Effect of a lattice quantum fluctuation in MX chains

5 August

1996

PHYSICS

ELSEYIER

LETTERS

A

Physics Letters A 2 18 ( 1996) 328-332

Effect of a lattice quantum fluctuation in MX chains Z.G. Yu a*b,C.Q. Wu”, X. Sun a*c,K. Nasu d a T.D. Lee Laboratory and Physics Department, Fudan University, Shanghai 200433, China ’ h National Laboratory of Infrared Physics, Academia Sinica, Shanghai 200083, China ’ International Center for Theoretical Physics, Trieste 34100. Italy ’ Photon Factory, National Laboratory for High Energy Physics, Tsukuba. Ibaraki 305, Japan Received

3 January

1996; revised manuscript

received

16 April 1996; accepted

for publication

29

April 1996

Communicatedby A.R. Bishop

Abstract

Based on renormalization-group arguments, the effect of a lattice quantum fluctuation on the ground state of an MX chain is discussed. It is shown that, when the fluctuation is included, the MX chain model is qualitatively different from the one-dimensional molecular-crystal model, although these two models are equivalent in the adiabatic approximation. In the limit o + CO, the former has the charge-density-wave ground state; the latter, however, has no long-range order. PACS: 7 1.38.+i; 65.3O.Kr

It is known that the zero-point lattice motion gives a notable impact on the Peierls ground state of quasi one-dimensional systems such as the conjugated polymer and the metal-halogen-mixed-valence chain (MX chain) complex [ I ,2]. The zero-point and the thermal lattice motion produce a tail of localized states below the energy gap and remove the inverse squareroot singularity in the electron density of states at the band edges. These features have been manifested in experiments of temperature-dependent optical absorption [ 3-51. Experiments on luminescence and resonance Raman spectra further reveal that the tail of localized states in the PtCl chain complex mainly originates from the zero-point lattice motion rather than the structural inhomogeneities [2]. In our previous work, we have investigated the lattice quantum fluctuation effect on the charge-density wave (CDW) amplitude of MX chains in a wide range of the electron-

phonon coupling by using a self-consistent renormalization method and found that the CDW can be seriously suppressed [ 61. To get a better and more complete appreciation of the quantum fluctuation effect, in this paper, we study some limiting cases and try to obtain the whole phase diagram of the MX system. Interesting enough, we find that, when the fluctuation is included, the MX chain model is qualitatively different from the one-dimensional molecular-crystal model, although these two models are equivalent in the adiabatic approximation. A typical crystal of the MX chain complex consists of weakly coupled linear chains of alternating metal (M) and halogen (X) atoms with the ligand groups attached to the metals. Its structure is illustrated in Fig. la. To make the calculation more transparent, we simply take a one-band model without the electronelectron interaction, though it has been pointed out that the two-band tight-binding model is more real-

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a -;-2

2

n,

+c”

x

I

.Y

.w

HMX

=

nr

+p’

I.

The skeleton

(3) irr and

14 L

structure of the MX

molecular crystal

chain

(a)

and the

(b)

istic to describe the MX chain properly Hamiltonian we study reads [ 9,101

HMX= x(;Mq:

H.c.)

PitI

2 1

one-dimensional

c $‘?:- t~(cj,ci+,u+

x

I+1

b

Fig.

329

Letters A 218 (1996) 328-332

Z.G. Yu et d/Physics

[7,8].

The

+ &Cq:) - t ~(c&+,o + H.c.1

I

lu

(1) where cf, (cl,,) denotes the creation (annihilation) operator of an electron of the M atom with spin (+ at site 1, and nj, z c~,c,~. z is the transfer energy of an electron between two neighboring metallic sites. ql is the coordinate of the X atom and describes its motion parallel to the chain. This MX chain model is very similar to the onedimensional molecular-crystal model, introduced by Holstein [ 111 to study polarons in molecular crystals, with the Hamiltonian

fhc = CC $4: + ;Cq:) -

r-j+j,cr+,, +H.c.) /CT

(2) In this model, the phonons are also taken to be dispersionless, as the MX chain model ( 1). The molecularcrystal model describes the vibration of the internal degree of freedom of a molecule, and the presence of electrons on the molecule modifies the equilibrium position of the oscillator. These two models not only look like each other, but also are equivalent to each other in the adiabatic limit. In the adiabatic approximation, i.e. M --+00 and the phonon frequency w = 0, the kinetic energy of the lattice is neglected. In the half-filling case, both models have a CDW ground state with a dimerized lattice in accordance with the Peierls theorem, i.e. ql = ( - 1) ‘mt, with mp the phonon staggered order parameter. We can rewrite the Hamiltonians (1) and (2) as

