Memory Function Approach to Correlated Electron Transport: A Comprehensive Review

0.24 MB·

by  Nabyendu Das

Memory Function Approach to Correlated Electron Transport:
A Comprehensive Review
Nabyendu Das1, Pankaj Bhalla1,2, Navinder Singh1
1Theoretical Physics Division, Physical Research Laboratory, Ahmedabad-380009, India.
2Indian Institute of Technology, Gandhinagar-382424, India.
(Dated: January 7, 2016)
Memory function formalism or projection operator technique is an extremely useful method to
studythetransportandopticalpropertiesofvariouscondensedmattersystems. Arecentrevivalof
itsusesin variouscorrelated electronic systemsis beingobserved. Itis beingused anddiscussed in
6 variouscontexts,rangingfromnon-equilibriumdynamicstotheopticalpropertiesofvariousstrongly
1 correlated systems such as high temperature superconductors. However, a detailed discussion on
0 this method, starting from its origin to its present day applications at one place is lacking. In this
2 articleweattemptacomprehensivereviewofthememoryfunctionapproachfocusingonitsusesin
n studyingthedynamics and thetransport properties of correlated electronic systems.
a
J PACSnumbers: 72.10.-d,72.15.-v

6

] I. MOTIVATION soft modes respectively[10].
l
e
-
r Condensed matter physics deals with the collective
st phenomena that emerge out of the mutual interactions
. between a large number of particles. Many of them are
t
a novel, i.e. are beyond the realm of pre-existent theories
m
and almost none of them can have first principle micro- A B
- scopic explanations. Understanding such novel complex
d
n cooperative phenomena requires new physicalideas such
o as spontaneous symmetry breaking, Goldstone modes, FIG. 1. In Fig. 1A we see viscous drag from the fluid
c renormalization of physical parameters etc.[1] The basic molecules during a streamline flow, i.e. when an external
[ particle of macroscopic size moves slowly inside the fluid. In
principle of all such descriptions is adapting an effective
this case the time scales corresponding to the macro-particle
1 description by separating the low energy slow degrees of
and the fluid particles are nicely separable. In Fig. 1B a
v freedom from the high energy faster degrees of freedom
turbulencesets in duringa faster motion of a particle within
7
in a system of macroscopically large number of particles fluid where such separation of scales is not possible.
2
to study its low energy and long wavelength properties.
1
1 Successes of suchtheories depend onhow accuratelyone To illustrate the idea of separating the scales, we can
0 can develop an effective description of the effects of a choose a simple example, a particle moving in a fluid.
1. largenumberoffasterdegreesoffreedomonafewslower In this case, when the particle moves, the fluid particles
0 degreesof freedomwhichare experimentally probedin a oppose its motion as depicted in Fig.1. An attempt to
6 certainphysicalsystem. Asaresultthesetheoriesarede- build up a microscopic theory for this motion will re-
1 pendentonsomecommonaspectsofthevarioussystems, quire a Hamiltonian that describes the Coulomb inter-
:
v e.g. energyandeachofthemworkswellforacertainclass actions between all the atoms and electrons present in
Xi of physical systems[2]. the total system. Then one can try to solve Poisson or
Advent of the memory function formalism is a major Schr¨odingerequationrespectively depending onwhether
r
a theoretical progress in this line of thought. It was de- the system is classical or quantum in nature. Such an
veloped and is being used to study the dynamics and microscopicattempt is not only impossible, also it is too
the transport properties of various complex many body complicatedto capture the essentialphysicaldescription
systems[3–8]. This technique relies on the idea of sep- of the system[1]. On the other hand we can build up a
arating the slow and the faster degrees of freedom in a simple description without compromising with the basic
physical system and to systematically calculate the ef- physics as follows. If the moving particle is macroscop-
fects of the later on the former. Separation of scales is ically large and if the velocity of the center of mass is
a very familiar and essential concept in studying various smallcomparedto the velocityofthe fluid molecules,we
physical systems. It is suitable in systems having finite can separate or project out the center of mass coordi-
number of slow modes related to the dynamics of con- nates fromthe restofthe degreesoffreedomof the total
served variables and or the broken symmetry variables. system. In such a situation, we can write its effective
Their studies are often termed as hydrodynamics and equation of motion where the effects of the rest of the
those slow modes are called hydrodynamic modes and degrees of freedom can be termed as “molecular drag”
2
on its motion. The later can be incorporated through a mathematicalsetupforcalculatingthememoryfunction.
