Universal amplitude ratios for scaling corrections on Ising strips with fixed boundary conditions
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A ug Universal amplitude ratios for scaling corrections on Ising stripswith fixed boundary conditions.
N. Sh. Izmailian ∗ Institute of Physics, Academia Sinica,Nankang, Taipei 11529, Taiwan andYerevan Physics Institute, Alikhanian Brothers 2, 375036 Yerevan, Armenia (Dated: July 30, 2018)
Abstract
We study the (analytic) finite-size corrections in the Ising model on the strip with fixed (+ − )boundary conditions. We find that subdominant finite-size corrections to scaling should be tothe form a k /N k − for the free energy f N and b ( n ) k /N k − for inverse correlation length ξ − n , withinteger value of k . We investigate the set { a k , b ( n ) k } by exact evaluation and their changes uponvarying anisotropy of coupling. We find that the amplitude ratios b ( n ) k /a k remain constant uponvarying coupling anisotropy. Such universal behavior are correctly reproduced by the conformalperturbative approach. PACS numbers: 05.50+q, 75.10-b ∗ Electronic address: [email protected] . INTRODUCTION Finite-size scaling and corrections for critical systems have attracted much attention inrecent decades. Although many theoretical results are now known about the critical expo-nents and universal relations among the leading critical amplitudes, not much informationis available on ratios among the amplitudes in finite-size correction terms [2]. New universalamplitude ratios have been recently presented for the Ising model [3–6]. Consider an Isingferromagnet on an N × M lattice. If Λ > Λ > Λ > Λ > ... are the eigenvalues of thetransfer matrix (TM), in the limit M → ∞ the free energy per spin, f N , and the inverselongitudinal spin-spin correlation length, ξ − n , are f N = 1 ζ N ln Λ and ξ − n = 1 ζ ln (Λ / Λ n ) . (1)Here ζ is a geometric factor, which is unity for the square lattice and, in triangular andhoneycomb geometries (also for the square lattice when the TM progresses along the diagonal[7]), corrects for the fact that the physical length added upon each application of the TMdiffers from one lattice spacing [8].At the critical point T c the asymptotic finite-size scaling behavior of the critical freeenergy ( f N ) and the inverse correlation lengths ( ξ − n ) of an infinitely long 2D strip of finitewidth N has the form [9, 10] lim N →∞ N ( f N − f ∞ ) − N f surf = A, (2)lim N →∞ N ξ − n = D n , (3)where f ∞ is the bulk free energy, f surf is the surface free energy and A and D n are theuniversal constants, but may depend on the boundary conditions. In some 2D geometries,the values of A and D n are known [9–11], to be related to the conformal anomaly number( c ), the conformal weight of the ground state (∆), and the scaling dimension of the n -thscaling field ( x n ) of the theory A = πζ (cid:16) c − ∆ (cid:17) , D n = πζ x n , (4)for strip geometry. Here ζ is anisotropy parameter. The principle of unitarity of the un-derlying field theory restricts through the Kac formula the possible values of c and for eachvalue of c only permits a finite number of possible values of ∆. For the 2D Ising model, wehave c = 1 / , / , / − ) and mixed in strip geometry. For fixed (+ − ) boundaryconditions the spins are fixed to the opposite values on two sides of the strip. The mixedboundary conditions corresponds to free boundary conditions on one side of the strip, andfixed boundary conditions on the other. In the terminology of surface critical phenomenathese three boundary universal classes: free, fixed (+ − ) and mixed correspond to ”ordinary”,”extraordinary” and ”special” surface critical behavior, respectively.The highest conformal weight ∆, and the scaling dimension x n depends on the boundaryconditions and for fixed (+ − ) boundary conditions they given