Entanglement entropy at infinite randomness fixed points in higher
  dimensions
Entanglement entropy at infinite randomness fixed points in higher dimensions
Yu-Cheng Lin1, Ferenc Iglói2,3 and Heiko Rieger1
Theoretische Physik, Universität des Saarlandes, 66041 Saarbrücken, Germany
Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary
Institute of Theoretical Physics, Szeged University, H-6720 Szeged, Hungary
(Dated: November 3, 2018)
The entanglement entropy of the two-dimensional random transverse Ising model is studied with
a numerical implementation of the strong disorder renormalization group. The asymptotic behavior
of the entropy per surface area diverges at, and only at, the quantum phase transition that is
governed by an infinite randomness fixed point. Here we identify a double-logarithmic multiplicative
correction to the area law for the entanglement entropy. This contrasts with the pure area law valid
at the infinite randomness fixed point in the diluted transverse Ising model in higher dimensions.
PACS numbers: Valid PACS appear here
Extensive studies have been devoted recently to under-
stand ground state entanglement in quantum many-body
systems [1]. In particular, the behavior of various entan-
glement measures at/near quantum phase transitions has
been of special interest. One of the widely used entan-
glement measures is the von Neumann entropy, which
quantifies entanglement of a pure quantum state in a bi-
partite system. Critical ground states in one dimension
(1D) are known to have entanglement entropy that di-
verges logarithmically in the subsystem size with a uni-
versal coefficient determined by the central charge of the
associated conformal field theory [2]. Away from the crit-
ical point, the entanglement entropy saturates to a finite
value, which is related to the finite correlation length.
In higher dimensions, the scaling behavior of the entan-
glement entropy is far less clear. A standard expectation
is that non-critical entanglement entropy scales as the
area of the boundary between the subsystems, known as
the ”area law” [3, 4]. This area-relationship is known to
be violated for gapless fermionic systems [5] in which a
logarithmic multiplicative correction is found. One might
suspect that whether the area law holds or not depends
on whether the correlation length is finite or diverges.
However, it has turned out that the situation is more
complex: numerical findings [7] and a recent analytical
study [8] have shown that the area law holds even for
critical bosonic systems, despite a divergent correlation
length. This indicates that the length scale associated
with entanglement may differ from the correlation length.
Another ongoing research activity for entanglement in
higher spatial dimensions is to understand topological
contributions to the entanglement entropy [9].
The nature of quantum phase transitions with
quenched randomness is in many systems quite different
from the pure case. For instance, in a class of systems
the critical behavior is governed by a so-called infinite-
randomness fixed point (IRFP), at which the energy scale
ǫ and the length scale L are related as: ln ǫ ∼ Lψ with
0 < ψ < 1. In these systems the off-critical regions
are also gapless and the excitation energies in these so-
called Griffiths phases scale as ǫ ∼ L−z with a nonuni-
versal dynamical exponent z <∞. Even so, certain ran-
dom critical points in 1D are shown to have logarith-
mic divergences of entanglement entropy with universal
coefficients, as in the pure case; these include infinite-
randomness fixed points in the random-singlet universal-
ity class [12, 13, 14, 15, 16] and a class of aperiodic singlet
phases [17].
In this paper we consider the random quantum Ising
model in two dimensions (2D), and examine the disorder-
averaged entanglement entropy. The critical behavior of
this system is governed by an IRFP [10, 11] implying that
the disorder strength grows without limit as the system is
coarse grained in the renormalization group (RG) sense.
In our study, the ground state of the system and the
entanglement entropy are numerically calculated using a
strong-disorder RG method [18, 19], which yields asymp-
totically exact results at an IRFP. To our knowledge this
is the first study of entanglement in higher dimensional
interacting quantum systems with disorder.
The random transverse Ising model is defined by the
Hamiltonian
H = −
〈i,j〉
i . (1)
Here the {σαi } are spin-1/2 Pauli matrices at site i of an
L × L square lattice with periodic boundary conditions.
