Resistivity of a 2d quantum critical metal

Physics Letters A 383 (2019) 2645–2651

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Physics Letters A www.elsevier.com/locate/pla

Resistivity of a 2d quantum critical metal Komal Kumari a,∗ , Raman Sharma a , Navinder Singh b a b

Department of Physics, Himachal Pradesh University, Shimla, 171005, India Theoretical Physics Division, Physical Research Laboratory, Ahmedabad, 380009, India

a r t i c l e

i n f o

Article history: Received 15 March 2019 Received in revised form 29 April 2019 Accepted 15 May 2019 Available online 21 May 2019 Communicated by L. Ghivelder Keywords: Metal near quantum critical point s − d Hamiltonian formalism Random phase approximation Resistivity near a magnetic instability

a b s t r a c t We calculate resistivity in the paramagnetic phase just above the curie temperature in a 2d ferromagnetic metal. The required dynamical susceptibility in the formalism of resistivity is calculated within the Random Phase Approximation (RPA). The mechanism of resistivity is magnetic scattering, in which s-band electrons are scattered off the magnetic spin fluctuations of d-band electrons. We use the s-d Hamiltonian 4

formalism. We find that near the quantum critical point the resistivity in 2d scales as T 3 and in the high temperature limit resistivity scales as proportional to T . In contrast to it, resistivity due to phonon scattering is given by T 5 in low temperature limit as is well known. Our RPA result agrees with the Self-Consistence Renormalization (SCR) theory result. © 2019 Elsevier B.V. All rights reserved.

4

1. Introduction Resistivity in metals is generally due to either impurity scattering or phonon scattering or both mechanism working together. In impurity scattering electrons are scattered off the immobile impurities leading to their momentum randomization thus resistivity. In phonon scattering electrons scatter by absorbing or emitting phonons or lattice vibrations [1]. This can be studied using BlochBoltzmann kinetic equation and one finds that resistivity is proportional to temperature (T ) when T   D , where  D is the Debye temperature. In the opposite limit T   D , ρ ∝ T 5 . Resistivity can also appear due to electron-electron scattering. However, system must either be a two-band system or Umklapp processes must be present as electron-electron scattering in a single band conserve momentum and cannot lead to resistivity. The above scenario is not applicable to magnetic materials tuned near their critical points. An alternative mechanism in which electrons scatter off magnetic spin fluctuations becomes important [14]. Currently, there is a renewed interest in the topic of magnetic critical points and physical properties near a magnetic instability [2–13]. It has been shown that electron-magnetic-spinfluctuation scattering in a 3d ferromagnetic metal [14] tuned near 5

to its critical point leads to a resistivity which scales as T 3 . This stands in sharp contrast to phonon scattering. It has also been shown within the Self-Consistence Renormalization (SCR) [12,13,

*

Corresponding author. E-mail addresses: [email protected], [email protected] (K. Kumari), [email protected] (R. Sharma), [email protected] (N. Singh). https://doi.org/10.1016/j.physleta.2019.05.028 0375-9601/© 2019 Elsevier B.V. All rights reserved.

15,16] theory that in 2d ferromagnetic case resistivity scales as T 3 near the critical point. In this paper we represent our calculation of resistivity in a 2d ferromagnetic metal near its critical point by using Random Phase Approximation (RPA) instead of SCR theory. We report that our RPA result agree with the SCR theory result, that is 4

ρ ∝ T 3 , and we give reason for this argument. This is discussed in the discussion section. Our calculation (within the s − d Hamiltonian formalism) is applicable to weakly ferromagnetic alloys such as Ni-Pd which has certain amount of quenched disorder. This is important as quenched disorder can lead to second-order phase transition near the critical point [17]. Our calculation is for second order ferromagnetic phase transition. In clean ferromagnets the presence of fermionic soft modes makes the transition first order as there is an extra entropy associated with collective excitations just above the ferromagnetic transition.1 It turns out that in the presence of disorder these fermionic soft modes becomes diffusive and the transition can become second order. For more details refer to [17]. However, in this investigation we do not study extra resistivity coming solely due to disorder. For disorder contribution to resistivity in quantum critical region reader is referred to [18]. In the next section we present the formalism and our calculation of resistivity.

1

If energy of a fermionic mode is h¯ ωq and density is nq , then the extra energy

needed to excite soft modes can be roughly given by δ Q = tropy is δ S =

δQ T

=

1 T

∞ 0

3

d q nq h¯ ω .

∞ 0

d3 q nq h¯ ω and en-

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K. Kumari et al. / Physics Letters A 383 (2019) 2645–2651

(5)

2. Formalism We calculate resistivity resulting from the scattering of mobile s-band electrons with the localized d-band electrons with in the standard s − d formalism. First, we calculate the scattering rate of s-band electrons from a given initial state to a given final state using Fermi golden rule. In the matrix element the spin parts of d-band electrons are written as transverse susceptibility which is separately calculated using RPA. Next, the resulting expression of the scattering rate is inserted into the standard expression of resistivity which is obtained from the variational solution of the Boltzmann equation. Then the calculations are performed as detailed in the following paragraphs. We start out with the s-d Hamiltonian [14]:

H int =

J  N



† ak ↑ ak↓ S − (k

k,k

† + (ak ↑ ak↑

† − k) + ak ↓ ak↑ S + (k

† − ak ↓ ak↓ ) S z (k



 − k)

(1)

(2)

The transition probability [20] that an s-electron with wave vector k will be scattered to be the state k + q can be written as

W k+q←k =

h¯

|d( F )|k + q| H int |k |d( I ) | ρ f .

=

2π J 2 h¯ N 2

(3)

×

f s (k )(1 − f s (k+q ))



+

< d( I ) S (−q)|d( F ) >< dF | S (q)|d( I ) > 

(4)

e −i H d ( I ) |d( I ) >= e −i d ( I ) |d( I ) >. Therefore we obtain:

2 h¯ N 2

∞

f s (k )(1 − f s (k+q ))

dte −i ωt [ S + (−q) S − (q, t ) +  S − (−q) S + (q, t ) ]

× 0

3

= 4

J2

2 N2

+∞ dω f s (k )(1 − f s (k+q ))(−n(−ω)) −∞

(6)

where n(−ω) = −β h¯1ω and χs−+ (q, ω) is the symmetric part of e −1 the complex susceptibility corresponding to a magnetic scattering of spins of d-band electrons with wave vector q and frequency ω . χ +− (q, ω) the susceptibility of d-electrons is defined by

χ

+−

(q, ω) = lim

i

ε →0 h ¯

∞

e −i ωt −εt [ S + (q, t ), S − (−q)] dt

(7)

0

This dynamical susceptibility is the Fourier transform of response function or retarded Green function defined with respect to spin densities of d-electrons post scattering. The susceptibility tensor is isotropic for paramagnetic system, therefore omitting anisotropy for the present system

2

χ +− (q, ω) = (χ xx + χ y y + χ zz )

(8)

The susceptibility explicitly defines that in an isotropic paramagnetic system s-electrons are equally scattered from all three components of magnetization. The standard transport theory [22] can now be used to calculate the transport property in terms of scattering probability W k+q←k

ρ

para

=



1 2k B T

para

(k − k+q ) W k+q←k dkdq  0 | ev k k ∂ f∂ (k ) dk|2 ∂ f 0 ( )

where f s (k ), is s-electron Fermi distribution function. The last term in Hamiltonian from the z-component of spin density of d-electrons get canceled due to scattering from the  same spin state of s-electrons. We employ the identity δ  = e i  t 2dtπ and

3 J2

para W k+q←k

(9)

h¯ k where e is the electronic charge, v k = m is the Fermi veloc-

−

+ < d( I )| S − (−q)|d( F ) >< d( F )| S + (q)|d( I ) >   × δ d ( F ) − d ( I ) + s (k + q) − s (k)

para

χ −+ (q, ω) as

k

I,F

W k+q←k =

lution of operators implies Heisenberg representation: S ± (q, t ) = e i H d t S ± e −i H d t . The integral term describes the Fourier transform of the correlation function of spin densities of d-band electrons. Using the Fluctuation-dissipation theorem [21], we express the transipara tion probability W k+q←k in terms of the dynamical susceptibility

3

2

The eigenstate of the system can be approximated by the product of the form |k |d( I ) , where the function |k describes the state of the s-electrons, and |d( I ) the states of the d-electrons system. By employing the Fermi golden rule, one finds that s-electron of wave vector k s with spin up/down is scattered into the state k + q with spin down/up at the rate [20] para W k+q←k

and < ... > denotes the thermal average. Time evo-

− k)

e −ik.r S (r )dr

2π

h¯ 2 k2 , 2ms

k =

× Imχs−+ (q, ω)δ(h¯ ω − k + k+q )

where J is s-d electron coupling constant, N is the number of atoms in the system, the a† and a are the creation and annihilation operators for s-electrons. S − (k − k) and S + (k − k) are lowering and raising spin density operators of d-band electrons, S z (k − k) represents z-component of spin density of d-band electrons and S (k) is the Fourier transform of spin density S (r ) of d-electrons and it is defined by

S (k) =

where h¯ ω = k − k+q is an energy transfer provided by s-electrons,

s

ity of s-electrons, and −k ∂  k is the measure of deviation k from the equilibrium in the electron distribution, the trial function k itself a measure of this deviation. If the usual assumption k = const . × q.u is made, and the variational integral in the denominator is solved by the assumption of isotropy in the electron distribution, then the resistivity expression (9) reduces to

para 2d

ρ

=

3 J 2 h¯ 2 N 2 k B T (ens )2

 ×

+∞

K F d2 k

dω −∞

0

d2 q(u .q)2 f 0 (k )(1 − f 0 (k+q ))(−n(−ω))

× Imχ −+ (q, ω) δ(h¯ ω − k + k+q )

(10)

where u is a unit vector parallel to the electric field and ns is the number of s-electrons per unit volume. Using property f (x)δ(x − ∞   2π a) = f (a)δ(x − a), and writing d2 k = 0 kdk 0 dφ

K. Kumari et al. / Physics Letters A 383 (2019) 2645–2651

para 2d

ρ

=

+∞

3(π ) J 2 h¯ 2 N 2 k B T (ens )2



q3 dqImχ −+ (q, ω)

dω (−n(−ω)) −∞

∞

Thus R(q, ω) is the real part of non-interacting d-electrons susceptibility with wave number q and frequency ω . I(q, ω) is the corresponding imaginary susceptibility



I(q, ω) = π

k dkf 0 (k )(1 − f 0 (k − h¯ ω))

×

2π dφδ(h¯ ω − k + k+q )

×

(11)

Simplifying the integral with respect to φ (Appendix A) and writq2

ing (u .q)2 = 2 for unit vector u, which is parallel to electric field and q. The expression gives

=

+∞

3(2π ) J 2 N 2 h¯ 2 k B T (ens )2

∞



dω (−n(−ω))

2

q dqImχ

−+

q0

− q20 (q,

(12)

ω)

here q0 = + mh¯ qω < k F , k F is Fermi wave vector of s-electron and on performing k integral (Appendix B) expression (12) reduces to

ρ

=

+∞

3(2π ) J 2

m

2μ N 2 h¯ 2 k B T (ens )2

2kd 2

×

q dqImχ 0

−+



dω

+

I N



†

q,k,k ,σ ,σ 

†

C k+q,σ C k,σ C k ,σ  C k +q,σ 

(13)

(14)

Here I is the exchange interaction parameter for d-band electrons, N is the number of lattice points. The susceptibility has been calculated in a famous paper by Izuyama et al. (1963) [21]. Here C and C † are the annihilation and creation operators for d-electrons, ε(k) is electron energy of Bloch state k of d-electrons. Using random phase approximation the transverse susceptibility [IKK] [21] is given by

χ −+ (q, ω) =

−+ (q, ω) 1 − I −+ (q, ω)

h¯ 2 k F q

(15)

f (εk ) − f (εk+q )

k

ε(k + q) − ε(k) − h¯ ω

p( 1a ) ∓ i π δ(a) to equation (16) the real part can be written as f (εk ) − f (εk+q )

k

ε(k + q) − ε(k) − h¯ ω

N (0) q¯  4

(

2

)2 q kF

(21)

=

q 2kd

is

(22)

kF

1

(23)

k F being the magnitude of wave number vector on the Fermi surface. The condition (23) is applicable since we are concerned with in the radius of Fermi surface of s-electron and the condition (22) comes from the fact that the energy change h¯ ω of s-electron in a hk q scattering process in order of ¯ mF (ms is the mass of s-electrons). s The imaginary part of susceptibility (I(q, ω)) can be solved as

   ∂ fk ∂ k (−q cos φ)δ q cos φ − h¯ ω ∂k ∂k k      ∂ f k ∂ εk ∂ εk − (24) cos φδ q cos φ − h¯ ω =π q ∂ εk ∂ k ∂k

I(q, ω) = π

k





 2π

1 Converting summation into integral as N (0)dε 0 dφ k = 2π and N (0) defines the density of states in two dimension. We re∂f place − ∂ εk = δ(ε − ε F ) and first order derivative of energy with k respect to k vector by h¯ v F . The integral equation becomes as follows:

I=

qh¯ v F

ε F N (0)δ(ε − ε F )dε

2





2π

dφ cos φ δ qh¯ v F cos φ − h¯ ω 0

(25) (16)

which the band structure function k is put to be constant. This can be seen if one’s set I = 0. Employing identity limη→0 a±1i η =



)2  N (0) +

2mv F

0



−+ (q, ω) is the susceptibility of non interacting electrons, for

R(q, ω) = P

qh¯

= v F q  |h¯ ω|,

where

−+ (q, ω) =

4

(

where N (0) is the density of states at Fermi level. q¯  = dimensionless wave-vector and h¯ k F = mv F .

q

(−n(−ω))ω (q, ω)( β h¯ ω ) e −1

N (0)

and

−∞

ε(k)C k† ,σ C k,σ

k,σ

x

R(q, ω)  N (0) +

m

To proceed further some assumption about χ −+ (q, ω) has to be made. We use Hamiltonian for d-electrons [14,21]

Hd =

(20)

q ∂∂k + 12 (q.∇)2 ε |k − h¯ ω

k

The expression can be approximated to

q 2

para 2d

 −q ∂∂kf − 12 (q.∇)2 f |k − 16 (q.∇)3 f |k x

(q, ω)

−∞

k2

(19)

We note that R(q, ω) is an even function of q. It can be expressed as

R(q, ω) =

k dkf 0 (k )(1 − f 0 (k − h¯ ω))



×



Thus the electron spin susceptibility of d-electron in terms of real and imaginary part is written as

−+ (q, ω) = R(q, ω) + i I(q, ω)

0

ρ

 f (εk ) − f (εk+q ) δ[ε (k + q) − ε (k) − h¯ ω] (18)

k

0

para 2d

2647

Using property of delta function

0

N (0)(ε )δ(ε − ε F )dε =

N (0) 2

for

energy integral and using δ(ax) = |1a| δ(x) for φ integral, we get

I=

N (0)



π

dφ cos φ δ cos φ −

2 0

(17)

 εF

Using

property

cos−1 ( ω

δ( f (θ)) =

) = θ0 , we obtain qv F

ω

 (26)

qv F



δ(θ−θ0 ) θ0 | f  (θ0 )| ,

and

setting

θ =

2648

I=

K. Kumari et al. / Physics Letters A 383 (2019) 2645–2651

N (0) 2



( qvωF )

4E d

(27)

1 − ( qvω )2

ρ

F

Apply the condition

I=

N (0) 2

Here q¯ = energy.

I=

(

ω qv F

q kd

para 2d

= p 0kd4 (

4k B T Ed

ωk−D 1 2q¯ v F

(28)

)

, kd is the d-electron Fermi vector,

h¯ 2 kd2 2md

Put ζ =

kB T Ed

Collecting the above information imaginary part of susceptibility (15) reduce to

I (1 − IR)2 + I 2 I2

(30)

Substituting real and imaginary part of transverse susceptibility from equations (21) and (27) into Imχ −+ (q, ω), we have

para ρ2d = p 0kd4 (4ζ ) 3

0

(35)

para 2d

ρ

(32)

¯

]2 + [ I4 q¯h¯Eω ]2

u du (e u − 1)(1 − e −u )

=

4 p 0 kd4 (4ζ ) 3

∞

4

u 3 du (e u − 1)(1 − e −u )

And we have an important result

d

N (0) h¯ ω 4 q¯ E d d

2 1 1 ( uζ ) 3 2 4

t3 t6 + 1

dt

(36)

∞ 0

t3 t6 + 1

dt

(37)

4

para ρ2d ∝T 3.

3.2. Case 2: high temperature limit (k B T >> E d ) For high temperature limit, the upper limit of the integral is 4E d << 1, the expression (35) becomes as k T

ξ=

B

para 2d

ρ

= C0 T

4 3

ξ 0

4

u 3 du u (e − 1)(1 − e −u )

(38)

4

B 3 where C 0 = p 0 kd4 ( 4k ) , integral over t (last factor) reduces to Ed some constant value in the high temperature limit. In the high temperature limit we have small u and expanding the exponent in the integrand leads to

4

2

para ρ2d  C 0 T 3 |u− 3 |u= ξ × ξ

3. Result



4 3

The above expression in low temperature ( T → 0) regime reduces to

+ I 2 [ N (40) q¯h¯Eω ]2

Here k20 = 1 − I N (0) = 1 − I¯ denotes the inverse of the RPA exchange enhancement factor for d-band. We are interested in the behavior of system for k20 = 0 i.e. c = c F . At c = c F one shift the critical point to a classical point to a desired low temperature regime. In other words nearness to a QCP is brought by chemical doping [14]. Here one can focus on the low temperature regime [23] for physical properties near the critical point. Therefore susceptibility takes the form:

43

dt

1

1 ( 2uζ )3

4

ζ

0

(31)

[

la

+1

, the expression (35) converts to

4

Imχ −+ (q, ω) = ¯ 2 Iq¯

t3 t6

d

(29)

[1 − (I N (0) +

lb

3.1. Case 1: low temperature limit (k B T << E d )

4 q¯ E d

N (0) h¯ ω 4 q¯ E d I N (0) q¯ 2 2 ( 4 ) )] 4



4

u 3 du (e u − 1)(1 − e −u )

1 −1 k B T 23 where la = 12 ( u4E ) , and lb = ( E d ) 3 (2uk B T ) 3 .

= E d its Fermi

N (0) h¯ ω

Imχ −+ (q, ω) =

0

ω << qv F , the I(q, ω) reduces to

) = N (0)(

Imχ −+ (q, ω) =

)

4 3

k B T

(39)

2

Writing 2d paramagnetic resistivity replacing Imχ −+ (q, ω) from equation (32) in expression (13) we have

para ρ2d =



2kd ×

0

ωd ω

dqq md ω h¯ kd

observe T

ω

(33)

6

[ q6 ( 4Eh¯ dω )2 + 1]

1 q ( E d ) 3 , q3 = t 3 kd3 ( 42Eh¯dω )−1 kd 42 h¯ ω 2 m (12π ) J N (0) E d , then 2μ N 2 h¯ 3 I¯ 2 (ens )2 kd

and write prefactor p 0 =

4E d

ρ

=

4h¯ 4 p 0 kd4 ( ) 3 Ed

para ρ2d ∝ T.

4 3

curvature at lower ζ and T -linear behavior at higher ζ .

4. Discussions

kd

put t =

para 2d

we get,

In the general case analytic solution is not feasible, so the numerical solution of eqn. (36) is plotted in Fig. 1, where one can

(e β h¯ ω − 1)(1 − e −β h¯ ω )

q 2

4E d kB T

3.3. Case 3: numerical solutions

4E d

 h¯

(12π ) J 2 N (0) E d 2 2μ N h¯ 3 I¯ 2 (ens )2 kd k B T m

As ξ =

h¯

4

ω 3 dω (e β h¯ ω

0

− 1)(1 − e −β h¯ ω )

lb

t3 t6

la 2

+1

dt , (34)

1

h¯ ω 3 where limits for t-integral change to la = 12 ( 4E ) and lb = ( 2h¯ dω ) 3 . d We put β h¯ ω = u to make integrals temperature independent. Then resistivity simplifies to E

In our calculation mobile s-electrons scatter via localized d-electrons. Important point is to be noticed is that in our model there is a clear separation of two subsystems of electrons in which momentum of one subsystem (s-electrons) is dissipated via fluctuations of the other electronic subsystem (d-band). This very important clear separation is not given in a somewhat related literature where this exponent is derived in the context of gauge field fluctuations in the context of RVB model of high T c superconductors [25,26]. In these references the model is manifestly one-band model and it is hard to understand where the dissipation going on, whereas in the present case we have s − d Hamiltonian which is a two-band system. This clear separation is required for momentum dissipation and resistivity. As electron-electron scattering conserve

K. Kumari et al. / Physics Letters A 383 (2019) 2645–2651

2649

5. Conclusion We have performed a calculation for electrical resistivity in a 2d metal which is tuned near to its ferromagnetic instability. The required dynamical susceptibility in the expression of resistivity is calculated using Random Phase approximation. In 2d we find that the real and imaginary parts of dynamical susceptibility are proportional to q2 and ω respectively. This is similar to 3d case. q However, we find that the resistivity calculated for 2d case scales 4

4

para

2d as ρ para ∝ T 3 . Our result (ρ2d ∝ T 3 ) agrees with the SCR theory and the reason is also clear. We have also calculated resistivity in the high temperature limit, which shows linear dependence with temperature.

Acknowledgements Fig. 1. Resistivity as a function of scaled temperature (ζ =

kB T Ed

).

momentum, a one band model with only electron-electron scattering cannot show resistivity. To show resistivity there must be either phonon scattering or Umklapp or impurity scattering or scattering via another band of electrons [19]. We believe that in the calculations of [24–26], this very important clear separation is not given. Next we compare our calculation with that in the SCR theory. As our result obtained using RPA susceptibility matches with that obtained using SCR theory, it is also important to consider physical arguments that go into the SCR theory [15,16]. In RPA, static susceptibility is written as

χ (T ) =

χ0

(40)

1 − 12 I χ0

I (Komal Kumari) thank Physical Research Laboratory (PRL), for providing me local hospitality during this work. We thank Prof. Dietrich Belitz for clarifying the importance of first order transitions in clean magnetic metals, and importance of disorder in bringing the second order transitions in such situations. Appendix A. Mathematical details of φ integral

1−

1 I 2

q 2k

χM+− I (q, ω) = i

∞ dte

βω 2

−i ωt



q

+ ). We use delta 2k

 

f (k,q,ω)

δ(x−xi ) i | F  (xi )| .

dφ δ(cos φ + f (k, q, ω))

2π

)G (ω)

 +− +− dI  {χ M I  (q, ω) + χ M0 (q, ω)}

0

h¯ kq

2π

=

 +∞

∂ G (ω) = −χ0 Im ∂ M2

mω

ω + We can put cos φ + m h¯ kq q q mω mω − 1 = 0, φ = cos (− h¯ kq − 2k ) = φ0 (q, k, ω) and here h¯ q + 2 < k.

(41)

where λ( T ) = 21π −∞ dω cot h(

I

dφ δ(cos φ +

0

χ0 χ0 + λ( T )

2

0

function property δ( F (x)) =

Whereas, in SCR theory it takes the form:

χ (T ) =

 2π

To solve the term

dφ 0

(42)

=

δ(φ − φ0 (q, k, ω)) | − sin φ|φ0 2π

1 sin φ0

q

= +

−

[ S (q, t ), S (−q)] M I

(43)

dφδ(φ − φ0 ) = 0

1 ω 1 − (m h¯ kq

1 sin φ0 1

+

q 2 ) 2k

0

=  2 mω q k − ( + )2  h q 2 ¯ 

 

(44)

q20

In eqn. (41) the term λ( T ) takes into account the renormalization of the thermodynamical canonical state due to spin fluctuations. Spin fluctuations increase the entropy of the paramagnetic state. This effect is completely neglected in the calculation of the RPA susceptibility. However, our result of the calculation of resistivity shows that this renormalization effects are not important in this

Appendix B. Mathematical details of k integral

∞ Ik =

4 3

2d specific case (that is ρ para ∝ T in both approaches). It is also to be noted that renormalization effects are very important for the calculation of the correct Curie temperature (in RPA, calculated Curie temperature is much greater than that calculated using SCR theory [15]). Thus we showed that although renormalization effects are important in correct calculation of T c , those are not important for the calculation of D.C. resistivity near the critical point. It para 2d

4 3

is understable as the result ρ ∝ T is valid near T → 0 and the quantum critical point is tuned by a non-thermal parameter i.e. doping in this case. At low temperature thermally excited magnetic fluctuations are negligible.

k dk

f 0 (k )(1 − f 0 (k − h¯ ω))



Converting k integral into energy k = lower limit for energy becomes infinite.

I =

m h¯

(45)

k2 − q20

q0

∞ d

2

0

0 =

h¯ 2 q20 2m

√

2m , h¯

√

md dk = √ and writing h¯ 2

and upperlimit for energy integral

f 0 (k )(1 − f 0 (k − h¯ ω))



2m h¯ 2

(46)

− q20

Replacing the value Fermi function f 0 (k ) =

1 e β( −μ) +1

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K. Kumari et al. / Physics Letters A 383 (2019) 2645–2651

√

m I = √ h¯ 2

=

1



h¯

∞

d



−

0

m 2

∞ 0

(

h¯ 2 q20 2m

1 e β( −μ) + 1

 ) 1−



1

R(q, ω) =

e β( −μ−¯hω) + 1

 β( −μ−¯hω)  d 1 e ( ) √  − 0 eβ( −μ) + 1 eβ( −μ−¯hω) + 1

I =

α β h¯

m

0

×

(47)

∞

2 0

du 1

μ + β log u − 0

(

1 u+1

)(

1

+

I =



β h¯

du

0

0

I =

=



β h¯

α

m

 ∞

2μ



0

m

du

1

(

+

(u + 1)(1 − α )

∞

α α−1

0

du



) log | |

(50)

α

α = e−β h¯ ω , we have  

α m − log(e−β h¯ ω ) m ω I = ( ) = β h¯ 2μ 2μ e β h¯ ω − 1 1 − e −β h¯ ω

0

 −(q.∇) f |k − 1 (q.∇)2 f |k − 1 (q.∇)3 f |k 2 6 q ∂∂k + 12 (q.∇)2 ε |k − h¯ ω

∂ f ∂  ∂k ∂ k ∂ kx

 −q ∂ 

q ∂∂k +

k

x

∂f

−

q ∂ f 2 ∂ k2x

q2 ∂ 2  2 ∂ k2x

(52)

R(q, ω) =

−

2

3

q ∂ f 6 ∂ k2x

− h¯ ω

(53)

R(q, ω) =

N (0)

π

qh¯ v F δ( −  F ) cos θ +

+

q3 6

q

2

2

 (h¯ v F )3 δ  ( −  F )

2m

Converting sum into integral, we have

(56)

¯ω cos2 θ − h¯ hqv

F

¯ ω , the above integral becomes and β = h¯ hqv F

cos φ dφ

(57)





I

N (0)

π

(

1

−α

1 )

√

−1

dx 1−

x x2

[(x −

1 2 ) 2α

(58)

− γ 2]

1

N (0)

π

(

1

−α

×

√

)

dx



( 

1

η+η x−η

1 − x2

−1

η

)+

η

1

( 

η+η x+η

 )  (59)

