close

Вход

Забыли?

вход по аккаунту

?

Electrons with correlated hopping interaction in one dimension.

код для вставкиСкачать
Ann. Physik 3 (1994) 163-180
Annalen
der Physik
0 Johann Ambrosius Barth 1994
Electrons with correlated hopping interaction
in one dimension1
G. Japaridze’ and E. Muller-Hartmann
Institute of Theoretical Physics, University of Cologne, D-50937 Kdln, Germany
Received 9 February 1994, accepted 25 February 1994
Abstract. The tD system of correlated electrons characterized in addition to the usual on-site (V)and
nearest-neighbour ( V ) repulsion by a correlated-hopping term ( t * ) is considered. The ground state
phase diagram is studied within the framework of the weak-coupling continuum-limit approach. At
filling v the effective interaction originating from the correlated-hopping term which appears in the
continuum-limit theory is given by t* cos (nv). Being repulsive for v< 1/2 and attractive for v > 1/2,
this interaction leads to a characteristic band-filling dependence of the phase diagram of the system.
For v f 1/2, depending on the relation between the bare values of the coupling constants
(U,V , t * cos (nv))and, hence, on the band-filling, the system shows three different phases in the infrared limit: a Luttinger metal, a nonmagnetic metal and a singlet superconducting phase. For v = 1/2,
the correlated-hopping interaction is dynamically trivial, leading only to a renormalization of the oneelectron hopping amplitude and the phase diagram of the model coincides with that of the extended
Hubbard model.
Keywords: Bond-charge repulsion; Correlated hopping; Superconductivity.
1 Introduction
The discovery of high temperature superconductivity [l] has renewed the interest in the
physics of correlated electrons in low dimensions. Although the main interest is centered
around the physics of two-dimensional highly correlated electron systems, the onedimensional analogues of models related to high temperature superconductivity are very
popular, mainly due to the conjecture [2] that properties of the 1D and 2D variants of
certain models have common aspects. Moreover, exact or analytical solutions of the
one-dimensional models usually serve as a good guide for understanding more complex
cases.
One of the challenging properties of the high-T, superconductors is the strong bandfilling dependence of the low-temperature properties of these materials. Within a quite
narrow range of values of the band-filling, controlled by doping, these materials
manifest properties of magnetic insulators, metals and superconductors [3 J.The complicated character of the low-energy phase diagram of the high-T, superconductors
-~
Research performed within the program of the Sonderforschungsbereich 341 supported by the
Deutsche Forschungsgemeinschaft.
Permanent address: Institute of Physics, Georgian Academy, TbiIisi, Georgia.
164
AM. Physik 3 (1994)
which exhibit (via doping) a sequence of phase transitions between states of various
symmetries led to an increased interest in models containing unusual correlation
mechanisms, which would produce strong band-filling dependence as well as superconducting instabilities.
Among other scenarios, the so called bond-charge repulsion is a subject of current
considerations [4, 5 , 61. This describes the interaction between charges located on
bonds and on sites. The simplest 1D tight-binding Hamiltonian including interaction
with charges located on bonds contains, in addition to the usual terms describing interaction between electrons on the same site (U)and on neighboring sites (V),a term
describing the modification of the electron hopping by the presence of other particles
on the sites [7]:
Here C ~ , ~ ( C ~is, the
~ ) creation (annihilation) operator for an electron with spin CJ at
site n and pn,a = C ~ , ~ C , , , . There are Np particles, L sites and the band filling
v = Np/2Lis controlled by the chemical potential p. In the standard case of Coulomb
repulsion all model parameters in (1) are positive, moreover in this article we will study
the weak-coupling case only, where to%U,V,t* >O will be assumed.
The correlated-hopping interaction, as a source for superconducting instabilities of
electron systems, was first discussed by Hirsch, two studied the three-dimensional version of the model (1) [4]. Hirsch pointed out that the correlated-hopping interaction
produces effective interactions of different sign at the bottom and the top of the band
which are most attractive near the top. Assuming proper band-fillings which ensure an
attractive character of the correlated-hopping interaction Hirsch found, within the
standard BCS type mean-field approach, that superconductivity exists in the system for
221 t* I > U+z V, where z is the number of nearest neighbours. However, the BCS-type
analysis is based on the assumption of Fermi liquid behaviour of the electron system
and completely ignores the interference between instabilities of different type, two
essential points possibly distinguishing the high-T, superconductors from the usual
ones [2]. Therefore, an exact or analytical solution of the 1D version of the correlatedhopping model, incorporating the non-Fermi-liquid character of the behaviour of the
electron system as well as the strong interference between correlations of different symmetries, is of great interest.
