Quantum critical behavior of the one-dimensional ionic Hubbard model

Abstract

We study the zero-temperature phase diagram of the half-filled one-dimensional ionic Hubbard model. This model is governed by the interplay of the on-site Coulomb repulsion and an alternating one-particle potential. Various many-body energy gaps, the charge-density-wave and bond-order parameters, the electric as well as the bond-order susceptibilities, and the density-density correlation function are calculated using the density-matrix renormalization group method. In order to obtain a comprehensive picture, we investigate systems with open as well as periodic boundary conditions and study the physical properties in different sectors of the phase diagram. A careful finite-size scaling analysis leads to results which give strong evidence in favor of a scenario with two quantum critical points and an intermediate spontaneously dimerized phase. Our results indicate that the phase transitions are continuous. Using a scaling ansatz we are able to read off critical exponents at the first critical point. In contrast to a bosonization approach, we do not find Ising critical exponents. We show that the low-energy physics of the strong coupling phase can only partly be understood in terms of the strong coupling behavior of the ordinary Hubbard model. Comment: 18 pages, 16 figures, submitted to Phys. Rev. B

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arXiv:cond-mat/0307741v1 [cond-mat.str-el] 30 Jul 2003
Quantum critical behavior of the one-dimensional ionic Hubbard model
S.R. Manmana,1,2V. Meden,2R.M. Noack,1and K. Sch¨ onhammer2
1Institut f¨ ur Theoretische Physik III, Universit¨ at Stuttgart,
Pfaffenwaldring 57, D-70550 Stuttgart, Germany
2Institut f¨ ur Theoretische Physik, Universit¨ at G¨ ottingen, Tammannstr. 1, D-37077 G¨ ottingen, Germany
(Dated: Preliminary version of February 2, 2008)
We study the zero-temperature phase diagram of the half-filled one-dimensional ionic Hubbard
model. This model is governed by the interplay of the on-site Coulomb repulsion and an alternating
one-particle potential. Various many-body energy gaps, the charge-density-wave and bond-order
parameters, the electric as well as the bond-order susceptibilities, and the density-density correlation
function are calculated using the density-matrix renormalization group method. In order to obtain
a comprehensive picture, we investigate systems with open as well as periodic boundary conditions
and study the physical properties in different sectors of the phase diagram. A careful finite-size
scaling analysis leads to results which give strong evidence in favor of a scenario with two quantum
critical points and an intermediate spontaneously dimerized phase. Our results indicate that the
phase transitions are continuous. Using a scaling ansatz we are able to read off critical exponents
at the first critical point. In contrast to a bosonization approach, we do not find Ising critical
exponents. We show that the low-energy physics of the strong coupling phase can only partly be
understood in terms of the strong coupling behavior of the ordinary Hubbard model.
PACS numbers: 71.10.-w, 71.10.Fd, 71.10.Hf, 71.30.+h
I.INTRODUCTION
A.Motivation
Theoretical studies of the ionic Hubbard model (IHM)
date back as far as the early seventies (see Ref. 1 and
references therein). The model consists of the usual
Hubbard model with on-site Coulomb repulsion U sup-
plemented by an alternating one-particle potential of
strength δ.It has been used to study the neutral to
ionic transition in organic charge-transfer salts1,2and
to understand the ferroelectric transition in perovskite
materials.3Based on results obtained from numerical4,5
and approximate methods,6,7it was generally believed
that at temperature T = 0 and for fixed δ a single phase
transition can be found if U is varied. This quantum
phase transition was also interpreted as an insulator-
insulator transition from a band insulator (U ≪ δ) to
a correlated insulator (U ≫ δ). In the present paper, we
discuss in detail how this transition occurs.
In 1999, Fabrizio, Gogolin, and Nersesyan used
bosonization to derive a field-theoretical model which
they argued to be the effective low-energy model of the
one-dimensional IHM.8Surprisingly, the authors found,
using various approximations, that the field-theoretical
model displays two quantum critical points as U is varied
for fixed δ. For U < Uc1the system is a band insulator
(with finite bosonic spin and charge gaps), as expected
from general arguments. At the first transition point Uc1,
they found Ising critical behavior as well as metallic be-
havior in the sense that the gap to the bosonic charge
modes goes to zero at the critical point only.
intermediate regime, Uc1 < U < Uc2, a spontaneously
dimerized insulator phase (in which the bosonic spin and
charge gaps are finite) with finite bond order (BO) pa-
In the
rameter was found. The authors argued that the system
goes over into a correlated insulator phase (in which the
bosonic charge gap is finite) with vanishing bond order
and bosonic spin gap at a second critical point Uc2which
is of Kosterlitz-Thouless (KT) type.
Several groups have attempted to verify this phase
diagram for the IHM using mainly numerical methods.
Variational and Green’s function Quantum Monte Carlo
(QMC) data obtained for the BO parameter, the electric
polarization, and the localization length were interpreted
in favor of a scenario with a single critical point Ucand fi-
nite BO for U > Uc.9In a different calculation using aux-
iliary field QMC, data for the one-particle spectral weight
were argued to show two critical points with an interme-
diate metallic phase.10Exact diagonalization studies of
the Berry phase11and energy gaps12,13,14have been in-
terpreted as favoring one critical point13or two points;11
in two investigations this issue was left unresolved.12,14
Several density-matrix renormalization group (DMRG)
studies have been performed focusing on different en-
ergy gaps, the localization length, the BO parameter,
the BO correlation function, different distribution func-
tions, and the optical conductivity.14,15,16,17Some of the
results have been interpreted to be consistent with a two-
critical-point scenario.15,16,17In Ref. 14 the signature of
only one phase transition was found and the possible ex-
istence of a second transition was left undetermined. The
phase diagram of the IHM has also been studied using ap-
proximate methods such as the self-consistent mean-field
approximation18,19,20, the slave-boson approximation,18
and a real space renormalization group method.19Al-
though these studies led to interesting insights, the va-
lidity of the approximations in the vicinity of the critical
region can be questioned on general grounds; therefore,
we do not focus on these approaches any further here.

