The isotropic XY model on the inhomogeneous periodic chain

0.62 MB·English

by  J. P de Lima

The isotropic XY model on the
5 inhomogeneous periodic chain
0
0
2

n
a
J. P. de Limaa,1, T. F. A. Alvesb, L. L. Gonc¸alvesa,2,
J ∗

1
2 aDepartamento de F´isica Geral, Instituto de F´isica, Universidade de Sa˜o Paulo,

C.P.66318, 05315-970, Sa˜o Paulo, SP, Brazil
]
h bDepartamento de F´isica, Universidade Federal do Ceara´, Campus do Pici,
c
e C.P.6030, 60451-970, Fortaleza, Ceara´, Brazil
m
-
t
a
t Abstract
s
.
t
a
The static and dynamic properties of the isotropic XY-model (s = 1/2) on the
m
inhomogeneous periodic chain, composed of N segments with n different exchange
-
d interactions and magnetic moments, in a transverse field h are obtained exactly at
n
arbitrary temperatures. The properties are determined by introducing the gener-
o
c alized Jordan-Wigner transformation and by reducing the problem to a diagonal-
[ ization of a finite matrix of n-th order. The diagonalization procedure is discussed

1 in detail and the critical behaviour induced by the transverse field, at T = 0, is
v
presented. The quantum transitions are determined by analyzing the behaviour of
5
the induced magnetization, defined as (1/n) n µ < Sz > where µ is the
2 m=1 m j,m m
5 magnetic moment at site m within the segment j, as a function of the field, and the
1 critical fields determined exactly. The dynamPic correlations, < Sz (t)Sz (0) >,
0 j,m j′,m′
5 and the dynamic susceptibility χzz(ω) are also obtained at arbitrary temperatures.
q
0
Explicit results are also presented in the limit T = 0, where the critical behaviour
/
t occurs,for the static susceptibility χzz(0) as afunction of thetransverse fieldh, and
a q
m for the frequency dependency of dynamic susceptibility χzz(ω). Also in this limit,
q
d- the transverse time-correlation < Sjx,m(t)Sjx′,m′(0) >, the dynamic and isothermal
n susceptibilities,χxqx(ω)andχxTx,areobtainedforthetransversefieldgreaterorequal
o than the saturation field.
c
:
v
Key words: XY model, quantum transition, inhomogeneous chain
i
X
PACS: 05.70.Fh, 05.70.Jk, 75.10.Jm, 75.10.Pq
r
a
Corresponding author: L. L. Gonc¸alves. Fax: [phone]
∗
E-mail address:lindberg@fisica.ufc.br
1 On a post-doctoral leave from Departamento de F´isica, Universidade Federal do
Piau´i, Campus Ministro Petroˆnio Portela, 64049-550, Teresina, Piau´i, Brazil.
2 On sabbatical leave from: Departamento de F´isica, Universidade Federal do
Preprint submitted to Elsevier Science 2 February 2008
1 Introduction
Models involving inhomogeneous spin chains have been subject of intensive
study in recent years motivated by various reasons. Amongst those, the most
remarkable one is the necessity to understand the unusual new properties pre-
sented by low dimensional magnetic materials [1,2,3,4] at low temperature,
which are described in terms of its many-body behaviour and its quantum
transitions [5]. As manifestation of these properties we can mention the ap-
pearance of magnetization plateaus as functions of the external magnetic field
[6,7], the existence of an energy gap between the ground state and first excited
state at zero field [8] and the presence of quantum critical behaviour [4,9,10].
The one-dimensional XY model (s = 1/2) introduced by Lieb et al. [11] plays
an important role in this context since it constitutes one of the few many-
body problems which can be exactly solved. The most recent results on the
inhomogeneous anisotropic model have been obtained by Derzhko et al.[12]
and are restricted to the thermodynamic properties. A good review of the
known results is also presented in his work.
For the isotropic model, the thermodynamic properties have also been ob-
tained by Derzhko[13] (see references therein for thermodynamic properties
of the alternating chain), and the study of the dynamics and of the quantum
critical behaviour has been restricted to the alternating chain [14,15,16], and
for the alternating superlattice[17].
In this paper we will also consider the isotropic XY model in a transverse
field on the inhomogeneous periodic chain consisting of N unit cells with n
sites, which has been studied by Derzkho [13]. Within the cells we can have
n different exchange constants as well as magnetic moments, and the model
corresponds toanextension ofthealternatingsuperlattice. Anextensive study
