The transverse susceptibility of the c-cyclic version of the one-dimensional XY model with respect to an infinitesimal magnetic field in the x-direction is investigated in more detail. Special attention is paid to the c-cyclic version of the one-dimensional Ising model. The c-cyclic susceptibility χxx is evaluated explicitly. The autocorrelation function of the magnetization Mx turns out to be time dependent in the c-cyclic Ising model

Capel, H.W.

Siskens, Th.J.

English

University of Twente Research Information

Physica 81A (1975) 214--220 © North-Holland Publishing Co. NOTE O N  THE DIFFERENCE BETWEEN THE a-CYCLIC A N D  c-CYCLIC VERSION OF THE X Y  M O D E L  H.W. CAPEL Instituut-Lorentz, Rijksuniversiteit te Leiden, Leiden, The Netherlands and TH. J. SISKENS * Department of Physics, Technological University of Twente, Enschede, The Netherlands Received 13 May 1975 The transverse susceptibility of the c-cyclic version of the one-dimensional X Y  model with respect to an infinitesimal magnetic field in the x-direction is investigated in more detail. Special attention is paid to the c-cyclic version of the one-dimensional/sing model. The c-cyclic suscep- tibility Zx~ is evaluated explicitly. The autocorrelation function of the magnetization M~ turns out to be time dependent in the c-cyclic Ising model. 1. Introduction The one-dimensional X Y  model has been introduced by Lieb, Schultz and Mat- tisl). Katsura 2) evaluated the free energy and the non-equilibrium properties were treated by Niemeyer3). Since then the X Y  model has been investigated extensively. Most treatments use the so-called c-cyclic version, which can be diagonalized exactly in terms of fermion operators. In many cases such as in the calculation of  time-dependent correlations between z-components of  spins, the c-cyclic model can be shown to produce exact results in the thermodynamic limit4-6). A more complicated quantity is the transverse susceptibility with respect to an infinitesimal magnetic field Bx in the x direction, i.e. (~2 Z =N~lim B~olim ( f iN ) -  t ~ In (e -~ ~e-BxM~> (la) = lim N - I  ~ S dr (0 e ~  S~ e- 'a r  S~>, (lb) N ~  k , j = l  0 * Present address: Instituut voor Theoretische Fysica, Universiteit van Amsterdam, The Netherlands. 214 a-CYCLIC AND c-CYCLIC VERSION OF X Y  MODEL 215 where 0 = e-#Se <e-#Se> -1 is the density operator corresponding to the hamiP tonian Yt ° in the absence of  Bx, Mx = ~ =  t S] is the x-component of  the magnetiza- tion and <0> = Tr 0 for an arbitrary operator 0. In ref. 7 we have given a high-temperature expansion up to order f16 for the transverse susceptibility Za of  the one-dimensional a-cyclic X Y  model described by the hamiltonian ~'~ = 9f',, where N J~t"~a = E {(1 4- 7) S~Sj~+~ 4- ( I  7) y y - -  SjSff+ 1 - -  BzS}} (SN+ 1 ~ Sl) .  (2) j = l  Here 7 is the anisotropy parameter and B~ is a magnetic field in the z-direction. Using the Jordan-Wigner transformation ~j = Pk S]x/2,  flj = -  Pk SIx/2, (3) t k = l  where Pr, = (2/0 c~fik = - 2 S ~ ,  (4) and where the ods and fl's are hermitean operators satisfying the anticommutation relations {~,,, ~A = {#,, #~} = ~,~, {o,,, #A = o; (5) the hamiltonian ~ a  can be written 5) ~ a  = ~'~c + l h  ( P  + 1). (6) Here Yfc is the c-cyclic hamiltonian given by N .