Basic statistical models

Transcription

1 Basic statistical models Valery Pokrovsky March 27, 2012 Part I Ising model 1 Definition and the basic roerties The Ising model (IM) was invented by Lenz. His student Ising has found the artition function of the 1d model and tried to solve 2d model but failed. The eact solution of 2d model was found by Onsager in It is one of the fundamental results in statistical hysics. The IM consist of sins σ = ±1 ascribed to sites of a lattice. The energy of the sin configuration is determined by the sum: H = J σ σ ' (1) where the summation is etended to all nearest neighbor airs (bonds) in the lattice. This is the isotroic homogeneous IM. A simlest generalization is the homogeneous, but anisotroic IM. On a square lattice it is defined as follows: H = i, j nn J h σ i, j σ i, j+1 + J v σ i, j σ i+1, j (2) Here (i; j) is the vector number of a site (i denotes the number of line, j denotes the number of column). The IM can be defined on different lattices: square, heagonal, regular triangular, Kagome etc. (see Fig.1) and also on multidimensional lattices. It also can be inhomogeneous, i.e. the couling constants J may be different for different bonds. Fig. 1a. Square lattice Fig. 1b. Heagonal lattice. Fig. 1c. Triangular Fig. 1d. Kagome Lattice 4 nearest neighbors 3 nearest neghbors lattice. 6 nn. 4 nn.

2 In a simlest case of the isotroic homogeneous IM (Eq. 1) the ground state deends on the sign of J. If it is ositive, all the sins in the ground state have the same value, either all +1 or all 1 (u or down). In both cases the energy of the ground state is equal to E G = b J, where b is the total number of bonds. This is ferromagnetic IM (FIM). If J is negative, generally the situation is more comlicated. However it is still simle for lattices that can de divided into two sublattices A and B in such a way that each site of the sublattice A has nearest neighbors belonging to B and vice versa and one sublattice turns into another at a symmetry oeration of the lattice (divisible lattices). The square and heagonal lattices satisfy these conditions, whereas the triangular and Kagome lattices do not. The energy of any configuration of Ising sins on divisible lattices is invariant with resect to changing sign of all couling constants (even in the inhomogenous case) and simultaneous change of sign of all sins in one of the two sublattices. Thus, the sectrum of AFIM coincides with the sectrum of the FIM, but corresonding states differ by the sign of sins of one of sublattices. on-divisible lattices with negative couling constant J have very degenerate ground state with finite entroy. Indeed, considering an elementary triangle in regular triangular lattice, we see that the minimal energy corresonds to two sins arallel each other and one antiarallel to them, i.e. E = b J / 3. If two sins are already chosen to be antiarallel, the third remains free. This freedom gives the entroy S = 1 3 ln2 er site (rove this) contradicting to the ernst theorem. Such a situation is called frustration. The FIM has the long-range order (LRO) at zero temerature. It means that the average value of the sin on a site is not zero. At very high temerature it is disordered. Indeed, consider the correlation function of two sins in one line G l ( ) = σ i, j σ i, j+l. It is monotonously decreasing function of l. At l = 1, it is equal to the average sin on a bond. The Boltzmann distribution gives for the average sin on a bond the value s b = tanh T J T J. Each net ste on the line changes the correlation function by the same factor since the dominant correlation is to the nearest neighbor and even it is very small. Therefore G l ( ) ( T J ) l. It tends to zero when l. Thus, there is the long-range order at zero temerature that vanishes at large temerature. There are two oortunities: the LRO may vanish at any arbitrarily small temerature, though on very large distances, or it can vanish at a finite temerature T c. To establish what haens let us consider thermal ecitations of the ground state. A simlest ecitation is the fli of one sin. Then 4 bond energies change sign for the square lattice (Fig. 2a). The energy of such an ecitation is 8J. A more general ecitation is the sin reversal inside some area A restricted by a contour C of the length L (Fig. 2b). The energy of such ecitation is E L free energy of such a contour is F L length L can be roughly estimated as S L ( ) = 2LJ. The ( ) = E TS( L). The entroy of the contour of the ( ) = L ln 3 assuming that after arriving at some oint the net unit eace of contour can occuy with the equal robability one of 3 ositions (it can not go back). This is obviously overestimated value of entroy since the contour cannot occuy any bond twice or more and the requirement of being closed also in not satisfied. However, the real entroy er cite should not differ too much. Acceting this estimate as a rough aroimation, we see that the free energy turns into zero at a

