Lower bounds for the conductivities of correlated quantum systems
Peter Jung and Achim Rosch
Institute for Theoretical Physics, University of Cologne, 50937 Cologne, Germany.
(Dated: October 30, 2018)
We show how one can obtain a lower bound for the electrical, spin or heat conductivity of cor-
related quantum systems described by Hamiltonians of the form H = H0 + gH1. Here H0 is an
interacting Hamiltonian characterized by conservation laws which lead to an infinite conductivity
for g = 0. The small perturbation gH1, however, renders the conductivity finite at finite temper-
atures. For example, H0 could be a continuum field theory, where momentum is conserved, or an
integrable one-dimensional model while H1 might describe the effects of weak disorder. In the limit
g → 0, we derive lower bounds for the relevant conductivities and show how they can be improved
systematically using the memory matrix formalism. Furthermore, we discuss various applications
and investigate under what conditions our lower bound may become exact.
PACS numbers: 72.10.Bg, 05.60.Gg, 75.40.Gb, 71.10.Pm
I. INTRODUCTION
Transport properties of complex materials are not only
important for many applications but are also of funda-
mental interest as their study can give insight into the
nature of the relevant quasi particles and their interac-
tions.
Compared to thermodynamic quantities, the transport
properties of interacting quantum systems are notori-
ously difficult to calculate even in situations where in-
teractions are weak. The reason is that conductivities
of non-interacting systems are usually infinite even at fi-
nite temperature, implying that even to lowest order in
perturbation theory an infinite resummation of a per-
turbative series is mandatory. To lowest order this im-
plies that one usually has to solve an integral equation,
often written in terms of (quantum-) Boltzmann equa-
tions or – within the Kubo formalism – in terms of vertex
equations. The situation becomes even more difficult if
the interactions are so strong that an expansion around
a non-interacting system is not possible. Also numeri-
cally, the calculation of zero-frequency conductivities of
strongly interacting clean systems is a serious challenge
and even for one-dimensional systems reliable calcula-
tions are available for high temperatures only1,2,3,4,5,6.
Variational estimates, e.g. for the ground state energy,
are powerful theoretical techniques to obtain rigorous
bounds on physical quantities. They can be used to
guide approximation schemes to obtain simple analytic
estimates and are sometimes the basis of sophisticated
numerical methods like the density matrix renormaliza-
tion group7.
Taking into account both the importance of transport
quantities and the difficulties involved in their calculation
it would be very useful to have general variational bounds
for transport coefficients.
A well known example where a bound for transport
quantities has been derived is the variational solution
of the Boltzmann equation, discussed extensively by
Ziman8. The linearized Boltzmann equation in the pres-
ence of a static electric field can be written in the form
Wk,k′Φk′ (1)
where Wk,k′ is the integral kernel describing the scatter-
ing of quasiparticles and we have linearized the Boltz-
mann equation around the Fermi (or Bose) distribution
= f0(ǫk) using fk = f
Φk. Therefore, the
current is given by I = −e
Φk and the dc con-
ductivity is determined from the inverse of the scattering
matrix W using σ = −e2
k,k′v
. It is
easy to see that this result can be obtained by maximiz-
ing a functional8,9,10,11 F [Φ] with
σ = e2max
F [Φ] ≥ e2 max
F [Φ] =
k,k′(Φk − Φk′)2Wk,k′
where we used that
Wk,k′ = 0 reflecting the conser-
vation of probability. The variational formula (2) is ac-
tually closely related8 to the famous H-theorem of Boltz-
mann which states that entropy always increases upon
scattering.
A lower bound for the conductivity can be obtained
by varying Φ only in a subspace of all possible func-
tions. This allows for example to obtain analytically
good estimates for conductivities without inverting an
infinite dimensional matrix or, euqivalently, solving an
integral equation, see Ziman’s book for a large number
of examples8.
The applicability of Eq. (2) is restricted to situations
where the Boltzmann equation is valid and bounds for
the conductivity in more general setups are not known.
However, for ballistic systems with infinite conductiv-
ity it is possible to get a lower bound for the so-called
Drude weight. Mazur12 and later Suzuki13 considered
situations where the presence of conservation laws pro-
hibits the decay of certain correlation functions in the
http://arxiv.org/abs/0704.0886v2
Re σ(ω) Re σ(ω)
σreg(ω)
πDδ(ω)
g != 0g = 0
FIG. 1: For g = 0 a Drude peak shows up in the conductivity,
resulting from exact conservation laws. For g 6= 0 the Drude
peak broadens and the dc conductivity becomes finite.
long time limit. In the context of transport theory their
result can be applied to systems (see Appendix A for de-
tails) where the finite-temperature conductivity σ(ω, T )
is infinite for ω = 0 and characterized by a finite Drude
weight D(T ) > 0 with
Re σ(ω, T ) = πD(T )δ(ω) + σreg(ω, T ). (3)
Such a Drude weight can arise only in the presence of
exact conservation laws Cj with [H,Cj ] = 0. Suzuki
showed that the Drude weight can be expressed as a sum
over all Cj
〈CjJ〉2
〈C2j 〉
〈CjJ〉2
〈C2j 〉
. (4)
where J is the current associated with σ. For conve-
nience a basis in the space of Ci has been chosen such
that 〈CiCj〉 = 0 for i 6= j. More useful than the equal-
ity in Eq. (4) is often the inequality12 which is obtained
when the sum is restricted to a finite subset of conser-
vation laws. Such a finite sum over simple expectation
values can often be calculated rather easily using either
analytical or numerical methods. The Mazur inequality
has recently been used heavily4,14,15,16,17 to discuss the
transport properties of one-dimensional systems.
Model systems, due to their simplicity, often exhibit
symmetries not shared by real materials. For exam-
ple, the heat conductivity of idealized one-dimensional
Heisenberg chains is infinite at arbitrary temperature as
the heat current is conserved. However, any additional
coupling (next-nearest neighbor, inter-chain, disorder,
phonon,...) renders the conductivity finite1,4,5,6,18,19,20.
If these perturbations are weak, the heat conductivity
is, however, large as observed experimentally21,22. For a
more general example, consider an arbitrary translation-
ally invariant continuum field theory. Here momentum
is conserved which usually implies that the conductivity
is infinite for this model. In real materials momentum
decays by Umklapp scattering or disorder rendering the
conductivity finite. It is obviously desirable to have a
reliable method to calculate transport in such situations.
In this work we consider systems with the Hamiltonian
H = H0 + gH1, (5)
where for g = 0 the relevant heat-, charge- or spin- con-
ductivity is infinite and characterized by a finite Drude
weight given by Eq. (4). As discussed above,H0 might be
an integrable one-dimensional model, a continuum field
theory, or just a non-interacting system. The term gH1
describes a (weak) perturbation which renders the con-
ductivity finite, e.g. due to Umklapp scattering or dis-
order, see Fig. 1. Our goal is to find a variational lower
bound for conductivities in the spirit of Eq. (2) for this
very general situation, without any requirement on the
existence of quasi particles. For technical reasons (see
below) we restrict our analysis to situations where H is
time reversal invariant.
In the following, we first describe the general setup and
introduce the memory matrix formalism, which allows
us to formulate an inequality for transport coefficients
for weakly perturbed systems. We will argue that the
inequality is valid under the conditions which we specify.
Finally, we investigate under which conditions the lower
bounds become exact and briefly discuss applications of
our formula.
II. SETUP
Consider the local density ρ(x) of an arbitrary phys-
ical quantity which is locally conserved, thus obeying a
continuity equation
∂tρ+∇j = 0.
Transport of that quantity is described by the dc con-
ductivity σ which is the response of the current to some
external field E coupling to the current,
〈J〉 = σE,
where J =
j(x) is the total current and 〈J〉 its expec-
tation value. Note that J can be an electrical-, spin-, or
heat current and E the corresponding conjugate field de-
pending on the context. The dynamic conductivity σ(z)
is given by Kubo’s formula, see Eq. (A1). We are inter-
ested in the dc conductivity σ = limω→0 σ(z = ω + i0).
Starting from the Hamiltonian (5) we consider a sys-
tem where H0 posesses a set of exact conservation laws
{Ci} of which at least one correlates with the current,
〈JC1〉 6= 0. Without loss of generality we assume
〈CiCj〉 = 0 for i 6= j. For g = 0 the Drude weight D
defined by Eq. (3) is given by Eq. (4). We can split up
the current under consideration into a part which is par-
allel to the Ci and one that is orthogonal,
J = J‖ + J⊥,
with J‖ =
〈CiJ〉
Ci, which results in a separation of
the conductivity,
σ(z) = σ‖(z) + σ⊥(z). (6)
Since the conductivity σ(z) is given by a current-current
correlation function and the current J‖ (J⊥) is diago-
nal (off-diagonal) in energy, cross-correlation functions
〈〈J‖; J⊥〉〉 vanish in Eq. (6).
