Finite-density phase diagram of a $(1+1)-d$ non-abelian lattice gauge theory with tensor networks

Pietro Silvi1,2, Enrique Rico3, Marcello Dalmonte4,5, Ferdinand Tschirsich1, and Simone Montangero1,6

1Institute for complex quantum systems & Center for Integrated Quantum Science and Technologies (IQST), Universität Ulm, D-89069 Ulm, Germany
2Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria
3Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, E-48080 Bilbao, Spain & IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, E-48013 Bilbao, Spain
4Institute for Theoretical Physics, University of Innsbruck, A-6020, Innsbruck, Austria & Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, A-6020 Innsbruck, Austria
5Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, Trieste, Italy
6Institute for Complex Quantum Systems & Center for Integrated Quantum Science and Technologies, Universität Ulm, D- 89069 Ulm, Germany

We investigate the finite-density phase diagram of a non-abelian $SU(2)$ lattice gauge theory in $(1+1)$-dimensions using tensor network methods. We numerically characterise the phase diagram as a function of the matter filling and of the matter-field coupling, identifying different phases, some of them appearing only at finite densities. For weak matter-field coupling we find a meson BCS liquid phase, which is confirmed by second-order analytical perturbation theory. At unit filling and for strong coupling, the system undergoes a phase transition to a charge density wave of single-site (spin-0) mesons via spontaneous chiral symmetry breaking. At finite densities, the chiral symmetry is restored almost everywhere, and the meson BCS liquid becomes a simple liquid at strong couplings, with the exception of filling two-thirds, where a charge density wave of mesons spreading over neighbouring sites appears. Finally, we identify two tri-critical points between the chiral and the two liquid phases which are compatible with a $SU(2)_2$ Wess-Zumino-Novikov-Witten model. Here we do not perform the continuum limit but we explicitly address the global $U(1)$ charge conservation symmetry.

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[2] Ashley Milsted and Tobias J. Osborne, "Quantum Yang-Mills theory: An overview of a program", Physical Review D 98 1, 014505 (2018).

[3] Erez Zohar and J. Ignacio Cirac, "Combining tensor networks with Monte Carlo methods for lattice gauge theories", Physical Review D 97 3, 034510 (2018).

[4] Ilya Kull, Andras Molnar, Erez Zohar, and J. Ignacio Cirac, "Classification of matrix product states with a local (gauge) symmetry", Annals of Physics 386, 199 (2017).

[5] Ferdinand Tschirsich, Simone Montangero, and Marcello Dalmonte, "Phase diagram and conformal string excitations of square ice using gauge invariant matrix product states", SciPost Physics 6 3, 028 (2019).

[6] Falk Bruckmann, Karl Jansen, and Stefan Kühn, "O(3) nonlinear sigma model in 1+1 dimensions with matrix product states", Physical Review D 99 7, 074501 (2019).

[7] Boye Buyens, