The effect of disorder on the quantum phase transitions induced by a
transverse field, anisotropy, and dimerization in XY spin chains is
investigated. The low-energy behavior near the critical point is described by a
Dirac-type equation with a random mass for which an exact analytic treatment is
possible. Results obtained for the dynamical critical exponent, the specific
heat, and transverse susceptibility agree with results recently obtained using
a real space renormalization group decimation technique, supporting Fisher's
claim that it is exact. A non-zero transverse field changes the universality
class of the anisotropy transition.Comment: 5 pages, RevTeX + epsf, 2 figures

A. A. Golub

A. Klein

A. P. Young

B. L. Altshuler

C. Dasgupta

C. Monthus

D. S. Fisher

E. Lieb

E. R. Smith

E. Westerberg

H. J. Fischbeck

I. M. Lifshits

J. Mertsching

J. P. Bouchaud

L. N. Bulaevskii

M. den Nijs

P. Pfeuty

R. A. Hyman

R. B. Griffiths

R. H. McKenzie

R. Hayn

R. Shankar

Ross H. McKenzie

S. Katsura

T. N. Nguyen

T. Senthil

English

arXiv

Crossref

Physical Review Letters

Exact Results for Quantum Phase Transitions in Random<i>XY</i>Spin Chains

The effect of disorder on the quantum phase transitions induced by a transverse field, anisotropy, and dimerization in XY spin chains is investigated. The low-energy behavior near the critical point is described by a Dirac-type equation for which an exact analytic treatment is possible. Results obtained for the dynamical critical exponent, the specific heat, and the transverse susceptibility agree with results recently obtained using a real space renormalization group decimation technique. A nonzero transverse field changes the universality class of the anisotropy transition