According to Eq. (4), the Drude peak of the unper-
turbed system, g = 0, arises solely from J‖:
Re σ‖(ω) = πDδ(ω), (7)
while σ⊥(z) appears in Eq. (3) as the regular part,
Re σ⊥(ω) = σreg(ω).
In this work we will focus on σ‖(ω), since the small
perturbation is not going to affect σ⊥(ω) much (which is
assumed to be free of singularities here, see section IV)
while σ‖(ω = 0) diverges for g → 0, see Fig. 1. As we
are interested in the small g asymptotics only, we may
neglect the contribution σ⊥(0) to the dc conductivity.
Hence we set J = J‖ and σ(ω) = σ‖(ω) in the following.
III. MEMORY MATRIX FORMALISM
We have seen that certain conservation laws ofH0 play
a crucial role in determining the conductivity of both the
unperturbed and the perturbed system. In the presence
of a small perturbation gH1, these modes are not con-
served anymore but at least some of them decay slowly.
Typically, the conductivity of the perturbed system will
be determined by the dynamics of these slow modes. To
separate the dynamics of the slow modes from the rest, it
is convenient to use a hydrodynamic approach based on
the projection of the dynamics onto these slow modes. In
this section we will therefore review the so called memory
matrix formalism23, introduced by Mori and Zwanzig24,25
for this purpose. In the next section we will show that
this approach can be used to obtain a lower bound for
the dc conductivity for small g.
We start by defining a scalar product in the space of
quantum mechanical operators,
(A|B) =
dλ〈A†B(iλ)〉 − β〈A†〉〈B〉 (8)
As the next step we choose a – for the moment – arbi-
trary set of operators {Ci}. In most applications, the Ci
are the relevant slow modes of the system. For notational
convenience, we assume that the {Ci} are orthonormal-
ized,
(Ci|Cj) = δij . (9)
In terms of these we may define the projector P onto (and
Q away from) the space spanned by these ‘slow’ modes
|Ci)(Ci| = 1−Q.
We assume that C1 is the current we are interested in,
|J) ≡ |C1).
The time evolution is given by the Liouville-
(super)operator
L = [H, .] = L0 + gL1
with (LA|B) = (A|LB) = (A|L|B), and the time evo-
lution of an operator may be expressed as |A(t)) =
|eiHtAe−iHt) = eiLt|A). With these notions, one obtains
the following simple, yet formal expression for the con-
ductivity:
σ(ω) =
ω − L
ω − L
Using a number of simple manipulations, one can
show23,24,25 that the conductivity can be expressed as
the (1, 1)-component of the inverse of a matrix
σ(ω) = (M(ω) + iK − iω)−1
, (10)
where
Mij(ω) =
ω − LQ
is the so-called memory matrix and
Kij =
Ċi|Cj
a frequency independent matrix. The formal expression
(10) for the conductivity is exact, and completely gen-
eral, i.e. valid for an arbitrary choice of the modes Ci
(they do not even have to be ‘slow’). Only C1 = J is re-
quired. However, due to the projection operators Q, the
memory matrix (11) is in general difficult to evaluate. It
is when one uses approximations to M that the choice of
the projectors becomes crucial (see below).
Obviously, the dc conductivity is given by the (1, 1)-
component of
(M(0) +K)−1. (13)
More generally, the (m,n)-component of Eq. (13) de-
scribes the response of the ‘current’ Cm to an external
field coupling solely to Cn. We note, that since a matrix
of transport coefficients has to be positive (semi)definite,
this also holds for the matrix M(0) +K.
To avoid technical complications associated with the
presence of K we restrict our analysis in the following to
time reversal invariant systems and choose the Ci such,
that they have either signature +1 or −1 under time
reversal34 Θ. In the dc limit, ω = 0, components of
Eq.(13) connecting modes of different signatures vanish.
Thus, M(0) + K is block-diagonal with respect to the
time reversal signature, and consequently we can restrict
our analysis to the subspace of slow modes with the same
signature as C1. However, if Cm and Cn have the same
signature, then (Cm|Ċn) = 0, and thus K vanishes on
this restricted space. The dc conductivity therefore takes
the form
σ = (M(0)−1)11. (14)
IV. CENTRAL CONJECTURE
To obtain a controlled approximation to the memory
matrix in the limit of small g, it is important to identify
the relevant slow modes of the system. For the Ci we
choose quantities which are conserved by H0, [H0, Ci] =
0, such that Ċi = ig[H1, Ci] is linear in the small coupling
g. As argued below, we require that the singularities
of correlation functions of the unperturbed system are
exclusively due to exact conservation laws Ci, i.e. that
the Drude peak appearing in Eq. (3) is the only singular
contribution. Furthermore, we choose J = J‖ = C1 and
consider only Ci with the same time reversal signature
as J , as discussed in the previous section.
To formulate our central conjecture we introduce the
following notions. We define Mn(ω) as the (exact) n× n
memory matrix obtained by setting up the memory ma-
trix formalism for the first n slow modes {Ci, i = 1, .., n}.
Note that the definitions of the relevant projectors P and
Q also depend on this choice, and that for any choice of
n one gets σ = (M−1n )11. We now introduce the approxi-
mate memory matrix M̃n motivated by the following ar-
guments: Ċi is already linear in g, therefore in Eq. (11)
we approximate L by L0 and replace (.|.) by (.|.)0 as
we evaluate the scalar product with respect to H0. As
L0|Ci) = 0 and (Cj |Ċi) = 0 due to time reversal symme-
try, one has L0Q = 1 and Q|Ċi) = |Ċi) and therefore the
projector Q does not contribute within this approxima-
tion. We thus define the n× n matrix M̃n by
M̃n,ij = lim
ω − L0
Note that M̃n is a submatrix of M̃m for m > n and
therefore the approximate expression for the conductiv-
ity σ ≈ (M̃−1n )11 does depend on n while (M−1n )11 is
independent of n. A much simpler, alternative deriva-
tion for M̃1 is given in Appendix B, where the validity of
this formula is also discussed.
The central conjecture of our paper is, that for small g
(M̃−1n )11 gives a lower bound to the dc conductivity, or,
more precisely,
1/g2 = (M̃
∞ )11 ≥ · · · ≥ (M̃−1n )11 ≥ · · · ≥ M̃−11 . (16)
Here σ|
1/g2 = (1/g
2) limg→0 g
2σ denotes the leading
term ∝ 1/g2 in the small-g expansion of σ. Note that
M̃n ∝ g2 by construction. M̃∞ is the approximate mem-
ory matrix where all35 conservation laws have been in-
cluded. In some special situations, discussed in Ref. 6,
one has σ ∼ 1/g4 and therefore σ|
1/g2 = ∞.
A special case of the inequality above is Eq. (B4) in
appedix B, as the scattering rate Γ̃/χ may be expressed
as Γ̃/χ2 = M̃1.
Two steps are necessary to prove Eq. (16). The simple
part is actually the inequalities in Eq. (16). They are
a consequence of the fact that the matrices M̃n are all
positive definite and that M̃n is a submatrix of M̃m for
m ≥ n. More difficult to prove is that the first equality
in (16) holds. To show this we will need an additional
assumption, namely, that the regular part of all correla-
tion functions (to be defined below) remains finite in the
limit g → 0, ω → 0. In this case, the perturbative ex-
pansion around M̃∞ in powers of g is free of singularities
at finite temperature (which is not the case for M̃n<∞).
This in turn implies that limg→0 M∞/g
2 = M̃∞/g
2 and
therefore σ|
1/g2 = (M̃
∞ )11.
Next, we present the two parts of the proof.
A. Inequalities
We start by investigating the (1,1)-component of the
inverse of the positive definite symmetric matrix M̃∞. It
is convenient to write the inverse as
(M̃−1∞ )11 = max
(ϕTe1)
ϕT M̃∞ϕ
where e1 is the first unit vector. The same method is
used to derive Eq. (2) in the context of the Boltzmann
equation. The maximum is obtained for ϕ = M̃−1∞ e1.
By restricting the variational space in (17) to the first n
components of ϕ we reproduce the submatrix M̃n of M̃∞
and obtain
(M̃−1∞ )11 ≥ max
(ϕTe1)
ϕT M̃∞ϕ
= (M̃−1m )11
≥ max
(ϕT e1)
ϕT M̃∞ϕ
= (M̃−1n<m)11
By choosing different values form and n < m, this proves
all inequalities appearing in (16).
