of the lattice of the model in coordinate plane (dottedlines). (b) Local R operator. The product of R matrices on each chain along x direc-tion on the lattice (see Fig.2) in the formulas of partitionfunction (2.7), after summation over the boundary states,constitutes a transfer matrix, τ . Taking into account con-ditions (2.16), we obtain the following representation forthe transfer matrices τ j = tr Y i = N − R ( i, j ) ≡ X α , j h α , j | Y i = N − R ( i, j ) | α , j i . (2.18)They act on the states of spins at sites { i + 1 , j − } i =0 ,N − , | Σ j i = | α , j − i| α , j − i · · · | α N − , j − i , and map them onto the states at sites { i + 1 , j +1 } i =0 ,N − , | Σ j +1 i = | α , j +1 i| α , j +1 i · · · | α N − , j +1 i . One can interpret the y direction, marked by integers j , as a time direction, while the transfer matrix τ j will bethe evolution operator for discrete lattice time. In termsof the transfer matrices, partition function (2.7) acquires the following form: Z = tr Y j = N τ j ≡ X { α i +1 , } i =0 ,N − h Σ | Y j = N τ j | Σ i . (2.19)In Eqs. (2.18 and 2.19) tr and tr represent sums ofthe states at the boundaries with (even, even) and (odd,odd) coordinates (dark and light circles in the figures),respectively. Borrowing the usual terminology of thetransfer-matrix theory, we can refer the states markedby white circles on the lattice as ”quantum” states, whilethe states marked with black circles as ”auxilary” states.
4. 2D classical model as a (1 + 1)
D quantumtheory.
Following to Ref. , one can introduce the limit J ∼ J ∆ t, e − J ∼ h ∆ t, ∆ t ≪ quantum one-dimensional model. In this limit R matrix(2.8) becomes R = Ih ∆ t + J/h
J/h J/h
J/h − + h ∆ t − − . (2.21)It acquires the following operator form: R = 1 h ∆ t (cid:8) ˆ1 ⊗ ˆ1+2∆ t ( Jσ ⊗ σ + h (1 ⊗ σ z + σ z ⊗ (cid:9) . (2.22)The coefficient of ∆ t in log τ , constructed with R -matrices (2.22), defines 1D quantum Hamiltonian for theIsing model on a chain in a transverse magnetic field h , H = X i ( Jσ ( i ) σ ( i + 1) + hσ z ( i )) . (2.23)
5. Finite magnetic field.
The Boltz-mann weights of the classical 2DIM in a uni-formly applied magnetic field B have the fol-lowing matrix representation: W B α ′ β ′ α β = e J (¯ σ α ¯ σ α ′ +¯ σ β ¯ σ β ′ )+ J (¯ σ α ¯ σ β +¯ σ α ′ ¯ σ β ′ )+ B (¯ σ α +¯ σ β +¯ σ α ′ +¯ σ β ′ ) . Operator representation of W B after unitary transfor-mation gets R B = U − ⊗ U − W B U ⊗ U (2.24)= e J J cosh [2 B ](ˆ1 ⊗ ˆ1+ σ ⊗ σ ) + e J − J (ˆ1 ⊗ ˆ1 − σ ⊗ σ )+ e J − J ( σ z ⊗ σ z − σ ⊗ σ )+ e − J J ( σ z ⊗ σ z + σ ⊗ σ )+ e J J sinh [2 B ](ˆ1 ⊗ σ + σ ⊗ ˆ1)+ sinh [ B ]( σ ⊗ σ z + σ z ⊗ σ ) + cosh [ B ](ˆ1 ⊗ σ z + σ z ⊗ ˆ1) . In the limit J ∼ J ∆ t, e − J ∼ h ∆ t, B ∼ B ∆ t, ∆ t ≪ , (2.25)operator R B obtains the form R B = 1 h ∆ t (cid:8) ˆ1 ⊗ ˆ1 + 2∆ t (cid:0) Jσ ⊗ σ + h (ˆ1 ⊗ σ z + σ z ⊗ ˆ1)+ B ⊗ σ + σ ⊗ ˆ1) (cid:19)(cid:27) , (2.26)which establishes an equivalence with the quantum 1DIMin the magnetic field B . In this case the operator t log τ ,when ∆ t →
0, defines the following structure of Hamil-tonian operator H B = X i (cid:18) Jσ ( i ) σ ( i + 1) + hσ z ( i ) + B σ ( i ) (cid:19) . III. FERMIONIC REALIZATION OFBOLTZMANN WEIGHTS
Besides the matrix formulation for Boltzmann weights(2.5) and (2.6), one can think of alternative representa-tions. Examples include representation of the R matrixin the Fock space of the scalar fermions and in a spacewith the basis of fermionic coherent-states, developed inRefs. . These reformulations are fully equivalent,and they allow developing a field theory corresponding tothe model. The latter simplifies calculations of physicalquantities (particularly, the free energy and the magne-tization) of the model, and can be extended for the com-putation of form factors too, which are problematic inthe standard scheme.Let us now consider the graded Fock space ofscalar fermions c + ( i, j ) and c ( i, j ), on the lattice,([ c + ( i, j ) , c ( i, j )] + = δ ik δ jr ), identifying the two-dimensional basis at each site, labelled by { i, j } , with | i i,j ( c ( i, j ) | i i,j = 0) and | i i,j = c + ( i, j ) | i i,j .Then it is not hard to construct fermionic representa-tion of the R operator. Note, that the R -operator definedin the previous section ”permutes” the arrangement ofthe spaces (as it is a ”check” R matrix), R : | α ′ i | β ′ i ⇒| β i | α i , so for graded spaces we have R = R β ′ α ′ α β | β i | α i h β ′ | h α ′ | (3.1)= R β ′ α ′ α β | β i h β ′ | | α i h α ′ | ( − p ( β ′ ) p ( α ) , where p ( α ) = α is the parity of the space | α i . Operators | α i i h α ′ | act on the Fock space with the basis {| i i , | i i } ,as | i i h | = 1 − c + i c i , | i i h | = c + i c i , | i i h | = c i , | i i h | = c + i . This means, that in terms of two scalar fermions { c + i , c j } = δ ij , { c i , c j } = { c + i , c + j } = 0 , i, j = 1 , R operator in the zero-field limit, B = 0, reads R ( c +1 , c ; c +2 , c ) = R + R c +1 c + R c +2 c +( R − R ) c +1 c + ( R − R ) c +2 c + R c +2 c +1 + R c c + ( R − R − R − R ) c +1 c c +2 c , (3.2) where R krij are the matrix elements of the R matrix.
