Singlet pair liquid, antiferromagnetism and superconductivity in a nearly half-filled narrow band

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Singlet pair liquid, antiferromagnetism and

superconductivity in a nearly half-filled narrow band

M. Héritier

To cite this version:

M. Héritier. Singlet pair liquid, antiferromagnetism and superconductivity in a nearly half-filled nar- row band. Journal de Physique, 1987, 48 (11), pp.1849-1854. �10.1051/jphys:0198700480110184900�.

�jpa-00210626�

Singlet pair liquid, antiferromagnetism ^and superconductivity ^{in a} nearly

half-filled narrow band

M. Héritier

Laboratoire de Physique ^des Solides, Université de Paris-Sud, ^Centre d’Orsay, ⁹¹⁴⁰⁵ Orsay, ^France

(Regu ^{le 29} juillet 1987, accepté ^{Ie 21} septembre 1987)

Résumé.2014 L’état de liaisons de valence résonnantes d’Anderson est discuté dans un modèle simple ^de liquide ^de paires singulets ^{dans une} bande de Hubbard étroite presqu’a ^moitié remplie. ^{Outre les solitons}

de spin, ^{les trous en} excès liés à une paire soliton-antisoliton forment un liquide ^{de Fermi} d’excitation,

dans lequel ^le nesting ^{induit une} instabilité antiferromagnetique près ^du demi-remplissage. L’échange

de fluctuations singulets produit entre trous en excès une forte interaction attractive conduisant à une

transition supraconductrice ^de temperature critique élevée, ^{dont le maximum se trouve} à la transition

liquide singulet-paramagnétique. L’application ^{aux nouveaux} oxydes supraconducteurs ^{est discutée.}

Abstract.- Anderson’s resonating ^{valence bond state is discussed in a} simple ^{model of} singlet pair liquid, ^{in a} nearly half-filled narrow Hubbard band. Besides the spin solitons, ^{the excess} holes bound to soliton-antisoliton pairs ^{form an} excitation Fermi liquid, ^{in which} nesting ^{induces an} antiferromagnetic instability ^near half-filling. ^A strong attractive interaction between excess holes arises from echange ^of singlet pairing fluctuations, leading ^{to a} superconducting transition with a high ^critical temperature, maximum at the singlet liquid-paramagnetic transition. Application ^{to the new} superconducting ^oxides

is discussed.

Classification

Physics ^Abstracts

74.20D - 74.70Y - 75.10J

In the profusion ^of high T, superconductivity theories, ^one usually distinguishes conventional

mechanisms, ^{in which} electron-phonon ^cou- pling ^meets highly favourable circumstances,

from unconventional ones. In the latter cate- gory, the most original ^{and the most} widely ^dis-

cussed theories invoke Coulomb and magnetic

correlation effects, either in the weak correla- tion or in the strong correlation limit. In fact,

weak correlation models, involving exchange ^of antiferromagnetic fluctuations between itinerant electrons in a wide band [1], ^have already ^been proposed ^{to discuss} heavy ^{fermion or} quasi-

one dimensional superconductors [2]. ^{They im-}

ply ^a maximum of the superconductivity ^criti-

cal temperature ^{at the} antiferromagnetic ^transi- tion, which is observed in Bechgaard ^salts [3], ^but

probably ^{not in the new} superconducting ^copper

oxides [4]. ^Anderson [5] ^was the first _{to pro-} pose to describe this new class of materials by

a Hubbard model in the narrow band limit. The

typical example ^is La2-.,,,, -xsrx CU04-y. Crystal

field splitting ^{leaves an} orbitally non-degenerate

copper dx2 -y2 band, strongly hybridizing ^with

the _oxygen p-band half-filled in the "pure ^com- pound" (x’ ^{= x = y =} 0), which is described by ^a

two-dimensional _square lattice Hubbard model :

Precise values of t and U are still unknown, ^but experimental ^{data seem to indicate} intermediate values for the ratio t N ^{1. Our basic hypothesis}

is that we are on the insulating side of the Mott

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198700480110184900

1850

transition, ^{as in} Anderson et ^al. [5] ^{and in Cyrot} [6], rather than in the itinerant ^case [1]. ^{In the}

following ^the typical figures t N ^0.5 eV, ^{U - 5 eV}

will be considered.

