About the thermodynamics of the phonon-induced

spin density waves

R. Szczesniak*

Institute of Physics, Czestochowa University of Technology, Al. Armii Krajowej 19, 42-200 Czestochowa, Poland

Received 23 February 2005; accepted 28 March 2005 by A.H. MacDonald

Available online 13 April 2005

Abstract

In the paper, the s-wave pseudogap, which originates from phonon-induced spin density wave (SDW) is considered. The

exact numerical solutions of the SDW Eliashberg equations are presented. In the self-consistent way, the SDW transition

temperature is calculated. The SDW order functions and the wave function renormalization factors as a function of the

Matsubara frequency for different values of the temperature are plotted. It is also shown that the ratio of the zero-temperature

SDW gap to the SDW critical temperature differs a little from the standard weak-coupling value.

q 2005 Elsevier Ltd. All rights reserved.

PACS: 74.72.Bk

Keywords: D. Spin density waves; D. Pseudogap

1. Introduction

From a theoretical point of view the existence of spin

density waves in high-Tc superconductors can be considered as

a possible scenario, which explains the anomalous property of

the normal state in this materials (the normal state pseudogap).

Experimental data, e.g. the nuclear quadrupole reson-

ance [1] and the nuclear magnetic resonance [2], show the

large value of the pseudogap critical temperature T* and the

very small isotope shift a*. For YBa2Cu4O8 the crossover

temperature T* equals 150 K and the isotope exponent a*Z0.061(8). Let’s notice, that the corresponding superconduct-

ing transition temperature in YBa2Cu4O8 is TSCZ81 K and

adequately the isotope exponent aSCZ0.056(12) [1]. In

high-Tc superconductors the ratio 2D*/kBT*, where D* is the

zero-temperature pseudogap is very large. For example for

Nd2KxCexCuO4 the ratio 2D*/kBT* approximately equals 10

0038-1098/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ssc.2005.03.058

* Tel./fax: C48 343250795.

E-mail address: szczesni@mim.pcz.czest.pl.

[3]. The experimental data in cuprates indicate that the

normal-state pseudogap has the d-wave symmetry [3]. On

the other hand, the nontrivial angular dependence of the

pseudogap suggests that the pairing fluctuations should

include also the s-wave component [4].

In the paper, we assume that the electron–phonon

induced SDW state can explain the s-wave part of the

normal state pseudogap in high-Tc superconductors. The

electron–phonon induced SDW is described by the SDW

Eliashberg equations [5]. In the framework of the SDW

Eliashberg theory, we consider the electrons in the two-

dimensional lattice, however, the electron–phonon inter-

action must occur in the three-dimensional lattice. This

assumption is connected with the strong anisotropy in the

internal structure of high-Tc superconductors [5]. In one

band calculation, the SDW Eliashberg equations create a

coupled set of six non-linear equations which determine the

SDW order functions P0(iun) and P1(iun), the wave

function renormalization factors Z0(iun) and Z1(iun) and

the energy shift functions c0(iun) and c1(iun), where the

symbol un is the Matsubara frequency. In Eliashberg

Solid State Communications 135 (2005) 38–43

www.elsevier.com/locate/ssc

R. Szczesniak / Solid State Communications 135 (2005) 38–43 39

formalism, presented in the paper, the functions P0(iun) and

P1(iun) describe the strength of the phonon-mediated

electron–electron potential, Z0(iun) and Z1(iun) determine

the enhancement of the electron mass due to the electron–

phonon interaction. The functions c0(iun) and c1(iun)

describe the renormalization of the energy band due to the

interaction effects. The SDW Eliashberg theory in contrast

to the weak-coupling SDW model handles momentum cut-

off and Matsubara frequency cut-off separately. In particular

the Matsubara frequency cut-off answers to the Balseiro–

Falicov energy cut-off [6].

So far the SDW Eliashberg equations has been analysed

only analytically. In the paper [5], the SDW transition

temperature TSDW and the SDW isotope coefficient aSDW

are calculated. The wave function renormalization factors

Z0(iun) and Z1(iun) were considered as one parameter Z.

Simultaneously the Matsubara frequency dependence of the

SDW order functions P0(iun) and P1(iun) was neglected.

The results included in the paper [5] suggest that the large

value of the SDW critical temperature and the small value of

the isotope coefficient are connected with the van Hove

singularity in the density of states, the large value of wave

function renormalization factor Z and the large value of the

electron–phonon coupling function.

In this article, we analyse the SDW Eliashberg equations in

the self-consistent way. We present the numerical results for

most important SDW parameters: the SDW transition

temperature TSDW and the ratio 2DSDW/kBTSDW, where

DSDW is the zero-temperature SDW gap. We also precisely

analyse the Matsubara frequency dependence of the SDW

order functions and the wave function renormalization factors

for different values of the temperature. We notice that within

the SDW Eliashberg theory the calculation of the SDW order

functions and the wave function renormalization factors are

very complicated. Nevertheless, this problem is possible to

solve by using a computer program because the SDW

Eliashberg equations are written on the imaginary frequency

axis thus the sums are discrete and the order functions and the

wave function renormalization factors are always real. The

numerical results presented in this paper are of great

importance also as criterion for the validity of approximations

used in applying analytical methods in the paper [5].

2. Model

2.1. SDW Elisahberg equations

The standard SDW Eliashberg equations we rewrite in

the form [5]:

P0ðiunÞZ1

b

Xkn

LnuDKðun KunÞðP0ðiunÞ

CP1ðiunÞÞDK11k ðiunÞ (1)

P1ðiunÞZP0ðiunÞC1

b

Xkn

ðLnW KL

nuDÞ

!Kðun KunÞP0ðiunÞDK12k ðiunÞ (2)

Z0ðiunÞZ 1C1

unb

Xkn

LnuDKðun KunÞunðZ0ðiunÞ

CZ1ðiunÞÞDK11k ðiunÞ (3)

Z1ðiunÞZ Z0ðiunÞC1

unb

Xkn

ðLnW KLn

uDÞ

!Kðun KunÞunZ0ðiunÞDK12k ðiunÞ (4)

c0ðiunÞZK1

b

Xkn

LnuDKðun KunÞð3k Cc0ðiunÞ

Cc1ðiunÞDK11k ðiunÞ (5)

c1ðiunÞZc0ðiunÞK1

b

Xkn

ðLnW KLn

uDÞKðun KunÞ

!ð3k Cc0ðiunÞÞDK12k ðiunÞ (6)

where:

D1kðiunÞh ðunðZ0ðiunÞCZ1ðiunÞÞÞ2 C ð3k Cc0ðiunÞ

Cc1ðiunÞÞ2 C ðP0ðiunÞCP1ðiunÞÞ

2 (7)

and

D2kðiunÞh ðunZ0ðiunÞÞ2 C ð3k Cc0ðiunÞÞ

2 CP20ðiunÞ (8)

We assume simple electron band energy 3kZKtg(k),

where: t is the nearest-neighbour hopping integral, gðkÞh2½cosðkxaÞCcosðkyaÞ�; ah ða; aÞ is the basis vector for

the two-dimensional square lattice. In the numerical

calculations we take t as energy unit. In our consideration

the chemical potential m equals zero (the half-filled band). In

this case, the energy shift functions c0(iun)Zc1(iun)Z0

fulfils the SDW Eliashberg equations. The symbol Lnx is

called the cut-off operator and is defined byXn

LnxFðiunÞh

Xjunj!x

FðiunÞ (9)

where: x is equal to uD or W; uD is the Debye frequency and

W is the half band width. The function F(iun) is appropriate

Matsubara frequency dependent function; unh(p/b)(2nC1), bh1/(kBT) is the inverse temperature and kB is the

Boltzmann constant. The function K(unKun) is written in

the form [7]:

Kðun KunÞZ ly2

ðnKnÞ2 Cy2(10)

Fig. 1. The SDW transition temperature TSDW as a function of the

Debye phonon frequency uD for different values of the electron–

phonon coupling function l. The inset shows the SDW transition

temperature calculated in analytical scheme (Eq. (20)).

R. Szczesniak / Solid State Communications 135 (2005) 38–4340

where:

yhbuD

2p(11)

and l is the electron–phonon coupling function. We notice

that in this case the phonons are considered in the harmonic

approximation and the internal structure of the Eliashberg

electron–phonon spectral function is neglected. We intro-

duce the density of states r(3) for two-dimensional square

lattice using our simple electron band energy function.

Then,

rð3ÞZ b1 ln3

b2

�������� (12)

where: b1ZK0.04687tK1 and b2Z21.17796t [8]. Thus we

study the strong-coupling SDW state in the van Hove

scenario.

The k-momentum dependent part of Eqs. (1)–(4) we can

calculate analytically. Then,

D*j ðiunÞh

Xk

DK1jk ðiunÞx

ðWKW

d3rð3ÞDK1j3 ðiunÞ

Z2

AjðiunÞrðWÞarctan

W

AjðiunÞ

� �Kb1 Im Li2

iW

AjðiunÞ

� �� �� �(13)

where: the index j is equal to 1 or 2 and

A1ðiunÞh ½ðunðZ0ðiunÞCZ1ðiunÞÞÞ2 C ðP0ðiunÞ

CP1ðiunÞÞ2�1=2 (14)

A2ðiunÞh ½ðunZ0ðiunÞÞ2 C ðP0ðiunÞÞ

2�1=2 (15)

The function Lin(z) is called the polylogarithm function and

LinðzÞhPCN

kZ1 zk=kn.

2.2. Numerical results

The starting point of a calculation by a numerical method

are the Eqs. (1)–(4). The set of Eqs. (1)–(4) must be solved

self-consistently when we want to describe the SDW critical

temperature TSDW, the order functions P0(iun), P1(iun) and

the wave function renormalization factors Z0(iun), Z1(iun).

Nevertheless, some discussion of the SDW Eliashberg

equations is possible.

In particular, the SDW order functions and the wave

function renormalization factors are completely symmetri-

cal in the Matsubara frequency:

P0ðiunÞZP0ðKiunÞZP0ðiuKðnC1ÞÞ

P1ðiunÞZP1ðKiunÞZP1ðiuKðnC1ÞÞ

Z0ðiunÞZZ0ðKiunÞZZ0ðiuKðnC1ÞÞ

Z1ðiunÞZZ1ðKiunÞZZ1ðiuKðnC1ÞÞ

This important property of these functions results in new

form of the SDW Eliashberg equations where we consider

only positive Matsubara frequency:

P0ðiunÞZ1

b

Xu0%un!uD

½Kðun KunÞCKðun CunÞ�

!ðP0ðiunÞCP1ðiunÞÞD*1 ðiunÞ (16)

P1ðiunÞZP0ðiunÞ

C1

b

Xu0%un!W

KX

u0%un!uD

" #½Kðun KunÞ

CKðun CunÞ�P0ðiunÞD*2 ðiunÞ (17)

Z0ðiunÞZ 1C1

unb

Xu0%un!uD

½Kðun KunÞ

KKðun CunÞ�unðZ0ðiunÞCZ1ðiunÞÞD*1 ðiunÞ

(18)

Z1ðiunÞZ Z0ðiunÞC1

unb

Xu0%un!W

KX

u0%un!uD

" #

!½Kðun KunÞKKðun CunÞ�unZ0ðiunÞD*2 ðiunÞ

(19)

The latter equations are much easier to implement

numerically than the SDW Eliashberg equations described

in the standard form.

Numerically calculated SDW critical temperature as a

function of the Debye phonon frequency uD for different

values of the electron–phonon coupling function l is shown in

the Fig. 1. The inset in Fig. 1 shows the transition temperature

TSDW for lZ2t and ZZZ0(iu0)CZ1(iu0)Z2.351 calculated

Fig. 2. The SDW order function P0(iun) as a function of the integer

number n for different values of the temperature. The inset shows

the order function P0(iu0) as a function of the temperature.

Fig. 3. The SDW order function P1(iun) as a function of the integer

number n for different values of the temperature. The inset shows

the order function P1(iu0) as a function of the temperature.

R. Szczesniak / Solid State Communications 135 (2005) 38–43 41

from [5]:

kBTSDW Zab2

ZeK 1

leff (20)

where the effective electron–phonon coupling function has the

form:

1

leff

h ln2 W

b2

� �C ln2ð2aÞK2K

2

l1Zb1

� �1=2

(21)

and 1/l1 is defined by

1

l1

hZ4

l½2Z2 Cl½HI1 KHI��KHI (22)

The function HI and HI1 are written as follows:

HI ZK2b1Z

pln

ZuD

b2

� �i1

W

ZuD

� �C i2

W

ZuD

� �� �(23)

HI1 ZK2b1Z

pln

ZW

b2

� �j1

1

Z

� �C j2

1

Z

� �� �(24)

The function i1(x)Kj2(x) can be expressed by series:

i1ðxÞhp

2lnðxÞ þ

XþNn¼0

ðK1Þn

ð2nþ 1Þ21

x2nþ1(25)

i2ðxÞhXCN

nZ0

ðK1Þn

ð2nC1Þ21

x2nC1lnðxÞC

1

2nC1

� �C

p

4ln2ðxÞC2K

(26)

j1ðxÞhXCN

nZ0

ðK1Þn

ð2nC1Þ2x2nC1 (27)

j2ðxÞhXCN

nZ0

ðK1Þn

ð2nC1Þ2x2nC1 lnðxÞK

1

2nC1

� �(28)

where K is the constant,

KhXCN

nZ0

ðK1ÞnC1

ð2nC1Þ3xK0:96894 (29)

We can compare the numerical results for the SDW

transition temperature TSDW with the corresponding analytical

results calculated from Eq. (20). One can see that the SDW

critical temperature is reduced when we calculate TSDW

numerically in the self-consistent way. The above results could

be understood by taking into account that Eq. (20) was derived

by neglecting the Matsubara frequency dependence of the

SDW order functions and substitute the wave function

renormalization factors for one parameter Z. But despite

several approximations the analytical equation for TSDW

reproduces qualitatively good feature typical for results

originating from advanced strong-coupling numerical

calculations.

In the Figs. 2–5, we show the SDW order functions

P0(iun), P1(iun) and the wave function renormalization

factors Z0(iun), Z1(iun) as a function of Matsubara

frequency for different values of the temperature. We take

602 Matsubara frequency un, where the index n ranges from

K301 to 300. One can see that the order functions and the

wave function renormalization factors decrease when the

positive Matsubara frequency increases. The maximum

value of the order parameters and the wave function

renormalization factors are always for nZ0. The tempera-

ture dependence of the order functions and the wave

function renormalization functions presented in the Figs. 2–

5 prove directly that in the vicinity of the critical

temperature only values of the order functions and the

wave function renormalization factors with the lowest

Matsubara frequencies play important role in the SDW

Eliashberg theory. Additionally, the insets in the Figs. 2 and

3 show the maximal value of the order functions P0(iu0)

and P1(iu0) as a function of the temperature calculated self-

consistently. We can see that the functions P0(iu0) and

Fig. 4. The wave function renormalization factor Z0(iun) as a

function of the integer number n for different values of the

temperature. The inset shows the value of the function Z0(iun) for

nZ0.

R. Szczesniak / Solid State Communications 135 (2005) 38–4342

P1(iu0) have typical temperature dependence of the order

parameter. On the contrary, the maximal value of the wave

function renormalization factors Z0(iu0) and Z1(iu0) are

very poorly dependent on the temperature. In our model,

Z0(iu0)x1.128 and Z1(iu0)x1.223 (Figs. 4 and 5). The

independence of Z0(iu0) and Z1(iu0) from the temperature

proves that the many-body effects connected with the

electron–phonon interaction play important role in the

whole range of the temperature where the spin density

waves exist.

The pseudogap state is characterized principally by two

energetic parameters. Those parameters are the pseudogap

at zero temperature D* and the energy connected with the

crossover temperature T*. Experimental data e.g. for

Nd2KxCexCuO4 show that the ratio 2D*/kBT* equals 10

[3]. We notice that the conventional SDW weak-coupling

Fig. 5. The wave function renormalization factor Z1(iun) as a

function of the integer number n for different values of the

temperature. The inset shows the value of the function Z1(iun) for

nZ0.

theory with a constant density of states predicts R(0)Z3.53

and the ratio is independent on the model parameters [9]. Let

us start with a discussion of the size of SDW gap DSDW(iun),

where

DSDWðiunÞhP0ðiunÞCP1ðiunÞ

Z0ðiunÞCZ1ðiunÞ(30)

We consider the SDW order parameter for nZ0 and

calculate the ratio of the SDW gap parameter to the SDW

transition temperature, R(T)hDSDW(iu0)/kBTSDW. In the

Fig. 6, we show the ratio R(T) as a function of the

temperature for uDZ1.5t and lZ2t. We obtain in our

model [R(0)]maxx3.685 by fitting. The small value of the

function R(0) in comparison with experimental data

obtained for the high-Tc superconductors suggests that our

model demands corrections. Most likely the most important

modification of the presented theory will consist in taking

into consideration Coulomb interaction between electrons in

the d-wave channel.

3. Concluding remarks

Concluding this paper, let us summaries the important

thermodynamic properties of the SDW strong-coupling

model. In the paper, we considered the simple electron band

energy for two-dimensional square lattice with the nearest-

neighbour hopping integral. We assumed half-filled electron

band energy. In this case, the van Hove singularity in the

electronic density of states is situated at the Fermi level and

the energy shift functions equal zero.

The physical quantities are calculated by numerical

solution of the SDW Eliashberg equations. First, we

calculated the SDW critical temperature TSDW. We gave

also the interpretation of the validity of approximations used

in applying analytical methods in the paper [5]. We showed

that the analytical formula for TSDW describes qualitatively

Fig. 6. The ratio R(T) as a function of the temperature. The inset

shows the maximal value of the ratio for TZ0.

R. Szczesniak / Solid State Communications 135 (2005) 38–43 43

good the strong-coupling limit of the SDW Eliashberg

theory.

In the paper, we also obtained the SDW order functions

P0(iun), P1(iun) and the renormalization functions Z0(iun),

Z1(iun) as a functions of the Matsubara frequency for

different value of the temperature.

We showed that the maximal value of the functions

P0(iu0), P1(iu0) have typical temperature dependence of

the order parameter. On the contrary the maximal value of

the renormalization functions Z0(iu0), Z1(iu0) are very

weakly dependent on the temperature. The independence

from the temperature of the large value of the sum Z0(iu0)CZ1(iu0) represents fact that the many-body effects of the

electron–phonon interaction influences significantly in the

value of the thermodynamic SDW parameters. From a

physical point of view these results demonstrate that the

change of the effective electron mass due to the electron–

phonon interaction can not be neglected.

In the paper, we could analysed the behaviour of the ratio

R(T) as a function of the temperature. For TZ0 K we

obtained [R(0)]maxx3.685 by fitting. In real superconduc-

tors, the value of the ratio is 2D*/kBT*x10. The small value

of R(0) in our model clearly shows that the SDW Eliashberg

model are not completed. We notice that to simplify we do

not take the strong Coulomb interactions between electrons

and d-wave symmetry of the pseudogap into account. So in

next step in the future, we incorporate the one-site Hubbard

interaction [10] into SDW Eliashberg theory.

Acknowledgements

The author would like to express the gratitude to Prof

M.B. Zapart and Prof W. Zapart for creating excellent

working conditions. Prof J. Czerwonko for the encouraging

discussion. Dr J. Solecki for the technical help.

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