ofthequantumcriticalbehaviourwillbepresentedandthedynamicproperties
in the field direction determined for arbitrary temperature. Within a new
formalism, we have been able to solve the model exactly, and present new
and more general results from those presented in the previous papers[17,18].
In particular, we have been able to obtain the dynamic correlation in the
xy-plane, at T = 0, for transverse field greater than the saturation field.
In section 2 we discuss in detail the diagonalization of the model and present
the analytical results for the chains composed of cells with two, three and
four sites. The excitation spectrum is also obtained, by a different method, in
Appendix A, and it is shown under which conditions the gap in the excitation
spectrum is suppressed.
Ceara´, Campus do Pici, C.P. 6030, 60451-970, Fortaleza, Ceara´, Brazil.
2
d=na
m=1 m=2 m=n
l−th cell
Fig. 1. Unit cell of the periodic chain.
The induced magnetization and the isothermal susceptibility χzz are obtained
T
in section 3 and we also determine the critical fields associated with the quan-
tum second order phase transitions. The static and dynamic correlations in
the field direction are obtained in section 4 and, in the section 5, we deter-
mine the longitudinal dynamic susceptibility χzz(ω). The dynamic correlation
q
Sx (t)Sx (0) is obtained, at T = 0 and for external fields greater than
1,m 1+l,m′
tDhesaturationfieldE,inappendix B.Under these conditions,theisothermal and
dynamic transverse susceptibility χxx are obtained in section 6, and finally in
section 7 we summarize the main results and present the conclusions.
2 The diagonalization of the Hamiltonian
We consider the isotropic XY model (s = 1/2) on the inhomogeneous periodic
chain with N cells, n sites per cell, and lattice parameter a, in a transverse
field, whose unit cell is shown Fig. 1. The Hamiltonian is given by
N n n−1
H= µ hSz + J Sx Sx +Sy Sy +
− ( m l,m m l,m l,m+1 l,m l,m+1
Xl=1 mX=1 mX=1 h i
+J Sx Sx +J Sy Sy , (1)
n l,n l+1,1 n l,n l+1,1
o
wheretheparametersJ aretheexchangecouplingbetweennearest-neighbour,
l,m
µ the magnetic moments, h the external field and we have assumed periodic
m
boundary conditions. If we introduce the ladder operators
S± = Sx iSy, (2)
±
and the generalized Jordan-Wigner transformation[19]
3
l−1 n m−1
Sl+,m=exp iπ c†l′,m′cl′,m′ +iπ c†l,m′cl,m′ c†l,m,
( l′=1m′=1 m′=1 )
X X X
1
Sz =c† c , (3)
l,m l,m l,m − 2
where c and c† are fermion annihilation and creation operators, we can
l,m l,m
write the Hamiltonian as[20]
H = H+P+ +H−P−, (4)
where
N n 1 n−1 J
H±= µ h c† c + m c† c +c† c
−Xl=1(mX=1 m (cid:18) l,m l,m − 2(cid:19) mX=1 2 (cid:16) l,m l,m+1 l,m+1 l,m(cid:17))−
N−1 J J
n c† c +c† c n c† c +c† c , (5)
− 2 l,n l+1,1 l+1,1 l,n ± 2 N,n 1,1 1,1 N,n
Xl=1 (cid:16) (cid:17) (cid:16) (cid:17)
and
I P
P± = ± , (6)
2
with P given by
N n
P = exp iπ c† c . (7)
l,m l,m
!
l=1m=1
X X
As it is well known [20,21,22], since the operator P commutes with the Hamil-
tonian, the eigenstates have definite parity, and P−(P+) corresponds to a
projector into a state of odd (even) parity.
Introducing periodic and anti-periodic boundary conditions on c′s for H− and
H+ respectively, the wave-vectors in the Fourier transform [23],
1
c = exp( iqdl)A ,
l,m q,m
√N −
q
X
1 N
A = exp(iqdl)c , (8)
q,m l,m
√N
l=1
X
are given by q− = 2rπ for periodic condition and q+ = π(2r+1),for anti-periodic
Nd Nd
condition, with r = 0, 1,...., N/2, and H− and H+ can be written in the
± ±
form
H± = Hq±, (9)
Xq±
where
4
n 1
Hq± =− µmh A†q±,mAq±,m − 2 −
m=1 (cid:18) (cid:19)
X
n−1 J
m † †
− 2 Aq±,mAq±,m+1 +Aq±,m+1Aq±,m −
mX=1 h i
J
− 2n A†q±,nAq±,1exp(−idq±)+A†q±,1Aq±,nexp(idq±) . (10)
h i
Although H− and H+ do not commute, it can be shown that in the thermo-
dynamic limit all the static properties of the system can be obtained in terms
of H− or H+. However, even in this limit, some dynamic properties depend
on H− and H+ [20,21,22]. Since
[Hq,Hq′] = 0, (11)
where we make the identification q q±, we can diagonalize the Hamiltonian
≡
by introducing the canonical transformation
n n
A = u ξ , A† = u∗ ξ† , (12)
q,m q,km q,k q,m q,km q,k
k=1 k=1
X X
and by imposing the condition
[ξ ,H ] = ε ξ , (13)
q,k q q,k q,k
which leads, for the coefficients u , to the equation
q,km
u u
q,k1 q,k1
   
