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Bulletin of the American Physical Society

Using Tensor Network States for Lattice Gauge Theories

Mari Carmen Bañuls, Krzysztof Cichy, J. I. Cirac, Karl Jansen, Stefan Kühn

Published Mar 7, 2018• 0 citations
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Mari Carmen Bañuls, Krzysztof Cichy, J. I. Cirac +2 more

The term Tensor Network States (TNS) refers to a number of families of states that represent different ansätze for the efficient description of the state of a quantum many-body system.Matrix Product States (MPS) are one particular case of TNS, and have become the most precise tool for the numerical study of one dimensional quantum many-body systems, as the basis of the Density Matrix Renormalization Group method.Lattice Gauge Theories (LGT), in their Hamiltonian version, offer a challenging scenario for these techniques.While the dimensions and sizes of the systems amenable to TNS studies are still far from those achievable by 4-dimensional LGT tools, Tensor Networks can be readily used for problems which more standard techniques, such as Markov chain Monte Carlo simulations, cannot easily tackle.Examples of such problems are the presence of a chemical potential or out-of-equilibrium dynamics.We have explored the performance of Matrix Product States in the case of the Schwinger model, as a widely used testbench for lattice techniques.Using finite-size, open boundary MPS, we are able to determine the low energy states of the model in a fully non-perturbative manner.The precision achieved by the method allows for accurate finite size and continuum limit extrapolations of the ground state energy, but also of the chiral condensate and the mass gaps, thus showing the feasibility of these techniques for gauge theory problems.

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A comprehensive stress test of truncated Hilbert space bases against Green’s function Monte Carlo in U(1) lattice Gauge theory

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A representation of Lattice Gauge Theories (LGT) suitable for simulations with tensor network state methods or with quantum computers requires a truncation of the Hilbert space to a finite dimensional approximation. In particular for U(1) LGTs, several such truncation schemes are known, which we compare with each other using tensor network states. We show that a functional basis obtained from single plaquette Hamiltonians – which we call plaquette state basis – outperforms the other schemes in two spatial dimensions for plaquette, ground state energy and mass gap, as it is delivering accurate results for a wide range of coupling strengths with a minimal number of basis states. We also show that this functional basis can be efficiently used in three spatial dimensions. Green’s function Monte Carlo appears to be a highly useful tool to verify tensor network states results, which deserves further investigation in the future.

•Jun 13, 2026•0 citations
We propose a novel tensor-network method to achieve accurate and efficient simulations of Abelian lattice gauge theories (LGTs) in (2+1)D for both ground-state and real-time dynamics. The first key is to identify a gauge canonical form of gauge-invariant tensor-network states, which already simplifies existing algorithms for (1+1)D LGTs. The second key is to employ the gauge canonical form of a projected entangled-pair state (PEPS) combining with variational Monte Carlo, enabling efficient computations for (2+1)D LGTs. We demonstrate the versatile capability of this approach for accurate ground-state simulation of pure Z_{2}, Z_{3}, and Z_{4} gauge theory, odd-Z_{2} gauge theories, and Z_{2} gauge theory coupled to hard-core bosons, on square lattices up to 32×32. Furthermore, we demonstrate that it allows for accurate simulations of real-time dynamics up to long time, exemplified by the dynamics of elementary excitations of the deconfined Z_{2} gauge field at 10×10. This is also the first example of simulating the real-time dynamics of PEPS with variational Monte Carlo, whose impact may extend beyond gauge theory. Our Letter establishes gauge-invariant PEPS as a powerful approach for both ground-state and dynamical simulations, opening up a new avenue for nonperturbatively studying (2+1)D LGTs.
The term Tensor Network States (TNS) refers to a number of families of states that represent different ansätze for the efficient description of the state of a quantum many-body system.Matrix Product States (MPS) are one particular case of TNS, and have become the most precise tool for the numerical study of one dimensional quantum many-body systems, as the basis of the Density Matrix Renormalization Group method.Lattice Gauge Theories (LGT), in their Hamiltonian version, offer a challenging scenario for these techniques.While the dimensions and sizes of the systems amenable to TNS studies are still far from those achievable by 4-dimensional LGT tools, Tensor Networks can be readily used for problems which more standard techniques, such as Markov chain Monte Carlo simulations, cannot easily tackle.Examples of such problems are the presence of a chemical potential or out-of-equilibrium dynamics.We have explored the performance of Matrix Product States in the case of the Schwinger model, as a widely used testbench for lattice techniques.Using finite-size, open boundary MPS, we are able to determine the low energy states of the model in a fully non-perturbative manner.The precision achieved by the method allows for accurate finite size and continuum limit extrapolations of the ground state energy, but also of the chiral condensate and the mass gaps, thus showing the feasibility of these techniques for gauge theory problems.
A representation of Lattice Gauge Theories (LGT) suitable for simulations with tensor network state methods or with quantum computers requires a truncation of the Hilbert space to a finite dimensional approximation. In particular for U(1) LGTs, several such truncation schemes are known, which we compare with each other using tensor network states. We show that a functional basis obtained from single plaquette Hamiltonians – which we call plaquette state basis – outperforms the other schemes in two spatial dimensions for plaquette, ground state energy and mass gap, as it is delivering accurate results for a wide range of coupling strengths with a minimal number of basis states. We also show that this functional basis can be efficiently used in three spatial dimensions. Green’s function Monte Carlo appears to be a highly useful tool to verify tensor network states results, which deserves further investigation in the future.