B. Expansion of the memory matrix
We proceed by expanding the exact memory matrix
Mn, where Pn = 1 − Qn is a projector on the first n
conservation laws, in powers of g. Using that LQn =
L0 + gL1Qn, we obtain the geometric series
Mn,ij(ω) =
ω − L0
ω − L0
Note that this is not a full expansion in g, as the scalar
product (8) is defined with respect to the full Hamilto-
nian H = H0 + gH1. We will turn to the discussion of
the remaining g-dependence later.
In general, one can expand
λmn|Am)(An|
in terms of some basis Am in the space of operators.
Therefore Eq. (18) can be written as a sum over products
of terms with the general structure
ω − L0
. (19)
In the following we would like to argue that such an ex-
pansion is regular for n = ∞ if all conservation laws
have been included in the definition of Q. As argued in
Appendix B, we have to investigate whether the series co-
efficients in Eq. (18) diverge for ω → 0. The basis of our
argument is the following: as Q∞ projects the dynam-
ics to the space perpendicular to all of the conservation
laws, the associated singularities are absent in Eq. (19)
and therefore the expansion of M∞ is regular.
To show this more formally, we split up B = B‖ +B⊥
in (19) into a component parallel and one perpendicular
to the space of all conserved quantities, |B‖) = P∞|B).
With this notation, the action of L0 becomes more trans-
parent:
ω − L0
|B) = 1
|B‖) +
ω − L0
|B⊥). (20)
As we assume that all divergencies can be traced back
to the conservation laws, we take the second term to be
regular. It is only the first term which leads in Eq. (19)
to a divergence for ω → 0, provided that (A|Qn|B‖) is fi-
nite. If we consider the perturbative expansion ofMn<∞,
where Pn = 1 − Qn projects only to a subset of con-
served quantities, then finite contributions of the form
(A|Qn|B‖) exist and the perturbative series in g will be
singular (see also Appendix B). Considering M∞, how-
ever, Q∞ projects out all conservation laws and therefore
by construction Q∞|B‖) = Q∞P∞|B) = 0. Thus the
first term in (20) does not contribute in (19) for n = ∞
and the expansion (18) of M∞ is therefore regular.
The only remaining part of our argument is to show
that in the limit g → 0 one can safely replace (.|.) by
(.|.)0. Here it is useful to realize that (A|B) can be in-
terpreted as a (generalized) static susceptibility. In the
absence of a phase transition and at finite temperatures,
susceptibilities are smooth, non-singular functions of the
coupling constants and therefore we do not expect any
further singularities from this step. If we define a phase
transition by a singularity in some generalized suscepti-
bility, then the statement that susceptibilities are regular
in the absence of phase transitions even becomes a mere
tautology.
Combining all arguments, the expansion (18) of
M∞(ω → 0) is regular, and using (Ċi|Q∞ = (Ċi| [see
discussion before Eq. (15)] its leading term, k = 0 is
given by M̃∞. We therefore have shown the missing first
equality of our central conjecture (16).
V. DISCUSSION
In this paper we have established that in the limit of
small perturbations, H = H0 + gH1, lower bounds to dc
conductivities may be calculated for situations where the
conductivity is infinite for g = 0. In the opposite case,
when the conductivity is finite for g = 0, one can use
naive perturbation theory to calculate small corrections
to σ without further complications.
The relevant lower bounds are directly obtained from
the memory matrix formalism. Typically26,27,28 one has
to evaluate a small number of correlation functions and
to invert small matrices. The quality of the lower bounds
depends decisively on whether one has been able to iden-
tify the ‘slowest’ modes in the system.
There are many possible applications for the results
presented in this paper. The mostly considered situ-
ation is the case where H0 describes a non-interacting
system26. For situations where the Boltzmann equation
can be applied, it has been pointed out a long time ago by
Belitz29 that there is a one-to-one relation of the memory
matrix calculation to a certain variational Ansatz to the
Boltzmann equation, see Eq. (2). In this paper we were
able to generalize this result to cases where a Boltzmann
description is not possible. For example, if H0 is the
Hamiltonian of a Luttinger liquid, i.e. a non-interacting
bosonic system, then typical perturbations are of the
form cosφ for which a simple transport theory in the
spirit of a Boltzmann or vertex equation does not exist
to our knowledge.
Another class of applications are systems where H0
describes an interacting system, e.g. an integrable one-
dimensional model6 or some non-trivial quantum-field
theory30. In these cases it can become difficult to cal-
culate the memory matrix and one has to resort to use
either numerical6 or field-theoretical methods30 to obtain
the relevant correlation functions.
An important special case are situations where H0
is characterized by a single conserved current with the
proper symmetries, i.e. with overlap to the (heat-, spin-
or charge-) current J . For example, in a non-trivial con-
tinuum field theory H0, interactions lead to the decay of
all modes with exception of the momentum P . In this
case the momentum relaxation and therefore the con-
ductivity at finite T is determined by small perturba-
tions gH1 like disorder or Umklapp scattering which are
present in almost any realistic system. As M̃∞ = M̃1 in
this case, our results suggest that for small g the conduc-
tivity is exactly determined by the momentum relaxation
rate M̃PP = limω→0 i(Ṗ |(ω − L0)−1|Ṗ ),
for g → 0. (21)
Here we used that J‖ = P (P |J)/(P |P ) with χPJ = (P |J)
and we have restored all factors which arise if the nor-
malization condition (9) is not used. In Appendix C, we
check numerically that this statement is valid for a real-
istic example within the Boltzmann equation approach.
A number of assumptions entered our arguments. The
strongest one is the restriction that all relevant singu-
larities arise from exact conservation laws of H0. We
assumed that the regular parts of correlation functions
are finite for ω = 0. There are two distinct scenarios
in which this assumption does not hold. First, in the
limit T → 0, often scattering rates vanish which can lead
to diverencies of the nominally regular parts of correla-
tion functions. Furthermore, at T = 0 even infinitesi-
mally small perturbations can induce phase transitions
– again a situation where our arguments fail. Therefore
our results are not applicable at T = 0. Second, finite
temperature transport may be plagued by additional di-
vergencies for ω → 0 not captured by the Drude weight.
In some special models, for instance, transport is singu-
lar even in the absence of exactly conserved quantities
(e.g. non-interacting phonons in a disordered crystal8).
In all cases known to us, these divergencies can be traced
back to the presence of some slow modes in the system
(e.g. phonons with very low momentum). While we have
not kept track of such divergencies in our arguments, we
nevertheless believe that they do not invalidate our main
inequality (16) as further slow modes not captured by ex-
act conservation laws will only increase the conductivity.
It is, however, likely that the equality (21) is not valid
for such situations. In Appendix C we analyze in some
detail within the Boltzmann equation formalism under
which conditions (21) holds. As an aside, we note that
the singular heat transport of non-interacting disordered
phonons, mentioned above, is well described within our
formalism if we model the clean system by H0 and the
disorder by H1, see the extensive discussion by Ziman
within the variational approach which can be directly
translated to the memory matrix language, see Ref. [29].
It would be interesting to generalize our results to cases
where time reversal symmetry is broken, e.g. by an exter-
nal magnetic field. As time reversal invariance entered
nontrivially in our arguments, this seems not to be sim-
ple. We nevertheless do not see any physical reason why
the inequality should not be valid in this case, too. One
example where no problems arise are spin chains in a
uniform magnetic field31 where one can map the field to
a chemical potential using a Jordan-Wigner transforma-
tion. Then one can directly apply our results to the time
reversal invariant system of Jordan-Wigner fermions.
Acknowledgments
We thank N. Andrei, E. Shimshoni, P. Wölfle and
X. Zotos for useful discussions. This work was partly sup-
ported by the Deutsche Forschungsgemeinschaft through
SFB 608 and the German Israeli Foundation.
APPENDIX A: DRUDE WEIGHT AND MAZUR
INEQUALITY
In this appendix we clarify the connection between the
Drude weight and the Mazur inequality, mentioned in the
introduction.
The Drude weight D is the singular part of the con-
ductivity at zero frequency, Re σ(ω) = πDδ(ω)+σreg(ω).
It can be calculated from the relation
D = lim
ω Im σ(ω).
It has been introduced by Kohn32 as a measure of ballistic
transport, indicated by D > 0.
Using Kubo formulas, conductivities can be expressed
in terms of the dynamic current susceptibilities33 Π(z)
using
σ(z) = − 1
ΠT −Π(z)
, (A1)
where Π(z) is the current response function
Π(z) =
dteizt〈[J(t), J(0)]〉 (A2)
Π′′(ω)
. (A3)
and ΠT is a current susceptibility. The conductivity may
be calculated by setting σ(ω) = σ(z = ω + i0). Relation
(A3) is a well known sum rule and for all regular corre-
lation functions one has ΠT = Π(0). In the presence of
a singular contribution to σ(ω), one easily identifies the
Drude weight with the expression ΠT−Π(0). For this dif-
ference Mazur12,13 derived a lower bound. Furthermore,
Suzuki13 has shown, that ΠT − Π(0) may be expressed
as a sum over all constants of the motion Ci present in
the system36,
D = ΠT −Π(0) =
〈CjJ〉2
〈C2j 〉
. (A4)
Thus, the Drude weight is intimately connected to the
presence of conservation laws: only components of the
current perpendicular to all conservation laws decay and
any conservation law with a component parallel to the
current (i.e. with a finite cross-correlation 〈CjJ〉) leads
to a finite Drude weight and thus ballistic transport. The
relation between the Drude weight and Mazur’s inequal-
ity has been first pointed out by Zotos14.