All recently proposed integrable models of strongly correlated electrons showing
superconducting transitions in the case of repulsive interaction [5, 61 contain as the
essential part the correlated-hopping interaction. The supersymmetric extended Hubbard model proposed by EBler, Korepin and Schoutens [ 5 ] (EKS-model) contains all
interactions of the correlated-hopping model (1) as well as the pair-hopping term [8]
and nearest-neighbour spin-spin interaction. Unfortunately, the symmetry conditions
ensuring integrability of the model, strongly restrict the values of the various coupling
constants and leave, besides the on-site (Hubbard) coupling constant U only a single
free model parameter in the theory. This restriction makes it impossible to separate the
various sources of correlation from each other and, hence, pinpoint the one which is
G. Japaridze, E. Miiller-Hartmann, Electrons with correlated hopping interaction
165
responsible for superconductivity. Moreover, since the number of electron pairs
located on a site is conserved in the EKS-model, the Hubbard interaction looses its role
as the source of various dynamical instabilities and simply serves as a chemical potential for local pairs.
Another integrable version of the correlated-hopping model, showing superconductivity, was recently proposed by Bariev, Kliimper, Schadschneider and Zittartz (BKSZmodel) (61. The Hamiltonian of the BKSZ-model has the form:
The exact solution of the model (2) [61 confirms Hirsch’s prediction about superconductivity originating from the correlated-hopping interaction and provides a critical
value of the particle concentration v, for the transition from a state characterized by
dominating density-density correlations into a state with most divergent correlations of
the superconducting type. For t : > O the superconducting transition takes place for
v > v,, where v, varies monotonically from 0.5 to 0.5858 with increasing coupling constant t:.
Unfortunately, the Hamiltonian (2) contains only half of the correlated-hopping interaction and thereby has a reduced U(1)-spin symmetry. This results, for any nonzero
t r , into the existence of a gap in the spin’excitation spectrum [6] and hence to different low-energy properties of the model. Moreover the BKSZ-model does not contain
the on-site (Hubbard) interaction and therefore does not shed light on the question of
the comDetition between on-site and correlated-hopping interactions.
One of the very profound tools for the investigation of correlations in one-dimensional systems is the field theory approach based on the construction of the corresponding continuum-limit Hamiltonian with subsequent use of the bosonization procedure
and renormalization group analysis. This approach was widely used for the investigation of one-dimensional systems of correlated electrons and various spin-chain models
(for reviews see [9- 1I]). Although the field theory treatment is based on the weakcoupling limit, where the bare values of coupling constants are assumed to be much
smaller than the bandwidth (t *, V ,Uez to), the essential features characterizing the
behaviour of the system, received within this description, can be expected to remain
valid even in the strong coupling limit.
In this paper we study the correlated-hopping model (1) using field theory techniques. In what follows we will briefly describe some qualitative features of the model
we will obtain. In the low energy limit, when the physics is controlled by states near
the Fermi points, our renormalization-group and bosonization methods indicate that
the strength and the sign of the effective interaction provided by the correlated-hopping
term strongly depend on the band-filling. The effective interaction originating from the
bond-charge repulsion which appears in the continuum-limit theory, is given by
t* cos (nv). This is repulsive for v < 1/2 and attractive for v > 1/2 and determines the
principal dependence of the ground state phase diagram of the system on the band-filling. At half-filling, v = 1/2, there is no cross coupling between the bond and site charge
densities [7] and the correlated-hopping interaction leads only to a renormalization of
166
Ann. Physik 3 (1994)
the one-electron hopping amplitude. In this case the phase diagram of the model (1)
coincides with that of the extended Hubbard model [9,121 corresponding to the case
t * = 0. For v # 1/2,depending on the relation between the bare values of the coupling
constants (U,
V,t* cos (xv)) and, hence, on the band-filling, the system shows three
different phases in the infrared limit: a Luttinger metal, a nonmagnetic metal and a
singlet superconducting phase.
For 4 t * > U + 6 V, depending on the band-filling, all these phases are realized: for
band-fillings v < vC1= arccos ( - (U-2 V)/4 t * ) / x the ground state corresponds to a
Luttinger metal phase (gapless charge and spin excitation spectra, dominating fluctuations of the charge-density wave (CDW) and spin-density wave (SDW) type), for v,,
€ v < vC2= arccos (- (U+6 V)/4t*)/7r to a nonmagnetic metal phase (gap in the spin
excitation spectrum and dominating CDW fluctuations) and for v > vC2to a singlet
superconducting phase (i.e. gap in the spin excitation spectrum and dominating singlet
superconducting fluctuations).
For U + 6 1/>4t* the superconducting transition is absent, but the transition from
the Luttinger metal into the nonmagnetic metal state remains for any I U-2 Vl < 4 t * .
For I U- 2 V l > 4 t * the phase diagram of the model (1) coincides with that of the extended Hubbard model.
The range of applicability of the continuum-theory description is restricted by the
condition U,V ,t * 4t sin (nv ) such that the linearization of the spectrum around the
Fermi points is justified. Nevertheless, qualitative features of the phase diagram obtained from the weak coupling theory are expected to remain valid also for U,V,t * t.
In a forthcoming paper we will consider, within the same weak-coupling approach, the
integrable version of the correlated-hopping model, the BKSZ-model governed by the
Hamiltonian (2). Comparison with the results of the exact (Bethe-ansatz) solution [6]
allows us to clarify the accuracy of our description and to identify the features of the
weak coupling theory surviving in the strong coupling limit.
Our paper is organized as follows: in the following section we will review the model
and its symmetries. In Section 3 we construct the continuum-limit version of the model
(1) and perform a renormalization-group analysis of the corresponding field theory. In
Section 4 we study the large-scale behaviour of the various correlation functions. In
Section 5 the ground state phase diagram of the model is presented. Finally, in Section
6 summary of results is given.
-
2 Review of the model and its symmetries
In the absence of the correlated-hopping interaction (t* = 0) the Hamiltonian (1) is the
Hamiltonian of the extended Hubbard model. The features of the extended Hubbard
model are well studied [9,121. Away from the half-filled case the charge excitation
spectrum of this model is always gapless, while the spin excitation spectrum is gapless
for U >2 V and has a gap for U <2 V. The line U = 2 V marks the crossover from the
Luttinger metal phase for U > 2 Vinto the nonmagnetic metal phase for U < 2 V. In the
half-filled case there is a gap in the charge excitation spectrum for any U,V > O and
the U = 2 V line serves as the boundary between the CDW insulating phase (Uc2V)
and the SDW insulating phase (U>2 V). As we see, in the case of repulsive interaction,
there is no trace of superconductivity in the phase diagram of the extended Hubbard
model. Moreover, except for the special case of the haif-filled band, features of the
model remain essentially the same for any value of the band-filling.
G. Japaridze, E. Miiller-Hartmann, Electrons with correlated hopping interaction
167
Let us analyse, from the symmetry point of view, the difference between the extended Hubbard and the correlated-hopping model. The extended Hubbard model has two
important symmetry properties, the S U(2)-spin invariance and invariance with respect
to particle-hole transformation
transforming &'(to,U,V) for the band-filling v (in what follows refered to as
X(to,U,V ,v ) ) into &'(to,U,V, 1 - v ) and thus indicating the essential equivalence of
descriptions in terms of particles or holes.
The three generators of the spin-SU(2) algebra
commute with the Hamiltonian (1) which shows its .SU(2)-spin invariance.
The particle-hole transformation (3) converts
&'(to,U,V,t*,v)-+&'(t0-2vt*,U,V,--tt*,l-v)
,
(5)
which indicates two different roles played by the correlated-hopping interaction. The
first is trivial - renormalization of the onelelectron hopping amplitude, the second is
dynamical - source of correlations. To separate these two effects it is convenient to
subtract out the Hartree part of the interaction, rewriting the Hamiltonian (I) in the
following way:
where
t(v) = t , - 2 v t *
(9)
is the renormalized hopping, p- is the renormalized chemical potential, and normal
ordering symbols :. . .: denote that the vacuum average has been subtracted out. For
the given value of the band-filling v, we obtain
168
Ann. Phvsik 3 (19941
The band-filling dependence of the one-particle hopping amplitude in (7) will introduce
an additional band-filling dependence of nonuniversal quantities, for instance, of the
conductivity of the system in the metallic state, but does not change the ground state
phase diagram, determined by the relationship between the small dimensionless coupling constants U/t, V / t ,t*/te1. In terms of the Hamiltonian (6),the result of the particle-hole transformation (3) reads
f l t ( v ) ,u, V , t * , v ) - + f l t ( l v- ) , u,V , - t*, 1 - v ) ,
(12)
from which one concludes that the particle-hole symmetry is violated by the correlatedhopping interaction, but the ground state phase diagrams of the correlated-hopping
model for t * >0 at band-filling v and for t * <0 at band-filling 1 - v coincide. As we
will show below, the effective interaction derived from the bond-charge repulsion in
the continuum-limit theory is given by t * cos (nv) which fulfills the symmetry condition.
