Universal Luttinger Liquid Relations in the 1D Hubbard Model

Abstract

We study the 1D extended Hubbard model with a weak repulsive short-range
interaction in the non-half-filled band case, using non-perturbative
Renormalization Group methods and Ward Identities. At the critical temperature,
T = 0, the response functions have anomalous power-law decay with
multiplicative logarithmic corrections. The critical exponents, the
susceptibility and the Drude weight verify the universal Luttinger liquid
relations. Borel summability and (a weak form of) Spin-Charge separation is
established.

Full text

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arXiv:1106.0356v1 [math-ph] 2 Jun 2011
Universal Luttinger Liquid Relations in the
1D Hubbard Model
G. Benfatto1
P. Falco2
V. Mastropietro1
June 3, 2011
Abstract
We study the 1D extended Hubbard model with a weak repulsive short-range interaction
in the non-half-filled band case, using non-perturbative Renormalization Group methods and
Ward Identities. At the critical temperature, T = 0, the response functions have anomalous
power-law decay with multiplicative logarithmic corrections. The critical exponents, the
susceptibility and the Drude weight verify the universal Luttinger liquid relations. Borel
summability and (a weak form of) Spin-Charge separation is established.
Contents
1 Main Results
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Extended Hubbard Model and Physical Observables . . . . . . . . . . . . . . . .
1.3 Anomalous exponents and logarithmic corrections . . . . . . . . . . . . . . . . . .
1.4 The Luttinger liquid relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Spin Charge separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Borel summability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7 Contents of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2
3
6
7
8
8
9
2 RG Analysis for the Hubbard Model
2.1 Functional integral representation . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2Multiscale analysis for the effective potential
2.3 The flow of the running coupling constants
2.4 The flow of renormalization constants . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Proof of Theorem 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
11
12
16
22
24
26
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . .
3 RG Analysis of the Effective Model
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2Ward Identities in the g1,⊥= 0 case
3.3 Schwinger-Dyson and Closed equations. . . . . . . . . . . . . . . . . . . . . . . .
3.4The two point function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 The four point functions and the densities correlations . . . . . . . . . . . . . . .
3.6Fine tuning of the parameters of the effective model . . . . . . . . . . . . . . . .
27
27
28
30
31
33
35
. . . . . . . . . . . . . . . . . . . . . . . . .
4 Spin-Charge Separation36
1Dipartimento di Matematica, Universit` a di Roma “Tor Vergata”, 00133 Roma, Italy.
2School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540
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5Susceptibility and Drude weight 37
A The g1map41
B Symmetries of the Effective Model and RG Flow 43
C Vanishing of the Beta Function46
1 Main Results
1.1Introduction
The Hubbard model, see e.g. [1], describing interacting spinning fermions on a lattice, plays the
same role in quantum many body theory as the Ising model in classical statistical mechanics,
that is it is the simplest model displaying many real world features: it is however much more
difficult to analyze. While our understanding of the Hubbard model in higher dimensions at
zero temperature is really poor (except for special choices of the lattice as in [2] and [3]), the
situation is better in d = 1, when the model furnishes an accurate description of real systems,
like quantum wires or carbon nanotubes [4].
The one dimensional Hubbard model (from now on the Hubbard model tout court) can
be exactly solved by Bethe ansatz, as shown by Lieb and Wu [5]: the system is insulating in
the half filled band case while it is a metal otherwise and the elementary excitations are not
electronlike, a phenomenon which is nowadays called electron fractionalization [6]. Recently in
[7] a strategy for a proof that the lowest energy state of Bethe ansatz form is really the ground
state has been outlined (see also [8]). This method is however of little utility for understanding
the asymptotic behavior of correlations; and does not apply in studying the ground state of a
slight generalization of the model, the extended Hubbard model, that consists in replacing the
local quartic interaction with a short-ranged one. Other approaches has been therefore developed
to get more insights into the physical properties of the Hubbard model.