3 temerature T c = 2 J ln 3 or equivalently 2K c = ln 3, where K = J / T. Eact answer is K c = ln 1+ 2 section). ( ) = ln2.41. It was obtained by Kramers and Wannier in 1940 (see the net Fig. 2a. One sin-fli. Fig. 2b. Domain and domain wall. 2. High Temerature and Low Temerature eansion. Dual transformation. High temerature eansion was roosed by a Dutch mathematician Van der Warden. The artition function for the Ising model reads: Z = e H { σ } T (3) { σ } where the summation roceeds over all ossible configurations { σ } of Ising sins on sites and H { σ } is the value of energy (1) or (2) for the configuration { }. For simlicity we consider the isotroic case and square lattice. Then the σ eq. (3) can be reresented as follows: ( ) Z = e Kσ b, (4) { } σ b where K = J / T, the roduct runs over all bonds and the sin on bond σ b = σ σ ' is the roduct of sins on the sites belonging to the bond. Let reresent the eonent in the latter equation as follows: e( Kσ b ) = cosh K +σ b sinh K = cosh 1+σ b tanh K Plugging this eression in eq. (4), one arrives at the eansion in owers of z = tanh K: ( ) (5) Z = 2 s ( cosh K ) b b q l z l, (6) where b is the number of bonds, s = b /2 is the number of sites (for square lattice); the coefficients q l are average values of corresonding clusters of l bonds: l=0

4 q l = 2 s σ b1 σ b2...σ bl (7) { σ } [ b 1,b 2...b l ] The second summation in equation (7) runs over all ossible sets of non-coinciding bonds, but after summation over all ossible configurations of sins non-zero results give only sets in which bonds form closed contours. The reason is that each sin on site must enter zero, two or four times, otherwise summation over oosite rojection of this sin gives zero. Thus, the coefficient q l is equal to the number c ( l) of different closed contours on the square lattice (including disconnected ones) of the length l. ow Eq. (6) can be rewritten in the following form: ( ) b Z ( K ) = 2 cosh K b c ( l)z l, (8) l=0 An imortant conclusion follows from Eqs. (6,8): there is no hase transition in any finite system. Indeed, the factor ( 2 cosh K ) b is a regular function of K at any real K. The remaining sum is a olynomial of a finite ower b and therefore also is a regular function of K. A singularity can aear only in the so-called thermodynamic limit b. ow the construction of low-temerature eansion is in order. As we established earlier, the ecitations are sin flis inside some finite area restricted by a closed contour of the length l. The energy of such ecitation is E (l) = 2Jl. Thus, the artition function can be written as follows: ( ) = e bk Z K b c ( l)ζ l, (9) l=0 where ζ = e( 2K ). ote that the domain walls are contours made from the bonds of the lattice dual to the initial one, whose sites are centers of elementary cells of the initial lattice (in 2 dimensions). In the considered case of a simle square lattice the dual lattice is again a simle square lattice. Therefore, the numbers c ( l) in equations (8) and (9) are identical. But it is not the case for the triangular lattice, whose dual lattice is heagonal (honeycombs). Let introduce a dual variable!k by the relationshi: e 2K Comaring equations (8) and (9), one finds: ( ) = tanh! K (10) Z ( K ) = Z (!K )( 2 cosh!k ) b e bk (11) The relation (10) transforms low-temerature (large K) into a high temerature (small K! ) and vice versa. It is easy to check that the relation (10) is mutual: e( 2!K ) = tanh K. Its elicitly symmetric form reads (derive all that): sinh2k sinh2!k = 1 (12) Kramers and Wannier assumed that there eists only one hase transition. Let the artition function Z(K) in the thermodynamic limit has only one singularity at K=K c. The equation (11) imlies K c =! K c and from eq. (12) one finds sinh2k c = 1 or equivalently