The nearest neighbor bonds Jij(≥ 0) are independent
random variables, while the transverse fields hi(≥ 0) are
random or constant. For a given realization of random-
ness we consider a square block A of linear size ℓ, and
calculate the entanglement between A and the rest of
the system B, which is quantified by the von Neumann
entropy of the reduced density matrix for either subsys-
tems:
S = −Tr(ρA log2 ρA) = −Tr(ρB log2 ρB). (2)
The basic idea of the strong disorder RG (SDRG) ap-
proach is as follows [18, 19]: The ground state of the sys-
tem is calculated by successively eliminating the largest
http://arxiv.org/abs/0704.0418v2
FIG. 1: (color online). An example of typical ground state
in the random quantum Ising model (a) in 1D, and (b) in 2D;
it contains a collection of spin clusters of various sizes, which
are formed and decimated during the RG. The entanglement
of a block (shaded area) is give by the number of decimated
clusters (indicated by red loops) that connect the block with
the rest of the system.
local terms in the Hamiltonian and by generating a new
effective Hamiltonian in the frame of the perturbation
theory. If the strongest bond is Jij , the two spins at i and
j are combined into a ferromagnetic cluster with an effec-
tive transverse field h̃(ij) =
. If, on the other hand,
the largest term is the field hi, the spin at i is decimated
and an effective bond is generated between its neighbor-
ing sites, say j and k, with strength J̃jk =
JijJik
. After
decimating all degrees of freedom, we obtain the ground
state of the system, consisting of a collection of indepen-
dent ferromagnetic clusters of various sizes; each cluster
of n spins is frozen in an entangled state of the form:
(| ↑↑ · · · ↑︸ ︷︷ ︸
n times
〉+ | ↓↓ · · · ↓︸ ︷︷ ︸
n times
〉). (3)
In this representation, the entanglement entropy of a
block is given by the number of clusters that connect
sites inside to sites outside the block [Fig. 1]. We note
that correlations between remote sites also contribute to
the entropy due to long-range effective bonds generated
under renormalization.
In 1D the RG calculation can be carried out analyti-
cally and the disorder-averaged entropy Sℓ of a segment
of length ℓ has been obtained as Sℓ =
log2 ℓ [12].
In higher dimensions the RG method can only be imple-
mented numerically. The major complication in this case
is that the model is not self-dual and thus the location of
the critical point is not exactly known. To locate the crit-
-1.5 -1 -0.5
-1.5 -1 -0.5 0
-0.15 -0.1 -0.05 0
L = 16
L = 32
L = 64
-1.5 -1 -0.5
-1.5 -1 -0.5 0
(a) (b)
(d) (e)
=1.175 h
=1.175
=1.175
=1.18 h0=1.17
PSfrag replacements
ln ˜h
ln ˜h
ln ˜J
ln ˜J
ln ˜h
/L0.55
FIG. 2: (color online). The distribution of the last deci-
mated effective log-fields lneh∞, and the distribution of the
last decimated effective log-bonds ln eJ∞ in the RG calcula-
tions. At h0 = 1.175, the distributions, shown in (a) and (b),
get broader with increasing system sizes, indicating the RG
flow towards infinite randomness, i.e. the system is critical.
A scaling plot of the data in (a) using energy-length scaling
lneh∞ ∼ L
ψ with ψ = 0.55 is presented in (c). The solid line
is just a guide to the eye. The subfigures (d) and (e) show
the log-field distribution at h0 = 1.18 and the log-bond dis-
tribution at h0 = 1.17, respectively; the distributions show
a power-law decaying tail in the low energy region, which is
clear evidence that the system is in the Griffiths phases.
ical point, we can make use of the fact that the excitation
energy of the system has the scaling behavior ln ǫ ∼ Lψ
at criticality, while it follows ǫ ∼ L−z in the off-critical
regions. In the numerical implementation of the SDRG
method, the low-energy excitations of a given sample can
be identified with the effective transverse field h̃∞ of the
last decimated spin cluster, or with the effective coupling
J̃∞ of the last decimated cluster-pair.