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The present situation can be summarized as being highly
controversial.
Here we refrain from giving a detailed discussion of the
merits and shortcomings of the various numerical meth-
ods used and the possible problems in interpretation of
numerical results in the literature. Instead, we present a
detailed study of the T = 0 phase diagram of the one-
dimensional IHM mainly based on DMRG calculations
on systems with both open and periodic boundary con-
ditions (OBC’s and PBC’s).
We have calculated a number of different many-body
energy gaps, including the spin gap, the one-particle gap
(the energy difference of the ground states with N + 1,
N, and N −1 electrons), and the gaps to the first (“exci-
ton”) and second excited states. A definition of the gaps
is given in Sec. IIA. Our results explicitly show that dif-
ferent gaps associated with charge degrees of freedom do
not coincide in the thermodynamic limit, although they
are often believed to in the literature (see also Refs. 16
and 14). Our data show that the exciton gap vanishes
at a coupling which depends on δ and which we define
as Uc1. At this critical point the spin gap remains finite.
The spin gap vanishes at a second critical coupling, which
defines our Uc2.
In addition to the energy gaps, we have determined the
BO parameter and susceptibility as well as the charge-
density-wave (CDW) order parameter. Since the single-
site translationalsymmetry is explicitly broken due to the
alternating potential, we will avoid using the term “order
parameter” in describing the CDW order and instead use
the term “ionicity” to refer to the difference in occupancy
between sites on the two sublattices ?nA− nB?. We find
that the ionicity is continuous and non-vanishing for all
values of the interaction strength.
From the finite-size scaling of the BO parameter, we
find a parameter regime with a non-vanishing dimeriza-
tion starting at Uc1and ending at Uc2. We find that the
transitions at both critical points are continuous. The
BO susceptibility shows one isolated divergence at Uc1
separated from a region of divergence starting at Uc2.
We have also investigated the electric susceptibility,
which is finite in the thermodynamic limit for U < Uc1
and diverges at the lower transition point Uc1.
U > Uc1, the behavior is less clear: there seems to be
a weak divergence with system size near Uc2 and for
U > Uc2. This behavior is consistent with that of the
density-density correlation function, which decays ex-
ponentially as expected in a band insulator phase for
U < Uc1, but surprisingly decays as a power law with
an exponent between 3 and 3.5 in the strong coupling
regime, U > Uc2.
Using a scaling ansatz for the BO and the electric sus-
ceptibility we can determine the critical exponents at
Uc1. In contrast to the bosonization approach8, we ob-
tain critical exponents different from those of the two-
dimensional Ising model.
For (almost) all observables, we find that a careful
finite-size scaling analysis is crucial to obtain reliable re-
For
sults in the thermodynamic limit. Furthermore, since it
is necessary to distinguish between fairly small, but finite,
gaps and order parameters and vanishing ones, a detailed
understanding of the accuracy of the DMRG data is es-
sential.
In order to obtain a comprehensive picture of the
ground-state phase diagram, we have studied the differ-
ent phases (as a function of U) for different δ’s which
cover a wide range of the parameter space. We also con-
sider the limit of large Coulomb repulsion U → ∞ (for
fixed δ and hopping matrix element t) and show that
some aspects of the physics of the model in this limit
can be understood in terms of an effective Heisenberg
model, as has been suggested earlier1but has recently
been questioned.14As a result of our investigations, we
are able to resolve many of the controversial issues and
present strong indications in favor of a scenario with two
quantum critical points. At the appropriate points in the
paper, we will briefly comment on the relationship of our
results with the ones obtained in earlier publications.
The remainder of the paper is organized as follows. In
Sec. IB, we introduce the model and discuss the limits
in which it can be treated exactly. In Sec. IC, we dis-
cuss the details of our DMRG procedure. In Sec. II, our
finite-size and extrapolated data for the energy gaps are
discussed. In Sec. III, we present our results for the ionic-
ity and show that in the large-U limit they are consistent
with analytical results obtained by mapping the IHM to
an effective Heisenberg model. The BO parameter and
the related susceptibility are investigated in Sec. IVA.
We present results for the electric susceptibility and the
density-density correlation function in Sec. IVB. In the
numerical calculations of Secs. III to IV we use OBC’s,
for which the DMRG algorithm performs best. To com-
plete our DMRG study in Sec. V, we present results for
the energy gaps calculated for PBC’s and summarize our
findings in Sec. VI.
B.Model and exactly solvable limits
The one-dimensional IHM is given by the Hamiltonian
H = −t
?
j,σ
?
c†
jσcj+1σ+ h.c.
?
+ U
?
j
nj↑nj↓
+δ
2
?
j,σ
(−1)jnjσ, (1)
where cjσ(c†
σ on lattice site j and njσ= c†
constant equal to 1 and denote the number of lattice
sites by L. Here we study the properties of the half-filled
system with N = L electrons.
The system corresponds to the usual Hubbard model
with an additional local alternating potential. It is useful
to consider various limiting cases in order to gain insight
into possible phases and phase transitions. For U = 0 and
jσ) destroys (creates) an electron with spin
jσcjσ. We set the lattice

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δ > 0, the model describes a conventional band insula-
tor with a band gap δ. Since the alternating one-particle
potential explicitly breaks the one-site translational sym-
metry, the ground state has finite ionicity.
The one-dimensional half-filled Hubbard model with-
out the alternating potential (δ = 0) and with U > 0
describes a correlated insulator with vanishing spin gap
∆HM
S
(U) and critical spin-spin and bond-bond correla-
tion functions.21All gaps associated with the charge de-
grees of freedom, such as the one-particle gap ∆HM
are finite.22(The gaps discussed here are defined in Sec.
IIA.) The ionicity and the dimerization are zero for all
values of U. These two limiting cases suggest that the
system will be in two qualitatively different phases in the
limits U ≪ δ and U ≫ δ.
In the atomic limit, t = 0, and for 0 < U < δ, every
second site of the lattice with on-site energy −δ/2 (A
sites) is occupied by two electrons while the sites with
energy δ/2 (B sites) are empty. The energy difference
between the ground state and the highly degenerate first
excited state is δ − U. For U > δ, both the A and B
sites are occupied by one electron and the energy gap is
U − δ. Thus for t = 0 a single critical point Uc(δ) = δ
with vanishing excitation gap can be found. One expects
similar critical behavior with at least one critical point
to persist for the full problem with finite t.
To describe the physics of the IHM in the limit U ≫
t,δ, an effective Heisenberg Hamiltonian
1
(U),
HHB= J
?
j
?
Sj· Sj+1−1
4
?
,J =
4t2U
U2− δ2
(2)
was derived in Ref. 1 analogously to the strong-coupling
perturbation expansion of the usual Hubbard model. It
has recently been pointed out that this strong-coupling
mapping does not take into account an explicitly broken
one-site translational symmetry.14However, it was shown
in Ref. 1 that the strong-coupling expansion preserves the
one-site translation symmetry in the effective spin Hamil-
tonian to all orders in the strong-coupling expansion. In
addition, the ionicity can be derived directly from the
effective spin Hamiltonian as follows. The symmetry of
the Hamiltonian [Eq. (1)] implies that after taking the
thermodynamic limit, njσ= nj+2σfor σ =↑,↓ and all j.
Using the Hamiltonian Eq. (1) and the Hellman-Feynman
theorem, the ionicity
?nA− nB? = −2
L
?
j,σ
(−1)j?njσ? (3)
can be determined via
?nA− nB? = −4
L
?∂H
∂δ
?
= −4
L
∂E0
∂δ
. (4)
The ground-state energy E0 of the effective Heisenberg
model [Eq. (2)] is known analytically27and, in terms of
U and δ, is given by
EHB
0
= L
4Ut2
U2− δ2
?
ln2 −1
4
?
(5)
in the thermodynamic limit. In the limit U ≫ δ, we can
thus derive an analytic expression for the ionicity
?nA− nB? = 32ln2
Uδt2
(U2− δ2)2. (6)
It implies that for any U < ∞, the ionicity of the IHM is
nonzero and for large U vanishes as 1/U3. Since CDW or-
der is explicitly favored by the Hamiltonian, it is not sur-
prising that the ionicity is non-vanishing for all finite U.
As will be shown in Sec. III, this expression shows excel-
lent agreement with our DMRG data for the IHM. This
gives us confidence that the effective Heisenberg model
indeed gives correctly at least certain aspects of the low-
energy physics. Since the Heisenberg model [Eq. (2)] has
a vanishing spin gap,23the mapping suggests that the
spin gap also vanishes in the large-U limit of the IHM.
Although the alternating potential breaks the one-site
translational symmetry explicitly, the model remains in-
variant to a translation by two lattice sites. This leads
to a site-inversion symmetry for closed-chain geometries
with periodic or antiperiodic boundary conditions, a
symmetry which is not present for OBC’s. As pointed
out in Ref. 12, the ground state of the effective Heisen-
berg model with periodic boundary conditions for sys-
tems with 4n lattice sites or antiperiodic boundary condi-
tions for systems with 4n+2 sites has a parity eigenvalue
of −1 whereas the ground state for U = 0 has a parity
eigenvalue of +1. This suggests that the IHM under-
goes at least one phase transition point with increasing
U for fixed δ. This level crossing will be replaced by level
repulsion and approximate symmetries for other bound-
ary conditions. In the thermodynamic limit, the effect
of the boundaries will disappear and the level repulsion
becomes vanishingly small. It is important to point out,
however, that a level crossing on small finite systems does
not necessarily lead to a first-order transition in the ther-
modynamic limit; careful finite-size scaling must be car-
ried out in order to determine the critical behavior.
From these considerations, one expects to find at least
one quantum phase transition from a phase with physical
properties similar to those of a non-interacting band in-
sulator to a phase with properties similar to those of the
strong coupling phase of the ordinary Hubbard model.
However, the details of the transition and the physical
properties of the different phases remain unclear from
these arguments. Furthermore, the behavior of the BO
parameter in the critical region cannot be estimated from
these simple limiting cases. Therefore, a detailed and
careful calculation of the characterizing gaps and order
parameters is necessary.Since no direct analytic ap-
proach is known to be able to treat the parameter values
in the critical regime, we restrict ourselves to numerical
calculations using the DMRG method with the details
described in the next section.
In the following, we measure energies in units of the
hopping matrix element t, i.e., set t = 1. In order to be
able to cover a significant part of the parameter space,
we have carried out calculations with δ = 1, δ = 4, and

