A Review Of The Linear Response Function In Condensed Matter Physics And Their Application In Some Elementary Processes

Abstract

Linear response theory in quantum theory with its linear response function and its quantization process has been formulated. The relation between the linear response function with its generalized susceptibility, its symmetry properties, and its analyticity has been studied. These properties produce the dispersion relation or Kramers-Kronig relation. The explicit form of the quantum response function and generalized susceptibility also been reviewed. Applications of linear response functions have been described for three elementary processes. The process discussed is the magnetic field disturbance in the magnetic system that generates magnetic susceptibility, and the electric field disturbance in the electrical system that generates electrical conductivity tensor and the ferromagnet Heisenberg that generates its generalized susceptibility.

Full text

Ibnu Jihad and Kamsul Abraha
ARTIKEL RISET
A Review Of The Linear Response Function In
Condensed Matter Physics And Their Application
In Some Elementary Processes
Ibnu Jihad*and Kamsul Abraha
Abstract
Linear response theory in quantum theory with its linear response function and its quantization process has
been formulated. The relation between the linear response function with its generalized susceptibility, its
symmetry properties, and its analyticity has been studied. These properties produce the dispersion relation or
Kramers-Kronig relation. The explicit form of the quantum response function and generalized susceptibility also
been reviewed. Applications of linear response functions have been described for three elementary processes. The
process discussed is the magnetic field disturbance in the magnetic system that generates magnetic susceptibility,
and the electric field disturbance in the electrical system that generates electrical conductivity tensor and the
ferromagnet Heisenberg that generates its generalized susceptibility.
Keywords: linear response theory; response function; general susceptibility; magnetic susceptibility; electric
conductivity tensor; Heisenberg ferromagnet
1 Introduction
Response functions describe the response of a material
due to the external field. The external field can be in
the form of an electric field and a magnetic field or
others. An example of a response function that is well
known is the susceptibility of matter, both due to the
electric field or magnetic field. Electric susceptibility of
an object illustrates the ease of object to be polarized
in response to an electric field applied to it. Similar to
the electric susceptibility, magnetic susceptibility is a
measure of the ease of objects affected by the external
magnetic field, and the influence of external fields
generate magnetic dipole moment per unit volume in
the field.
Electric susceptibility gives an overview of the
electric polarization, and magnetic susceptibility
objects give an idea of the formation of the magnetic
dipole moment. Both describe the response of objects
on the external field (either electric or magnetic)
applied to the objects. The response of the material
due to the external field determines the properties
of objects that are very important to know. As an
*Correspondence: ibnu.jihad@ugm.ac.id
Departemen Fisika, Fakultas Matematika dan Ilmu Pengetahuan Alam
Universitas Gadjah Mada, Yogyakarta, Indonesia
Full list of author information is available at the end of the article
†Equal contributor
example, in the material that has a positive magnetic
susceptibility (called paramagnetic material), the
magnetic field inside the material will be gained by
the magnetic dipole moment formed by an external
magnetic field. On objects with negative susceptibility
(called diamagnetic material), the reverse process
occurs. The magnetic field inside the material will be
weakened. This weakening is caused by the formation
of magnetic dipole moments in the opposite direction
of the external magnetic field.
Generally, the response function can provide an
overview of the properties of the considered material.
Therefore, knowledge about the response function
is becoming essential in science and technology.
Previously, research on the formulation of the response
function is using the classical theory, without the use
of quantum theory. As a result, the validity of the
formulation of response function has a limited scope,
only in classical cases. These limitations, for instance,
are for materials in large quantities and large size only.
Therefore, the formulation of response functions using
quantum theory is needed to support the development
of science and technology began to spread to those
quantum cases. Thus, in this study will be redefined
response function in quantum theory, along with its
application to some elementary processes.

