SELF-MATCHING PROPERTY OF CORRELATED ELECTRONS:CHARGE–DENSITY WAVE ENHANCES SPIN ORDERING

S.I. MUKHIN†*Lorentz Institute for Theoretical Physics, Leiden University, 2300 RA Leiden, The Netherlands

Abstract—The local minima of the free energy of a quasi one-dimensional electron system with short-rangerepulsion on a lattice were found which correspond with the symmetry-allowed matching pairs of the precursorCDW and SDW modulation periods. Possible relation of the theory to the ‘staircase’ instabilities in the dopednickelates with perovskite structure is discussed.q 1998 Elsevier Science Ltd. All rights reserved

Keywords:doped nickelates, perovskite structure, spin ordering

In this paper we approach analytically a problem ofinhomogeneous charge and spin-density states in thecorrelated electron system, which has drawn much atten-tion recently [1–7]. Starting from the ‘weak-couplingside’ we find a class of discrete symmetry relationsbetween the spin and charge density spatial variationperiods, which, we believe, may be valid as well in themoderately strong coupling limit.

The simplest tight binding Hamiltonian on the latticeincludes the nearest neighbour hopping integral,t, and theon-site repulsive interaction of the strengthU. It providesthe planar nested parts of the Fermi surface in the one-dimensional case, which are the most importantingredients of our theory, though a generalization to themulti-dimensional case is straightforward.

It is well known [8] that repulsive interaction com-bined with the nested planar parts of the Fermi surfacemay lead to the SDW formation at some temperature,TSDW. In this paper we show that besides this ‘common’situation, one also may expect precursor CDW’s at thetemperatures far aboveTSDW. In the mean-fieldapproximation with respect to the precursor CDW theHamiltonian reads:

H ¼ ¹ t∑

,i, j.,jc†

i,jcj,j þ U∑

i

ri(ni ¹ ni)2

¹ (S3i )2

� �þ

U4

∑i

r2i ¹ m

∑i

ni : ð1Þ

To derive eqn (1) we have used an identity:ni↑ni↓ ¼ 1=4n2

i ¹ (S3i )2, where S

3i is the operator of the

z-component of electronic spin on sitei, andni ↑ (ni ↓ )is the density operator for spin-up (spin-down) electrons;

finally, ni ↑ þ ni ↓ ; ni. The mean-field variation of thecharge density is defined as follows:

, ni ¹ n . ¼ ri ; di ¼ nocos(dcx), (2)

wheren is the average number of electrons per site (, 1),and , … . means an average taken over a (meta)stablestate. When writing eqn (1) we have neglected the chargefluctuation term, (ni ¹ ni)

2, which is supposed to berelatively small in the repulsive case,U . 0. Thedc, thewave vector of CDW, is supposed to be small:dca, 1. Athalf-filling the Fermi momentum is:kF ¼ p/(2a), (thePlank’s constant is taken as unity). Away from half-filling

kF ¼p

2a¹

d

2;d

2; xd

p

2a, (3)

wherexd is the doping concentration. Resulting from thelattice periodicity a spin-density wave may be written inthe form:

S(x) ¼12

sinp

a¹ ds

� �x

h iþ sin

p

aþ ds

� �x

h in o: (4)

The first term is caused by electron ‘back-scattering’ withthe momentum change of 23 (p/(2a) ¹ ds /2), while thesecond term is caused by umklapp skattering of electronswith the total momentum change 2p/a ¹ 2 3 (p/(2a) ¹

ds /2). In the presence of the CDW periodic potential thequasi-particle momentum is conserved only modulom3

dc, m¼ 6 1, 6 2,…. Hence, a wave-vector of the SDWmay now differ from 23 kF. The symmetry of eqn (4) ispreserved by the wave-vectorsds and dc, which obey‘matching relations’ below (d is fixed by the dopingconcentration according to eqn (3)):

dc ¼2d

n 7 m

ds ¼(n 6 m)d

n 7 mm, n; m¼ 0, 1, 2, …; n¼ 1, 2, …

,

8>>>>><>>>>>:(5)

with the upper(lower) signs in the case ofds . ( , )d.