McKenzie, RH

University of Queensland eSpace

VOLUME 77, NUMBER 23 P HY S I CA L REV I EW LE T T ER S 2 DECEMBER 1996
4804Exact Results for Quantum Phase Transitions in Random XY Spin Chains
Ross H. McKenzie*
School of Physics, University of New South Wales, Sydney 2052, Australia
(Received 18 September 1996)
The effect of disorder on the quantum phase transitions induced by a transverse field, anisotropy,
and dimerization in XY spin chains is investigated. The low-energy behavior near the critical point is
described by a Dirac-type equation for which an exact analytic treatment is possible. Results obtained
for the dynamical critical exponent, the specific heat, and the transverse susceptibility agree with results
recently obtained using a real space renormalization group decimation technique. A nonzero transverse
field changes the universality class of the anisotropy transition. [S0031-9007(96)01787-5]
PACS numbers: 75.10.Jm, 68.35.Rh, 75.10.Nr, 75.50.EeTheoretical studies of quantum phase transitions in
the presence of quenched (i.e., time-independent) dis-
order have been stimulated by recent experiments
on spin glasses, 4He absorbed in porous media, and
superconductor-insulator transitions in dirty thin films.
Compared to thermal phase transitions in disorder-free
systems these transitions are poorly understood because
many of the theoretical methods (e.g., exact solutions,
renormalization group, e expansions) that have been
so useful for pure systems are difficult to implement
for disordered systems. Yet these phase transitions are
associated with particularly rich physics such as large
differences between average and typical behavior, new
universality classes, logarithmic scaling, Griffiths phases
[1] (in which susceptibilities diverge although there
are only short-range correlations), and the breakdown
of folklore such as “the correlation length is inversely
proportional to the energy gap” [2]. This Letter considers
a simple exactly soluble model which has many of these
interesting properties.
Fisher recently performed an exhaustive study of the
effect of randomness on the simplest spin model to
undergo a quantum phase transition: the transverse field
Ising spin chain [3]. He used a real space renormalization
group decimation technique (RSRGDT) [4] which he
claims is exact near the critical point. Fisher found
the phase diagram (which included Griffiths phases near
the critical point), all the critical exponents, and scaling
forms for the magnetization and correlation functions in
an external field. The latter have never been derived for0031-9007y96y77(23)y4804(4)$10.00the disorder-free case but can be derived in the presence
of disorder because distributions become extremely broad
near the critical point. The same model was recently
studied numerically by Young and Rieger [5], who
found results consistent with Fisher. The RSRGDT
has also been used to study the effect of disorder on
dimerized [6] and anisotropic [7] spin chains, chains
with random spin sizes [8], and quantum Potts and
clock chains [9]. Possible experimental realizations of
random spin chains are quinoloniumsTCNQd2 [10] and
SR3CuPt12xIrxO6 [11].
This Letter considers quantum phase transitions in XY
spin chains in the presence of disorder. The spin chains
are mapped onto a fermion model, the continuum limit
of which is a Dirac-type equation with random mass, for
which exact analytic results can be derived. The results
agree with those of the RSRGDT [3,6,7], supporting
Fisher’s claim that it is exact near the critical point [12].
New results are obtained for the anisotropy transition in
a nonzero average transverse field. It is in a different
universality class to the Ising transition.
The Hamiltonian to be considered is
H ­ 2
LX
i­1
fJxi s
x
i s
x
i11 1 J
y
i s
y
i s
y
i11 1 his
z
i g . (1)
The hsai j are Pauli spin matrices, and the interactions Jxi ,
J
y
i , and transverse fields hi may be independent random
variables (see Table I) with a Gaussian distribution. The
average values will be denoted kJxi l ; Jx , kJ
y
i l ; Jy , and
khil ; h, and will all be assumed to be positive.TABLE I. Different parameters for the continuum limit of three different transitions. Near the critical point all are described by
the low-energy effective Hamiltonian (3). The lattice constant is set to unity. The Fermi wave vector kF determines whether the
problem is commensurate or not. yF is the Fermi velocity. D is a measure of the deviation from criticality. The variance of the
random variable equals g, which enters Eq. (4).
Transition cos kF yF D Csx ­ ndy Random variable
Ising 1 jJx 2 Jyj h 2 Jx 2 Jy scyn , cnd hi
Anisotropy h
Jx 1 Jy
sJx 1 Jyd sin kF sJx 2 Jyd sin kF scyn e2inkF , cne2inkF d J
x
i 2 J
y
i
Dimerization 0 2J D scyn i2n , cyn ind J
x
i ­ J
y
i© 1996 The American Physical Society
VOLUME 77, NUMBER 23 P HY S I CA L REV I EW LE T T ER S 2 DECEMBER 1996At zero temperature and in the absence of disorder the
model undergoes two distinct quantum phase transitions
[13]. Both transitions are second order. The transition
at Jx 1 Jy ­ h from a paramagnetic to a ferromagnetic
phase will be referred to as the Ising transition [14]. The
transition at Jx ­ Jy for h , sJx 1 Jyd from an Ising
ferromagnet with magnetization in the x direction to one
with magnetization in the y direction will be referred
to as the anisotropy transition [7,15]. The dimerization
transition involves the Hamiltonian (1) with kJxi l ­
kJyi l ­ J 1 s21diD and khil ­ 0. At D ­ 0 there is a
transition from a spin liquid to a gapped phase with long-
range topological order [6].