1. The fermionic representation (3.2) of theIsing model’s R matrix (2.8) has the following ma-trix elements: R = 2(cosh [2 J ] cosh [2 J ] + 1) ,R = 2(cosh [2 J ] cosh [2 J ] − , (3.3) R = R = 2 sinh [2 J ] cosh [2 J ] ,R = R = 2 sinh [2 J ] sinh [2 J ] ,R = R = 2 cosh [2 J ] sinh [2 J ] . In the following we shall operate in the coherent-state ba-sis. For that purpose we need to represent R operator inthe normal ordered form. For zero magnetic field the op-erator R can be expressed as an exponent of a quadraticform (a consequence of property (2.9)), R = R : exp A ( c +1 , c +2 , c , c ) : , (3.4)where A ( c +1 , c +2 , c , c ) = ( c − c +1 c + c +2 c )+ b ( c +1 c + c +2 c ) + d ( c +2 c +1 + c c ) , (3.5) a = R , b = R /R ,c = R /R , d = R /R . (3.6)For a finite magnetic field B the fermionic realization R B of the R B operator (2.24) can be obtained in the sameway, substituting the corresponding matrix elements informulas (3.1). Then the normal ordered form of the R B can be represented as follows: R B = r B : e A B ( c +1 ,c +2 ,c ,c ) : , (3.7) A B ( c +1 , c +2 , c , c ) = c B ( c +1 c + c +2 c ) + b B ( c +1 c + c +2 c )+ d B ( c +2 c +1 + c c ) + h B ( c + c + c +1 + c +2 )+( η c + η c +1 ) c +2 c + ( η c + η c +2 ) c +1 c , (3.8)with r B = 2 cosh [ B ] + 2 cosh [2 J ] cosh [2 J ]+ e J + J ) (sinh [ B ]) ,c B = − r B (cosh [ B ] + cosh [2( J − J )]) ,b B = 12 r B (cid:16) e J + J ) (sinh [ B ]) + 2 cosh [2 J ] sinh [2 J ] (cid:17) ,d B = 12 r B (cid:16) e J + J ) (sinh [ B ]) + 2 cosh [2 J ] sinh [2 J ] (cid:17) ,h B = 1 r B (cid:16) sinh [ B ] + sinh [2 B ] e J + J ) (cid:17) ,η = 4 r B (cid:0) e J cosh [ B ] + cosh [2 J ] (cid:1) sinh [ B ] sinh [2 J ] ,η = − r B sinh [ B ] sinh [2 J ] sinh [2 J ] . (3.9)
2. General R matrix and XYZ model. The fermionicrepresentation (3.2) is justified as well for arbitrary R matrix, which has form R = R R R R R R R R . (3.10)Now the A ( c +1 , c +2 , c , c ) in the normal ordered form(3.4) has a quartic term also, A ( c +1 , c +2 , c , c ) = ( c − c +1 c + ( c ′ − c +2 c (3.11)+ b c +1 c + b ′ c +2 c + d c +2 c +1 + d ′ c c + ∆ c +1 c c +2 c ,a = R , b = R R , b ′ = R R , c = R R , c ′ = R R , (3.12) d = R R , d ′ = R R , ∆ = R R + R R − R R − R R R . For the
XY Z model’s general R matrix, given in Eq.(2.15), the ∆ parameter writes as∆ = 2 sn[ i λ + u , k ]cn[ iλ, k ]dn[ iλ, k ]sn[ i λ − u , k ](sn[ iλ, k ]) . (3.13)The limit ∆ = 0 corresponds to the free-fermionic XYmodel. It fulfills, whencn[ iλ, k ]dn[ iλ, k ] = 0 . (3.14)Possible solutions are cn[ I, k ] = 0 and dn[ I + iI ′ k ] = 0.Here I, I ′ are the half-periods of the elliptic functions.Note that the Ising limit derived in the Ref. correspondsto the values λ = I ′ / , ∆ = 2 k / (1 + k )sn[ i λ + u , k ]sn[ i λ − u , k ] . (3.15) IV. QUANTUM FIELD THEORYREPRESENTATION ON THE LATTICE: WITHGENERAL R -MATRIX AND 2DIM In this section we will introduce fermionic fields ¯ ψ ( i, j )and ψ ( i, j ), corresponding to the coherent-states of scalarfermions c + ( i, j ) and c ( i, j ). By definition, coherent-states are the eigenstates of annihilation operators ofscalar fermions. For a set of the scalar fermions c i and c + i , they are defined by the following relations c i | ψ i i = ψ i | ψ i i , h ¯ ψ i | c + i = h ¯ ψ i | ¯ ψ i . (4.1)Because of the Fermi statistics, namely, anticommuta-tion relations, these eigenvalues are Grassmann variablesdenoted in Eq. (4.1) by ψ i and ¯ ψ i . They fulfill (i) or-thonormality and (ii) completeness relations: h ¯ ψ i | ψ j i = δ ij e ¯ ψ i ψ i , Z d ¯ ψ i dψ i e − ¯ ψ i ψ i | ψ i ih ¯ ψ i | = I. (4.2) The kernel of any normal ordered operator K ( { c + i , c j } )in terms of coherent-states can be obtained simply byreplacing creation-annihilation operators c i , c + i by theireigenvalues and multiplying by e P i ¯ ψ i ψ i , K ( { ¯ ψ i , ψ j } ) ≡ h Y ¯ ψ i | K ( { c + i , c j } ) | Y ψ j i = (4.3) e P i ¯ ψ i ψ i K ( { ¯ ψ i , ψ j } ) . The trace of the operator K ( { c + i , c j } ) in coherent-statesis an integral over the Grassmann variables,tr K ( { c + i , c j } ) = Z DψD ¯ ψe P i ¯ ψ i ψ i K ( { ¯ ψ i , ψ j } ) ,DψD ¯ ψ = Y i dψ i d ¯ ψ i . (4.4)In order to obtain the form of the partition function Z [Eq. (2.7)] in the basis of coherent-states, let us at eachcircled vertex of the lattice, between the R ( i, j ) operators(see Fig. 2), insert the following identity operators I = Z d ¯ ψ ( i, j ) dψ ( i, j ) e − ¯ ψ ( i,j ) ψ ( i,j ) | ψ ( i, j ) ih ¯ ψ ( i, j ) | . With the properties of coherent-states representedabove, we can easily calculate the matrix elements of the R -operator (3.2), using the normal ordered form repre-sentations (3.4) and (3.11), in terms of Grassmann fields, h ¯ ψ (2 i +1 , j +1) |h ¯ ψ (2 i +2 , j ) | (4.5) R ( i, j ) | ψ (2 i, j ) i| ψ (2 i +1 , j − i = R exp n A h ¯ ψ (2 i +2 , j ) , ¯ ψ (2 i +1 , j +1) ,ψ (2 i, j ) , ψ (2 i +1 , j − i +¯ ψ (2 i +2 , j ) ψ (2 i, j )+ ¯ ψ (2 i +1 , j +1) ψ (2 i +1 , j − o . Then the partition function Z for large N can be writtenas a path integral, Z = tr Q Nj =1 Q i = N − R ( i, j ) = ( R ) N R D ¯ ψDψ e − A (cid:0) ¯ ψ, ψ (cid:1) , (4.6)with action A ( ¯ ψ, ψ ): − A ( ¯ ψ, ψ ) = P i,j { b ¯ ψ (2 i +2 , j ) ψ (2 i +1 , j − b ′ ¯ ψ (2 i +1 , j +1) ψ (2 i, j )+ c ¯ ψ (2 i +2 , j ) ψ (2 i, j )+ c ′ ¯ ψ (2 i +1 , j +1) ψ (2 i +1 , j − d ¯ ψ (2 i +2 , j ) ¯ ψ (2 i +1 , j +1)+ d ′ ψ (2 i, j ) ψ (2 i +1 , j − ψ (2 i +2 , j ) ψ (2 i +1 , j −
1) ¯ ψ (2 i +1 , j +1) ψ (2 i, j ) − ¯ ψ (2 i, j ) ψ (2 i, j ) − ¯ ψ (2 i +1 , j +1) ψ (2 i +1 , j +1) } + P j ¯ ψ (2 N, j ) ψ (0 , j )+ P i ¯ ψ (2 i +1 , N +1) ψ (2 i +1 , . (4.7)In sum (4.7) the last two terms come from the trace.So, the partition function of a model, defined by R matrix (3.10), has fermionic path-integral representationwith local action (4.7) on the two-dimensional lattice.As it is apparent, the 2DIM has fermionic represen-tation with local quadratic action (see Eqs. (3.5) and(3.3)). It is true also for the partition function of the XY model, defined with the Eqs. (3.14), as well, since theGaussian quadratic form is a consequence of the ”free-fermionic” property, given by Eq. (2.9), and the formulaof the coefficient at quartic term (3.12). On the otherhand, the Ising limit of the eight-vertex ( XY Z ) modeldoes not correspond to a quadratic action, as it is fol-lowed from the Eqs. (3.15). However it is well knownthat the two limits of the