1

1

where η = 21α + ( α + 4α1 2 ) 2 and η = − 21α + ( α + 4α1 2 ) 2 are q dependent parameters. In limit q → 0, α and β are very small values, so that η >> 1 and η << 1. Thus we have

(h¯ v F )2 δ  ( −  F )

  −1 h¯ 2 q2 × qh¯ v F cos θ + cos2 θ − h¯ ω

qh¯ 2mv F

cos φ − α cos2 φ + β

β

 k

cos θ +

qh¯ 2mv F

α=

1

=

takes the form

cos θ dθ

π

1

∂f

∂ ∂f ∂ kx ( ∂ kx )

R(q, ω) =

2π

N (0)

γ = ( αβ + 4α1 2 ) 2 . The integral reduces to

∂ ∂ f ∂  ∂k ∂ kx ( ∂  ∂ k ∂ kx ), therefore the double derivative of Fermi function becomes −(¯h v F )2 cos2 θδ  ( −  F ). The real part

=

(55)

cos2 θ − h¯ h¯qvω F

Set cos φ = x, thus we obtain

writing ∂ k = ∂  ∂∂ k ∂∂kk = −¯h v F δ( −  F ) cos θ and kkx = cos θ . The x x higher derivative of Fermi function can be written in the form ∂2 f ∂ k2x

qh¯ 2mv F



here

R(q, ω) =

cos θ +

(51)

simplifies to 2



cos3 θ dθ

0

x

2

N 2d (0)δ  ( −  F )d

2π

1

put θ = φ + π ,

R(q, ω) =

Appendix C. Mathematical details of real part of susceptibility

k

×

0

writing

R(q, ω) =

6

cos2 θ − h¯ h¯qvω F

0

R(q, ω) =

αu + 1

1

2μ 1 − α

β h¯

∞

(qh¯ v F )2

+

qh¯ 2mv F

the expression further gives

This elementary integral reduces to

α

cos2 θ dθ cos θ +

0

(49)

(u + 1)(α u + 1)

2μ

cos2 θ − h¯ h¯qvω F

N 2d (0)δ  ( −  F )d

2π

∞

m

2

qh¯ 2mv F

2π

(48)

)

∞

qh¯ v F

×

αu + 1

cos θ dθ cos θ +

0

Using the above defined condition of low temperature limit, we √ can reduce the square root term as μ + β1 log u − 0  μ.

α

N 2d (0)δ( −  F )d

π 2π

To simplify it further we take α = e −β h¯ ω , u = e β( −μ) , and its log gives  = μ + β1 log u. In low temperature case if we apply condition 0  μ the lower limit for u becomes zero and higher limit goes to infinity. The above expression converts in new form as



 ∞

1

R(q, ω) =

N (0)

π

(

1

)

η

−α η + η

β

1 √ −1

dx 1−

1

x2

(− )

1

η (1 − ηx )

(54)

(60) Using series expansion method to solve integral, we get

K. Kumari et al. / Physics Letters A 383 (2019) 2645–2651

R(q, ω) =

N (0) (η + η )−1

π +

 1

α 1

1 2η 2

√ −1

x2 dx

√ −1

dx 1 − x2

+

1

η

1 √ −1

xdx 1 − x2

 (61)

1 − x2

for small α and β sum of η and η gives (α )−1 . To solve integral terms we set cos θ = x, which reduces the expression as

π R(q, ω) = [π + 2 ] π 4η N (0)

setting

η=

1 α =

2mv F qh¯

R(q, ω) = N (0) +

(62)

, the real part of susceptibility becomes

N (0) 4

(

qh¯ 2mv F

)2

(63)

References [1] Navinder Singh, Electronic Transport Theories from Weakly to Strongly Correlated Materials, CRC Press, 2016. [2] M. Brando, D. Belitz, F.M. Grosche, T.R. Kirkpatrick, Rev. Mod. Phys. 88 (2016) 025006. [3] G.R. Stewart, Rev. Mod. Phys. 73 (2011).

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2651

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