Linear Response and Dynamical Correlation Functions
    5.1  Dynamics of the Statistical Operator: von Neumann's Equation
    5.2  Linear Response: Dynamical Susceptibility
    5.3  Spectral Representation
    5.4  Symmetries
    5.5  Mori's Scalar Product
    5.6  The Liouville Operator

List of Figures

    1  Dynamical processes in physical systems
    Integration contour for the Θ-function

5  Linear Response and Dynamical Correlation Functions

Many dynamical processes induced by external perturbations of physical systems can be divided into three categories, namely (i) driving by external forces, (ii) relaxation from a partial to a global equilibrium, and (iii) probing by scattered 'particles'. See fig. 1
Figure 1: Dynamical processes in physical systems
All of these experimentally relevant situation can be described in terms of certain so-called correlation functions all of which are intimately related to each other.

For simplicity of notation we set = kB =1 in the remainder of this chapter.

5.1  Dynamics of the Statistical Operator: von Neumann's Equation

Consider a quantum system
H ~ =H+W(t) (1)
and an arbitrary ensemble, described by a statistical operator
ρS (t)= n ρn |n,tn,t|, (2)
where |n,t set the
microstates in the Schrödinger picture with probabilities ρn . The time evolution of the microstates is governed by Schrödigner's equation
i t |n,t= H ~ |n,t. (3)
From this we get
von Neumann's equation of motion for ρ(t) in the Schrödinger picture
t ρS (t)=-i n ρn [( H ~ |n,t)n,t|-|n,t(n,t| H ~ )]=-i[ H ~ , ρS (t)]. (4)
This can be translated into
Heisenberg's picture AH (t)= ei H ~ t A(t) e-i H ~ t , |n,t S = ei H ~ t |n,t=|n,0 S easily
ρH (t)= ρS (0), (5)
and finally in the
Dirac or interaction picture AD (t)= eiHt A(t) e-iHt , |n,t D = eiHt |n,t
t ρD (t) = iH ρD (t)-i ρD (t)H-i[H+ WD (t), ρD (t)] = -i[ WD (t), ρD (t)],       (6)

where WD (t)= eiHt W(t) e-iHt is the interaction in Dirac's picture.
Eqn. (6) seem particularly suited for perturbative approximations since the r.h.s. is proportional to W. We consider a case, in which W(t) is adiabatically switched in starting from t=-
W(t-)=0   öff" . (7)
Therefore at t=- the system is in
equilibrium and
[ ρS (-),H]=0, (8)
with a
canonical distribution function
ρ ρS (-)= 1 Z e-βH = ρD (-), (9)
where the last equality follows from eqn. (8) and AD (t)= eiHt A(t) e-iHt . From eqn. (6) we therefore get
ρD (t) = ρ-i - t [ WD (t'), ρD (t)]dt= = ρ-i - t [ WD (t'),ρ]dt+O( W2 )       (10)

5.2  Linear Response: Dynamical Susceptibility

Let us consider a situation in which the perturbation W(t) is due to generalized external forces
W(t)=- ν Aν fν (t), (11)
where the fν (t) correspond to the forces and the operators Aν refer to the generalized coordinates. Many such pairs of ( Aν , fν (t)) are relevant in practical applications: (real space, momentum), (magnetization, magnetic field), (electric dipole, electric field), (electric current density, vector potential), (charge density, electric field), and so on. We now consider the response of the expectation value of Aμ + (t) to linear order in the W
δ Aμ + (t) = Aμ + (t)- Aμ + (-) = Tr{ AμD + (t)( ρD (t)- ρD (-))} = -i - t Tr{ AμD + (t)[ WD (t'),ρ]}dt+O( W2 ) = i ν - t Tr{ρ[ AμD + (t), AνD (t')]} fν (t')dt' = i ν - t [ AμD + (t), AνD (t')] ρ fν (t')dt',       (12)

where we have dropped terms of O( W2 ).

In the remainder of section 5 time dependencies will usually refer to the Dirac picture, or speaking differently the Heisenberg picture for W=0. Moreover statistical averages will be formed w.r.t. to the equilibrium statistical operator ρ at t=- from eqn. (9). We will therefore drop the subscripts ' D' and ' ρ' in expressions like eqn. (12).