XY Z model - Ising limit ( XZ )and free-fermionic limit ( XY ) are equivalent and can bebrought one to another by redefinition of the model pa-rameters. A. Path integral representation of partitionfunction for the case of finite magnetic field
For construction of partition function of 2DIM in anon-zero magnetic field we make use of fermionic expres-sion (3.7). In order to take into account the graded char-acter of the R ( B )-operators (recall, that they have nodefinite parity), we are led to include non local opera-tors, Z ( B ) = tr N Y j =1 0 Y i = N − {R ( even ) B ( i, j )+ R ( odd ) B ( i, j ) J ( i, j ) } . (4.8) In Eq. (4.8) operators R ( even ) B and R ( odd ) B represent theparts of the R ( B )-operator (3.7) that have even and oddgradings (or their series expansions consist of even/oddpowers of B or fermionic operators), correspondingly.Operator J ( i, j ) = Q (1 − n ) is the Jordan-Wigner non-local operator (see Eq. (5.5) and the Appendix).Let us introduce formal definitions R ′ ( even ) / ( odd ) B ( i, j ) = R ( even ) / ( odd ) B ( i, j )[1 − n (2 i, j )] , R ′′ ( even ) / ( odd ) B ( i, j ) =[1 − n (2 i +1 , j +1)] R ( even ) / ( odd ) B ( i, j ) . (4.9)Then we can expand the product in Eq. (4.8) and rewriteit as Z ( B ) = tr X C { k,r } Y i,j R C kr B ( i, j ) , (4.10)where the sum goes over all lattice sites denoted by C { k,r } .Operator R C kr B ( i, j ) is attached to the square ( i, j ) (seeFig. 2). It is equal to R ( odd ) B ( i, j ) or R ′ / ′′ ( odd ) B ( i, j ),if { i, j } = { k, r } , and is equal to R ( even ) B ( i, j ) or R ′ / ′′ ( even ) B ( i, j ) otherwise.Each summand in Eq. (4.10) can be written in thebasis of coherent-states in the same way as it was donein previous subsection. Finally we find Z ( B ) = Z D ¯ ψDψe − P i,j ¯ ψ ( i,j ) ψ ( i,j )+ P j ¯ ψ (2 N, j ) ψ (0 , j )+ P i ¯ ψ (2 i +1 , N +1) ψ (2 i +1 , I B ( ¯ ψ,ψ ) , I B ( ¯ ψ, ψ ) = ln X C { k,r } Y i,j h ¯ ψ (2 i +1 , j +1) |h ¯ ψ (2 i +2 , j ) | R C kr B ( i, j ) | ψ (2 i, j ) i| ψ (2 i +1 , j − i . (4.11)Because we operate with local fermionic R B matrices,form (4.11) of the partition function will be held for thecase of inhomogeneous magnetic field as well. B. Partition function
For calculation of partition function (4.6) in the ”free-fermionic” case ∆ = 0 we need to diagonalize the action A ( ¯ ψ, ψ ). Taking into account antiperiodic boundary con- ditions imposed on the Grassmann fields,¯ ψ (2 N, j ) = − ¯ ψ (0 , j ) , ψ (2 N, j ) = − ψ (0 , j ) , ¯ ψ (2 N +1 , j +1) = − ¯ ψ (1 , j +1) ,ψ (2 N +1 , j +1) = − ψ (1 , j +1) , ¯ ψ (2 i, N ) = − ¯ ψ (2 i, , ψ (2 i, N ) = − ψ (2 i, , ¯ ψ (2 i +1 , N +1) = − ¯ ψ (2 i +1 , ,ψ (2 i +1 , N +1) = ψ (2 i +1 , , (4.12)we can perform the Fourier transformation with odd mo-menta ψ ( r, k ) = 1 N N − X n r ,n k =0 e − iπ N ((2 n r +1) r +(2 n k +1) k ) × ψ α h π N (2 n r + 1) , π N (2 n k + 1) i . (4.13)Here α = 1 for even coordinates ( r, k ), and α = 2 forodd coordinates. After defining new Grassman fields ψ ( p, q ) , ψ ( p, q ) as ψ ( π − p, π − q ) ≡ − ¯ ψ ( p, q ) , ψ ( π − p, π − q ) ≡ − ¯ ψ ( p, q ) , ¯ ψ ( π − p, π − q ) ≡ ψ ( p, q ) , ¯ ψ ( π − p, π − q ) ≡ ψ ( p, q ) , (4.14)we will come to the following simple form for the action A (4.7) in the momentum space − A ( ¯ ψ, ψ ) = X p,q X k,r A kr ( p, q ) ¯ ψ k ( p, q ) ψ r ( p, q ) . (4.15)In Eq. (4.15) we have introduced the notationsA( p, q ) = c e i p − b e i ( p + q ) − d e i ( p − q ) b ′ e i ( p + q ) c ′ e iq − d e − i ( p − q ) d ′ e − i ( p − q ) c e − ip − b ′ e − i ( p + q ) − d ′ e i ( p − q ) b e − i ( p + q ) c ′ e − iq − , (4.16)and p = π N (2 n p + 1) , q = π N (2 n q + 1) ,n p = 1 , ..., N/ − , n q = 1 , ..., N − . (4.17)Then the partition function acquires form of a productof determinants Z = ( R ) N N − ,N − Y n p ,n q =0 Det (cid:20) A (cid:18) π (2 n p + 1)2 N , π (2 n q + 1)2 N (cid:19)(cid:21) . (4.18)The determinants are found asDet[A( p, q )] = n (cid:18) R R (cid:19) + (cid:18) R R (cid:19) − R ( R − R )( R ) (cos[2 p ] + cos[2 q ])+4 (cid:18) R R (cid:19) cos[2 p ] cos[2 q ] − (cid:18) R R (cid:19) cos[2( p + q )] − (cid:18) R R (cid:19) cos[2( p − q )] o . (4.19)Here we have assumed that R = R and R = R . Ising model.
Substituting the matrix elements givenin Eqs. (3.3), we are arriving atDet[A( p, q )] { J ] cosh[2 J ] } =2 n J ] cosh[2 J ]) + (sinh[2 J ] sinh[2 J ]) − J ] sinh[2 J ](cos[2 p ] + cos[2 q ]) − (sinh[2 J ]) cos[2( p + q )] − (sinh[2 J ]) cos[2( p − q )] o . (4.20)In the limit p → , q →
0, it takes place Det[A( p, q )] → (Det[A ]) , withDet[A ] = 2 (cid:18) − sinh[2 J ] sinh[2 J ]1 + cosh[2 J ] cosh[2 J ] (cid:19) . (4.21) Equations (4.20) and (4.21) suggest that Det[A( p, q )] > { p, q } 6 = { , } , and only exactly at { p, q } = { , } and { − sinh[2 J c ] sinh[2 J c ] = 0 } wehave Det[A ] = 0, corresponding to the point of the sec-ond order phase transition. XY model.
Free-fermionic limit of the eight-vertexmodel corresponds to relation (3.14). The solutions ofthat relations are (to within the periods of the ellipticfunctions) iλ = I or iλ = I + iI ′ . (4.22)When iλ = I , then the expression in Eq. (4.19) writes asDet[A xy ( p, q )] = (4.23)2 ( (cid:16) sn [ i u ′ , k ] dn [ i u, k ] cn [ i u ′ , k ] (cid:17) (1 + 2 cos[2 p ] cos[2 q ]) − dn [ i u ′ , k ] cn [ i u ′ , k ] cos[2( p + q )] − ( k sn[ i u ′ , k ]) cos[2( p − q )] ) . Here there is redefinition of the parameter u of the Eq.(2.15), u ′ = λ + u . When k = 0 XY model goes to the XX model,Det[A xx ( p, q )] = 2(cos[ u ′ ]) (4.24) × n u ′ ]) cos[2 p ] cos[2 q ] − cos[2( p + q )] o . As we can see this expression goes to the value 0, when p = − q = π , for all the values of parameter u ′ , which isa hint of the known fact , that the region − ≤ ∆ ≤ XY Z model.
