2 e or not 2 e : Flux Quantization in the Resonating Valence Bond State

The *resonating valence bond. (RVB) state has been proposed as the basis for an explanation of high-temperature superconductivity. Recently, we have described the charge and spin excitations about this state, and have shown that they are solitons, precisely analogous to those found in polyacetylene. Since the charged solitons are + e bosons, it is natural to ask whether flux quantization will occur in units of hcl2e, as in traditional BCS superconductivity, or will come only in larger units of hcle. We show here that flux quantization in units of hcl2e will occur unless a condensation of cooperative ring exchanges occurs analogous to that found in the fractional quantized Hall effect. Resonance, the description of the quantum ground state of a system (say benzene) as a superposition of several bond configurations, has been the chemist's way of incorporating some of the delocalization energy which is naturally described using electronic energy bands. L. Pauling originally introduced the .resonating valence bond. (RVB) state in the hope of describing simple metals. This state, a quantum liquid of valence bonds, is kept from crystallizing into a Peierls state by its < M sites. Since we are interested in topological properties (rather than energetics), we need not examine the true ground state-any state that is adiabatically connected to the ground state wil l do. The states we will examine will be coherent superpositions of nearestneighbor singlet bond configurations, where each electron participates in a singlet bond with one neighboring electron. We define RVB states as those which can be constructed perturbatively from one of these superpositions (I). (*) IBM T. J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598. (l) We should caution that this definition probably excludes the states discussed by Anderson and collaborators, which have no gap to spin excitations. Such states will have a large density of neutral solitons, while the unperturbed states we discuss have none. 354 EUROPHYSICS LETTERS There are two ways to describe the valence bond configurations: one can pay attention to either the singlet bonds or the empty space between them. The latter description becomes preferable near half-filling M = N , where the unoccupied sites are dilute: we have shown [2] that these sites act as charge + e spinless bosons. Because of the close analogy between these excitations and the domain walls in polyacetylene, we call them charged solitons. There are disadvantages to each of these points of view. The soliton description is incomplete: many bond configurations are compatible with a given arrangement of solitons. The description in terms of singlet bonds is complicated, because the various valence bond configurations are not orthogonal. (For example, on a 2 x 2 square, the overlap between the state with two horizontal bonds and the state with two vertical bonds is 11.2.) This paper will use both of these descriptions. First, we shall use the singlet bond description to investigate the Aharonov-Bohm (microscopic) periodicity in the energy. Since the elementary units in this description have charge 2 e , one might imagine that the energy must be rigorously periodic in the enclosed flux with periodicity hcl2e. We shall see that the lack of orthogonality (') between configurations makes the Aharonov-Bohm energy periodicity hcle. The Aharonov-Bohm effect is microscopic in the sense that the magnitude of the periodic terms decays with the size of the ring. (The argument applies equally to metals and insulators.) Second, we shall use the soliton description to investigate flux quantization. Flux quantization is given by the macroscopic energy periodicity in large annuli, i . e . the stiffness caused by the off-diagonal long-ranged order. The Aharonov-Bohm flux periodicity is only an upper bound on the flux quantum: even in ordinary superconductors, extremely small annuli will have energy periodic only with period hcle. It may be possible to prove (3) that flux quantization in a system of electrons must occur in even multiples of e , but we do not know of a compelling argument. (The flux quantum has been measured by several groups to be hd2e in both Laz-,SrzCu04-y and in YBa2Cu07-,(4).) We shall show that a condensation of cooperative ring exchanges [5] is a necessary condition for flux quantization in units of hcle. Aharonov and Bohm showed that in a system of particles of charge q contained in an annulus with magnetic flux @ passing through, the total free energy is periodic in @ with period hclq. The standard argument is made by constructing wave functions with the same energy at fluxes @ and @ + hc/q. Choose the vector potential to lie in the azimuthal direction: A = Ve(@8/2x). Since the Hamiltonian depends only on (hli) V j ( q / c ) A , the wave function has the same energy at flux @ + hc/q as !P@ had at flux @. (The €ij are the angular coordinates of the particle positions Rj.) This argument also works if the particles are molecules with (2) After completing this work, we received