Under certain drastic approximations, like replacing the sinusoidal dispersion relation with a
linear relativistic one and neglecting certain terms called backward and umklapp interactions (see
after (2.25) for their definition), one obtains the spinning Luttinger Model, which is exactly solv-
able in a stronger sense, [9], [10]: all its Schwinger functions, at distinct points, can be explicitly
computed. This model, describing interacting fermions, can be exactly mapped in a model of
two kinds of free bosons, describing the propagation of charge or spin degrees of freedom and
with different velocities (spin-charge separation); again, a phenomenon of electron fractional-
ization which has received a considerable attention in the context of high Tc superconductors
[11]. Moreover, as in spinless Luttinger model, the correlations have a power law decay rate with
interaction dependent exponents.
However, neglecting the lattice effects and backscattering or umklapp interactions is not
safe, and indeed the mapping to free bosons is not expected to be true in the Hubbard model.
A somewhat more realistic effective description can be obtained by including the backward
interaction in the spinning Luttinger Model, so obtaining the g-ology model. This system is no
more solvable; however, a perturbative Renormalization Group (RG) analysis, [12], shows that,
in the repulsive case, such extra coupling is marginally irrelevant, i.e. becomes negligible over
large space-time scales. In [13] the necessity of implementing Ward Identities in a RG approach
was emphasized, in order to go beyond purely perturbative results, but the analysis was limited
to the Luttinger model and no attempt was done to include the effects of nonlinear bands. In
[14] it was observed that the correlations of the repulsive g-ology model would qualitatively differ
from the Luttinger model ones for the presence of multiplicative logarithmic corrections.
A new point of view, that extended previous ideas of Kadanoff, [15], and Luther and Peschel,
[16], was proposed by Haldane, [17], and is nowadays known as Luttinger Liquid Conjecture. The
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idea is to exploit the concept of universality, a basic notion in statistical physics saying that the
critical properties are largely independent from the details of the model, at least inside a certain
class of models. In the present case, as the exponents are non trivial functions of the coupling,
universality has a meaning more subtle than usual; it does not mean that the exponents are the
same (the exponents do depend on the details of the model), but that the exponents and certain
thermodynamic quantities verify a set of universal relations which are identical to the Luttinger
model ones. Such relations give an exact determination of physical quantities in terms of a
few measurable parameters. The validity of such relations has been checked in certain special
solvable spinless fermionic lattice models [17], but a proof of their validity in the Hubbard model
(or in the non solvable extended Hubbbard model) is an open problem. It should be remarked
that, even thought the Hubbard model differs from the spinning Luttinger model for irrelevant
or marginally irrelevant terms in the Renormalization Group sense (in the weak non half filled
band case), this would not imply at all the validity of the same relations as in the Luttinger
model; irrelevant terms do renormalize the exponents and the thermodynamic quantities.
Starting from the 90’s, the methods developed in constructive Quantum Field Theory (QFT)
for the analysis of QFT models in d = 1 + 1 [18, 19] were applied to interacting non relativistic
spinless fermionic systems in the continuum [20], so establishing the anomalous dimension in a
non solvable model, by combining Renormalization Group methods with non perturbative infor-
mation coming from the exact solution of the Luttinger model; the extension to spinning fermions
was done in [21]. An important technical advance was achieved in [22, 23], by implementing Ward
Identities based on local symmetries with Renormalization Group methods. A well known diffi-
culty in any Wilsonian Rermalization Group approach is that the momentum cut-offs break the
local symmetries on which Ward Identities are based; in [22, 23] it was developed a technique
allowing to rigorously take into account the extra terms produced in the Ward Identities by the
cut-offs, so that interacting non relativistic fermions in d = 1 were constructed without any use
of exact solutions [23, 24, 25]. The main outcome, with respect to the early Renormalization
Group analysis [12], is that the results are exact (the lattice and non linear bands are fully taking
into account) and non-perturbative; and physical observables are written in terms of convergent
expansions so that they can be computed with arbitrary precision. The complexity of such ex-
pansions made however impossible to verify explicitly the universal Luttinger Liquid relations
in [26]; they have finally been proven for interacting spinless fermions on the lattice (the XXZ
chain and extended versions) in [27, 28] through Ward identities; this fact appears to be related
to a non-perturbative version of the Adler-Bardeen theorem of the non renormalization of the
anomalies, [29]. In this paper we will extend such ideas to spinning fermions in the Hubbard
model; as we will see, the extension is rather non trivial and new phenomena take place.