u u
q,k2 q,k2
Aq ..  = εq,k .. , (14)
 .   . 
   
   
u  u 
 q,kn  q,kn
   
   
where A is given by
q
h J1 0 0 Jn exp( iqd)
1 2 ··· 2 −
 
J1 h J2 0
2 2 2
 . 
A  0 J22 h3 J23 .. , (15)
q ≡ − ... J3 ... ... 0 
 2 
 
 0 ... h Jn−1 
 n−1 2 
 
J2n exp(iqd) 0 ··· 0 Jn2−1 hn 
 
and the u′s satisfy the orthogonality relations
5
n
uq,kmu∗q,k′m = δkk′, (16)
m=1
X
n
uq,kmu∗q,km′ = δmm′. (17)
k=1
X
Therefore the Hamiltonian can be written in the diagonal form
1
†
H = ε (ξ ξ ), (18)
q q,k q,k q,k − 2
k
X
where the spectrum ε of H is determined from the determinantal equation
q q
det(A ε I) = 0. (19)
q q
−
In passing, we would like to note that for uniform magnetic moments, µ µ,
m
theterm n µhSz commutes withtheHamiltonian,andconsequently≡the
− l,m l,m
effect of the field is to shift the spectrum.
P
By using (16) we can express the operators ξ′s in terms of A′s which are given
by
n n
ξ = u∗ A , ξ† = u A† , (20)
q,k q,km q,m q,k q,km q,m
m=1 m=1
X X
and from eq.(8) we obtain,
1 N n
ξ = exp(iqdl)u∗ c ,
q,k √N q,km l,m
l=1m=1
X X
1 N n
ξ† = exp( iqdl)u c† , (21)
q,k √N − q,km l,m
l=1m=1
X X
and their inverse, as
1
c = exp( iqdl)u ξ ,
l,m √N − q,km q,k
q k
XX
1
c† = exp(iqdl)u∗ ξ† . (22)
l,m √N q,km q,k
q k
XX
The solution of eqs.(14) and (19) can be obtained analytically for n 4, and
≤
numerically for n > 4. In particular, the exact dispersion relations for the
cases n = 2,3,4 are given below.
For n = 2 is given by
1 1
ε = (µ +µ )h (µ µ )2h2 +J2 +J2 +2J J cos(2q), (23)
q −2 1 2 ± 2 1 − 2 1 2 1 2
q
6
and for n = 3 by [24]
1 θ +pπ
ε = (µ +µ +µ )h+2√ Rcos q , (24)
q 1 2 3
−3 − 3 !
where p = 0,2,4 and θ = arccos R /√ R3 , with R and R given by
q q q
−
(cid:16) (cid:17)
3a a2 9a a 27a 2a3
R = 2 − 1, R = 2 1 − q − 1, (25)
q
9 54
where
a =(µ +µ +µ )h,
1 1 2 3
1
a =(µ µ +µ µ +µ µ )h2 J2 +J2 +J2 ,
2 1 2 1 3 2 3 − 4 1 2 3
1 (cid:16) 1 (cid:17)
a =µ µ µ h3 µ J2 +µ J2 +µ J2 h+ J J J cos(3q). (26)
q 1 2 3 − 4 1 2 2 3 3 1 4 1 2 3
(cid:16) (cid:17)
For n = 4, the four branches of the dispersion relation are obtained from the
expression [24]
A +2Y (A +2Y ) 2(A B / A +2Y )
A 1 q 1 q 1 1 1 q
∓ ± − − ∓
ε = + r , (27)
q q q
−4 2
by considering the sign combinations (+ + +, , + , + +), and
− − − − − −
where
3A2 A3 AB
A = +B, B = +C,
1 1
− 8 8 − 2
5 P θ +2π S 27
q q q
Y = A 2 − cos( ), θ = arccos ,
q −6 1 − s 3 3 q  2 v P3
u− q
u
A 2 A3 A G B2 t 
P = 1 G , S = 1 + 1 q 1,
q q q
− 12 − −108 3 − 8
3A4 BA2 AC
G = + +D , (28)
q q
−256 16 − 4
and
7
A=(µ +µ +µ +µ )h,
1 2 3 4
B=(µ µ +µ µ +µ µ +µ µ +µ µ +µ µ )h2
2 1 4 1 3 1 4 2 3 2 4 3
−
(J 2 +J 2 +J 2 +J 2)
1 2 3 4
,
− 4
C=(µ µ µ +µ µ µ +µ µ µ +µ µ µ )h3
4 3 1 3 2 1 4 3 2 4 2 1
−
(J2µ +J2µ +J2µ +J2µ +J2µ +J2µ +J2µ +J2µ )h
4 3 2 1 1 3 4 2 2 4 1 4 3 1 3 2 ,
− 4
(J2µ µ +J2µ µ +J2µ µ +J2µ µ )h2