Ibnu Jihad and Kamsul Abraha Page 2 of 3
The linear response function of a system due
to external stimuli is associated with a Green’s
function, as stated by Rickayzen (1980)[1] and
Cottam-Tiley (1989)[2]. In the Green’s function was
found that the correlation function has a dominant
role. It can be said that the Green’s function is a
generalization that is appropriate for the concept of
correlation functions. The correlation function is a
function of the distribution in statistical mechanics,
which describes the relationship between the two
operators[3]. Therefore, the response function of the
system can be determined if the correlation function
can be determined.
The formulation of the linear response function
involves the Green’s function, and the correlation
function is given the prominence of the method that
is easy and simple, even though for many-particle
systems. Green’s function contains complete and
sufficient information about the many-particle
systems. Calculations involving correlation function
involving the boundary conditions can be facilitated
using the spectral theorem[3].
Some essential applications of the linear response
theory include the formulation of a common
susceptibility to the quantum system, as has been
stated by Jensen-Mackintosh (1991)[4] in his book
on rare earth magnetism. In the same book,
Jensen-Mackintosh also gives another example that
is applied to Heisenberg ferromagnet. Another vital
application is the study of spin waves in objects
antiferromagnet and ferrimagnet[5]. We also discuss
other applications of the theory of linear function in
magnetic susceptibility, electrical conductivity[3], and
linear response in the Heisenberg ferromagnet.
2 Basic Definition and Properties Of
Linear Response Function
In general, a physical system will respond if stimulus
or interference is given. The higher the stimulus is
given, the system response is higher. This is called
alinear response. Most physical systems fall into
this category, although this applies only to relatively
small disturbances. If the given interference is large
enough, the response produced by the system is no
longer linear. The response also depends on the time
behavior of the stimuli. For example, if the stimulus is
in the form of an oscillating voltage (AC voltage), then
the system’s response will be different when compared
to systems stimulated with DC voltage. The response
system will depend on the time dependency of the
stimulus, which in general is an arbitrary function
of time. This problem will be more easily analyzed
if the initial stimulus is broken down according to
its frequency components using Fourier transform in
Fourier analysis. Each component of this stimulus will
be sought for the response. Then these responses are
summed (superposition principle applies considering
linear response). This is the basis of reasoning that
will be used.
Consider the system response is written as R(t), and
stimulation or ”forces” as f(t) which is given to the
system. The assumptions are used:
1 The system will start to respond if the stimulus
started. In this case, causality applied. Responses
should not precede stimulation.
2 The response is linear, so it applies the principle
of superposition.
The general relationship between responses at time t
to the stimuli f(t0) which is working from time t=−∞
to tis in the form of a superposition of f(t0)
R(t) = Zt
−∞
φ(t, t0)f(t0)dt0(1)
with φ(t, t0) is the response per unit stimulus that is
called as response function. Equation (1) shows that
the response at time tcan only be caused by previous
stimulation. Therefore, this response is called a causal
response. Stimulation at a time greater than t, will not
give any influence to the current response R(t).
A change needs to be made to our equation,
considering the response does not depend on absolute
time, but only depends on the time difference between
the stimulus and the response measurements, so
φ(t, t0) = φ(t−t; ) and
R(t) = Zt
−∞
φ(t−t0)f(t0)dt0(2)
The form of the response function φ(t−t0) will conform
with the stimulus and the response. If they are scalar,
then the response function is a scalar function. If they
are vectors, then the response function is a 2-degree
tensor. In general, if the stimulus and the response
is an n-degree tensor, then the response function is a
2n-degree tensor.
Response function should behave as a decaying
function with time. If the new force just worked,
then the response was great, at a later time after
the force worked, the response will be decreased and
became more stable. A response function can oscillate,
or at some time have negative value, as long as the
function shape is progressively decayed towards zero
or a specific value. Clearly, a response function also
should not increase over time if no more force applied.

Ibnu Jihad and Kamsul Abraha Page 3 of 3
2.1 Generalized Susceptibility
The response equation in Eq. (2) can be transform to
frequancy domain using standard Fourier transform
Z∞
−∞
eiωt ¯
R(ω)dω =Zt
−∞
φ(t−t0)Z∞
−∞
eiωt0¯
f(ω)dωdt0
(3)
Then by changing the order of integration and
introducing τ=t−t0we get
˜
R(ω) = 

∞
Z
0
eiωτ φ(τ)dτ 
˜
f(ω),∀ω∈ < (4)
with <is set of real number and by defining
˜
R(ω)≡χ(ω)˜
f(ω),∀ω∈ < (5)
This χ(ω) is called as generalized susceptibility.
This term is analogous with electric or magnetic
susceptibility, which is a measure of the response of
a material in an external field.
2.2 The Symmetry and Analytic Properties of
Generalized Susceptibility
This is the equation that relates generalized
susceptibility to its response function:
χ(ω) =
∞
Z
0
eiωτ φ(τ)dτ with τ≥0.(6)
This equation is also called the Fourier-Laplace
transform of the response function. Function φ(τ) is
a real function, so in general χ(ω) is a complex-valued
function. The equation above can be seen that
Re χ(−ω) = Re χ(ω),and (7)
Im χ(−ω) = −Im χ(ω) (8)
for real ω. This shows that generalized susceptibility
is symmetric.
The function χ(ω) is an analytic function in the
upper half-plane of the complex plane, as well as the
real line (based on the initial assumption). Thus, for
any closed loop in the upper half-plane ω, the integral
contour obey
IC
χ(ω)dω = 0 (Cauchy’s Integral Theorem).(9)
2.3 The Dispersion Relation
Analytic properties of χ(ω) permit the formulation
χ(ω0)(with ω0is a real physical value) that can be
expressed in another real number ω, and this is called
as a dispersion relation. We start by defining
f(ω0) = χ(ω0)
ω0−ω(10)
The equation has a simple pole on the real axis in
ω-plane.
Gambar 1: Contour integration on ω-plane.
The function in Eq. (10) have contour shape of the as
shown in Fig. 1, and by applying the Cauchy Integral
theorem, then we have
χ(ω) = P
iπ Z∞
−∞
χ(ω0)dω0
ω0−ω=−iP
πZ∞
−∞
χ(ω0)dω0
ω0−ω(11)
which is an integration of all real ω0from minus infinity
towards infinity. If the real and imaginary components
are separated, it become
Re χ(ω) = P
πZ∞
−∞
Im χ(ω0)dω0
ω0−ω(12)
and
Im χ(ω) = −P
πZ∞
−∞
Re χ(ω0)dω0
ω0−ω.(13)
This form of both equations is called a Hilbert
transform pair.
Equation (12) can be rewritten into
Re χ(ω) = 2P
πZ∞
0
dω0ω0Im χ(ω0)
ω02−ω2.(14)
Equation (13) can be written in a similar way
Im χ(ω) = −2P ω
πZ∞
0
dω0Re χ(ω0)
ω02−ω2.(15)