5 2K c = ln( 1+ 2 ). Modification for the anisotroic IM is straightforward: the closed contours must be characterized by their horizontal and vertical lengths l h and l v and horizontal and vertical high-temerature variables z h;v = tanhk h;v and low-temerature variables ( ), ζ v = e( 2K h ) (note that the vertical bond of a domain wall searates ζ h = e 2K v the sins on the horizontal bond of the initial lattice and vice versa). Thus, the duality relation that relaces (10) on anisotroic lattice reads: e( 2K h ) = tanh K! v (13) The second relation e( 2K v ) = tanh! K h follows from the first one eq. (13) and mutuality of the dual equation (10). Again, the assumtion that there eists only one hase transition imlies K h,v =! K h,v. The hase transitions roceed at the line in the lane (K h, K v ) determined by equation: sinh2k h sinh2k v = 1 (14) At fied values of J h, J v the transition takes lace at the crossing oint of the curve (14) and the straight line K v J h K h J v = 0 (Fig. 3). The dual transformation as in the isotroic case connects the values of artition function at low and high temeratures: Z ( K h,k v ) = e vk h + h K v ( 2 cosh K! h ) h ( 2 cosh K! v ) Z K! v h, K! v ( ) (15) Emloying the duality relations (13) and an analogous one with h and v ermuted it is ossible to rewrite eq. (15 in symmetric form: ( sinh2 K! h ) h /4 ( sinh2 Kv! ) v /4 Z ( Kh,K v ) = ( sinh2k h ) v /4 ( sinh2k v ) h /4 Z K! h, K! v In the net section the dual transformation will be derived in a more general contet. ( ) (16) Fig. 3. Phase transition line in the lane K h -K v according to duality relation. 3 Dual transformation and self-dual models Here we consider the dual transformation in a class of lattice statistical models with the symmetry Z or C of integers n with resect to addition in mod or the root ω n = e( i 2πn ) of the -th ower from 1 with resect to multilication. They are called clock models. The statistical weight W b of a lattice bond b deends on the variable ω b

6 that is equal to the roduct of ω ω ' on the sites belonging to the bond. Equivalently, W b deends on the variable n b = n n ' that is a discrete gradient field. ote that W b (n b ) may be different functions at different bonds. The IM is a secial case of the clock model with = 2. The artition function of the clock model can be reresented as follows: Z = W b ( n b ) δ n b' b (17) { } n b The summation in Eq. (16) runs over all ossible values of the variables n b indeendently for each bond b with the statistical weight W b ( n b ). The inde b at the statistical weight indicates that it may be different for different bonds. The δ-functions enforce the circulation of the discrete vector field n b to be zero on each elementary laquette and enable indeendent summation over all n b. The centers of laquetts form a lattice dual to the initial one (Fig. 4). For a square lattice the dual lattice is also a square lattice. But the lattice dual to a heagonal one is regular triangular. It is obvious that duality of lattices is a mutual roerty. Each bond b of the initial lattice crosses only one bond b! of the dual lattice (see Fig. 4). These bonds can be considered as dual each other. b' Fig. 4. Dual lattices, bonds, laquettes The urose of the following calculation is to show that the artition function (16) can be reresented as analogous artition function over the dual lattice. For this urose it is necessary to use the elicit reresentation of delta-function over the modulus : δ ( k) = 1 1 e i 2πkm (18) Let us substitute Eq. (17) into the eression for Z (16) with k = m=0 n b' b' for a laquette and m = m different and indeendent for each laquette. First we erform summation over n b corresonding to a fied bond b. The sum reads: 1 1 W b ( n b )e i 2πn m b b! =! m=0 ( ), W!b m!b

7 (19) where m!b is a dual to n b variable on a dual bond! b determined as m!b = m ' m if the dual bond goes from the center of laquette to the center '. We see that the dually transformed statistical weight W!b m!b n b weight W b after the dual transformation we find: ( ) is the Fourier transform of the initial statistical ( ) and that the dual bond vector field m!b is again a discrete gradient. Thus, Z = W! ( m!b ) δ m!b'!b'! (20) { m!b }! This transformation establishes statistical equivalence of two generally different models on different lattices. However, in some cases the dual statistical weights are the same function of the bond sin and differ only by arameters. If simultaneously the dual lattices are of the same tye, the models are called self-dual. A simlest eamle of the self-dual model is the so-called -state Potts model on a square lattice. The Potts model is defined as a clock model with the statistical weight on the bond W ( n) = Aδ n,0 + B where A and B are some constants. Its Fourier-transform is W! m ( ) = A + Bδ m,0. Introducing a ratio R = A / B as a characteristic of the statistical weight, we find R!R =. The hase transition roceeds at R =!R =. In the case of the isotroic IM ( = 2), n = 0 corresonds to arallel sins and n = 1 to antiarallel. Therefore, A + B = e K ; B = e K ; R = A / B = e 2K 1. The duality relation reads e 2K 1 ( ) = 2 ( ) e 2!K 1 that is equivalent to e 2K = tanh!k. Please, analyze the duality relation for = 3 ositional clock model. ote that in 2 dimensions the model dual to a model of vector field on bonds belongs to the same class. In 3 dimensions such a dual model has variables from the grou Z ascribed to a laquette erendicular to the initial bond, i.e. quite different toology. 4 Ising model with magnetic field. Lattice gas. The magnetic field h can be introduced into the Ising model so that the total Hamiltonian becomes: H = J σ σ ' h σ (21) b At non-zero magnetic field h the hase transition disaears even in the thermodynamic limit. The reason is that the magnetization aears at any temerature as a result of violation of the Z 2 symmetry by magnetic field. At zero magnetic field and T < T c the magnetization takes one of two equilibrium values ±m T ( ) that runs from 0 at T = T c to ±1at T = 0. The hase transition roceeds only at h = 0. The variable σ = ±1 can be eressed in terms of Z 2 variable n = 0; 1 as σ = 1 2n. Then the Hamiltonian (20) turns into the Hamiltonian for the lattice gas:

8 H LG = U n n ' µ n, (22) b where U = 4J and µ = 4J + h. The model can be interreted as a model of articles with a short-range interaction (they interact only on nearest neighbor sites). Each site can be filled with a article (n = 1) or emty (n = 0). The number of articles is conserved. The value µ is the chemical otential of the gas. The hase transition roceeds at T = T c and <n >= n c = 1/2 that corresonds to zero magnetization of the IM. At T < T c the average density takes one of two equilibrium values n( T ) = n c ± m( T ) / 2. The smaller of them can be interreted as the density of a gas, whereas the larger should be the density of a liquid in equilibrium with the gas. Thus, the line n = 1/2, T < T c is the line of the first order hase transitions or liquid-gas equilibrium. Its uer oint T = T c, n = n c = 1/2 is the critical oint liquid-gas. The model of Lattice gas ossesses an additional symmetry of reflection with resect to n = 1/2 that the real gas-liquid system does not have. evertheless, it reroduces correctly the critical behavior near the critical oint due to universality. It means that real critical oint has the asymtotic symmetry of the IM. 5 Transfer-matri. Kramers and Wannier in the same famous work of 1940 roosed a new aroach to the Ising roblem: the transfer-matri method. The idea is to consider one of directions in the lattice as discrete time. More accurately, let us consider the square IM with M lines and rows and introduce the artition function Z M (Θ) as function of a sin configuration ( ) in the last line. There are 2 indeendent sin configurations Θ. The ( Θ) is related to Z M (Θ) by a Θ = σ 1,σ 2...σ artition function with number of lines larger by one Z M +1 linear equation: Z M +1 (Θ) = T ΘΘ' Z M (Θ') (23) Θ' The matri T ΘΘ' is called transfer-matri. For the anisotroic 2d IM its matri element reads: T ΘΘ' = e K h σ ' ' ' ( j σ j+1 + K v σ j σ j ) (24) j=1 The eriodic boundary condition σ j+ = σ j is used. Reeating this oeration, one finds: Z M = T M Z 0 (25) Identifying configurations Θ 0 = Θ M and erforming summation over them, one arrives at the result: Z = TrT M (26) All matri elements (23) are ositive. They are not symmetric, but can be easily symmetrized by a minor modification of Eq. (23): 1 T ΘΘ' = e 2 K h σ ' ' j σ j ' ( 2 K h σ j σ j+1 + K v σ j σ j ) (27) j=1 A real symmetric matri can be diagonalized by an orthogonal transformation. A theorem

9 of algebra states that such a matri with all ositive matri elements has only ositive eigenvalues. Let the transfer-matri eigenvalues are equal to t 1,t 2...t 2. Eq. (25) can be rewritten as follows: Z = ( t j ) M j=1 2 (28) In the thermodynamic limit M, the contribution of the maimal eigenvalue in this sum is dominant, others can be neglected. Thus, in the thermodynamic limit Z = ( t ma ) M (29) and the Helmholtz free energy reads: F = T ln Z = MT lnt ma (30) The roblem is reduced to calculation of the maimal eigenvalue of the transfermatri. Problem Find the free energy and magnetization of one-dimensional IM in eternal magnetic field. 6 Onsager solution. The transfer- matri (24) can be reresented as roduct of two matrices T = T2T1, where T1 is diagonal matri with matri elements: ( T 1 ) ΘΘ' = δ ΘΘ' e K h σ j σ j+1 j=1 and non-diagonal matri T 2 has matri elements: (31) ' ( T 2 ) ΘΘ' = e K v σ j σ j j=1 (32) The matri T 2 can be reresented as direct roduct T 2 = j=1 ϒ j of matrices of the same form acting in 2-dimensional subsaces R j : where σ j ϒ j = e K v e K v e K v e K v = ek v I + e 2K v σ j ( ), (33) are Pauli matrices acting in the sace R j. Emloying the dual transformation, the same matri can be reresented as ϒ j = rewritten as follows: 2 / sinh2!k h e! K h σ j. Thus, the matri T2 can be 2 T 2 = sinh2!k h e!k h σ j j=1 (34) The matri T 1 acting in the same 2 dimensional sace R 1 R 2... R can be reresented also in terms of the Pauli matrices: /2

