Incommmensurability and unconventional superconductor to insulator transition in
the Hubbard model with bond-charge interaction
A. A. Aligia,1 A. Anfossi,2, 3 L. Arrachea,4, 3 C. Degli Esposti Boschi,5 A.
O. Dobry,6 C. Gazza,6 A. Montorsi,2 F. Ortolani,7 and M. E. Torio6
1Comisión Nacional de Enerǵıa Atómica, Centro Atómico Bariloche and Instituto Balseiro, 8400 S.C. de Bariloche, Argentina
2Dipartimento di Fisica del Politecnico and CNISM,
corso Duca degli Abruzzi 24, I-10129, Torino, Italy
3BIFI, Universidad de Zaragoza, Corona de Aragón 42, 5009 Zaragoza, Spain
4Departamento de F́ısica de la Materia Condensada, Universidad de Zaragoza, 5009 Zaragoza
5Unità CNISM and Dipartimento di Fisica dell’Università di Bologna, viale Berti-Pichat 6/2, I-40127, Bologna, Italy
6Instituto de F́ısica Rosario, CONICET-UNR, Bv. 27 de Febrero 210 bis, 2000 Rosario, Argentina.
7Dipartimento di Fisica dell’Università di Bologna and INFN, viale Berti-Pichat 6/2, I-40127, Bologna, Italy
(Dated: November 1, 2018)
We determine the quantum phase diagram of the one-dimensional Hubbard model with bond-
charge interaction X in addition to the usual Coulomb repulsion U > 0 at half-filling. For large
enoughX < t the model shows three phases. For large U the system is in the spin-density wave phase
as in the usual Hubbard model. As U decreases, there is first a spin transition to a spontaneously
dimerized bond-ordered wave phase and then a charge transition to a novel phase in which the
dominant correlations at large distances correspond to an incommensurate singlet superconductor.
PACS numbers: 71.10.Fd,71.10.Hf,71.10.Pm,71.30.+h
The Hubbard model has been originally proposed to
describe the effect of the Coulomb interaction in tran-
sition metals, which usually contain localized orbitals.
Other real compounds containing more extended orbitals
cannot in general be properly described by this simple
Hamiltonian. Well-known examples are several quasi-
one-dimensional (1D) materials that have been recently
investigated [1], which exhibit a variety of phases that
cannot be explained with the usual Hubbard model. Ad-
ditional interactions should be included. A natural in-
teraction that arises in systems with extended orbitals is
the bond-charge interaction X [2]. In fact, it is natural
to assume that the charge in the bond affects screening
and the effective potential acting on valence electrons,
and therefore the extension of the Wannier orbitals and
the hopping between them should vary with the charge.
This leads to the U −X Hamiltonian:
H = −t
σ=↑,↓,〈ij〉
iσcjσ +H.c.) + U
ni↑ni↓
σ,〈ij〉
cjσ +H.c.)(ni−σ + nj−σ). (1)
This model has been studied in two dimensions, moti-
vated by a theory of hole superconductivity [3]. A mod-
ified version of it has been derived as an effective model
for the cuprates and shows enhanced d-wave supercon-
ducting correlations [4]. Recently, this model has been
paramount to broader audiences, and its relevance has
been discussed in the context of mesoscopic transport [5]
and quantum information [6, 7].
In 1D, there are bosonization [8, 9] and numerical [9]
results available. However, at half-filling, the effect of X
disappears in the standard bosonization treatment and a
behavior different from the usual Hubbard model was not
expected in these studies. For X = t, an exact solution
is available [10]. In this case the ground state is highly
degenerate: the transition to a metallic state takes place
at Uc = 4t > 0, but the response of the system to an
applied magnetic flux indicates that it is not supercon-
ducting [11]. In view of the previous studies, the recent
evidence of an insulator-metal transition driven by X < t
at finite Uc > 0 at half-filling comes as a surprise [12].
The nature of the metallic phase and the character of
the transition have not been fully elucidated, though the
possibility of superconductivity has been suggested.
In this Letter we employ several analytical and numer-
ical techniques to calculate accurately the phase diagram
of the model at half-filling in 1D and to determine the na-
ture of each phase. We establish that the insulator-metal
transition is of commensurate-incommensurate (CIC)
type to a phase with dominating singlet superconducting
(SS) correlations. Remarkably, unlike other CIC tran-
sitions [13, 14], it is not driven by one-body effects like
chemical potential or the emergence of more than two
Fermi points in the noninteracting dispersion relation,
but by strong correlations induced by large enough X . In
addition, we unveil that inside the insulating phase there
is a spin transition separating the expected spin-density
wave (SDW) for U > Us from a spontaneously dimerized
bond-ordered wave (BOW) phase for Uc < U < Us. This
transition is of Kosterlitz-Thouless (KT) type and a spin
gap opens in the BOW phase.