Ibnu Jihad and Kamsul Abraha Page 4 of 3
Both of equations is called dispersion relation or also
called Kramers-Kronig relation.
2.4 Changing from Classical Linear Response Theory
toward Quantum Mechanics
Changing from the classical to the quantum theory
can begin by reviewing a classical system and its
response function, followed by the first quantization
to obtain quantum theory. This discussion refers to
Kubo (1957)[6].
In a closed system, the system Hamiltonian is
written as H. Dynamical motion of a system
determined by His called a natural motion. Consider
an external force imposed on the system. Its energy
can represent the impact of this perturbation,
H0(t) = −AF (t).(16)
In this discussion, the perturbation is limited to a
weak force only. The response system will be sought
at the linear rate approach. System response is in the
changes the physical quantities ∆B(t) from its original
B. Next, ∆B(t) will be stated in terms of the natural
motion of the system.
At first, consider the classical mechanics perspective.
Suppose the system is in a statistical ensemble
represented by the distribution function f(p, q) in
phase space. The equations of motion describe the
natural motion of the system is
∂f
∂t =−X∂f
∂q
∂H
∂p −∂f
∂p
∂H
∂q = (H, f ) (17)
with pand qis the set of canonical momentum
and canonical coordinate, while the parenthesis is the
Poisson bracket
(A, B) = X∂A
∂q
∂B
∂p −∂A
∂p
∂B
∂q .(18)
Assume the distribution function is fat t=−∞.
Suppose that system is in a state of equilibrium, so
(H, f ) = 0. Then disturbance in Eq. (16) is given to the
system adiabatically at t=−∞(with F(t=−∞) =
0). The distribution function satisfies
∂f 0(t)
∂t = (H, f 0)+(H0(t), f ).(19)
a linear approach causes
f0(t) = f+ ∆f
which can replace (19) with
∂∆f
∂t = (H,∆f)−F(t)(A, f ).(20)
The solution is
∆f(t) = −Zt
−∞
ei(t−t0)L(A, f )F(t0)dt0,(21)
where the operator Lis defined as an operation
iLg ≡(H, g).(22)
So the change in ∆B(t) to a dynamic quantity Bis
given statistically by
∆B(t) = Z∆f(t)·B(p, q)dΓ
=−ZdΓZt
−∞ nei(t−t0)L(A, f )o·F(t0)B(dt0)
=−ZdΓZt
−∞
(A, f )B(t−t0)F(t0)dt0
(23)
where dΓ is the volume element in the phase space[7].
The last equation is obtained from the second equation
by remembering that the transformation of eiLt is a
natural motion that conserves the size of the phase
space and B(t) is the dynamic motion of the phase
function B(p, q) that satisfies the equation
˙
B(p, q)=(B, H).
The above equation is related to the Heisenberg
equation of motion in quantum mechanics. The
equation (23) means that the response ∆B(t) is a
superposition of the impact of the stimulus F(t0)dt0,
for −∞ < t0< t. The response for each stimulus unit
is referred to as response function or after-effect
function φBA(t). The form of this response function
can be known from Equation (23)
φAB(t) = −ZdΓ(A, f )B(t) (24)
which explains the response of ∆Bwhen tafter the
stimulus is given. The response ∆B(t) to the equation
(23) is written as
∆B(t) = Zt
−∞
φAB(t−t0)F(t0)dt0.(25)
The above review applies even to the very sharp
initial distribution, as long as the disturbance is
quite small. Applied response functions are needed
for macro-scale systems, which in these cases, the
statistical average requires real meaning. If ˆ
Aand
ˆ
Bare both macro-scale quantities, then the average