drag force. This leads to a very simple and well known Letusstartwithamanybodysystemhavingmacroscopi-
equationofmotionofthedraggedparticleoftheform[9], callylargenumber,i.e. anAvogradonumber(∼1023/cc)
of degrees of freedom and look for its macroscopic prop-
R¨ −γR˙ +F=0. (1)
erties. Our systemcanbe both classicalandquantumin
HereRisthe positionvectorofthe centerofmassofthe nature. A classical system is described by a set of vari-
macroscopic particle of unit mass, R˙ and R¨ represent ables comprise of the particles position and momentum
its time derivative or the velocity and the acceleration variables. Suchasetofpositionandmomentumvariables
respectively and F is the external force. This is indeed is called phase space. In the systems obeying quantum
a major simplification of a very complex system. The mechanics a set of linearly independent operatorsdo the
parameter γ is termed as friction coefficient, viscous co- same job. In that case a mathematical space containing
efficientetc. depending onthe contexts. Itdescribesdis- the set of operators forms a Hilbert space as depicted
sipationortheflowofenergyandormomentumfromthe pictorially in the big green circle in Fig.2. Now under-
coherent to the incoherent degrees of freedom in a sys-
tem. It can also be space and time dependent. However
when the velocity of the particle becomes comparable to (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) Projected Space
that of the fluid molecules, as seen in the part B of the (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) of
Fig.1, turbulence sets in and the idea of separation of (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)M(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)an(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)y (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)Bo(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)dy(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) O(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)p(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)era(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)to(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)r (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) Conserved Variables
scales does not remain obvious. (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)H(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)ilb(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)er(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)t S(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)pa(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)ce(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
Manysuchexamplescanbefoundinthevastliterature (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
on complex systems both in the classical and quantum (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)Q(cid:0)(cid:0)(cid:1)(cid:1) =(cid:0)(cid:0)(cid:1)(cid:1) 1(cid:0)(cid:0)(cid:1)(cid:1)−P(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)P(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
domains. It is to be noted that there is no concept of (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
dissipation in the microscopic principles. In an effective (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
description of a physical system, we observe the system (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
within our desired or convenient time scale and length
FIG.2. Aschematicrepresentationoftheideaofprojectionin
scale. We thus ignore the complete distribution of en-
thememoryfunctionformalism. Herethefullbigcircleisthe
ergy which occurs over a larger time and length scales.
total many body operator Hilbert Space and is equivalent to
Hence we effectively observe dissipation of the momen- identity. TheProjection ofthefullmanybodystatesdefined
tum or energy of the particle as a result of the interac- by few operators representing conserved variables residing in
tion with other fast degrees of freedom. The same pic- the region P. On the other hand the incoherent degrees of
ture emerges in various effective descriptions of interact- freedomlivesinthepartoftheHilbertspacedefinedbyI−Q.
ing systems and the study of the low energy properties
becomes synonymous to finding out the generalized dis- standing the lowenergy consequencesof the interactions
sipative constant or the scattering rates of the collective between such a large number of variables or operators is
excitations. We will see in the later sections that the extremelycomplicated,ifnotimpossible. Weneedmeth-
Memory function approach deals with systematic evalu- ods whicheliminate the technicaldifficulties but capture
ations of the generalized dissipative constant(γ in Eqn. the correct low energy physics.
1) related to the dynamics of generalized slow variables. Basic principle of the memory function formalism is
In this review, we will first address the general as- as follows. Suppose we are interested in studying the
pects of the Memory function formalism in section II. center ofmass motion ofa systemof N number of parti-
We elaborate it further in section III and in section IV. cles. Then we separate or project out the center of mass
Inthosesectionswepresentderivationofthe generalized variable from the others. Here the center of mass vari-
Langevin equation within this formalism and present a able is a macroscopic variable and is defined as a linear
continuedfractionrepresentationofthedynamiccorrela- combination of the microscopic variables. Now in mem-
tionrespectively. InsectionVwediscussitsapplications ory function formalism, it is shown that the effects of
in various electronic systems. Finally in section VI, we the rest of the microscopic variables on the dynamics of
conclude. themacroscopicvariablecanbeestimatedsystematically
and is cast in a so called Memory function in different
systems[12–16]. The reason behind the use of the term
II. PROJECTORS AND MEMORY FUNCTIONS “memory” will be discussed in details in the next sec-
tion. The above discussion is applicable to the quantum
MemoryfunctiontechniquewasintroducedbyZwanzig systems also, except the fact that the classical variables
andMori[3–7]. Themathematicalformalismusedforsys- will be replaced by operators. Since we discuss this for-
tematic calculationofthe memoryfunction is alsocalled malism in context of the electronic systems, we invoke
projection operator method. Following Ref.[10] and [11] quantum mechanics from the very beginning and work
wesummarizetheideaofusingprojectorsandthegeneral with operator language henceforth.
3
Letusconsideranobservablerepresentedbyanopera- Therefore the expression for the correlator can be re-
tor A obeying the Hamiltonian dynamics. To determine written as,
its dynamics, we define the Liouville operatorassociated
1 1
withtheHamiltonian. Mathematicallythelateriscalled R (z)= χ + hA | LA iχ−1R . (10)