HMC

=

c

+cq;

-

t

c

(&cl+,cr

+

H.c.) (4)

Irr respectively. Clearly, these two Hamiltonians are equivalent in the ground state with A = 2/3. The properties of the ground states are well known and can be readily obtained by minimizing the total ground state energy with respect to mp [ 121. One has the consistent equation

(5) where A = Am, and K(x) is the complete elliptic integral of the first kind. When h is very small, one finds A = 8te-rc#. The electrons form a CDW and the electronic parameter is defined as me = k C(-~)‘(G), Irr

(6) order

(7)

which is related to the phonon order parameter by m, = (C/A)m,. From Eq. (6), we can see that, in the o = 0 limit, both the MX chain and the one-dimensional molecular crystal have dimerization of long-range order for arbitrary nonzero electron-phonon coupling constant. When the kinetic energy of the lattice is included, our self-consistent renormalization calculation showed that the dimerization or the CDW amplitude of the MX chain will be reduced for the finite phonon frequency w [ 61. Variational [ 131 and Monte Carlo [ 121 studies of the molecular-crystal model also indicated the reduction of CDW for the finite frequency. But what will happen if we go to some limiting cases? To answer this question, first, we rewrite Hamiltonian ( 1) as

330

Z.G. Yu et al./Physics

Letters A 218 (19%) 328-332

&H =

+BCqlh,

-w-l,),

-rxC C/,CI+I~ + Kc.)

(8)

la

-t~(c/&+,rr+ H.c.)

then we discuss the path integral form of the partition function of Hamiltonian (8))

lo

P

z=

T,exp -

z)ql(~)

dr s

+ Pq1(7)

~b,&)

-

c Having integrated have

@g:(r)

-t

I( 1

0

%1cT(~)l . )

c (cf,cl+,n+ H.c.1

(9) This is nothing but the extended Hubbard model with

out the freedom of the phonon, we

l-J=_2p2 “J? c )

C’

This model cannot be solved exactly; however some features of the ground state can be understood by following the renormalization-group arguments of Emery [ 141. Using his notation,

P

Z =TrT,exp-

-rc[ciJr) 0

ci+~o(r)

lrr

VII =U,_ =U-2V

w_L =u-2Y

+ H.c.] >

WI1=U+6Y

and since U = -2V in the present situation, 0, WJ_ = -4v< WI1= 4v> 0,

(111= I71 = -4v<

0,

and IWl(sgn(UII) and

G(r,r') = <-a,’ +&)-I

=

&,_w,,-,,.

When we adopt the anti-adiabatic 0, i.e., w + co, and note that u2G(r,r’)

approximation

“~m~(r-r’),

we obtain the effective electronic MX chain,

(11) M +

(12) Hamiltonian

of the

= -4Y

WII=4V>

0.

Fig. 2 is the phase diagram from the renormalization group [ 141, showing regions in which various longrange correlations will develop at zero temperature. SP, TP, and SDW represent singlet-pairing, tripletpairing, and the spin-density wave, respectively. From this phase diagram, we know that the ground state of the MX chain, in the anti-adiabatic limit, is the CDW. The CDW survives a quantum fluctuation even for infinite phonon frequency in the MX chain. For the molecular-crystal model, in the limit w + 00, as shown by Horsch and Fradkin [ 121, the effective electronic Hamiltonian after integrating out the freedom of the phonon is

Z.G. Yu et al./Physics

Letters A 218 11996) 328-332

331

\ and 6(1u> = 2/3( -l)‘q,;

SDW

then

\ SP TP SP 0

CDW

where

the dimensionless

coupling

constant

A =

2P2/Kt, and (+; are the Pauli matrices, Fig. 2. Phase diagram from the renormalization group [ 141. SP, TP and SDW mean singlet-pairing, triplet-pairing and the spin-density wave, respectively. In the limit w + 00, the effective electronic Hamiltonian of the molecular crystal (solid circle) lies on the boundary between the CDW and SP regions and that of the MX chain (open circle) lies in the CDW region.