~¢ = ~ {½ (1 + 7) ~Xj+lflJ + ½ (1 -- 7) ogjflj+~ -- iB~o~jflj}, g = l  (~xN+ , = oq,  f iN+,  ----- f l , ) -  (7) The operator P is the product  of  all Jordan-Wigner factors P = l--I~: 1 Pj and the operator h contains the operators (3) relative to site ! and N, i.e., h = - i  (1 + 7)  oqflu - i (1 -~ 7)  0~Nflt = {2 (1 + 7)  StS~ + 2 (1 7)  y ~' ~ . s , s ; , }  e .  (8) Using (6), (8) and the trivial relation p2 = 1, ~*ffc can be expressed by x x y y ~'~ = ~ - {(1 + 7) S~Sfv + (1 - 7) SaSh}  (1 + P).  (9) 216 H.W. CAPEL AND TH.J. SISKENS In ref. 7, the averages in eq. (16) with respect to the hamiltonian ~ a  were ex- pressed in terms of  averages with respect to the c-cyclic hamil tonian ogzc (and also its c-anticyclic counter  part)  using a project ion operator  technique as in ref. 5. A difficulty was the occurrence o f  an opera tor  containing the c-cyclic as well as the c-anticyclic hamil tonian Yao -= J¢~ + ti. The c-cyclic averages were evaluated using the the rmodynamic  Wick theorem due to Bloch and de DominicisS). We also investigated the spin correlation functions in eq. ( lb) with respect to the c-cyclic hamil tonian ~ .  As a result o f  a high-temperature expansion ~e  obtained the relation, c f  eq. (7.19) o f  ref. 7, ~ j  e ° S j + p )  = (Oa ~ o j  S j+p)  • (0¢ CJeo e-~(ze°+,~), (10) where ~'c and ~o a are the density operators corresponding to ~%,°c and ~%fa resp. Eq. (10) is valid, up to an arbi t rary order in /3, for sufficiently large values o f j  and N - ]  - p. F rom (10) one can expect the inequality z~ -¢ z . ,  (11) where Z~ and Za are the c-cyclic and a-cyclic susceptibilities, which can be obtained f rom eq. ( lb) by substituting ~¢f = ~ c  and o~ = 9f',  respectively. (In fact, Z¢ was evaluated up to o rde r /~4  cf. eq. (7.23) o f  ref. 7.) The inequality (11) is only valid, if one first takes the limit B~ ~ 0 and the the rmodynamic  limit N ~  oo afterwards. I f  one defines " the rmodynamic"  sus- ceptibilities 2¢ and Za by interchanging the two limits in eq. (la),  then both  suscep- tibilities should be equal. This is obvious from the relation . f (~ '~  - BxM~) = f(.NtO~ - BxM~) (12) for the l'ree energy per part iclef( , .~)  = lira N -  '~ F (W), F ( W )  = - / 3 -  ~ In ( e - t ~ ) ,  corresponding to the hamil tonians ~ = -~t~:¢ - B~M~ and Jt ° = J'/{~ - BxMx .  Eq. (12) is an immediate consequence o f  a special case o f  the Bogol iubov in- equality 9) F ( - ~ o )  - ] l~ali  < r ( ~ o  + . ,~ , )  < F(~ZZo) + i l -~, l l ,  (iN) where Ii0Ll = sup I(x. 0x)l (x, x ) -  ~ is the operator  norm. Eq. (12) is obvious from (9) and (13), since the norm [l(,,Y{'~ - ":/#,)ll is finite. Since there is no reason to doubt  the validity o f  the relation Z, = 2~ for the a-cyclic model, eq. (11) implies that  Z~ # 2¢, i.e. in the calculation of  the suscep- tibility Z~  for the c-cyclic chain the two limits B~ --* 0 and N - *  oo cannot  be interchanged. Note  that  for a direct evaluat ion o f  2a = 2¢ one should know the free energy of  an X Y  chain in the presence o f  B~. In ref. 7 we calculated Z~  for a finite chain in the limiting case B~ --* 0, taking the the rmodynamic  limit after- wards. The inequality (11) shows that  the difference between the transverse sus- a-CYCLIC AND c-CYCLIC VERSION OF XY MODEL 217 ceptibilities per particle in the absence of a field B~ for finite a-cyclic and c-cyclic chains tends to a non-vanishing value in the thermodynamic limit. As another consequence of (10) one can expect a difference between the c-cyclic and a-cyclic autocorrelation functions of the x component of the magnetization, i.e., R¢(t) g= R.