Using eqn. (12) we define the dynamical susceptibility
χμν (t,t')=iΘ(t-t')[ Aμ + (t), Aν (t')]2iΘ(t-t') χμν '' (t,t')., (13)
where we have introduced to so-called spectral function χμν '' (t,t'). Because of the Θ-function we may write
δ Aμ + (t)= ν - χμν (t,t') fν (t')dt'. (14)
Obviously this extends the meaning of a susceptibility from static thermodynamics to dynamical phenomena. If we assume that the coordinates Aμ are time-independent in the Schrödinger picture, then we have
[ Aμ + (t), Aν (t')] = Tr{ρ[ eiHt Aμ + e-iHt , eiHt' Aν e-iHt ]} = Tr{ρ[ eiH(t-t') Aμ + e-iH(t-t') , Aν ]} = [ Aμ + (t-t'), Aν ],       (15)

i.e. the dynamical susceptibility is a function of the time-difference only
χμν ('') (t,t')= χμν ('') (t-t',0) χμν ('') (t-t'). (16)
Example 1 Let us consider a harmonic oscillator under the influence of an external driving force which acts on its displacement coordinate
H ~ = ω0 b+ b+( b+ +b)f(t)       (17) =     H     +       W(t)       .

For bosons we ave [b, b+ ]=1. Note that
b+ +b r ^ , (18)
where r ^ = r ^ + is the displacement coordinate of the oscillator. The time dependence in the Dirac picture is obtained from bD (t)= eiHt be-iHt , or
t bD (t) = ieiHt [H,b] e-iHt =i ω0 eiHt [ b+ b,b] e-iHt = i ω0 eiHt ( b+ [b,b]+[ b+ ,b]b) e-iHt = -i ω0 eiHt be-iHt =-i ω0 bD (t).       (19)

This DEQN can be solved by direct integration and taking into account that bD (0)=b
bD (t) = e-i ω0 t b       (20) bD + (t) = ei ω0 t b+       (21)

where the hermitean conjugate has yield a corresponding equation for the creation operator. Inserting this into eqn. (15) we get an expression for the displacement susceptibility
χrr (t) = iΘ(t)[ ei ω0 t b+ + e-i ω0 t b, b+ +b] = iΘ(t)( ei ω0 t [ b+ ,b]+ e-i ω0 t [b, b+ ]) = iΘ(t)( e-i ω0 t - ei ω0 t ).       (22)

Note in passing that it is due to the particular simplicity of this example, that the dynamical susceptibility is completely temperature independent.

5.3  Spectral Representation

The description of dynamical phenomena involves many different types of functions, mathematically similar to eqn. (13). As we continue we will find, that these function are all determined by the spectral function. Let us consider the Laplace transform of χμν (t), dropping the subscripts for simplicity
χ(z) = 0 dt eizt χ(t)= - dt2 ieizt Θ(t) χ'' (t),       (23)

where Im(z)>0. Note that the following applies to
any pair of functions χ(t) and χ'' (t) related trough eqn. (13). Now, from the residue theorem (see fig. 2)
Figure 2: Integration contour for the Θ-function

- dω π eiωt ω-z ={ 2 ieizt t>0 0t<0 , (24)
i.e. we may write
χ(z)= - dt χ'' (t) - dω π eiωt ω-z = - dω π χ'' (ω) ω-z , (25)
where χ'' (ω) is the Fourier transform of χ'' (t). The preceding equation is called a
spectral representation of χ(z). It is used to define the retarted / advanced dynamical susceptibilities
χR/A (u)=χ(u±i 0+ )=[P - dω π χ'' (ω) ω-u ]±i χ'' (u) χ' (u)±i χ'' (u), (26)
where we have used 1/(x±i 0+ )=P1/x±iδ(x), moreover we have defined the real part χ' (u). Eqn. (26) is called the
Kramer-Kronig transformation(representation) of the retarded / advanced susceptibility. Note that while in the derivation of eqn. (25) we needed that Im(z)>0, we may nevertheless 'use' eqn. (25) also as a definition in for the case of Im(z)<0. That is how we mathematically extend its usage in the definition of χA (u).
With eqn. (16), eqn. (14) is of convolution form and suggest a simple Fourier transform. However some care has to be exercised because of the Θ-function which renders the direct Fourier transform of χ(t) delicate. Consider first
- dt eiωt 2iΘ(t) χ'' (t)= - eiωt - dω' π eiω't ω'-i 0+ χ'' (t)dt       (27) = - dω' π 1 ω'-i 0+ - ei(ω+ω')t χ'' (t)dt= - dω' π χ'' (ω') ω'-(ω+i 0+ ) = χR (ω).