Ibnu Jihad and Kamsul Abraha Page 5 of 3
ensemble ∆ ˆ
B(t) can be observed. It is because a
macro-scale system can be considered to consist of
smaller systems so that the observed value of ∆ ˆ
Bis
the number of components, and the change is relatively
minimal.
The change to quantum mechanics can also be
done, based on the classic formulation above. The
distribution function in the classical phase space is
then replaced by the ρdensity operator matrix.
The original ensemble that statistically represents the
initial state of the system is determined by this density
operator and satisfies [ ˆ
H, ρ] = 0, while the ensemble
dynamic is affected by interference (16 ) is represented
by ρ0(t), which satisfies the equation
d
dtρ0(t) = 1
i~[ˆ
H+ˆ
H0(t), ρ0(t)].(26)
The initial conditions are
ρ0(−∞) = ρ
and ρ0(t) are listed as
ρ0(t) = ρ+ ∆ρ(t).
The same steps as (20) direct the results to
∆ρ(t) = −1
i~Zt
−∞
e−i(t−t0)ˆ
H/~[ˆ
A, ρ]ei(t−t0)ˆ
H/~F(t0)dt0.
(27)
Then, ease of work can be obtained by introducing the
a×operator that works on other operators bwith the
following definition
a×b= [a, b],(28)
which from the definition applies
ea×b=eabe−a(29)
Equations (26) and (27) can be written with this new
notation to
d
dtρ0(t) = 1
i~(ˆ
H×+ˆ
H0(t)×)ρ0(t) (30)
∆ρ(t) = −1
i~Zt
−∞
e−i(t−t0)ˆ
H×/~[ˆ
A, ρ]F(t0)dt0.(31)
This form will later show clearly the similarity between
(21) and (27).
The response ∆ ˆ
B(t) to the amount of ˆ
Bstatistically
is
∆ˆ
B(t) = Tr ∆ρ(t)ˆ
B
=−1
i~Tr Zt
−∞
e−i(t−t0)ˆ
H/~[ˆ
A, ρ]×
ei(t−t0)ˆ
H/~ˆ
BF (t0)dt0
=−1
i~Tr Zt
−∞
[ˆ
A, ρ]ˆ
B(t−t0)F(t0)dt0(32)
where ˆ
B(t) is the Heisenberg representation for the ˆ
B
operator that satisfies the equation
dˆ
B(t)
dt =1
i~[ˆ
B(t),ˆ
H],ˆ
B(0) = ˆ
B. (33)
The equation (32) is a quantum mechanical version of
(23).
So, the response function is
φBA(t) = 1
i~Tr[ ˆ
A, ρ]ˆ
B(t) (34)
or can be written (because of the trace invariance
nature) as
φBA(t) = 1
i~Tr[ ˆ
A, ρ]ˆ
B(t) (35)
which is the quantum mechanical version of Equation
(24). The General susceptibility (according to the
equation (6)) is
χBA(ω) = Z∞
0
e−iωt 1
i~Tr[ρ, ˆ
A]ˆ
B(t)dt. (36)
2.5 Linear Responses in Quantum Theory
A response function for the macroscopic order
associates the physical changes in the ensemble means
of the physical observables hB(t)iwith an external
force f(t). The application of linear response theory
is limited to cases where hB(t)ichanges linearly with
force. Therefore assume that f(t) is weak enough to
ensure that the response is linear. Assume that the
system is in thermal equilibrium before the external
forces work.
In the thermal balanced state, the system is
characterized by a density operator
ρ0=1
Ze−βˆ
H0;Z= Tr e−βˆ
H0,(37)
where ˆ
H0is Hamiltonan (effective), Zis a large
canonical ensemble partition function, and β=

Ibnu Jihad and Kamsul Abraha Page 6 of 3
q/kBT. The review is only in the linear part of the
response, so it can be considered a form of f(t) which
results in a linear time-time disorder in the total
Hamiltonian ˆ
His:
ˆ
H1=−ˆ
Af(t) ; ˆ
H=ˆ
H0+ˆ
H1,(38)
where ˆ
Ais a fixed operator. As a result, the operator
density ρ(t) becomes a time-catcher, and so does the
ensemble average of the operator ˆ
B:
hˆ
B(t)i= Tr{ρ(t)ˆ
B}.(39)
Linear response (as in Eq. (2)) is
hˆ
B(t)i−hˆ
Bi=
t
Z
−∞
dt0φBA(t−t0)f(t0),(40)
with hˆ
Bi=hˆ
B(t=−∞)i= Tr{ρ0ˆ
B},f(t) considered
vanishing on t→ ∞. The nature of this equation is
the same as the equation (2).
The general susceptibility (after undergoing the
Fourier transform in the equation (40)) can be
expressed as
χBA(z) = 1
2πZ∞
0
φBA(t)eiztdt. (41)
where z=z1+iz2is a complex variable. If
R∞
0|φBA|e−tdt is considered to be finite (finite) at
the →0+limit, the opposite relationship is
φBA(t) = Z∞+i
−∞+i
χBA(z)e−iztdz; > 0.(42)
This equation shows that χBA(z) is an analytic
function in the upper half of the complex plane z.
The changes of the system can be uniquely
determined by ρ0=ρ(−∞) and f(t), provided the
external interference is given slowly or adiabatically.
This can be done by replacing f(t0) in (40) with
f(t0)et0,  > 0. This style disappears at t0→ −∞,
and additional unwanted effects can be removed by
taking the →0+limit. So, in terms of this ’general’
Fourier transformation
hˆ
B(ω)i= lim
→0+
1
2πZ∞
−∞ hˆ
B(t)i−hˆ
Bieiωte−t dt,
(43)
equation (40) transforms into
hˆ
B(ω)i=χBA(ω)f(ω),(44)
where χBA(ω) is the boundary condition for the
analytic function χBA(z) on the real axis:
χBA(ω) = lim
→0+χBA(z=ω+i).(45)
This is the general susceptibility[4].
2.6 Quantum Response Function
The equation for the response function in terms of
operator ˆ
Band ˆ
A, and unperturbed Hamiltonian ˆ
H0,
is
hˆ
B(t)i−hˆ
Bi= Tr{(ρ(t)−ρ0)ˆ
B}
=i
~Tr Zt
−∞
[ˆ
A0(t0−t), ρ0]ˆ
Bf (t0)dt0
and by using the invariance on track experiencing
cyclic permutation, then obtained the low-order terms
hˆ
B(t)i−hˆ
Bi=i
~Zt
−∞
Tr{ρ0[ˆ
B, ˆ
A0(t0−t)]}f(t0)dt0
=i
~Zt
−∞
h[ˆ
B0(t),ˆ
A0(t0)]i0f(t0)dt0
(46)
analogous with response function defition, which
produce
φBA(t−t0) = i
~θ(t−t0)h[ˆ
B(t),ˆ
A(t0)]i,(47)
with a unit step function are introduced, namely for
and for t= 0. This equation, which is expressed in
microscopic quantity, called the Kubo formula for the
response function[8].
2.7 Quantum Green Function
The Green function is defined as:
GBA(t−t0)≡ hh ˆ
B(t); ˆ
A(t0)ii
≡ − i
~θ(t−t0)h[ˆ
B(t),ˆ
A(t0)]i
=−φBA(t−t0).(48)
This Green function is also called double-time
Green function or retarded Green function,
which is the previous response function, but with the
opposite sign. Laplace transform of the function using
(41), is
GBA(ω)≡ hh ˆ
B;ˆ
Aiiω= lim
→0+GBA(z=ω+i)
= lim
→0+GBA(t)ei(ω+i)tdt =−χBA(ω).
(49)