D =µ µ µ µ h4 4 3 2 3 2 1 2 4 1 1 4 3 +
q 4 3 2 1
− 4
(J2J2 +J2J2) J J J J
+ 1 3 4 2 1 2 3 4 cos(4q). (29)
16 − 8
The excitation spectrum, given by eq.(19), can also be obtained by a transfer
matrix technique which leads to a different, but equivalent, expression. This
calculation is presented in appendix A, and it is expressed as
(ω,h)
ℑ trace[T (ω,h)] = 2cos(dq), (30)
cell
J J ...J ≡
1 2 n
where T (ω,h) is given by eq.(A.15). Although it is not shown in Appendix
cell
A, we have verified that (ω,h) depends on the square of the exchange con-
ℑ
stants, J2,J2,....,J2 . This means that the effect of the change of the signs
{ 1 2 n}
of the J′s is to introduce a shift of π/d in the spectrum wave-vectors, as can
be seen in eq.(30).
AsitisalsoshowninAppendixA,forzeroexternalfield,thereisnoenergygap
between the ground state and the first excited state, for n odd and arbitrary
J′s. However, for n even, the gap in the spectrum is only suppressed when the
J′s satisfy the condition
J J ...J = J J ...J . (31)
1 3 n−1 2 4 n
This latter result can also be obtained from the criticality condition for 1D
random Ising model in a transverse field [25] and the equivalence between
XY chain and two decoupled transverse field Ising chains[26,27].
The excitation spectrum for a chain with n = 8, equal and different magnetic
moments, is shown in Fig. 2. As can be seen, when the J′s satisfy eq.(31)
(continuous line) there is no gap at zero field, whereas a gap is present when
eq.(31) is not satified (dot-dashed line). Although these results are presented
for equal µ′s, they are still valid when the µ′s are different.
The dotted line represents the spectrum for nonzero field, different µ′s and
the J′s also satisfying eq.(31). As can be seen, the spectrum shifts and opens
a gap at zero and boundary wave-vectors, which is a consequence of the non-
comutativity, in this case, of the field term with the Hamiltonian.
8
3 The induced magnetization and isothermal susceptibility χzz
T
From eqs.(3) and (22) we can express the local induced magnetization Mz
l,m
as
µ µ
Mz µ Sz = m u∗ u n m, (32)
l,m ≡ m l,m N q,km q,km q,k − 2
D E Xq,k
where the occupation number n is given by
q,k
1
n = , (33)
q,k 1+eβεq,k
and the calculation can be done by considering H = H− [20,21,22], since we
are interested in the thermodynamic limit.
We can define an average cell magnetization operator in the z direction, τz,
l
as
1 n 1 n 1
τz µ Sz = µ c† c , (34)
l ≡ n m l,m n m l,m l,m − 2
m=1 m=1 (cid:18) (cid:19)
X X
which corresponds to a generalization of the cell spin operator defined in the
study of the alternating superlattice[18]. By using eqs. (32) and (34) we can
write the induced magnetization per site as
1 n 1 1
Mz τz = µ u∗ u n , (35)
≡ h l i n mN q,km q,km q,k − 2
m=1 q,k
X X
 