by∆ = 12 , x = 1 , x = 2 , x = 3 , ..., x n = n. (5)Quite recently, Izmailian and Hu [3] studied the finite size correction terms for the freeenergy per spin and the inverse correlation lengths of critical 2D Ising models on N × ∞ lattice and 1D quantum Ising chain with periodic boundary conditions. They obtainedanalytic expressions for the finite-size correction coefficients a k and b ( n ) k in the expansions N [ f N − f ∞ ] = f s + ∞ X k =1 a k N k − , (6) ξ − n = ∞ X k =1 b ( n ) k N k − , (7)and find that although the finite-size correction coefficients a k , b ( n ) k (for n = 1 ,
2) themselvesare non-universal (except for a and b ( n )1 ), the amplitude ratios for the coefficients of theseseries b ( n ) k /a k are universal.Later that result has been extended for the quantum Ising chain for antiperiodic andfree boundary conditions [4] and for the two-dimensional (2D) Ising models on M × ∞ lattice with the special boundary conditions studied by Brascamp and Kunz (BK) [5]. Itwas shown that Brascamp and Kunz (BK) boundary conditions [12] belong to the mixedboundary condition universality class although the mixed boundary condition and the BKboundary condition are different on one side of the long strip.In the present paper we will present exact calculations for a set of universal amplituderatios for the Ising model with fixed (+ − ) boundary conditions. We obtain analytic equa-3 ' +1=121 21 L ++++ (cid:1)(cid:1)(cid:1)(cid:1) FIG. 1: The lattice L tions for a k and b ( n ) k (for n = 1 , , , ... ) in the expansions given by Eqs. (6) and (7) and findthat amplitude ratios b ( n ) k /a k are universal. We will show that such universal behavior arecorrectly reproduced by the conformal perturbative approach. II. THE ISING MODEL ON CYLINDER
The Ising model on a cylinder, with width N and height M and free, fixed and mixedboundary condition along diagonal was formulated in [7] in terms of a double-row transfermatrix D(u). They define the lattice L as a square lattice rotated by 45 degrees, in whichthe rows have alternately L − L faces. Vertically the lattice have columns of L ′ faces,and the periodic boundary conditions imposed in this direction by identifying the first and( L ′ + 1)th rows of faces. The lattice L consists of N (zigzagging) columns edges and M (zigzagging) rows edges, respectively, and N and M are given by N = 2 L M = 2 L ′ (8)The lattice L is shown in Fig. 1. We are concerned with the fixed (+ − ) type of boundaryconditions. For fixed (+ − ) boundary conditions we choose the spins at the left and rightboundaries of the lattice L to be +1 and − Z NM = X { σ } exp J X σ i σ j + K X σ i σ j ! (9)where we define the set of spins not fixed by the boundary conditions by { σ } . For fixedboundary conditions, the first sum is over edges in odd columns and the second sum overedges in even columns of the lattice.Consider the critical Ising model on a cylindrical lattice of N columns and M rows asdescribed previously. The finite-size partition function Z NM can be written as Z NM ( u ) = T r (cid:0) D ( u ) M (cid:1) = X n e − ME n ( u ) (10)where the sum is over all eigenvalues of a transfer matrix D ( u ), written as e − E n ( u ) . Since werestrict ourselves to the critical Ising model, we have sinh(2 J ) sinh(2 K ) = 1. This conditioncan be conveniently parameterized by introducing a so-called spectral parameter u , so thatsinh(2 J ) = cot(2 u ) , sinh(2 K ) = tan(2 u ), with 0 < u < π/
4. The anisotropy parameter ζ related to the spectral parameter u through ζ = sin 4 u (11)For isotropic system ( K = J ) we have u = π/ ζ = 1.For fixed (+ − ) boundary conditions all eigenvalues have been determined in [7], for anyfinite value of N. The energy E n associated with the eigenvalues of a transfer matrix D ( u )are given by E n ( u ) = −
