Specific heat and “susceptibility” in the Yu and Anderson model

Specific heat and “susceptibility” in the Yu and Anderson model

Volume 128, number 3,4 PHYSICS LETTERS A 28 March 1988 SPECIFIC HEAT AND “SUSCEPTIBILITY” IN THE YU AND ANDERSON MODEL Franco NAPOLI Is:ituto di Fi...
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Volume 128, number 3,4

PHYSICS LETTERS A

28 March 1988

SPECIFIC HEAT AND “SUSCEPTIBILITY” IN THE YU AND ANDERSON MODEL Franco NAPOLI Is:ituto di Fisica di Ingegneria, Università di Genova and Unità C.LS.M, Genoa, Italy

Maura SASSETTI Dipartimento di Fisica, Università di Genova and Unità C.LS.M, Genoa, Italy

and Enrico GALLEANI D’AGLIANO Istituto di Fisica di Ingegneria, Università di Genova and Unità C.I.S.M., Genoa, Italy Received 9 November 1987; revised manuscript received 22 January 1988;accepted for publication 25 January 1988 Communicatedby A.R. Bishop

Starting from the expression ofthe partition function Z for a local oscillator interacting with an electron gas expressed in terms ofthat ofa alassical “Coulomb” gas, we have calculated the specific heat and the displacive responseby an approximated evaluation ofsuch Z. Analogies of the present problem with the Kondo effect are briefly discussed.

A new approach to the problem of electron—phonon interaction in metals in the strong coupling regime has been proposed by Yu and Anderson [1}. In such a niodel one studies a gas of conduction electrons strongly interacting with a local oscillator (LO). The coupling with the electron gas provides

charge q interacting with a logarithmic potential, whose strength y< 2 [1] measures the non-adiabatic coupling between LO and electrons. Z is then the partition function of the classical gas in the grand canonical ensemble at temperature T~ = ‘,volume V5=fJ and fugacity y. We remind here that such a re-

for the (LO) ax~effective potential, together with a retarded self-interaction. For this model the partition function at temperature T=fl —‘is simply related [1,2] to the quantity 2~Z Z= ~ y 2,,(fi, y), (1)

sult applies when: (a) the interaction between the electrons and the LO is strong enough as to change the LO potential into a double well potential. (b) the fugacity y is much less than unity, which means small tunneling rateof the LO through the energy barrier between the two wells. We are thus bound to consider a very dilute gas. Such a gas is a somewhat peculiar “Coulomb” gas: indeed despite of its one-dimensionality the potential is logarithmic as in two dimensions and the particles are ordered in a given sequence (r,
where

J ~L J ~ $

fi/2

z2,, =

—p12

fi/2

+~

f~E~ to

to

/ 2n ~ Xex~y~ (—l~~~ln ‘~ ~ —

)

...

(2)

represents the partition function of a charged classical one-dimensional gas with 2n particles of size r0,

0375-9601/88/s 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

—

151

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PHYSICS LETTERS A

tion function (1) applies to an anisotropic Kondo problem (J± #J~)for general values of the parameters y and y, while the isotropic case (J÷ =J~)requires y= l/2—y/4. Because of condition (b) this means that only for y ~ 2 our results could compare with the solution of the isotropic Kondo problem. On the other hand, the results that are obtained in the present in they
28 March 1988

make a virial expansion of ln Z in powers ofy. As a first step, using eq. (3), we rewrite eq. (1) as Z= ~ j~2~’12~(y),

(5)

where we have introduced an “effective fugacity” 2 (6) j~y(fl/t0 ) 1 Y/expansion of ln Z gives The virtial ln Z= ~ ~P2”J 2~(y)

(7)

,

rect evaluation of Z, physical quantities as the specific heat C~and the generalised susceptibility x representing the displacement of the oscillator from its equilibrium position due to an external force. Let us limit for the moment to the range 0 ~ y~1, where Z2~is finite also when the cut-off t0 is set to zero; the effect of this cut-off is to introduce corrections 0 (t0/fl) which are negligible over a wide range of temperature. Setting to zero the cut-off and introducing dimensionless integration variables, Z2~can be written as =

~

$ ~$~ $

n(2—y)

\ 10!

1/2

1/2

/

—1/2

(—

1)

xj

‘~-‘

ln IX

n(

‘2n

12,

J4 14

—

~

(8)

...

represent the virial coefficients. The expansion (7) may be stopped after the first few terms only for j~4~ 1, which requires that fl~azfl,. where (9)

10

in fact We reach here thea conclusion high temperature that theapproximation. virial expansionWe is notice that Tk =fl~’represents the analogous of the

2n

X2,- I

—X1

)

Kondo temperature for this problem, because as it is evident from eqs. (6) and (7) it provides a natural temperature scale for ln Z. For y= 0 one has

2—y)

(~)

J2

1/2

2n

x exp(~i>j—_ ~ 1

where the quantities J2,, given by

(Y).