C. Continuum limit: IM
At the critical line, which is described by the param-eters J c and J c , correlation length of the system goesto infinity. All the relevant distances become large atcriticality and it is natural at that limit to be interestedin large distances compared to the lattice constant. Itis well known that in the continuum limit, at the pointof the second order phase transition, 2DIM is describedby free massless Majorana fermions. Below, for achiev-ing it, we are going to expand the action near the point J = J = J c (considering for simplicity homogeneouscase) with small values of momenta p, q .Diagonalization of matrix (4.16) brings the action tothe form − A ( ¯ ψ, ψ ) = X p,q E k ( p, q ) ¯ ψ ′ k ( p, q ) ψ ′ k ( p, q ) , (4.25)with the eigenvalues E k ( p, q ) of A( p, q ) being E ( p, q ) = e − e − e , E ( p, q ) = e − e + e , E ( p, q ) = e + e − e , E ( p, q ) = e + e + e , (4.26) e = c (cos [2 p ]+cos [2 q ]) − , e = (cid:16) c (cos [2 p ] − cos [2 q ]) − ( d sin [ p − q ]) − ( b sin [ p + q ]) (cid:17) , e = (cid:16) c (cos [4 p ]+cos [4 q ] −
2) + ( d cos [ p − q ]) +( b cos [ p + q ]) − c (cos [2 p ]+cos [2 q ]) e (cid:1) , e = (cid:16) c (cos [4 p ]+cos [4 q ] −
2) + ( d cos [ p − q ]) + ( b cos [ p + q ]) + c (cos [2 p ]+cos [2 q ]) e (cid:1) . We see from Eq. (4.26) that at the critical value of cou-pling J , J = J c , and at the momenta { p, q } = { , } (or { p, q } = { , π } ) two eigenvalues E ( p, q ) and E ( p, q ) be-come 0, whereas the remaining two eigenvalues E ( p, q )and E ( p, q ) take the value − . As we have mentioned,taking the continuum limit at the point of second or-der phase transition is justified, as the lattice constantcan be neglected compared to the correlation length,and the latter is proportional to the inverse of mass.Thus, we expand the action for the massless fermions¯ ψ ′ k ( p, q ) , ψ ′ k ( p, q ), k = 2 ,
4, at the critical point. Ex-pansion of the eigenvalues gives E ( p, q ) = √ J − J c ) − p − p − q , (4.27) E ( p, q ) = √ J − J c ) + p − p − q . After a linear transformation of the field variables ψ ′ ( p, q ) and ψ ′ ( p, q ), the sum (cid:16) √ J − J c ) − p − p − q (cid:17) ¯ ψ ′ ( p, q ) ψ ′ ( p, q )+ (cid:16) √ J − J c ) + p − p − q (cid:17) ¯ ψ ′ ( p, q ) ψ ′ ( p, q )takes the form (cid:0) ¯ ψ + ( p, q ) , ¯ ψ − ( p, q ) (cid:1) (cid:18) m iq − piq + p m (cid:19) (cid:18) ψ + ( p, q ) ψ − ( p, q ) (cid:19) , (4.28) m = √ J − J c ) . Continuum action of 2DIM can be conveniently writtenupon introducing two-dimensional gamma matrices γ = σ and γ = iσ , as − A ( ¯ ψ, ψ ) = Z ¯ ψ ( p, q ) ( m − iγ µ p µ ) ψ ( p, q ) . (4.29)Here ψ ( p, q ) = (cid:18) ψ + ( p, q ) ψ − ( p, q ) (cid:19) , ¯ ψ ( p, q ) = (cid:0) ¯ ψ + ( p, q ) ¯ ψ − ( p, q ) (cid:1) , p = iq and p = p . D. Thermal capacity
Determinant representation of partition function(4.18) leads to the following expression for free energy, F = − ( T ln Z ) /N , per site: F = − T /N P p,q ln 2 { J ] cosh[2 J ]) +(sinh[2 J ] sinh[2 J ]) (4.30) − J ] sinh[2 J ](cos[2 p ]+cos[2 q ]) − (sinh[2 J ]) cos[2( p + q )] − (sinh[2 J ]) cos[2( p − q )] } . The thermal capacity is related to the second derivativeof the free energy with respect to the temperature asfollows: C = − T ∂ F∂T . (4.31)In order to obtain the temperature dependence of thefree energy, one has to replace parameters ( J , J ) with( J /T, J /T ). Then the result for thermal capacity C follows upon performing this replacement in Eq. (4.31)and substituting F into Eq. (4.31). The result has asimple form in homogeneous case, J = J = J , andreads C = 4 1 N (cid:18) JT (cid:19) N,N/ X p,q (4.32) ( JT ] + cosh[8 JT ] − p ] cos[ q ]) cosh[4 JT ](cosh[2 JT ]) − p ] cos[ q ] sinh[2 JT ]) − (1 + cosh[4 JT ] − p ] cos[ q ]) ) sinh[4 JT ](cosh[2 JT ]) − p ] cos[ q ]) ! . In the thermodynamic limit, N → ∞ , the sum in Eq.(4.32) should be replaced by the integral as1 N N/ ,N X p, q → π Z π/ Z π dp dq. (4.33)Then, after performing the integration, we obtain C = 4 π (cid:18) JT csch h JT i(cid:19) ( − (cid:18) cosh h JT i(cid:19) (4.34) × π + (cid:16) h JT i(cid:17) E " (cid:18) sech h JT i tanh h JT i(cid:19) + (cid:18)
15 + cosh h JT i(cid:19) K " (cid:18) sech h JT i tanh h JT i(cid:19) . Here the functions E and K are the complete elliptic in-tegrals of the second and the first kinds. Equation (4.34)reproduces the expression for the thermal capacity ob-tained from Onsager’s solution . The consequence ofthe factorization property of the determinants in expres-sion Eq. (4.18) for the partition function,Det[A( p, q )](1 + cosh[2 J ] cosh[2 J ]) / J ] cosh[2 J ] − cos[ p + q ] sinh[2 J ] − cos[ p − q ] sinh[2 J ]) × (cosh[2 J ] cosh[2 J ]+cos[ p + q ] sinh[2 J ]+cos[ p − q ] sinh[2 J ]) , . Note, thatthe first and the second terms in the product on right-hand side of Eq. (4.35) differ only by shifts π − ¯ p and π − ¯ q , where ¯ p = p + q and ¯ q = p − q . Therefore, ex-pression (4.18) for the partition function can be writtenas a product of the first terms in Eq. (4.35) only, where¯ p = p + q and ¯ q = p − q take values in the interval fromzero to π . V. CORRELATION FUNCTIONS: IM, B = 0 Fermionic approach formulated above is very conve-nient for calculation of correlation functions and sponta-neous magnetization. Let us first analyze vacuum expec-tation value of the spin variable, ¯ σ α , h ¯ σ α ( i, j ) i = 1 Z X { ¯ σ } { ¯ σ α ( i, j ) Y k,r W α ′′ β ′′ α ′ β ′ ( k, r ) } . (5.1)Here, as in the beginning, ¯ σ α ( i, j ) are classical spin vari-ables attached to the vertex ( i, j ).Our recipe for further evaluation is simple. For calcu-lation of the average of any quantity, say ¯ g ( { ¯ σ α ( i, j ) } ),first we represent it in the spin operator form (as itwas done for the Boltzmann weights in Sec. I) as afunction of Pauli operators, g ( { σ k ( i, j ) } ). Then we de-termine corresponding fermionic realization of g in thenormal ordered form N (cid:2) g f (cid:0) { c + ( i, j ) , c ( i, j ) } (cid:1)(cid:3) . Aver-age h ¯ g ( { ¯ σ α ( i, j ) } ) i then will be equivalent to the Green’sfunction h N [ g f ( { ¯ ψ ( i, j ) , ψ ( i, j ) } )] i in the correspondingfermionic field theory with local quadratic action (4.7)on the lattice.The average of a spin variable in the Eq. (5.1) can beexpressed via operator forms of Boltzmann weights (2.5)and R -matrices (2.6), h ¯ σ α ( i, j ) i = Z tr Q { k, r>j } W ( k, r ) Q { k>i } W ( k, j ) σ z ( i, j ) Q { k, r ≤ j } W ( k, r ) Q { k ≤ i } W ( k, j )= Z tr Q { k, r>j } R ( k, r ) Q { k>i } R ( k, j ) σ ( i, j ) Q { k, r ≤ j } R ( k, r ) Q { k ≤ i } R ( k, j ) . (5.2)Here the trace is understood as the composition of tr a defined in Eqs. (2.18) and (2.19): tr = tr tr . Bytaking into account Jordan-Wigner non-local operator, J = Q (1 − n ) (for details, see the Appendix), we canrepresent the single spin operators on the lattice viafermionic creation-annihilation operators, σ ( i , 2 j ) i = [ c ( i , 2 j ) + c + ( i , 2 j )] J ( i, j ) , (5.3) σ ( i +1, 2 j +1 ) i = [ c ( i +1, 2 j +1 )+ c + ( i +1, 2 j +1 )] J ( i, j ) , (5.4) J ( i, j ) = Y kj [1 − n (