12 ln − N/ N/ Y k =1 (cid:20) cosec (cid:18) k − / N + 1 π (cid:19) + µ k sin(4 u ) (cid:21) (12)Using the identity L Y k =1 sin (cid:18) k − / L + 1 π (cid:19) = 2 − L (13)the Eq. (12) can be simplified to the following form E n ( u ) = − N/ X k =1 ln (cid:20) µ k sin(4 u ) sin (cid:18) k − / N + 1 π (cid:19)(cid:21) (14)where µ k = ±
1. For fixed (+ − ) boundary conditions µ k should satisfied the conditions Q Lk =1 µ k = −
1. This correctly yields 2 L − eigenvalues for the fixed (+ − ) boundary cases.5he ground state E correspond to all µ k = 1 except µ = − E n are given by Eq. (14) with µ n = − µ k are equal to 1: µ k = 1( k = n ). Thus for the critical free energy ( N f N = − E ) and the inverse correlation lengths( ξ − r = E r − E ) (for r = 1 , , ..., n ) we obtain N f N = N X k =1 ω (cid:18) k − / N + 1 π (cid:19) + ω (cid:18) − π N + 1) (cid:19) − ω (cid:18) π N + 1) (cid:19) , (15) ξ − = ω (cid:18) π N + 1) (cid:19) − ω (cid:18) π N + 1) (cid:19) + ω (cid:18) − π N + 1) (cid:19) − ω (cid:18) − π N + 1) (cid:19) , (16) ξ − = ω (cid:18) π N + 1) (cid:19) − ω (cid:18) π N + 1) (cid:19) + ω (cid:18) − π N + 1) (cid:19) − ω (cid:18) − π N + 1) (cid:19) , (17) ξ − = ω (cid:18) π N + 1) (cid:19) − ω (cid:18) π N + 1) (cid:19) + ω (cid:18) − π N + 1) (cid:19) − ω (cid:18) − π N + 1) (cid:19) , (18)... ξ − n = ω (cid:18) (2 n + 1) π N + 1) (cid:19) − ω (cid:18) π N + 1) (cid:19) + ω (cid:18) − π N + 1) (cid:19) − ω (cid:18) − (2 n + 1) π N + 1) (cid:19) , (19)where ω ( x ) is given by ω ( x ) = 12 ln [1 + sin(4 u ) sin x ] (20)The sum in Eq. (15) can be transformed as N X k =1 ω (cid:18) k − / N + 1 π (cid:19) = − ω (cid:16) π (cid:17) + 12 N X k =0 ω (cid:18) k + 1 / N + 1 π (cid:19) , (21)and can be handled by using the Euler-Maclaurin summation formula [13]. Suppose that F ( x ) together with its derivatives is continuous within the interval ( a, b ). Then the generalEuler-Maclaurin summation formula states N − X n =0 F ( a + nh + αh ) = 1 h Z ba F ( τ ) d τ + ∞ X k =1 h k − k ! B k ( α ) (cid:0) F ( k − ( b ) − F ( k − ( a ) (cid:1) (22)where 0 ≤ α ≤ h = ( b − a ) /N and B k ( α ) are so-called Bernoulli polynomials defined interms of the Bernoulli numbers B p byB k ( α ) = k X p =0 B p k !( k − p )! p ! α k − p (23)Indeed, B n (0) = B n . Bernoulli polynomials satisfy the identity: B n (1 /
2) = (cid:0) − n − (cid:1) B n (24)6sing the Euler-Maclaurin summation formula given by Eq. (22) with a = 0 , b = π, α = 1 / F ( x ) = ω ( x ) the asymptotic expansion of the critical free energy f N can be written inthe following form N ( f N − f ∞ ) = f surf − ∞ X p =1 λ p − (2 p − (cid:18) B p (1 / p + 2 − p +2 (cid:19) (cid:18) πN + 1 (cid:19) p − , (25)= f surf − π sin 4 u N + 1) − u − sin 4 u )5760 (cid:18) πN + 1 (cid:19) + . . . , where f ∞ = 1 π Z π ω ( x ) dx (26) f surf = −
14 ln (1 + sin 4 u ) + 1 π Z π ω ( x ) dx (27)and λ k is the coefficients in the Taylor expansion of the ω ( x ): ω ( x ) = ∞ X p =0 λ p p ! x p , (28)with λ = 0 , λ = sin 4 u, λ = − sin 4 u , λ = sin u − sin 4 u, ... .Using the Taylor expansion of the ω ( x ) the asymptotic expansion of the critical inversecorrelation lengths ξ − r for r = 1 , , , ..., n can be written in the following form ξ − = ∞ X p =1 p − − λ p − p − (2 p − (cid:18) πN + 1 (cid:19) p − , (29)= π sin 4 uN + 1 + 13(sin u − sin 4 u )12 (cid:18) πN + 1 (cid:19) + . . . ,ξ − = ∞ X p =1 p − − λ p − p − (2 p − (cid:18) πN + 1 (cid:19) p − , (30)= 2 π sin 4 uN + 1 + 31(sin u − sin 4 u )6 (cid:18) πN + 1 (cid:19) + . . . ,ξ − = ∞ X p =1 p − − λ p − p − (2 p − (cid:18) πN + 1 (cid:19) p − , (31)= 3 π sin 4 uN + 1 + 57(sin u − sin 4 u )4 (cid:18) πN + 1 (cid:19) + . . . , ... ξ − n = ∞ X p =1 n + 1) p − − λ p − p − (2 p − (cid:18) πN + 1 (cid:19) p − , (32)= nπ sin 4 uN + 1 + [(2 n + 1) − u − sin 4 u )24 (cid:18) πN + 1 (cid:19) + . . . , N + 1) − (2 p − correctionterms in the free energy and the inverse correlation lengths expansions, i.e., b ( n ) p /a p shouldnot depend on the spectral parameter u and given by b ( n ) p a p = − (2 n + 1) p − − p − B p (1 / p + 1 . (33)For p = 1 we have b ( n )1 a = − n (34)and for p = 2 we have b ( n )2 a = − (cid:2) (2 n + 1) − (cid:3) = − n (4 n + 6 n + 3) . (35) III. PERTURBATED CONFORMAL FIELD THEORY