ij z ij i z−LQ l lm mj
a super-operator as it acts on operators and produces a lm
X
newoperator. ForasystemwithagivenHamiltonianH,
Above expression can be cast in a matrix notation as
it is defined as,
follows,
dA
LA=[H,A]=−i . (2) zI−Kχ−1 R=χ. (11)
dt
Here [ , ] is the commutator between two operators. The matrix K has(cid:0)the element(cid:1)s defined as,
From Eqn.2, we see that an operator evolves with time z
K =hA | LA i. (12)
as, il i z−LQ l
A(t)=eiLtA(0). (3) Elements of K can be decomposed as,
1
In a many body system we need to quantify the correla- K =hA |LA i+hA |LQ LA i. (13)
il i l i l
tion between various physical quantities represented by z−LQ
variousoperatorsAi. Wecanexpresstheircorrelationin The first part of the above expression is called the fre-
terms of a correlation function matrix R(z), defined in quency matrix and is given as,
frequency space with matrix elements as,
L =hA |LA i. (14)
il i l
R (z)=i dteizthA (t)|A i=i dtei(z−L)thA |A i
ij i j i j Remaining part contains the effects of the faster degrees
Z 1 Z offreedomresidingintheun-projectedpartoftheHilbert
=hAi| Aji, z =ω+iη. (4) space and is termed as the memory matrix. It is defined
z−L
as,
Here η → 0+ is a small positive number, which assures
1
causality. Evaluation of Rij(z) is a many body problem Mil =hAi|LQ QLAli. (15)
and is in general complicated. To simplify the evalua- z−QLQ
tion of the above expression, we invoke the principle of The relation Q2 = Q is used to write it in a symmetric
memory function formalism and introduce a projection form. Thisformisveryinstructive. Theaboveexpression
operator defined as follows[10, 11], tells that the memory function is defined in terms of the
P = |A iχ−1hA |, χ =hA |A i, un-projectedpartofthe|A˙i=L|Aiandtheun-projected
i ij j ij i j partoftheLiouvilleoperatorL,i.e. QLQ. Theprojected
ij
X degrees of freedom during their slow dynamics, can not
=I−Q. (5)
keeptrackofthemovementsofthefastunprojectedpart
Here P separates the operator A s, corresponding to the and treat the later as incoherent excitations. Since the
i
observed macroscopic quantity, from the rest of the mi- memory function consists of the unprojected degrees of
croscopic degrees of freedom and the act of Q is just the freedomonly,itdescribestheeffectsoftheincoherentex-
opposite. A generic projection operator should have the citations on the low energy excitations in a system and
following properties. accounts for the dissipation in the slow degrees of free-
dom. Usingtheaboveexpressionsthecorrelatorbetween
P2 =P, PQ=QP =0, etc. (6)
different components of A can be written in a compact
We also introduce a decomposition L = LP +LQ and notation as,
introduce the identity 1
R(z)= χ. (16)
1 1 1 1 zI−[L+M(z)]χ−1
= − Y . (7)
X+Y X X X +Y Writting in terms of the matrix elements, it takes the
Using the above identity the expression for the time de- form,
pendent correlation function becomes
zδ − [L +M ]χ−1 R (z)=χ . (17)
1 1 1 il is is sl lj ij
Rij(z)=hAi| + LP |Aji. (8) l s !
z−LQ z−LQ z−L X X
(cid:26) (cid:27)
Here we see that any given correlation function can be
Since Q|A i = 0, we can simplify the first term in the
j written in terms of the corresponding memory matrix.
right hand side of the above expression as,
This completes the general description of the memory
1 1 1 function formalism. The use of it to study the electronic
hA | |A i= hA |A i= χ . (9)
i z−LQ j z i j z ij transport will be discussed in later sections.
4
III. GENERALIZED LANGEVIN EQUATION is given as[20],
d
Beforeusing this formalismin caseofelectronictrans- A(t)=iLA(t), A(t)=eiLtA(0). (19)
dt
port,let us elaborateits physicalcontents inmoredetail
in context of an well known physical system. Consider The time derivative of the operator A(t) can be written
the dynamics of a system where few macroscopic slow in terms of its initial value A(0) as,
degrees of freedom are immersed in and interacting with
d
a soup of fast microscopic degrees of freedom. The fa- A(t)=iLeiLtA(0). (20)
mous Brownian motion is such an example[10, 17]. Here dt
a particle is suspended in a fluid and it collides with the
Now we introduce the projection operator acting on an
fluid molecules. As a result it follows a zigzag trajectory
operator B as,
as shown in Fig.3.
hA,Bi
PB = A, P2 =P. (21)
hA,Ai
We insert the identity I=P +Q in Eqn.19, and get,
d
A(t)=ieiLt(P +Q)LA(0). (22)
dt
Nowonwardswedroptheargument0fromA(0)forcon-
venience and thus the first term in the above expression
can be evaluated as,
hLA,Ai
eiLtPLA= eiLtA=ΩeiLtA. (23)
hA,Ai
FIG.3. Motion ofaBrownian particle(Big circle) movingin Here Ω = hLA,Ai is the frequency matrix. To evaluate
hA,Ai
afluid. Reddotsrepresentssmallfluidmoleculesandtheyex- the second term we introduce the identity,
hibit faster movements. The Brownian particle collides with
thefluid particles and follows a zigzag path. eiLt =ei(P+Q)Lt
t
=eiQLt+ dτeiL(t−τ)iPLeiQLt. (24)
To explain such a phenomena, in the classical limit a
Z0
scenario that the particle is experiencing “some random
Applying it to the second term of the Eqn. 22, we get,
force”, is adopted. As a result, for the particle a Newto-
nian equation of motion with phenomenological random eiLtiQLA=eiQLtiQLA
force,mimickingthe kicksfromthefluidparticlescanbe
t
written. Such an equation of motion is called Langevin + dτeiL(t−τ)iPLeiQLtiQLA. (25)
equation[18] and for the simplest case, in one dimension Z0
it takes the following form. We call the first term of the above equation a “force”
which is given as,
d2 d
R(t)=γ R(t)+f(t). (18) f(t)=eiQLtiQLA=eiQLtf(0). (26)
dt2 dt
Here f(0) is a “force” inserted on the slow variable by
Here f(t) is a random force of “white noise” type, i.e.
the incoherent degrees of freedom and f(t) is formers
with correlation hf(t)f(t′)i = γδ(t − t′) and the mass
time propagation. The other term can be evaluated as
of the particle is assumed to be unity. This correlation
follows,
has a delta function in time structure and thus it is fre-
quency independent. Such a random force description t
I = dτeiL(t−τ)iPLeiQLtiQLA
is used quite often and is highly successful in explaining 2
Z0
various complex phenomena. How such a probabilistic t
picture emerges from the microscopic interactions which = dτeiL(t−τ)iPLf(t)
are deterministic in nature, can be addressed within the Z0
memory function formalism[19]. = tdτeiL(t−τ)ihLf(t),AiA
hA,Ai
Let us consider a time dependent operator A(t) which Z0
mimic the velocity of a Brownian particle and consider t hLQf(t),Ai
= dτeiL(t−τ)i A. (27)
its time evolution. The correspondingLiouville equation hA,Ai
Z0
5
Since both Q and L are Hermitian, the above integral continued fraction form. It was first shown by Mori, in
can be written as, his seminal work[7]. From Eqn. 29 we see that, the time
dependence of a general dynamical variable say A(t) is
I =− tdτeiL(t−τ)ihf(t),LQAiA dictated by the correlations of its time derivative A˙(t).