Doping ^and non-stoichiometry ^{introduce a}

concentration 6 ⁼ x+3x’-2y ^{of excess holes in}

the half-filled hand. Note that x’ is generally badly ^{known. A} value x’ N 0.02 in the "pure compound" ^without oxygen vacancies (x ^{= y =} 0) gives 6 =0.06, ^{to be} compared ^{with the doped}

case z ⁼ 0.15, ^{x’ =} 0, ^{y - 0.04} corresponding

to 6 ⁼ 0.07. Therefore similar superconducting properties [7] ^{for x = 0 and x = 0.15 might well}

be ascribed to compensating effects of lanthanum and _oxygen vacancies.

Perfect nesting ^{of the U = 0} half-filled tight binding ^{band of} equation (1) implies ^{an antiferro-} magnetic instability ^of the Hubbard model either in the small U or in the large ^{U limit.} However,

Anderson et al. [5] ^{have proposed that quan-}

tum fluctuations stabilize a state of Resonating

Valence Bonds [8] (RVB).

We shall adopt ^{the same} hypothesis, ^con- sidering ^{the RVB state as} the reference state in which antiferromagnetic ^and superconducting ^in-

stabilities _may occur. More precisely, ^we approx- imate the RVB state by ^a liquid ^of singlet pairs

(SPL) ^{defined as} follow : consider a partition

of the 2N lattice sites into N nearest neighbour pairs. ^In the half-filled _case, a singlet ^{is created}

on each pair (ij)

thus defining ^a singlet pair configuration. ^The

SPL is a coherent superposition of all the possible singlet pair configurations corresponding ^{to the} possible pair partitions.

Anderson et al. [5] and Kivelson et al. [9]

have dicussed the nature of such a state : it does not possess ^a broken symmetry ^{and a} long range order. However, ^a topological long-range ^order

can be defined. Topological excitations are neu-

tral spin ^solitons (or antisolitons), ^{i.e. dangling}

bonds obeying ^Fermi statistics, ^{formed in} pair by breaking ^a singlet pair, ^and charged ^boson

formed by adding ^{an excess} hole which bind to a

free spin, leaving ^a charged soliton, ^with charge

+e and spin ^0.

In the large ^U limit, the Hubbard Hamilto- nian (1) ^can be written in the sub-space ^without

double _occupancy of atomic sites :

where J = 4t2/U, 8i+ = c£r ^{Ci, Pi = ErCc,}

and H’ describes the kinetic terms :

We define the singlet pair ^creation operator

The relative phases ^between neighbouring ^sin- glet bonds determine the nature of the singlet pairing. ^{In the} following, ^we consider the case

of extended s-wave singlet pairing and of d-wave

singlet pairing ^described by ^a gap function

in which A is the mean-field "order parameter"

defined by

and 7k ^{= cos} kxa + ^cos kya, ^{where the sign +}

stands for the extended s-wave and sign - ^{for the}

d-wave.

The Hamiltonian reads

Singlet pairing is favoured by ^the exchange

term but inhibited by the kinetic term.

The latter is not simple ^to handle because of the Gutzwiller projection. Up to now, the effects of this projection ^{have not} been taken into account properly. This work is the first at-

tempt, ^{as far as we} know, ^{to treat them} seriouly.

We shall see below that the effects neglected ^so far, ^i.e. principally ^a Nagaoka ferromagnetic ^ex- change ^induced by the kinetic _energy of the ex- cess holes, ^{have a crucial} importance : first, they

influence the RVB long ^{or short} range order and the nature of the quasiparticle spectrum ; may be even more importantly, they provide ^{a new} mechanism, ^{which has not} been considered so far

[13], ^for Cooper pairing and therefore _supercon-

ducting ordering by exchange ^of magnetic ^fluc-

tuations between excess holes in the strong ^cor- relation limit.

First, ^we consider the limit Ult ^{» 1, using}

the results of Nagaoka [10] in the limit U/t - ^oo.