(t) ,  (14) where R~(t) - lim N-*  {(o~M:, e 'ae~t Mx e-'ae°t> - (0~M~) 2} (e = c, a). (15) N--+ cc In section 2 both inequalities ( l l )  and (14) will be made more explicit for the simple case of an Ising model, where the c-cyclic quantities )re and R~(t) can be evaluated exactly without using high-temperature expansions or expansions in powers of t. (Of course for this particular case the free energy per spin of the a-cyclic model in the presence of  a magnetic field B~ is well known. Our purpose, however, is to show that the use of the c-cyclic version leads to an expression for Z~ different from g, = 2, = 2~.) 2. Ising model In this section we shall evaluate the c-cyclic correlation functions Cjk(7~ ) ~ ( 0  c e "C'/g°c S;  e-~.*eo S~) (16) in the special case y -- l, B, = 0. The a-cyclic hamiltonian is then given by N J/a = 2 Z S~'S~+I (S~+1 = S~). (17) j = t  Since the spin components S~' commute with ~ a ,  the a-cyclic time xx  spin cor- relation functions and consequently the correlation function of Mx do not depend on time. For the a-cyclic model we have (gaS~:Si, ~) = ¼ { ( - t a n h  ½fl)lk-jI + ( - t a n h  ½fl)x-,k-j,} x {1 + (--tanh½fi)N} -1,  (18) N - 1  Za = l i m  ~ f l ( o a S ~ S ~ + , )  = ¼ f l e  -~ =;~, = ~ c ,  (19) N--+ ot~ p = O  R,(t) = ¼e-~. (20) 218 H.W. CAPEL AND TH.J.  SISKENS The equality 2a = ¼fle -B is well known;  the equality 2a = 2c has been justified in section 1. The hamil tonian o f  the c-cyclic version o f  the 1sing model reads ,~c = ~ a  - -  2SIS~ (1 + P).  (21) In  the evaluation of  the r.h.s, o f  (16) use will be made of  the (an t i )commuta t ion  relations / {P, S~} = 0, [ ~ a ,  S~] = 0, (22) [s~s~ (~ + P),  ~ ]  = o and eq. (8) for the special case 7 = 1, B = 0. Then Cjk(z) = (o~ e - 2*s~xs~(l +e) e ~*~ Sj ~ e -**° S~ e =¢s*~s~*(*-e~) = (0~ e-4~sds~'*J" S f S ; )  = (e¢ e-h* S]S~> = Ckj(r). (23) The operator  e -he, for Y = 1, B = 0, can be written e -h~ = cosh ~ , 4S1S~,P sinh ~ = cosh r + 2i0¢~/3N sinh z. (24) Using the Jo rdan-Wigner  t ransformat ion (3), the ant i -commutat ion  relations (5) and eq. (24) we obtain,  (p _> 0), j + p - - 1  Ca a+ . = ¼ (oc (cosh z + 2i~xlfl u sinh ~) (2/i)" I-I (t3k~Xk+,))" (25) k = j  The r.h.s, o f  (25) can be evaluated using the the rmodynamic  Wick theorem and the relations, c f  eqs. (2.25), (2.29), (3.17)-(3.19) of  ref. 7, (26) As a result we obtain Cjj+p(z) = ¼ ( - t a n h  ½fi)P (cosh z - sinh r tanh ½fi). (27) The second factor in eq. (27), which is equal to (¢¢e - ~ )  can also be obtained f rom the second factor on the r.h.s, o f  eq. (10) in the special case that  7 = 1, B = 0, so that  [aeg¢, h] = 0. Note  that  eq. (27) is also valid for a finite chain and a-CYCLIC AND c-CYCLIC VERSION OF XY MODEL 219 that the correlation function is independent o f j .  