Under a Jordan-Wigner transformation which maps
spins onto spinless fermions, (1) becomes [15]
H ­ 2
LX
i­1
fsJxi 1 J
y
i d sc
y
i ci11 1 c
y
i11cid
1 sJxi 2 J
y
i d sc
y
i c
y
i11 2 cici11d 1 his2c
y
i ci 2 1dg ,
(2)
where boundary terms have been neglected and cyi and ci
denote creation and annihilation operators for a fermion
on site i. In the disorder-free case (2) can be diagonalized
by a Bogoliubov transformation [13,15].
The continuum limit of (2) must be taken relative to
the Fermi wave vector kF , the wave vector at which
the energy gap in the fermionic spectrum occurs for the
disorder-free case [13]. This wave vector depends on
which transition is being considered (see Table I). The
low-energy properties of all the transitions in Table I are
described by [16]
H ­
Z
dxCy
h
2iyFs
3 ›
›x
1 V sxds1 1 V sxdps2
i
C ,
(3)
where sa (a ­ 1, 2, 3) and s6 ; 12 ss1 6 s2d are Pauli
matrices. Csxd is a spinor, with (modulo a transformation
in spinor space) the components given in Table I. V sxd is
Gaussian white-noise potential with
kV sxdl ­ D, kV sxdV sx0dpl ­ gdsx 2 x0d . (4)
D measures the deviation from the critical point, and jDj
is the energy gap in the absence of disorder. Table I lists
values of D, the Fermi velocity yF , and the random vari-
ables whose variance equals g, for the Ising, anisotropy,
and dimerization transitions. kF is commensurate with the
lattice and V sxd is real for the Ising transition (kF ­ 0)
and the anisotropy and dimerization transitions in zero
transverse field (kF ­ py2). It will be seen that the effect
of disorder is significantly different for commensurate and
incommensurate cases. The continuum limit describes the
low-energy properties (E ¿ yF) when yF À D, pg, i.e.,
arbitrarily close to the critical point and for weak disorder.
It is useful to define an energy D and a dimensionless
parameter d which are measures of the disorder strengthand the deviation from criticality, respectively,
D ;
g
yF
, d ;
jDj
D
. (5)
Note that for the Ising transition with Jy ­ 0, to leading
order in DyJx , the parameter d defined by Fisher [3] and
Young and Rieger [5] is DyD.
The advantage of casting the problem in the form of the
Hamiltonian (3) is that the latter has been studied exten-
sively previously, and exact analytic expressions given for
the energy dependence of the disorder-averaged density
of states krsEdl and the localization length lsEd [17–23].
(Because of the one-dimensionality all the states are local-
ized by the disorder.) The exact results have been found
by Fokker-Planck equations [17], supersymmetry [19,20],
the replica trick [23], S-matrix summation [18], and the
Dyson-Schmidt method [21]. The localization length can
be found because in one dimension it is related to the real
part of the one-fermion Green’s function [19]. The den-
sity of states and the localization length can be written in
terms of f 0dsud, the derivative of a dimensionless function
fdsud given in Table II,
d
dE
1
lsEd
1 iprsEd ­ pr0f 0dsEyDd , (6)
where r0 ; 1ypyF is the value of the density of states at
high energies (jEj À D, Dd.
The low-energy (jEj ¿ D) behavior of the density of
states is given in Table II. For the commensurate case
the density of states diverges at E ­ 0 for d , 1y2 and
is zero at E ­ 0 for d . 1y2. These two cases lead
to qualitatively very different behavior. In the former
case some susceptibilities will diverge as the temperature
approaches zero. This corresponds to a Griffiths or
weakly ordered phase [1]. Hence for the Ising transition
there will be four phases: ferromagnet, weakly ordered
ferromagnet, weakly ordered paramagnet, and paramagnet
[3]. The full energy dependence of the density of states
for various values of d is shown in Fig. 1.
Distinctly different behavior occurs for the incommen-
surate case. The density of states is always finite at zero
energy. There is a smooth crossover from gapless behav-
ior (krs0dl , r0 for d , 1) to effectively gapped behav-
ior (krs0dl ¿ r0 for d À 1). Hence the Griffiths phases
still exist but no longer have clearly defined boundaries.
Similarly, for the commensurate case more than one type
of disorder (e.g., the anisotropy transition with both a ran-
dom transverse field and random anisotropy) removes the
singularity in the density of states [17].
The specific heat.—Because the eigenstates of the
Hamiltonian (3) are noninteracting fermions the low-
temperature behavior of the specific heat and the trans-
verse susceptibility (for the dimerization and anisotropy
transitions) follows from the energy dependence of the
disorder-averaged density of states [10,24] and are given4805
VOLUME 77, NUMBER 23 P HY S I CA L REV I EW LE T T ER S 2 DECEMBER 1996TABLE II. Summary of the low-energy behavior of different physical quantities. The commensurate case corresponds to the
Ising transition and to the anisotropy and dimerization transitions in zero transverse field. The incommensurate case corresponds
to the anisotropy transition in a nonzero transverse field. For the commensurate case qualitatively different behavior occurs at the
critical point (d ­ 0) and away from it (d Þ 0). The function fdsud determines the full energy dependence of the density of states
and the localization length [compare Eq. (6)]. I0sxd and I1sxd are the zeroth and first order modified Bessel functions, respectively.
Gsdd is the gamma function. Hs2dd sud is a Hankel function of index d. Iiysxd is a modified Bessel function with imaginary index.
For small x, Gsxd , x21, I0sxd , 1 1 x2y4, and I1sxd , xy2.
Quantity Symbol Commensurate Incommensurate
d Þ 0 d ­ 0
Density
of states
k rsEdlyr0 d
Gsdd2