1.2Extended Hubbard Model and Physical Observables
Let β > 0 be the inverse temperature, −µ the chemical potential and C = {−[L/2],...,|[(L −
1)/2]L} a one dimensional lattice of L sites. The extended Hubbard model [30] describes fermions
hopping on C with a short-range density-density interaction; the Hamiltonian plus the chemical
potential term is
H = −1
2
?
x∈C
s=±
(a+
x,sa−
x+1,s+a+
x,sa−
x−1,s)+µ
?
x∈C
s=±1
a+
x,sa−
x,s+λ
?
x,y∈C
s,s′=±1
v(x−y)a+
x,sa−
x,sa+
y,s′a−
y,s′ (1.1)
where a±
a smooth, even potential such that |v(x)| ≤ Ce−κ|x|(short range condition); periodic boundary
conditions are assumed: aε
1,s.
The set-up of the Grand Canonical Ensemble is standard, and we remind it concisely; more
details are, for example, in [31]. If Ox is a monomial in the operators aε
x,sare fermionic creation and annihilation operators at site x with ’spin’ s, and v(x) is
L+1,s= aε
x,s, ε,s = ±, or in
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the density operators aε
eHx0Oxe−Hx0(so that x0 has the meaning of imaginary time); then, given the observables
Ox1,...,Oxn, their Grand Canonical correlation is
x,saε′
x,s′, ε,ε′,s,s′= ±, given x0∈ [0,β], define x = (x,x0) and Ox:=
?Ox1···Oxn?L,β:=Tr[e−βHT(Ox1···Oxn)]
Tr[e−βH]
(1.2)
where T is the time order product. Similarly, ?Ox1;··· ;Oxn?T;L,βdenotes the corresponding
truncated correlations. We are interested in the correlations in the thermodynamic limit L → ∞
and at the critical temperature β−1= 0; the limit L,β → ∞ will be indicated by dropping the
labels L,β.
Define ¯ pF∈ [0,π], the free Fermi momentum, and ¯ vF, the free Fermi velocity, such that
cos ¯ pF= µ¯ vF= sin ¯ pF .
In this paper we have three main assumptions on the parameters:
¯ pF?= 0,π/2,π ,ˆ v(2¯ pF) > 0 ,λ ≥ 0(1.3)
The condition ¯ pF ?= 0,π means that the empty band and the filled band cases are not included;
the reason of such exclusion is that, if ¯ vF = 0, the scaling of the model would be very different
and would depend in a critical way on the interaction. The condition ¯ pF ?= π/2 excludes the
half-filled band case; it will have the effect to make the Umklapp interaction (see §2.1) irrelevant
(in the RG language). The two other conditions can be loosely called the repulsive condition on
the interaction; they indeed imply that one of the contribution to the effective interaction (in
the RG language) is strictly positive at all scales.
The model is SU(2) symmetric, as the Hamiltonian is invariant under transformation a±
?
respectively: in the former case the repulsive condition is U ≥ 0.
By definition, the T = 0 free energy is
x,s→
s′Ms,s′a±
corresponding to the interactions λv(x − y) = Uδx,y and λv(x − y) = Uδx,y+1
x,s′ with M ∈ SU(2); and includes the standard and the U-V Hubbard models,
2V δ|x−y|,1,
E(λ) := − lim
L,β→∞(Lβ)−1logTr[e−βH] ,(1.4)
and the 2-point Schwinger function is
S2,β,L(x − y) := ?a−
x,+a+
y,+?β,L= ?a−
x,−a+
y,−?β,L. (1.5)
The connection with experimental physics is through the response functions, defined as Fourier
transforms of the following truncated correlations:
Ωα,β,L(x − y) := ?ρα
xis one of the following densities (see pagg. 54, 55 of [4]):
?