The nature of each phase and the qualitative aspects of
the phase diagram can be understood by a weak coupling
bosonization analysis provided it includes vertex correc-
tions of second order in X to the coupling constants and
http://arxiv.org/abs/0704.0717v2
one term of order a2 in the bosonization of the bond-
charge interaction as described below, where a is the lat-
tice constant. A bosonized version of (1) is given by the
following Hamiltonian density:
H = H0σ +H0ρ +
(2πα)2
8φσ)−
(2πα)2
(2πα)2
8φσ)∂xφρ, (2)
where H0σ and H0ρ are the usual known quadratic forms
and α is a short range cutoff in the bosonization proce-
dure. The first line of (2) has the structure of the pre-
viously studied bosonized theory [8], which corresponds
to two decoupled sine-Gordon field theories, one for the
spin (φσ) and the other for the charge (φρ). In order
to take into account the effect of the bond-charge inter-
action on the phase diagram of the system, we included
vertex corrections of second order in X in the definition
of the the coupling constants gi, due to virtual processes
involving states far from the Fermi energy [15]. In addi-
tion, we took into account the usually neglected gσρ term
that couples spin and charge degrees of freedom. The lat-
ter is ∝ a2. It arises including spatial derivatives of the
fermionic fields in the representation of (1) in terms of
a low energy field theory. All of these terms have naive
scaling dimension 3 and are usually neglected. However,
one term that bosonize as the second line of (2) becomes
relevant for large enough X and provides a mechanism
for an incommensurate transition, as discussed below.
Explicitly, the effective parameters read g1⊥ = g2⊥ =
(U− 8X
π(t−X)
)a and gσρ =
2a2X . The forward and umk-
lapp processes are the same as in the Hubbard model,
g3⊥ = g4⊥ = Ua. The Luttinger liquid parameters (Kρ
and Kσ) and the charge and spin wave velocity (uρ and
uσ) in terms of gi are given by known expressions [16].
Neglecting the gσρ term, the renormalization-group (RG)
flow diagrams are of KT type. A spin gap opens when
g1⊥ < 0, i.e., when the flow of RG, which takes place
on the separatrix of the KT diagram due to spin SU(2)
symmetry, goes to strong coupling. Therefore, the spin
gapped phase appears when U < Us =
π(t−X)
. As for
the behavior of the charge modes, a gap opens when the
g3⊥ term becomes relevant. The charge gapped phase
takes place for U > Uc, with Uc < Us. The gσρ term
becomes relevant for Kσ < 1/2 (X > 0.6t for U = 0).
In the spin gapped phase the cos(
8φσ) is frozen at its
mean value. This term could be interpreted as a chemical
potential [µ =
(2πα)2
〈cos(
8φσ)〉] times a charge density
operator. The effects of such a term are known [16].
If we start the analysis from a situation where there is
also a charge gap (∆c) smaller than the spin one (∆s),
and we then increase the value of X , the effect of this
term is to close ∆c, leading to a metallic phase when
µ > ∆c. The effective Fermi level is shifted with respect
to the original one and the system develops incommen-
0 0,2 0,4 0,6
Bosonization S
Bosonization C
0 0,2 0,4 0,6 0,8
0 0,2 0,4 0,6 0,8 1
Exact
Top. Ph S
Top. Ph C
DMRG S
DMRG C
Figure 1: (Color online). Phase diagram. Left: Bosoniza-
tion (top), and real space renormalization-group (bottom)
predictions. Right: Numerical results as obtained by DMRG
(circles-squares) and topological phase (crosses) methods.
surate correlations. A numerical analysis discussed be-
low shows that the system has dominant SS correlations.
Thus, this phase can be characterized as incommensurate
singlet superconducting (ICSS).
For a qualitative localization of the boundary transi-
tion line between the insulator and the ICSS phase, we
have implemented a procedure as follows: (i) We start
from a parameter regime where the spin gap is open.
(ii) We follow the RG flow up to a length scale where
|g1⊥|/(πUs)| ∼ 1. (iii) At this point the gσρ term is de-
coupled by a mean field approach similar to that used
by Nersesyan et al. to show incommensurability in the
anisotropic zigzag chain [17]. The value of 〈cos(
8φσ)〉
is exactly obtained at the LE point (Kσ = 1/2). (iv) For
vanishing gσρ, ∆c is obtained by rescaling the problem
to the LE point of the charge sector, by using the RG
equations of the sine-Gordon theory. (v) The CIC tran-
sition takes place when
(2πα)2
〈cos(
8φσ)〉 = ∆c [16]. In
the top left panel of Fig. 1 we show the phase diagram
of the model predicted by this approach. For each value
of X , there are two transition points Uc and Us corre-
sponding to the charge and spin transition, respectively.
Each phase is characterized by the gapped modes and the
relevant order parameter. For U > Uc the system is an
insulator. For U > Us, the slowest decaying correlation
functions are the spin-spin ones. The system is in a SDW
phase. For Uc < U < Us a fully gapped (spin and charge)
phase is developed. The fields φσ and φρ are located at
the minimum of the potential, and the translation sym-
metry is spontaneously broken. The BOW parameter,
defined below, acquires a nonzero value. For U < Uc
the charge gap closes and the dominant correlations at
large distances are the SS ones. While the nature of each
phase has been identified, the phase boundaries predicted
by bosonization are not quantitatively valid, particularly
for large values of the interactions. In the right panel of
Fig. 1 we show the phase diagram of the model, as ob-
tained by accurate numerical techniques. One of them,
used to determine the charge transition line, consists in
studying singularities of single-site entanglement [12] by
means of density-matrix renormalization group (DMRG)
[18]. Another method is based on topological numbers,
or jumps of Berry phases [19], which was successfully ap-
plied to a similar model [8] (b). The value of Uc (Us) is
determined in this case by the jump of the charge (spin)
Berry phase. The corresponding values of Uc and Us in
systems up to L = 14 sites, extrapolated to the thermo-
dynamic limit using a parabola in 1/L2, are also shown
in Fig. 1.
DMRG evaluations of ∆c and ∆s confirm these pre-
dictions. The charge gap was calculated in [12] from
the definition 2∆c = E0(N + 2) + E0(N − 2)− 2E0(N),
E0(N) being the ground-state energy of the chain with
N particles. Similarly, the spin gap is here determined
through ∆s = E0(Sz = 1) − E0(Sz = 0), being E0(Sz),
the ground-state energy of the half-filled system within
the subspace with a given total Sz. We can see in Fig. 1
that the closing of ∆c, ∆s do not take place simultane-
ously for small U and X . The critical lines for the closing
of both gaps obtained by extrapolations to the thermo-
dynamic limit are in reasonable quantitative agreement
with the ones determined by the method of the topolog-
ical phases.
We have verified that the spin transition is of KT
type, calculating the scaling dimensions of the singlet and
triplet operators as described in [19]. In order to identify
the universality class of the charge transition, we em-
ployed the finite-size crossing method [20]. The study of
the dependence of 〈ni↑ni↓〉 = ∂eL/∂U on the size L (eL
being the ground-state energy density) provides a loca-
tion of the critical points in agreement with the methods
discussed above. In addition, the divergence that devel-
ops ∂e2L/∂U
2 with increasing L indicates that the gap ex-
ponent ν remains close to 1/2 (the value that can be com-
puted exactly at the point X = t) for X/t = 0.6, 0.7, 0.8
with a possible increase for X/t → 0.5; below this point,
our numerical analysis suggests that the charge transi-
tion becomes of KT type, with “ν = ∞”. The estimate
ν = 1/2 relies upon the assumption that the dynamic
exponent ζ (through which gap and correlation length ξ
are related, ∆c ∝ ξζ) is still ζ = 2, as in the exactly
solvable case X = t [7]. As already noted in [12], the
behavior of ∆c ∝ L−2 along the transition line is consis-
tent with this exponent. We stress that such feature is in
agreement with the CIC character of the metal-insulator
transition [16]. Instead, within the metallic phase, the
finite-size scaling suggests ∆c ∝ L−1, although the data
are rather noisy due to incommmensurability.
In Fig. 2 we show numerical results supporting the in-
commensurate character of the metallic phase. We report
the density distributions in real space for the local charge
8 16 24 32 40 48 56 64
U=4.0 t
U=3.05 t
U=2.5 t
U=0.5 t
8 16 24 32 40 48 56 64
X=0.3 t
X=0.58 t
X=0.8 t
Figure 2: (Color online). Charge distribution 〈ni〉 evaluated
by DMRG. Left: X = 0.8t. Right: U = 1.5t.
density ni = ni↑ + ni↓ in the ground state in an open
chain with L = 64 sites. The incommensurate character
of the metallic phase manifests itself also in the behavior
of the charge and spin correlation functions, whose cor-
responding structure factors show peaks away from the
commensurate reciprocal vector q = π (not shown). The
left panel of Fig. 2 corresponds to X = 0.8t as U is var-
ied. The behavior is similar to the one observed within
the incommensurate phase of the Hubbard model includ-
ing next-nearest-neighbor hopping (t−t′−U model) [13].
For U > Uc = 3.05t, the commensurate charge distribu-
tion characterizing the insulating phase is reached within
a few lattice sites from the edge. The insulator-metal
transition shows up via the appearance of incommensu-
rate modulations in the charge distribution, whose wave-
length increases within the metallic phase. The right
part of the figure shows the results obtained by varying
X at U = 1.5t. Interestingly, a first modulation appears
already for Xs < X < Xc ( Xs ≈ 0.5t, and Xc ≈ 0.6t).
Again, for X > Xc further incommensurate modulations
appear in the LE phase.
Within the charge sector U < Uc, the dominating cor-
relations at large distance are superconducting pair-pair
ones if the correlation exponent Kρ > 1 or charge-charge
ones otherwise. We calculatedKρ employing the method-
ology described in [9]. This study casts extrapolated val-
ues Kρ ∼ 1.3 for U = 0 and X = 0.8t. To provide
stronger evidence for the SS character of the incommen-
surate phase, we have calculated on-site pairing correla-
tions 〈P †i Pj〉 with Pi = c
i↓ and charge-charge correla-
tions |〈ninj〉 − 〈ni〉〈nj〉| in an open chain with 100 sites
and using the sites 30 to 70 to avoid boundary effects.
The results are displayed in Fig. 3. A fitting of the pair-
ing correlations at distances between 8 and 40 sites gives
Kρ = 1.32± 0.01. This value is also consistent with the
long distance behavior of the charge-charge correlations.
The inset also shows the tendency of the system to show
the anomalous flux quantization characteristic of super-
0.0 0.5 1.0 1.5
0.0 0.5 1.0
-5.060
-5.055
-5.050
Φ / Φ
 pair-pair
 charge-charge
 |i-j|
Figure 3: Pair-pair and charge-charge correlation functions
for U = t and X = 0.8t. Full (dashed) line corresponds to
a power law with exponent 1/Kρ (Kρ). The inset shows the
ground state energy as a function of an applied magnetic flux.
conductivity [11], which is more pronounced as the size
of the system increases.
An additional argument suggesting superconducting
correlations within this phase is provided by the real
space renormalization-group method, used before for the
standard Hubbard model [21]. Different from that case,
the recursive equations for the renormalized parameters
in the positive U regime, depending on X and U , exhibit
three different fixed points for the nth step renormalized
Coulomb interaction U (n) in the large n limit: U (n) > 0
for U > Urc, U
(n) = 0 for U = Urc, and U
(n) < 0 for
U < Urc. In the latter case, the effective Coulomb inter-
action becomes attractive. In the bottom left insert of
Fig. 1 Urc obtained in this way is reported.
To support the bosonization predictions, which char-
acterize the intermediate phase as a BOW, we have eval-
uated with DMRG the BOW order parameter OBOW =
(−1)i〈c†i+1σciσ +H.c.〉]/(L− 1) in chains with open
boundary conditions, following the same procedure as
Manmana et al. for the ionic Hubbard model [22] in
chains up to 400 sites. In spite of the large systems used,
finite-size effects are still important and do not allow an
accurate extrapolation. In any case, the qualitative be-
havior of our results (not shown) is similar to that found
by Manmana et al. showing a clear maximum inside the
BOW phase, an abrupt fall for U ∼ Uc as the system en-
ters the SS phase and a slower decay for larger U ∼ Us,
which for finite systems extends inside the SDW phase.
To conclude, we have presented compelling evidence,
based on bosonization as well as on other analytical and
numerical techniques, of the existence of a narrow bond-
ordered wave phase and a transition to an unconventional
incommensurate metallic one with dominant singlet su-
perconducting correlations in the phase diagram of the
U − X model. The appearance of superconductivity in
a model with repulsive on-site interactions at half fill-
ing, and of incommensurate correlations induced by in-
teraction are both unusual features. Their emergence can
be understood from the structure of the exactly solvable
case X = t. There the number Nd of doubly occupied
sites (doublons) becomes a conserved quantity; holes and
doublons play an identical role regarding the kinetic en-
ergy ǫ(kF ), which can be mapped into that of a spinless
fermion system, with Fermi momentum kF . The compe-
tition of ǫ(kF ) and UNd fixes the Fermi level of the re-
sulting effective model. The presence of doublons in the
ground state (U < 4t) simultaneously drives the spinless
fermions away from half-filling (kF 6= π), and switches on
the doublons role in the kinetic energy. The latter ceases
to be identical to that of holes as soon as X 6= t, gener-
ating incommensurability within the system. Moreover
superconducting correlations can dominate away from
half-filling [8]. Thus, a nonvanishing number of doublons
provides the scenario for both incommensurability and
superconductivity for X . t.
We thank D. Cabra for useful discussions. We acknowl-
edge support from PICT’s No. 03-11609, No. 03-12742,
and No. 05-33775 of ANPCyT and PIP’s No. 5254
and No. 5306 of CONICET, Argentina, No. FIS2006-
08533-C03-02, and the “Ramon y Cajal” program from
MCEyC of Spain, Angelo Della Riccia Foundation, and
PRIN 2005021773 Italy.
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  We determine the quantum phase diagram of the one-dimensional Hubbard model
with bond-charge interaction X in addition to the usual Coulomb repulsion U at
half-filling. For large enough X and positive U the model shows three phases.
For large U the system is in the spin-density wave phase already known in the
usual Hubbard model. As U decreases, there is first a spin transition to a
spontaneously dimerized bond-ordered wave phase and then a charge transition to
a novel phase in which the dominant correlations at large distances correspond
to an incommensurate singlet superconductor.

Incommmensurability and unconventional superconductor to insulator transition in
the Hubbard model with bond-charge interaction
A. A. Aligia,1 A. Anfossi,2, 3 L. Arrachea,4, 3 C. Degli Esposti Boschi,5 A.
O. Dobry,6 C. Gazza,6 A. Montorsi,2 F. Ortolani,7 and M. E. Torio6
1Comisión Nacional de Enerǵıa Atómica, Centro Atómico Bariloche and Instituto Balseiro, 8400 S.C. de Bariloche, Argentina
2Dipartimento di Fisica del Politecnico and CNISM,
corso Duca degli Abruzzi 24, I-10129, Torino, Italy
3BIFI, Universidad de Zaragoza, Corona de Aragón 42, 5009 Zaragoza, Spain
4Departamento de F́ısica de la Materia Condensada, Universidad de Zaragoza, 5009 Zaragoza
5Unità CNISM and Dipartimento di Fisica dell’Università di Bologna, viale Berti-Pichat 6/2, I-40127, Bologna, Italy
6Instituto de F́ısica Rosario, CONICET-UNR, Bv. 27 de Febrero 210 bis, 2000 Rosario, Argentina.
7Dipartimento di Fisica dell’Università di Bologna and INFN, viale Berti-Pichat 6/2, I-40127, Bologna, Italy
(Dated: November 1, 2018)
We determine the quantum phase diagram of the one-dimensional Hubbard model with bond-
charge interaction X in addition to the usual Coulomb repulsion U > 0 at half-filling. For large
enoughX < t the model shows three phases. For large U the system is in the spin-density wave phase
as in the usual Hubbard model. As U decreases, there is first a spin transition to a spontaneously
dimerized bond-ordered wave phase and then a charge transition to a novel phase in which the
dominant correlations at large distances correspond to an incommensurate singlet superconductor.
PACS numbers: 71.10.Fd,71.10.Hf,71.10.Pm,71.30.+h
The Hubbard model has been originally proposed to
describe the effect of the Coulomb interaction in tran-
sition metals, which usually contain localized orbitals.
Other real compounds containing more extended orbitals
cannot in general be properly described by this simple
Hamiltonian. Well-known examples are several quasi-
one-dimensional (1D) materials that have been recently
investigated [1], which exhibit a variety of phases that
cannot be explained with the usual Hubbard model. Ad-
ditional interactions should be included. A natural in-
teraction that arises in systems with extended orbitals is
the bond-charge interaction X [2]. In fact, it is natural
to assume that the charge in the bond affects screening
and the effective potential acting on valence electrons,
and therefore the extension of the Wannier orbitals and
the hopping between them should vary with the charge.
This leads to the U −X Hamiltonian:
H = −t
σ=↑,↓,〈ij〉
iσcjσ +H.c.) + U
ni↑ni↓
σ,〈ij〉
cjσ +H.c.)(ni−σ + nj−σ). (1)
This model has been studied in two dimensions, moti-
vated by a theory of hole superconductivity [3]. A mod-
ified version of it has been derived as an effective model
for the cuprates and shows enhanced d-wave supercon-
ducting correlations [4]. Recently, this model has been
paramount to broader audiences, and its relevance has
been discussed in the context of mesoscopic transport [5]
and quantum information [6, 7].
In 1D, there are bosonization [8, 9] and numerical [9]
results available. However, at half-filling, the effect of X
disappears in the standard bosonization treatment and a
behavior different from the usual Hubbard model was not
expected in these studies. For X = t, an exact solution
is available [10]. In this case the ground state is highly
degenerate: the transition to a metallic state takes place
at Uc = 4t > 0, but the response of the system to an
applied magnetic flux indicates that it is not supercon-
ducting [11]. In view of the previous studies, the recent
evidence of an insulator-metal transition driven by X < t
at finite Uc > 0 at half-filling comes as a surprise [12].
The nature of the metallic phase and the character of
the transition have not been fully elucidated, though the
possibility of superconductivity has been suggested.
In this Letter we employ several analytical and numer-
ical techniques to calculate accurately the phase diagram
of the model at half-filling in 1D and to determine the na-
ture of each phase. We establish that the insulator-metal
transition is of commensurate-incommensurate (CIC)
type to a phase with dominating singlet superconducting
(SS) correlations. Remarkably, unlike other CIC tran-
sitions [13, 14], it is not driven by one-body effects like
chemical potential or the emergence of more than two
Fermi points in the noninteracting dispersion relation,
but by strong correlations induced by large enough X . In
addition, we unveil that inside the insulating phase there
is a spin transition separating the expected spin-density
wave (SDW) for U > Us from a spontaneously dimerized
bond-ordered wave (BOW) phase for Uc < U < Us. This
transition is of Kosterlitz-Thouless (KT) type and a spin
gap opens in the BOW phase.
The nature of each phase and the qualitative aspects of
the phase diagram can be understood by a weak coupling
bosonization analysis provided it includes vertex correc-
tions of second order in X to the coupling constants and
http://arxiv.org/abs/0704.0717v2
one term of order a2 in the bosonization of the bond-
charge interaction as described below, where a is the lat-
tice constant. A bosonized version of (1) is given by the
following Hamiltonian density:
H = H0σ +H0ρ +
(2πα)2
8φσ)−
(2πα)2
(2πα)2
8φσ)∂xφρ, (2)
where H0σ and H0ρ are the usual known quadratic forms
and α is a short range cutoff in the bosonization proce-
dure. The first line of (2) has the structure of the pre-
viously studied bosonized theory [8], which corresponds
to two decoupled sine-Gordon field theories, one for the
spin (φσ) and the other for the charge (φρ). In order
to take into account the effect of the bond-charge inter-
action on the phase diagram of the system, we included
vertex corrections of second order in X in the definition
of the the coupling constants gi, due to virtual processes
involving states far from the Fermi energy [15]. In addi-
tion, we took into account the usually neglected gσρ term
that couples spin and charge degrees of freedom. The lat-
ter is ∝ a2. It arises including spatial derivatives of the
fermionic fields in the representation of (1) in terms of
a low energy field theory. All of these terms have naive
scaling dimension 3 and are usually neglected. However,
one term that bosonize as the second line of (2) becomes
relevant for large enough X and provides a mechanism
for an incommensurate transition, as discussed below.
Explicitly, the effective parameters read g1⊥ = g2⊥ =
(U− 8X
π(t−X)
)a and gσρ =
2a2X . The forward and umk-
lapp processes are the same as in the Hubbard model,
g3⊥ = g4⊥ = Ua. The Luttinger liquid parameters (Kρ
and Kσ) and the charge and spin wave velocity (uρ and
uσ) in terms of gi are given by known expressions [16].
Neglecting the gσρ term, the renormalization-group (RG)
flow diagrams are of KT type. A spin gap opens when
g1⊥ < 0, i.e., when the flow of RG, which takes place
on the separatrix of the KT diagram due to spin SU(2)
symmetry, goes to strong coupling. Therefore, the spin
gapped phase appears when U < Us =
π(t−X)
. As for
the behavior of the charge modes, a gap opens when the
g3⊥ term becomes relevant. The charge gapped phase
takes place for U > Uc, with Uc < Us. The gσρ term
becomes relevant for Kσ < 1/2 (X > 0.6t for U = 0).
In the spin gapped phase the cos(
8φσ) is frozen at its
mean value. This term could be interpreted as a chemical
potential [µ =
(2πα)2
〈cos(
8φσ)〉] times a charge density
operator. The effects of such a term are known [16].
If we start the analysis from a situation where there is
also a charge gap (∆c) smaller than the spin one (∆s),
and we then increase the value of X , the effect of this
term is to close ∆c, leading to a metallic phase when
µ > ∆c. The effective Fermi level is shifted with respect
to the original one and the system develops incommen-
0 0,2 0,4 0,6
Bosonization S
Bosonization C
0 0,2 0,4 0,6 0,8
0 0,2 0,4 0,6 0,8 1
Exact
Top. Ph S
Top. Ph C
DMRG S
DMRG C
Figure 1: (Color online). Phase diagram. Left: Bosoniza-
tion (top), and real space renormalization-group (bottom)
predictions. Right: Numerical results as obtained by DMRG
(circles-squares) and topological phase (crosses) methods.
surate correlations. A numerical analysis discussed be-
low shows that the system has dominant SS correlations.
Thus, this phase can be characterized as incommensurate
singlet superconducting (ICSS).
For a qualitative localization of the boundary transi-
tion line between the insulator and the ICSS phase, we
have implemented a procedure as follows: (i) We start
from a parameter regime where the spin gap is open.
(ii) We follow the RG flow up to a length scale where
|g1⊥|/(πUs)| ∼ 1. (iii) At this point the gσρ term is de-
coupled by a mean field approach similar to that used
by Nersesyan et al. to show incommensurability in the
anisotropic zigzag chain [17]. The value of 〈cos(
8φσ)〉
is exactly obtained at the LE point (Kσ = 1/2). (iv) For
vanishing gσρ, ∆c is obtained by rescaling the problem
to the LE point of the charge sector, by using the RG
equations of the sine-Gordon theory. (v) The CIC tran-
sition takes place when
(2πα)2
〈cos(
8φσ)〉 = ∆c [16]. In
the top left panel of Fig. 1 we show the phase diagram
of the model predicted by this approach. For each value
of X , there are two transition points Uc and Us corre-
sponding to the charge and spin transition, respectively.
Each phase is characterized by the gapped modes and the
relevant order parameter. For U > Uc the system is an
insulator. For U > Us, the slowest decaying correlation
functions are the spin-spin ones. The system is in a SDW
phase. For Uc < U < Us a fully gapped (spin and charge)
phase is developed. The fields φσ and φρ are located at
the minimum of the potential, and the translation sym-
metry is spontaneously broken. The BOW parameter,
defined below, acquires a nonzero value. For U < Uc
the charge gap closes and the dominant correlations at
large distances are the SS ones. While the nature of each
phase has been identified, the phase boundaries predicted
by bosonization are not quantitatively valid, particularly
for large values of the interactions. In the right panel of
Fig. 1 we show the phase diagram of the model, as ob-
tained by accurate numerical techniques. One of them,
used to determine the charge transition line, consists in
studying singularities of single-site entanglement [12] by
means of density-matrix renormalization group (DMRG)
[18]. Another method is based on topological numbers,
or jumps of Berry phases [19], which was successfully ap-
plied to a similar model [8] (b). The value of Uc (Us) is
determined in this case by the jump of the charge (spin)
Berry phase. The corresponding values of Uc and Us in
systems up to L = 14 sites, extrapolated to the thermo-
dynamic limit using a parabola in 1/L2, are also shown
in Fig. 1.
DMRG evaluations of ∆c and ∆s confirm these pre-
dictions. The charge gap was calculated in [12] from
the definition 2∆c = E0(N + 2) + E0(N − 2)− 2E0(N),
E0(N) being the ground-state energy of the chain with
N particles. Similarly, the spin gap is here determined
through ∆s = E0(Sz = 1) − E0(Sz = 0), being E0(Sz),
the ground-state energy of the half-filled system within
the subspace with a given total Sz. We can see in Fig. 1
that the closing of ∆c, ∆s do not take place simultane-
ously for small U and X . The critical lines for the closing
of both gaps obtained by extrapolations to the thermo-
dynamic limit are in reasonable quantitative agreement
with the ones determined by the method of the topolog-
ical phases.
We have verified that the spin transition is of KT
type, calculating the scaling dimensions of the singlet and
triplet operators as described in [19]. In order to identify
the universality class of the charge transition, we em-
ployed the finite-size crossing method [20]. The study of
the dependence of 〈ni↑ni↓〉 = ∂eL/∂U on the size L (eL
being the ground-state energy density) provides a loca-
tion of the critical points in agreement with the methods
discussed above. In addition, the divergence that devel-
ops ∂e2L/∂U
2 with increasing L indicates that the gap ex-
ponent ν remains close to 1/2 (the value that can be com-
puted exactly at the point X = t) for X/t = 0.6, 0.7, 0.8
with a possible increase for X/t → 0.5; below this point,
our numerical analysis suggests that the charge transi-
tion becomes of KT type, with “ν = ∞”. The estimate
ν = 1/2 relies upon the assumption that the dynamic
exponent ζ (through which gap and correlation length ξ
are related, ∆c ∝ ξζ) is still ζ = 2, as in the exactly
solvable case X = t [7]. As already noted in [12], the
behavior of ∆c ∝ L−2 along the transition line is consis-
tent with this exponent. We stress that such feature is in
agreement with the CIC character of the metal-insulator
transition [16]. Instead, within the metallic phase, the
finite-size scaling suggests ∆c ∝ L−1, although the data
are rather noisy due to incommmensurability.
In Fig. 2 we show numerical results supporting the in-
commensurate character of the metallic phase. We report
the density distributions in real space for the local charge
8 16 24 32 40 48 56 64
U=4.0 t
U=3.05 t
U=2.5 t
U=0.5 t
8 16 24 32 40 48 56 64
X=0.3 t
X=0.58 t
X=0.8 t
Figure 2: (Color online). Charge distribution 〈ni〉 evaluated
by DMRG. Left: X = 0.8t. Right: U = 1.5t.
density ni = ni↑ + ni↓ in the ground state in an open
chain with L = 64 sites. The incommensurate character
of the metallic phase manifests itself also in the behavior
of the charge and spin correlation functions, whose cor-
responding structure factors show peaks away from the
commensurate reciprocal vector q = π (not shown). The
left panel of Fig. 2 corresponds to X = 0.8t as U is var-
ied. The behavior is similar to the one observed within
the incommensurate phase of the Hubbard model includ-
ing next-nearest-neighbor hopping (t−t′−U model) [13].
For U > Uc = 3.05t, the commensurate charge distribu-
tion characterizing the insulating phase is reached within
a few lattice sites from the edge. The insulator-metal
transition shows up via the appearance of incommensu-
rate modulations in the charge distribution, whose wave-
length increases within the metallic phase. The right
part of the figure shows the results obtained by varying
X at U = 1.5t. Interestingly, a first modulation appears
already for Xs < X < Xc ( Xs ≈ 0.5t, and Xc ≈ 0.6t).