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δ = 20 for weak interaction values U ≪ δ, for strong
coupling U ≫ δ and in the intermediate critical regime
U ≈ δ. For the sake of compactness, we will mostly focus
on δ = 20 when presenting results that are generic to all
three δ-regimes.
C. DMRG method
We have carried out our calculations using the finite-
system DMRG algorithm. Our investigation focuses on
the ground-state properties for systems with OBC’s, i.e.,
we have performed DMRG runs mostly with OBC’s and
one target state, the case in which the DMRG algorithm
is most efficient.In order to perform the demanding
finite-size scaling necessary, we have performed calcula-
tions for systems with up to L = 768 sites, much larger
than in an earlier work.24
In order to investigate the low-lying excitations, we
have also performed calculations targeting up to three
states simultaneously on systems with OBC’s. These nu-
merically more demanding calculations were carried out
for systems with up to L = 256 sites for three target
states and with up to L = 450 sites for two target states.
In order to compare with exact diagonalization calcula-
tions and to extend its finite-size scaling to larger sys-
tems, we have performed calculations for PBC’s with up
to L = 64 sites and one to three target states. In this
case, the maximum system size is limited by the relatively
poor convergence.
The DMRG calculations for OBC’s with one target
state were carried out performing up to six finite-system
sweeps keeping up to m = 800 states. For more mul-
tiple states and for PBC’s up to 12 sweeps were per-
formed, keeping up to m = 900 states. In order to test
the convergence of the DMRG runs, the sum of the dis-
carded density-matrix eigenvalues and the convergence
of the ground-state energy were monitored. For OBC’s,
the discarded weight is of order 10−6in the worst case
and the ground-state energy is converged to an absolute
error of 10−3but in most cases the absolute error is 10−5
or better. This accuracy in both the energy and the dis-
carded weight gives us confidence that the wave function
is also well-converged and that local quantities are quite
accurate.
For PBC’s, the discarded weight is of the order 10−5
in the worst case and the convergence of the ground-
state energy for most runs is up to an absolute error
of 10−3or better, but for extreme cases such as L =
64 and three target states for parameter values near the
phase transition points, the convergence in the energy
is sometimes reduced to an absolute error of only 10−1.
However, we believe that this accuracy is high enough for
the purposes of the discussion in section V.
In general, we find that our data are sufficiently accu-
rate so that extrapolation in the number of states m kept
in the DMRG procedure does not bring about significant
improvement in the results (at least for OBC’s). Details
of the extrapolations and error estimates for particular
calculated quantities are given in the corresponding sec-
tions.
II.ENERGY GAPS
One important way to characterize the different phases
of the IHM are the energy differences between many-
body eigenstates. Gaps to excited states can be used to
characterize phases by making contact with the gaps ob-
tained in bosonization calculations and also form the ba-
sis for experimentally measurable excitation gaps, found,
for example, in inelastic neutron scattering, optical con-
ductivity, or photoemission experiments. In addition to
the gaps themselves, however, matrix elements between
ground and excited states as well as the density of ex-
cited states are important in forming the full experimen-
tally relevant dynamical quantities. An example is the
matrix element of the current operator that comes into
calculations of the optical conductivity. We have investi-
gated the behavior of the matrix elements for the dynam-
ical spin and charge structure factors and for the optical
conductivity using exact diagonalization on systems with
both PBC’s and OBC’s.
In the following, we present DMRG calculations of the
gaps to first and second excited states, the spin gap, and
the one-particle gap in which a careful finite-size scaling
on systems of up to 512 sites is carried out. As we shall
see, this is necessary in order to resolve the behavior of
the gaps in the transition regime and to distinguish be-
tween scenarios with one or two critical points.
A. Definition of the gaps
In this section, we study excitations between a non-
degenerate S = 0 ground state and various excited states.
In the numerical calculations, we have found that for
OBC’s the ground state is non-degenerate with total spin
S = 0 for all parameter values studied here. We define
the exciton gap
∆E= E1(N,S) − E0(N,S = 0) (7)
as the gap to the first excited state in the sector with the
same particle number N and with Sz = 0, where Sz is
the z-component of the total spin. We also calculate the
expectation value of the total spin operator ?S2? so that
S is known.
The spin gap is defined as the energy difference be-
tween the ground state and the lowest lying energy eigen-
state in the S = 1 subspace
∆S = E0(N,S = 1) − E0(N,S = 0) .(8)
When the first excited state E1(N,S) in the Sz= 0 sub-
space is a spin triplet with S = 1, ∆S= ∆E. Within the
DMRG, this gap can be calculated by determining the

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ground-state energies in different Sz subspaces in two
different DMRG runs.
If ∆E < ∆S, we call the lowest excitation a charge
excitation. In fact, exact diagonalization calculations
for system with PBC’s suggest that the gap ∆E corre-
sponds to the gap in the optical conductivity.14We have
carried out additional exact diagonalization calculations
that show that the corresponding matrix elements of the
current operator are also nonzero for OBC’s. We there-
fore expect that ∆E (for excitations with S = 0 and
when ∆E < ∆S) corresponds to the optical gap in the
thermodynamic limit.25To obtain a deeper understand-
ing of the excitation spectrum in the critical region, we
also calculate the gap to the second excited state
∆SE= E2(N,S) − E0(N,S = 0)(9)
for selected parameters.
In the literature, gaps to excitations which can be clas-
sified as charge excitations are often calculated by tak-
ing differences between ground-state energies in sectors
with different numbers of particles (this gap is commonly
called the “charge gap”). In particular, one can define a
p-particle gap
∆p = [E0(N + p,Sz
−2E0(N,S = 0)]/p
min) + E0(N − p,Sz
min)
(10)
which is essentially the difference in chemical potential
for adding and subtracting p particles. The spin Sz
the minimal value, 1/2 or 0 for p odd and even, respec-
tively. Either the one particle gap ∆1or the two particle
gap ∆2 are commonly used. The calculation of ∆1 or
∆2is numerically less demanding than that of ∆Esince
it is sufficient to calculate the ground-state energies in
the subspaces with the corresponding particle numbers.
However, since these gaps involve changing the particle
number and, for p = 1, the spin quantum number, it is
not a priori clear if they can be used to characterize pos-
sible phase transition points of the N-particle system. In
many cases of interest, the difference between ∆1, ∆2,
and ∆E vanishes for L → ∞, but in other systems (an
example is the Hubbard chain with an attractive inter-
action), their behavior differs. As we shall see, ∆1 and
∆Edo behave differently near Uc1. In this work we focus
our investigation on ∆1. We have also calculated ∆2and
find that it behaves similarly to ∆1, although it generally
takes on slightly larger values for finite systems.
Gaps are also used to characterize the phase diagram
within the bosonization approach.8,14It is generally be-
lieved that the bosonic charge gap defined there can be
identified with the gap to the first excited state with spin
quantum number S = 0 (i.e. the exciton gap ∆E[Eq. (7)]
as long as ∆E< ∆S) and the bosonic spin gap with ∆S
[Eq. (8)], although a formal proof is missing.
Based on ∆E, ∆S, and ∆SEand the very limited knowl-
edge on matrix elements due to the small system sizes
available to exact diagonalization, no reliable character-
ization of the metallic or insulating behavior of different
phases and transition points can be given.
minis
B. Gaps to excited states
In this section, we calculate excited states within the
Sz= 0 sector. Due to the additional numerical difficulty
of calculating excited states in the same quantum number
sector, we are restricted to systems of L = 450 lattice
sites for ∆Eand L = 256 sites for ∆SE.
In Fig. 1(a), ∆E as a function of U is presented for
δ = 20 and various L. For comparison, the spin gap
for L = 300 is also shown. The exciton gap develops
a local minimum around U = 21.38, which, for increas-
ing L, becomes sharper. Furthermore, the value at the
minimum becomes smaller and seems to approach zero.
There is a cusp in ∆E for all system sizes shown here
at a certain U to the right of the minimum, and then a
smooth decay towards zero gap with further increasing
U. As illustrated for L = 300, this corresponds to a level
crossing with the first triplet (S = 1) excitation, which
becomes the first excited state for all larger U values, i.e.,
∆E= ∆S. The data for δ = 1 and δ = 4 behave similarly,
but the increase to the right of the minimum (up to the
cusp) is substantially steeper as a function of δ, so that
it approaches a jump.
In Fig. 1(b), we display ∆E, the gap to the second ex-
cited state ∆SE, and the spin gap ∆S (calculated using
the ground state in the Sz = 1 sector) for L = 128. It
can be seen that ∆SE< ∆Sfor U-values to the left of the
minimum in ∆E. A similar behavior is found for δ = 4
and δ = 1. This means that there is more than one S = 0
excitation below the lowest lying S = 1 excitation, con-
sistent with a scenario in which a continuum of S = 0
excitations becomes gapless at Uc1. This is the scenario
predicted to occur at the first quantum critical point in
the bosonization approach.8Since system sizes for calcu-
lations of ∆SEwere limited to L = 128 (L = 256 for some
parameter values), we did not attempt to systematically
extrapolate ∆SEto the thermodynamic limit.
We next discuss the finite-size scaling for ∆Eto the left
of the cusp. For U sufficiently far from the critical region
(i.e., the minimum), the finite-size corrections are small
and the data can safely be extrapolated to the thermody-
namic limit using a quadratic polynomial in 1/L, leading
to a finite exciton gap. Close to the minimum, the scal-
ing becomes more complicated. At smaller system sizes,
we find ∆E = ∆S and the scaling is nonlinear. How-
ever, at larger system sizes, there is a crossover to lin-
ear scaling with ∆E(L) ?= ∆S(L). The crossover length
scale becomes larger as U approaches the position of the
minimum. As a consequence, a reliable finite-size extrap-
olation in the critical region requires very large system
sizes.
To investigate the behavior as L → ∞, we interpolate
∆Eas a function of U for fixed L close to the minimum
with cubic splines. From the interpolation we can read
off the minimal value of the gap ∆min(L) and the position
Umin(L) for the different system sizes. Fig. 2 shows the
resulting ∆min(L) as a function of 1/L for δ = 1, 4, and
20. A linear extrapolation of the data gives ∆min(L =

