Numerical estimation of critical parameters using the Bond entropy
Numerical estimation of critical parameters using the Bond entropy
Rafael A. Molina1 and Peter Schmitteckert2
Instituto de Estructura de la Materia - CSIC, Serrano 123, 28006, Madrid, Spain
Institut für Theorie der Kondensierten Materie,
Universität Karlsruhe, 76128 Karlsruhe, Germany
Using a model of spinless fermions in a lattice with nearest neighbor and next-nearest neighbor
interaction we show that the entropy of the reduced two site density matrix (the bond entropy) can
be used as an extremely accurate and easy to calculate numerical indicator for the critical parameters
of the quantum phase transition when the basic ordering pattern has a two-site periodicity. The
actual behavior of the bond entropy depends on the particular characteristics of the transition under
study. For the Kosterlitz-Thouless type phase transition from a Luttinger liquid phase to a charge
density wave state the bond entropy has a local maximum while in the transition from the Luttinger
liquid to the phase separated state the derivative of the bond entropy has a divergence due to the
cancelation of the third eigenvalue of the two-site reduced density matrix.
PACS numbers: 03.67.Mn,75.10.Jm
Keywords:
I. INTRODUCTION
A Quantum Phase Transitions (QPT) is a qualitative
change in the ground state of a quantum system as some
parameter is varied [1, 2]. Contrary to classical phase
transitions, QPTs occur at zero temperature and are due
to the effect of quantum fluctuations and not of ther-
mal fluctuations. The previous definition is very gen-
eral, however, the abrupt change in the structure of the
ground state that define the phase transition can have
different consequences depending on the different cases.
The ground state energy may become non-analytic when
approaching the critical parameter. The energy gap be-
tween the ground state and the first excited state may go
to zero in the critical point. The correlations at the crit-
ical point may decay as power laws instead of exponen-
tially indicating a diverging correlation length. It is pos-
sible to find quantum systems which have some of these
indications of the QPT and not others [3, 4]. For this rea-
son, alternative ways for the classification of QPTs and
for the numerical investigation of the critical parameters
in a QPT can be very helpful.
In recent years quantum information concepts have
started to be applied to the study of QPTs. One cen-
tral concept in quantum information theory is the con-
cept of entanglement [5]. Two quantum systems in a
pure state are entangled if their state cannot be writ-
ten as the product of two separate pure states for each
of the quantum systems. Entanglement measures quan-
tum correlations and as correlations are typically maxi-
mum at the critical points of QPTs it was realized that
some entanglement measures may have a singularity or
a maximum at the critical point. The amount of en-
tanglement has been shown to be a very sensitive quan-
tity to the value of the critical parameter governing the
phase transition [6, 7, 8, 9, 10]. In particular, concur-
rence [11] has been used to investigate spin models and
this quantity shows an extreme or singular behavior at
the corresponding critical points [7]. The block-block
entanglement between two parts of the system has also
been used, establishing connections with conformal field
theory [8, 12, 13].
In a recent work Gu et al. analyzed the local en-
tanglement and its relationship with phase transitions
in the one-dimensional and two-dimensional XXZ spin
models [14]. The local entanglement was measured with
the Von-Neumann entropy of the two-site density matrix.
It is more convenient numerically than the block-block
entanglement as the size of the density matrix needed
for the latter quantity grows exponentially with the size
of the block. Using Bethe ansatz results for the one-
dimensional XXZ model Gu and coworkers showed that
the local information obtained from the entanglement en-
tropy of the two-site density matrix is enough to study
the phase transitions that occur in this model. Consider-
ing blocks larger than the characteristic length scales of
the system (that in the critical point diverge) was shown
to be unnecessary as the entanglement between a block
of two spins and the rest of the system is sufficient to re-
veal the most important information about the system.
Their results hint to the possibility of using these prop-
erties for the numerical study of phase transitions with
small systems.