10 T 1 = e K h σ z z j σ j+1 j=1 (35) The dual transformation to a new set of Pauli matrices is defined on half-integer (dual) sites as follows: τ j+1/2 = σ z z z j σ j+1 ; τ j+1/2 j = σ k (36) z at a constraint τ 1/2 = 1, interchange T 1 and T 2. Though they do not commute, the artition function (26) being a trace does not change ecet of the reeonential factor in T 2 (34). Thus we find that the dual transformation does not change the artition function with recision of a regular function: ( ) = ( sinh2k h sinh2k v ) M Z K h,k 2 v Z K! h, K! v For further calculation it is convenient to interchange z and in the transfer-matri so that k=1 ( ) (37) T 1 = e K h σ j σ j+1 ; j=1 ( ) /2 z e!k h σ j T 2 = 2sinh2K v j+1 (38) et we aly Wigner-Jordan transformation from sin to Fermi oerators: j 1 a j = 1 z 2 σ k σ + j ; a j = 1 z 2 σ k σ j, (39) k=1 where σ j ± = σ j ± iσ j y. It is easy to check that oerators a j and a j anticommute at different sites and obey a system of anticommuting relations: { a j,a j ' } = 0; a j,a j ' From Eqs. (39) and (40) it follows: σ z j = 1 2a j a j ;σ j σ j+1 j 1 k=1 { } = 0; { a j,a j ' } = δ jj ' (40) z = ( a j + a j )σ j ( a j+1 + a j+1 ) = a j a j ( )( a j+1 + a j+1 ) (41) Plugging these equations into (38), we find the fermion reresentation of the oerators T 1, T 2 : T 1 = e K h T 2 = 2cosh K v It is natural to use the Fourier-transformed oerators j=1 a ( j a j )( a j+1 + a j+1 ) (42) ( ) e 2 K! h a j a j j=1 (43)

11 a = 1/2 e ij a j j ; a = 1/2 e ij z a j j. The eriodic boundary conditions for σ j that are quadratic functions of a j and a j turn into either eriodic or antieriodic boundary conditions for the latter: a j+n = ±a j. The sign deends on the value of the oerator Γ = σ 1 z σ 2 z σ z. This oerator commutes with the transfer-matri since it anticommutes with any a j and a j. Therefore, all eigenstates of the transfer-matri are characterized either by Γ = 1 or Γ= -1. On the other hand, a j+ = σ z 1 σ z 2 σ z z σ +1 we have used the eriodic boundary conditions for σ j,y,z z σ + j σ j + = Γa j. Here and anticommutation of any a j and Γ. Thus, for even states a j+ = a j and for odd states a j+ = a j. The discrete values of momentum are k = π( 2k 1) and k = 2πk for even and odd states, resectively. In both cases the matrices T 1 and T 2 in Fourier-reresentation are factorized to the roduct of matrices T 1 and T 2 ( > 0), where with recision of regular factors: T 1 = e 2K h ( a a + a a 1 )cos + i a ( a + a a )sin { } (44) ( a a + a a ) T 2 = e 2 K! h (45) Thus, the roblem is reduced to the diagonalization of the matrices in a four-dimensional sace sanned onto states 0,0, 1,0, 0,1 and 1,1, where the numbers 0 and 1 denote the fermion occuation numbers in the states and. This roblem is further reduced to diagonalization in 2-dimensional subsaces sanned onto airs of vectors 0,0, 1,1 and 1,0, 0,1 since all oerators in T 1 and T 2 conserve the arity of occuation numbers. For each oerator T 1 and T 2 the states 1,0, 0,1 are eigenstates with the eigenvalue 1. Therefore, only the remaining 2-d subsace sanned on the states 0,0, 1,1 should be considered. In this subsace the oerator a a + a a 1 acts as ( ) acts as σ y. Multilying one of this state by a σ z and the oerator i a a + a a hase factor i, it is ossible to relace σ y by σ. Thus, the equivalent form of the matrices (43,44) reads: T 1 = e 2K h ( σ z cos +σ sin ) = cosh2k + σ h ( cos +σ sin z )sinh2k h (46) or in the matri form: T 2 = e 2!K h σ z ( ) (47) T 1 = cosh2k h + sinh2k h cos sinh2k h sin sinh2k h sin cosh2k h sinh2k h cos (48) e T 2 = 2!K h 0 0 e 2!K h (49) The eigenvalues of the roduct of these two matrices are:

12 where t ± = cosh2k h cosh2! K h sinh2k h sinh2! K h cos ± ε, (50) ( ) 2 ε = cosh2k h cosh2!k h sinh2k h sinh2!k h cos + sinh2k h sin 1/2 (51) It is obvious that t + > 1 > t. Therefore, the maimum eigenvalue of the total transfermatri is: The Helmholtz free energy reads: + t ma = t (52) + F = TM lnt ma = TM lnt (53) In the thermodynamic limit the sum in Eq. (51) turns into integral: F = TM d + lnt 2π π π (54) The integral has singularity at K h =! K h. At this relation fulfilled and = 0, the eression under square root turns into zero. To elucidate the character of singularity, let us estimate t + at Kh!K h K h and 1 : t +! 1+ ( ) 2 + sinh2k h 4 K h " K h Thus, near the singularity the following aroimation is valid: lnt +! τ 2 + sinh2k h ( ) 2 2 (55) ( ) 2, (56) ( ) can be acceted as a temerature deviation from the where the value τ = 2 K h!k h hase transition oint. Then the secific heat near hase transition is: M C = lnτ (57) π sinh2k h This is singular behavior, the famous Onsager s result. Still the hysics behind the singularity is not visible. Who are the fermions laying so imortant role in the roblem also is not clear. The second critical eonent of the magnetization requires even more difficult calculations. Onsager first resented it at a Conference in 1948 without derivation, four years after his large article was ublished and 6 years after his first short communication in Phys. Rev. Letters. C.. Yang ublished the derivation of this second Onsager great achievement in 1952 and it took about 20 ages in a Phys. Rev. article. It shows that a simler aroach with more visible hysics is much areciated. Such an aroach will be resented in the net section.

13 7 Strongly anisotroic IM. Domain walls as fermions. In this section we will show that the fermions can be visualized as domain walls. We will derive their Hamiltonian and obtain the correlation length and singularity of secific heat from it. We also will find the critical eonent (1/8) for the magnetization. In a strongly anisotroic IM ( J h J v ) the energy er unit length of the horizontal domain wall (DW) is much larger than that of the vertical one. Therefore, the DW at temeratures between J h and J h in average have long vertical ieces and short horizontal ones. Since the statistical weight of the unit interval is eonentially small e( 2K v ), the robability of longer horizontal intervals can be neglected. On the other hand, the contribution to the "action" (eonent in statistical weight) of a unit vertical interval in the same range of temeratures is 2K h 1. Therefore, the discretness of vertical coordinate ("time") can be neglected, whereas the discretness of the horizontal coordinate still remains substantial at any temerature ecet of a small vicinity of the critical temerature T c. We will show that it is located in the same range of temerature: J h T c T v. Everywhere in the chosen range of temerature the DW roagate dominantly in the vertical direction. The unit horizontal intervals aear on the average distance e( 2K v ) directed with equal robability left or right. There are no crossings of two DWs in one oint. However, two domain walls that roagate vertically at distance 1 can annihilate and two domain walls can aear at some moment of time" at the unit distance each to other (Fig. 5). Therefore it is ossible to interret the DWs as world lines of fermions whose number is not conserved. The Hamiltonian for these fermions is a generator of "time" evolution. Elementary rocesses included in the evolution on the net ste are shown on Fig. 5. The statistical weight of a ste without dislacement of coordinate is 2K h ; for a ste with horizontal dislacement the statistical weight is e( 2K v ) =!K h. The same statistical weight can be ascribed to the rocesses of creation and annihilation of a air of domain wall lines closing the domain wall loos. Thus, the Hamiltonian of the DW as fermions reads: h H = 2K h a i a i 2K v ( a i+1 + a i 1 )a i + a i+1 a i + a i a i+1 (58) i=1 h i=1 The Hamiltonian (55) can be reduced to a set of indeendent Hamitonians H for airs of oosite momenta by the Fourier transform: a j = 1 a e ij ; a j = 1 a e ij (59) h h