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6
0
0.25
0.5
21 21.2 21.4 21.6 21.8
∆
U
(a)
∆E, L = 128
L = 200
L = 300
L = 400
∆S, L = 300
0
0.25
0.5
2121.221.421.621.8
∆
U
(b)
∆E
∆SE
∆S
FIG. 1: (a) The exciton gap ∆E for finite system sizes L and δ = 20. The spin gap ∆S for L = 300 is also shown for comparison.
(b) The exciton gap, the spin gap ∆S and the gap to the second excited state ∆SE for L = 128.
∞,δ = 1) = 3 × 10−3, ∆min(L = ∞,δ = 4) = 5 ×
10−4, and ∆min(L = ∞,δ = 20) = −1 × 10−4. Within
the accuracy of our data and our extrapolation, these
minimal gaps can be considered to be zero. In analogy
with the atomic limit, we interpret the vanishing of the
exciton gap as defining a critical point.26The critical
coupling Uc1can be determined from fitting Umin(L) to
a linear function in 1/L, as shown for δ = 20 in Fig. 3.
The extrapolation is similar for the other δ-values and
we obtain Uc1(δ = 1) ≈ 2.71, Uc1(δ = 4) ≈ 5.61, and
Uc1(δ = 20) ≈ 21.39. As will be discussed in Sec. IVB,
the vanishing of the exciton gap is accompanied by a
diverging electric susceptibility.
In Sec. IVA, we will present strong evidence in favor
of a spontaneously dimerized phase for Uc1< U < Uc2.
Since the dimerized phase has an Ising-like symmetry,
as L → ∞ the ground state in this phase is expected
to be two-fold degenerate and the exciton gap ∆Eis ex-
pected to vanish - at least if the thermodynamic limit is
taken using PBC’s. At first glance this appears to be at
odds with the increase of ∆Eas a function of U to the
right of Uc1 (but before the cusp is reached) as can be
observed in Fig. 1(a). For finite systems, the OBC’s lift
the degeneracy between the states with the two possi-
ble bond alternation patterns (strong, weak, strong, ...
and weak, strong, weak, ...), energetically favoring one
of them which becomes the ground state. We have calcu-
lated the bond expectation values (see Sec. IVA) of the
ground state and the first excited state on systems of up
to L = 450 (the largest size we were able to reach) and
find that the first excited state does not have the oppo-
site alternation pattern. Instead, the alternation pattern
is the same as in the ground state near the ends, but
reverses itself in the middle of the chain. This change in
0
0.02
0.04
0.06
0 0.001 0.002
1/L
0.003 0.004
∆min(L)
δ = 1
δ = 4
δ = 20
FIG. 2: Finite-size scaling analysis of the minimal value of
the exciton gap ∆E. The solid lines are linear fits through
the four system sizes shown, L = 256, 300, 350, 400.
the alternating BO parameter is evenly spread over the
chain so that it has a cosine-like form with two nodes.
It is difficult to perform finite-size extrapolation on ∆E
in this region since there are few system sizes and only
a very limited range of U available. However, one might
speculate that ∆E will remain finite as L → ∞ due to
the pinning of the BO parameter at the ends.
Sufficiently far from Uc1, the data presented in Fig.
1(a) suggest a linear closing of the exciton gap, which
gets rounded off in the critical region for finite systems.

Page 7

7
21.32
21.34
21.36
21.38
21.4
0 0.002 0.004
1/L
0.006 0.008
Umin(L)
Uc1 ≈ 21.39
FIG. 3: Finite-size scaling analysis of the U-value at the min-
imum of the exciton gap ∆E for δ = 20. The solid line repre-
sents a linear least-squares extrapolation of the data yielding
Uc1 ≈ 21.39.
The larger L, the closer to Uc1the deviation from linear
behavior sets in. This suggests that ∆E∼ Uc1− U close
to but below the first critical point. It implies that the
product of the critical exponents z1ν1= 1 at the first crit-
ical point,26where z1is the dynamical critical exponent
and ν1is the exponent associated with the divergence of
the correlation length.
Our finding of a vanishing exciton gap at the coupling
Uc1 for OBC’s is consistent with results obtained using
PBC’s and L = 4n. For this case, a ground-state level
crossing of two spin singlets at U = Ux(L,δ) (implying a
zero exciton gap) was found using exact diagonalization
of small systems.11,12,14A change of the site inversion
symmetry at U = Uxwas also observed. In Sec. V, we
will argue that Ux(L → ∞,δ) coincides with Uc1(δ).11
The presence of the ground-state level crossings might
lead one to speculate that discontinuous behavior will
persist in the thermodynamic limit, implying a first order
phase transition at Ux(L = ∞,δ). However, we find no
discontinuous behavior for systems with OBC’s, either
on finite systems or in the L → ∞ extrapolations. In
order to agree with the results obtained for OBC’s in
the thermodynamic limit, the discontinuous behavior for
PBC’s must become progressively smoothed out as L →
∞.
C. The spin gap ∆S
The spin gap ∆Sis shown in Fig. 4 as a function of U
for δ = 20 and system sizes between L = 16 and 512. In
Fig. 4(a), one can see that the spin gap systematically
scales towards zero above a certain U value. However, it
is crucial that the finite-size scaling is carried out care-
fully and systematically in order to determine the behav-
ior in the thermodynamic limit. As can be seen in the
scaling as a function of 1/L for representative U values in
Fig. 4(b), and as was pointed out in Ref. 16, there is non-
monotonic behavior as a function of 1/L for U < Uc1. In
addition, the minimum of ∆Sas a function of 1/L shifts
to larger system sizes as the critical region is approached.
This makes an extrapolation to the thermodynamic limit
in the critical region a difficult task which requires fairly
large system sizes. In order to carry out an accurate
extrapolation, we fit to a cubic polynomial in 1/L.
Fig. 5 shows the extrapolated spin gap for δ = 20 pre-
sented together with the extrapolated values for ∆1and
∆E. All three gaps are approximately equal for U ≪ Uc1
(see the inset). Close to the transition, as can be seen
on the expanded scale in the main plot, ∆Egoes to zero
at Uc1, while ∆Sand ∆1stay finite and are (almost; see
below) equal. For U > Uc1, ∆Eincreases until it reaches
the spin gap ∆S. We find a region of U > Uc1in which
∆S(L = ∞) has a value that is clearly nonzero, well above
the accuracy of the data which is of the order of the sym-
bol size. The behavior is similar for δ = 4 (not shown).
For even smaller values of δ, ∆Sclose to Uc1becomes sig-
nificantly smaller. As a consequence, the region in which
∆S is non-vanishing for U > Uc1 is less pronounced at
δ = 1. In this case, ∆S at Uc1 is only a factor of six
larger than the estimated accuracy of our data (this has
to be compared to the factor of 20 for δ = 4 and 40 for
δ = 20) with a fast decrease for U > Uc1. We take the
estimate of accuracy from comparison of DMRG calcu-
lations for the one-particle gap of the usual 1D Hubbard
model with Bethe ansatz results. We find that the dif-
ference is about
???∆HM,DMRG
to be finite for δ = 1 and in a small region of U ≥ Uc1.
For δ substantially smaller than 1, it is impossible to re-
solve a non-vanishing ∆Sat U ≥ Uc1using the DMRG.
The spin gap data in Fig. 5 indicate that ∆Sgoes to
zero very smoothly between 21.55 and 21.8 and remains
zero from there on. We here define Uc2as the coupling
at which ∆Sgoes to zero. As we have argued in Sec. IB,
the mapping onto a Heisenberg model at strong coupling
[Eq. (2)] suggests that the spin gap should vanish at
sufficiently large U. However, we cannot strictly speaking
exclude that Uc2= ∞ from the spin gap data. We give
further evidence in support of two transition points at
finite U below.
Note that the extrapolated (Fig. 5) as well as the large-
L data (Fig. 4) for ∆Sdisplay an inflection point in the
vicinity of Uc1. This might be an indication of a non-
analyticity related to the phase transition at Uc1.
1
− ∆HM,exact
1
??? = 0.003 in the
worst case. We nevertheless interpret this small spin gap
D.The one-particle gap ∆1
In Fig. 6(a), ∆1as a function of U is shown for δ = 20
and different L. Away from the critical region (which