(3)

\to1

From eqs. (3) and (1) we immediately obtain the equation of state of the gas as

T~°~ =y/to=A, A LO being the well splitting between the lowest levels ofthe double potential induced by the tunneling through the energy barrier [2]. The specific heat a 2ln Z Cvfl2 (10)

nZ PgvgTgTg~~v =(l—ly)<2n>T 5,

g

(4)

where <2n> is the averaged particle number of the gas in the grand canonical ensemble. We remark thatoffor 0 eq. reduces to the equation of state an y= ideal gas (4) of <2n> particles, while for y—÷1 eq. (4), as it stands, would correspond to the equation of state ofan ideal gas of <,> dimerized particles. Yet the 1 limit requires more care [4]. In order to evaluate the partition function we will now exploit the conditionyctzl (very dilute gas), and ~

152

usingeq.(7)isgivenby

c~=(2—y)

2”J

n[n(2—y)— l]y~ 2~(y) (11) 4)this expression becomes Neglecting terms of 0(~ ~ (2...~) (1 —y)J 2~, (12) 2 (y) (Tk/T) and since J 2(y) may be evaluated exactly as 1 J2(y)= (2—y)(l—y) (13) one obtains to the lowest order in Tk/T

.

Volume 128, number 3,4

PHYSICS LETTERS A

~14 We notice that the behaviour of C~as T2 for y= 0 (adiabatic limit) corresponds to the tail of the Schottky anomaly in the specific heat of the LO, C~ (T~/T)

.

28 March 1988

good ofphase the values the interaction over arepresentative large portion of space.of We will therefore approximate the partition function (2) as p,,~2 dt p/2 dt p/2 dt Z 2,, exp( ~flgV~,,) j ~! j ~ j _~

which in such a limitbecomes a simple two level systern [2]. Turning now to the low temperature case we remarkthatasflgrowstheexpansioneq. (7) isnomore rapidly converging, and then one is obliged to consider the contribution of any number ofparticles, despite of the low density of the gas. Then at very low temperature we must follow a different approach. In the limit fl—*oo the interaction potential V2fl (tI...t2) for 2n gas particles has a stationary point corresponding to the configuration obtained distributing all the particles equally spaced between each other at 1P1‘2 fl~ ‘‘15’/ For such a configuration the interaction potential of the 2n particles is —

•~

—

2,, =

—q2 ~

—p/2

=

exp(flgv2n)

~)

1 (2n)!

(19)

Within the approximation (19) the internal energy and the entropy of the gas are given by 8 ln Z2 U2,, = = 1)2,,, (20) —

“

8 ln Z2,, S2,, = ln Z2,, —fi8

oa

(fl/to

=ln

Pg

)2fl

(2 ~ ‘~

(21)

,.

We notice the expression (21) for thecase, entropy is the samethat as that of the non-interacting because within our approximation all the phase space becomes equally acessible to the system. Inserting eqs. (18) and (19) into eq. (1) we obtain n~O

j~=,

n,,-

2

(fi \“

-

(—1 )‘~‘ ln(~_~ ~-)

q2(nln2n+

~

(—lY~’3lnIi—jI

=

2n (p\fl(2_V) \~t0)

(p\fl(2_Y)

(pj~)flY (2n)!

(n~)~1

(22)

(2n)! Wenoticethatthesumineq. (22) isconvergingonly n~O’\fik)

fi’\ _nln~’).

(16)

One finds that asymptotically as n-+~the sum in eq. (16) behaves as 2n

•~

(—l)’~~l1~Ii—fI n~nln~

(17)

In fact eq. (17) is well satisfied also for small n, and in the following we will use for any n the approximation 2

v-

2,,~—q

nln(nir’ro/fl)

.

(18)

For small enough y and low fugacity, v2,, (r1 ...12) becomes a very flat functjon’~around such a stationary point, and therefore we can argue that v2,, is a ~ An approximate evaluation ofthe curvature of the interaction potential with respect to the displacements ofthe chargesfrom the positions r, shows that such a curvature is small,

for 0~y<2.The divergence of the sum (22) for y~2 is a sign of the Kosterlitz—Thouless [5] ~ transition from the plasma regime (y< 2) to the insulator one (y> 2), in which as is well known [4] the particles are bound in pairs. Although the sum (22) is converging for 0~y<2,we expect that it will be a good approximation for Z only for 0 ~ y< 1, because for y>l the entropy will be drastically modified by the interaction. We have calculated numerically the low temperature specific heat C~,from eqs. (10) and (22), and we report it in fig. 1 as a function of T/ Tk for some values of y. When y is different from zero, C,, presents a peak 52

We remark here that in our problem the K—T transition associated to the logarithmic potential is at 7=2 and not at y=4 as in the K—T paper, because our gas is one dimensional.