14 Fig. 5. Domain walls as fermions. The closed domain loo is treated as the world line of a fermion. Elmentary stes on trajectory corresond to indicated arts of the Hamiltonian. Then the Hamiltonian H reads: H = 2 K h!k h cos ( )( a a + a a ) + 2i!K h sin a ( a a a ) (60) By a simle hase transformation α e iπ /4 α ; α e iπ /4 α takes the following form: H = 2 K h!k h cos the same Hamiltonian H ( )( a a + a a ) + 2!K h sin a ( a + a a ) (60 ) In order to diagonalize this Hamiltonian we aly the Bogoliubov transformation: a = u α + v α ; a = v α + u α, (61) with real coefficients u,v satisfying the requirements of unitarity u 2 + v 2 = 1 and orthogonality u = u, v = v. Then α,α satisfy canonical anticommutation { } = { α,α ' } = 0; { α,α ' } = δ '. Plugging the Bogoliubov relations: α,α ' transformation (61) into the Hamiltonian (60 ) and requiring the coefficient at α α to be zero, one obtains: ( K h!k h cos )2u v =!K h sin u 2 2 ( v ) (62) Together with the unitarity condition u 2 + v 2 = 1 it leads to the following result: u 2 = ξ ; ε 2 = ξ (63) v ε

15 where ξ = K h K! h cos ; ε = ξ 2 + K! 2 h ( sin ) 2 (64) The Hamiltonian eressed in terms of the transformed oerators reads: The ground state energy E G is equal to H = E G + ε α α (65) ( ) E G = ε ξ (66) In the thermodynamic limit, M the ground state energy reads: E G = π π ( ) (67) d ε ξ 2π The hase transition roceeds at K h =! K h. In the hase transition oint the fermion mass π d 2π π ( ) 1. It turns into zero as K h K! h and the correlation length turns to infinity as K h K! h shows that the critical eonent of the correlation length ν = 1. The free energy can be calculated according to general equation (52) as: F = TME G = TM ( ε ξ ) (68) ear critical oint its singular art behaves as τ 2 ln τ, where τ = K h! K h. Thus, we reroduce the Onsager s result on secific heat singularity. It is straightforward to calculate the correlation function of fermions in critical oint. First of all, note that eactly at critical curve K h =! K h and at small 1 the Bogoliubov coefficients obey the relations u 2 = v 2 = 1 2. Therefore, the simultaneous correlation functions at large distances ' 1 are: a a ' = 1 e i( ') 2 v = 1 2 e i( ') = 1 2 δ ( ' ) (69) a a ' = a ' a = i e i( ' ) u v = i e i( ' ) 2 sign = 1 2π ( ' ) (70) The last result was obtained in the thermodynamic limit that allows transition from summation to integration according to the rule 1 π d = π. Since only small matters the limits in the integral can be relaced by ± and u v by 1 2 sign(). For the calculation of the magnetization critical eonent the key observation is that the roduct of two Ising variables in one line σ σ ' is equal to ( 1) dw(,' ), where dw (, ' ) is the number of domain walls crossing the line (of constant "time") between and. According to our interretation, it is the number of fermions f (, ' ) simultaneously located in the interval of coordinate (, ' ): 2π

16 f The average of the roduct σ σ ' can be written as: ' (, ' ) = a y a y (69) y= σ σ ' = e iπ dw(,' ) Here! dw (, ' ) = dw, ' = e i dw(,' ) e π 2 2 2! dw (, ' ) (70) ( ) dw (, ' ) is the fluctuating art of domain wall number. (, ' ) 1 In equation (70) we have used the law of large numbers assuming that dw and the fluctuation of the number of domain walls is Gaussian distributed. Thus the roblem is reduced to the calculation of the average: 2 dw ' = a! y a yay'! a y' = a y a y' a y a y' a y a y' a y a y' (71) y,y'= ' y,y'= We have used the Wick theorem according to which the average of roduct of Fermioerators can be reresented as sum of all ossible their airing taken with the sign of a ermutation of these oerators that was necessary to ut the oerators from their initial ositions in the roduct to the ositions they took at a airing. Since we are interested in long distances, we relace the summation over y and y' by integration. For the averages we will use their critical oint values (67, 68). The first term in the square brackets of eq. (71) is formally equal to ' dy dy' δ y y' (72) ' ( ) Square of the delta-function is not well-defined. In reality what we consider as δ function is a function that has a shar eak with the width about the lattice constant where our aroimation fails. Therefore, let us ut one of δ functions equal to1/ a, where a is the cut-off length (of the order of lattice constant), and integrate the second δ function. The result is ( ' ) / 4a : Emloying Eq. (68), we find for the integral of the second term: 1 ' ' 1 dy 4π dy' (73) 2 ( y y' ) 2 The integral in Eq. (70) diverges at y = y'. For regularization the integration over y' will be erformed from till y b and from y + b till '. The result reads: 2 (! dw ) = ' 4a 1 ' 2 4π 2 b 1 y 1 ' y (74) The first linear in ' term in eq. (74) is comensated by the term indeendent on y under the integral atb = 2a / π 2. The linear term in the eonent would mean the finite correlation length, whereas we know that the correlation length at the hase transition oint becomes infinite. However, as it is seen from our calculation, the adjustment of a constant at linear term in the eonent roceeds on short distances and therefore is beyond of recision of our long-scale aroimation. In the eact Onsager s solution this

