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arXiv:hep-lat/0509045 · Proceedings of XXIIIrd International Symposium on Lattice Field Theory — PoS(LAT2005)

Gauge invariance in a Z_2 hamiltonian lattice gauge theory

Taka Sugihara

Published Dec 14, 2005• 1 citation
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Abstract

We propose an efficient variational method for $Z_2$ lattice gauge theory based on the matrix product ansatz. The method is applied to ladder and square lattices. The Gauss law needs to be imposed on quantum states to guarantee gauge invariance when one studies gauge theory in hamiltonian formalism. On the ladder lattice, we identify gauge invariant low-lying states by evaluating expectation values of the Gauss law operator after numerical diagonalization of the gauge hamiltonian. On the square lattice, the second order phase transition is well reproduced.

DOI
10.22323/1.020.0181

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We propose an efficient variational method for Z2Z_2Z2​ lattice gauge theory based on the matrix product ansatz. The method is applied to ladder and square lattices. The Gauss law needs to be imposed on quantum states to guarantee gauge invariance when one studies gauge theory in hamiltonian formalism. On the ladder lattice, we identify gauge invariant low-lying states by evaluating expectation values of the Gauss law operator after numerical diagonalization of the gauge hamiltonian. On the square lattice, the second order phase transition is well reproduced.
We propose an efficient variational method for Z{sub 2} lattice gauge theory based on the matrix product ansatz. The method is applied to ladder and square lattices. The Gauss law needs to be imposed on quantum states to guarantee gauge invariance when one studies gauge theory in hamiltonian formalism. On the ladder lattice, we identify gauge invariant low-lying states by evaluating expectation values of the Gauss law operator after numerical diagonalization of the gauge hamiltonian. On the square lattice, the second order phase transition is well reproduced.
We study the lattice Schwinger model by combining the variational uniform matrix product state (VUMPS) algorithm with a gauge-invariant matrix product ansatz that locally enforces the Gauss law constraint. Both the continuum and lattice versions of the Schwinger model with θ = π are known to exhibit first-order phase transitions for the values of the fermion mass above a critical value, where a second-order phase transition occurs. Our algorithm enables a precise determination of the critical point in the continuum theory. We further analyze the scaling in the simultaneous critical and continuum limits and confirm that the data collapse aligns with the Ising universality class to remarkable precision.
We numerically analyze the critical behaviors of the lattice Schwinger model by applying the variational uniform matrix product state (VUMPS) algorithm combined with a gauge-invariant matrix product ansatz that locally enforces the Gauss law constraint. Both the continuum and lattice versions of the Schwinger model with θ=π\theta=\piθ=π are known to exhibit first-order phase transitions for the values of the fermion mass above a critical value, where a second-order phase transition occurs. Our algorithm enables a precise determination of the critical point in the continuum theory. We further analyze the scaling in the simultaneous critical and continuum limits and confirm that the data collapse aligns with the Ising universality class with high precision.
In this work, we implement a variational quantum eigensolver (VQE) suitable for investigating ground states and static string breaking in a Z 2 lattice gauge theory (LGT). We consider a two-leg ladder lattice with Kogut-Susskind staggered fermions and verify the results of the VQE simulations using tensor network methods. We find that for varying Hamiltonian parameter regimes and in the presence of external charges, the VQE is able to arrive at the gauge-invariant ground state without explicitly enforcing gauge invariance through penalty terms. Additionally, experiments showing string breaking are performed on IBM’s quantum platform. Thus, VQEs are seen to be a promising tool for Z 2 LGTs, and could serve as a stepping stone toward studies of other gauge groups. We find that the scaling of gradients with the number of qubits is favorable for avoiding barren plateaus. Furthermore, strategies that avoid barren plateaus arise naturally as features of LGTs, such as choosing the initialization by setting the Gauss law sector and restricting the Hilbert space to the gauge-invariant subspace.
Quantum computation represents an emerging framework to solve lattice gauge theories (LGTs) with arbitrary gauge groups, a general and long-standing problem in computational physics. While quantum computers may encode LGTs using only polynomially increasing resources, a major open issue concerns the violation of gauge invariance during the dynamics and the search for ground states. Here, we propose a class of parametrized quantum circuits that can represent states belonging only to the physical sector of the total Hilbert space. This class of circuits is compact yet flexible enough to be used as a variational Ansatz to study ground-state properties, as well as representing states originating from a real-time dynamics. Concerning the first application, the structure of the wavefunction Ansatz guarantees the preservation of physical constraints such as the Gauss law along the entire optimization process, enabling reliable variational calculations. As for the second application, this class of quantum circuits can be used in combination with time-dependent variational quantum algorithms, thus drastically reducing the resource requirements to access dynamical properties.
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