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arXiv:2208.04623 · Physical review. B./Physical review. B

U(1)-symmetric Gaussian fermionic projected entangled paired states and their Gutzwiller projection

Jheng-Wei Li, Jan von Delft, Hong-Hao Tu

Published Feb 27, 2023• 10 citations
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Abstract

We develop a formalism for constructing particle-number-conserving Gaussian fermionic projected entangled pair states [U(1)-GfPEPSs] and show that these states can describe ground states of band insulators and gapless fermions with band touching points. When using them as variational Ans\"atze for two Dirac fermion systems (the $\ensuremath{\pi}$-flux model on the square lattice and the $[0,\ensuremath{\pi}]$-flux model on the kagome lattice), we find that the U(1)-GfPEPSs, even with a relatively small bond dimension, can accurately approximate the Dirac Fermi sea ground states. By applying Gutzwiller projectors on top of these U(1)-GfPEPSs, we obtain a PEPS representation of U(1)-Dirac spin liquid states for spin-1/2 systems. With state-of-the-art tensor network numerics, the critical exponent in the spin-spin correlation function of the Gutzwiller-projected $\ensuremath{\pi}$-flux state is estimated to be $\ensuremath{\eta}\ensuremath{\approx}1.7$.

DOI
10.1103/physrevb.107.085148

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We develop a formalism for constructing particle-number-conserving Gaussian fermionic projected entangled pair states [U(1)-GfPEPSs] and show that these states can describe ground states of band insulators and gapless fermions with band touching points. When using them as variational Ans"atze for two Dirac fermion systems (the \ensuremathπ\ensuremath{\pi}\ensuremathπ-flux model on the square lattice and the [0,\ensuremathπ][0,\ensuremath{\pi}][0,\ensuremathπ]-flux model on the kagome lattice), we find that the U(1)-GfPEPSs, even with a relatively small bond dimension, can accurately approximate the Dirac Fermi sea ground states. By applying Gutzwiller projectors on top of these U(1)-GfPEPSs, we obtain a PEPS representation of U(1)-Dirac spin liquid states for spin-1/2 systems. With state-of-the-art tensor network numerics, the critical exponent in the spin-spin correlation function of the Gutzwiller-projected \ensuremathπ\ensuremath{\pi}\ensuremathπ-flux state is estimated to be \ensuremathη\ensuremath≈1.7\ensuremath{\eta}\ensuremath{\approx}1.7\ensuremathη\ensuremath≈1.7.
We study the parton construction of chiral spin liquids (CSLs) using projected Gaussian fermionic entangled pair states (GfPEPSs). First, we show that GfPEPSs can represent generic spinless Chern insulators faithfully with finite bond dimensions. Then by applying the Gutzwiller projection to a bilayer GfPEPSs, spin-12\frac{1}{2}21​ Abelian and non-Abelian CSLs are obtained for Chern number C=1C=1C=1 and 2, respectively. As a consequence of the topological obstruction for GfPEPSs, very weak Gossamer tails are observed in the correlation functions of the fermionic projected entangled pair state (PEPS) Ans\"atze, suggesting that the no-go theorem for chiral PEPS is universal but does not bring any practical limitation. Remarkably, without finetuning, all topological sectors can be constructed showing the expected number of chiral branches in the respective entanglement spectra, providing a sharp improvement with respect to the known bosonic PEPS approach.
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We explain how to implement, in the context of projected entangled-pair states (PEPSs), the general procedure of fermionization of a tensor network introduced in P. Corboz and G. Vidal, Phys. Rev. B 80, 165129 (2009). The resulting fermionic PEPS, similar to previous proposals, can be used to study the ground state of interacting fermions on a two-dimensional lattice. As in the bosonic case, the cost of simulations depends on the amount of entanglement in the ground state and not directly on the strength of interactions. The present formulation of fermionic PEPS leads to a straightforward numerical implementation that allowed us to recycle much of the code for bosonic PEPS. We demonstrate that fermionic PEPS are a useful variational ansatz for interacting fermion systems by computing approximations to the ground state of several models on an infinite lattice. For a model of interacting spinless fermions, ground state energies lower than Hartree-Fock results are obtained, shifting the boundary between the metal and charge-density wave phases. For the $t\text{\ensuremath{-}}J$ model, energies comparable with those of a specialized Gutzwiller-projected ansatz are also obtained.