E
D
2d21 D
jEj fln j ED jg3
1
I0s2dd2
Specific heat kCsT dl d
Gsdd2
ˆ
T
D
!2d 1
flns TD dg3
T
I0s2dd2
Dynamical z ‘ 1
critical exponent
1
2d
Localization lsEd
length
yF
D
yF
D
ln

D
E

4yF
D
"
1 1
8dI1s2dd
I0s2dd
#21
Transverse kxzzsT dl T 2d21
susceptibility
1
Tflns TD dg2
1
I0s2dd2
fdsud 2u
›
›u
lnfH s2dd sudg d
›
›d
lnfI22ius2ddgin Table II. The transverse susceptibility clearly shows
a Griffiths singularity. The results for the commensurate
case agree with the RSRGDT [3,6,7].
FIG. 1. Energy dependence of the disorder-averaged density
of states for the commensurate case for various values of the
dimensionless parameter d [see Eq. (5)], which is a measure of
the deviation from criticality. The density of states is singular
at the Fermi energy sE ­ 0d when d , 12 . This parameter
range corresponds to a Griffiths phase. Note that only far from
criticality (d À 1) is there effectively a gap in the system. This
contrasts with the disorder-free case, for which there is always
a gap except at the critical point.4806Dynamical critical exponent z.—This relates the scal-
ing of energy (or time) scales to length scales. A crude
scaling argument [25] implies that krsEdl , E1yz21. Thus
for the commensurate case, to leading order in d, z ­
1y2d, in agreement with Fisher [3] and Young and Rieger
[5]. This is a particularly striking result because it shows
that (i) z is not universal, and (ii) z diverges at the critical
point. The latter implies logarithmic scaling and activated
dynamics [2]. In contrast for the incommensurate case,
the density of states is finite and constant at low energies
and so z ­ 1, as in the absence of disorder.
Finite size scaling.—Monthus et al. [26] studied an
equation equivalent to (3) with V sxd real and D ­ 0 [23].
They have shown that on a line of length L, for a typical
potential V sxd the lowest eigenvalue E0 scales like E20 ,
exps2cL1y2d, where c is a constant. This is consistent
with the scaling of lnE0 with L1y2 at the critical point
found numerically [5]. The average kE20l , exps2dL1y3d
where d is a constant [26], showing the discrepancy
between average and typical values.
Correlation lengths.—Fisher [3] stressed the distinc-
tion between average and typical correlations. If Cij ;
kAiAjl denotes a correlation function of a variable Ai then
the average correlation function Cav srd ; 1L
PL
i­1 Ci,i1r
is what is measured experimentally. Away from the criti-
cal point Cav srd , exps2ryjavd, where jav is the aver-
age correlation length. However, Cav srd is dominated by
rare pairs of spins with Cij , 1. In contrast, with proba-
bility one Ci,i1r , exps2ryjtypd where jtyp denotes the
typical correlation length. It is distinctly different from
jav (jtyp ¿ jav ), having a different critical exponent.
VOLUME 77, NUMBER 23 P HY S I CA L REV I EW LE T T ER S 2 DECEMBER 1996The localization length is useful because it is proportional
to the typical correlation length for quantities that are di-
agonal in the fermion representation [27]. Consequently,
the results in Table II imply that for the commensurate
case the typical correlation length jtyp , ls0d21 , D21
is consistent with previous work [3,28]. This critical ex-
ponent is not modified by the presence of disorder. In
contrast, for the incommensurate case, ls0d is finite at the
critical point.
Distribution functions.—Both Fisher [3] and Young
and Rieger [5] considered the distribution functions for
various quantities. The present approach can be used
to obtain exact results by using known results for the
distribution function for the density of states of one-
dimensional conductors [29]. The cumulants (or irre-
ducible moments) of the transverse susceptibility for the