Ibnu Jihad and Kamsul Abraha Page 7 of 3
If ˆ
Aand ˆ
Bare dimensionless operators, then GBA (ω)
or χBA(ω) has the inverse dimension of energy .
At t0= 0, the derivative of the Green function above
to tis
d
dtGBA (t) = −i
~δ(t)h[ˆ
B(t),ˆ
A]i+
θ(t)h[dˆ
B(t)/s, ˆ
A]i
=−i
~δ(t)h[ˆ
B, ˆ
A]i−
i
~θ(t)h[[ ˆ
B(t),ˆ
H],ˆ
A]i.(50)
The Fourier transform of the above equation into the
Green function motion equation:
~ωhh ˆ
B;ˆ
Aiiω− hh[ˆ
B, ˆ
H]; ˆ
Aiiω=h[ˆ
B, ˆ
A]i.(51)
The ωlabels indicate the Fourier transform (49),
and ~ωis short for ~(ω+i) with →0+.
In many applications, ˆ
Aand ˆ
Bare the same
Hermitian operator, so the right side of the equation
(51) above disappears, and the equation can be
derived a second time. Because h[[[ ˆ
A(t),ˆ
H],ˆ
H],ˆ
A]i
equals −h[[ ˆ
A(t),ˆ
H],ˆ
A]i, Green equation of motion for
hh[ˆ
A, ˆ
H]; ˆ
Aiiωto be
(~ω)2hh ˆ
A;ˆ
Aiiω+hh[ˆ
A, ˆ
H]; [ ˆ
A, ˆ
H]iiω=h[[ ˆ
A, ˆ
H],ˆ
A]i.
(52)
The pair of equations (51) and (52) will underlie the
applied linear response theory.
In accordance with the understanding about the
response function KBA (t),
KBA(t) = i
~h[ˆ
B(t),ˆ
A]i=i
~h[ˆ
B, ˆ
A(−t)]i(53)
and KBA(z) = 2iχ00
BA(z) we get
KBA(ω) = 2iχ00
BA(ω) = −2iG00
BA(ω).(54)
This equation can be written down
2iZ∞
−∞
χ00BA(ω)e−iωt dω =i
~h[ˆ
B(t),ˆ
A]i(55)
and by selecting t= 0, we will get the following
addition rule
2~Z∞
−∞
χ00
BA(ω)dω =h[ˆ
B, ˆ
A]i,(56)
The Green function in (51) must meet these
summation rules, and the thermal equalization of (51)
and (56) must be the same. The equation (56) is only
the first term of this whole series of addition rules.
The n-time derivative of ˆ
B(t) can be written as
dn
dtnˆ
B(t) = i
~n
Lnˆ
B(t) with Lˆ
B(t)≡[ˆ
H,ˆ
B(t)].
(57)
Anderivative on both sides of the equation (55)
results
i
πZ∞
−∞
(−iω)nχ00
BA(ω)e−iωtdω =i
~n+1
h[Lnˆ
B(t),ˆ
A]i.
(58)
Then the normalized ”spectral weight function” was
introduced
FBA(ω) = 1
χ0
BA(0)
1
π
χ00
BA(ω)
ω,Z∞
−∞
FBA(ω)dω = 1.
(59)
The FBA(ω) normalization is the result of the
Kramers-Kronig relationship. The n-order of ωof the
FBA(ω) spectral weight function, is defined
hωniBA =Z∞
−∞
ωnFBA(ω)dω. (60)
which permits a link between the n-derivative at t= 0
is written
χ0
BA(ω)h(~ω)n+1iBA = (−1)nh[Lnˆ
B, ˆ
A]i.(61)
This is an addition rule that associates spectral
frequencies with thermal expectation values of
operators obtained from ˆ
B, ˆ
A, and ˆ
H. If ˆ
B=ˆ
A=ˆ
A†,
then FBA(ω) is an even function in ω, and all the odd
parts are gone. In this case, the even part is
χ0
AA(0)h(~ω)2niAA =−h[L2n−1ˆ
A, ˆ
A]i.(62)
2.8 Relaxation Function
This section will discuss the behavior of the φBA (t)
response function in the t→ ∞ approach. It can go
to a certain value or it can not. If it goes to a certain
value, then according to Kubo (1957)[6], it must fulfill
lim
t→∞ φBA(t) = 0 (if there is a limit). (63)
If the application of continuous interruption (fis
constant) from t=−∞ to t= 0 is stopped at t= 0,