153

Volume 128, number 3,4 0.5

PHYSICS LETTERS A

non-adiabatic terms of the interaction with the

-

electrons Let us now briefly consider the case 1 ~y< 2. Here one Z keep the cut-off thesimilar expression 2~.must A virial expansion of in ln Z to eq.(2) (7)for is

io’ Cv 0.4

-

•

~i 0.3

~

2.

•

—

1. \

•

\\

~

0.

\\

/~‘\

0.2

o

ilo

210

io

4’O

1CI2 T/Tk

‘

___________________________________ 0.

28 March 1988

1.0

2.0

3.0

T/1k40

5.0

6.0

Fig. I. The specific heat is shown as a function of T/Tk for y=0 (dotted curve), 7=0.2 (dash-dotted curve), 7=0.4 (dashed curve), 7=0.6 (full curve). In the insert the same curves are reproduced on a larger scale.

for T—~Tk reminding of the Schottky anomaly which is present for T—~4mT~°~ in the y=O case; the height ofthe peak and the entropy under the peak decrease with increasing y. On the hand at very high temperatures, where the virial expansion is valid, the specific heat given by eq. (14) goes to zero with a rate decreasing with y. This explains the reduction of the peak at low temperatures, since the high temperature entropy must be in any case equal to ln 2. In the insert of fig. 1 is reported, in a larger scale, the region of very low temperature (0< T< 0.05 Tk) in which C~presentsfor y~Oa linear dependence on

still possible, with fl-dependent virial coefficients. Following the same steps as before we find that at just as in the sufficiently high2_~, temperatures theprevious specific case, heat the behaves as (Tk/T) being that here the role of Tk as a only difference characteristic temperature is more dubious, since the viral expansion is less controlled with respect to fi. It is interesting to consider also the response function x relative to the displacement of the Oscillator caused by a constant external force F, analogous to the magnetic susceptibility in the Kondo effect:

a < Q> I IF= 0

$ .e

=fl—’

p

dt$ dt’

o

154

(23)

In terms of the instantonic paths used in ref. [2] to obtain eq. (1) for Z, eq. (23) becomes x=131 g~ (24) where D=2Q,,,

(~fl_~

(—1

)i 1.)

T, with a coefficient increasing with y. When y= 0 and in the same interval of temperature the specific heat is practically zero, since in this case it has a purely Schottky behaviour and therefore it decreases exponentially for T / Tk—I’0. We thus conclude that the non-adiabatic part of the interaction between the electrons and the LO introduces at low temperatures a term linear in T in the specific heat, which can be interpreted as a correction to the electronic contribution to C,,-, analogous to that obtained in the isotropic Kondo effect [6,7]. Such a correction may be attributed to the broadening of the levels of the oscillator due to the

.

o

(25)

with Qm being the position of the minima of the LO double well potential, and < > g the average in the grand canonical ensemble defined by the partition function (1). Eq. (24) allows to establish a simple correspondence between ~ and the dielectric susceptibility of our “Coulomb” gas. Indeed the quantity P

~

(—1

)‘t~

(26)

We remark that a similar correction to the specific heat is already present in the weak coupling case in which the oscillator potential has a single well, as can be shown using the results of appendix A of ref. [2]. This supports the interpretation that such a correction is due to a broadening ofthe oscillatorlevels.

Volume 128, number 3,4

PHYSICS LETTERS A

represents the total electric dipole moment of the gas, which for the classical fluctuation—dissipation theorem is related to the dielectric susceptibility through the equation Xg =

L~

2 > g~

(27)

1.
On the other hand, exploiting the time translational invariance, x is also given by s~ X

Qrn

x=~—

—

~E’

2~1 I fTk\ (2—y)(3—~’)~T)

j.

(34)

The dominant term of x at large Thas a Curie-like behaviour, with a y-independent “Curie” constant; this is the expected result because at high temperatures the tunneling between the two wells is uneffective and the oscillator behaves as a two degenerate levels system. In the low temperature case using the same ap-

5 = 2Q~(~fi~
(28)

where the term proportional to

g is due to the tunneling of the oscillator from one side to the other of the (27) double well potential. A to comparison of eqs. (24), and (28) allows us write

.