1846

J. Phys. Chem SolidsVol 59, No. 10–12, pp. 1846–1848, 19980022-3697/98/$ - see front matter

q 1998 Elsevier Science Ltd. All rights reservedPII: S0022-3697(98)00114-0Pergamon

*Present address: University of Maryland, Physics Depart-ment, c/o Prof. R.A. Farrell, College Park MD 20742, USA.Tel: +1 301 405 6148; fax: +1 301 314 9465

†Permanent address: Moscow State Institute for Steel andAlloys, Theoretical Physics Department, Leninskii prospect 4117936 Moscow, Russia

It is amazing that the simple symmetry relations ineqn (5) also follow directly from the Fourier componentof the magnetic susceptibility,xq, calculated in thepresence of a CDW potential, eqn (2), using the well-known properties of the Bessel functions of integer order,Jn(x):

xq ¼14t

lnt

2pT

3 J20

no

tdc

� �dq, 6d þ

∑n¼ 1

J2n

no

tdc

� �dq6d, 6ndc

" #,

ð6Þ

wheredqd, 6 ndc are the Kronecker’s symbols. Only for apair of the wave-vectors {d,dc} which obey the symmetryconditions in eqn (5) are two of the Kronecker symbolsnon-zero simultaneously:dq 6 d, 6 ndc ¼ 1 anddq 6 d ¼ 1.Because all the terms in the sum in eqn (6) are non-negative, we conclude that matching pairs of {ds,dc}mark the dominant spin-fluctuation contributions to thefree energy of the system. Fig. 1 shows free-energy as afunction of CDW’s amplitude no, calculated usingeqn (6). The solid, dotted and dashed lines correspondto the lowest order matching pairs from eqn (5): {ds ¼ d,dc ¼ 2d}, { ds ¼ d/3, dc ¼ 2d/3}, and {ds ¼ 3d, dc ¼ 2d}.

The dashed–dotted line represents the non-matching pair{ ds ¼ 0, dc ¼ d} taken for comparison. Thin lines arerelated to the same set of pairs as the thick ones but at aten times greater value of the ‘coupling strength para-meter’,dt/TSDW. The temperature for the curves in Fig. 1is taken far aboveTSDW: T ¼ 6.7 3 TSDW. The threeminima of the thick curves are nearly degenerate. Thismay cause the ‘staircase’ transitions, which wereobserved in the doped nickelates [6, 7]. It is importantthat the two pairs:ds ¼ d, dc ¼ 2d, andds ¼ d/3,dc ¼ 2d/3found above, have the same constant value of theratio dc/ds ¼ 2 as observed experimentally.

The author highly profited from the enlightening dis-cussions with Jan Zaanen during his stay at the LorentzInstitute for Theoretical Physics. Useful discussions withV.L. Pokrovskii, Wim van Saarlos, Richard Ferrell andW. Lawrence, as well as warm hospitality of Jos de Jonghis acknowledged. This work was supported in part byNWO and F0M (Dutch Foundation for FundamentalResearch).

REFERENCES

1. Zaanen, J. and Gunnarson, O.,Phys. Rev., 1989,B40, 7391.2. Schulz, H. J.,J. Phys. (Paris), 1989,50, 2833.

1847Self-matching property of correlated electrons

Fig. 1. Free energy as function of CDW’s amplitudeU/Ef ¼ 0.3(3), T/TSDW¼ 6.7, TSDW/Ef ¼ 1.84d (thick lines), TSDW/Ef ¼ 0.25d

(thin lines).

3. Schulz, H. J.,Phys. Rev. Lett., 1990,64, 1445.4. Emery, V. J. and Kivelson, S. A.,Physica, 1993,C209, 597.5. Emery, V. J. and Kivelson, S. A.,Physica, 1994,C235, 189.6. Tranquada, J. M., Sternlieb, B. J., Axe, J. D., Nakamura, Y.

and Uchida, S.,Nature (London), 1995,375, 561.

7. Tranquada, J. M., Buttrey, D. J. and Sachan, V.,Phys. Rev.,1996,B54, 12318.

8. Horovitz, B., Gutfreund, M. and Weger, M.,Phys. Rev.,1975,B12, 3174.

1848 S. I. MUKHIN