APPENDIX B: PERTURBATION THEORY FOR
Let us give an example of a naive perturbative deriva-
tion (see also Ref. [6]) to gain some insight about what
problems can turn up in a perturbative derivation as the
one presented in this work. According to our assump-
tions, the conductivity is diverging for g → 0 and there-
fore it is useful to consider the scattering rate Γ(ω)/χ
(with the current susceptibility χ) defined by
σ(ω) =
Γ(ω)/χ− iω
. (B1)
If J is conserved for g = 0 (i.e. for J = J‖, see above),
the scattering rate vanishes, Γ(ω) = 0, for g = 0, which
results in a finite Drude weight. A perturbation around
this singular point results in a finite Γ(ω). In the limit
g → 0 we can expand (B1) for any finite frequency ω in
Γ to obtain
ω2Re σ(ω) = Re Γ(ω) +O(Γ2/ω). (B2)
We can read this as an equation for the leading order
contribution to Γ(ω), which now is expressed through
the Kubo formula for the conductivity. By partially in-
tegrating twice in time we can write Γ(ω) = Γ̃(ω)+O(g3)
Re Γ̃(ω) = Re
dteizt〈[J̇(t), J̇(0)]〉0
z=ω+i0
where J̇ = i[H, J ] = ig[H1, J ] is linear in g and therefore
the expectation value 〈...〉0 can be evaluated with respect
to H0 (which may describe an interacting system). Thus
we have expressed the scattering rate via a simple corre-
lation function of the time derivative of the current.
To determine the dc conductivity one is interested in
the limit ω → 0 and it is tempting to set ω = 0 in
Eq. (B3). We have, however, derived Eq. (B3) in the
limit g → 0 at finite ω and not in the limit ω → 0 at
finite g. The series Eq. (B2) is well defined for finite
ω 6= 0 only and in the limit ω → 0 the series shows
singularities to arbitrarily high orders in 1/ω.
At first sight this makes Eq. (B3) useless for calculating
the dc conductivity. One of the main results of this paper
is that, nevertheless, Γ̃(ω = 0) can be used to obtain a
lower bound to the dc conductivity
σ(ω = 0) ≥ χ
Γ̃(0)
for g → 0. (B4)
APPENDIX C: SINGLE SLOW MODE
In this appendix we check whether in the presence of
a single conservation law with finite cross correlations
with the current the inequality (16) can be replaced by
the equality (21). This requires us to compare the true
conductivity, which in general is hard to determine, to
the result given by M̃1. Thus we restrict ourselves to
the discussion of models for which a Boltzmann equation
can be formulated and the expression for the conductivity
can be calculated at least numerically. In the following
we first show numerically that the equality (21) holds for
a realistic model. In a second step we discuss the precise
regularity requirement of the scattering matrix such that
Eq. (21) holds.
To simplify numerics, we consider a simple one-
dimensional Boltzmann equation of interacting and
weakly disordered Fermions. Clearly, the Boltzmann ap-
proach breaks down close to the Fermi surface due to
singularities associated with the formation of a Luttinger
liquid, but in the present context we are not interested
in this physics as we only want to investigate properties
of the Boltzmann equation. To avoid the restrictions as-
sociated with momentum and energy conservation in one
dimension we consider a dispersion with two minima and
four Fermi points,
ǫk = −
. (C1)
The Boltzmann equation reads
k′qq′
fkfk′(1− fq)(1 − fq′)
− fqfq′(1 − fk)(1 − fk′)
δ(ǫk − ǫk′)
fk(1 − fk′)− fk′(1 − fk)
Wkk′Φk′ (C2)
where the inelastic scattering term S
kk′ = δ(ǫk + ǫk′ −
ǫq − ǫq′)δ(k+ k′ − q− q′) conserves both energy and mo-
mentum. In the last line we have linearized the right
hand side using the definitions of the introductory chap-
ter. The velocity vk is given by vk =
ǫk. The scat-
tering matrix splits up into an interaction component
and a disorder component, Wkk′ = W
kk′ + g
2W 1kk′ . As
we do not consider Umklapp scattering, W 0kk′ conserves
momentum,
′ = 0, and one expects that mo-
mentum relaxation will determine the conductivity for
small g.
For the numerical calculation we discretize momentum
in the interval [−π/2, π/2], kn = nδk = nπ/N with inte-
ger n. (At the boundaries the energy is already too high
to play any role in transport.) The delta function aris-
ing from energy conservation is replaced by a gaussian of
width δ. The proper thermodynamic limit can for exam-
ple be obtained by choosing δ = 0.3/
N . The numerics
shows small finite size effects.
In Fig. 2 we compare the numerical solution of the
Boltzmann equation to the single mode memory matrix
calculation or, equivalently29, to the variational bound
obtained by setting Φk = k in Eq. (2)
k,k′ kWkk′k
k,k′ kW
. (C3)
As can be seen from the inset, in the limit of small g
one obtains the exact value for the conductivity, which is
what we intended to demonstrate.
0 0.05 0.1 0.15 0.2 0.25 0.3
0 0.1 0.2 0.3
FIG. 2: Comparison of the result of a single mode memory
matrix calculation (solid line), Eq. (C3), to the full numerical
solution of the Boltzmann equation (dotted line) for T = 0.05
and N = 500. The memory matrix is always a lower bound to
the Boltzmann result and converges towards it as the disorder
strength g is reduced, as shown in the inset (ratio of the single
mode approximation to the Boltzmann result).
Next we turn to an analysis of regularity conditions
which have to be met in general by the scattering matrix
Wkk′ such that convergence is guaranteed in the limit
g → 0. According to the assumptions of this appendix,
for g = 0 the variational form of the Boltzmann equa-
tion (2) has a unique solution Φ̄k (up to a multiplica-
tive constant), with F (Φ̄k) = ∞,
kk′ Φ̄k′ = 0 and
k vkΦ̄kdf
0/dǫk > 0.
In the presence of a finite, but small g we write the solu-
tion of the Boltzmann equation as Φ = Φ̄+Φ⊥, where Φ⊥
has no component parallel to Φ̄ (i.e.
k Φ̄kΦ
0/dǫk =
0 ). On the basis of the two inequalities
F [Φ̄] ≤ F [Φ] (C4)
ΦWΦ = Φ̄g2W 1Φ̄ + Φ⊥WΦ⊥ ≥ Φ̄g2W 1Φ̄ (C5)
one concludes that Eq. (21) is valid, i.e. that
F [Φ̄]
F [Φ]
under the condition that
vkΦ̄k
or, equivalently,
vkΦ⊥,k
= 0. (C6)
We therefore have to check whether Φ⊥ becomes small
in the limit of small g.
Expanding the saddlepoint equation for (2) we obtain
W 0kk′Φ
k′ = vk
k′k′′ Φ̄k′g
2W 1k′k′′ Φ̄k′′
k′ vk′
g2W 1kk′ Φ̄k′ +O(g2W1Φ⊥,Φ⊥W0Φ⊥)
As by definition Φ⊥ has no component parallel to Φ̄, we
can insert the projector Q which projects out the con-
servation law in front of Φ⊥k on the left hand side. We
therefore conclude that if the inverse of W 0Q exists, then
Φ⊥ is of order g
2, Eq. (C6) is valid and therefore also
Eq. (21). In our numerical examples these conditions are
all met.
Under what conditions can one expect that Eq. (C6) is
not valid? Within the assumptions of this appendix we
have excluded the presence of other zero modes of W 0
(i.e. conservation laws) with finite overlap with the cur-
rent. But it may happen that W 0 has many eigenvalues
which are arbitrarily small such that the sum in Eq. (C6)
diverges. In such a situation the presence of slow modes
which cannot be identified with conservation laws of the
unperturbed system invalidates Eq. (21).
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http://arxiv.org/abs/cond-mat/0607837

  We show how one can obtain a lower bound for the electrical, spin or heat
conductivity of correlated quantum systems described by Hamiltonians of the
form H = H0 + g H1. Here H0 is an interacting Hamiltonian characterized by
conservation laws which lead to an infinite conductivity for g=0. The small
perturbation g H1, however, renders the conductivity finite at finite
temperatures. For example, H0 could be a continuum field theory, where momentum
is conserved, or an integrable one-dimensional model while H1 might describe
the effects of weak disorder. In the limit g to 0, we derive lower bounds for
the relevant conductivities and show how they can be improved systematically
using the memory matrix formalism. Furthermore, we discuss various applications
and investigate under what conditions our lower bound may become exact.