(141

This is the Hubbard model with U = -h*/C; as is well known, its ground state has no long-range order. In the phase diagram Fig. 2, it lies on the boundary between the CDW and SP regions. So if w --+ co, the CDW will be destroyed by the quantum fluctuation and the ground state is disordered. The very different behavior of the two models in the anti-adiabatic limit is understandable. In the molecular-crystal model the electron couples to the local displacement of the lattice and, in the limit w = 00, this coupling leads to an instantaneous interaction for the electrons at the same site. In the MX chain, however, the electron couples to the relative displacement between the neighboring X atoms, resulting in interactions for the electrons at the adjacent sites as well as the same site. Thus the effective electronic Hamiltonian in the molecular-crystal model is the Hubbard model; whereas in the MX model, it is the extended Hubbard model. The behavior of the MX chain Hamiltonian in the w - cc limit can also be realized if we pass from the discrete model ( 1) to its continuum version, by introducing [lo]

UF = 2ta and will be set to 1 in the following sions. If we make the unitary transformation

the Hamiltonian

?_I,,=

dx

J

discus-

( 15) will be written as

(z+$%.$ 1

In the M ---t 0 limit, by making the substitution d”(x) -+

J-

&7rh J(x) ,

the Lagrangian

s

in this system can be written as

\

where 4 = qtyc = qt,z. After the bosons are integrated out, an effective fermion Lagrangian is

(18) s

the SU( 2) Gross-Neveu model. Renormalizationgroup properties of this model also indicate that (I&/J)

332

Z.G. Yu er al./Physics

acquires a nonvanishing expectation value for any A f 0, and the ground state has the CDW long-range order for all nonzero couplings [ 15,161. For the molecular-crystal model, it is not very easy to define a simple continuum version appropriately because the electron couples to the local displacement and the relative phase between the neighboring displacements of the lattice is random. This is in contrast with the MX model, in which the electron couples to the relative displacement between the neighboring X atoms and only the optical piece of the phonon spectrum is important for the ground state and low-lying spectrum. The acoustic piece, in the terminology of the renormalization group, is irrelevant to the physics of the ground state. In the molecular-crystal model, however, both acoustic and optical phonons must be taken into account so that we cannot obtain a simple and appropriate continuum Hamiltonian. Thus we can conclude that, in MX chains, the CDW ground state is stable against the lattice quantum fluctuation no matter how large the phonon frequency is, even if it goes to infinity; in the one-dimensional molecular crystal, however, the CDW ground state is stable only for the finite phonon frequency and the quantum fluctuation will destroy the CDW in the limit o ---f 00. So the quantum fluctuation exerts different effects on the MX chain and molecular-crystal models, and leads to qualitative dissimilarities of the ground state properties in these systems, though in the adiabatic limit these two models are the same.

Letters A 218 (1996) 328-332

This work was partly supported by the National Natural Science Foundation and the State Education Commission of China.

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121EH. Long, SP Love, B.I. Swanson and R.H. McKenzie, Phys. Rev. Lett. 71 (1993) 762. 131 B.R. Weinberger et al., Phys. Rev. Lett. 53 ( 1984) 86. [41 Z. Vardeny et al., Phys. Rev. Len. 54 ( 1985) 75. [51 H. Kuzmany, Pure Appl. Chem. 57 ( 1985) 235. [61 Z.G. Yu, Y.S. Ma, X. Sun and Y. Takahashi, Commun. Theor. Phys., in press; H.Y. Chu and Z. Ye, Solid State Commun. 91 ( 1994) 387. [71 J.T. Gammel, A. Saxena, 1. Batistic, A.R. Bishop and S.R. Phillpot, Phys. Rev. B 45 ( 1992) 6408. J.T. Gammel, A.R. Bishop and E.Y. [81 SM. Weber-Milbrodt, Lor Jr., Phys. Rev. B 45 (1992) 6435. 191 D. Baeriswyl and A.R. Bishop, J. Phys. C 21 (1988) 339. IlO1 Y. Onodera, J. Phys. Sot. Jpn. 56 (1987) 250. III1 T. Holstein, Ann. Phys. 8 (1959) 325. [I21 J.E. Hirsch and E. Fradkin, Phys. Rev. B 27 ( 1983) 4302. [I31 H. Zheng, D. Feinberg and M. Avignon, Phys. Rev. B 39 (1989) 9405. solids, eds. 1141 V.J. Emery, in: High conducting one-dimensional J. Devteese, R. Evrard and V. van Doren (Plenum, New York, 1979). iI51 E. Fradkin and J.E. Hirsch, Phys. Rev. B 27 ( 1983) 1680. 1161 D. Gross and A. Neveu, Phys. Rev. D 10 (1974) 3235.