In our implementation we set for convenience the
transverse fields to be a constant h0 and the random
bond variables were taken from a rectangular distribu-
tion centered at J = 1 with a width ∆ = 0.5. The critical
point was approached by varying the single control pa-
rameter h0. Although this initial disorder appears to be
 = 1.170
 = 1.175
 = 1.180
 = 1.185
 = 1.190
L = 16
L = 24
L = 32
L = 40
L = 64
 = 1.175
PSfrag replacements
ln ℓℓ
FIG. 3: (color online). Left panel: The disorder averaged
block entropy per surface unit Sℓ/ℓ vs. the linear size of the
block ℓ for a system size L = 64 for various values of h0. We
observe that the entropy for ℓ = L/2 reaches its maximum
at the critical point hc = 1.175 (cf. Fig 2). Right panel:
The block entropy per surface area vs. ln ℓ on a log-scale for
different system sizes L at the critical point. The data show
a straight line (guided by the dashed line), corresponding to
the scaling obeying the area law with a double-logarithmic
correction, as given in Eq. (4).
weak, the renormalized field and bond distributions be-
come extremely broad even on a logarithmic scale [Fig. 2]
at the critical point h0 = hc = 1.175. This indicates the
RG flow towards infinite randomness. Slightly away from
the critical point, both in the disordered Griffiths-phase
with h0 = 1.18 and in the ordered Griffiths-phase with
h0 = 1.17, the distributions have a finite width and obey
quantum-Griffiths scaling h∞ ∼ L−z. At the critical
point one has IRFP scaling lnh∞ ∼ Lψ and we estimate
the scaling exponent as ψ = 0.55, quite close to the value
ψ = 0.5 for the 1D case [18].
Now we consider the entanglement entropy near the in-
finite randomness critical point. To obtain the disorder-
averaged entanglement entropy Sℓ of a square block of
size ℓ, we averaged the entropies over blocks in different
positions of the whole system for a given disorder real-
ization and then averaged over a few thousand samples.
In Fig. 3 we show the entropy per surface unit Sℓ/ℓ = sℓ
for different values of h0. This average entropy density
is found to be saturated outside the critical point, which
corresponds to the area law. At the critical point sℓ in-
creases monotonously with ℓ, and the numerical data are
consistent with a log-log dependence:
Sℓ ∼ ℓ log2 log2 ℓ (4)
as illustrated in Fig. 3. In this way we have identified an
alternative route to locate the infinite randomness criti-
cal point: it is given by the field h0 for which the average
block entropy at ℓ = L/2 is maximal. Indeed the nu-
merical results in Fig. 3 predict the same value of hc as
obtained from the scaling of the gaps. We note that the
same quantity, the position of the maxima of the average
entropy, can be used for the random quantum Ising chain
to locate finite-size transition points [21].