Lower bounds for the conductivities of correlated quantum systems
Peter Jung and Achim Rosch
Institute for Theoretical Physics, University of Cologne, 50937 Cologne, Germany.
(Dated: October 30, 2018)
We show how one can obtain a lower bound for the electrical, spin or heat conductivity of cor-
related quantum systems described by Hamiltonians of the form H = H0 + gH1. Here H0 is an
interacting Hamiltonian characterized by conservation laws which lead to an infinite conductivity
for g = 0. The small perturbation gH1, however, renders the conductivity finite at finite temper-
atures. For example, H0 could be a continuum field theory, where momentum is conserved, or an
integrable one-dimensional model while H1 might describe the effects of weak disorder. In the limit
g → 0, we derive lower bounds for the relevant conductivities and show how they can be improved
systematically using the memory matrix formalism. Furthermore, we discuss various applications
and investigate under what conditions our lower bound may become exact.
PACS numbers: 72.10.Bg, 05.60.Gg, 75.40.Gb, 71.10.Pm
I. INTRODUCTION
Transport properties of complex materials are not only
important for many applications but are also of funda-
mental interest as their study can give insight into the
nature of the relevant quasi particles and their interac-
tions.
Compared to thermodynamic quantities, the transport
properties of interacting quantum systems are notori-
ously difficult to calculate even in situations where in-
teractions are weak. The reason is that conductivities
of non-interacting systems are usually infinite even at fi-
nite temperature, implying that even to lowest order in
perturbation theory an infinite resummation of a per-
turbative series is mandatory. To lowest order this im-
plies that one usually has to solve an integral equation,
often written in terms of (quantum-) Boltzmann equa-
tions or – within the Kubo formalism – in terms of vertex
equations. The situation becomes even more difficult if
the interactions are so strong that an expansion around
a non-interacting system is not possible. Also numeri-
cally, the calculation of zero-frequency conductivities of
strongly interacting clean systems is a serious challenge
and even for one-dimensional systems reliable calcula-
tions are available for high temperatures only1,2,3,4,5,6.
Variational estimates, e.g. for the ground state energy,
are powerful theoretical techniques to obtain rigorous
bounds on physical quantities. They can be used to
guide approximation schemes to obtain simple analytic
estimates and are sometimes the basis of sophisticated
numerical methods like the density matrix renormaliza-
tion group7.
Taking into account both the importance of transport
quantities and the difficulties involved in their calculation
it would be very useful to have general variational bounds
for transport coefficients.
A well known example where a bound for transport
quantities has been derived is the variational solution
of the Boltzmann equation, discussed extensively by
Ziman8. The linearized Boltzmann equation in the pres-
ence of a static electric field can be written in the form
Wk,k′Φk′ (1)
where Wk,k′ is the integral kernel describing the scatter-
ing of quasiparticles and we have linearized the Boltz-
mann equation around the Fermi (or Bose) distribution
= f0(ǫk) using fk = f
Φk. Therefore, the
current is given by I = −e
Φk and the dc con-
ductivity is determined from the inverse of the scattering
matrix W using σ = −e2
k,k′v
. It is
easy to see that this result can be obtained by maximiz-
ing a functional8,9,10,11 F [Φ] with
σ = e2max
F [Φ] ≥ e2 max
F [Φ] =
k,k′(Φk − Φk′)2Wk,k′
where we used that
Wk,k′ = 0 reflecting the conser-
vation of probability. The variational formula (2) is ac-
tually closely related8 to the famous H-theorem of Boltz-
mann which states that entropy always increases upon
scattering.
A lower bound for the conductivity can be obtained
by varying Φ only in a subspace of all possible func-
tions. This allows for example to obtain analytically
good estimates for conductivities without inverting an
infinite dimensional matrix or, euqivalently, solving an
integral equation, see Ziman’s book for a large number
of examples8.
The applicability of Eq. (2) is restricted to situations
where the Boltzmann equation is valid and bounds for
the conductivity in more general setups are not known.
However, for ballistic systems with infinite conductiv-
ity it is possible to get a lower bound for the so-called
Drude weight. Mazur12 and later Suzuki13 considered
situations where the presence of conservation laws pro-
hibits the decay of certain correlation functions in the
http://arxiv.org/abs/0704.0886v2
Re σ(ω) Re σ(ω)
σreg(ω)
πDδ(ω)
g != 0g = 0
FIG. 1: For g = 0 a Drude peak shows up in the conductivity,
resulting from exact conservation laws. For g 6= 0 the Drude
peak broadens and the dc conductivity becomes finite.
long time limit. In the context of transport theory their
result can be applied to systems (see Appendix A for de-
tails) where the finite-temperature conductivity σ(ω, T )
is infinite for ω = 0 and characterized by a finite Drude
weight D(T ) > 0 with
Re σ(ω, T ) = πD(T )δ(ω) + σreg(ω, T ). (3)
Such a Drude weight can arise only in the presence of
exact conservation laws Cj with [H,Cj ] = 0. Suzuki
showed that the Drude weight can be expressed as a sum
over all Cj
〈CjJ〉2
〈C2j 〉
〈CjJ〉2
〈C2j 〉
. (4)
where J is the current associated with σ. For conve-
nience a basis in the space of Ci has been chosen such
that 〈CiCj〉 = 0 for i 6= j. More useful than the equal-
ity in Eq. (4) is often the inequality12 which is obtained
when the sum is restricted to a finite subset of conser-
vation laws. Such a finite sum over simple expectation
values can often be calculated rather easily using either
analytical or numerical methods. The Mazur inequality
has recently been used heavily4,14,15,16,17 to discuss the
transport properties of one-dimensional systems.
Model systems, due to their simplicity, often exhibit
symmetries not shared by real materials. For exam-
ple, the heat conductivity of idealized one-dimensional
Heisenberg chains is infinite at arbitrary temperature as
the heat current is conserved. However, any additional
coupling (next-nearest neighbor, inter-chain, disorder,
phonon,...) renders the conductivity finite1,4,5,6,18,19,20.
If these perturbations are weak, the heat conductivity
is, however, large as observed experimentally21,22. For a
more general example, consider an arbitrary translation-
ally invariant continuum field theory. Here momentum
is conserved which usually implies that the conductivity
is infinite for this model. In real materials momentum
decays by Umklapp scattering or disorder rendering the
conductivity finite. It is obviously desirable to have a
reliable method to calculate transport in such situations.
In this work we consider systems with the Hamiltonian
H = H0 + gH1, (5)
where for g = 0 the relevant heat-, charge- or spin- con-
ductivity is infinite and characterized by a finite Drude
weight given by Eq. (4). As discussed above,H0 might be
an integrable one-dimensional model, a continuum field
theory, or just a non-interacting system. The term gH1
describes a (weak) perturbation which renders the con-
ductivity finite, e.g. due to Umklapp scattering or dis-
order, see Fig. 1. Our goal is to find a variational lower
bound for conductivities in the spirit of Eq. (2) for this
very general situation, without any requirement on the
existence of quasi particles. For technical reasons (see
below) we restrict our analysis to situations where H is
time reversal invariant.
In the following, we first describe the general setup and
introduce the memory matrix formalism, which allows
us to formulate an inequality for transport coefficients
for weakly perturbed systems. We will argue that the
inequality is valid under the conditions which we specify.
Finally, we investigate under which conditions the lower
bounds become exact and briefly discuss applications of
our formula.
II. SETUP
Consider the local density ρ(x) of an arbitrary phys-
ical quantity which is locally conserved, thus obeying a
continuity equation
∂tρ+∇j = 0.
Transport of that quantity is described by the dc con-
ductivity σ which is the response of the current to some
external field E coupling to the current,
〈J〉 = σE,
where J =
j(x) is the total current and 〈J〉 its expec-
tation value. Note that J can be an electrical-, spin-, or
heat current and E the corresponding conjugate field de-
pending on the context. The dynamic conductivity σ(z)
is given by Kubo’s formula, see Eq. (A1). We are inter-
ested in the dc conductivity σ = limω→0 σ(z = ω + i0).