The propagation ^{of an excess hole through the}

lattice perturbs ^the spin configuration. Although singlet pairing ^is preserved [5], ^{the singlet bonds}

are displaced by ^{the hole} tunnelling, ^{thus de-} stroving ^{the wave function} phase coherence : the

wave vector is no longer ^a good quantum ^num- ber [11]. ^{Using the} techniques ^of Nagaoka [10]

and Brinkman and Rice [11], ^{the self energy of}

an excess hole described by H’, ^{localized on site}

i for a spin configuration {Q}, ^{can be written :}

where the An ^{I s are the} weighted ^{number of} diagonal n-step walks of the excess hole through

the lattice. A diagonal ^{walk is a} walk in which the initial spin configuration (ol is restored when the hole return to the site i. In the ferro-

magnetic configuration, all the walks are diago-

nal : this property minimizes the kinetic _energy

[10]. ^{The excess hole density of states has the}

full band width 2xt of the non interacting ^case.

In a random spin configuration, the contribution of the closed loops ^{to the} An ^{s are strongly re-}

duced, ^{and even more in the} antiferromagnetic configuration [11]. In the latter _case, to a good approximation, ^the diagonal walks reduce to the

self-retracing ^walks [11] : the contribution of the closed loops ^{are almost} completely supressed.

This tends to localize the hole, increases its ki- netic _energy and narrows the bandwith.

In the SPL configuration, ^{in the case of a}

"bare" excess hole or of a "charged boson", ^the

contribution of the closed loops ^are negative [12] :

the hole return to the origin ^{with a} phase ^shift

x. The result is a destructive interference effect,

which tends to localize the hole and increases its kinetic _energy compared ^{to the} antiferromag-

netic case. For example, ^the smallest closed loop

on the _square lattice (this could be extended to any alternate lattice), ^{the 4-step square, has a}

weight - 2 in the SPL. However, creating ^{an ex-}

cess hole in a perfect ^SPL necessarily ^{breaks a} singlet pair, forming ^a soliton-antisoliton pair.

The "charged boson" is a soliton-hole bound sta- te. It is also possible, ^without spending ^{any more} magnetic energy ^to bind the charged ^{boson to}

the anti-soliton, ^{i.e. to the} dangling bond. This is _very efficient in suppressing the closed loops

available for the hole propagation [12]. Then,

the excess hole bound ^to the soliton-antisoliton

pair ^{has a kinetic} energy comparable ^{to the bare}

hole in the antiferromagnetic ^lattice. Using ^the

moment technique of Brinkman and Rice [11],

we have calculated the lower band edge energy of the excess hole in the _square lattice, ^{for var-}

ious spin configurations. The results are : -4 t in the ferromagnetic case, -3.5 t in the antifer-

romagnetic case, -3.3 t for the charged ^boson

in the SPL and -3.5 t for the hole bound to the soliton-antisoliton pair ^{in the SPL. There-} fore, the latter quasiparticle should be stable at

temperature lower than a binding energy - 0.2 t

(-1000-2000K) and should behave as a "charged

fermion" .

The picture ^discussed by ^{Anderson et al.}

[5] ^{and by} Kivelson et al. [9] ^was the follow-

ing : introducing ^excess holes in the half-filled band by doping ^{does not} change ^the spin ^soli-

ton Fermi surface but creates charged ^{boson ex-}

citations which can undergo ^a Bose-Einstein con-

densation leading ^to superfluidity. ^Our point ^of

view is different. We recover the more intuitive

picture ^{in which} doping introduces excess holes which behave as charged ^fermions strongly ^cou- pled ^to magnetic fluctuations by many body ^ef-

fects.

The results show the importance ^{of the cou-} pling between the excess hole and the singlet pairing correlation, ^{which can} be described by

an interaction Hamiltonian of the form :

in which akq is ^a numerical factor of order unity.

First, consider the half-filled case 6 ⁼ 0. The Hamiltonian (6) ^is easily solved in the mean-field

approximation. ^{A mean} field critical tempera-

ture for SPL formation kBTe(6 ⁼ 0) =t2/U ^is

found for extended s-wave as well as for d-wave

pairing.