incommensurate case at criticality are
ksxzzdnlc ­ xn0
Gs 32 dGsnd
2
Gsn 1 12 d
µ
pD
16T
¶n21
, (7)
where x0 is the susceptibility in the absence of disor-
der. These moments completely determine the distribu-
tion function which has been extensively analyzed in
Ref. [29]. For T À D the distribution is Gaussian and
centered on the mean. In contrast, for T ¿ D the distri-
bution is extremely broad and asymmetric. The maximum
of the distribution then occurs near x0 exps21ysdy2s,
where s ­ 16TyDp , which is much less than the mean
x0. This shows the large discrepancy between typical and
average values. Hopefully (7) can be generalized to the
commensurate case and away from criticality.
This work was supported by the Australian Research
Council. I thank D. D. Betts, R. J. Bursill, D. S. Fisher,
M. Gulacsi, G. Honner, and V. Kotov for very helpful
discussions. D. Scarratt produced the figure.
*Electronic address: ross@newt.phys.unsw.edu.au
[1] R. B. Griffiths, Phys. Rev. Lett. 23, 17 (1969).
[2] H. Rieger and A. P. Young, Report No. cond-maty
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[3] D. S. Fisher, Phys. Rev. Lett. 69, 534 (1992); Phys. Rev. B
51, 6411 (1995).
[4] C. Dasgupta and S.-k. Ma, Phys. Rev. B 22, 1305 (1980).[5] A. P. Young and H. Rieger, Phys. Rev. B 53, 8486 (1996).
[6] R. A. Hyman et al., Phys. Rev. Lett. 76, 839 (1996).
[7] D. S. Fisher, Phys. Rev. B 50, 3799 (1994).
[8] E. Westerberg et al., Phys. Rev. Lett. 75, 4302 (1995).
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(1996).
[10] L. N. Bulaevskii et al., Zh. Eksp. Teor. Fiz. 62, 725 (1972)
[Sov. Phys. JETP 35, 384 (1972)].
[11] T. N. Nguyen, P.A. Lee, and H.-C. zur Loye, Science 271,
489 (1996).
[12] However, it should be noted that in the fermion rep-
resentation, results involving the magnetization and its
correlations cannot be derived because of the nonlocal
relationship between the spin and the fermion operators.
[13] M. den Nijs, in Phase Transitions and Critical Phenom-
ena, edited by C. Domb and J. L. Lebowitz (Academic,
London, 1988), Vol. 12, p. 264.
[14] P. Pfeuty, Ann. Phys. (N.Y.) 27, 79 (1970).
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407 (1961); S. Katsura, Phys. Rev. 127, 1508 (1962).
[16] An easy way to derive (3) for the transverse Ising model
in a random transverse field is to take the continuum limit
of Eq. (39) in Ref. [5]. For the disorder-free case, see
R. Shankar, Acta Phys. Pol. B 26, 1835 (1995).
[17] I.M. Lifshits, S. A. Gredeskul, and L.A. Pastur, Introduc-
tion to the Theory of Disordered Systems (Wiley, New
York, 1988), p. 109.
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[21] J. Mertsching, Phys. Status Solidi B 174, 129 (1992).
[22] For another application of this model, see R. H. McKenzie
and J.W. Wilkins, Phys. Rev. Lett. 69, 1085 (1992).
[23] A mathematically equivalent model describes diffusion of
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et al., Ann. Phys. (N.Y.) 201, 285 (1990).
[24] E. R. Smith, J. Phys. C 3, 1419 (1970).
[25] The total number of states (per unit length) with energy
less than E, NsEd ;
RE
0 rsE0ddE0 scales with the inverse
of any length scale ,. By definition E , ,2z .
[26] C. Monthus et al., Phys. Rev. E 54, 231 (1996).
[27] A. Klein and J. F. Perez, Commun. Math. Phys. 128, 99
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[28] R. Shankar and G. Murphy, Phys. Rev. B 36, 536 (1987).
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95, 348 (1989) [Sov. Phys. JETP 68, 198 (1989)].4807


Exact results for quantum phase transitions in random XY spin chains

Exact results for quantum phase transitions in random XY spin chains

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