VOLUME 86, NUMBER 22 PHYSICAL REVIEW LETTERS 28 MAY 2001
Dynamical Response of Quantum Spin-Glass Models at T 5 0
L. Arrachea and M. J. Rozenberg
Departamento de Física, FCEN, Universidad de Buenos Aires,
Ciudad Universitaria Pabellón I, (1428) Buenos Aires, Argentina
(Received 29 August 2000)
We study the behavior of two archetypal quantum spin glasses at T 0 by exact diagonalization
techniques: the random Ising model in a transverse field and the random Heisenberg model. The behavior
of the dynamical spin response is obtained in the spin-glass ordered phase. In both models it is gapless
and has the general form x
00
v qdv 1x
00
reg
v, with x
00
reg
v v for the Ising and x
00
reg
v
const for the Heisenberg, at low frequencies. The method provides new insight to the physical nature of
the low-lying excitations.
DOI: 10.1103/PhysRevLett.86.5172 PACS numbers: 75.10.Nr, 75.10.Jm, 75.40.Gb
Quantum random magnets are fascinating systems that
continuously challenged our understanding for more than
30 years [1–4]. Despite much theoretical effort, many
fundamental questions remain unsolved.
Among the variety of theoretical models of quantum
spin glasses we can single out two which have received
continuous attention for many years now, namely, the
infinite-range random Ising model in a transverse field
(RITF) and the random Heisenberg (RH) model. The for-
mer is believed to be directly relevant to the physics of
the Li
12x
Ho
x
YF
4
compound which is a dipolar coupled
random magnet [5] and was the focus of recent beauti-
ful experiments [6]. On the other hand, the second model
can be taken as the natural starting point to study the ef-
fects of disorder in the magnetic behavior of transition met-
als and rare earth compounds. In both cases, the physics
within the disordered paramagnetic phase is rather well
known [7–11]. In contrast, there are not many theoretical
results in the literature on the behavior of these models in-
side the ordered phase. This is mainly due to the fact that
most of the theoretical treatment relies on the replica trick
[2,12], which usually becomes impractical in the glassy
phase when replica symmetry breaking occurs. Among
the available results, we should mention a Landau theory
for a class of disordered models that provided valuable in-
sight into the critical region of the RITF [13], and also the
study of SUM extensions of the RH model treated in the
limit of M ! ` [14,15].
In this paper we concentrate on these models at zero
temperature T 0 and obtain detailed behavior inside
the spin-glass phase, which has eluded an analytic solu-
tion so far. We employ the method of exact diagonaliza-
tion of a large number of finite-size clusters with magnetic
interactions drawn from a random probability distribution
function [16]. This method avoids the need for the replica
trick as one can perform direct averages over the disor-
der. Our main result is the numerical solution of both
models in their glassy phases. Various equilibrium ob-
servables are computed, and compelling evidence is found
that the dynamical spin response of both models behaves as
x
00
v qdv 1x
00
reg
v in the thermodynamic limit.
The method also provides a transparent physical picture
for the structure of the ground state and the nature of its
elemental excitations.
The general model Hamiltonian reads
H
1
p
N
N
X
i ,j1
J
ij
S
z
i
S
z
j
1aS
x
i
S
x
j
1 S
y
i
S
y
j
2G
N
X
i 1
S
x
i
, (1)
where S
m
i
, m x , y , z are components of a spin
1
2
operator
at site i , J
ij
denote the infinite-range magnetic interactions
between sites i and j which are normally distributed with
variance J
2
that we set to unity, N is the number of sites, G
is an applied transverse field acting at every site, and 0 #
a# 1. For G 0 and a 0, one has the well-known
Sherrington-Kirkpatrick (SK) spin-glass model [17]. Start-
ing from this limit, the effects of quantum fluctuations are
introduced by G or a. At T 0 and a 0 the model
is the RITF and has a quantum critical point for G G
c
[7]. Alternatively, for G 0 and a 1, one has the RH
model which is SU(2) rotationally invariant.
These types of infinite-range quantum spin models are
usually treated within an approach based on the replica
trick and formulated on the imaginary time axis [10]. In
contrast, our treatment avoids the need for the trick as the
average over the random ensemble is directly performed.
In addition, one also avoids the problem of analytic con-
tinuation to compute the dynamical response on the real
frequency axis. We take systems of size N from the ran-
dom ensemble and exactly diagonalize the Hamiltonians
to obtain averaged quantities over a large number of
samples. The exact ground state (GS) and dynamical cor-
relation functions are calculated by the Lanczos method
[18]. We then extrapolate our results to the infinite-size
system. Systems of N # 17 are solved and averages over
several thousands of disorder realizations are typically
performed. Although one deals with systems of finite size
that have a finite number of poles, the average over the
disorder naturally produces smooth response functions
5172 0031-9007 01 86(22) 5172(4)$15.00 © 2001 The American Physical Society