I.e., the because the Θ-function is a distribution (like the δ-function), the Fourier transform of χ(t) is
not simply χ(ω) but χR (ω). Using this, eqn. (14,16), and the convolution-theorem, and re-introducing the subscripts we get
δ Aμ + (ω)= ν χμν R (ω) fν (ω). (28)
Apart from the case of a driving force at frequency ω, as in eqn. (28), another important case is that of a perturbation switched on adiabatically i.e. infinitesimally slow up to t=0
fν (t)=Θ(-t) e 0+ t fν . (29)
In that case, one is interested in the response at t=0
δ Aμ + (t=0) = ν - 0 χμν (0-t') e 0+ t' fν dt'= ν 0 χμν (t') e- 0+ t' dt' fν = ν χμν R (ω=0) fν ν χμν Iso fν       (30)

χμν Iso is called the
isolated/adiabatic susceptibility. It is very tempting to associate this susceptibility with the isothermal susceptibility
χμν T = Aμ + fν . (31)
from static thermodynamics. However this is wrong, as χμν Iso χμν T in general, as we will see later.
Example 2 Let us apply the preceding to the results from example 1. From eqn. (22) we get
χrr '' (t)= 1 2 ( e-i ω0 t - ei ω0 t ), (32)
and from that
χrr '' (ω) = 1 2 - eiωt ( e-i ω0 t - ei ω0 t )dt = π[δ(ω- ω0 )-δ(ω+ ω0 )],       (33)

i.e., the Fourier transform of the spectral function has 'peaks' at the excitation energy of the system. Therefore it is frequently called 'the spectrum' of the system. The retarded / advanced susceptibility follows from the spectral representation eqn. (25)
χrr R/A (ω) = - δ(ω- ω0 )-δ(ω+ ω0 ) ω'-ωi 0+ dω'= = 2 ω0 ω0 2 -(ωi 0+ )2 ,       (34)

from which we can deduce the isolated susceptibility
χrr Iso = 2 ω0 . (35)
For ω0 0 the system gets 'soft', i.e. it is physically clear, that in this situation its response to an adiabatic perturbation will diverge - as in eqn. (35).
Exercise 3 Assume χ'' (ω)=21- ω2 ·Θ(1- ω2 ). Calculate χ(z) from eqn. (25). Verify, that eqn. (26) is satisfied, regarding χ'' (ω). Plot χ'('') (ω). Discuss the large- ω behavior of χR (ω).

5.4  Symmetries

The symmetries of χμν ('') (ω) are of great importance. For eg., they serve as consistency checks, both for experiments and theories, they restrict the allowed forms that susceptibilities can take on, and they lead to reciprocity relations between various types of responses, just to name a few. To simplify the notation we define
Aμ + = A μ (36)
Complex conjugation
χμν '' (ω ) =- χ μ ν '' (-ω). (37)
χμν '' (ω ) = [ - eiωt 1 2 [ Aμ + (t), Aν ]dt] = - e-iωt 1 2 [ Aμ + (t), Aν ] dt       (38) = - e-iωt 1 2 [ Aν + , Aμ ++ (t)]dt=- - e-iωt 1 2 [ Aμ ++ (t), Aν + ]dt=- χ μ ν '' (-ω)

where we have used A = A+ and ( Aμ + (t ))+ = Aμ ++ (t).
χμν '' (ω)=- χ ν μ '' (-ω). (39)
χμν '' (ω) = - eiωt 1 2 [ Aμ + (t), Aν ]dt=- - eiωt 1 2 [ Aν , Aμ + (t)]dt       (40) = - - eiωt 1 2 [ Aν ++ (-t), Aμ + ]dt=- - e-iωt 1 2 [ Aν ++ (t), Aμ + ]dt=- χ ν μ '' (-ω).