which can be written in the form
1 βε
Mz = µ u∗ u tanh q,k . (36)
−2nN m q,km q,km 2 !
q,k,m
X
The isothermal susceptibility can be obtained from the expression
1∂Mz 1 ∂u∗ βε
χzz = µ q,kmu tanh q,k +
T ≡ n ∂h −2nN  m ∂h q,km 2 !
qX,k,m
∂u βε
+ µ u∗ q,km tanh q,k +
m q,km ∂h 2 !
q,k,m
X
β βε ∂ε
+ µ u∗ u sech2 q,k q,k . (37)
2 m q,km q,km 2 ! ∂h 
qX,k,m 
At T = 0, where the system presents quantum transitions, the induced mag-
netization, obtained from eq. (36) in the limit T 0, is given by
→
1
Mz = µ u∗ u sign(ε ), (38)
−2nN m q,km q,km q,k
q,k,m
X
9
and from eq.(37) we obtain χzz, which diverges at the critical fields h . We
T c
identify the largest critical field as h , since for h > h the induced magneti-
s s
zation is saturated.
For identical magnetic moments, the average cell magnetization operator, τz,
l
is proportional to the average cell spin operator, 1 n Sz , and eqs (36)
n m=1 l,m
and (38) can be written, respectively, as
P
1 βε
Mz = µtanh q,k , (39)
−2nN 2 !
q,k,m
X
1
Mz = µsign(ε ), (40)
q,k
−2nN
q,k,m
X
where µ = µ for any m.
m
The results for Mz and χzz, at T = 0, are presented in Fig. 3 as functions of
T
the field h, for a chain with n = 8 and identical µ′s. The continuous line and
the dashed line correspond to the cases where the exchange constants satisfy
and do not satisfy the condition shown in eq.(31), respectively. As we have
shown for the alternating superlattice [18], the magnetization also presents
plateaus which are limited by critical fields h , where the isothermal suscep-
c
tibility diverges, which correspond to quantum phase transitions induced by
the field. The regions of plateaus, which we associate with disordered regions
[18], correspond to the gaps in the excitation spectrum, and the critical fields
are associated with the zero-energy mode with the wave-vectors q = 0 and
π/d.
As it can also be seen, when there is no gap in the excitation spectrum at
zero field (continuous line), the zero magnetization plateau, which is present
when there is a gap (dashed line), is suppressed, and consequently the total
number of transitions induced by the field is n 1. It should be noted that
−
the local magnetization also presents plateaus and non-analytic behaviour at
the critical fields.
For different magnetic moments, the critical fields are obtained as in the pre-
vious case, and the results for Mz and χzz, also at T = 0, are shown in Fig. 4.
T
The main difference from the previous case is the fact that the magnetization
does not present plateaus, and the susceptibility χzz is different from zero at
T
both sides of the transition, although it diverges at one side only. The local
magnetizationpresentssimilarbehaviourasthetotalmagnetization,andthere
is no suppression of the transition at zero field when there is no gap in the
excitation spectrum (continuous line). The regions between two critical fields
where the susceptibility χzz is finite correspond to the gaps in the spectrum.
T
As in the case of equal magnetic moments, we associated these regions with
disordered regions.
10