3 Continuum-limit theory and renormalization-groupanalysis
Although the derivation of the continuum-limit versions of ID lattice Hamiltonians is
quite standard and has been reviewed in many places [9, 10, 111 the special case of the
correlated-hopping interaction has not been described before. We will stress here the
main points only. For details of the method we refer the reader to the excellent paper
of Shankar [ i l l .
The field theory treatment of the model (6) is based on the weak-coupling limit
t*, V, U g t . The free particle Hamiltonian A$ is diagonalized by Fourier transform,
to give the bare particle spectrum ~ ( k=) - t [cos ( k a )- cos (k F a )],where a is the lattice spacing and kF = n v / a . Assuming that the low energy physics is controlled by
states near the Fermi points f kF we linearize the spectrum around these points and
obtain two species (for each spin projection a), y1(2),,(n), which describe excitations
= k uFp. The Fermi velocity is uF = ta sin (nv), and
with dispersion relations El,cz)
the momentum p is measured from the two Fermi points. More explicitly, one decomposes the momentum expansion in (13) into two parts, centered around +kF,
where the Fermion fields yl,,(n) and y2,,(n)describe right-moving and left-moving
particles, respectively, and are assumed to be smooth on the scale of the lattice spacing.
This allows us to define the continuum fields
G. Japaridze, E. Milller-Hartmann, Electrons with correlated hopping interaction
169
The effective Hamiltonian describing the low-energy excitations with linear spectrum,
X ) , the form
written in terms of continuum fields W ~ ( ~ ) , ~ ( has
which is recognized as the Hamiltonian of a free massless Dirac field.
Now let us turn to qnt.
Using Eqs. (14) and (15) we show that
t
: c n , u ~ n , u := :wT,u(n)
+ ei2nnv
cy,,,(n):
v2Jn):
-
w:,u(n>W2,u(n)+e
-+ab,,u(x)
+
+ :w;&)
- i2nnv
+P2,U(X)
i2nnv
wzt,,(n)y/,,,(n)
+ ei2nnv v:,,(X)
wI,u(x) wl,u(x)l
W2,AX)
(17)
?
and
:ctn,ucn+,,u:
= einv:v:,u(n)vl,u(n+1 ) : + e - ' * ~ : ~ / t 2 , ~ ( n ) ~ ~1):
,~(n+
+ ei(2n
3
a
+ 1)nv
einv
( [
w:,,(n)w2,,(n+
l)+e
P1,u(X)+e-iXvP2,u(X)+e
+,-i(2n+t)nv
-i(2n+l)nv
+ 1)
i(2ni1 ) ~ v
w:.u(x)w2,u(x)
+O(a2)1
v;,u(x)vl,u(x)l
wt2,u(4 Wl,u(n
(18)
7
where
P1(2),u
t
(19)
= :v1(2),uW1(2),u(X):
Using (17) and (18) we now will rewrite A&t in terms of continuum fields w ~ ( ~ ) , ~ ( x ) .
The whole information about the lattice structure of the initial model (6) is contained
in the oscillating phase factors eiinV". On the grounds that the fields I , u ~ , ~ ~are
,(~)
dx. All terms in
smooth on the lattice, one can convert sums into integrals:
Xntthat contain the rapidly oscillating factors e *""' and e*14nnV will drop out.
However, in the case of a half-filled band, the Umklapp terms, which go as
* i 4 n n v = 1, are not oscillating and have to be kept. Performing this procedure, and
keeping only first order terms with respect to a, one arrives at:
qnt =a
1dx C (A,
<p:,u(x>
+~ 3 , u ( x ) +
) ~0
.j.
@ i , u ( ~ ) ~ --u(x)
,,
t- P ~ , ~ , ( x )-,(XI>
A,
U
+ ~ l , , P l , U ( ~ ) P 2 , U ( X ) +A1 L P l , U ( X ) P 2 .
t
t
-&I
t
+ A B S , , ( v 1 , a v 2 , u vv,,uw2,u + h.c.)+ 'ZBSL (VLU
t
v2,- u v1,- u v 2 , u + h.c.1
170
- .
YI
Y2
--p---->-
vz
*I
.Y2
Forward scattering in the
vicinity of one Fermi point
--<;
7 Y
--+-*2
7.T--*)+-
--*)->-
*2
91
YT
?*
-2Pz
*I
*I
Backward scattering
Yl
--
--
YI
Forward scattering in the vicinity of different Fermi points
= -*?