0, 2 r +2 )] . (5.5) For a finite lattice the expectation value given by Eq.(5.2) always acquires value 0 due to the Z symmetry ofthe model. In fermionic approach this is quite apparent,as it corresponds to an integration of a polynomial overodd Grassmann variables (see Eqs. (5.3) and (5.4)), whilethe integration goes by even number of variables, Eq.(4.4). The case of infinite lattice will be specified in thenext section.Now it is convenient to rewrite operators (1 − n ) as1 − n = ( c + + c )( c + − c ) , (5.6)which brings expressions (5.3) and (5.4) to the form( c + + c ) ( Q ( c + + c )( c + − c )). Then we insert the re-sulting formulas of Eqs. (5.3) and (5.4) into Eq. (5.2).In the previous section we included fermionic fields foreach R operator locally (or for each dashed square inthe lattice on Fig. 2), later represented them in thenormal ordered form and finally switched to the coher-ent basis. In order to escape complications in the fur-ther calculations, we shall always attach ”even-even”[ c + (2 i, j ) ± c (2 i, j )] fermionic operators to R ( i, j ) ma-trix, [Fig. 3 a], and the ”odd-odd” fermionic operators[ c + (2 k + 1 , r + 1) ± c (2 k + 1 , r + 1)] to R ( k, r )-matrix[Fig. 3 b]. In Fig. 3 operators ( c + ± c ) are shown bylarge circles on the vertices. This choice, which of coursewill not affect the result of the calculation of expectationvalues, has a simple explanation. ❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅❅(cid:0)(cid:0)(cid:0) ❅❅❅ (cid:0)(cid:0)(cid:0)❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅❅(cid:0)(cid:0)(cid:0) ❅❅❅ (cid:0)(cid:0)(cid:0) ✉ ✉❡❡tt❡❡tt✉ ✲ ❤ ( ) R(i,j)R(i-1,j) a ❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅❅(cid:0)(cid:0)(cid:0) ❅❅❅ (cid:0)(cid:0)(cid:0)❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅❅(cid:0)(cid:0)(cid:0) ❅❅❅ (cid:0)(cid:0)(cid:0) ✉ ✉ ❄ ❡t❤ ( ) ✉ ✉❡❡tt R(k,r)R(k,r+1) b FIG. 3: (a) R ( i, j )[ c + (2 i, j ) ± c (2 i, j )](b) [ c + (2 k + 1 , r + 1) ± c (2 k + 1 , r + 1)] R ( i, j ) Let us consider normal ordered forms of the operators R ( i, j )[ c + (2 i, j ) ± c (2 i, j )] and [ c + (2 k +1 , r +1) ± c (2 k +1 , r + 1)] R ( i, j ), where R is the fermionic R operatorgiven by Eq. (3.2). They are particular cases of a generalexpression R : [ x c + x c + x c +1 + x c +2 ] e A ( c +1 , c +2 , c , c ) : , (5.7)with different choice of x , x , x and x depending onparameters b, c, d [see Eq. (3.6)]. A is defined by Eq.(3.5).1While for operators [ c + (2 i, j ) ± c (2 i, j )] R ( i − , j )and R ( i, j + 1)[ c + (2 k + 1 , r + 1) ± c (2 k + 1 , r + 1)],fermionic normal ordered form belongs to the followinggeneral expression R : [ x ( c + c +1 + c + c +2 ) + ( x c + x c +1 ) c +2 c +( x c + x c +2 ) c +1 c ] e A ( c +1 , c +2 , c , c ) : . (5.8)Eqs. (5.7) and (5.8) show that in the latter case we haveadditional powers of Grassmann fields.In expressions of the two-point operators σ ( i, j ) σ ( k, r ) some of (1 − n ) operators coincideand cancel each other due to Eq. (A.5). The remainingoperators between two points, ( i, j ) and ( k, r ), form apath, which can be deformed using feature (A.16). Forexample, if i > i , j > j , we have σ (2 i +1 , j +1) σ (2 i , j ) (5.9)= [ c (2 i +1 , j +1)+ c + (2 i +1 , j +1)] × [1 − n (2 i +1 , j +1)] Q j r = j +1 [1 − n (2 i +2 , r )] × Q i − k = i +1 [1 − n (2 k +1 , j +1)][ c (2 i , j )+ c + (2 i , j )]= [ c + (2 i +1 , j +1) − c (2 i +1 , j +1)] × Q j r = j +1 [1 − n (2 i +2 , r )] × Q i − k = i +1 [1 − n (2 k +1 , j +1)][ c (2 i , j )+ c + (2 i , j )] . Making use of expression (5.6), we bring the correlationfunctions h σ ( i , j ) σ ( i , j ) i to the form h [ c + − c ] hY ( c + + c )( c + − c ) i [ c + + c ] i . (5.10)With the help of the Wick’s theorem, we can representaverage (5.10) in terms of the Pfaffian form with elements h [ c + ( i, j ) ± c ( i, j )][ c + ( k, r ) ± c ( k, r )] i . Let us consider in details the cases, when two spins arearranged along a direct line on the lattice, in horizontalor in vertical directions. Since there is a translationalinvariance in both directions we shall restrict ourselves bytwo cases: h σ (0 , σ (0 , k ) i and h σ (2 k + 1 , σ (1 , i .For the vertically arranged spins we have G ( k ) ≡ h σ (0 , k ) σ (0 , i = h [ c + (0 , k )+ c (0 , k )] × Q k − r =0 [1 − n (0 , r )][ c + (0 , c (0 , i = h [ c + (0 , k )+ c (0 , k )] × (cid:16)Q k − r =1 [ c (0 , r ) − c + (0 , r )][ c + (0 , r )+ c (0 , r )] (cid:17) × [ c (0 , − c + (0 , i . (5.11)Wick’s rules allow representing the last expression in Eq.(5.11) as a square root of a determinant, and hence G ( k )has the following representation by means of a Gaussianpath integral: G ( k ) = Z Dχ e P ki,j =1 G ij χ ( i ) χ ( j ) − P k − i =0 χ (2 i +1) χ (2 i ) . (5.12) Here χ ( i )’s are Grassmann variables. Antisymmetric ma-trix elements G ij are defined as G i +1 2 j +1 = h [ c + (0 , i ) − c (0 , i )][ c + (0 , j ) − c (0 , j )] i , G i j = h [ c + (0 , i )+ c (0 , i )][ c + (0 , j )+ c (0 , j )] i , (5.13) G i j +1 = h [ c + (0 , i )+ c (0 , i )][ c (0 , i ) − c + (0 , j )] i . All the expressions in (5.13) can be easily derived in thebasis of coherent-states (4.1). As it was stated earlier, thenormal ordered form of the operator R ( i, j )[ c + (2 i, j ) ± c (2 i, j )] has form (5.7). The parameters in this case aregiven by { x , x , x , x } = {± , d, c, b } . (5.14)Let { x ′ , x ′ , x ′ , x ′ } and { x ′′ , x ′′ , x ′′ , x ′′ } be the parame-ters corresponding to the operator ( c + ± c ) at (0 , i ) and(0 , j ) points, respectively. Then we can rewrite all theexpressions in Eq. (5.13) as a general function of theseparameters, namely G ( i, j, { x ′ } , { x ′′ } ), which has the fol-lowing integral form in coherent-state basis: G ( i, j, { x ′ } , { x ′′ } ) = ( R ) N Z R D ¯ ψDψ e A ( ¯ ψ,ψ ) (5.15) × (cid:2) x ′ ψ (0 , i )+ x ′ ψ (1 , i −
1) + x ′ ¯ ψ (2 , i )+ x ′ ¯ ψ (1 , i +1) (cid:3) × (cid:2) x ′′ ψ (0 , j )+ x ′′ ψ (1 , j −
1) + x ′′ ¯ ψ (2 , j )+ x ′′ ¯ ψ (1 , j +1) (cid:3) , where A and Z are defined in Eqs. (4.7) and (4.6).The second sum ( − P k − i =0 χ (2 i + 1) χ (2 i ) ) in Eq.(5.12), and hence the additional unity elements with G i +1 2 i ( −G i, i +1 ), are conditioned by the normal or-dered version of relation 1 − n = ( c + + c )( c + − c ), i.e.: 1 − n := 1+ : ( c + + c )( c + − c ) :.Then straightforward calculations lead to the followingexpression for G ( r, j, { x ′ } , { x ′′ } ): G ( r, j, { x ′ } , { x ′′ } ) = 1 /N N/ ,N X n =1 ,n =1 (cid:26)(cid:18) K k x ′ x ′′ + kK x ′ x ′′ − kK x ′ x ′′ − K k x ′ x ′′ (cid:19) (A − ) + (cid:18) K x ′ x ′′ − kk K x ′ x ′′ + K x ′ x ′′ − kk K x ′ x ′′ (cid:19) (A − ) + (cid:18) K k x ′ x ′′ + k K x ′ x ′′ − k K x ′ x ′′ − K k x ′ x ′′ (cid:19) (A − ) + (cid:18) K x ′ x ′′ − K kk x ′ x ′′ + 1 K x ′ x ′′ − K kk x ′ x ′′ (cid:19) (A − ) + (cid:18) K − K (cid:19) (cid:16) ( x ′ x ′′ − x ′ x ′′ )(A − ) + ( x ′ x ′′ − x ′ x ′′ )(A − ) (cid:17) + k (cid:18) Kx ′ x ′′ − x ′ x ′′ K (cid:19) (A − ) + k (cid:18) Kx ′ x ′′ − x ′ x ′′ K (cid:19) (A − ) +1 k (cid:18) x ′′ x ′ K − Kx ′ x ′′ (cid:19) (A − ) + 1 k (cid:18) x ′ x ′′ K − Kx ′ x ′′ (cid:19) (A − ) (cid:27) (cid:12)(cid:12)(cid:12) p =2 π n N ,q =2 π n N . (5.16)2Here A is the 4 × K = e i (2 rp +2 jq ) , K = e i ((2 r +1) p +(2 j +1) q ) , (5.17) k = e i p , k = e i q , r, j = 1 · · · N. Similar expressions can be obtained for the horizon-tally arranged spins too, G ′ ( k ) ≡ h σ (2 k +1 , σ (1 , i = (5.18)= h [ c + (2 k +1 , − c (2 k +1 , × k − Y r =1 [1 − n (2 r +1 , c + (1 ,