2
Z0 hA,Ai SameshouldholdtrueforA˙(t)anditshighertimederiva-
t hf(t),f(0)i tivesalso[24]. Thisleadstoanewtypeofmomentexpan-
=− dτeiL(t−τ) A
hA,Ai sionas follows. We canrewritethe generalizedLangevin
Z0
t equation (Eqn. 29) as,
=− dτκ(t)A(t−τ). (28)
Z0 dA(t)=iΩ A(t)− tdτκ (t)A(t−τ)+A (t). (30)
0 1 1
This term relates the dissipation in A with the fluc- dt
Z0
tuations in other fast incoherent degrees of freedom.
HereA is termedasthe randomforceandhasthe same
Such a relation is often termed as fluctuation dissipa- 1
symmetry as A˙. If we write its equation of motion, it
tion theorem[21,22]. ItleadstothegeneralizedLangevin
willalsofollowageneralizedLangevinequationinvolving
equation which can be written as,
higher time derivative of A˙. The equation of motion for
d t a nth order “force” An becomes,
A(t)=iΩA(t)− dτγ(t)A(t−τ)+f(t). (29)
dt Z0 A˙ (t)=iΩ A (t)− tdτκ (t)A (t−τ)+A (t),
The force-force correlator γ(t) and its Fourier transform n n n n+1 n n+1
Z0
have in general, complicated time and hence frequency (31)
dependence respectively. Thus it carries the information withn-thorderfrequencyandthen+1-thordermemory
ofthememoryorthehistoryofthepastscatteringevents kernel
andaretermedasnon-Markovianprocesses. Thisiswhy
hLA ,A i hA (t),A i
n n n+1 n+1
the kernel describing the effects of the unprojected or Ω = , κ (t)= (32)
n n+1
hA ,A i hA ,A i
incoherentdegreesoffreedomistermedasmemory func- n n n n
tion. For a specific case when γ(t) = γ0δ(t), i.e. when respectively. This recurrence formula can be cast in a
its Fourier transform is constant, the dynamics becomes singlecontinuedfractionformofthe correlationfunction
memory less and are called Markovian process in statis- as follows,
tical mechanics literature[23]. Let us now compare the
∞
above expression which is obtained from the exact mi- dte−izthA(t);Ai
croscopic description, with that of the Langevin Equa- Z0
tion used froma phenomenologicalconsideration. In the hA;Ai
= . (33)
later case the force f(t) is considered as random with a i(z−Ω )+ ∆1
variance assumed phenomenologically. In principle, f(t) 0 i(z−Ω1)+i(z−Ω∆22)+....
follows deterministic equation of motion and its exact In the above expression ∆ = κ (0). Here the dynamic
n n
evolutionrequiressolutionsofinfinite setofequationsas propertyofasystemiscompletelydescribedbyitsstatic
we will discuss in the next section. This is computation- correlations. The above result is in principle exact. But
ally impossible as no system is completely isolated and the exact evaluation needs the knowledge of the static
thus the “total system” implies the whole universe! To correlations upto infinite order[25]. However depending
a good approximation, it is justified to consider force of on the situation one can truncate the continued fraction
some suitable order as random variables and solving the at some suitable order and get sensible results[26]. A
above equation to get an effective understanding about detailed discussions of its use in various systems are be-
the system dynamics. Findings from such probabilistic yond the scope of this article. Interested readers may
descriptionfitsnicelywithexperimentalfindings. Thisis look at the references [27], [28] and [29] where dynamic
how an effective random or probabilistic description out correlationsincaseofsimple metal, HubbardModeland
of deterministic microscopic principles can emerge in a spin 1 XYZ model respectivelyare castin the continued
2
complex system. However,the originofrandomness in a fraction form.
purely deterministic system is a subtle issue. For more
discussions interested readers can consult[5].
V. APPLICATION TO THE ELECTRONIC
TRANSPORT
IV. CONTINUED FRACTION DESCRIPTION
In the previous sections we described various aspects
To elaborate on the memory function description fur- of the memory function approach. Thus we set up the
ther we discuss how any dynamical correlation function stageforusingittostudythe dynamicaltransportprop-
can be expressed in terms of the static correlations in a erties of various electronic systems. Here our focus is on
6
the time evolutionofthe currentoperatorandthe corre- simplified picture, it takes the following form,
lation of its various components in a generic many body
1
system. In our discussions on electronic systems, first M(z)=hJ|LQ QLJi
z−QLQ
we assume that the momentum is the only nearly con-
1
served quantity in the system1. Thus there is only one =hJ˙|Q QJ˙i. (35)
z−QLQ
slow mode associated with this conservation law. We
study the momentum relaxation of a charged particle To get some qualitative idea, we can opt for a “high fre-
under external perturbation. In this case the projector quency expansion”of the above expression. We consider
operator is defined solely in terms of the current oper- anenergyscalez =hJ˙|QLQJ˙i/χ. Aslongasz <<|z|,
0 0
ator. This assumption holds good if there is no other we can expand inverse operator 1 in a series and
z−QLQ
slow modes associated with any other conservation law can rewrite the memory function as follows,
orbrokensymmetrythatcouplestothechargedegreesof
1 1 1
freedom[30–33]. However for simplicity we stick to this M = hJ˙|Q 1+ QLQ+ QLQQLQ+··· QJ˙i
z z z2
picture for the time being and will generalize it in the (cid:18) (cid:19)
1 1
later subsections. = hJ˙|QJ˙i+ hJ˙|QLQJ˙i
z z2
Nowwecanstartwiththeexpressionformemoryfunc-
1
tion as defined in Eqn.15. In certain situations, we can + hJ˙|QLQLQJ˙i+··· (36)
z3
evaluate the expressionin the spirit of perturbation the-
ory. Memory function can be viewed as “the self en- Here we use the fact Q2 = Q and the above expansion
ergy” of the current-current correlation function. It has can be termed as a high frequency expansion. Its va-
an added advantage that such “self energy” calculation lidity will depend on how small or large the z /|z| is.