~
'f-*
~
- - *-
--*-
---p*I
- - *- - -
*I
~
*Z
YZ
*Z
Ann. Physik 3 (1994)
12
Y2
Umklapp processes
'I
Fig. 1 Classification of the
scattering processes found
when interaction terms are expressed in a basis of left (" 1")
and right ("2") moving Fermions.
As we see qnt
contains eight scattering terms (see Fig. 1) including forward scattering
in the vicinity of one Fermi point (Aoll,.), forward scattering in the vicinity of different Fermi points (Alll,L), backward scattering (ABsll,L) and also, in the case of a
half-filled band, Umklapp processes (Aul~,.). The bare values of the coupling constants are given by:
1
2
Lo. = - A * .
ABS.
1
=-(V+2V+4t*cos(nv)) ,
2
1
1
=--(U--2Y+4t*cos(7tv)) , A " . = Z ( u - 2 1 / ) .
2
As we see the lattice features of the initial model are reflected in the continuum-limit
Hamiltonian in two ways:
a) via the selection of scattering processes relevant for the physics of the system at
large distances at the given value of the band-filling; and
b) via the bare values of the coupling constants of the corresponding scattering processes, i.e. via normalization conditions for the effective interactions.
Contrary to the extended Hubbard model case (t* = 0), when the band-filling
dependence appears in the continuum theory only through the selection of the scattering processes (Umklapp processes, responsible for the dynamical generation of a gap
in the charge excitation spectrum for any V, Y> 0, appear in the continuum theory only
for the commensurate case of a half-filled band) we see from (18) and (21), that the
site off-diagonal nature of the correlated-hopping interaction results in a band-filling
dependence of the bare coupling constants of the effective continuum-limit Hamiltonian. Only in the special case of a half-filled band the role of the correlated-hopping
interaction is reduced to a trivial renormalization of the electron hopping amplitude
given by (9) and the dynamical features of the model coincide with those of the extended Hubbard model. In what follows, if not indicated specifically, we will assume bandfillings not close to 1/2 and neglect Umklapp processes. Moreover, the terms corre-
G. Japaridze, E. Miiller-Hartmann, Electrons with correlated hopping interaction
171
sponding to the forward scattering in the vicinity of a single Fermi point, (Aorl,+)
leading only to a renormalization of the Fermi velocity, will be omitted. Taking thls
into account, we deal with the following Hamiltonian, easily recognized as the LutherEmery backscattering model [14]:
Here
1
Pl(2)
=f
i
c
W:(2),aW1(2),u
7
g
vF is the Fermi velocity and the small dimensionless coupling constants are given by
1
g,=
--
(
4
1
1+ L l L - - 2 L B S ( ( ) = --
(O+6 V + 4 t * cos (nv)) ,
? VF
71 VF
go=
1
1
-- 1 (All[ - A * , - 2 & q ) = -(U-2
nVF
g, =-21,
1
72 OF
Y + 4 t * cos (7cv)) ,
AVF
1
=-(LI-2V+4t*cos(7cv))
.
7.c VF
The model (22) is characterized by a complete decoupling of charge (p = p1+p2)
and spin (a = o1+02)degrees of freedom. The most convenient way to analyze it is
to use the bosonization procedure by which this decoupling becomes more transparent.
[I31
In the boson representation of the Fermi fields ~1(2),o(x)
(here a is a cut-off parameter of the order of the lattice constant) the Hamiltonian (22)
splits into two decoupled parts, describing, respectively, charge- and spin-density degrees of freedom [14]:
Let us first consider the charge sector. As seen in (31), away from half-filling the
charge degrees of freedom of the system are described in terms of the Tomonaga model
[15]. The coupling constants in the Tomonaga model are not renormalized and 8 is
easily diagonalized by use of the Bogoliubov transformations 1161 leading to the free
bose field Hamiltonian
Here up is the sound velocity and p,,(x) and P,(x) are a scalar field and its conjugate
momentum, respectively, related to the charge and current density in following way:
with
G=
8 n (l++gp)1/2(l-;gp)
- 1/2
- (I++.>
*
Thus, it follows from (33) that for arbitrary values of the coupling constants the model
is characterized by a gapless charge excitation spectrum. The effective interaction remains weak and, since it is included in (33) via renormalization of the sound velocity
and via rescaled values of the p,(x) and P,(x) fields, it manifests itself in a slight
renormalization of the charge excitation spectrum and in the renormalization of the
critical indices describing the power-law decay of the various charge-charge and current-current correlation functions.
Let us now consider the spin channel. In analogy to the case of the charge channel,
we rewrite 2, in the standard form of a quantum Sine-Gordon Hamiltonian
where v, is the spin wave velocity, p,(x) and P,(x) are a scalar field and its conjugate
momentum, respectively, related to the spin and spin-current density in the following
way:
G. Japaridze, E. Miiller-Hartmann, Electrons with correlated hopping interaction
173
with
In contrast to the case of the charge sector, the coupling constants in the Sine-Gordon
theory are renormalized. Fortunately, in the weak-coupling limit (lg,l, Jg, 1 Q 1) the
properties of the Sine-Gordon model are well known [17, 181. The infrared behaviour
of the Sine-Gordon Hamiltonian 2, is described by the corresponding pair of renormalization group equations for the effective coupling constants I'i :
where L = In ( a ) and rj(0)= gi [17]. The Eq. (41) describe a Kosterlitz-Thouless transition [19] with a flow diagram as given in Fig. 2. The flow lines lie on the hyperbolae
r2,-r: = g2,- g t = constant
,
(42)
and, depending on the relation between the bare coupling constants gu and g,, show
two different regimes of behaviour.
For g, 2 I g 1 we have a weak coupling regime; the effective mass f I 40, and the
fixed-point Hamiltonian, describing the low-energy behaviour of the system, coincides
with the free bose field Hamiltonian. The spin excitation spectrum in this case is gapless
and the values of the corresponding critical indices describing the power-law decay of
,
Fig. 2 The renormalization-group flow
diagram; the arrows denote the direction of
flow with increasing length scale.
I
174
Ann. Physik 3 (1994)
the spin density-density and spin current-current correlations are determined by the
fixed-point value of the effective coupling constant r,.
For g,< lg, 1 the system scales into a strong coupling regime; depending on the
sign of the bare mass mu,the effective mass f
+m , which signals the cross-over into a strong coupling regime and indicates the dynamical generation of a commensurability gap in the spin excitation spectrum. The field pu gets ordered with the
vacuum expectation values
,
-+
In the case of the correlated-hopping model (6) which we are considering here due
to the S U(2)-spin symmetry the flow lines are restricted to moving along the bisectrix
gu = g,.
A) For g,>O i.e. U-2 V + 4 t * cos (nv)>O the system renormalizes into the weakcoupling regime, with fixed-point values of the effective coupling constants
ru=r,=o.
(45)
B) For gu<O i.e. 11-2 V + 4 t * cos (nv)<O it moves towards the strong-coupling
regime in the spin-channel, accompanied by the ordering of the field pUwith vacuum
expectation value
(p,)=O
*
(46)
The ordering of the p0 field strongly influences the behaviour of the correlation functions and determines the symmetry properties of the ordered ground states of the Fermionic system.
4 Correlation functions
To clarify the symmetry properties of the various ground states of the system let us introduce the following order parameters describing the short wave-length fluctuations
of the charge density
and of the spin density
as well as two superconducting order parameters corresponding to singlet (Ass) and
triplet (ATS)superconductivity:
G. Japaridze, E. Mliller-Hartmann, Electrons with correlated hopping interaction
A&)
=
c v:,u(x)wtz,-&I - exp (iBp@)*COS
(p,(4,/2) ,
U
175
(49)
where
Pp
=g w p
9
and QP is the field dual to pp:
= Pp,aXpp= Pp.
Let us first consider the sector of the coupling constants
U-2V+4t* c0s(lrv)~0
(A)
corresponding to the case of the weak-coupling regime in the spin channel.
In this sector, charge and spin channel are both gapless and the behaviour of the corresponding degrees of freedom is governed by effective free bose field Hamiltonians.
Using the expressions for free bose field correlators
(exp
w,,,, [vo, ( x ) - v,,,
WI/W
- Ix -x‘ I - v
~n, ,/
~
(52)
and
and the fixed-point value of the effective coupling constant p$,, = 87c, one can easily
obtain the following behaviour of the correlations at large distances:
176
Ann. Physik 3 (1994)
where
gp = (U+6V + 4 t * cos (~cv))/TcvF>O
.
(58)
Let us now consider the sector of the coupling constants
u-2 V + 4 t * cos(nv)<O
(B)
corresponding to the case of the strong-coupling regime in the spin-channel.