For Aμ = Aν A and A+ =A we get χ'' (ω)=- χ'' (-ω), i.e. in that case the spectral function is
real and odd in frequency.
Time inversion is a linear operation which needs some introduction, which for completeness will be included here. In its coordinate representation any solution ψ(r,t) of the one-particle Schrödinger equation has the following symmetry
i t ψ(r,t)=[ p2 2m +V(r)]ψ(r,t)      i t ψ (r,-t)=[ p2 2m +V(r)] ψ (r,-t), (41)
which can easily be generalized to many particles. I.e. for states "inversion of time + complex conjugation" is a symmetry of the Schrödinger equation. Under time inversion the operators 'space' r and 'momentum' p change according to
Πr Π-1 =r      Πp Π-1 =-p. (42)
For [r,p] to be invariant under Π, the following has to hold
-i = [r,-p]=[Πr Π-1 ,Πp Π-1 ]=Π[r,p] Π-1 =Πi Π-1 -iΠ = Πi,       (43)

which is generalized into the requirement
a Π=Πa, (44)
i.e. time inversion is
While the adjoint of any linear operator is given by
μ|Aν=ν| A+ μ = A+ μ|ν, (45)
(where we have used von-Neumann's notation for scalar products instead of Dirac's) it will turn out to be reasonable to
define the so-called anti-adjoint
Π+ μ|ν μ|Πν. (46)
From this and the requirement that the norm is conserved under Π - which is
consistent eqn. (45) we get
Πμ|Πμ = Π+ Πμ|μ μ|μ =μ|μ       (47) Π+ Π=1 Π-1 Π=Π Π-1 = Π+ Π=Π Π+ =1.       (48)

It is important to realize that in view of eqn. (41) it is the
2nd equal sign in eqn. (47) which is consistent with eqn. (46) but not with eqn. (45). This is the reason for introducing the anti-adjoint. In turn time inversion is anti-unitary.
Now we assume, that H and ρ are invariant under Π
[H, Π(-1,+) ]=[ρ, Π(-1,+) ]=0. (49)
Applying this on the spectral function in the time-domain and dropping the subscripts μ,ν for this
Tr{ρ[A(t),B(t')]}= n n| Π+ Πρ[A(t),B(t')]n= n Πn|Πρ[A(t),B(t')]n = n Πn|Πρ[A(t),B(t')] Π+ Πn = n Πn|ρ[ΠA(t) Π+ ,ΠB(t') Π+ ]Πn .       (50)

Since |n in the preceding is a complete orthonormal set (COS), |Πn is also and we may replace it by |n in the trace. Many operators have a so-called
parity ε w.r.t. certain operations like time-reversal Π or (real-space) inversion P, i.e.
ΠA(t) Π-1 = εt A(-t)       (51) PA(t) P-1 = εP A(t).       (52)

Typical examples are coordinate r with εP =-1 and εt =1, or velocity v with εP =-1 and εt =-1, or (electric/)magnetic field ( E/) B with εP =(-1/)-1 and εt =(1/)-1. In the following we restrict ourselves to such. Then eqn. (50) continues as
= εA t εB t n n|ρ[A(-t),B(-t')]n = εA t εB t χAB (-t,-t' ) =- εA t εB t χ A B (-t,-t'). (53)
In the frequency domain assuming eqn. (16) and returning to the subscripts we get
χμν (ω)=- εμ t εν t χ μ ν (-ω). (54)
Example 4 Consider the spectral function from examples 1 and 2
χrr '' (ω)=π[δ(ω- ω0 )-δ(ω+ ω0 )]. (55)
Within the notation of eqn. (36) we have r = r+ =r. From this

The commutator rule
χrr '' (ω)=- χ r r '' (-ω)=- χrr '' (-ω) (56)
is obviously satisfied. The
complex conjugation rule
χrr '' (ω ) = χrr '' (ω ) =- χrr '' (-ω) (57)
sets χrr '' (ω) to be
real and odd in ω. Finally, the t-reversal rule together with εr t =+1
χrr '' (ω)=-(+1)(+1) χrr '' (-ω) (58)
leads to no additional insight.