Starting from the Hamiltonian (5) we consider a sys-
tem where H0 posesses a set of exact conservation laws
{Ci} of which at least one correlates with the current,
〈JC1〉 6= 0. Without loss of generality we assume
〈CiCj〉 = 0 for i 6= j. For g = 0 the Drude weight D
defined by Eq. (3) is given by Eq. (4). We can split up
the current under consideration into a part which is par-
allel to the Ci and one that is orthogonal,
J = J‖ + J⊥,
with J‖ =
〈CiJ〉
Ci, which results in a separation of
the conductivity,
σ(z) = σ‖(z) + σ⊥(z). (6)
Since the conductivity σ(z) is given by a current-current
correlation function and the current J‖ (J⊥) is diago-
nal (off-diagonal) in energy, cross-correlation functions
〈〈J‖; J⊥〉〉 vanish in Eq. (6).
According to Eq. (4), the Drude peak of the unper-
turbed system, g = 0, arises solely from J‖:
Re σ‖(ω) = πDδ(ω), (7)
while σ⊥(z) appears in Eq. (3) as the regular part,
Re σ⊥(ω) = σreg(ω).
In this work we will focus on σ‖(ω), since the small
perturbation is not going to affect σ⊥(ω) much (which is
assumed to be free of singularities here, see section IV)
while σ‖(ω = 0) diverges for g → 0, see Fig. 1. As we
are interested in the small g asymptotics only, we may
neglect the contribution σ⊥(0) to the dc conductivity.
Hence we set J = J‖ and σ(ω) = σ‖(ω) in the following.
III. MEMORY MATRIX FORMALISM
We have seen that certain conservation laws ofH0 play
a crucial role in determining the conductivity of both the
unperturbed and the perturbed system. In the presence
of a small perturbation gH1, these modes are not con-
served anymore but at least some of them decay slowly.
Typically, the conductivity of the perturbed system will
be determined by the dynamics of these slow modes. To
separate the dynamics of the slow modes from the rest, it
is convenient to use a hydrodynamic approach based on
the projection of the dynamics onto these slow modes. In
this section we will therefore review the so called memory
matrix formalism23, introduced by Mori and Zwanzig24,25
for this purpose. In the next section we will show that
this approach can be used to obtain a lower bound for
the dc conductivity for small g.
We start by defining a scalar product in the space of
quantum mechanical operators,
(A|B) =
dλ〈A†B(iλ)〉 − β〈A†〉〈B〉 (8)
As the next step we choose a – for the moment – arbi-
trary set of operators {Ci}. In most applications, the Ci
are the relevant slow modes of the system. For notational
convenience, we assume that the {Ci} are orthonormal-
ized,
(Ci|Cj) = δij . (9)
In terms of these we may define the projector P onto (and
Q away from) the space spanned by these ‘slow’ modes
|Ci)(Ci| = 1−Q.
We assume that C1 is the current we are interested in,
|J) ≡ |C1).
The time evolution is given by the Liouville-
(super)operator
L = [H, .] = L0 + gL1
with (LA|B) = (A|LB) = (A|L|B), and the time evo-
lution of an operator may be expressed as |A(t)) =
|eiHtAe−iHt) = eiLt|A). With these notions, one obtains
the following simple, yet formal expression for the con-
ductivity:
σ(ω) =
ω − L
ω − L
Using a number of simple manipulations, one can
show23,24,25 that the conductivity can be expressed as
the (1, 1)-component of the inverse of a matrix
σ(ω) = (M(ω) + iK − iω)−1
, (10)
where
Mij(ω) =
ω − LQ
is the so-called memory matrix and
Kij =
Ċi|Cj
a frequency independent matrix. The formal expression
(10) for the conductivity is exact, and completely gen-
eral, i.e. valid for an arbitrary choice of the modes Ci
(they do not even have to be ‘slow’). Only C1 = J is re-
quired. However, due to the projection operators Q, the
memory matrix (11) is in general difficult to evaluate. It
is when one uses approximations to M that the choice of
the projectors becomes crucial (see below).
Obviously, the dc conductivity is given by the (1, 1)-
component of
(M(0) +K)−1. (13)
More generally, the (m,n)-component of Eq. (13) de-
scribes the response of the ‘current’ Cm to an external
field coupling solely to Cn. We note, that since a matrix
of transport coefficients has to be positive (semi)definite,
this also holds for the matrix M(0) +K.
To avoid technical complications associated with the
presence of K we restrict our analysis in the following to
time reversal invariant systems and choose the Ci such,
that they have either signature +1 or −1 under time
reversal34 Θ. In the dc limit, ω = 0, components of
Eq.(13) connecting modes of different signatures vanish.
Thus, M(0) + K is block-diagonal with respect to the
time reversal signature, and consequently we can restrict
our analysis to the subspace of slow modes with the same
signature as C1. However, if Cm and Cn have the same
signature, then (Cm|Ċn) = 0, and thus K vanishes on
this restricted space. The dc conductivity therefore takes
the form
σ = (M(0)−1)11. (14)
IV. CENTRAL CONJECTURE
To obtain a controlled approximation to the memory
matrix in the limit of small g, it is important to identify
the relevant slow modes of the system. For the Ci we
choose quantities which are conserved by H0, [H0, Ci] =
0, such that Ċi = ig[H1, Ci] is linear in the small coupling
g. As argued below, we require that the singularities
of correlation functions of the unperturbed system are
exclusively due to exact conservation laws Ci, i.e. that
the Drude peak appearing in Eq. (3) is the only singular
contribution. Furthermore, we choose J = J‖ = C1 and
consider only Ci with the same time reversal signature
as J , as discussed in the previous section.
To formulate our central conjecture we introduce the
following notions. We define Mn(ω) as the (exact) n× n
memory matrix obtained by setting up the memory ma-
trix formalism for the first n slow modes {Ci, i = 1, .., n}.
Note that the definitions of the relevant projectors P and
Q also depend on this choice, and that for any choice of
n one gets σ = (M−1n )11. We now introduce the approxi-
mate memory matrix M̃n motivated by the following ar-
guments: Ċi is already linear in g, therefore in Eq. (11)
we approximate L by L0 and replace (.|.) by (.|.)0 as
we evaluate the scalar product with respect to H0. As
L0|Ci) = 0 and (Cj |Ċi) = 0 due to time reversal symme-
try, one has L0Q = 1 and Q|Ċi) = |Ċi) and therefore the
projector Q does not contribute within this approxima-
tion. We thus define the n× n matrix M̃n by
M̃n,ij = lim
ω − L0
Note that M̃n is a submatrix of M̃m for m > n and
therefore the approximate expression for the conductiv-
ity σ ≈ (M̃−1n )11 does depend on n while (M−1n )11 is
independent of n. A much simpler, alternative deriva-
tion for M̃1 is given in Appendix B, where the validity of
this formula is also discussed.
The central conjecture of our paper is, that for small g
(M̃−1n )11 gives a lower bound to the dc conductivity, or,
more precisely,
1/g2 = (M̃
∞ )11 ≥ · · · ≥ (M̃−1n )11 ≥ · · · ≥ M̃−11 . (16)
Here σ|
1/g2 = (1/g
2) limg→0 g
2σ denotes the leading
term ∝ 1/g2 in the small-g expansion of σ. Note that
M̃n ∝ g2 by construction. M̃∞ is the approximate mem-
ory matrix where all35 conservation laws have been in-
cluded. In some special situations, discussed in Ref. 6,
one has σ ∼ 1/g4 and therefore σ|
1/g2 = ∞.
A special case of the inequality above is Eq. (B4) in
appedix B, as the scattering rate Γ̃/χ may be expressed
as Γ̃/χ2 = M̃1.
Two steps are necessary to prove Eq. (16). The simple
part is actually the inequalities in Eq. (16). They are
a consequence of the fact that the matrices M̃n are all
positive definite and that M̃n is a submatrix of M̃m for
m ≥ n. More difficult to prove is that the first equality
in (16) holds. To show this we will need an additional
assumption, namely, that the regular part of all correla-
tion functions (to be defined below) remains finite in the
limit g → 0, ω → 0. In this case, the perturbative ex-
pansion around M̃∞ in powers of g is free of singularities
at finite temperature (which is not the case for M̃n<∞).
This in turn implies that limg→0 M∞/g
2 = M̃∞/g
2 and
therefore σ|
1/g2 = (M̃
∞ )11.
Next, we present the two parts of the proof.
A. Inequalities
We start by investigating the (1,1)-component of the
inverse of the positive definite symmetric matrix M̃∞. It
is convenient to write the inverse as
(M̃−1∞ )11 = max
(ϕTe1)
ϕT M̃∞ϕ
where e1 is the first unit vector. The same method is
used to derive Eq. (2) in the context of the Boltzmann
equation. The maximum is obtained for ϕ = M̃−1∞ e1.
By restricting the variational space in (17) to the first n
components of ϕ we reproduce the submatrix M̃n of M̃∞
and obtain
(M̃−1∞ )11 ≥ max
(ϕTe1)
ϕT M̃∞ϕ
= (M̃−1m )11
≥ max
(ϕT e1)
ϕT M̃∞ϕ
= (M̃−1n<m)11
By choosing different values form and n < m, this proves
all inequalities appearing in (16).