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8
0
0.25
0.5
2121.221.4 21.6 21.8
∆S
U
(a)
L=16
L=32
L=64
L=128
L=512
0
0.25
0.5
0 0.01
1/L
0.02
∆S
(b)
U = 21.05
U = 21.25
U = 21.34
U = 21.39
U = 21.45
U = 21.85
FIG. 4: (a) The spin gap ∆S for finite systems L as a function of U and (b) the finite-size scaling analysis for δ = 20 for chosen
U-values. The system sizes L = 64, 128, 200, 256, 300, 350, 400, 450, and 512 are shown in (b) and are used for a least-squares
fit to a third-order polynomial in 1/L (solid lines). The dashed line in (b) shows the value of ∆S at the largest system size in
order to illustrate the non-monotonic behavior.
0
0.25
0.5
0.75
1
21 21.2 21.4 21.6 21.8
∆
U
∆1
∆S
∆E
0
20
0 20
FIG. 5: The exciton gap ∆E, the spin gap ∆S, and the one-
particle gap ∆1 for δ = 20 after extrapolating to the thermo-
dynamic limit L → ∞. The inset shows the result for a larger
range of U.
is between U ≈ 21.15 and U ≈ 22), the finite-L data
rapidly approach the thermodynamic limit and accurate
results for ∆1(L = ∞) can easily be obtained by fitting
to a polynomial in 1/L. Close to Uc1, the data for large L
develop a minimum. As L increases, the position of the
minimum shifts to larger U-values. The shape is quite
rounded for the small system sizes, but becomes sharper
for the largest sizes.
In the critical region, the finite-size scaling is again del-
icate. We examine ∆1as a function of 1/L for a number
of U-values near Uc1for δ = 20 in Fig. 6(b). The data
sufficiently away from the minimum (on both sites) shows
linear behavior in 1/L for smaller system sizes, but then
deviates from linear behavior and saturates at a finite
value for larger L-values. This behavior is directly re-
lated to the L-dependence of the minimum of ∆1, which
shifts to larger U and becomes sharper with increasing
system size. The scale on which a deviation from the lin-
ear behavior can be observed shifts to larger system sizes
as U approaches Uc1. In order to perform the finite-size
scaling analysis, we fit to cubic polynomials in 1/L, as we
did for the spin gap. We have carried out this procedure
for δ = 1 and 4 and find that ∆1(L,U) behaves similarly.
We have extracted the position and value at the min-
ima by interpolating the data for fixed L with cubic
splines and then extrapolating to L → ∞ with a fit to
a quadratic polynomial. We obtain Umin(δ = 1) ≈ 2.71,
Umin(δ = 4) ≈ 5.63, and Umin(δ = 20) ≈ 21.40 for the
positions and ∆1(δ = 1,Umin) ≈ 0.02, ∆1(δ = 4,Umin) ≈
0.05, and ∆1(δ = 20,Umin) ≈ 0.08. The minimal values
are finite to within the resolution of the data and the
extrapolation, although the values are small, especially
at small δ. Therefore, ∆1 is finite in the critical region
and is certainly larger than ∆E which vanishes at Uc1.
The positions of the minima are very close to, but at a
slightly larger U-value than Uc1. The largest difference
Umin(δ)−Uc1(δ) turns out to be 0.02 (for δ = 4). In Ref.
16, calculation were carried out for δ = 0.6 (in our units),
this difference was found to be 0.04, and ∆1(δ,Umin) was

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9
0
0.25
0.5
0.75
1
1.25
21 21.2 21.421.621.8
∆1
U
(a)
L=16
L=32
L=64
L=128
L=512
0
0.25
0.5
0.75
1
1.25
0 0.01
1/L
0.02
∆1
(b)
U = 21.05
U = 21.25
U = 21.39
U = 21.45
U = 21.85
FIG. 6: The one-particle gap ∆1 for δ = 20. (a) Results for finite systems with L = 16 through 512. (b) The finite-size scaling
behavior for L = 64, 128, 200, 256, 300, 350, 400, 450, 512. The solid lines in (b) show least-squares fits to a third-order
polynomial in 1/L.
concluded to be zero. The authors interpreted this as an
indication of a second transition point (in addition to Uc1
which they determined from the vanishing of the exciton
gap). While we have not carried out calculations at this
value of δ, our results suggest that ∆1(δ = 0.6,Umin) is
(perhaps unresolvably) small, but nonzero. Therefore, we
believe that Uminis not associated with a second phase
transition. In fact, as we have seen in Sec. IIC, the spin
gap goes to zero at a substantially higher value of U than
Umin, and we associate this value with Uc2.
Up to a small difference (see Fig. 5) ∆1(L = ∞) and
∆S(L = ∞) are equal for U < Uc1. In fact, the values
are virtually identical for the largest few system sizes
and deviate only at smaller sizes. We therefore believe
that the difference in the extrapolated gaps stems from
differences in the fitting to the scaling function at smaller
system sizes and that ∆1(L = ∞) = ∆S(L = ∞) for
U < Umin≈ Uc1is consistent with our results. At this
coupling, ∆1(L = ∞) starts to become larger than ∆S
and as U further increases, grows approximately linearly
in U as one would expect in a Mott insulator.
To summarize the behavior of the finite-size extrapo-
lated gaps, we find that for U ≪ Uc1, ∆E= ∆S = ∆1
as in a non-interacting band insulator. As Uc1 is ap-
proached, the gaps to two (or more) S = 0 excitations
drop below ∆Sand at least one of them goes to zero at
Uc1. The one-particle gap ∆1reaches a finite minimum
around Uc1and then increases (linearly for large U), and
the spin gap ∆S goes to zero smoothly at Uc2 > Uc1.
This smooth decay of the spin gap makes it difficult to
quantitatively estimate Uc2. Since the above behavior is
similar for the widely different potential strengths stud-
ied here, δ = 1, 4, and 20, we believe that it is generic
for all δ.
10-5
10-4
10-3
10-2
10-1
100
20 40 60 80 100
〈nA - nB〉
U
δ=1
δ=4
δ=20
FIG. 7: The ionicity ?nA − nB? for δ = 1,4,20. The solid
lines indicate analytical results from Eq. (6) and the symbols
numerical DMRG results for L = 32 sites.
III.IONICITY
As argued in Sec. IB, the effective strong-coupling
model (2) predicts that the ionicity ?nA−nB? ∼ 1/U3for
large U. For t = 0, on the other hand, one expects a dis-
continuous jump from ?nA− nB? = 2 to ?nA− nB? = 0
at the single transition point Uc. Here we explore the
behavior of ?nA− nB? for all U calculated within the
DMRG.