Xg=~fig(flX/Q~n)

In the high temperature regime we can perform a virial expansion~for gjust as we did for ln Z. We thus write

2,,

(30)

,

with L2=K2, L4=K4—K212,

(31)

...,

and

$

1/2

1/2

dx2

dX,

P

2Y)

<~>~~ and

1 “.flkl

(~~fl(

$

(~

/ 2n

dx2n

•

(

1) ‘X1

)

n(nx)~ (2n+l)!

_______

n(2—y)

x~flQ~= [l4,,~~

~)

n (nit) (2n+l)!]’

(35)

(36)

In the adiabatic limit (y= 0) we obtain from eq. (36) the simple result Xad=

tanh.

(37)

Eq. (37) shows that

gfl ~ ji~L

1/2

proximation which led us to the expression (22) for Z we get

(29)

We will evaluate now~givenby eq. (28) in the range 0 ~ y< 1, following the same procedures as we did for the specific heat.

K2,, =

28 March 1988

X~(T)

in finite for T—~0.This

is due to the fact that the ground state of the system is non-degenerate, because ofthe tunjieling of the LO between the two minima of the double well potential. Such a non-degenerate ground state is the analogous of the single ground state in the antiferromagnetic Kondo problem. Furthermore we note that, in analogy with the isotropic Kondo problem, X~(0) scales as l/T~°~. At y 0 the evaluation of eq. (36) for fl-boo gives

\i”l xi

—1/2

the result

X2,,—l

2n

xexp(y ~ \

(_l)1+ilnIxi_xjl).

(32)

X(O)=X~(0)() icey

(33)

which shows that

i>j=l

To the lowest order in Tk/ T we find 2~ 1 = __________ (2—y) (3—y) ~s”~) (Tk\ ~‘

and inserting eq. (33) into eq. (28) we obtain “

In eq. (28) the quantity ~is different from zero because, as wehave already stated, the average is referred to a particular sequence ofalternating positive and negative charges.

=Xad(0) (2)~2~

x(O)

T~i, (38) Tk remains finite also in pres-

ence of the non-adiabatic coupling. The effect of y, as shown by eq. (38) and for y sufficiently small, is to increase X(0), which is consistent with Tk being decreasing with y (see eq. (9)). Physically, this is due to the fact that the corre155

Volume 128, number 3,4

PHYSICS LETTERS A

28 March 1988

behaviour the fact that x(0) remains finite also for y~0implies (see eq. (29)) that

2.0

1.6

p~v

8-.00

t4

lim ~ = + no, (39) which is the expected result for the dielectric susceptibility and the results in theareplasma summarized region. in fig. 2. A remarkWe have numerically evaluated x given by eq. (36)

/f~~’\\

t2

0.8

able

0.6 0.4

•

0.2 I

I

I

I

T/Tk 1.5 2.0 Fig. 2. The displactive susceptibility, normalized to X~(T0) (see eq. (37)) and with y=0.l, is plotted as a function of T / T~°~ for 7=0 (dotted curve), 7=0.2 (dash-dotted), 7=0.4 (dashed curve), 7=0.6 (full curve). 0.

0.5

1.0

lation between jumps caused by the non-adiabatic coupling y tends to stabilize the LO within each well, so reducing the splitting A, and therefore Tk. This agrees with the findings of the Yu and Anderson scaling [1], and it is a very natural result because as y increases we approach the cross-over at y= 2 between antiferromagnetic and ferromagnetic Kondo behaviour, where x( T) becomes Curie-like at low temperatures. On the other hand, in terms of the Coulomb gas

156

feature ofthese results is the peak of x( T) at low temperatures, which is presented only for y <0, and whose height increases with y. From eq. (28) this implies that at large gas volumes the quantity V~

5 is a more rapidly increasing function of V5 ~fi,due to the switching on of the interaction between the charges. In terms of the oscillator dynamics this means that the tunneling contribution to x~ as a consequence of the jumps correlations, becomes a more rapidly increasing function of temperature.

References [I] C.C. Yu and P.W. Anderson, Phys. Rev. B 29 (1984) 6165. [2] M. Sassetti, E. Galleani d’Agliano and F. Napoli, Nuovo Cimento D 9 (1987) 361. [3] G. Yuval and P.W. Anderson, Phys. Rev. B 1 (1970) 1522. [4] J. Zittartz, Z. Phys. B 31(1978) 63, 79, 89. [51J.M. Kosterlitz and D.J. Thouless, J. Phys. C 6 (1973) 1181. [6] K. Wilson, Rev. Mod. Phys. 47 (1975) 773. [7] N. Andrei, K. Furuya and J.H. Lowenstein, Rev. Mod. Phys.

55(1983)331.