It is the purpose of this work to study numerically
the local entanglement as a function of the size of the
system. In particular, we will concentrate in the crit-
ical points of the phase transitions. We will study a
one-dimensional model of spinless fermions with nearest-
neighbor interactions that can be transformed through
the Jordan-Wigner transformation to the XXZ model
with spins at each site Sj = 1/2 [15]. For repulsive inter-
action V1 = 2.0 the ground state performs a Kosterlitz-
Thouless type phase transition from a Luttinger Liquid
to a charge density wave, which corresponds to an antifer-
romagnetically ordered state in the spin picture. For at-
tractive interaction V1 = −2.0 there is a phase transition
to a phase separated state, which corresponds to a ferro-
magnet in the spin picture. We will show that the bond
http://arxiv.org/abs/0704.1597v1
entropy Sbond enables us to determine the critical points
of both quantum phase transitions with an astonishing
accuracy, albeit they present very different characteris-
tics and symmetries. We will show how these differences
are reflected in the behavior of the bond entropy. Finally
we include next-nearest-neighbor interaction to test the
generality of our finding. The effect of longer ranged in-
teraction was studied before in the context of multiple
umklapp scattering [16]. In this case the interaction can
now lead to phase transitions at fillings different from 1/2
and to ordering patterns with increased unit cell.
II. BOND-ENTROPY FOR THE
NEAREST-NEIGHBOR INTERACTION MODEL
We shall consider the one-dimensional spinless
fermions model with next-neighbor interactions.
Ĥ = −t
i ĉi−1 + ĉ
i−1ĉi
V1(n̂i−
)(n̂i−1−
where the operators appearing in the formula are the
usual fermionic creation, annihilation, and number oper-
ator at site i, L is the total number of sites, t is the hop-
ping matrix element between neighboring sites, and V1 is
the nearest-neighbor interaction strength. An important
property of this model is that it can be transformed into
the XXZ spin S = 1/2 model through the Jordan-Wigner
transformation [15, 17].
Ŝ−j = exp
cj , (2)
Ŝzj = n̂j − 1/2. (3)
For an even number of particles a phase term appears
in the boundary condtion when we apply the Jordan-
Wigner transformation to Hamiltonian (1). As we are
not interested in this even-odd effect we will consider
periodic (anti-periodic) boundary conditions, c0 ≡ cM
(c0 ≡ −cM ), for N odd (even) and both models will be
equivalent in our examples. Although we will mainly
consider the spinless fermion model we will make com-
ments regarding the equivalent behavior of both models
when we believe it will be useful to clarify some situation
(specially in the “ferromagnetic” phase). The Hamilto-
nian (1) commutes with the total N̂ =
n̂i operator
so the total number of particles is a conserved quantity
and we will consider subspaces with a definite number of
fermions, equivalent to consider subspaces with a definite
value of Sz in the XXZ model.
This model has an interesting phase diagram depend-
ing on the value of the ratio of the interaction parameter
and the hopping term V1/t and also on the number of
particles N . Without loss of generality we can consider
t = 1 and consider the phases as we change V1. For
V1 < −2 the equivalent spin system is ferromagnetic and
the ground state is fully spin polarized. When we cross
the first transition point Vca = −2 the ground state of
the system can be shown to be non-degenerate and with
spin S = 0 [18]. Only in the half-filled case, N = L/2,
there is another transition point at Vcb = 2. For V1 > 2
the system is a charge density wave type insulator, the
transition is of the Kosterlitz-Thouless type and the or-
der parameter depends exponentially on the difference
V − Vcb making an accurate numerical determination of
the transition point notoriously hard.
We will define the bond-entropy as the Von Neumann
entropy of the reduced density matrix of two-neighboring
sites ρ̂ii+1. As a result of the conservation of N the re-
duced density matrix can be written as a 4 × 4 matrix
with three sectors of N = 0, N = 1, and N = 2. In the
two-site basis |00〉, |01〉, |10〉, |11〉 it can be represented
ρ̂ii+1 =
u− 0 0 0
0 ω z 0
0 z∗ ω 0
0 0 0 u+
. (4)
For this particular model using its invariance under
translations it can be shown that we can write this matrix
elements in terms of certain correlation functions [19, 20],
u− = 1+ 〈n̂in̂i+1〉 −
〈n̂i〉 −
〈n̂i+1〉 . (5)
u+ = 〈n̂in̂i+1〉+
〈n̂i〉 −
〈n̂i+1〉 . (6)
〈n̂i〉+
〈n̂i+1〉 − 〈n̂in̂i+1〉 . (7)
i ĉi+1
i+1ĉi
. (8)
We define the bond entropy Si
as the Von Neu-
mann entropy of the two-site density matrix ρ̂i i+1. In
the model under study, the invariance under translations
also implies that Si
does not depend on the site i.