As shown by ^{Anderson et al.} [5] ^and by

Kivelson et al. [9], ^the quasi-particles in ^the half- filled ^{case 6 = 0 are} spin ^soliton fermions, ^with

a dispersion Eo(k) ⁼ J d I lie I exhibiting perfect

nesting ^{for the wave vector} Q = (E, E) . ^These

1852

quasiparticles ^{can be described} by ^{a Fermi} liq-

uid theory ^for kBT J. As noted by ^Anderson

et at. [5], ^perfect Fermi surface nesting implies

an antiferromagnetic instability, ^{if one takes into}

account the quasiparticle interactions, neglected

in the Hartree-Fock Hamiltonian but of course

present. If A is the molecular field constant for these interactions, ^{the Neel} temperature ^{is of the}

form :

where Ao = 1 ^{£k 1-yk I is the low} temperature SPL order parameter. ^This picture ^is only ^valid

for Tun « Tel’ ^which requires ^A « J, ^{a condition}

which might ^{not be} fulfilled. Nevertheless, ^{we be-}

lieve that perfect nesting ^would imply ^{an insta-} bility ⁱⁿ any case, ^even though ^the correct treat- ment would be much less simple.

Adding ^a concentration 6 « t/U ^{of excess}

holes, they ^form ferromagnetic spin polarons, ^as

a result of the balance between kinetic _energy and magnetic ^free energy, propagating ^{in the an-} tiferromagnetic medium. The SPL energy spec- trum may be approximated by

where Ep ^{is the} spin polaron ^{energy. The pres-}

ence of excess holes rapidly spoils ^{the Fermi}

surface nesting, ^and antiferromagnetism ^{is de-} stroyed by ^a small value of 6. In fact, ^{in real} systems, ^a small violation of perfect nesting ^ex-

ists even at half-filling. ^Best nesting, and, thus,

maximum of TN might ^occur slightly away from

half-filling. ^{In some} cases, nesting may be too

imperfect ^{to induce an} instability and antiferro-

magnetism may ^{not occur at} all. This might ^be

the case of the YBa2Cu307-y ^compounds.

However, ^this picture corresponds ^{to the li-}

mit U/t ^» 1, ^while the real situation seems

rather U/zt £5 ^{1. As} Ult decreases, ^the polaron

energy increases, ^and eventually, ^{for a critical ra-}

tio (Ult)c - ^{the polaron state merges with the}

band of diffusive motion of the bare hole. The

precise determination of this critical value de-

pends very much on the details of the variational treatment, but the Cu oxide situation seems to

correspond ^to this limit in which the polaron

state is no longer stable : the excess hole is no

longer surrounded by ^a long ^life-time ferromag-

netic bubble, but it is strongly coupled ^to large amplitude ^local spin fluctuations. The complete description ^{of such a} system ^is very difficult, ^but

the simplest approximation ^{is a mean field} ap-

proach in which the spin fluctuations _are, first, neglected. ^{In this} sheme, because of the coupling

(8), ^we replace ^{the kinetic term Hr in} (2) ^{by an}

effective kinetic Hamiltonian for the excess hole

f, I :

in which _wo (A) is the Brinkman-Rice bandwidth

narrowing ^factor [11] ^{and ek is the dispersion}

relation for non-interacting ^electrons (U ⁼ 0).

Moreover, ^{the kinetic} energy dependence ^{on A is} responsible ^{for a} molecular field for singlet pair- ing, ^{which we} write in the form :

in which a is a numerical factor of order unity,

and _p is the mean field _average (ct cj) ^{on nearest}

neighbour ^bond (ij). ^{We must} emphasize ^that

this is only ^a rough approximation because the

wave vector is not a good quantum number. The

excess hole should be described, ⁱⁿ principle, by

a broad wave packet.

Using (9) ^and (10) ^our Hartree-Fock Hamil- tonian for the SPL transition reads :

in which p is the chemical potential.

The Bogolubov transformation gives ^the quasiparticle energy :

The _gap and the chemical potential ^are given by

the self-consistence equations :

As 6 increases from _zero, Te ^decreases rapidly. ^It

is given by :

where ek ⁼ 6wo (A)ek - ^p.

The decrease is much larger for extended s-wave

pairing, where -y2 ^{vanishes on} the Fermi surface

ek ⁼ 0, than for d-wave pairing ^{where the aver-}

age of i£ ^on the Fermi surface is 1. In the latter case, for 8 « t

where a’ is a numerical factor of order unity.

