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arXiv:cond-mat/9212019 · Physical review. B, Condensed matter

1/Nexpansion and spin correlations in constrained wave functions

Maxim Raykin, Assa Auerbach

Published Mar 1, 1993• 15 citations
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Abstract

We develop a large-N expansion for Gutzwiller projected spin states. We consider valence-bond singlets, constructed by Schwinger bosons or fermions, which are variational ground states for quantum antiferromagnets. This expansion is simpler than the familiar expansions of the quantum Heisenberg model, and thus more instructive. The diagrammatic rules of this expansion allow us to prove certain identities to all orders in 1/N. We derive the on-site spin-fluctuations sum rule for arbitrary N. We calculate the correlations of the one-dimensional valence-bond solid states and the Gutzwiller projected Fermi gas up to order 1/N. For the boson case, we are surprised to find that the mean-field, the order-1/N, and the exact correlations are simply proportional. For the fermion case, the 1/N correction enhances the zone-edge singularity. The comparison of our leading-order terms to known results for N=2 enhances our understanding of large-N approximations in general.

DOI
10.1103/physrevb.47.5118

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We develop a large-N expansion for Gutzwiller projected spin states. We consider valence-bond singlets, constructed by Schwinger bosons or fermions, which are variational ground states for quantum antiferromagnets. This expansion is simpler than the familiar expansions of the quantum Heisenberg model, and thus more instructive. The diagrammatic rules of this expansion allow us to prove certain identities to all orders in 1/N. We derive the on-site spin-fluctuations sum rule for arbitrary N. We calculate the correlations of the one-dimensional valence-bond solid states and the Gutzwiller projected Fermi gas up to order 1/N. For the boson case, we are surprised to find that the mean-field, the order-1/N, and the exact correlations are simply proportional. For the fermion case, the 1/N correction enhances the zone-edge singularity. The comparison of our leading-order terms to known results for N=2 enhances our understanding of large-N approximations in general.
SU ( N ) quantum spin systems may be realized in a variety of physical systems including ultracold atoms in optical lattices. The study of such models also leads to insights into possible novel quantum phases and phase transitions of SU (2) spin models. Here we use Gutzwiller projected fermionic variational wavefunctions to explore the phase diagram and correlation functions of SU ( N ) quantum spin models in the self-conjugate representation. In one dimension, the ground state of the SU (4) spin chain with Heisenberg bilinear and biquadratic interactions is studied by examining instabilities of the Gutzwiller projected free fermion ground state to various broken symmetries. The variational phase diagram so obtained agrees well with exact results. The spin–spin and dimer–dimer correlation functions of the Gutzwiller projected free fermion state with N flavours of fermions are in good agreement with exact and 1/ N calculations for the critical points of SU ( N ) spin chains. In two dimensions, the phase diagram of the antiferromagnetic Heisenberg model on the square lattice is obtained by finding instabilities of the Gutzwiller projected π-flux state. In the absence of biquadratic interactions, the model exhibits long-range Néel order for N = 2 and 4, and spin Peierls (columnar dimer) order for N >4. Upon including biquadratic interactions in the SU (4) model (with a sign appropriate to a fermionic Hubbard model), the Néel order diminishes and eventually disappears, giving way to an extended valence bond crystal. In the case of the SU (6) model, the dimerized ground state melts at sufficiently large biquadratic interaction, yielding a projected π-flux spin liquid phase which in turn undergoes a transition into an extended valence bond crystal at even larger biquadratic interaction. The spin correlations of the projected π-flux state at N = 4 are in good agreement with 1/ N calculations. We find that the state shows strongly enhanced dimer correlations, in qualitative agreement with recent theoretical predictions. We also compare our results with a recent quantum Monte Carlo study of the SU (4) Heisenberg model.
We present a systematic formalism for the variational evaluation of ground-state properties of Hubbard-type models in finite dimensions d. The formalism starts from generalized Gutzwiller correlated wave functions, which are then studied in a systematic (1/d) expansion around the limit of high dimensions (d=\ensuremath{\infty}). The limit of d\ensuremath{\rightarrow}\ensuremath{\infty} has recently been introduced by Metzner and Vollhardt (MV) for itinerant lattice fermion systems. The approach, presented in this paper, is particularly efficient since results in d=\ensuremath{\infty} are obtained without having to calculate a single graph. Our results confirm the finding of MV that counting approximations in the spirit of the Gutzwiller approximation become exact in d=\ensuremath{\infty} for translationally invariant wave functions. This type of approximation is no longer exact for more complicated (e.g., antiferromagnetic) wave functions. In addition, we completely reproduce the results of the Kotliar-Ruckenstein path-integral approach to the Hubbard model. Performing a (1/d) expansion for the Gutzwiller wave function, we show that the lowest orders in (1/d) are sufficient to reproduce all numerical findings in d=2,3 quantitatively. We therefore conclude that the limit of d=\ensuremath{\infty} is a very fruitful starting point for the study of finite-dimensional systems. On the basis of our study we propose new variational wave functions for the numerical investigation of antiferromagnetism in the Hubbard, the t-J, and the spin-1/2 Heisenberg model. For the first two models we calculate in d=\ensuremath{\infty} only, for the Heisenberg model we also derive corrections up to order (1/d). To this order we obtain complete agreement with linear spin-wave theory. Since our trial state is based on a fermionic description of the Heisenberg model, we interpret this analytically determined wave function as Fermi sea of spin-1/2 quasiparticles (``spinons'').