Sbond = −
λj lnλj , (9)
where λj are the four eigenvalues of the reduced two-site
density matrix ρ̂i i+1.
III. NUMERICAL RESULTS
In this section we will show the numerical results for
the bond entropy as a function of the size of the system
using the DMRG algorithm [21].
In Figure 1 we show the results of Sbond at half fill-
ing for different number of sites L. The behavior of the
-2 -1 0 1 2 3 4 5
L = 10 N = 5
L = 14 N = 7
L = 18 N = 9
L = 22 N = 11
L = 26 N = 13
L = 30 N = 15
FIG. 1: (Color online) Bond entropy Sbond as a function of
the interaction for different sizes of the spinless fermions ring
L at half filling N = L/2. The results for L = 26 and L = 30
are hardly distinguishable. We have a maximum of Sbond at
V1 = 2 with extremely good approximation, see next figure,
that marks the CDW insulator transition. The slope of the
entropy diverges at V1 = −2 marking the appearance of the
ferromagnetic transition.
bond entropy around the two critical points is very dif-
ferent, reflecting the different changes in the symmetries
and correlations of the ground state. The slope of Sbond
diverges around Vca = −2 (we will explain that in more
detail in the next section), while Sbond is continuous but
has a local maximum in the proximity of the second crit-
ical point Vcb = 2. In both cases one can understand the
behavior of Sbond from the behavior of the correlation
functions in the different phases [14]. More importantly,
one can estimate with extraordinary precision the value
of the critical parameter from very small system sizes
even in the case of the transition to the CDW phase. In
addition, Sbond has a minimum for V1 = 0.
In Figure 2 we see a zoom of the last figure 1 for one
particular case with parameter L = 30, N = 15 in the
region around V1 = 2. When we calculate numerically
the position of the maximum with DMRG we obtain
the value of V1 with a precision better than 10
−5. Of
course, in order to take full advantage of these proper-
ties of Sbond we need a very accurate algorithm such as
DMRG. We used at least 500 states per block for the
L ≤ 30 sites leading to an discarded entropy below 10−9,
and 1400 states per block for the 36 site system (see be-
low), including the five lowest lying states in order to
treat the degeneracies correctly, leading to a discarded
entropy typically below 10−10, and up to 2 · 10−8 close
to the phase transitions and applied always eleven finite
lattice sweeps. Close to the CDW-I – Luttinger liquid
transition (see below) we used at least 2050 states per
block and only two low lying states to check our results.
This resulted in a discarded entropy below 10−12. We’d
like to note that despite the large number of states the
DMRG runs are much cheaper as compared to the cal-
culations in [16], since no resolvent has to be computed,
e.g. the largest run took about a hundred CPU minutes.
 0.9519
 0.951902
 0.951904
 0.951906
 0.951908
 0.95191
 0.951912
 0.951914
 1.95  1.96  1.97  1.98  1.99  2  2.01  2.02  2.03  2.04  2.05
FIG. 2: (Color online) Behavior of Sbond as a function of V1 in
the neighborhood of the phase transition at V1 = 2 in the case
L = 30, N = 15. We can numerically pinpoint the maximum
of the curve at V1 = 2 with a precision better than 10
The function f(x) is the second order polynomial fit used to
obtain the maximum of the curve.
-2 0 2 4
L = 26 N = 13
L = 26 N = 9
L = 22 N = 9
FIG. 3: (Color online) Behavior of Sbond as a function of V for
L = 26, N = 13 (half-filling) and L = 26, N = 9, and L = 22,
N = 9 (outside half-filling). We observe the same behavior in
the ferromagnetic transition around Vca = −2 but a complete
different one in the transition around Vcb = 2.