Ibnu Jihad and Kamsul Abraha Page 8 of 3
then the ∆Bresponse will stretch according to the
following equation
∆B(t) = Z0
−∞
φBA(t−t0)fdt0
=fZ∞
t
φBA(t0)dt0, t > 0.(64)
So functions can be defined
ΦBA(t) = lim
→0+ Z∞
t
φBA(t0)e−t0dt0(65)
which is called relaxation function. This function
explains the relaxation of ∆Bafter the outside has
been removed.
Theorem and form of this relaxation function,
according to Kubo (1957)[6] is: The χBA (ω)
acceptability can be calculated using the following
formula:
χBA(ω) = lim
→0+ Zmax
min
φBA(t)e−iωt−td
= lim
→0+
1
iω +n˙
φBA(0) +
Z∞
0
φBA(t)e−iωt−tdt
=ΦBA(0) −iω Z∞
0
ΦBA(t)e−iωtdt. (66)
The form of the relaxation function is:
ΦBA(t) = i
~Z∞
t
Tr{ρ[B(t0), A]}dt0(67a)
=Zβ
0
Tr{ρA(−i~λ)B(t)dλ} − βTr{ρA0B0}
(67b)
3 Application of Linear Response Theory
3.1 Magnetic Susceptibility
A uniform magnetic field of H(t) is applied to
a magnetic system, the total magnetic moment is
written as M. The energy perturbation caused by H(t)
is
ˆ
H0(t) = −MH(t).(68)
The natural motion of the magnetic moment when
there is no external field is represented by M(t). Then,
the φµν (t) response function for magnetization in the
µ-direction when the external field H(t) points in the
ν-direction, according to (46) is
φµν (t) = i
~h[Mµ(t), Mν]i
=Zβ
0
h˙
Mν(−i~λ)Mµ(t)idλ, (69)
and according to (67), the relaxation function is
Φµν (t) = Zβ
0
h(Mµ(−i~λ)−M0
µ)(Mµ(t)−M0
µ)idλ (70)
where M0
νand M0
µare the diagonal portion of Mνand
Mµof unperturbed Hamiltonan ˆ
H.
If a system is measured in one volume, this
admittance will become a magnetic susceptibility. So
the magnetic susceptibility tensor can be expressed in
either φµν (t) or Φµν (t). The simplest equation is to use
the equation (66)
χµν (ω) = Φµν (0) −iω Z∞
0
Φµν (t)e−iωtdt. (71)
The static susceptibility is obtained in the form
χµν (0) = Zβ
0
h(Mν(−i~λ)−M0
ν)(Mµ−M0
µ)idλ (72)
which is a susceptibility for an isolated system, and
does not have to be the same as an isothermal
susceptibility in the form
χT
µν =Zβ
0
h(Mν(−i~λ)Mµ)idλ −βhMνihMµi.(73)
In the equation χµν (0), the diagonal portion of Mν
and Mµis subtracted. This is related to the fact
that magnetization in an isolated system takes place
adiabatically, the chance of occupation from its energy
levels remains unchanged. But in the isothermal
process these levels change to χT
µν . This difference will
diminish if the environmental role is taken into account
on a larger scale.
In the classic approach, Equation (72) becomes
χµν (0) = 1
kT h(Mν−M0
ν)(Mµ−M0
µ)i.(74)
meanwhile Equation (73) becomes
χT
µν =1
kT h(MνhMνi)(Mµ− hMµi)i.(75)
The last equation was first revealed by Kirkwood
(Kirkwood, 1939)[9] for the classical theory of

Ibnu Jihad and Kamsul Abraha Page 9 of 3
dielectric polarization. Equation (74) is its extension to
adiabatic susceptibility, and Equation (71) for complex
susceptibility which is common for unequilibrium
states. This is proof that similar equations are also
obtained for dielectric polarization.
3.2 Electric Conductivity Tensor
As an applied example, the link between the electrical
conductivity tensor and the Green function will be
reviewed. Assume that the electric field E(t) is given
adiabatically, which is uniform in space and changes
periodically with a frequency of ω
E(t) = Ecos ωt.
The perturbation operator is
ˆ
H1
t=−Xej(ˆ
Exj) cos ωt et (76)
(where ejis the jth particle load, and the sum includes
all particles with coordinates xj). The perturbation
(76) causes an electric current to appear in the system
ˆ
jµ(t) =
∞
Z
−∞
hhjµ(t); H1
τ(τ)iidτ, (77)
with
ˆ
H1
τ(τ) = ˆ
H1(τ) cos ωτ eτ ,
ˆ
H1(t) = −X
j,µ
ejEµ˙
ˆxµ(τ),
ˆ
jµ(t) = X
j
ej˙
ˆxµ(t),
(78)
ˆ
jµis the current density operator, if the system volume
is considered one unit volume. Integration of each part
makes the equation (77) writeable in form
jµ(t) = −Re 