1) + c (1 , i , which also admits integral representation (5.12), in thiscase with the matrix elements G i +1 , j +1 = h [ c + (2 i +1 , − c (2 i +1 , c + (2 j +1 , − c (2 j +1 , i , G i, j = h [ c + (2 i +1 , c (2 i +1 , c + (2 j +1 , c (2 j +1 , i , G i, j +1 = h [ c + (2 i +1 , − c (2 i +1 , c + (2 j +1 , c (2 j +1 , i . (5.19)The elements given by Eq. (5.19) can also be ex-pressed by the function G ( i, j, { x ′ } , { x ′′ } ) [Eqs. (5.15)and (5.16)], but here the parameters defined by the nor-mal ordered form (5.7) of the operator [ c + (2 i + 1 , j +1) ± c (2 i + 1 , j + 1)] R ( i, j ) read { x , x , x , x } = { b, c, d, ± } . (5.20)The above relations enable us to find correlation func-tions for all the statistical models with weights, whichcan be written in the matrix form (3.10), with condition(2.9), letting R = R , R = R . Inserting the parameters { x ′ } , { x ′′ } and theelements of the inverse matrix A − defined for the 2DIM(3.3) into Eq. (5.16), we find the following expression forEq. (5.19) G i j = G i +1 2 j +1 = − N N/ − X n =1 N − X n =1 sin[2( i − j ) p ] a Det[ A ( p, q )] × n cosh [2 J ] sin [2( p − q )] − cosh [2 J ] sin [2( p + q )]+2 sin [2 q ](cos [2 p ] − J ] sinh [ J ]) o , G k r +1 = −G r +1 2 k = 2 N N/ − X n =1 N − X n =1 a Det[ A ( p, q )] × n cos [2( r − k ) p ] “ J ] cosh [2 J ] − J ] cosh [ J ]+2 cos [2 p ] cos [2 q ] − cos [2( p + q )] cosh [2 J ] − cos [2( p − q )] cosh [2 J ] − p ]+cos [2 q ]) sinh [ J ] sinh [ J ] ” +cos [2( r − k − p ] sinh [2 J ] sinh [2 J ] o , (5.21) p = π N (2 n +1) , q = π N (2 n +1) . The elements G ij in Eq. (5.13) for the vertical casecan be obtained from expressions (5.21) simply by inter-changing the coupling constants J and J .In the homogeneous case J = J = J , we have G i j =0 and G i +1 2 j +1 = 0, and the expression for G ( i ) [ G ′ ( i )]simplifies to the determinant G ( i ) = Det ¯ G i ¯ G i · · · ¯ G ii − ¯ G i − ¯ G i − · · · ¯ G i − i − − G ¯ G − · · · ¯ G i − , (5.22)¯ G k + i k ≡ ¯ G i ≡ G k + i ) 2 k +1 . (5.23)Therefore we can rewrite the Gaussian integral represen-tation (5.12) in the following way: G ( i ) = Z D ¯ χ ′ Dχ e P i,i − k =1 ,r =0 ¯ G k − r ¯ χ ′ k χ r − P i − k =1 ¯ χ ′ k χ k . (5.24)After the replacement ¯ χ ′ k = ¯ χ k − , Eq. (5.24) reads G ( i ) = Z D ¯ χDχ e P i − k,r =0 ¯ G k − r +1 ¯ χ k χ r − P i − k =1 ¯ χ k − χ k . (5.25)Of course, the expressions for correlation functions canbe caught as well from the logarithmic derivatives of thepartition function Z ( B ) [Eqs. (4.8) and (4.11)] with re-spect to inhomogeneous field B ( i, j ), taken at B ( i, j ) = 0. A. Limit of an infinite lattice and large distances.Magnetization
In the limit of an infinite lattice, N → ∞ , one canreplace the sums in Eq. (5.21) by integrals in accordancewith Eq. (4.33). After evaluation of the integral over q ,one will obtain3 G i j +1 = δ i,j + 2 π Z π/ cos [2( i − j − p ] sinh [ J ] sinh [ J ] − cos [2( i − j ) p ] q J ] sinh [2 J ]) − p ] sinh [2 J ] sinh [2 J ] d p, G i j = G i +1 2 j +1 = 0 . (5.26)The last expression in Eq. (5.26) shows, that as in the homogeneous case, on the infinite lattice in the inhomogeneouscase J = J also we can use determinant representation (5.22) instead of Eq. (5.12), G ( i ) = Z D ¯ χDχ e P i − ,i − k,r =0 ¯ G ′ k − r +1 ¯ χ k χ r = Det[ G ′ ( i )] , (5.27)[ G ′ ( i )] k, r ≡ ¯ G ′ k +1 − r , k, r = 0 , i − , with ¯ G ′ k − r +1 = 2 π Z π/ cos [2( k − r − p ] sinh [ J ] sinh [ J ] − cos [2( k − r ) p ] q J ] sinh [2 J ]) − p ] sinh [2 J ] sinh [2 J ] d p. (5.28)It is easy to see that the integral for the matrix elements in Eq. (5.28) can be transformed into the form¯ G ′ n +1 = 2 π Z π/ cos [2( n − p ] sinh [2 J ] sinh [2 J ] − cos [2 np ] q J ] sinh [2 J ]) − p ] sinh [2 J ] sinh [2 J ] d p = 1 π Z π cos [( n − p ] sinh [2 J ] sinh [2 J ] − cos [ np ] q J ] sinh [2 J ]) − p ] sinh [2 J ] sinh [2 J ] d p = 12 π Z π e i np ( e − ip sinh [2 J ] sinh [2 J ] − e − i np ( e ip sinh [2 J ] sinh [2 J ] − p ( e ip sinh [2 J ] sinh [2 J ] − e − ip sinh [2 J ] sinh [2 J ] −
1) d p = 12 π Z π − π e i ( np ) p e − ip sinh [2 J ] sinh [2 J ] − p e ip sinh [2 J ] sinh [2 J ] − p . (5.29)It is well known, that one can investigate the magneti-zation h σ ( i, j ) i by analyzing the large distance asymp-totes of two spin-correlation function on an infinite lat-tice. Namely,( h ¯ σ α ( i, j ) i ) = lim K →∞ ( lim N →∞ h ¯ σ α (0 , σ α ′ ( K, K ) i ) (5.30)= lim K →∞ ( lim N →∞ h ¯ σ α (0 , σ α ′ (0 , K ) i ) , where N is the linear size of the square lattice.In the Ref. it was shown that spin-spin correlationfunctions h ¯ σ α (0 , σ α ′ ( i, i ) i and h ¯ σ α (0 , σ α ′ (0 , i ) i (for T < T c ) have a determinant representation. These corre-lation functions have been represented as a determinantof an i × i matrix, C i , of the Toeplitz type, C i = c c − · · · c − i +1 c c · · · c − i +2 ... ... ... ... c i − c i − · · · c . (5.31) In Eq. (5.31) the matrix elements are given by c n = 12 π Z π dθe − inθ C( e iθ ) , (5.32)C( e iθ ) = (cid:18) (1 − α e iθ )(1 − α e − iθ )(1 − α e − iθ )(1 − α e iθ ) (cid:19) / . For the caselim N →∞ h ¯ σ α (0 , σ α ′ ( i, i ) i = Det[ C i ] , (5.33)( α i )’s are defined as follows: α = 0 , α = (sinh [2 J ] sinh [2 J ]) − . (5.34)Careful analysis of the matrix G ′ ( i ), given by Eq.(5.27), shows, that due to Eqs. (5.28 and 5.29) andafter some rearrangement of its rows, which leave thedeterminant invariant, G ′ ( i ) coincides with C i . Notethat in our notations the coordinate plane on the lat-tice is 45 rotated with respect to the coordinate planein Ref. , so the correlation function of the spins arrangedin the horizontal or vertical lines in our case coincide with h ¯ σ α (0 , σ α ′ ( i, i ) i derived in Ref. .4Now one can follow the technique developed in Ref. ,based on the Szeg¨o’s theorem, and find the solution forthe magnetization. The theorem can be applied when T < T c and directly reproduces the known result for themagnetization originally derived by Yang in article :lim i →∞ h ¯ σ α (0 , σ α ′ (0 , i ) i = lim i →∞ h ¯ σ α (0 , σ α ′ ( i, i ) i (5.35)= (cid:18) (1 − α )(1 − α )(1 − α α ) (cid:19) = (cid:0) − (sinh [2 J ] sinh [2 J ]) − (cid:1) . VI. RELATED ONE-DIMENSIONALQUANTUM PROBLEM
It is possible to connect the partition function ofthe quantum 1DIM to the partition function of two-dimensional classical system (2DIM) using limit (2.20)and the Trotter formula, see Ref. .As it was stated in the second section, the transfermatrix of two-dimensional model, which is defined as aproduct of R matrices, plays a role of the discrete timeevolution operator defined on a 1D chain.In this section we investigate the transfer matrixgiven by Eq. (2.18) and express it via one-dimensionalfermionic fields defined on a chain. By the convention,the trace of the transfer matrix can be connected with thepartition function of the quantum chain model, definedby Hamiltonian operator H ,tr τ = tr e −H . The trace in the definition of the transfer matrix, τ j =tr Q i R ( i, j ), in Eq. (2.18) is taken over the variableswhich have even-even lattice coordinates (denoted byblack circles on the figures); R ( i, j ) matrices are ar-ranged along the horizontal chain with N vertices (whitecircles on the figures). In the following we shall omitthe coordinate indices and will use only indices denot-ing the vertices on the chain. Using R operators repre-sented in terms of fermionic creation-annihilation opera-tors, R ( c +1 , c ; c +2 , c ) [Eqs. (3.2) and (3.11)], one easilycomes to the transfer matrix τ ( { c + n , c n } ) = tr N Y i =1 R ( c +1 , c ; c + i , c i ) . (6.1)We can evaluate the trace in Eq. (6.1) passing to thecoherent basis with Grassman variables for the fermionicoperators c +1 , c and { c + i , c i } . After integration by thevariables corresponding to the operators c +1 , c , we shallarrive at [we have chosen the homogeneous case b = b ′ , c = c ′ , d = d ′ , (3.11)] t ( { ¯ ψ, ψ } ) = Y i h ¯ ψ i | τ ( { c + n , c n } ) Y i | ψ i i (6.2)= ( R N + R N ) e − H ( ¯ ψ i ,ψ i ) , − H ( ¯ ψ i , ψ i ) (6.3)= N X k =1 ( − k c N − k ∆ k (1 + c N ) k X i < ··· j − ( c − ∆ n ) · · · ( c − ∆ n i − )( c − ∆ n j ) · · · ( c − ∆ n N ) i < j Here n i = ¯ ψ i ψ i . Correspondingly the normal orderedexpression of τ ( { c + n , c n } ) is τ ( { c + n , c n } ) = ( R N + R N ) : e − H ( c + i ,c i ) − P i c + i c i : . (6.5)The expression of H in (6.3) simplifies if c = 0. In case of∆ = 0 the function H is a quadratic function and admitsdiagonalization by means of Fourier transformation. IM, XY.