The log-log size dependence of the average entropy in
Eq.(4) at criticality is completely new; it differs from the
scaling behavior observed in 2D pure systems, like the
area law, Sℓ ∼ ℓ, for critical bosonic systems [7, 8], or
a logarithmic multiplicative correction to the area law,
Sℓ ∼ ℓ log2 ℓ, as found in free fermions [5, 6, 7, 8]. This
double-logarithmic correction can be understood via a
SDRG argument: In the 1D case a characteristic length
scale r at a given RG step is identified with the aver-
age length of the effective bonds, i.e. the average size
of the effective clusters. At the scale r(< ℓ) the frac-
tion of the total number of spins, nr, that have not been
decimated is given by nr ∼ 1/r [18]; these active (i.e.,
undecimated) spins have a finite probability to form a
cluster across the boundary of the block (a segment ℓ in
the 1D case) and thus to give contributions to the en-
tanglement entropy. Repeating the renormalization until
the scale r ∼ ℓ, the contributions to the entropy are
summed up: Sℓ ∼
dr nr ∼ ln ℓ, leading to the log-
arithmic dependence of the 1D model [12]. For the 2D
case with the same type of RG transformation with a
length scale r < ℓ, the fraction of active spins in the
renormalized surface layer of the block is nr ∼ ℓ/r. Here
we have to consider the situation in which some of these
active surface spins would form clusters within the sur-
face layer and thus contribute zero entanglement entropy;
the number of the active spins that are already engaged
in clusters on the surface at RG scale r is proportional
to ln r, as known from the 1D case, and only O(1) of
the active surface spins would form clusters connecting
the block with the rest of the system. Consequently,
the entropy contribution in 2D can be estimated as:
dr nr/ ln r ∼ ℓ ln ln ℓ, i.e. a double-logarithmic ℓ-
dependence, as reflected by the numerical data in Fig. 3.
Based on the SDRG argument described above, the
double-logarithmic correction to the area law appears to
be applicable for a broad class of critical points in 2D with
infinite randomness. For instance, the critical points of
quantum Ising spin glasses are believed to belong to the
same universality class as ferromagnets since the frustra-
tion becomes irrelevant under RG transformation, and
the same type of cluster formations as observed in our
numerics for the ferromagnet is expected to be generated
during the action of the RG. The entanglement entropy
at the IRFP is completely determined by the cluster ge-
ometries occurring during the SDRG.
Another type of IRFP in higher dimensions occurs
in the bond-diluted quantum Ising ferromagnet: The
Hamiltonian is again given by (1), but now Jij = 0 with
probability p and Jij = J > 0 with probability 1− p. At
percolation threshold p = pc there is a quantum critical
line along small nonzero transverse fields, which is con-
p  = 0.49
 = 0.50
p  = 0.52
1 10 10010
L = 128
L = 256
L = 512
 = 0.5
PSfrag replacements
FIG. 4: (color online). The entropy per surface area
Sℓ/ℓ = sℓ vs. ℓ near the percolation threshold pc = 0.5 for
the 2D bond-diluted Ising model at small transverse fields for
L = 512. The curves converge to finite values for ℓ → ∞,
corresponding to the area law. The inset shows sℓ − s∞ as
a function of ℓ. s∞ is estimated from sL/2 at L = 512. The
dashed line corresponds to ℓ−1.
trolled by the classical percolation fixed point, and the
energy scaling across this transition line obeys ln ǫ ∼ Lψ,
implying an IRFP [20]. The ground state of the system
is given by a set of ordered clusters in the same geom-
etry as in the classical percolation model – only nearest
neighboring sites are combined into a cluster. In this
cluster structure, the block entropy, determined by the
number of the clusters connecting the block and the rest
of the system, is bounded by the area of the block, i.e.
Sℓ ∼ ℓd−1 with d being the dimensionality of the sys-
tem. To examine this, we determined the entanglement
entropy by analyzing the cluster geometry of the bond-
diluted transverse Ising model. Fig. 4 shows our results
for the square lattice, which follow a pure area-law with
an additive constant: Sℓ = aℓ+ b+O(1/ℓ).
To summarize, we have found that the entanglement
properties at quantum phase transitions of disordered
systems in dimensions larger than one can behave quite
differently. Generalizing our arguments for the 2D case,
we expect for the random bond transverse Ising systems
a multiplicative d-fold logarithmic correction to the area
law in d dimensions at the critical point, whereas for
diluted Ising model at small transverse fields the area
law will hold in any dimension d > 1 at the percolation
threshold. Although both critical points are described
by infinite randomness fixed points, the structure of the
strongly coupled clusters in both cases is fundamentally
different, reflecting the different degrees of quantum me-
chanical entanglement in the ground state of the two sys-
tems. This behavior appears to be in contrast to one-
dimensional systems governed by IRFPs [12].