B. Expansion of the memory matrix
We proceed by expanding the exact memory matrix
Mn, where Pn = 1 − Qn is a projector on the first n
conservation laws, in powers of g. Using that LQn =
L0 + gL1Qn, we obtain the geometric series
Mn,ij(ω) =
ω − L0
ω − L0
Note that this is not a full expansion in g, as the scalar
product (8) is defined with respect to the full Hamilto-
nian H = H0 + gH1. We will turn to the discussion of
the remaining g-dependence later.
In general, one can expand
λmn|Am)(An|
in terms of some basis Am in the space of operators.
Therefore Eq. (18) can be written as a sum over products
of terms with the general structure
ω − L0
. (19)
In the following we would like to argue that such an ex-
pansion is regular for n = ∞ if all conservation laws
have been included in the definition of Q. As argued in
Appendix B, we have to investigate whether the series co-
efficients in Eq. (18) diverge for ω → 0. The basis of our
argument is the following: as Q∞ projects the dynam-
ics to the space perpendicular to all of the conservation
laws, the associated singularities are absent in Eq. (19)
and therefore the expansion of M∞ is regular.
To show this more formally, we split up B = B‖ +B⊥
in (19) into a component parallel and one perpendicular
to the space of all conserved quantities, |B‖) = P∞|B).
With this notation, the action of L0 becomes more trans-
parent:
ω − L0
|B) = 1
|B‖) +
ω − L0
|B⊥). (20)
As we assume that all divergencies can be traced back
to the conservation laws, we take the second term to be
regular. It is only the first term which leads in Eq. (19)
to a divergence for ω → 0, provided that (A|Qn|B‖) is fi-
nite. If we consider the perturbative expansion ofMn<∞,
where Pn = 1 − Qn projects only to a subset of con-
served quantities, then finite contributions of the form
(A|Qn|B‖) exist and the perturbative series in g will be
singular (see also Appendix B). Considering M∞, how-
ever, Q∞ projects out all conservation laws and therefore
by construction Q∞|B‖) = Q∞P∞|B) = 0. Thus the
first term in (20) does not contribute in (19) for n = ∞
and the expansion (18) of M∞ is therefore regular.
The only remaining part of our argument is to show
that in the limit g → 0 one can safely replace (.|.) by
(.|.)0. Here it is useful to realize that (A|B) can be in-
terpreted as a (generalized) static susceptibility. In the
absence of a phase transition and at finite temperatures,
susceptibilities are smooth, non-singular functions of the
coupling constants and therefore we do not expect any
further singularities from this step. If we define a phase
transition by a singularity in some generalized suscepti-
bility, then the statement that susceptibilities are regular
in the absence of phase transitions even becomes a mere
tautology.
Combining all arguments, the expansion (18) of
M∞(ω → 0) is regular, and using (Ċi|Q∞ = (Ċi| [see
discussion before Eq. (15)] its leading term, k = 0 is
given by M̃∞. We therefore have shown the missing first
equality of our central conjecture (16).
V. DISCUSSION
In this paper we have established that in the limit of
small perturbations, H = H0 + gH1, lower bounds to dc
conductivities may be calculated for situations where the
conductivity is infinite for g = 0. In the opposite case,
when the conductivity is finite for g = 0, one can use
naive perturbation theory to calculate small corrections
to σ without further complications.
The relevant lower bounds are directly obtained from
the memory matrix formalism. Typically26,27,28 one has
to evaluate a small number of correlation functions and
to invert small matrices. The quality of the lower bounds
depends decisively on whether one has been able to iden-
tify the ‘slowest’ modes in the system.
There are many possible applications for the results
presented in this paper. The mostly considered situ-
ation is the case where H0 describes a non-interacting
system26. For situations where the Boltzmann equation
can be applied, it has been pointed out a long time ago by
Belitz29 that there is a one-to-one relation of the memory
matrix calculation to a certain variational Ansatz to the
Boltzmann equation, see Eq. (2). In this paper we were
able to generalize this result to cases where a Boltzmann
description is not possible. For example, if H0 is the
Hamiltonian of a Luttinger liquid, i.e. a non-interacting
bosonic system, then typical perturbations are of the
form cosφ for which a simple transport theory in the
spirit of a Boltzmann or vertex equation does not exist
to our knowledge.
Another class of applications are systems where H0
describes an interacting system, e.g. an integrable one-
dimensional model6 or some non-trivial quantum-field
theory30. In these cases it can become difficult to cal-
culate the memory matrix and one has to resort to use
either numerical6 or field-theoretical methods30 to obtain
the relevant correlation functions.
An important special case are situations where H0
is characterized by a single conserved current with the
proper symmetries, i.e. with overlap to the (heat-, spin-
or charge-) current J . For example, in a non-trivial con-
tinuum field theory H0, interactions lead to the decay of
all modes with exception of the momentum P . In this
case the momentum relaxation and therefore the con-
ductivity at finite T is determined by small perturba-
tions gH1 like disorder or Umklapp scattering which are
present in almost any realistic system. As M̃∞ = M̃1 in
this case, our results suggest that for small g the conduc-
tivity is exactly determined by the momentum relaxation
rate M̃PP = limω→0 i(Ṗ |(ω − L0)−1|Ṗ ),
for g → 0. (21)
Here we used that J‖ = P (P |J)/(P |P ) with χPJ = (P |J)
and we have restored all factors which arise if the nor-
malization condition (9) is not used. In Appendix C, we
check numerically that this statement is valid for a real-
istic example within the Boltzmann equation approach.
A number of assumptions entered our arguments. The
strongest one is the restriction that all relevant singu-
larities arise from exact conservation laws of H0. We
assumed that the regular parts of correlation functions
are finite for ω = 0. There are two distinct scenarios
in which this assumption does not hold. First, in the
limit T → 0, often scattering rates vanish which can lead
to diverencies of the nominally regular parts of correla-
tion functions. Furthermore, at T = 0 even infinitesi-
mally small perturbations can induce phase transitions
– again a situation where our arguments fail. Therefore
our results are not applicable at T = 0. Second, finite
temperature transport may be plagued by additional di-
vergencies for ω → 0 not captured by the Drude weight.
In some special models, for instance, transport is singu-
lar even in the absence of exactly conserved quantities
(e.g. non-interacting phonons in a disordered crystal8).
In all cases known to us, these divergencies can be traced
back to the presence of some slow modes in the system
(e.g. phonons with very low momentum). While we have
not kept track of such divergencies in our arguments, we
nevertheless believe that they do not invalidate our main
inequality (16) as further slow modes not captured by ex-
act conservation laws will only increase the conductivity.
It is, however, likely that the equality (21) is not valid
for such situations. In Appendix C we analyze in some
detail within the Boltzmann equation formalism under
which conditions (21) holds. As an aside, we note that
the singular heat transport of non-interacting disordered
phonons, mentioned above, is well described within our
formalism if we model the clean system by H0 and the
disorder by H1, see the extensive discussion by Ziman
within the variational approach which can be directly
translated to the memory matrix language, see Ref. [29].
It would be interesting to generalize our results to cases
where time reversal symmetry is broken, e.g. by an exter-
nal magnetic field. As time reversal invariance entered
nontrivially in our arguments, this seems not to be sim-
ple. We nevertheless do not see any physical reason why
the inequality should not be valid in this case, too. One
example where no problems arise are spin chains in a
uniform magnetic field31 where one can map the field to
a chemical potential using a Jordan-Wigner transforma-
tion. Then one can directly apply our results to the time
reversal invariant system of Jordan-Wigner fermions.
Acknowledgments
We thank N. Andrei, E. Shimshoni, P. Wölfle and
X. Zotos for useful discussions. This work was partly sup-
ported by the Deutsche Forschungsgemeinschaft through
SFB 608 and the German Israeli Foundation.
APPENDIX A: DRUDE WEIGHT AND MAZUR
INEQUALITY
In this appendix we clarify the connection between the
Drude weight and the Mazur inequality, mentioned in the
introduction.
The Drude weight D is the singular part of the con-
ductivity at zero frequency, Re σ(ω) = πDδ(ω)+σreg(ω).
It can be calculated from the relation
D = lim
ω Im σ(ω).
It has been introduced by Kohn32 as a measure of ballistic
transport, indicated by D > 0.