∞
Z
−∞
hhjµ;˙
ˆ
H1(τ)iieiωt+τ dτ
iω +
+h[jµ(0), H1(0)]ieiωt+t 1
ω−i.(79)
It should be noted that
˙
ˆ
H1(τ) = −(ˆ
Ej(τ)),and
[˙
ˆxµ1,ˆxνj2] = −i
mδµν δj1j2(~= 1) (80)
so from this equation is obtained
jµ(t) = Re{σµν (ω)Eνeiωt+t},(81)
σµν (ω) = −ie2n
mω δµν +
∞
Z
−∞
hhjµ(0); jν(τ)iieiωt+τ
iω +dτ
(82)
where σµν (ω) is the tensor of electrical conductivity,
and nis the number of electrons per unit volume.
The first term in (82) is related to the electrical
conductivity of a system that has a free charge and
does not interact with other particles. As ω→ ∞
increases the second term decreases faster than the
first term ( lim
ω→∞ Im ωσµν (ω) = −e2nδµν /m), and the
system behaves as a collection of free loads. Equation
(82) can be written in a different but equivalent form,
and integrating it byτ, it becomes
σµν (ω) = −ie2n
mω δµν +e−ω
β−1
ω
∞
Z
−∞
hjµ(0)jν(t)ie−iωtdt.
(83)
This equation is known as Nyquist theorem, which
is then generalized by Callen and Welton for the
case of quantum mechanics[10]; this equation connects
electrical conductivity with changes in current.
3.3 Linear Responses to Heisenberg Ferromagnet
In this section, we will review the linear response
function which is applied to the Heisenberg
ferromagnet case in three dimensions, with
Hamiltonian
ˆ
H=−1
2X
i6=j
J(ij)Si·Sj,(84)
where Siis the spin on the iion, which is placed on
a Bravais lattice at the position of Ri. The Fourier
transform of the exchange coupling, provided J(ii)≡
0, is
J(q) = 1
NX
ij
e−iq·(Ri−Rj)=X
j
J(ij)e−iq·(Ri−Rj),
(85)

Ibnu Jihad and Kamsul Abraha Page 10 of 3
and vice versa
J(ij) = 1
NX
q
J(q)eiq·(RiRj)
=V
N(2π)3ZJ(q)eiq·(Ri−Rj)dq,(86)
relies on the review of q(which is defined in the
primitive Brillouin zone) as a discrete or continuous
variable, and will be considered as a discrete variable.
Nis the number of spins, Vis the volume, and the
reversal of symmetry from the Bravais lattice causes
that J(q) = J(−q) = J∗(q).The maximum value of
J(q) is considered to be J(q = 0), which in that state
the equilibrium state occurs at zero temperature, or in
other words at the ground state, i.e. the ferromagnet:
hSii=Sˆ
zon T= 0,(87)
where ˆ
zis the unit vector on the z-axis, which is
constructed as the direction of magnetization caused
by an infinitesimal magnetic field. This result is
appropriate, but as the temperature rises above zero,
a number of approaches are needed. As a first step, the
hSii=hSithermal expectation value can be written
(after the rearrangement of the terms)
ˆ
H=X
i
ˆ
Hi−1
2X
i6=j
J(ij)(Si− hSi)·(Sj− hSi),(88a)
with
ˆ
Hi=−Sz
iJ(0)hSzi+1
2J(0)hSzi2,(88b)
and hSi=hSziˆ
z. The second term is ignored based
on the average field approach, which results in the
original multi-fold Hamiltonian being the sum of the
NHamiltonian free single spin (88b). In this approach,
hSziis determined by self-consistence equations is
hSzi=
+S
X
M=−S
M eβM J(0)hSzi/
+S
X
M=−S
eβM J(0)hSzi
(89a)
(the last term in (88b) does not affect thermal
averages). The equation (89a) in the low temperature
approach is
hSzi ' S−e−βSJ(0).(89b)
Review the Green function, to include initial order
effects on two-place correlations, with
G±(ii0, t) = hhS+
i(t); S−
i0ii.(90)
According to (51), the change in time according to
G±(ii0, t) depends on the operator
[S+
i,ˆ
H] = −1
2X
j
J(ij)(−2S+
iSz
j+ 2Sz
iS+
j).(91)
The inclusion of this commutator in the equation
(51) leads to the link between the original Green
function and the new Green function more clearly.
Through this equation of motion the new function can
be stated with further new functions, thus forming
infinite coupled equations. The answer to the approach
can be obtained by using the condition that the
expected value of Sz
iis close to the saturation value
when it is at a low temperature. So in this approach,
Sz
ihas to be almost timeless, in other words Sz
i'
hSzi.In ’this phase-random approach’ (random phase
approximation (RPA)) the commutator shrank to
[S+
i,ˆ
H]' − X
j
J(ij)hSzi(S+
j−S+
i) (92)
and the equation of motion leads to the following set
of linear equations:
~ωG±(ii0, ω) + X
j
J(ij)hSzi{G±(ji0, ω)−
G±(ii0, ω)}=h[S+
i, S−
i0]i= 2hSziδii0.(93)
4 Conclusion
4.1 Linear Response Theory
1 The response R(t) given by an order because the
external force f(t0) given to him at time t0can be
described by the response function φ(t−t0) with
the equation :
R(t) = Zt
−infty
φ(t−t0)f(t0)dt0(94)
assuming the style and response meet the rule of
causality and the assumption of linearity the
response to its style.
2 The link between general susceptibility and
the response function is
χ(ω) = Z∞
0
φ(τ)eiωτ dτ (95)
3 The properties possessed by general susceptibility
are:
(a) The real and imaginary parts have properties
Reχ(−ω) = Reχ(ω),and (96)
Imχ(−ω) = −Imχ(ω) (97)