In this case the ordering of the field pa with vacuum expectation value (p,) = 0
completely suppresses the SDW and TS instabilities and changes the power-law decay
of the CDW and SS correlations into
and
Here only the gapless charge degrees of freedom were relevant.
5 Phase diagram
Using the results of the previous section about the excitation spectrum of the system
and about the large.distance behaviour of the correlation functions, we will now discuss
the ground state phase diagram of the model (1).
Let us first consider the case U = V = 0, where the system is characterized only by
the correlated-hopping interaction with coupling constant t* >0. The effective interaction appearing in the continuum-limit theory is given by t* cos (nv) and, depending on
the band-filling, leads to two different regimes of behaviour of the system: for v< 1/2,
the effective interaction is repulsive, both charge and spin excitations are gapless and
the system manifests properties of a Luttinger liquid. The ground state of the system
can be characterized as a paramagnetic metal with strongest instabilities of the densitydensity type, characterized by the critical indices
2t*
acD,=asD,-2--cos(nv)
.
ZVF
The superconducting correlations, characterized by critical indices
- + -cos (nv)
ass= aTS 2
&b
XVF
,
G. Japaridze, E. Muller-Hartmann, Electrons with correlated hopping interaction
177
decay faster. We call this phase the Luttinger metal phase.
With increasing band-filling v the effective interaction decreases. For v = 1/2 the
correlated-hopping interaction becomes dynamically trivial, leading only to a renormalization of the electron hopping amplitude (9). At this point all critical indices are
equal to 2 and coincide with those of the free electron system.
For v > 1/2 the effective interaction is attractive and a transition into the state characterized by a gap in the spin excitation spectrum occurs. The freezing out of the spin
excitations, accompanied by the ordering of the field pGwith vacuum expectation
value (p,) = 0, completely suppresses the SDW and TS instabilities. The power-law
decay of the CDW and SS correlations is changed. The corresponding critical indices
are given by
2t*
1+-cos(xv)
1
ass=-=
aCDW
,
ZUF
which implies that the SS correlations are most divergent for v > 1/2. Thus, for bandfillings v > v, = 1/2, the correlated-hopping model manifests properties of a nonmagnetic metal with strongly developed instabilities of the singlet superconducting type
(see Fig. 3a).
1
I
Luttinger
I
I
I
I
I
metal
CDW
Luttinger
I
I
I
SDW
SDW
ss
I
I
I
I
I
I
I
I
I
I
'V
I
Nonmagnetic
metal
I
CDW
superconductor
I
I
metal
Singlet
CD W
I
I
Singlet
I
superconductor
I
I
I
I
I
I
I
ss
178
Ann. Physik 3 (1994)
Let us now turn on the on-site repulsion U >0. In this case the role of the correlatedhopping interaction remains nontrivial only for U <4 t* and in the regime
v > v c =1 - ~ c c o ~ ( - ~ ),
71
where the system is in a superconducting state (see Fig. 3 b). Outside of this regime, the
phase diagram ,coincides with that of the Hubbard model, with the coupling constant
replaced 'by an effective coupling constant Ueff= U + 4 t * cos (nv ) >0.
The nearest-neighbour interaction V leads to a competition with the on-site term and
makes the phase diagram more rich. Let us first consider the case
4t*> U + 6 V ,
(65)
where, depending on the band-filling, the system shows three different regimes of
behaviour corresponding to a Luttinger metal, a nonmagnetic metal and a singlet
superconducting state.
For
1
u-2v ,
v<vcl=-arccos(--;;;-)
n
we have a Luttinger metal phase. The critical indices of the correlations are
a C D W = aSD W
-
-
ass = aTs 2 +
- I/+ 6 V + 4t* cos (nv) ,
2nvF
U+ 6 V + 4t* cos (nv)
2nvp
At the band-filling v = vcl a transition into a nonmagnetic metal phase takes place.
This state is characterized by a gap in the spin excitation spectrum and most divergent
CD W fluctuations. The corresponding critical indices are
1
aco,=--=lass
U + 6 V + 4 t * cos (nv)
2nvF
As follows from (69), for
U + 6 V + 4 t * cos (nv)<O ,
(70)
the SS fluctuations become most divergent and thus, at the band-filling
vc2 = Iarccos
(-7 )
U+6V
,
R
a crossover from the state with dominating CDW fluctuations (vc,<v<vc2) into a
state with dominating SS fluctuations (v > vc2) occurs (see Fig. 3 c). Outside the regime
G. Japaridze, E. Miiller-Hartmann, Electrons with correlated hopping interaction
179
of coupling constants given by (70) the superconducting transition does not occur, but
the transition from the Luttinger metal into the nonmagnetic metal state exists as long
as
IU-2VI < 4 t *
.