At larger ^values of 6, Tc ^decreases very rapidly

and vanishes for 6c given, still in the d-wave, ap-

proximately by ^the equation

with

so that 6, til t

^{u .}

Therefore, ^{in this} picture, ^we expect ^{a d-wave} SPL. However, this result depends heavily ^{on the} approximation ^{made in} (9), which describes the

excess hole by ^a well defined wave vector. The

wave packet broadening ^should strongly ^reduce

the _energy difference between s-wave and d-wave

pairing. ^In fact, ^a mixing ^{of these two} states, ^as allowed by ^the symmetry ^of the orthorhombic or

tetragonal lattice, ^seems plausible.

As discussed above, ^this simple ^treatment neglects ^the large spin fluctuations which sur-

round each excess hole. As in _any intermedi- ate coupling problem, ^{it seems difficult to} give

a precise ^{treatment of these} fluctuations, ^but they probably destroy ^{the small} energy gap -

61, which, itself, requires ^the presence of ex- cess holes. Therefore the SPL long range order is probably destroyed, ^{but short range} magnetic

correlations still persist ^as long ^{as 6 «} (t/U)1/2.

As above, ^{a small} concentrations of excess holes

spoils ^the nesting ^and rapidly destroys ^antifer-

romagnetism. ^{We are} left with a short _range SPL order below the mean field SPL "transition line". It does not seem possible ^{to invoke a Bose-}

Einstein condensation of the "charged bosons",

since they ^are trapped by spin antisolitons.

However, ^a Cooper pairing ^{of the} charged

fermions can be induced by ^the strong coupling

to the singlet fluctuations described by equation (8). ^{Exchange of} singlet pairing fluctuations gi-

ves rise to an effective interaction between fer-

mions, ^{the low} frequency behaviour of which is attractive. Keeping only ^{the terms in which}

the Cooper pairs ^{have zero} momentum, ^{the hole} interaction takes the forms :

where wm (q) ^{is the} singlet pairing fluctuation en-

ergy, and A(k, k’) ^{is a} numerical factor. The or-

der of magnitude of this attractive coupling, ^is

the low frequency limit, is - J - ^{-U, and of}

the cut-off _energy beyond which the interaction becomes repulsive ^{J. One can} replace ^{the attrac-}

tive potential by ^a constant at frequency ^lower

than J, ^as in BCS. The dimension-less coupling

parameter is g - I ^{- T 81/2. For 6 «} (If) 2 , t ^the

weak coupling ^limit gives ^{a BCS} ordering ^tem- perature

However, ^this approximation breaks down when

one approaches ^{the SPL mean} field transition.

The trace over the singlet pairing operators ^in- volves the correlation function for singlet pairing

The attractive potential between fermions can be written

which diverges ^on the SPL transition line, where, therefore, ^we expect the maximum superconduc- ting ^critical temperature. ^Of course, the same

mechanism applies ^{in the} paramagnetic phase,

1854

with the same orders of magnitude. ^The qualita-

tive origin ^{of this} strong ^attrative interaction be- tween excess holes is _{easy to} understand : an ex- cess hole destroys ^the singlet correlation around itself to lower its kinetic _energy. This is energet- ically favourable to the second excess hole reab-

sorbing ^the magnetic fluctuations.

The expected phase diagram is schematized in figure ^1.

The main features are the following :

i) antiferomagnetism appears ^near half-filling ^for good enough nesting ^{and is} rapidly destroyed by doping. Antiferromagnetic ^{order is} not necessary for the occurrence of superconductivity ^and may be absent if nesting ^{is not} good enough.

ii) ^The superconductivity transition temperature

is maximum on the SPL transition line for 6

b, ^{away from the} antiferromagnetic transition.

Such a model seems to well describe the main properties ^{of the new} superconduct- ing ^oxides L82-:.cSr:.cCu04-y, ^{but also YBa2Cu3} 07-y, in which less good nesting might ^not

allow antiferromagnetism.

I gratefully ^{thank G.} Collin, ^J. Joffrin, ^G.

Montambaux, ^C. Noguera ^{and J.-P.} Pouget ^for helpful discussions.

Fig.l.- ^Phase diagram of the Hubbard model for a

concentration 6 of excess holes. Antiferromagnetic instability ^{of the} singlet pair liquid ^occurs very ^near

half-filling. The shaded area represents ^the supercon-

ducting phase, ^{with a} maximum of the critical tem-

perature ^{on the SPL} transition line.

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