Page 10

10
In Fig. 7 we compare Eq. (6) for δ = 1, 4, 20 and
various U to results obtained from DMRG with OBC’s
and L = 32. By also considering larger system sizes (up
to L = 512) and PBC’s (up to L = 64), we have verified
that the L = 32 results shown are already quite close
to the thermodynamic limit for U ≫ δ. On the scale
of the figure the difference between L = 32 and L = ∞
is negligible. For large U, the DMRG data agree quite
well with the analytical prediction, Eq. (6). This gives a
strong indication that the large-U mapping of the IHM
onto an effective Heisenberg model1is applicable at large
but finite U. It is therefore tempting to conclude that
Uc2 < ∞. One should nevertheless keep in mind that
the excellent agreement of the numerical data and the
analytical prediction for the ionicity does not constitute
a proof of this statement. We will return to this issue.
The DMRG data for ?nA− nB? for L = 32 shown in
Fig. 7 are continuous as a function of U for all U. We
examine ?nA−nB? more carefully as a function of system
size in the vicinity of the first phase transition at Uc1for
δ = 20 in Fig. 8. The main plot shows DMRG data for
various L as a function of U for δ = 20. While the data
are continuous as a function of U for all sizes, there is
significant size dependence between U = 21.2 and 21.5,
near the first critical point at Uc1. We have extrapolated
the data to the thermodynamic limit using a second order
polynomial in 1/L and have checked that other extrap-
olation schemes do not lead to significant differences in
the extrapolated values. The L = ∞ extrapolated curve
is shown in the inset. While the curve is still continuous,
an inflection point can be observed close to Uc1. This
might be related to non-analytic behavior at Uc1. We
have found similar behavior of ?nA− nB? for δ = 1 and
4.
The behavior of ?nA− nB? for PBC’s (and L = 4n),
which we have checked using the DMRG for up to L = 64,
is quite different. For finite L, the data display a jump
discontinuity in the critical region which decreases in size
for increasing L. The origin of this jump is the ground-
state level crossing at Ux(L,δ). Since we do not observe
any discontinuity in the ionicity calculated for OBC’s for
δ = 1, 4, 20 and up to L = 512, and since the jump
obtained for PBC’s becomes smaller with system size,
we expect that the jump vanishes in the thermodynamic
limit and ?nA− nB? becomes a continuous function.
IV.ORDER PARAMETERS AND
SUSCEPTIBILITIES
A. The bond order parameter and susceptibility
The energy gaps have given us indications for two crit-
ical points. To study the nature of the intervening phase
and the possibility of dimerization in more detail, we cal-
0.55
0.6
0.65
0.7
0.75
0.8
21.3 21.4
U
21.5
〈nA-nB〉
L=16
L=32
L=64
L=128
L=256
L=512
0.6
0.8
21.3 21.4 21.5
FIG. 8: The ionicity ?nA− nB? for finite systems with L =
16, 32, ... , 512 for δ = 20. The inset shows the L → ∞
extrapolated value.
culate the BO parameter
?B? =
1
L − 1
?
j,σ
(−1)j?
c†
j+1σcjσ+ c†
jσcj+1σ
?
. (11)
Since the OBC’s break the symmetry between even and
odd bonds, ?B? ?= 0 for all finite systems. Therefore,
a spontaneous dimerization can be obtained directly by
extrapolating ?B? to L → ∞, i.e., without adding a
symmetry-breaking field explicitly.
corresponding BO susceptibility χBOby adding a term
One can form the
Hdim= ρ
?
j,σ
(−1)j?
c†
j+1σcjσ+ c†
jσcj+1σ
?
(12)
to the Hamiltonian (1) and taking
χBO=∂?B?(ρ)
∂ρ
????
ρ=0
.(13)
In practice, the derivative is discretized as [?B?(ρ) −
?B?(−ρ)]/(2ρ) where ρ is taken to be small enough so
that the system remains in the linear response regime.28
Due to the additional symmetry breaking by the exter-
nal dimerization field ρ, the DMRG runs converge more
rapidly than in the ρ = 0 case, making it easier to reach
larger system sizes. Thus we were able to calculate χBO
on lattices of up to L = 768 sites.
Fig. 9(a) shows ?B? as a function of U for δ = 20
and different L. The data develop a well-defined max-
imum near Uc1 for large L. The width of the “peak”
for L = 512 gives a first indication that there is a re-
gion in which the dimerization is non-vanishing. Typical
results for the finite-size scaling of ?B? are presented in

Page 11

11
Fig. 9(b). For U ≪ Uc1, the data extrapolate linearly
to zero in 1/L.In the opposite limit, U ≫ Uc1, we
find ?B? ∼ 1/Lκwith κ ≈ 0.5 − 0.6. A similar slow
decay of the BO parameter has also been found in the
standard and extended Hubbard models at half-filling.29
The substantial finite-size corrections thus require very
large systems to distinguish between scaling to zero with
a slow power-law and scaling to a finite L → ∞ limit.
Below, but close to Uc1, the data for small L initially dis-
play power-law-like finite-size scaling with κ < 1, but for
larger system size, one finds a crossover to a linear scaling
of the BO parameter (to zero) as L → ∞. There is also
a crossover in the behavior for U-values near but above
Uc1. One again finds a crossover from a power law with
κ < 1 for smaller system sizes to linear behavior that
can be extrapolated to finite values of ?B?∞ for larger
system sizes. The crossover length scale increases as U
approaches Uc1until it becomes larger than the largest
system size considered here. This length scale Lccan be
used to estimate the correlation length, which diverges at
the first (continuous) critical point. We have been able
to calculate Lcfor U-values on both sides of Uc1and find
that it diverges approximately linearly in |U −Uc1|. This
implies ν1= 1 (see also below). Taking into account that
z1ν1= 1 as extracted from the linear closing of ∆E, one
finds z1= 1 for the dynamical critical exponent.
This diverging crossover length scale makes it essential
to treat system sizes that are significantly larger than the
scale Lc, even close to the critical point Uc1. In order
to obtain reliable results, we have calculated ?B?Lfor a
number of system sizes L > 200. In carrying out the
finite-size extrapolation, we fit to a linear form for the
largest system sizes if it is clear that Lchas been reached,
as can be seen in the inset of Fig. 9(b).
In Fig. 10, the finite-size extrapolation ?B?∞is shown
as a function of U for δ = 1, 4, and 20. As can be seen,
?B?∞= 0 to well within the error of the extrapolation
for U < Uc1. For U > Uc1, we find a region of width be-
tween 0.2 and 0.4 (i.e., a factor of 5 to 10 larger than the
extent of the dimerized phase claimed to be found in Ref.
16) in U in which ?B?∞is distinctly finite. The onset of
finite ?B?∞at Uc1is rather steep for all three values of δ,
but seems to be continuous. This steep onset suggests a
critical exponent of the order parameter that is substan-
tially smaller than 1. Within bosonization the first crit-
ical point was predicted to be Ising-like with β1= 1/8.8
The fall-off to zero as U increases, on the other hand,
is slow, with a small or vanishing slope. This behavior
would be consistent with a second critical point at which
the critical exponent for the order parameter is larger
than one or at which a higher order phase transition such
as a Kosterlitz-Thouless transition takes place.8As can
be seen by comparing Fig. 10(a), (b), and (c), the height
of the maximum increases with increasing δ. For δ signif-
icantly smaller than 1, the BO parameter is so small that
it cannot be concluded to be finite within the numerical
accuracy of the DMRG. For the couplings at which the
finite dimerization sets in we obtain Uc1(δ = 20) ≈ 21.39
and Uc1(δ = 4) ≈ 5.61, which are in excellent agreement
with the results obtained from the vanishing of ∆E. The
value obtained for δ = 1, Uc1(δ = 1) ≈ 2.67, is also
in reasonably good agreement with the results obtained
from the analysis of the gaps.
While our data suggest that a critical coupling˜Uc2,
with ?B?∞= 0 for U >˜Uc2, exists, no reliable quanti-
tative estimate of˜Uc2can be given based on the DMRG
data for the BO parameter. Due the close proximity of
the two critical points, we were not able to obtain quan-
titative results for the critical exponents β1and β2at the
critical points, either by a direct fit of the L = ∞ results
or by a scaling plot of the finite-size data. As discussed
next, accurate exponents at Uc1 can be extracted from
both the BO and the electric susceptibilities, and a more
accurate estimate of˜Uc2 can be obtained from the BO
susceptibility.
In order to understand the behavior of the BO suscep-
tibility, it is useful to first examine the behavior of the
BO parameter ?B? as a function of the applied dimer-
ization field ρ.This quantity is shown in Fig. 11 for
δ = 20, three representative values of U, and different
system sizes. For U = 19 < Uc1, the system is in a phase
with vanishing BO parameter, and the slope at ρ = 0 re-
mains finite for all system sizes, corresponding to a finite
susceptibility. The value U = 21.42 is in the intermedi-
ate regime where we have found a finite BO parameter
in the thermodynamic limit. As can be seen in the main
part of the figure, a jump in ?B?(ρ) develops. As the
system size increases, the absolute value of dimerization
field at which the jump occurs becomes smaller. This
is the behavior expected in a dimerized phase in a sys-
tem with OBC’s. Therefore, the jump in ?B?(ρ) provides
additional evidence in support of an intermediate phase
with finite dimerization. For the approximate calculation
of the susceptibility χBO≈ [?B?(ρ) − ?B?(−ρ)]/(2ρ), we
have taken ρ = 10−4which is small enough to stay to the
right of the jump for all system sizes considered. Finally,
for U = 50 ≫ Uc2, ?B?(ρ) goes to zero for |ρ| → 0 and
increasing system size indicating a phase without spon-
taneous dimerization. However, the slope at small |ρ|
becomes steeper with increasing system size, indicating
a divergence of χBO.
In Fig. 12, the BO susceptibility as a function of U
is shown for δ = 1, 4, 20 and different L. For all δ val-
ues, one observes a two-peak structure that becomes pro-
gressively more well-defined with increasing system size.
There is a narrow peak at a U-value that agrees well with
Uc1determined earlier whose height grows rapidly with
system size. It signals the onset of spontaneous dimeriza-
tion. For somewhat larger U there is a minimum in χBO,
surrounded by narrow region in which its value seems to
saturate with system size. For still larger U-values, a
second, broad peak develops. The position of this sec-
ond maximum is roughly at˜Uc2, the U-value at which
the BO parameter vanishes. We argue that the second
peak is related to the second phase transition from the
dimerized phase into an undimerized phase. To the right

