Variational study of U(1) and SU(2) lattice gauge theories with Gaussian states in $1+1$ dimensions

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Variational study of U(1) and SU(2) lattice gauge theories with Gaussian states in $1 + 1$ dimensions

P. Sala, T. Shi, S. Kühn, M. C. Bañuls, E. Demler, and J. I. Cirac

Phys. Rev. D 98, 034505 – Published 27 August 2018

Abstract

We introduce a method to investigate the static and dynamic properties of both Abelian and non-Abelian lattice gauge models in $1 + 1$ dimensions. Specifically, we identify a set of transformations that disentangle different degrees of freedom, and apply a simple Gaussian variational ansatz to the resulting Hamiltonian. To demonstrate the suitability of the method, we analyze both static and dynamic aspects of string breaking for the U(1) and SU(2) gauge models. We benchmark our results against tensor network simulations and observe excellent agreement, although the number of variational parameters in the Gaussian ansatz is much smaller.

Received 31 May 2018
Corrected 20 December 2018

DOI:https://doi.org/10.1103/PhysRevD.98.034505

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Lattice gauge theory

Variational approach

Condensed Matter, Materials & Applied PhysicsParticles & Fields

Corrections

20 December 2018

Correction: Conversion errors in the typesetting of bold script Q letters throughout the paper have been fixed.

Authors & Affiliations

P. Sala^1,*, T. Shi^2,†, S. Kühn³, M. C. Bañuls⁴, E. Demler⁵, and J. I. Cirac⁴

¹Department of Physics, T42, Technische Universität München, James-Franck-Straße 1, D-85748 Garching, Germany
²Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100190, China
³Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5, Canada
⁴Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany
⁵Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

^*pablo.sala@tum.de
^†tshi@itp.ac.cn

Article Text

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Issue

Vol. 98, Iss. 3 — 1 August 2018

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Images

Figure 1
Comparison between the static potential $V_{Q} (L) / g$ obtained from the Gaussian method and the MPS calculation from Ref. [30]. (a) Deviation from the continuum analytical result for the massless case $m / g = 0$ (black solid line) for $x = 100$ (red solid line for the Gaussian ansatz and green crosses for the MPS) and $x = 400$ (blue solid line for the Gaussian ansatz and cyan asterisks for the MPS). (b) Static potential for the massive case with $Q / g = 1$ and $x = 100$ from the Gaussian ansatz (solid lines) and the MPS results (crosses). (c) Partial string breaking for noninteger values of $Q / g$ and $m / g = 1$ , $x = 400$ . Solid lines again represent the result from the Gaussian ansatz, crosses the MPS results from Ref. [30].
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Figure 2
Evolution of the electric flux distribution $< L_{n} >$ for an initial static string of length $L / a = 19$ and $Q / g = 1$ imposed on top of the interacting vacuum. (a) The noninteracting case $m = g = 0$ . (b) The appearance of string and antistring configurations for $m = 0.1$ , $g = 1$ . (c) The strong coupling limit $m = 3$ , $g = 3.5$ .
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Figure 3
Evolution of the electric flux distribution $⟨ L_{n} ⟩$ for an initial free string of length $L / a = 19$ and $Q / g = 1$ , the columns corresponds to $m = g = 0$ (first column), $m = 0.1$ , $g = 1$ (second column), and $m = 3$ , $g = 3.5$ (third column). Panels (a)–(c) show the evolution of a string imposed on top of the interacting vacuum, where no waves are coming from the boundaries. Panels (d)–(f) show the evolution of the string on imposed on top of the bare vacuum, leading to waves propagating from the boundaries and destructively interfering with the propagating string. The parameter regime $m = g = 0$ used in panels (a) and (d) can be exactly solved with the Gaussian ansatz. Moreover, panels (b) and (e) show the Schwinger mechanism.
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Figure 4
Comparison between the equilibrium properties of the SU(2) LGT from the non-Gaussian ansatz and the MPS simulations. The columns correspond to parameters $m = 1$ , $g = 1$ (first column), $m = 0.5$ , $g = 1$ (second column), and $m = 0.75$ , $g = 1.5$ (third column). Panels (a)-(c) in the first row show the static potential $V_{Q} (L) / ϵ$ . Additionally, in panel (a) we compare the results for different choices of $s_{1}$ and $s_{2}$ for the non-Gaussian ansatz in Eq. (34) and verify the result computing the expectation values for the parity operators $P_{1}$ and $P_{2}$ in the global ground state determined via a MPS simulation. Panels (d)-(f) show the color-electric flux profiles at various separations $L / a$ between the external charges yielding a string ground state. Analogously, panels (g)-(i) show the flux profiles for separations where the flux string is broken. Finally, the panels (j)-(l) in the last row show the correlation function $C_{2} (L)$ between static charges in the ground state as a function of the distance.
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Figure 5
Correlation function $C_{dyn}^{α} (n_{1}, n)$ between the first static charge and dynamical fermions for $m = 1$ , $g = 1$ and distances $L / a = 5$ (a) and $L / a = 23$ (b) between static charges.
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Figure 6
Evolution of a color-flux string for an initial spin-singlet state between the static charges. Panels (a)–(c) show the evolution of the flux profile, panels (d)–(f) the evolution of the correlation function $C_{2} (t)$ . The different columns correspond to the parameters $m = 0.5$ , $g = 1$ , $L / a = 25$ (first column), $m = 0.75$ , $g = 1.5$ , $L / a = 25$ (second column) and $m = 0.75$ , $g = 1.5$ , $L / a = 15$ (third column).
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