0
doesnotrequirevertexcorrection. Thelaterisextremely Since the time derivative of two different orders are un-
importantandproblematicwhenthecurrent-currentcor- correlated in a system having time reversal symmetry,
relationisexpressedthroughtherenormalizedsinglepar- i.e. hJ|J˙i,hJ˙|J¨i=0, as proved in the Appendix A,
ticle propagators [34]. Now, for further simplicity, we
1 1
consider the case of an one component current operator M(z)= hJ˙|J˙i+ hJ¨|J¨i+··· (37)
z z3
J and replace |Ai by |Ji. To clarify more, in an elec-
Clearly this expansion will hold good in high frequen-
tronic system |Ji ≡ J|Φ i, where |Φ i is the electronic
0 0
cies and will breakdown below certain energy scale set
ground state. The correlation denoted by hJ(t)|Ji , can
by the incoherentpartofthe Hamiltonian. Here J is the
be used to represent correlation of the form h[J(t),J]i
current operator. Now its time derivative J˙ = [J,H =
withoutchangingthe formoftheEqn.16. We choosethe
H +H′]=[J,H′]isproportionaltothecouplingstrength
later form, as it is used to describe the response func- 0
g (say) with the dimension of energy of different inter-
tion. We focus on the response of an electronic system
actions. Thus the above expansion can be viewed as an
under anexternalelectric fieldand the relevantquantity
is dynamic conductivity σ(z) and is given in terms of a expansionintermsof g2. Forveryweakinteractionsone
z2
commutator correlationof the current operators. In this can truncate the above expressionat the first term itself
case, Eqn.16 can be written as, and calculate the conductivity. However this perturba-
tion theory is different form the diagrammatic perturba-
1 tion theory that incorporates interaction effects through
σ(z)= χ. (34)
z−M(z)/χ single particle self energy and vertex corrections[36].
It is to be notedthat we assumetime reversalinvariance
so that hJ|L|Ji = hJ˙|Ji = 0, i.e. the generalized fre- A. Weak coupling theory
quency vanishes. Here we see that using memory func-
tion formalism the dynamic susceptibility can be writ- The memory function formalism was first used in a
ten in an Extended Drude form, frequently used by the systematicwaytocalculatetheelectricalconductivityin
experimentalists[35] to explain any non-Drude dynamic case of simple metals with various interactions by G¨otze
conductivity. Here we see that the memory function or andW¨olfle[36]. Similarapproachisusedbymanyothers
the generalized many particle(two particle in this case) in this context[37–43]. Their approach can be summa-
selfenergydefinedbyEqn.15isthemostimportantquan- rizedasfollows. Accordingto the linearresponsetheory,
tity to determine the dynamic conductivity. Within our the dynamical conductivity is defined as[34, 44–46],
1 ω2
p
σ(z)=−i χ(z)+i . (38)
z 4πz
1 It means that the related correlation function has a very slow Here ω2 = 4πN e2/m is the square of plasma frequency
decay,e.g. oftheforme−τt,withtherelaxationtimeτ →∞. whereepelectroneiccharge,m electronmassandN is the
e
7
electron density, z is the complex frequency and χ(z) is Hamiltonian has two parts, H the free part or the ki-
0
the current-currentcorrelation function defined as, netic part and an interaction part. While the first part
commutes with the current operator, the later part does
∞
χ(z)=hhJ;Jiiz =i eizth[J(t),J]i, (39) not. Thus A= J˙ = [J,H0+H′]= [J,H′] is determined
Z0 by the interaction part only and is different for differ-
ent types of interactions. Within this approach, they
where J = kev(k)c†k,σck,σ is the current density and calculated the frequency dependent conductivity with
v(k) is the velocity dispersion. Here [J(t),J] denotes various interactions such as electron-phonon, electron-
P
the commutator, h...i denotes the ensemble average at impurity, electron-magnetic impurity, scattering with lo-
temperature T and hh...ii denotes the Laplace transform calizedmodesetc. Toillustratetheirworkfurtherwewill
of the ensemble average. discussthesimplestcaseofelectron-impurityinteraction.