Here we are presenting the transfer matri-ces, which correspond to the free-fermionic cases: ∆ = 0in Eq. (3.11), i.e., IM [Eq. (2.11)] and XY model[Eqs. (2.15 and 4.22)]. Now logarithm of Eq. (6.1) isa quadratic function over N pairs of fermion operators, { c + n , c n } , due to the Eqs. (3.4 and 3.5). After performingFourier transformation for operators c + n , c n in Eq. (6.3),the transfer matrix takes the form τ = ( R ) N (cid:0) − c N (cid:1) : exp { N/ − X p =0 H ( p ) } :; (6.6) H ( p ) = c − b e iπ p +1 N − c e iπ p +1 N + d e − iπ p +1 N − c e − iπ p +1 N ! c + p c p + c − b e − iπ p +1 N − c e − iπ p +1 N + d e iπ p +1 N − c e iπ p +1 N ! c + N − p − c N − p − + 2 i b d sin (cid:2) π p +1 N (cid:3) c − c cos (cid:2) π p +1 N (cid:3) ( c + p c + N − p − + c p c N − p − ) . (6.7)In the course of calculation of the partition functionin Sec. III we have diagonalized this type of quadraticexpression by a simple change of basis (4.14). Recallthat here c + p , c p are not Grassmann variables but ratherfermionic operators and any transformation must keepanticommutation relations . So we distinguish two kindof fermion fields, defined as c α p , α = 1 , c +1 p = c + p , c p = c p , c +2 p = c N − p − , c p = c + N − p − . (6.8)These replacements bring the operator P N/ − p =0 H ( p ) tothe form N/ − X p =0 X α, β =1 , H ′ α β ( p ) c + α p c β p . (6.9)5The task now is to diagonalize the matrix H ′ ( p ) = (cid:18) r ( p ) r ( p ) − r ( p ) − r ( − p ) (cid:19) , (6.10)where r ( p ) = c − b e iπ p +1 N − c e iπ p +1 N + d e − iπ p +1 N − c e − iπ p +1 N ,r ( p ) = 2 i b d sin (cid:2) π p +1 N (cid:3) c − c cos (cid:2) π p +1 N (cid:3) . (6.11)We can represent the transfer matrix given by Eq. (6.6)in the following diagonal form: τ ≈ e P N/ − p =0 ( a ′ + ( p ) c ′ +1 p c ′ p + a ′− ( p ) c ′ +2 p c ′ p ) . (6.12)with the eigenvalues of matrix (6.10), a ′± ( p ) = 12 (cid:16) r ( p ) − r ( − p ) ± p ( r ( p )+ r ( − p )) − r ( p )) (cid:17) . (6.13)Thus, we arrive at a 1D quantum system defined withHamiltonian operator H = − N/ − X p =0 (cid:0) a ′ + ( p ) c ′ +1 p c ′ p + a ′− ( p ) c ′ +2 p c ′ p (cid:1) . (6.14)Particularly, for the IM [where b, c, d are defined as in Eq.(3.6)], in the homogeneous case, J = J = J , we have r ( p ) = r ( − p ) and eigenvalues (6.13) acquire the form a ′± ( p ) = ± p | r ( p ) | + | r ( p ) | . (6.15)The ground state of the system is composed by thenegative-energy modes. In the thermodynamic limit, N → ∞ , the gap between two spectral curves, a ′± ( p ),is found at the Fermi points with momenta 0 , π and isequal to[ a ′ + (0) − a ′− (0)] ≡ r (0) = 2 (cid:18) − sinh[ 2 JT ] (cid:19) (cid:18) JT ] (cid:19) . (6.16)We see that r (0) vanishes at the critical temperature T c of 2DIM, given by sinh[2 J/T c ] = 1, as[ a ′ + (0) − a ′− (0)] ∼ ( T − T c ) , (6.17)demonstrating that at T = T c the 1D system is gaplessand has no massive excitations. Behavior (6.17) holdstrue for the inhomogeneous case J = J also. VII. SUMMARY
In this work we have presented an approach to the in-vestigation of two-dimensional statistical models, basingon the fermionic formulation of the vertex R matrices (Boltzmann weights). If the operator form of the R ma-trix in terms of scalar fermionic creation and annihilationoperators has definite even grading [for XY Z model and2DIM see Eq. (3.2)], then fermionic representation of R ( i, j ) on the lattice acquires local character. If the op-erators have indefinite grading [models in the presenceof an external magnetic field, see Eq. (3.7)], then onemust take into account Jordan-Wigner non-local opera-tor, as in Eq. (4.8), which is discussed in details in theAppendix.For the models under consideration we derive partitionfunctions as continual integrals with corresponding fieldtheoretical actions on the square lattice: Eq. (4.7) givesthe fermionic action corresponding to the general eight-vertex model, which includes both XY Z model and two-dimensional Ising model. Although there is a correspon-dence between 2DIM and XZ models, we straightfor-wardly presented the R matrix of the 2DIM in Eq. (2.22)as a solution of Yang-Baxter equation which ensures theintegrability of the model. For the free-fermionic case thedirect calculation of the partition function and correla-tion functions is performed [Eqs (4.18) and (5.16]. In caseof the 2DIM the continuum limit of the two-dimensionalaction is presented in Eq. (4.29) and the known thermo-dynamic and magnetic characteristics are reproduced [seeEqs. (4.20), (4.34) and (5.35)]. We also consider 2DIM inthe presence of a finite magnetic field and correspondingnonlocal fermionic action is evaluated [Eq. (4.11)].In light of correspondence of two-dimensional classicalstatistical models and one-dimensional quantum modelswe obtain one-dimensional quantum fermionic Hamilto-nian operator (6.3) for eight-vertex model. For free-fermionic cases the Hamiltonian operators are brought tothe diagonal form (6.14), the spectral analysis of whichreflects the critical behavior of the underling models. Acknowledgements
Sh. Kh. thanks the Volkswagen Foundation for thepartial financial support.
Appendix
Jordan-Wigner transformation.