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12
0
0.2
0.4
2121.2 21.421.621.8
|〈B〉|
U
(a)
L=16
L=32
L=64
L=128
L=512
0
0.1
0.2
0.3
0.4
0 0.02 0.04
1/L
0.06 0.08
|〈B〉|
(b)
U = 21.05
U = 21.35
U = 21.40
U = 21.42
U = 21.45
U = 21.55
U = 21.95
0.18
0.2
0.22
0.24
0 0.005 0.01
FIG. 9: (a) The bond order parameter ?B? for δ = 20 for finite systems as a function of U for various system sizes. (b) The
scaling of the data as a function of the inverse system size 1/L. The solid lines are least-squares fits to the data as described
in the text. The inset shows an expanded view of the scaling for U values near the critical point.
of the second peak, χBOdoes not seem to saturate for in-
creasing system size, implying that χBOis divergent for
all U ≥ Uc2. One can understand this divergent behavior
by studying the BO susceptibility for the ordinary Hub-
bard model χHM
all U > 0 because the bond-bond correlation function is
critical.21A finite-size extrapolation of χBOis shown in
Fig. 13 for large U-values for both δ = 0 and δ = 20. We
find a power law divergence, χBO(L) ∼ Lζ, with ζ ≈ 0.68
for the ordinary Hubbard model and ζ ≈ 0.65 for the
IHM. These values are in good agreement, considering
the accuracy of the fit and additional finite-size effects.
Since χBO diverges for all U to the right of the sec-
ond peak, it is difficult to accurately determine the crit-
ical coupling˜Uc2. However, two different ways of esti-
mating˜Uc2(δ) under- and overestimate its value. In the
first method,˜Uc2is estimated as the lowest U-value for
which χBO seems to diverge for increasing L and the
available system sizes. It is then still possible that there
is a crossover above a length scale unreachable by us and
χBOscales to a finite value.
This tends to underestimate ˜Uc2.
method,˜Uc2 is taken to be the position of the second
peak at fixed L, extrapolated to L → ∞. Since the peak
position decreases for increasing L, this method tends
to overestimate˜Uc2. From these two procedures, we ob-
tain the bounds 21.55 <˜Uc2(δ = 20) < 21.69. For the
other values of δ, it is very difficult to accurately de-
termine the lower bound with the data available. We
therefore only give the upper bound˜Uc2(δ = 1) < 2.95
and˜Uc2(δ = 4) < 5.86.
It is generally believed that a quantum critical point is
accompanied by a vanishing characteristic energy scale.26
BO. One finds that χHM
BOis divergent for
In the second
At˜Uc2the most obvious candidate is ∆S, consistent with
our numerical data (see Figs. 4 and 5) and implying that
˜Uc2= Uc2. This is assumed in the following discussion.
Since the peak in χBOat Uc1is well-defined and has a
clear growth with system size, it is reasonable to perform
a finite-size scaling analysis. We use a scaling ansatz of
the form
χ(U,L) = L2−η˜ χ(L/ξ) ,(14)
with ξ ∼ |U − Uc|−ν.
data for δ = 20 and system sizes of L = 128 and greater
collapse onto one curve. The best fit is obtained with
Uc1 = 21.385 and the critical exponents η1 = 0.45 and
ν1 = 1. The latter value is consistent with the value
ν1= 1 extracted from the divergence of the length scale
discussed above. Note that the value of η1is not in agree-
ment with the value expected in the two-dimensional
Ising transition, η = 1/4.8We have also applied the scal-
ing ansatz for δ = 1 and 4. For decreasing δ, the quality
of the collapse of the data for the available systems sizes
becomes poorer and the extracted exponents therefore
become less reliable. The best fit is again obtained with
ν1= 1 for both δ, η1(δ = 4) ≈ 0.55 and η1(δ = 1) ≈ 0.65.
For the critical coupling we obtain Uc1(δ = 1) ≈ 2.7 and
Uc1(δ = 4) ≈ 5.6, in excellent agreement with the values
found by other means.
It is also possible to collapse the finite-size data onto
one curve at the second transition point using the scaling
ansatz (14). We find that the best results are obtained for
ξ ∼ exp(A/(U − U′
of the susceptibility at Uc2 may indeed be exponential
as expected for a KT-like transition. However, fitting
the limited amount of data available to this form does
As can be seen in Fig. 12(d),
c2)B), indicating that the divergence

Page 13

13
0
0.05
2.4 2.6 2.8
U
3 3.2
|〈B〉|∞
(a)
0
0.05
0.1
0.15
5.4 5.6 5.8 6
|〈B〉|∞
U
(b)
0
0.05
0.1
0.15
0.2
0.25
21.4 21.6 21.8 22
|〈B〉|∞
U
(c)
FIG. 10: The bond-order parameter ?B?∞ in the thermody-
namic limit for (a) δ = 1, (b) δ = 4, and (c) δ = 20 plotted
as a function of U near the transition points.
not produce completely unambiguous results for all fit
parameters.
Therefore, we have not further attempted to obtain
results for A, B, U′
c2, and η2with this method.
B. The electric susceptibility and the
density-density correlation function
In order to further investigate the physical properties
of the different phases and transition points, we calculate
the electric polarization and susceptibility.30The polar-
ization is given by
?P? =1
L
?
j
xj?nj↑+ nj↓? ,(15)
where xj= j −L/2−1/2 is the position along the chain,
measured from the center. The polarization is the re-
-0.3
-0.15
0
0.15
0.3
-0.01-0.005 0
ρ
0.005 0.01
〈B〉
L = 64
L = 128
L = 256
L = 512
-0.02
0
0.02
0.04
-0.01 0 0.01
-0.02
0
0.02
-0.01 0 0.01
FIG. 11:
dimerization field ρ for δ = 20 and U = 21.42. The upper
inset shows data for U = 19 and the lower inset data for
U = 50.
The BO parameter ?B? as a function of applied
ponse due to a linear electrostatic potential
Hel= −E
?
j
xj(nj↑+ nj↓) (16)
which is added to the Hamiltonian (1). The electric sus-
ceptibility
χel =
∂?P?(E)
∂E
????
E=0
(17)
is the susceptibility associated with this field.
The electric susceptibility has been used to investi-
gate the metal-insulator transition in the t-t′-Hubbard
model.30In this model, both a phase in which χeldiverges
as L2(a perfect metal) and a phase in which for increas-
ing system size χelscales to a finite value (an insulator)
were found when varying U for fixed nearest-neighbor
hopping t and next-nearest-neighbor hopping t′.
In contrast to the ordinary Hubbard model, the po-
larization does not always vanish at field E = 0 in the
IHM. For U = 0, δ > 0, one finds ?P? = −1/2. This
is due to the alternating ionic potential which induces a
charge displacement to the sites with lower potential en-
ergy. Due to the OBC’s, a chain with even length L starts
and ends with a different potential, inducing a dipole mo-
ment. This is a boundary effect. In the strong coupling
limit, U ≫ δ, we find that ?P? → 0, as expected. The
electric suceptibility χelcan be calculated by discretizing
the derivative as [?P?(E) − ?P?(E = 0)]/E. The field
E must be taken to be small enough so that the system
remains in the linear response regime.28Note that it is
necessary to subtract ?P?(E = 0) since it is nonzero in
general.