17 adjustment roceeds automatically. The remaining terms after integration give logarithms diverging on the ends of the integration interval. This divergence is surious since and is the result of the failure of continuous aroimation on the scale of the lattice constant. It must be removed by the integration cut-off on the scale of lattice constant (unity). Thus, we find: (! dw (, ' )) 2 = 1 ln ' (75) 2 2π Substituting it into the eression for the sin correlation function (70) we find: σ σ ' = e i dw(,' ) ' 1/4 (76) Thus, we have found a new critical eonent 1/ 4 for decay of the sin correlation in critical oint. However, equation (76) contains also the oscillating eonential factor ( ). Indeed, dw, ' e i dw,' ( ) = a a ( ' ) = n dw ( ' ), where n dw is the domain wall density whose average value remains finite (about 1/3) in the critical oint. In reality these oscillations are surious. They aear as a consequence of alication of the longwave aroimation to the short scale where our aroimation fails. Indeed the oscillations roceed on the scale n dw 1 3. The value n dw with overwhelming robability is irrational. Therefore, the average hase from this wave-vector in the integer oints of the lattice is zero by the modulus 2π. On the other hand, the ower-like decay of the correlation amlitude is a consequence of the long-range fluctuations where the alied aroimation is valid. The function ¼ ln ' is slowly varying and is not sensitive to the discreteness of the lattice. The hysical henomenon behind this henomenon is as follows. Each sin creates near itself an effective field favoring the same olarization of neighbors. It therefore favors a longer domain of arallel sins near itself. 8 Scaling and other critical eonents Since the correlation length turns into infinity in the critical oint, there aears scale invariance or scaling of critical henomena. We have seen already that the correlation length is inverse roortional to the deviation of temerature from critical R c τ 1. We also have seen that the correlation of two sins is roortional to the length in ower -1/4. It means that sin (or magnetization) varies with the length scale as L 1/8. Therefore, at temerature slightly below the hase transition, the sontaneous magnetization m τ varies with temerature as ( τ ) 1/8. This result can be obtained by direct calculation from the domain-fermion reresentation. The sin correlation function can be obtained as we did before, but near the transition oint the fermion correlation functions (67, 68) are no more ower-like. They acquire an additional eonential factor e ' /R c as a consequence of finite mass of relativistic quasi-articles ε m with m = K h K! h Therefore the logarithmic integration now in eq. (74) is effectively cut off not only at short, but also at long distances at the scale R c = m 1. ( ) ( ) = τ.

18 Let us use scaling to calculate the critical behavior of the magnetic suscetibility χ on temerature near hase transition. The magnetic suscetibility is the derivative of magnetization by magnetic field: χ = dm (77) dh From eq. (21) it follows that ( ln Z ) m = σ = T 1 M h Differentiating this equation one more time, we find: χ = T 2 ln Z = 1 M h 2 M (78) σ σ ' (79) In this sum and ' denote oints in 2d sace rather on line. Eactly in the critical oint this sum diverges since the correlation of two sins decays too slowly with distance between them. Beyond the critical oint, but close to it the integration is effectively cut at the distance ' R c τ 1. Thus χ R c 7/4 τ 7/4 (80) Simultaneously we have found that the magnetic field, as it follows from eq. (77) scales as m / χ L 1/8 7/4 = L 15/4,' or as τ 15/4. The reader can try to aly these ideas to find how the magnetization behaves on magnetic field at T = T c or, equivalently, at τ h. Another interesting ossibility is to find how the finite size of system smears the hase transition. It haens when the correlation length becomes of the same order of magnitude as the size of system. For eamle, the secific heat will have a finite maimum instead of singularity. How does it deend on the size of system? At what deviation from critical temerature it stos to grow? If you have addressed this question, try to do the same for suscetibility.