We develop an improved variant of U(1)U(1)U(1)-symmetric infinite projected entangled-pair states (iPEPS) ansatz to investigate the ground-state phase diagram of the spin-1/21/21/2 square ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model. In order to improve the accuracy of the ansatz, we discuss a simple strategy to select automatically relevant symmetric sectors and also introduce an optimization method to treat second-neighbor interactions more efficiently. We show that variational ground-state energies of the model obtained by the U(1)U(1)U(1)-symmetric iPEPS ansatz (for a fixed bond dimension D)D)D) set a better upper bound, improving previous tensor-network-based results. By studying the finite-DDD scaling of the magnetically order parameter, we find a N\'eel phase for J2/J1<0.53{J}_{2}/{J}_{1}<0.53J2​/J1​<0.53. For 0.53<J2/J1<0.610.53<{J}_{2}/{J}_{1}<0.610.53<J2​/J1​<0.61, a nonmagnetic columnar valence bond solid (VBS) state is established as observed by the pattern of local bond energy. The divergent behavior of correlation length $\ensuremath{\xi}\ensuremath{\sim}{D}^{1.2}$ and vanishing order parameters are consistent with a deconfined N\'eel-to-VBS transition at J2c1/J1=0.530(5){J}_{2}^{{c}_{1}}/{J}_{1}=0.530(5)J2c1​​/J1​=0.530(5), where estimated critical anomalous exponents are ${\ensuremath{\eta}}_{s}\ensuremath{\sim}0.6$ and ${\ensuremath{\eta}}_{d}\ensuremath{\sim}1.9$ for spin and dimer correlations, respectively. We show that the associated VBS order parameter monotonically increases with J2/J1{J}_{2}/{J}_{1}J2​/J1​ and finally a first-order quantum phase transition takes place at J2c2/J1=0.610(2){J}_{2}^{{c}_{2}}/{J}_{1}=0.610(2)J2c2​​/J1​=0.610(2) to the conventional Stripe phase. We compare our results with earlier DMRG and PEPS studies and suggest future directions for resolving remaining issues.
Infinite projected entangled-pair states (iPEPSs) provide a powerful variational framework for two-dimensional quantum matter and have been widely used to capture bosonic topological order, including chiral spin liquids. Here, we extend this approach to fermionic topological order by variationally optimizing U(1)-symmetric fermionic iPEPSs for a fractional Chern insulator (FCI), with bond dimensions up to D=9. We find evidence for a critical bond dimension, above which the ansatz faithfully represents the FCI phase. The FCI state is characterized using bulk observables, including the equal-time single-particle Green's function and the pair-correlation function, as well as the momentum-resolved edge entanglement spectrum. To enable entanglement-spectrum calculations for large iPEPS unit cells, we introduce a compression scheme and show that the low-lying part of the spectrum is already well converged at relatively small cutoff dimensions.
Infinite projected entangled-pair states (iPEPS) provide a powerful variational framework for two-dimensional quantum matter and have been widely used to capture bosonic topological order, including chiral spin liquids. Here we extend this approach to \emph{fermionic} topological order by variationally optimizing U(1)U(1)U(1)-symmetric fermionic iPEPS for a fractional Chern insulator (FCI), with bond dimensions up to D=9D=9D=9. We find evidence for a critical bond dimension, above which the ansatz faithfully represents the FCI phase. The FCI state is characterized using bulk observables, including the equal-time single-particle Green's function and the pair-correlation function, as well as the momentum-resolved edge entanglement spectrum. To enable entanglement-spectrum calculations for large iPEPS unit cells, we introduce a compression scheme and show that the low-lying part of the spectrum is already well converged at relatively small cutoff dimensions.