We make use of a recently developed diagrammatic theory to calculate correlation functions for interacting fermions of Hubbard-type models in terms of the Gutzwiller wave function. Of the eleven nontrivial correlation functions involving the spin, density, empty and doubly occupied sites, and local Cooper pairs, four are shown to be independent. They are expressed as a power series in a suitably chosen correlation parameter, whose terms are represented diagrammatically. In one dimension these terms may be evaluated to arbitrary order by employing symmetry relations. This allows for an analytic, approximation-free calculation of the correlation functions for arbitrary momentum, particle density, and interaction strength. In the atomic limit the momentum-dependent spin-correlation function shows an antiferromagnetic divergence at half filling in all dimensions. In one dimension the behavior is in very good agreement with all exact analytic and numerical results for the antiferromagnetic Heisenberg chain. Hole-hole correlations also compare very well with exact results. However, correlations between holes and doubly occupied sites appear insufficient. Superconducting correlations involving on-site, singlet Cooper pairs are suppressed. The results allow for an analytic evaluation of the ground-state energy of a large class of extended Hubbard models in terms of the Gutzwiller wave function. Thus they provide exact upper bounds for their ground-state energies.
A new analytic approach to the evaluation of ground-state properties of Hubbard-type models in terms of the Gutzwiller variational wave function is presented. It is based on the observation that expectation values in terms of this wave function may be expressed by sums over different lattice sites. This makes the application of Wick's theorem and the resulting contractions extremely simple, since the latter involve only anticommuting numbers as in a Grassmann algebra. Expressions for the momentum distribution ${n}_{\mathrm{k}\ensuremath{\sigma}}$ and the Hubbard interaction in terms of a power series in a particular correlation parameter are derived which are valid for all dimensions. An explicit diagrammatic evaluation of the coefficients is described. In one dimension these coefficients may be determined to all orders which yields an approximation-free calculation of nk{n}_{\mathrm{k}}nk​ and ground-state energy E for arbitrary density n and interaction strength U. In the case of a half-filled band and large U, hitherto unexpected nonanalyticities are found. The results allow for the first approximation-free assessment of the properties of the Gutzwiller wave function. It is shown that the well-known Gutzwiller approximation may be derived diagrammatically, too. The approximation is seen to yield the exact results for expectation values in terms of this wave function in the limit of infinite dimensionality.
We investigate Gutzwiller-correlated wave functions as variational ground states for the periodic Anderson model. We extend a recently developed technique for the Hubbard model to the case of two hybridized bands with degeneracy N. We calculate expectation values in the following limits: dilute system (single-impurity case), large degeneracy N, and large dimension d. The mean-field results using the slave-boson approach of Coleman and of Kotliar and Ruckenstein are reproduced in the large-N and large-d limit, respectively. These techniques, therefore, obey variational principles at T=0 in the respective limits and their applicability for T>0 is naturally limited to the Fermi-liquid regime. Evaluations of Gutzwiller-correlated wave functions using the ``Gutzwiller approximation'' are seen to become exact in the limit of large dimensions. A systematic expansion in 1/d can be performed for N=2. In general, corrections are small, of order 1/(dN).
We study generalizations of the singlet-sector amplitude-product (AP) states in the valence-bond basis of S=1/2S=1/2S=1/2 quantum spin systems. In the standard AP states, the weight of a tiling of the system into valence bonds (singlets of two spins) is a product of amplitudes depending on the length of the bonds. We here introduce correlated AP (CAP) states, in which the AP is further multiplied by factors depending on two bonds connected to a pair of sites (here nearest neighbors). While the standard AP states can describe a phase transition between an antiferromagnetic (N\'eel) state and a valence-bond solid (VBS) in one dimension (which we also study here), in two dimensions it cannot describe VBS order. With the CAP states, N\'eel-VBS transitions are realized as a function of some parameter describing the bond correlations. We here study such phase transitions of CAP wave functions on the square lattice. We find examples of direct first-order N\'eel-VBS transitions, as well as cases where there is an extended U(1) spin liquid phase intervening between the N\'eel and VBS states. In the latter case the transitions are continuous and we extract critical exponents and address the issue of a possible emergent U(1) symmetry in the near-critical VBS. We also consider variationally optimized CAP states for the standard Heisenberg model in one and two dimensions and the JJJ-QQQ model in two dimensions, with the latter including four-spin interactions (QQQ) in addition to the Heisenberg exchange (JJJ) and harboring VBS order for large Q/JQ/JQ/J. The optimized CAP states lead to significantly lower variational energies than the simple AP states for these models.