Ibnu Jihad and Kamsul Abraha Page 11 of 3
(b) χ(ω) is analytic in the upper half of the
complex plane ω, which is caused by the rule
of causality.
(c) Dispersion relation / Kramers-Kronig
relation connecting real and imaginary
parts to general susceptibility is
Reχ(ω) = 2P
πZ∞
0
dω0ω0Imχ(ω0)
ω02−ω2(98)
and
Imχ(ω) = −2P ω
πZ∞
0
dω0Reχ(ω0)
ω02−ω2.(99)
4.2 Linear Responses in Quantum Theory
1 The form of the link between responses and
disturbances in quantum theory is
hˆ
B(t)i−hˆ
Bi=i
~Zt
−∞
Tr{ρ0[ˆ
B, ˆ
A0(t0−t)]}×
f(t0)dt0
=i
~Zt
−∞
h[ˆ
B0(t),ˆ
A0(t0)]i0f(t0)dt0
(100)
2 The form of the quantum response function is
φBA(t−t0) = i
~θ(t−t0)h[ˆ
B(t),ˆ
A(t0)]i,(101)
3 The form of the fluctuation-dissipation theorem
in this theory viz
SBA(ω) = 2~1
1−eβ~ωχ00
BA(ω),(102)
4 The link between the snooze function and the
response function is
GBA(t−t0)≡ hh ˆ
B(t); ˆ
A(t0)ii
≡ − i
~θ(t−t0)h[ˆ
B(t),ˆ
A(t0)]i
=−φBA(t−t0).(103)
5 The link between the Green delay function and
general susceptibility is
GBA(ω)≡ hh ˆ
B;ˆ
Aiiω= lim
→0+GBA(z=ω+i)
= lim
→0+GBA(t)ei(ω+i)tdt =−χBA(ω).
(104)
4.3 Application of Response Function in Some
Elementary Processes
1 Magnetic susceptibility to systems with a total
magnetic moment of Mthat has external
interference H(t) is:
(a) The static susceptibility is:
χµν (0) = Zβ
0
h(Mν(−i~λ)−M0
ν)(Mµ−M0
µ)idλ
(105)
(b) The isothermal susceptibility is
χT
µν =Zβ
0
h(Mν(−i~λ)Mµ)idλ −βhMνihMµi.
(106)
2 The form of the electrical conductor tensor that
is experiencing outside interference in the form of
an electric field E(t) = Ecos ωt. is
σµν (ω) = −ie2n
mω δµν +
∞
Z
−∞
hhjµ(0); jν(τ)iieiωt+τ
iω +dτ
(107)
or it can also be written in a different form, known
as the Nyquist theorem, viz
σµν (ω) = −ie2n
mω δµν +e−ω
β−1
ω
∞
Z
−∞
hjµ(0)jν(t)ie−iωtdt.
(108)
3 Linear response to Heisenberg Ferromagnet, with
the general form of susceptibility is
χxx(q, ω) = χyy (q, ω) = 1
4{χ+−(q, ω)+χ−+(q, ω )}.
(109)
with
χ+−(q, ω) = 2hSzi
Eq−~ω+iπ2hSziδ(~ω−Eq).
(110a)
and
χ−+(q, ω) = 2hSzi
Eq+~ω−iπ2hSziδ(~ω+Eq),
(110b)

Ibnu Jihad and Kamsul Abraha Page 12 of 3
4.4 Future Study
The discussion that has been carried out in this article
is limited to a linear process. Further research can be
developed to examine non-linear response functions,
the more general response functions. This non-linear
response function is expected to have a wider scope
of coverage and a better level of accuracy than the
response function, which is limited to linear areas only.
Acknowledgment
This article is a summary of the thesis worked on by
the author[11]. Our gratitude goes to all colleagues
and lecturers in the Physics Master Program of
Universitas Gadjah Mada. Our gratitude also go to
the Faculty of Mathematics and Natural Sciences,
Universitas Gadjah Mada, which has funded this
activity through the ”Dana Masyarakat Alokasi
Fakultas” 2020 (117/J01.1.28/Pl.06.02/2020 ).
Author
1 Ibnu Jihad
Dari :
(1) Departemen Fisika, Fakultas Matematika
dan Ilmu Pengetahuan Alam, Universitas Gadjah
Mada
2 Kamsul Abraha
Dari :
(1) Departemen Fisika, Fakultas Matematika
dan Ilmu Pengetahuan Alam, Universitas Gadjah
Mada
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