Fermionic represen-tation of spin states naturally introduces grading for bothstates and operators. ¯ σ α , α = 0 , | i , | i fermionic states with zero and onefermions. Single fermion states are anticommuting atdifferent points of the lattice. The same property takesplace for the odd operators in terms of fermionic creationand annihilation operators. This property does not holdfor spin states and operators. Therefore, if one would liketo represent the action of odd number of spin operators σ ( k )1 defined in the space of spins (nongraded space) { ˆ1 (1) ⊗ ˆ1 (2) · · ·⊗ σ ( k )1 ⊗ · · · ⊗ ˆ1 ( n ) } : | α i| α i· · ·| α n i , (A.1)6in terms of fermionic operators ( c + c + ) ( k ) , which acton graded states | α k i , one has to take into account thegraded behavior of all states | α i i , i < k , placed beforethe state | α k i . This can be done with the help of theoperator 1 − n , action of which on the state | α i dependson the parity, p ( α ) = α , as follows:(1 − n ) | α i = ( − p ( α ) | α i . (A.2)Using these operators, one can represent the action of aspin operator σ ( k )1 , as { ˆ1 (1) ⊗ ˆ1 (2) · · · ⊗ σ ( k )1 ⊗ · · · ⊗ ˆ1 ( n ) }⇒ ( c + c + ) ( k ) (1 − c + c ) (1) · · · (1 − c + c ) ( k − . (A.3)This expression constitutes the inverse Jordan-Wignerspin-fermion nonlocal transformation.It is clear that for the product of two odd operators atdifferent points one needs to take into account only thestates between them, {· · · ˆ1 ( i − ⊗ σ ( i )1 ⊗ ˆ1 ( i +1) · · · ˆ1 ( k − ⊗ σ ( k )1 ⊗ ˆ1 ( k +1) · · · }⇒ ( c + c + ) ( i ) Q k − r = i (1 − c + c ) ( r ) ( c + c + ) ( k ) , (A.4)which is a consequence of the property(1 − c + c ) ( i ) (1 − c + c ) ( i ) = 1 . (A.5)Note, that operator (1 − c + c ) is the fermionic form cor-responding to the Pauli matrix σ z . This means that ifwe place the operators σ ( i ) z instead of unity 1 ( i ) in Eq.(A.3) for all i < k , we shall have { σ (1) z ⊗ σ (2) z · · · ⊗ σ ( k )1 ⊗ · · · ⊗ ˆ1 ( n ) } ⇒ ( c + c + ) ( k ) . (A.6)Similarly, we have { . . . ˆ1 ( i − ⊗ ( σ σ z ) ( i ) ⊗ σ ( i +1) z . . .. . . σ ( k − z ⊗ σ ( k )1 ⊗ ˆ1 ( k +1) . . . } (A.7) ⇒ ( c + c + ) ( i ) ( c + c + ) ( k ) . Jordan-Wigner spin-fermion transformation on thetwo-dimensional lattice.
In Sec. II the partition func-tion Eq. (2.19) was defined as an expectation value of theproducts of R operators. These products can be rewrit-ten as Z = X { α i +1 , } i =0 ,N − X { α , j } j =1 ,N (A.8) h Σ | Y j = N Y i = N − R ( i, j ) | Σ i , where the trace is taken over both ”auxiliary” and ”quan-tum” states, | Σ i = | α , N i · · · | α , i| α , i| α , i| α , i · · · | α N − , i . In fermionic representation described in Sec. III, thestates | α i,j i acquire grading and the arrangement in | Σ i becomes significant. Fermionic R operator given by Eq.(3.2) has zero parity, which ensures the local ”fermion-ization” of the partition function: each R operator inEq. (A.8) can be replaced with its fermionic counter-part without any ”tail”. But the formulas of spin-spincorrelation functions contain the spin operator σ ( k, r ),which in the fermionic formulation has odd parity. Fromthe inverse Jordan-Wigner transformation in Eq. (A.4)it follows that the fermionic operator corresponding to σ ( k, r ) should contain non-local operator Q (1 − n ( i, j )),where the product runs over sites ( i, j ), arranged beforethe site ( k, r ).Recall, that the mean value of the operator σ (2 k, r )is defined by h σ (2 k, r ) i = 1 Z X (A.9) h Σ | ( R · · · R ( k, r ) σ (2 k, r ) R ( k − , r ) · · · R ) | Σ i . And due to the conventions adopted in the previous sec-tions, the R ( i, j ) operator acts as R ( i, j ) | α i, j i| α i +1 , j − i (A.10)= R α i +1 , j +1 α i +2 , j +2 α i, j α i +1 , j − | α i +1 , j +1 i| α i +2 , j +2 i , with the matrix elements defined by Eq. (3.3). Thenone can notice, that the action of R operators, placedon the right side of σ (2 k, r ) in the right hand side ofEq. (A.9), on the state | Σ i , transforms it to the followingstate: Y i = k − R ( i, r ) Y j = r Y i = N − R ( i, j ) | Σ i (A.11) ⇒ | α , N i · · · | α , r +2 i| α , r +1 i · · ·| α k − , r +1 i| α k, r i| α k +1 , r − i · · · | α N − , r − i . Hence, according to Eq. (A.3), operator σ (2 i, j ) in itsfermionic formulation reads[ c (2 k, r )+ c + (2 k, r )] Y i = k − (1 − n (2 i +1 , r +1)) × N Y j = r +1 (1 − n (0 , j )) . (A.12)Similarly, in expression for the vacuum average value ofspin operators σ (2 k +1 , r +1), defined at odd-odd sites, h σ (2 k +1 , r +1) i = 1 Z X (A.13) h Σ | ( R · · · R ( k +1 , r ) σ (2 k +1 , r +1) R ( k, r ) · · · R ) | Σ i , the R operators on the right-hand side of σ (2 k +1 , r +1)transform the state | Σ i into Y i = k R ( i, r ) Y j = r Y i = N − R ( i, j ) | Σ i ⇒ (A.14) | α , N i · · · | α , r +2 i| α , r +1 i · · ·| α k +1 , r +1 i| α k +2 , r i| α k +3 , r − i · · · | α N − , r − i , σ (2 k +1 , r + 1) is equipped with the same non-local operatoras Eq. (A.12)[ c (2 k + 1 , r + 1) + c + (2 k + , r + 1)] (A.15) Y i = k − (1 − n (2 i +1 , r +1)) N Y j = r +1 (1 − n (0 , j )) . As an example, consider the spin operators on the ver-tices (5 ,
3) and (6 ,
2) in Fig. 2. There the positions of the1 − n operators are marked by arrows at the correspond-ing sites. If spin operators are placed on the edges of thelattice, σ (2 i + 1 ,
1) and σ (0 , j ), they immediately acton | Σ i and can be replaced by the fermionic operators,[ c (2 i +1 , c + (2 i +1 , N Y r =1 (1 − n (0 , r )) i − Y k =1 (1 − n (2 k +1 , c (0 , j ) + c + (0 , j )] N Y r = j +1 (1 − n (0 , r )) , respectively, in accordance with general expressions inEqs. (A.12) and (A.15). In order to calculate the correlation function h σ ( i, j ) σ ( k, r ) i in the fermionic operator form, it is nec-essary to replace σ ( i, j ) by corresponding fermionic op-erators (A.12) and (A.15). As it is shown in the first partof this section, coinciding operators 1 − n in fermioniccounterparts of σ ( i, j ) , σ ( k, r ) operators cancel eachother and only operators placed on a path, which con-nects points ( i, j ) and ( k, r ), will be left. The choice ofthe path is arbitrary, which is a result of property (A.5),1 − n = σ z and( σ z ⊗ σ z ) R ( σ z ⊗ σ z ) = R , (A.16)with R operator defined in Eq. (2.6).More precisely, a fermionic realization for the prod-uct of two spin operators, σ ( i, j ) and σ ( k, r ), when i < k, j < k , has the form presented in Eq. (5.9).It looks as if one inserts into the vertices on a path be-tween ( i, j ) and ( k, r ) points, operators σ z (= 1 − n ),instead of unity operators in the spin representation andvice versa: it is a simple task to derive the correla-tion function h σ ( i, j ) ( Q σ z ( i ′ , j ′ )) σ ( k, r ) i on the two-dimensional lattice, where the operators σ z are placed ona path of vertices connecting points ( i, j ) and ( k, r ). Itcan be done by replacing operators σ by ( c + c + ) andfinding corresponding Green’s functions [see Eq. (A.7)]. E. Ising, Zs. Phys. , 253 (1925). H. A. Kramers, G. H. Wannier, Phys. Rev. , 252 (1941). L. Onsager, Phys. Rev. , 117 (1944). B. Kaufmann, Phys. Rev. , 1232 (1949). C. N. Yang, Phys. Rev. , 808 (1952). M. Kac, J. Ward, Phys. Rev. 88, p.1332-1337 (1952). K. Huang,
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