ρSi
x=
s,s′=±
=1
2
s=±
ε=±
=1
2
s,s′=±
ε=±
xρα
y?T;β,L:= ?ρα
xρα
y?β,L− ?ρα
x?β,L?ρα
y?β,L
(1.6)
where ρα
ρC
x=
s=±
?
a+
x,sa−
x,s
(charge density)
a+
x,sσ(i)
s,s′a−
x,s′
(spin densities)
(1.7)
ρSC
x
?
?
saε
x,saε
x,−s
(singlet Cooper density)
ρTCi
x
aε
x,s? σ(i)
s,s′aε
x+e,s′ ,e = (1,0)(triplet Cooper densities)
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where i = 1,2,3 and
σ(1)=
?0
?1
1
01
?
?
σ(2)=
?0
?0
−i
0
?
i
?
σ(3)=
?1
?0
0
0
−1
0
1
?
? σ(1)=
0
00
? σ(2)=
1
01
? σ(3)=
0
?
When the interaction is off (λ = 0), all functions Ωα(x−y) have power law decay for large |x−y|
with the same exponent; as we will see, turning on the interaction removes such degeneracy: some
correlations are enhanced by the presence of interaction and others are depressed, so that the
exponents become different.
Define the Fourier transform of the response functions as
?β/2
where p = (p,p0), with p ∈2π
κ := lim
p→0
ˆΩα(p) := lim
β,L→∞
−β/2
dx0
?
x∈C
eipxΩα,β,L(x) (1.8)
LC and p0∈2π
βZ. The susceptibility is given by1
ˆΩC(p,0) . (1.9)
The paramagnetic part of the current Jxis defined as
Jx=1
2i
?
s=±
[a+
x+1,sa−
x,s− a+
x,sa−
x+1,s] (1.10)
while the diamagnetic part is
τx= −1
2
?
s=±
[a+
x,sa−
x+1,s+ a+
x+1,sa−
x,s](1.11)
The Drude weight is defined as
D = −?τx? − lim
p0→0lim
p→0
lim
β,L→∞
?β/2
−β/2
dx0
?
x∈Λ
eipx?JxJ0?T;L,β≡ lim
p0→0lim
p→0
ˆD(p) (1.12)
where the first term is a constant independent from x. If one assumes analytic continuation in
p0 around p0= 0, one can compute the conductivity in the linear response approximation by
the Kubo formula, that is σ = limω→0limδ→0
infinite conductivity.
The conservation law
∂ρC
x
∂x0
ˆ D(−iω+δ,0)
−iω+δ
. Therefore, a nonvanishing D indicates
= eHx0[H,ρx]e−Hx0= −i∂(1)
xJx≡ −i[Jx,x0− Jx−1,x0] , (1.13)
where ∂(1)
between the Schwinger functions, the density correlations and the vertex functions, defined as
G2,1
play an important role in the following, are
x
denotes the lattice derivative, implies exact relations, called Ward identities (WI),
ρ(x,y,z) = ?ρ(C)
x a−
ya+
z?Tand G2,1
j(x,y,z) = ?Jxa−
ya+
z?T. Some Ward Identities, which will
−ip0ˆG2,1
ρ(k,k + p) − i(1 − e−ip)ˆG2,1
−ip0ˆΩC(p) − i(1 − e−ip)ˆΩj,ρ(p) = 0
−ip0ˆΩρ,j(p) − i(1 − e−ip)ˆD(p) = 0
where Ωj,ρ(x,y) = ?ρC
1in a fermion system, κ = κcρ2, where κc is the compressibility and ρ the density, see e.g. (2.83) of [32]
j(k,k + p) =ˆS2(k) − S2(k + p) (1.14)
(1.15)
(1.16)
xJy?T,β,L.
5