Other disordered quantum systems in higher dimen-
sions might also display interesting entanglement prop-
erties: For instance, the numerical SDRG has also been
applied to higher dimensional random Heisenberg anti-
ferromagnets which do not display an IRFP [22]. The
ground states involve both singlet spins and clusters with
larger moments; therefore, we expect the correction to
the area law to be weaker than a multiplicative loga-
rithm and different from the valence bond entanglement
entropy in the Néel Phase [23].
Useful discussions with Cécile Monthus are gratefully
acknowledged. This work has been supported by the Na-
tional Office of Research and Technology under Grant
No. ASEP1111, by a German-Hungarian exchange pro-
gram (DAAD-MÖB), by the Hungarian National Re-
search Fund under grant No OTKA TO48721, K62588,
MO45596.
[1] For a review see: L. Amico et al., quant-ph/0703044.
[2] P. Calabrese and J. Cardy, J. Stat. Mech. Theor. Exp.
2004, P06002 (2004).
[3] M. Srednicki, Phys. Rev. Lett. 71, 666 (1993).
[4] M.B. Plenio et al., Phys. Rev. Lett. 94, 060503 (2005);
M. Cramer et al., Phys. Rev. A 73, 012309 (2006);
M. Cramer and J. Eisert, New J. Phys 8 71 (2006).
[5] M.M. Wolf, Phys. Rev. Lett. 96, 010404 (2006); D. Gioev
and I. Klich, Phys. Rev. Lett. 96, 100503 (2006).
[6] W. Li et al, Phys. Rev. B 74, 073103 (2006).
[7] T. Barthel, M.-C. Chung, and U. Schollwöck, Phys. Rev.
A 74, 022329 (2006).
[8] M. Cramer, J. Eisert, and M.B. Plenio, Phys. Rev. Lett.
98, 220603 (2007).
[9] A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404
(2006); M. Levin and X.-G. Wen, Phys. Rev. Lett. 96,
110405 (2006); E. Fradkin and J.E. Moore, Phys. Rev.
Lett. 97, 050404 (2006).
[10] C. Pich et al., Phys. Rev. Lett. 81, 5916 (1998).
[11] O. Motrunich et al., Phys. Rev. B 61, 1160 (2000); Y.-
C. Lin et al., Prog. Theor. Phys. Suppl. 138, 479 (2000).
[12] G. Refael and J.E. Moore, Phys. Rev. Lett. 93, 260602
(2004).
[13] G. Refael and J.E. Moore, Phys. Rev. B 76, 024419
(2007).
[14] R. Santachiara, J. Stat. Mech. Theor. Exp. 2006, L06002
(2006).
[15] N.E. Bonesteel and K. Yang, cond-mat/0612503.
[16] A. Saguia et al., Phys. Rev. A 75, 052329 (2007).
[17] F. Iglói, R. Juhász, and Z. Zimborás, Europhys. Lett. 79,
37001 (2007); R. Juhász and Z. Zimborás, J. Stat. Mech.
Theor. Exp. 2007, P04004 (2007).
[18] D.S. Fisher, Phys. Rev. B 50, 3799 (1994). D.S. Fisher,
Phys. Rev. B 51, 6411 (1995).
[19] F. Iglói and C. Monthus, Phys. Rep. 412, 277 (2005).
[20] T. Senthil and S. Sachdev, Phys. Rev. Lett. 77, 5292
(1996).
[21] F. Iglói, Y.-C. Lin, H. Rieger, and C. Monthus, Phys.
Rev. B 76, 064421 (2007).
[22] Y.-C. Lin et al, Phys. Rev. B 68, 024424 (2003); Y.-
C. Lin et al, Phys. Rev. B 74, 024427 (2006).
[23] F. Alet et al, cond-mat/0703027.
http://arxiv.org/abs/quant-ph/0703044
http://arxiv.org/abs/cond-mat/0612503
http://arxiv.org/abs/cond-mat/0703027
