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arXiv:1702.08838 · Physical review. D/Physical review. D.

Finite-representation approximation of lattice gauge theories at the continuum limit with tensor networks

Boye Buyens, Simone Montangero, Jutho Haegeman, Frank Verstraete, Karel Van Acoleyen

Published May 24, 2017• 84 citations
arXivHTMLPDF

Abstract

It has been established that matrix product states can be used to compute the ground state and single-particle excitations and their properties of lattice gauge theories at the continuum limit. However, by construction, in this formalism the Hilbert space of the gauge fields is truncated to a finite number of irreducible representations of the gauge group. We investigate quantitatively the influence of the truncation of the infinite number of representations in the Schwinger model, one-flavor ${\mathrm{QED}}_{2}$, with a uniform electric background field. We compute the two-site reduced density matrix of the ground state and the weight of each of the representations. We find that this weight decays exponentially with the quadratic Casimir invariant of the representation which justifies the approach of truncating the Hilbert space of the gauge fields. Finally, we compute the single-particle spectrum of the model as a function of the electric background field.

DOI
10.1103/physrevd.95.094509

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It has been established that matrix product states can be used to compute the ground state and single-particle excitations and their properties of lattice gauge theories at the continuum limit. However, by construction, in this formalism the Hilbert space of the gauge fields is truncated to a finite number of irreducible representations of the gauge group. We investigate quantitatively the influence of the truncation of the infinite number of representations in the Schwinger model, one-flavor QED2{\mathrm{QED}}_{2}QED2​, with a uniform electric background field. We compute the two-site reduced density matrix of the ground state and the weight of each of the representations. We find that this weight decays exponentially with the quadratic Casimir invariant of the representation which justifies the approach of truncating the Hilbert space of the gauge fields. Finally, we compute the single-particle spectrum of the model as a function of the electric background field.
Over the last decade tensor network states (TNS) have emerged as a powerful tool for the study of quantum many body systems. The matrix product states (MPS) are one particular case of TNS and are used for the simulation of 1+1 dimensional systems. In [1] we considered the MPS formalism for the simulation of the Hamiltonian lattice gauge formulation of 1+1 dimensional one flavor quantum electrodynamics, also known as the massive Schwinger model. We deduced the ground state and lowest lying excitations. Furthermore, we performed a full quantum real-time simulation for a quench with a uniform background electric field. In this proceeding we continue our work on the Schwinger model. We demonstrate the advantage of working with gauge invariant MPS by comparing with MPS simulations on the full Hilbert space, that includes numerous non-physical gauge variant states. Furthermore, we compute the chiral condensate and recover the predicted UV-divergent behavior.
The matrix product state formalism is used to simulate Hamiltonian lattice gauge theories. To this end, we define matrix product state manifolds which are manifestly gauge invariant. As an application, we study (1+1)-dimensional one flavor quantum electrodynamics, also known as the massive Schwinger model, and are able to determine very accurately the ground-state properties and elementary one-particle excitations in the continuum limit. In particular, a novel particle excitation in the form of a heavy vector boson is uncovered, compatible with the strong coupling expansion in the continuum. We also study full quantum nonequilibrium dynamics by simulating the real-time evolution of the system induced by a quench in the form of a uniform background electric field.
Quantum simulations of lattice gauge theories with continuous gauge groups require a choice of basis (how states of the gauge theory map to the qubit computational basis) and truncation scheme (which of the infinitely many gauge theory basis states should be omitted in the map to the finite dimensional qubit Hilbert space.) The most common choices are adapted to an electric basis for the gauge theory, where the basis states for each link are associated with irreducible representations of the gauge group. Electric bases and truncations are generally advantageous for preserving gauge symmetries and simulating the strong coupling regime, but their effectiveness in the weak coupling regimes of physical interest are less well understood. In this work, we compute relative probabilities p(R′)/p(R)p(R')/p(R)p(R′)/p(R) for finding a link in the irreps R,R′R,R'R,R′ in the ground state of SU(3) lattice gauge theory using Euclidean Monte Carlo methods and weak-coupling lattice perturbation theory. The observables that compute these relative probabilities are constructed by introducing seams into an anisotropic lattice, and by a similar construction we are also able to measure relative probabilities of different site singlets (ways that electric flux braids together into a gauge singlet) at each vertex. With these methods we can quantify, for example, the importance of higher electric flux states in the weak coupling ground state. These results directly inform what truncations are needed to accurately capture non-abelian gauge dynamics near the continuum limit.
Over the last decade tensor network states (TNS) have emerged as a powerful tool for the study of quantum many body systems.The matrix product states (MPS) are one particular case of TNS and are used for the simulation of 1+1 dimensional systems.In [1] we considered the MPS formalism for the simulation of the Hamiltonian lattice gauge formulation of 1+1 dimensional one flavor quantum electrodynamics, also known as the massive Schwinger model.We deduced the ground state and lowest lying excitations.Furthermore, we performed a full quantum real-time simulation for a quench with a uniform background electric field.In this proceeding we continue our work on the Schwinger model.We demonstrate the advantage of working with gauge invariant MPS by comparing with MPS simulations on the full Hilbert space, that includes numerous non-physical gauge variant states.Furthermore, we compute the chiral condensate and recover the predicted UV-divergent behavior.
Over the last decade tensor network states (TNS) have emerged as a powerful tool for the study of quantum many body systems.The matrix product states (MPS) are one particular class of TNS and are used for the simulation of 1+1 dimensional systems.In this proceeding we use MPS to determine the elementary excitations of the Schwinger model in the presence of an electric background field.We obtain an estimate for the value of the background field where the oneparticle excitation with the largest energy becomes unstable and decays into two other elementary particles with smaller energy.
Over the last decade tensor network states (TNS) have emerged as a powerful tool for the study of quantum many body systems.The matrix product states (MPS) are one particular class of TNS and are used for the simulation of (1+1)-dimensional systems.In this proceeding we use MPS to determine the elementary excitations of the Schwinger model in the presence of an electric background field.We obtain an estimate for the value of the background field where the oneparticle excitation with the largest energy becomes unstable and decays into two other elementary particles with smaller energy.