Again, for X > Xc further incommensurate modulations
appear in the LE phase.
Within the charge sector U < Uc, the dominating cor-
relations at large distance are superconducting pair-pair
ones if the correlation exponent Kρ > 1 or charge-charge
ones otherwise. We calculatedKρ employing the method-
ology described in [9]. This study casts extrapolated val-
ues Kρ ∼ 1.3 for U = 0 and X = 0.8t. To provide
stronger evidence for the SS character of the incommen-
surate phase, we have calculated on-site pairing correla-
tions 〈P †i Pj〉 with Pi = c
i↓ and charge-charge correla-
tions |〈ninj〉 − 〈ni〉〈nj〉| in an open chain with 100 sites
and using the sites 30 to 70 to avoid boundary effects.
The results are displayed in Fig. 3. A fitting of the pair-
ing correlations at distances between 8 and 40 sites gives
Kρ = 1.32± 0.01. This value is also consistent with the
long distance behavior of the charge-charge correlations.
The inset also shows the tendency of the system to show
the anomalous flux quantization characteristic of super-
0.0 0.5 1.0 1.5
0.0 0.5 1.0
-5.060
-5.055
-5.050
Φ / Φ
 pair-pair
 charge-charge
 |i-j|
Figure 3: Pair-pair and charge-charge correlation functions
for U = t and X = 0.8t. Full (dashed) line corresponds to
a power law with exponent 1/Kρ (Kρ). The inset shows the
ground state energy as a function of an applied magnetic flux.
conductivity [11], which is more pronounced as the size
of the system increases.
An additional argument suggesting superconducting
correlations within this phase is provided by the real
space renormalization-group method, used before for the
standard Hubbard model [21]. Different from that case,
the recursive equations for the renormalized parameters
in the positive U regime, depending on X and U , exhibit
three different fixed points for the nth step renormalized
Coulomb interaction U (n) in the large n limit: U (n) > 0
for U > Urc, U
(n) = 0 for U = Urc, and U
(n) < 0 for
U < Urc. In the latter case, the effective Coulomb inter-
action becomes attractive. In the bottom left insert of
Fig. 1 Urc obtained in this way is reported.
To support the bosonization predictions, which char-
acterize the intermediate phase as a BOW, we have eval-
uated with DMRG the BOW order parameter OBOW =
(−1)i〈c†i+1σciσ +H.c.〉]/(L− 1) in chains with open
boundary conditions, following the same procedure as
Manmana et al. for the ionic Hubbard model [22] in
chains up to 400 sites. In spite of the large systems used,
finite-size effects are still important and do not allow an
accurate extrapolation. In any case, the qualitative be-
havior of our results (not shown) is similar to that found
by Manmana et al. showing a clear maximum inside the
BOW phase, an abrupt fall for U ∼ Uc as the system en-
ters the SS phase and a slower decay for larger U ∼ Us,
which for finite systems extends inside the SDW phase.
To conclude, we have presented compelling evidence,
based on bosonization as well as on other analytical and
numerical techniques, of the existence of a narrow bond-
ordered wave phase and a transition to an unconventional
incommensurate metallic one with dominant singlet su-
perconducting correlations in the phase diagram of the
U − X model. The appearance of superconductivity in
a model with repulsive on-site interactions at half fill-
ing, and of incommensurate correlations induced by in-
teraction are both unusual features. Their emergence can
be understood from the structure of the exactly solvable
case X = t. There the number Nd of doubly occupied
sites (doublons) becomes a conserved quantity; holes and
doublons play an identical role regarding the kinetic en-
ergy ǫ(kF ), which can be mapped into that of a spinless
fermion system, with Fermi momentum kF . The compe-
tition of ǫ(kF ) and UNd fixes the Fermi level of the re-
sulting effective model. The presence of doublons in the
ground state (U < 4t) simultaneously drives the spinless
fermions away from half-filling (kF 6= π), and switches on
the doublons role in the kinetic energy. The latter ceases
to be identical to that of holes as soon as X 6= t, gener-
ating incommensurability within the system. Moreover
superconducting correlations can dominate away from
half-filling [8]. Thus, a nonvanishing number of doublons
provides the scenario for both incommensurability and
superconductivity for X . t.
We thank D. Cabra for useful discussions. We acknowl-
edge support from PICT’s No. 03-11609, No. 03-12742,
and No. 05-33775 of ANPCyT and PIP’s No. 5254
and No. 5306 of CONICET, Argentina, No. FIS2006-
08533-C03-02, and the “Ramon y Cajal” program from
MCEyC of Spain, Angelo Della Riccia Foundation, and
PRIN 2005021773 Italy.
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