In Figure 3 we see some examples comparing results
at half-filling and outside half-filling. The qualitative be-
havior of the bond entropy is exactly the same around
the first critical point Vca = −2, but the maximum of
the bond entropy around Vcb = 2 disappears as soon as
we move outside half-filling reflecting the absence of the
CDW transition for fillings different from 1/2.
IV. FERROMAGNETISM AND THE TWO-SITE
DENSITY MATRIX
As we have mentioned before the slope of Sbond di-
verges at V1 = Vca. In Fig. 4 we show the results for
the value of the third eigenvalue of the two-site density
matrix λ3 for different sizes. We can reach a very high
numerical precision in the determination of the ferromag-
-0.0001
 0.0001
 0.0002
 0.0003
 0.0004
 0.0005
 0.0006
 0.0007
-2.05 -2.04 -2.03 -2.02 -2.01 -2 -1.99 -1.98 -1.97 -1.96 -1.95
FIG. 4: (Color online) Behavior of the third eigenvalue of the
two-site density matrix λ3 as a function of V in the neighbor-
hood of the phase transition at V1 = −2 for different sizes at
half filling. λ3 vanishes at V1 = −2 with very high precision.
netic critical point studying the cancellation of the third
eigenvalue of the two-site density matrix.
In the thermodynamic limit at V1 = −2, ω = z = 1/4
in Eq. (4). One can immediately see that the equality of
both matrix elements implies that one of the eigenvalues
of the density matrix is zero, which leads to the singular-
ity of Sbond. If one tries a direct numerical examination
of the values of the correlation functions independently
one does not get a very accurate estimation of the critical
parameter. However, the examination of the particular
combination appearing in the Von Neumann entropy of
the two-site density matrix allows a very accurate calcu-
lation even with very small system sizes due to the fact
that although ω and z converge slowly to the thermo-
dynamic limit value 1/4, their difference converges very
quickly to zero at the critical value of the interaction.
V. NEXT NEAREST-NEIGHBOR
INTERACTION MODEL
In order to test the generality of our conclusions and
to obtain a critical parameter of a phase transition in
a model not solvable with Bethe ansatz we add next-
nearest neighbors interaction,
Ĥ = −t
i ĉi−1 + ĉ
i−1ĉi
n̂i −
n̂i−1 −
n̂i −
n̂i−2 −
, (10)
where V2 is the strength of the interaction between sites
separated by two lattice spacings. This Hamiltonian has
been used to study the physics of materials that exhibit
0 0,25 0,5 0,75 1
0,951
0,952
0,953
FIG. 5: Bond entropy as a function of V2 in the line V2 =
5− 2V1 for L = 36 and N = 18. The maximum can be used
to estimate the critical point at V2,c = 0.280 with very high
precision.
multiple phase transitions. Usually one considers V2 < V1
as one expects the interaction to reduce with distance.
However, there can be exceptions if the nearest-neighbor
interaction is supressed by the lattice geometry.
The phase diagram of the model represented by the
Hamiltonian (10) has been studied as a function of V1 and
V2 by Schmitteckert and Werner [16]. In this paper the
authors used DMRG to calculate the ground state curva-
ture. The phase diagram depends on the filling, commen-
surability effects are extremely important due to the mul-
tiple umklapp scattering. If we concentrate on half-filling
and repulsive interactions we have a charge density wave
(CDW) phase in which the ground state is twofold de-
generate with ordering pattern (•◦•◦) and (◦•◦•). Here ◦
denotes a vacant and • denotes an occupied site. In phase
CDW II the ground state is fourfold degenerate with or-
dering pattern (••◦◦), (◦••◦), (◦◦••), and (•◦◦•). We
will follow reference [16] and study the critical parameters
along the line V2 = 5 − 2V . For example, studying sys-
tems of sizes up to L = 60 they obtained a critical point
for the transition between the CDW I phase and the Lut-
tinger Liquid phase as (V1,c, V2,c) = (2.4±0.05, 0.2±0.1)}.