Using Kubo formulas, conductivities can be expressed
in terms of the dynamic current susceptibilities33 Π(z)
using
σ(z) = − 1
ΠT −Π(z)
, (A1)
where Π(z) is the current response function
Π(z) =
dteizt〈[J(t), J(0)]〉 (A2)
Π′′(ω)
. (A3)
and ΠT is a current susceptibility. The conductivity may
be calculated by setting σ(ω) = σ(z = ω + i0). Relation
(A3) is a well known sum rule and for all regular corre-
lation functions one has ΠT = Π(0). In the presence of
a singular contribution to σ(ω), one easily identifies the
Drude weight with the expression ΠT−Π(0). For this dif-
ference Mazur12,13 derived a lower bound. Furthermore,
Suzuki13 has shown, that ΠT − Π(0) may be expressed
as a sum over all constants of the motion Ci present in
the system36,
D = ΠT −Π(0) =
〈CjJ〉2
〈C2j 〉
. (A4)
Thus, the Drude weight is intimately connected to the
presence of conservation laws: only components of the
current perpendicular to all conservation laws decay and
any conservation law with a component parallel to the
current (i.e. with a finite cross-correlation 〈CjJ〉) leads
to a finite Drude weight and thus ballistic transport. The
relation between the Drude weight and Mazur’s inequal-
ity has been first pointed out by Zotos14.
APPENDIX B: PERTURBATION THEORY FOR
Let us give an example of a naive perturbative deriva-
tion (see also Ref. [6]) to gain some insight about what
problems can turn up in a perturbative derivation as the
one presented in this work. According to our assump-
tions, the conductivity is diverging for g → 0 and there-
fore it is useful to consider the scattering rate Γ(ω)/χ
(with the current susceptibility χ) defined by
σ(ω) =
Γ(ω)/χ− iω
. (B1)
If J is conserved for g = 0 (i.e. for J = J‖, see above),
the scattering rate vanishes, Γ(ω) = 0, for g = 0, which
results in a finite Drude weight. A perturbation around
this singular point results in a finite Γ(ω). In the limit
g → 0 we can expand (B1) for any finite frequency ω in
Γ to obtain
ω2Re σ(ω) = Re Γ(ω) +O(Γ2/ω). (B2)
We can read this as an equation for the leading order
contribution to Γ(ω), which now is expressed through
the Kubo formula for the conductivity. By partially in-
tegrating twice in time we can write Γ(ω) = Γ̃(ω)+O(g3)
Re Γ̃(ω) = Re
dteizt〈[J̇(t), J̇(0)]〉0
z=ω+i0
where J̇ = i[H, J ] = ig[H1, J ] is linear in g and therefore
the expectation value 〈...〉0 can be evaluated with respect
to H0 (which may describe an interacting system). Thus
we have expressed the scattering rate via a simple corre-
lation function of the time derivative of the current.
To determine the dc conductivity one is interested in
the limit ω → 0 and it is tempting to set ω = 0 in
Eq. (B3). We have, however, derived Eq. (B3) in the
limit g → 0 at finite ω and not in the limit ω → 0 at
finite g. The series Eq. (B2) is well defined for finite
ω 6= 0 only and in the limit ω → 0 the series shows
singularities to arbitrarily high orders in 1/ω.
At first sight this makes Eq. (B3) useless for calculating
the dc conductivity. One of the main results of this paper
is that, nevertheless, Γ̃(ω = 0) can be used to obtain a
lower bound to the dc conductivity
σ(ω = 0) ≥ χ
Γ̃(0)
for g → 0. (B4)
APPENDIX C: SINGLE SLOW MODE
In this appendix we check whether in the presence of
a single conservation law with finite cross correlations
with the current the inequality (16) can be replaced by
the equality (21). This requires us to compare the true
conductivity, which in general is hard to determine, to
the result given by M̃1. Thus we restrict ourselves to
the discussion of models for which a Boltzmann equation
can be formulated and the expression for the conductivity
can be calculated at least numerically. In the following
we first show numerically that the equality (21) holds for
a realistic model. In a second step we discuss the precise
regularity requirement of the scattering matrix such that
Eq. (21) holds.
To simplify numerics, we consider a simple one-
dimensional Boltzmann equation of interacting and
weakly disordered Fermions. Clearly, the Boltzmann ap-
proach breaks down close to the Fermi surface due to
singularities associated with the formation of a Luttinger
liquid, but in the present context we are not interested
in this physics as we only want to investigate properties
of the Boltzmann equation. To avoid the restrictions as-
sociated with momentum and energy conservation in one
dimension we consider a dispersion with two minima and
four Fermi points,
ǫk = −
. (C1)
The Boltzmann equation reads
k′qq′
fkfk′(1− fq)(1 − fq′)
− fqfq′(1 − fk)(1 − fk′)
δ(ǫk − ǫk′)
fk(1 − fk′)− fk′(1 − fk)
Wkk′Φk′ (C2)
where the inelastic scattering term S
kk′ = δ(ǫk + ǫk′ −
ǫq − ǫq′)δ(k+ k′ − q− q′) conserves both energy and mo-
mentum. In the last line we have linearized the right
hand side using the definitions of the introductory chap-
ter. The velocity vk is given by vk =
ǫk. The scat-
tering matrix splits up into an interaction component
and a disorder component, Wkk′ = W
kk′ + g
2W 1kk′ . As
we do not consider Umklapp scattering, W 0kk′ conserves
momentum,
′ = 0, and one expects that mo-
mentum relaxation will determine the conductivity for
small g.
For the numerical calculation we discretize momentum
in the interval [−π/2, π/2], kn = nδk = nπ/N with inte-
ger n. (At the boundaries the energy is already too high
to play any role in transport.) The delta function aris-
ing from energy conservation is replaced by a gaussian of
width δ. The proper thermodynamic limit can for exam-
ple be obtained by choosing δ = 0.3/
N . The numerics
shows small finite size effects.
In Fig. 2 we compare the numerical solution of the
Boltzmann equation to the single mode memory matrix
calculation or, equivalently29, to the variational bound
obtained by setting Φk = k in Eq. (2)
k,k′ kWkk′k
k,k′ kW
. (C3)
As can be seen from the inset, in the limit of small g
one obtains the exact value for the conductivity, which is
what we intended to demonstrate.
0 0.05 0.1 0.15 0.2 0.25 0.3
0 0.1 0.2 0.3
FIG. 2: Comparison of the result of a single mode memory
matrix calculation (solid line), Eq. (C3), to the full numerical
solution of the Boltzmann equation (dotted line) for T = 0.05
and N = 500. The memory matrix is always a lower bound to
the Boltzmann result and converges towards it as the disorder
strength g is reduced, as shown in the inset (ratio of the single
mode approximation to the Boltzmann result).
Next we turn to an analysis of regularity conditions
which have to be met in general by the scattering matrix
Wkk′ such that convergence is guaranteed in the limit
g → 0. According to the assumptions of this appendix,
for g = 0 the variational form of the Boltzmann equa-
tion (2) has a unique solution Φ̄k (up to a multiplica-
tive constant), with F (Φ̄k) = ∞,
kk′ Φ̄k′ = 0 and
k vkΦ̄kdf
0/dǫk > 0.
In the presence of a finite, but small g we write the solu-
tion of the Boltzmann equation as Φ = Φ̄+Φ⊥, where Φ⊥
has no component parallel to Φ̄ (i.e.
k Φ̄kΦ
0/dǫk =
0 ). On the basis of the two inequalities
F [Φ̄] ≤ F [Φ] (C4)
ΦWΦ = Φ̄g2W 1Φ̄ + Φ⊥WΦ⊥ ≥ Φ̄g2W 1Φ̄ (C5)
one concludes that Eq. (21) is valid, i.e. that
F [Φ̄]
F [Φ]
under the condition that
vkΦ̄k
or, equivalently,
vkΦ⊥,k
= 0. (C6)
We therefore have to check whether Φ⊥ becomes small
in the limit of small g.
Expanding the saddlepoint equation for (2) we obtain
W 0kk′Φ
k′ = vk
k′k′′ Φ̄k′g
2W 1k′k′′ Φ̄k′′
k′ vk′
g2W 1kk′ Φ̄k′ +O(g2W1Φ⊥,Φ⊥W0Φ⊥)
As by definition Φ⊥ has no component parallel to Φ̄, we
can insert the projector Q which projects out the con-
servation law in front of Φ⊥k on the left hand side. We
therefore conclude that if the inverse of W 0Q exists, then
Φ⊥ is of order g
2, Eq. (C6) is valid and therefore also
Eq. (21). In our numerical examples these conditions are
all met.
Under what conditions can one expect that Eq. (C6) is
not valid? Within the assumptions of this appendix we
have excluded the presence of other zero modes of W 0
(i.e. conservation laws) with finite overlap with the cur-
rent. But it may happen that W 0 has many eigenvalues
which are arbitrarily small such that the sum in Eq. (C6)
diverges. In such a situation the presence of slow modes
which cannot be identified with conservation laws of the
unperturbed system invalidates Eq. (21).
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http://arxiv.org/abs/cond-mat/0607837