According to the G¨otze and W¨olfle approach[36], the
In this case the interaction part of the Hamiltonian is
memory function is defined as
given as,
χ(z) 1
M(z)=zχ0−χ(z), (40) H′ = N hk|U|k′ic†kσck′σ. (47)
j
X
where χ corresponds to the static limit of correlation
0
Here U denotes the impurity potential. In this case the
function(i.e. χ =N /m)[36]. Usingthis,theexpression
0 e
j-th component of the force operator A is given as,
fordynamicalconductivity inEqn.(38)canbe castin an
Extended Drude form as,
1
i ω2 Aj = N hk|U|k′i[vj(k)−vj(k′)]c†kσck′σ. (48)
σ(z)= p . (41) X
4πz+M(z) Herev (k)isthevelocityoftheparticlewithmomentum
j
k in the j−th direction. The force-force correlation in
In Ref.[36], an expansion for M(z) =
the case of free electron is estimated as,
zχ(z) 1+ χ(z) −··· is used. Basis of this assumption
χ0 χ0 φ(z)=hhA;Aii
is the(cid:16)smallness of th(cid:17)e contribution from the interaction z
part as compared to the kinetic energy of free electrons. =(2c/3m2N) |hk|U|k′i|2(k−k′)2
Using this expansion and on keeping the leading order kk′
X
term, the memory function M(z) can be written as f(ǫk)−f(ǫk′)
× . (49)
z−ǫk+ǫk′
χ(z) hhJ;Jii
z
M(z)=z =z . (42)
χ χ Fromthisexpressiontheimaginaryortheabsorptivepart
0 0
of the memory function can be estimated as,
To compute memory function, we need hhJ;Jii which
z
1
by using equation of motion is M′′(ω)=C× |hk|U|k′i|2(k−k′)2
N2
kk′
zhhJ;Jii =h[J,J]i+hh[J,H′];Jii . (43) X
z z ×[f(ǫk)−f(ǫk′)]δ(ω−ǫk+ǫk′)/ω. (50)
As the firstterm of r.h.s is zero, hence the above expres- For momentum independent U, i.e point impurity
sion is equivalent to second term which can be further
calculated by applying equation of motion. M′′(ω)=C′(Uρ )2ǫ , (51)
F F
zhh[J,H′];Jiiz =h[[J,H′],J]i−hh[J,H′];[J,H′]iiz. where ρF and ǫF are the density of states at the Fermi
(44) surfaceand the Fermienergy respectively. For ω <<ǫ ,
F
For z = 0, h[[J,H′],J]i = hh[J,H′];[J,H′]iiz=0. Thus, the imaginary part of the memory function is indepen-
the memory function M(z) becomes dent of the frequency and the result is identical to the
Drude result. On the other hand, if the impurities are
φ(0)−φ(z)
M(z)= . (45) spatially extended,
zχ
0
1
Here φ(z) (called as correlationfunction) is defined as M′′(ω)≡ ∼vF dΩσsc(1−cosθ). (52)
τ
Z Z
φ(z)=hh[J,H′];[J,H′]iiz. (46) In the above expression the differential scattering cross
section is defined as,
Here the current operator Ji = vi(k)c†kσckσ and its
derivative or “force” A = J˙ = [JP,H]. Here the total σsc(Ω)=(πkF)2|ρFU(kF−k′F)|2. (53)
8
For interactions with non-magnetic impurity we see that On the other hand Kubo Mori relaxation function is de-
theresultsareidenticaltothesingleparticlecalculations fined as,
with vertexcorrections. This is indeeda benchmark and
Φ (t−t′)≡((A(t)|B(t′)))=−iΘ(t−t′)(A(t)|B(t′)).
major success of this formalism. In other cases there AB
(56)
are deviations from the Drude formula. They argued
The last expressionis called the Kubo-Moriscalar prod-
that these discrepancies are because of spin-flip scatter-
uct and is defined as[9],
ing in a magnetic field, because of resonance scattering,
because of phonon creation at low temperatures, and β
because of breaking of the screening cloud attached to (A(t)|B)= dλhA(t−iλ)Bi. (57)
chargedimpurities respectively. However,this versionof Z0
the memory function approach to calculate the dynamic It can be shown that [60],
conductivity is somewhat limited. It is designed for sim-
ω((A|B)) =(A|B)+hhA|Bii
ple metal, i.e. for weakly-interacting electrons with very ω ω
weakelectron-phononorelectron-impurityorotherinter- =−hhA|Bii0+hhA|Biiω. (58)
actionsandnonexpandabletothecasesofstronginterac-
Now we apply projection operator technique to the
tions. Lifting these limitations, as required for more ex-
Green’s function,
oticsystemslikestrangemetalphaseincupratesnearop-
timaldoping,andothersneedsubstantialimprovements. Gk,k′(t−t′)= Ak(t)|A†k′(t′) . (59)
DD EE
It has an equation of motion
d
B. Strong coupling extension idt Ak(t)|A†k′(t′) =δ(t−t′) [Ak,Bk′]η
DD EE D E
+ A˙ (t)|A† (t′) . (60)
k k′
A large volume of works[47–59] on the applications
DD EE
of the memory function in the case of strongly inter- HereA˙ (t)=[A (t),H]. Nowweextractthe linearterm
k k
acting electronic systems is being done by Plakida and in the equation of motion as
his collaborators based on the mathematical formalism
develop by Tserkovnikov[60]. We see in the G¨otze and iA˙k(t)=[Ak(t),H]= Ek,qAq+Zkir. (61)
W¨olfleformalism,thatthedynamicalconductivityorthe q
X
current-currentcorrelationhJ;Ji can be calculated with The irreducible part Zir is defined by the orthogonality
theknowledgeofhJ˙;J˙iwhichiscalculatedbysimpleper- condition k
turbation theory. However it will be seen in this section
that,theirapproachmissessomesubtlepointswhichmay Zir,A† =0. (62)
k k′
notaffecttheresultsintheperturbativelimitbutcanbe (cid:28)h iη(cid:29)
problematic in the case of strongly correlated systems. This defines the frequency matrix
Plakida et al. refined the relation between the memory
function and the J˙- J˙ correlation as follows. First, a re- Ekq = [[Ak(t),H],Ak′]η Ik−′1q, Ik′q = [Ak,Ak′]η .
lation between two time retarded Green’s function and Xk′ D E D E
(63)
the Kubo-Morirelaxationfunction can be established as
Upon Fourier transform, Eqn.60 gives,
follows. The two time retarded Green’s function for two
Heisenberg operators A and B are defined as,
Gk,k′(ω)=G0k,k′(ω)+ G0k,k′(ω)Iq−q1′ Zqir′|A†k′ .