In Fig. (5) we show results for the ground state bond
entropy as a function of V2 for along the previous men-
tioned line for L = 36 and N = 18. Even from the small
size used we can accurately determine the critical V2 as
V2,c = 0.280, which is within the error bars previously
given by Schmitteckert and Werner [16]. The determina-
tion of the critical parameter for the transition between
CDW I and the Luttinger liquid is done with much less
numerical work as compared to the finite size analysis
of excitation gaps and the ground state curvate in [16].
Notably, the finite size corrections are smaller, e.g. the
results of the same analysis with L = 18 already gives a
critical parameter of V2,c = 0.277. In Fig. 6 we show the
numerical results for the value of the next-neighbor inter-
action V2 in which we have a local maximum of Sbond (
along the same line as before) as a function of the inverse
0 0,025 0,05 0,075
0,275
0 0,04 0,08
0,945
FIG. 6: Finite size scaling for the position of the maximum of
the bond entropy V2,c as a function of 1/L. The continuous
line is a second order polynomial fit. It is used to extrapolate
the value for L = ∞. In the inset we show the value of the
local maximum of Sbond as a function of 1/L, a second order
polynomial also fits very well the numerical results.
of the total length of the system 1/L. We have calculated
numerically the bond entropy at each size with an inter-
val of 0.001 in V2 in the region around the maximum of
Sbond, except in the case of L = 36 where we have used
an interval of = 0.0002. The values of V2 in the maxi-
mum where obtained through a second-order polynomial
fit of the numerical results for Sbond. The actual value
of the interval used was not very critical as the fits were
very good. With another second-order polynomial fit we
can extrapolate the calculated values to obtain the result
for the thermodynamic limit V2,c = 0.2814± 0.0001. In
the inset of the figure we can see the numerical results
used in the extrapolation of the value of the maximum of
the bond entropy in the critical point. The extrapolated
value being Smax = 0.95385± 0.00001.
In this case the bond entropy and the two-site den-
sity matrix does not give very useful information about
the phase transitions to charge density wave phases with
ordering patterns with basic sizes bigger than two. We
obtain no clear signature in the bond entropy for the
quantum phase transition to CDW II. One may have to
study entropies of density matrices of blocks with at least
the size of the basic ordered block of the phases we are
looking at. Also, one could try to study the bond entropy
for the excited states. The numerical determination of
critical parameters in this case is out of the scope of this
study of the bond entropy and will be subject of future
work. We note that in the case of CDW II the ground
state is fourfold degenerate in the thermodynamic limit.
However, the degeneracy of the two lowest lying states is
lifted by finite size effects.
VI. CONCLUSIONS
The bond entropy defined as the Von Neumann en-
tropy of the two-site density matrix can be a very effec-
tive tool for the study of phase transitions and critical
parameters. Its behavior depends on the correlations in
the ground state of the system.
We have studied the bond entropy for a model of spin-
less fermions with nearest-neighbor interactions and pe-
riodic boundary conditions. The size dependence of its
behavior near the two critical points in the model has
been studied in detail, showing an amazing precision in
the estimation of the critical parameter. We have also
studied a model with next-nearest neighbor interactions.
We could determine the critical point of the phase tran-
sition from the Luttinger liquid to the CDW I state with
an ordering pattern of period two. If the fundamental
block contains only two sites we show that the bond en-
tropy displays a clear signature of the quantum phase
transitons and allows for the determination of the crit-
ical parameters. The bond entropy of the ground state
could not be used for the transition to CDW II with or-
dering pattern of period four. In this case we may have
to turn to a block entropy of higher size. In general, we
can say that the bond entropy can be used as a numerical
indicator for phase transitions but the actual behavior of
the bond entropy is not universal and will depend on the
QPT under study. Our results should open the way to
the numerical study of phase transitions with small sized
systems.
Acknowledgments
RAM wishes to acknowledge useful discussions with
J. Dukelsky. He also acknowledges finantial support at
the Instituto de Estructura de la Materia-CSIC by an
I3P contract funded by the European Social Fund. This
work is supported in part by Spanish Government Grant
FIS2006-12783-C03-01.
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