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arXiv:2208.04566 · Physical review. B./Physical review. B

Projected d -wave superconducting state: A fermionic projected entangled pair state study

Qi Yang, Xing-Yu Zhang, Haijun Liao, Hong-Hao Tu, Lei Wang

Published Mar 13, 2023• 9 citations
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Abstract

We investigate the physics of projected $d$-wave pairing states using their fermionic projected entangled pair state (fPEPS) representation. First, we approximate a $d$-wave Bardeen-Cooper-Schrieffer state using the Gaussian fPEPS. Next, we translate the resulting state into fPEPS tensors and implement the Gutzwiller projection which removes double occupancy by modifying the local tensor elements. The tensor network representation of the projected $d$-wave pairing state allows us to evaluate physical quantities in the thermodynamic limit without employing the Gutzwiller approximation. Despite having very few variational parameters, such physically motivated tensor network states are shown to exhibit competitive energies for the doped $t\text{\ensuremath{-}}J$ model. We expect that such construction offers useful initial states and guidance for variational tensor network calculations.

DOI
10.1103/physrevb.107.125128

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We investigate the physics of projected ddd-wave pairing states using their fermionic projected entangled pair state (fPEPS) representation. First, we approximate a ddd-wave Bardeen-Cooper-Schrieffer state using the Gaussian fPEPS. Next, we translate the resulting state into fPEPS tensors and implement the Gutzwiller projection which removes double occupancy by modifying the local tensor elements. The tensor network representation of the projected ddd-wave pairing state allows us to evaluate physical quantities in the thermodynamic limit without employing the Gutzwiller approximation. Despite having very few variational parameters, such physically motivated tensor network states are shown to exhibit competitive energies for the doped t\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J model. We expect that such construction offers useful initial states and guidance for variational tensor network calculations.
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 $\ensuremath{\eta}\ensuremath{\approx}1.7$.
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 construct a family of simple fermionic projected entangled pair states (fPEPS) on the square lattice with bond dimension D=3D=3D=3 which are exactly hole-doped resonating valence bond (RVB) wave functions with short-range singlet bonds. Under doping the insulating RVB spin liquid evolves immediately into a superconductor with mixed d+isd+isd+is pairing symmetry whose pair amplitude grows as the square root of the doping. The relative weight between sss-wave and ddd-wave components can be controlled by a single variational parameter ccc. We optimize our ansatz with respect to ccc for the frustrated $t\text{\ensuremath{-}}{J}_{1}\text{\ensuremath{-}}{J}_{2}$ model (including both nearest and next-nearest neighbor antiferromagnetic interactions J1{J}_{1}J1​ and J2{J}_{2}J2​, respectively) for ${J}_{2}\ensuremath{\simeq}{J}_{1}/2$ and obtain an energy very close to the infinite-PEPS state (using full update optimization and same bond dimension). The orbital symmetry of the optimized RVB superconductor has predominant ddd-wave character, although we argue a residual (complex sss-wave) time reversal symmetry breaking component should always be present. Connections of the results to the physics of superconducting cuprates and pnictides are outlined.
Gutzwiller-projected Bardeen--Cooper--Schrieffer (BCS) wave functions of Abrikosov fermions are widely used to describe quantum many-body states. Taking the one-dimensional transverse-field Ising model as an example, we exactly embed any even-parity spinless fermionic Gaussian state representable as a paired exponential in the chosen particle basis into a projected BCS state of spinful Abrikosov fermions. The construction provides controlled initial states for variational Monte Carlo studies of nonintegrable models.
Recent work by Wu et al. [arXiv:1910.11011] proposed a numerical method, so-called matrix product operator--matrix product state (MPO-MPS) method, by which several types of quantum many-body wave functions, in particular, the projected Fermi sea state, can be efficiently represented as a tensor network. In this paper, we generalize the MPO-MPS method to study Gutzwiller projected paired states of fermions, where the maximally localized Wannier orbitals for Bogoliubov quasiparticles/quasiholes have been adapted to improve the computational performance. The study of SO(3)SO(3)SO(3)-symmetric spin-1 chains reveals that this new method has better performance than variational Monte Carlo for gapped states and similar performance for gapless states. Moreover, we demonstrate that dynamic correlation functions can be easily evaluated by this method cooperating with other MPS-based accurate approaches, such as the Chebyshev MPS method.
Fermionic Gaussian projected entangled pair states (PEPS) are fermionic tensor network state constructions that describe the physics of ground states of noninteracting fermionic Hamiltonians. As noninteracting states, one may study and analyze them very efficiently, in both analytical and numerical means. Recently it was shown that they may be used as the starting point---after applying so-called PEPS gauging mechanisms---for variational study of nontrivial, interacting states of lattice gauge theories. This is done using sign-problem free variational Monte Carlo techniques. In this work we show how to generalize such states from two to three spatial dimensions, focusing on spin representations and requirements of lattice rotations. We present constructions that are crucial for the application of the above-mentioned variational Monte Carlo techniques for studying nonperturbative lattice gauge theory physics, with fermionic matter, in 2+1D2+1\mathrm{D}2+1D and 3+1D3+1\mathrm{D}3+1D models. Thus, the constructions presented here are crucial for the study of nontrivial lattice gauge theories with fermionic tensor network states.