Symmetries allow practically relevant conclusions about the vanishing of certain susceptibilities. Consider eg. the isolated (momentum p,real-space r)-susceptibility
χpr Iso . (59)
From t-reversal we get
χpr '' (ω)=-(-1)(+1) χpr '' (-ω)= χpr '' (-ω). (60)
Contrast this against eqn. (58) ! I.e. χpr '' (ω) is even in ω. Using eqn. (27,30) we get
χpr Iso = - dω' π χpr '' (ω') ω' =0. (61)
A similar argument leads to χME Iso =0 and χPB Iso =0, for a time-inversion symmetric H and ρ, where E(B) and P(M) are the electric(magnetic) field and the electric polarization (magnetization).

At first sight, generating a magnetic moment with an electric field seems counter-intuitive anyway. However, to allow for χME Iso 0 or χPB Iso 0 a system simply has to break time-inversion symmetry. By the famous P(arity) C(harge conjugation) T(ime inversion)-theorem, this is synonymous to the breaking of parity (i.e. space inversion symmetry) in most cases. Solids which display such symmetry loss are also called non-centro-symmetric.
Exercise 5 (Real space) Inversion Consider the r-dependent operators
i) density n ^ (r)= l δ(r- R ^ l )
ii) momentum density p ^ (r)= l δ(r- R ^ l ) p ^ l
iii) spin density S ^ (r)= l δ(r- R ^ l ) S ^ l
of an N-particle system, with l=1N, where the '      ^ ' refers to operators and r is a real space coordinate.
a) Use that under inversion P R ^ l P-1 =- R ^ l , P p ^ l P-1 =- p ^ l , and P S ^ l P-1 = S ^ l to show that
PA(r) P-1 = εA P A(-r), (62)
for A(r)= n ^ (r), p ^ (r) and S ^ (r), with εn P =1, εp P =-1, and εS P =1.
b) Assume a statistical operator ρ invariant under inversion [ρ,H]=0. Defining the the real-space dependent spectral function χμν (r,t;r',t')iΘ(t-t')[ Aμ + (r,t), Aν (r',t')], show that in this case
χμν (r,t;r',t')= εμ P εν P χμν (-r,t;-r',t'). (63)
As a consequence, assuming translational invariance and a time independent Hamilton operator, i.e. χμν (r,t;r',t')= χμν (r-r',t-t'), show that the Fourier transform into wave vector (i.e. q) and frequency (i.e. ω) space satisfies
χμν (q,ω)= εμ P εν P χμν (-q,ω). (64)
c) Use eqn. (30) to and eqn. (64) to derive the isolated susceptibilities χnp Iso (r=0) and χsp Iso (r=0). What is the physical significance of this.

5.5  Mori's Scalar Product

We may define the following mapping from the vector space of linear operators Aμ on the scalar space of complex numbers into the complex numbers
Aμ | Aν = 0 β Δ Aμ + (τ)Δ Aν dτ       (65) β = 1/T ΔA = A-A       (66) A(τ) = eτH Ae-τH   τRe.       (67)

Here τ is called
'imaginary' time. Note, that unlike for real time [A(τ )]+ A+ (τ). The brackets on the r.h.s are usual thermal averages. The brackets |on the l.h.s. of eqn. (65) should not be confused with a scalar product of quantum mechanical states. Yet, this so-called Mori product satisfies all properties of a conventional scalar product, i.e. it is:
i) Bilinear: this is trivial
ii) Positive semi-definite: (w.l.o.g. set A=0)
A|A = 0 β A+ (τ)Adτ= 1 Z lm 0 β e-β El l| A+ (τ)|mm|A|ldτ = 1 Z lm e-β El l| A+ |mm|A|l 0 β eτ( El - Em ) dτ = 1 Z Em En |m|A|l |2 e-β Em - e-β El El - Em + β Z Em = En |m|A|l |2 .       (68)

