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arXiv:cond-mat/0204043 · Physical review. B, Condensed matter

Projected wave functions for fractionalized phases of quantum spin systems

Dmitri A. Ivanov, T. Senthil

Published Sep 17, 2002• 33 citations
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Abstract

The Gutzwiller projection allows the construction of an assortment of variational wave functions for strongly correlated systems. For quantum spin $S=1/2$ models, Gutzwiller-projected wave functions have resonating-valence-bond structure and may represent states with fractional quantum numbers for the excitations. Using insights obtained from field-theoretical descriptions of fractionalization in two dimensions, we construct candidate wave functions of fractionalized states by projecting specific superconducting states. We explicitly demonstrate the presence of topological order in these states.

DOI
10.1103/physrevb.66.115111

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The Gutzwiller projection allows the construction of an assortment of variational wave functions for strongly correlated systems. For quantum spin S=1/2S=1/2S=1/2 models, Gutzwiller-projected wave functions have resonating-valence-bond structure and may represent states with fractional quantum numbers for the excitations. Using insights obtained from field-theoretical descriptions of fractionalization in two dimensions, we construct candidate wave functions of fractionalized states by projecting specific superconducting states. We explicitly demonstrate the presence of topological order in these states.
Variational wave functions have enabled exceptional scientific breakthroughs related to the understanding of novel phases of matter. Examples include the Bardeen-Cooper-Schrieffer theory of superconductivity, the description of the fractional quantum Hall effect through the Laughlin state, and Feynman's variational understanding of large-scale quantum effects in liquid helium. More recently, Gutzwiller-projected wave functions, typically constructed from fermionic degrees of freedom, have been employed to examine quantum spin models in the presence of competing interactions, where exotic phases with no spontaneous symmetry breaking and fractional excitations may exist. In this work, we investigate the aforementioned fermionic wave functions supplemented with neural networks, specifically with the so-called restricted Boltzmann machine (RBM), to boost their accuracy and obtain reliable approximations to the ground state of generic spin models. In particular, we apply our neural augmented fermionic construction to the description of both magnetically ordered and disordered phases of increasing complexity, including cases where the ground state displays a nontrivial sign structure. Even though the RBM state is by far more effective for N\'eel states endowed with a particularly simple sign structure, it provides a significant improvement over the original fermionic state in highly frustrated regimes where a complex sign structure is anticipated, thus marking the path to an understanding of strongly correlated spin models on the lattice via neural Gutzwiller-projected variational wave functions.
Gutzwiller-projected (GP) wave functions have been widely used for describing spin-liquid physics in frustrated magnets and in high-temperature superconductors. Such wave functions are known to represent states of the resonating-valence-bond (RVB) type. In the present work I discuss the RVB structure of a GP singlet superconducting state with nodes in the spectrum. The resulting state for the undoped spin system may be described in terms of a ``path integral'' over loop coverings of the lattice, thus extending the known construction for RVB states. The problem of topological order in GP states may be reformulated in terms of the statistical behavior of loops. The simple example of the projected ddd-wave state on the square lattice demonstrates that the statistical behavior of loops is renormalized in a nontrivial manner by the projection.
We study the topological order in a resonating valence bond (RVB) state derived from the Gutzwiller projection of the BCS-like mean-field state. We propose the construction of the topological excitation on the projected RVB state through the Gutzwiller projection of the mean-field state with inserted Z2{Z}_{2}Z2​ flux tube. We prove that all projected RVB states derived from bipartite effective theories, no matter what the gauge structure in the mean-field ansatz is, are positive definite in the sense of the Marshall sign rule, which provides a universal origin for the absence of topological order in such an RVB state.
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.
The unconventional low-lying spin excitations, recently observed in neutron scattering experiments on Cs2CuCl4\mathrm{C}{\mathrm{s}}_{\mathrm{2}}\mathrm{CuC}{\mathrm{l}}_{\mathrm{4}}Cs2​CuCl4​, are explained with a spin liquid wave function. The dispersion relation as well as the wave vector of the incommensurate spin correlations are well reproduced within a projected BCS wave function with gapless and fractionalized spin-1/21/21/2 excitations around the nodes of the BCS gap function. The proposed wave function is shown to be very accurate for one-dimensional spin-1/21/21/2 systems and remains similarly accurate in the two-dimensional model corresponding to Cs2CuCl4\mathrm{C}{\mathrm{s}}_{\mathrm{2}}\mathrm{CuC}{\mathrm{l}}_{\mathrm{4}}Cs2​CuCl4​, thus representing a good ansatz for describing spin fractionalization in two dimensions.
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.