The finite-size corrections to Eqs. (2) and (3) can be calculated by the means of aperturbated conformal field theory [14, 15]. In general, any lattice Hamiltonian will containcorrection terms to the critical Hamiltonian H c H = H c + X p g p Z N/ − N/ φ p ( v ) dv, (36)where g p is a non-universal constant and φ p ( v ) is a perturbative conformal field. Belowwe will consider the case with only one perturbative conformal field, say φ l ( v ). Then theeigenvalues of H are E n = E n,c + g l Z N/ − N/ < n | φ l ( v ) | n > dv + . . . , (37)where E n,c are the critical eigenvalues of H . The matrix element < n | φ l ( v ) | n > can be com-puted in terms of the universal structure constants ( C nln ) of the operator product expansion[14]: < n | φ l ( v ) | n > = (2 π/N ) x l C nln , where x l is the scaling dimension of the conformal field φ l ( v ). The energy gaps ( E n − E ) and the ground-state energy ( E ) can be written as E n − E = 2 πN x n + 2 πg l ( C nln − C l ) (cid:18) πN (cid:19) x l − + . . . , (38) E = E ,c + 2 πg l C l (cid:18) πN (cid:19) x l − + . . . . (39)8ote, that the ground state energy E and the energy gaps ( E n − E ) of a quantum spinchain are, respectively, the quantum analogies of the free energy f N and inverse correlationlengths ξ − n for the Ising model; that is, N f N ⇔ − E , and ξ − n ⇔ E n − E . (40)For the 2D Ising model, one finds [16] that the leading finite-size corrections (1 /N ) can bedescribed by the Hamiltonian given by Eq. (36) with a single perturbative conformal field φ l ( v ) = L − ( v ) with scaling dimension x l = 4 .In order to obtain the corrections we need the matrix elements < n | L − ( v ) | n > , whichhave already been computed by Reinicke [17]. < ∆ + r | L − | ∆ + r > = (cid:18) πN (cid:19) (cid:20) r ) (cid:18) ∆ −
524 + r (2∆ + r )(5∆ + 1)(∆ + 1)(2∆ + 1) (cid:19)(cid:21) (41)The universal structure constants C l , C l and C l can be obtained from the matrix element < n | L − ( v ) | n > = (2 π/N ) x l C nln , where x l = 4 is the scaling dimension of the conformal field L − ( v ).At the critical point the spectra of the Hamiltonian with fixed (+ − ) boundary conditionscan be understood in terms of irreducible representations ∆ of a single Virasoro algebra withvalues of ∆ is .For fixed (+ − ) boundary conditions the ground state | > and the excited states | r > ( r = 1 , , ..., n ) are given by | r > = | ∆ = 12 , r = k >, for k = 1 , , ..., n (42)After reaching this point, one can easily compute the universal structure constants C rlr ( r = 0 , , , ..., n ) for fixed (+ − ) boundary conditions. The values of C rlr can be obtainedfrom Eqs. (41) and (42) and given by: C rlr = 172911520 + 7 r (4 r + 6 r + 3)24 C l = 1729 / , C l = 45409 / , C l = 210049 / , .... (43)Equations (38) and (39) implies that the ratios of first-order corrections amplitudes for E n − E ( ξ − n ) and − E ( f N ) is universal and equal to ( C l − C nln ) /C l , which is consistentwith Eq. (35) r n (2) = C l − C nln C l = − n (4 n + 6 n + 3) (44)9 V. SUMMARY
In this paper we present exact calculations for all coefficients in the asymptotic expansionsgiven by Eqs. (6) and (7) for an infinite number of specific levels E n ( n = 0 , , , ... ), forwhich we find that the ratios b ( n ) p /a p are indeed universal and doesn’t depend on the spectralparameter u . We find that such universal behavior are correctly reproduced by the conformalperturbative approach. V. ACKNOWLEDGEMENTS
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