GrAB(t−t′)≡hhA(t)|B(t′)ii Xqq′ DD EE(64)
=−iΘ(t−t′)hA(t)B(t′)−ηB(t′)A(t)i. Here the zeroth order Green’s function is given as,
(54) G0k,k′(ω)= ωδ Iq−k′E . (65)
kq kq
q
Here the step function Θ(t) = 1 for t > 0 and Θ(t) = 0 X
for t < 0. The h........i represents the thermal average Thisdefinestheexcitationspectruminthemeanfieldap-
i.e. Tre−βH(..) with β as the inverse temperature and proximations. Inordertodeterminethemanybodypart
η =± for Bosons and Fermions respectively. The above of the Green’s function Zir(t)|A† (t′) , one needs to
q′ k′
Green’s function follows an equation of motion, differentiateitwithrespDeDcttot′ andafteErEtakingFourier
transform one obtains,
d
i hhA(t)|B(t′)ii=δ(t−t′) [A,B] + A(˙t)|B(t′)
dt η Gk,k′(ω)=G0k,k′(ω)+ G0k,q(ω)Tqq′(ω)G0q′,k′(ω). (66)
D E DD (5E5E) qq′
X
9
The scattering matrix appeared above is defined as iM′′(ω). Again following the general procedure as dis-
cussed earlier, the time derivative of the response func-
Tkk′(ω)= Ik−,q1 Zqir|(Zir)†q′ Iq−′k1′. (67) tion,
Xqq′ DD EE
Φ (t−t′)=((J(t)|J(t′))), (72)
JJ
Now if we define the self energy as,
followed by the Fourier transform gives,
Tkk′(ω)=Σkk′(ω)+ Σkq(ω)G0q,q′(ω)Tq′k′(ω). (68)
Xqq′ ΦJJ(ω)=Φ0JJ(ω)+Φ0JJ(ω)TJJ(ω)Φ0JJ(ω). (73)
Then the Green’s function can be cast in a Dyson form
Here Φ0 (ω)= χ0 andin this casethe scatteringmartix
as, JJ ω
is given as,
Gk,k′(ω)=G0k,k′(ω)+ G0k,q(ω)Σqq′(ω)Gq′,k′(ω). (69) 1 1
qq′ TJJ(ω)= J˙|J˙ . (74)
X χ0 ω χ0
(cid:16)(cid:16) (cid:17)(cid:17)
This tellsthatthe generalizedself-energyorthe memory
In order to express the Eqn.73 in the form of Eqn.71 we
function is given by the proper part of the scattering
needthefollowingrelationbetweenthememoryfunction
matrix. Thus it can be written as,
and the scattering matrix.
proper
Σkk′(ω)=Xqq′ Ik−,q1DDZqir|(Zir)†q′EE Iq−′k1′. (70) TJJ(ω)=−χ10 M(ω)+M(ω)Φ0JJ(ω)TJJ(ω) . (75)
IfwerecalltheDysonequationsintheelectronicGreen’s (cid:2) (cid:3)
The above equation tells that the memory function for
function[34], we can easily identify that the T(ω) is the
theelectricalconductivityisequivalenttotheirreducible
generalized many body or multi-particle scattering ma-
or the proper part of the force-force correlator,i.e.,
trix while the memory function or the generalizedmulti-
particle self energy Σ(ω) is given by the proper part of 1 proper 1
M(ω)= J˙|J˙ . (76)
T(ω). Diagrammatically it means that the part of the χ0 ω χ0
scattering matrix not connected by a single relaxation (cid:16)(cid:16) (cid:17)(cid:17)
function. To clarify this statement, a diagrammatic de- These authors use this improved definition of the mem-
scriptionforthe samein caseofsingle particleelectronic ory function to calculate dynamical quantities in various
Green’s function or propagator in a coupled electron- systemsuch as t-J model in contextof high temperature
Boson system is presented in Fig.4. superconductors and many others in the references cited
at the beginning of this subsection.
As we see, the evaluation of memory function requires
evaluation of J˙ = [J,H′] and its correlation. Thus any
Σ T
= scheme based on the memory function depends on the
interaction part of the Hamiltonian and the algebra fol-
lowed by the operators corresponding to the observables
in a system. Also an evaluation of correlatorof the form
T = + T shown in Eqn.76 needs some approximations with suit-
able justifications.
Plakida et al. consider the t−J model[61, 62] for the
FIG. 4. A diagrammatic description of the relation between
strongly correlated normal phase of the cuprate super-
the single particle self energy and the T-martix for electron-
conductors to apply their formalism. The authors use
Bosoninteractionisshownhere. Thesolidlinesrepresentsthe
electronpropagatorwhilethedashedlinesrepresentsBosonic it to calculate both the optical conductivity and the dy-
propagators. namic spin susceptibility in this model. Due to the in-
volved mathematical complexity, the detail description
Now to calculate the conductivity, we need to focus on of the application in this model is beyond the scope of
the current current correlations. In this case the rele- thisreview. However,thefindingsfromthisapproachare
vantresponsefunctionΦ (ω)andthe memoryfunction comparable with the other analytical methods and also
JJ
or the corresponding multi-particle self energy M(ω) is comparable with the experiments. Interested reader can
related to each other as, look at the references [52] and [57], where Plakida has
nicely reviewed the related works.