This is not true in the general case of the X Y  model. From eq. (27) we obtain immediately the c-cyclic susceptibility in the thermo- dynamic limit # p = l  = 2 (tanh ½ f l ) { l +  k ~, ( - t a n h  {fl)P} = Za2fl -~  tanh ½fl, p = l  (28) which is clearly different from ~a = ~ a  = Zc"  Substituting r = - i t  in eq. (27), we find (e¢S] e l'ae° S~: e -l*~e°) = ¼ ( - t a n h  ½fl)lk-jI (COS t + i sin t tanh lfl). (29) The validity for k < j can be seen by taking the hermitean conjugate of both members of the corresponding equation for k > j. As a result the auto-correlation function of the x component of the magnetization is given by Rc(t) = ¼ e -a (cos t + i sin t tanh ½fl). (30) As a result of  using the c-cyclic version the time auto-correlation function of Mx is no longer a constant but depends on time. The time average of  Re(t) vanishes (also in the case of  the finite c-cyclic Ising model). Hence, the magnetization Mx is an ergodic operator in the c-cyclic Ising model. (Both for finite N and in the limit N ~ oo.) In the finite and infinite a-cyclic Ising chain Mx is not an ergodic operator, since in the absence of Bx the microcanonical and canonical averages of M~ vanish, whereas the time average of  Ra(t) is given by (20). 3. Remark So far we have restricted ourselves to the c-cyclic version of  the Ising model. For  the c-anticyclic version, which is defined by the hamiltonian ~aac =-- M'¢ + h = ~ - 2S~S~, (1 -- P) ,  (31) the correlation functions C~j+p are also given by the r.h.s, of  (27), i.e., Cjj+p(z) = <~a~ e ~a° S f  e -~ra° Sf+p> = Cij+p(z ). (32) 220 H.W. CAPEL AND TH. J. SISKENS This can be seen by  the rep lacement  P--+ - P  in eq. (23), which leads to the c-ant icycl ic  ana logue  of  (25), i.e., / j + p - - I  '\ Cja+v(z) = ¼ ( 0 ~ c ( c o s h T -  2 i~ ;qf iNsinhr) (2 / i )  v 1~ ( f lk~k+,)>-  (33) \ k = j  / N o w  (32) is obvious  f rom (33) not ing  that  <Oac~'l fiN> = --  <eo~'l G > ,  (34) (Oac~jflj-i> = ( 0 c a j & - ,  > ( j  = 2 . . . . .  N) .  F r o m  (32) we have the re la t ions  Zac = Zc, Rac(/ ) = Rc(t  ).  (35) The suscept ibi l i ty  and  the au tocor re la t ion  funct ion of  Mx in the c-cyclic and  c-ant icycl ic  version are equal .  No te  tha t  this is true even for  the finite chain.  In  general ,  the p roper t ies  o f  the c-cyclic and  c-ant icycl ic  X Y  model  are equal  only  in the t h e r m o d y n a m i c  l imit .  Acknowledgement The au thors  are indeb ted  to Professor  P. Mazur  for interest ing discussions and read ing  the manuscr ip t .  References 1) E. Lieb, T.Schultz and D. Mattis, Ann. Physics 16 (1961) 407. 2) S. Katsura, Phys. Rev. 127 (1962) 1508. 3) Th.Niemeyer, Physica 36 (1967) 377. 4) P. Mazur and Th.J. Siskens, Physica 69 (1973) 259. 5) Th. J. Siskens and P. Mazur, Physica 71 (1974) 560. 6) Th. J. Siskens, Physica 72 (1974) 123. 7) H. W. Capel, E. J. van Dongen and Th. J. Siskens, Physica 76 (1974) 445. 8) C.Bloch and C.de Dominicis, Nucl. Phys. 7 (1958) 459. 9) R.B.Griffiths, J. Math. Phys. 5 (1964) 1215. 

Note on the difference between the a-cyclic and c-cyclic version of the XY model

8) C.Bloch and C.de Dominicis,

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Note on the difference between the a-cyclic and c-cyclic version of the XY model

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