(72)
For I U - 2 VI >4 t * the phase diagram of the model (6) coincides with that of the
extended Hubbard model: for arbitrary v # 1/2 and
u-2 V>4t*
the system shows the properties of a Luttinger metal state and for
u-2v<-4t*
of a nonmagnetic metal state.
6 Summary
To summarize, we have considered the one-dimensional correIated-hopping model (1)
which describes a ID system of correlated electrons incorporating, in addition to the
usual repulsive on-site (U)and nearest-neighbour (V) interactions, an interaction between charges located on sites and on bonds (the correlated-hopping interaction). The
ground state phase diagram was studied within the framework of the weak-coupling
continuum-limit approach. The effective interaction originating from the correlatedhopping term which appears in the continuum-limit theory is given by t* cos ( K V ) .
Repulsive for v < 1/2 and attractive for v > 1/2, this interaction leads to a characteristic
band-filling dependence of the ground state phase diagram of the system. It was shown
that for v # 1/2, depending on the relation between the bare values of the coupling constants (U,V ,t* cos (nv))and, hence, on the band-filling, the system shows three different regimes of behaviour in the infrared limit. These regimes correspond to a Luttinger metal, a nonmagnetic metal and a singlet superconducting state. For v = 1/2, the
correlated-hopping interaction becomes dynamically trivial, leading only to a renormalization of the one-electron hopping amplitude and the phase diagram of the model
coincides with that of the extended Hubbard model. The excitation spectrum of the
system as well as the behaviour of the various correlation functions was studied and
the detailed ground state phase diagram was constructed.
One of the authors (G.J.) greatfully acknowledges S. b t l u n d and H. Johannesson for their kind
hospitality and support during his stay at Chalmers University of Technology, where this work was
started. He also would like to thank R. Bariev, H. Johannesson and A. Schadschneider for many
helpful discussions.
180
Ann. Physik 3 (1994)
References
[l] J.G. Bednorz, K.A. Miiller, Z. Phys. B64 (1986) 189
[2] P.W. Anderson, Science 235 (1987) 1196; Phys. Rev. Letters 64 (1990) 1839
13) B. Batlogg, in: High Temperature Superconductivity, K. S. Bedell (ed.), Addison-Wesley, Redwood City 1990
[4] J.E. Hirsch, Physica C158 (1989) 326; Phys. Letters A138 (1989) 83
[5] F.B. Essler, V.E. Korepin, K. Schoutens, Phys. Rev. Letters 68 (1992) 2960
[6] R.Z. Bariev, A. Kltimper, A. Schadschneider, J. Zittartz, J. Phys. A: Math. Gen. 26 (1993) 1249
[7] S. Kivelson, W.-P. Su, J.R. Schrieffer, A.J. Heeger, Phys. Rev. Letters 58 (1987) 1899
[8] K.A. Penson, M. Kolb, Phys. Rev. B33 (1986) 1633
[9] V. J. Emergy, in: Highly Conducting One-Dimensional Solids, J. T. De Vreese, R. P. Evrard,
V.E. Van Doren (eds.), Plenum Press, New York 1979
[lo] J. Solyom, Adv. Phys. 28 (1979) 201
[ l l ] R. Shankar, Int. J. Mod. Phys. B4 (1990) 2371; ICTP Summer Lecture Course, Trieste (1992)
[I21 J.W. Cannon, E. Fradkin, Phys. Rev. B41 (1990) 9435
[13] K.D. Schotte, U. Schotte, Phys. Rev. 182 (1969) 479; A. Luther, 1. Peschel, Phys. Rev. B9
(1974) 2911
[14] A. Luther, V.J. Emery, Phys. Rev. Letters 33 (1974) 589
1151 S. Tomonaga, Prog. Theor. Phys. 5 (1950) 349
[16] D.C. Mattis, E.H. Lieb, J. Math. Phys. 6 (1965) 304
(171 P. Wiegmann, J. Phys. C11 (1978) 1583
[lS] D.Boyanovsky, J. Phys. A22 (1989) 2601
[19] J.M. Kosterlitz, D. Thouless, J. Phys. C: Solid State Phys. 6 (1973) 1181; J. Phys. C:Solid State
Phys. 7 (1974) 1046
Документ
Категория
Без категории
Просмотров
0
Размер файла
916 Кб
Теги
dimensions, electrons, one, correlates, interactiv, hopping
1/--страниц
Пожаловаться на содержимое документа