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arXiv:2008.11176 · Physical Review Letters

Tensor Networks Can Resolve Fermi Surfaces

Quinten Mortier, Norbert Schuch, Frank Verstraete, Jutho Haegeman

Published Nov 8, 2022• 19 citations
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Abstract

We demonstrate that projected entangled-pair states are able to represent ground states of critical, fermionic systems exhibiting both 1d and 0d Fermi surfaces on a 2D lattice with an efficient scaling of the bond dimension. Extrapolating finite size results for the Gaussian restriction of fermionic projected entangled-pair states to the thermodynamic limit, the energy precision as a function of the bond dimension is found to improve as a power law, illustrating that an arbitrary precision can be obtained by increasing the bond dimension in a controlled manner. In this process, boundary conditions and system sizes have to be chosen carefully so that nonanalyticities of the Ansatz, rooted in its nontrivial topology, are avoided.

DOI
10.1103/physrevlett.129.206401

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We demonstrate that projected entangled-pair states are able to represent ground states of critical, fermionic systems exhibiting both 1d and 0d Fermi surfaces on a 2D lattice with an efficient scaling of the bond dimension. Extrapolating finite size results for the Gaussian restriction of fermionic projected entangled-pair states to the thermodynamic limit, the energy precision as a function of the bond dimension is found to improve as a power law, illustrating that an arbitrary precision can be obtained by increasing the bond dimension in a controlled manner. In this process, boundary conditions and system sizes have to be chosen carefully so that nonanalyticities of the Ansatz, rooted in its nontrivial topology, are avoided.
We show that Projected Entangled-Pair States (PEPS) are able to describe critical, fermionic systems exhibiting both 1d and 0d Fermi surfaces on a 2d lattice. In the thermodynamic limit, the energy precision as a function of the bond dimension improves as a power law, illustrating that an arbitrary precision can be obtained by increasing the bond dimension in a controlled manner. We also identify a non-trivial obstruction in the Gaussian and fermionic variant of PEPS, rooted in its topology and restricting its efficient applicability to models with a matching parity configuration.
We demonstrate the use of finite-size fermionic projected entangled pair states, in conjunction with variational Monte Carlo, to perform accurate simulations of the ground state of the 2D Hubbard model. Using bond dimensions of up to D=28, we show that we can surpass state-of-the-art density matrix renormalization group energies that use up to m=32 000 SU(2) multiplets on eight-leg ladders. We further apply our methodology to 10×16, 12×16, and 16×16 lattices at 1/8 hole doping and observe the dimensional crossover between stripe orientations. Our Letter shows the power of finite-size fermionic tensor networks to resolve the physics of the 2D Hubbard model and related problems.
We extend the study of finite-entanglement scaling from one-dimensional gapless models to two-dimensional systems with a Fermi surface. In particular, we show that the entanglement entropy of a contractible spatial region with linear size L scales as S∼Llog[ξf(L/ξ)] in the optimal tensor network, and hence area-law entangled, state approximation to a metallic state, where f(x) is a scaling function which depends on the shape of the Fermi surface and ξ is a finite correlation length induced by the restricted entanglement. Crucially, the scaling regime can be realized with numerically tractable bond dimensions. We also discuss the implications of the Lieb-Schultz-Mattis theorem at fractional filling for tensor network state approximations of metallic states.
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 $\ensuremath{\eta}\ensuremath{\approx}1.7$.
Tensor network states, and in particular projected entangled pair states (PEPS) have been a strong ansatz for the variational study of complicated quantum many-body systems, thanks to their built-in entanglement entropy area law. In this work, we use a special kind of PEPS---gauged Gaussian fermionic PEPS (GGFPEPS)---to find the ground state of 2+12+12+1 dimensional pure Z2{\mathbb{Z}}_{2}Z2​ lattice gauge theories for a wide range of coupling constants. We do so by combining PEPS methods with Monte-Carlo computations, allowing for efficient contraction of the PEPS and computation of correlation functions. Previously, such numerical computations involved the calculation of the Pfaffian of a matrix scaling with the system size, forming a severe bottleneck; in this work we show how to overcome this problem. This paves the way for applying the method we propose and benchmark here to other gauge groups, higher dimensions, and models with fermionic matter, in an efficient, sign-problem-free way.
An infinite projected entangled-pair state (iPEPS) is a variational tensor network ansatz for two-dimensional wave functions in the thermodynamic limit where the accuracy can be systematically controlled by the bond dimension DDD. We show that for the doped Hubbard model in the strongly correlated regime (U/t=8,n=0.875)(U/t=8,n=0.875)(U/t=8,n=0.875) the iPEPS yields lower variational energies than state-of-the-art variational methods in the large two-dimensional limit, which demonstrates the competitiveness of the method. In order to obtain an accurate estimate of the energy in the exact infinite-DDD limit we introduce and test an extrapolation technique based on a truncation error computed in the iPEPS imaginary time evolution algorithm. The extrapolated energies are compared with accurate quantum Monte Carlo results at half-filling and with various other methods in the doped, strongly correlated regime.