Page 14

14
0
50
100
2.4 2.6 2.8 3
χBO
U
(a)
L = 16
L = 32
L = 64
L = 128
L = 256
L = 512
L = 768
0
50
100
150
5.2 5.4 5.6 5.8
χBO
U
(b)
L = 16
L = 32
L = 64
L = 128
L = 256
L = 512
0
50
100
150
200
250
21.2 21.4 21.6 21.8
χBO
U
(c)
L = 16
L = 32
L = 64
L = 128
L = 256
L = 512
0
0.005
0.01
0.015
0.02
0.025
0.03
-100-50050100
χBO(L (U-Uc1))/Lµ
L (U-Uc1)
µ=1.55, Uc1=21.395
(d)
L = 64
L = 128
L = 256
L = 512
FIG. 12: The BO susceptibility χBO as a function of U for (a) δ = 1, (b) δ = 4, and (c) δ = 20 and different L. (d) A scaling
analysis of the δ = 20 data from (c).
A plot of χelas a function of U for various system sizes
is shown in Fig. 14(a) for δ = 20. For U ≪ Uc1and in-
creasing L, χelconverges to a finite value, similar to the
behavior in a non-interacting band insulator and in the
correlated insulator phase of the t-t′-Hubbard model.30
The data clearly develop a maximum at Uc1whose height
increases markedly with system size, indicating a diver-
gence at the first critical point. The finite-size scaling
of this height is consistent with a power-law increase,
L2−η1, with η1 ≈ 0.46.This increase is weaker than
the L2divergence (which implies η = 0) found in Ref.
30 and associated with a perfect metal. For U slightly
larger than Uc1, the data again seem to saturate with
system size. Assuming the scaling form of Eq. (14), the
data close to Uc1 can be collapsed on a single curve as
demonstrated in Fig. 14(b). The best fit is obtained for
ν1 = 1 and η1 ≈ 0.45.Both of these exponents are
in excellent agreement with those found in the scaling
analysis for χBO. We have carried out a finite-size scal-
ing analysis for δ = 4 and δ = 1 and also find diverg-
ing peaks at Uc1, as well as collapse of the data onto a
single curve using the scaling form (14) with exponents
η1(δ = 1) = 0.52, η1(δ = 4) = 0.45, and ν1 = 1 (for
both δ). The critical U-values obtained from this scaling
procedure are Uc1(δ = 1) = 2.68, Uc1(δ = 4) = 5.59, and
Uc1(δ = 20) = 21.38, which compare well to the values
for the critical coupling obtained from the gaps and from
the BO parameter and susceptibility.
The data for δ = 20 and δ = 4 for the largest sys-
tem sizes, L = 256 and L = 512, suggest that a second

Page 15

15
10-1
100
101
102
10-3
10-2
1/L
10-1
χBO
δ = 0, U = 10
0.36 L0.65
δ = 20, U = 50
0.08 L0.68
FIG. 13: The BO susceptibility χBO as a function of 1/L
for the ordinary Hubbard model (δ = 0) and U = 10 and the
ionic Hubbard model for δ = 20 and U = 50. DMRG data are
indicated by the corresponding symbols and the solid curves
represent a least-squares fit to the indicated forms.
peak may develop around Uc2. In order to investigate
the behavior of χel(L) more precisely in this region, we
fit a quadratic polynomial to ?P?(E) through several data
points and then take the derivative of this fit function at
E = 0. This procedure should eliminate errors caused
by a small linear response regime. Results obtained from
this procedure for δ = 20 indicate a weak divergence at
U = 21.65, corresponding to a U-value near Uc2. In addi-
tion, we find an even weaker divergence for all U > 21.65.
The larger the U-value, the smaller the coefficient of the
diverging part, so that the divergence is very difficult to
observe numerically deep in the strong-coupling-phase.
One generally expects the divergence of χel to be con-
nected to the closing of a gap to excited states which
possess at least some “charge character” (in the sense
discussed below). At Uc1the divergence is accompanied
by the closing of the exciton gap, leading to a consistent
picture.
The situation is less clear for U ≥ Uc2. This issue can
further be investigated by examining the behavior of the
density-density correlation function
Cden(r) = ?nini+r? − ?ni??ni+r? ,(18)
shown in Fig. 15 for δ = 1 and different U > Uc2. Here
we have averaged over a number of i-values (typically
six) for each r and have performed the calculation on an
L = 256 lattice. For each value of U, it is evident that
the correlation function behaves linearly on the log-log
scale above some value of r, indicating that the domi-
nant long-distance behavior is a power law. (For r close
to the system size, finite-size effects from the open bound-
aries also appear.) Note that the sign of the correlation
function is negative for r > 0, so that the negative is
plotted. A least-squares fit to the linear portion of the
curve yields an exponent of approximately 3−3.5 for all
values of U > Uc2. This behavior is markedly different
from the behavior for U < Uc1, where we find a clear
exponential decay as in a non-interacting band insulator,
and from the behavior at Uc1, where we find a power law
decay with an exponent of ≈ 2. Note that if the decay
were exponential for U > Uc2, we would expect the cor-
relation length to change quickly with U, leading to a
marked variation in the slope. We have ruled out finite-
size effects as an origin of the power-law tails as well as
possible symmetry breaking due to the OBC by compar-
ing calculations for L = 128 and L = 256 with OBC and
L = 64 with PBC, which yield identical values except for
distances r near the lattice size (or half the lattice size
for PBC’s).
We have performed calculations for δ = 20 and find
similar behavior. The exponent of the power-law tails has
a comparable value to the ones given above, even at very
large U-values such as U = 50, where the prefactor of the
power-law part is ≈ 2×10−6. It therefore seems justified
to conclude that this power-law decay is a generic feature
of the strong-coupling phase for all δ.
Our findings for χel and Cden(r) are consistent with
a scenario in which there is a continuum of gapless ex-
citations for U > Uc2, where matrix elements of charge
operators such as the density nj = nj↑+ nj↓, are non-
vanishing for some of the states belonging to this con-
tinuum.These are the states mentioned above which
possess charge character. To further confirm this idea,
we have calculated matrix elements ?m|nj|0?, where |m?
denotes the m-th excited state and |0? the ground state,
for up to m = 4, δ = 20, U > Uc2, and L = 32. We
find that the third excited state is the first S = 0 state,
both for the ordinary Hubbard model and the IHM (the
fourth state as well as the m = 1,2 states have S = 1).
For the ordinary Hubbard model, ?3|nj|0? vanishes for
all j to within the accuracy of our data and this S = 0
state can be classified as a spin excited state since its
excitation energy is well below the charge gap. In con-
trast, ?3|nj|0? is nonvanishing for the IHM and shows
a non-trivial dependence on j which has a wavelength
of approximately the lattice size, implying that the wave
vector characterizing the excitation is near q = 0.
As a consequence, this state contributes to the dy-
namical charge structure factor in the IHM but not in
the ordinary Hubbard model. This shows that although
several similarities between the strong coupling phase of
the IHM and the Hubbard model were found, low-lying
excitations in both models are of quite different nature.
As we have verified, the energy of |3? becomes smaller
for increasing U, in contrast to the behavior of the one-
particle gap which increases linearly with U. Due to the
numerical effort necessary to target such a large number
of states, we were unable to perform these calculations
on larger lattices in order to carry out a finite-size scaling
analysis of the matrix elements.