χ
0
ΦJJ(ω)=((J|J))ω = ω+M(ω). (71) Similarapproachisalsousedbysomeotherresearchers
to calculate the spin susceptibility in strongly correlated
Here χ = χ (0) and the memory function has both electronic systems[63–70]. Since we focus on the electri-
0 JJ
the real and imaginary parts, i.e., M(ω+ıδ)=M′(ω)+ cal conductivity here, we skip those discussions here.
10
C. Effects of additional slow modes insteadofstandardperturbationtheorytodeterminethe
relevant response functions. In their approach they in-
Untilnowweconsidersystemswheretheelectricalcur- clude the effects of other slow modes such as charge dif-
rentofthemomentumwastheonlyslowmode. However fusions and heat diffusions in their formalism. Within
in many systems we need to consider many other slow this formalismthey presentsomeexplanationofrecently
modes which couple to the electrical current. For ex- observed anomalous behavior in the hall angle in the
ample in a system where there is charge conservation, strangemetal phase[71]. In summary this approachcon-
alongwiththedriftcurrent(relatedtothemomentumof tains the essential complexity of a typical non-Fermi liq-
the chargeparticles), there is another slowmode namely uid. It provides a systematic way of including various
charge diffusion or diffusion current mode[30–33]. The slow modes within this approach and thus very promis-
later is also a slow mode and couples to the electric cur- ing.
rent. In such a system, the projected space along with
theelectricalcurrentalsocontainsdensityfluctuationop-
erators. Following Ref.[30] we define, D. Comments on few recent works
J (q)=ρ(q) and J (q)=J(q). (77)
0 1 In this subsection we will qualitatively discuss few of
Due to the charge conservation,the density and the lon- the presentdayactivities basedonmemoryfunction for-
gitudinal part of the current operator are related as, malism. We aim to give a flavor of the present day im-
portance of this formalism and keep the discussion very
LJ =−qJL. (78)
0 brief. For details, readers are advised to look at the ar-
ticles cited in appropriate paragraphs.
Sincethenumberofslowmodesismorethanone,inthis
Holographic approach: Since the memory function for-
case,the memoryfunction takes a matrixformas shown
malism does not invoke single particle picture to calcu-
in Eqn.15. Here we consider two slow modes, hence the
late electrical transport. In principle it can be used to
memory matrix has a 2×2 structure and is defined by
calculate transport properties even when quasi-particle
four correlation functions, namely R = ρ−ρ, R =
00 01
picture is not valid. Recently many researchers used
ρ − J, R = J − ρ and R = J − J. However, the
10 11
both the holographic ADS-CFT principle to study elec-
density fluctuations couple only to the longitudinal part
tronic transportin a similar situation, namely 2dmetals
ofthecurrentandthusinthiscaseEqn.17canbewritten
near quantum critical point[31]. These models include
in a set of two decoupled equations as follows,
the coupling between various slow modes and thus pro-
1
duces various non-Fermi liquid transport behavior. To
zδ − L +ML χ−1 RL(z)=χ ,
il is is sl ij ij check the consistency, results are often compared with
" #
Xl=0 Xs (cid:2) (cid:3) thecomplimentarymemoryfunctioncalculationsandin-
z− L +MT χ−1 RT (z)=χT . (79)
11 11 11 11 11 terestingly they have very good agreements.
Hereχij =hJi|(cid:2)Jjiis(cid:2)thestaticc(cid:3)orrel(cid:3)ationfunction. Now Non-equilibrium steady state: Some of the present
the continuity equation 78 and the time reversal invari- authors[72] used memory function formalism to study
ance in the system tell that J |JL ∼ J |J˙ = 0. electronic transport in non-equilibrium steady state,
0 1 0 0 wherethe electronandthe phonontemperaturesaredif-
Moreover, since LJ0 ∝ J1L, th(cid:10)e first(cid:11)comDponentEof the ferent. They consider non-equilibriumrelaxationof elec-
“four force” lies within the projected space and thus its trons due to their coupling with phonons in a simple
unprojected part i.e. QLJ0 is identically zero. This sim- metal. In their model electrons are living at a higher
plifies this picture drastically as it leads to temperature than that of the phonon bath, mimicking a
M =ML =ML =0. (80) non-equilibrium steady state situation. They show that
00 01 10 the dc scattering rate at high temperatures and optical
ThusonlyM =M survivesanditcanbecalculatedus- scattering rate at high frequencies, are independent of
11
ing Eqn.15 and assuming Q≈1[30, 31]. Once the mem- thetemperaturedifferencebetweentheelectronsandthe
oryfunctionisdeterminedsuitably,thecoupledequation phonons is found in this work. The present formalism
can be solved and the effects of charge diffusion can be forms a basis which can also be extended to study hot-
discussed in varieties of systems as done in references electron relaxation in more complex situations
cited in this section. Scattering rates in the gapped system: In an another
In this connection we can mention the works of Lu- work two of the present authors[73] calculated General-
cas and Sachdev[31, 33]. In their work they focus on izedDrudescattering(GDS)rateforthecaseofelectron-
the magneto-transport in strange metals. Their system phonon scattering in metals with a gap in the electronic
is a 2D quantum critical metal under an external mag- density of states at the Fermi energy. The resulting
neticfield. Theyconsidersystemswhereelectronicquasi- GDS is compared with a recent one by Sharapov and
particles are absent. Thus they use memory formalism Carbotte [74] obtained through a different setup. They