Both sums on the last line contain only positive addends. Moreover if, and only if all matrix elements of A are zero, i.e. A=0, then A|A=0. q.e.d.

iii) Complex conjugation:
A|B = 0 β A+ (τ)B dτ= 0 β Tr{ρ A+ (τ)B } dτ= 0 β Tr{ B+ [ A+ (τ )]+ ρ}dτ = 0 β Tr{ B+ A(-τ)ρ}dτ= 0 β Tr{ B+ (τ)Aρ}dτ=B|A.       (69)

Mori's product has many applications in statistical mechanics. The first one applies to the isothermal susceptibility
χμν T = Aμ + fν = Aμ | Aν . (70)
The high-temperature limit β0 and also the classical case, i.e. [H, Aμ ]=0, of this equation can be read of easily from eqn. (65)
χμν T β0or [H, Aμ ]=0 Δ Aμ + Δ Aν T . (71)
This is the famous relation proportionality between fluctuations and susceptibilities. Eg. for a magnetic system in this limit we have χT =(ΔS )2 /T, which is simply
Curie's law. The general proof of eqn. (70) is left to exercise 8.
Exercise 6 Given two operators with Aμ + = Aμ and Aν + = Aν . Show that Aμ | Aν = Aν | Aμ . I.e. for hermitean operators, and because of eqn. (69), Mori's product is real.
Example 7 Let's continue with the harmonic oscillator. Analogous to eqn. (19) we get from eqn. (67)
b(τ) = e- ω0 τ b b+ (τ) = e ω0 τ b+ .       (72)

Using that b= b+ =0, and inserting into eqn. (65) we get
r|r = 0 β ( e ω0 τ b+ + e- ω0 τ b)( b+ +b)dτ = 0 β e ω0 τ b+ bdτ+ 0 β e- ω0 τ bb+ dτ = n(β)( 1 ω0 )( e ω0 β -1)+(1+n(β))(- 1 ω0 )( e- ω0 β -1) = 1 ω0 + 1 ω0 = 2 ω0       (73)

where we have used bb= b+ b+ =0, and b+ b=n(β)=1/( eβ ω0 -1), and [b, b+ ]=1.

From this and eqns. (35,71) we can read off that
χrr Iso = χrr T (74)
Exercise 8 Thermodynamic perturbation theory. Prove eqn. (70)

5.6  The Liouville Operator

The time-evolution resulting from the equations of motion for operators in the Heisenberg picture can be brought into a form formally similar to that of states in the Schrödinger picture by introducing a so-called super operator, the Liouville operator L, which by definition is an operator acting on operators
LAμ (t) [H, Aμ (t)]       (75) d dt Aμ (t) = iLAμ (t)       (76) Aμ (t) = eiLt Aμ .       (77)

The Liouville operator is hermitean w.r.t. to the Mori product (w.l.o.g. A=B=0)
A|LB = 0 β A+ (τ)LBdτ= 0 β eτH A+ e-τH [H,B]dτ= 0 β eτH [H,A ]+ e-τH Bdτ = 0 β e-τH BeτH [H,A ]+ dτ= 0 β B+ (τ)[H,A] dτ=B|LA       (78)

Index (showing section)

anti-adjoint, 5.4
anti-linear, 5.4
anti-unitary, 5.4

canonical distribution function, 5.1
Curie's law, 5.5

dynamical susceptibility, 5.2

imaginary time, 5.5
inversion, 5.4
isolated/adiabatic susceptibility, 5.3
isothermal susceptibility, 5.3

Kramer-Kronig, 5.3

microstates, 5.1
Mori product, 5.5

non-centro-symmetric, 5.4

on Neumann's equation, 5.1

parity, 5.4
PCT-theorem, 5.4

retarded/advanced, 5.3

spectral function, 5.2
spectral representation, 5.3
super operator, 5.6

time inversion, 5.4

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On 24 Jan 2010, 21:36.