a preprint from David Thouless on the same topic; he finds the energy has period hd2e regardless of the size of the system. In his two-dimensional models, his treatment ignores the nonorthogonality of the bond configurations; his work should be interpreted as a generalization of the topological constraint to nonbipartite lattices. We agree with his conclusion that strictly localized singlet bonds in single chain annuli have period hd2e; however, realistic models will have exponentially falling tails to the singlet bond wave functions which produce a small contribution to the energy with period hcle. (3) There are hints of a general argument in Yang's work on off-diagonal long-range order [3]. (4) Many of these measurements were of periodicities in Josephson junctions, which could disrupt the bonds. Recently, the flux lattice period has been measured by P. L. Gammel et al. [4]. s. A. KIVELSON et al.: 2e OR NOT 2e: FLUX QUANTIZATION ETC. 355 internal degrees of freedom, so long as their extent is small compared to the hole in the annulus (so that their angular center of charge 6, is well defined). What goes wrong when we apply this argument to valence bond configurations? We can imagine writing a many-body wave function Y$ for the bonds as a superposition of bond configurations, and label each configuration in terms of the center 6, and orientation of each of the singlet bonds. Can we use eq. (1) to create a new wave function for flux @ + hclze? The lack of orthogonality between the bond configuration .position eigenstates. prevents us from doing so. Changing the relative phase of two overlapping pieces of the wave function changes the normalization and the potential energy as well as the kinetic energy. Two bond configurations are orthogonal if the solitons are in different places, but the center of charge can move (and the relative phase can change) without moving the solitons: as shown in fig. 1, rearrangements of bonds with enclose the hole in the annulus can change the center of charge Ce, by Z. Thus exchange rings which encircle the hole (fig. 1) change the microscopic AharonovBohm flux periodicity of the energy from hc12e to hcle. Does this extend to flux quantization? For large annuli, the rearrangements which span the hole have exponentially small overlaps with the original configuration ( -2 -L , where L is the circumference); each of their contributions to the flux-dependent energy is also exponentially small (5). There are, however, many distinct rearrangements which encircle the hole-if they are important to the wave function, then the periodicity of the macroscopic energy could also be hcle. This is precisely analogous to the physics of the fractional quantum Hall effect, where exponentially small contributions from an exponentially large number of exchange loops add coherently to the energy. To make this analogy more precise, we turn to the soliton description of the resonating valence bond state. First, let us establish some conventions. It is useful to distinguish a red and a black sublattice with a checkerboard convention: our singlet bonds always connect a red site to a neighboring black site. Vacant sites can lie on either sublattice, so there are red and black charged solitons. We will be interested in ordered pairs {A, B } of singlet bond configurations; the first element can be considered as a bra and the second as a ket in a matrix element (AIdIB). It is also useful to give a direction(6) to each bond: the bonds in the A configuration are directed from red to black, in the B configuration from black to red. Suppose first that the empty sites (charged solitons) are in the same places in the two configurations. Draw both configurations on the same lattice. Since any occupied site shares exactly one A bond and one B bond, the drawing will decompose into nonintersecting loops [61. (If a particular bond is part of both configurations, it will form a trivial loop of no enclosed area.) The directions of the bonds give orientations to each of the loops, and allow one to reconstruct the original pair of configurations from the oriented loops. These loops represent the electronic degrees of freedom left after the positions of the charge solitons are set. We would like to define a quasi-particle wave function P(R1, ..., RN-M) for the solitons, where the electronic degrees of freedom are considered 3 ( 5 ) In the large-U limit, it is easy to see that it goes like tL/UL-l. (6) Many of the arguments presented here, distinguishing red and black sublattices and directions of bonds, work only on bipartite lattices (e.g. square and cubic). The conclusions presented here are also valid for triangular lattices; however, the arguments are a bit differen-ince the loops are no longer oriented, the winding number is only defined modulo two. Also, we should note that an annular region with an edge dislocation through the hole also has winding number defined modulo two (if there are an odd number of sites on a loop, it can change a red soliton to a black one). Again, simple modifications of the arguments give the same answer. 356 EUROPHYSICS LETTERS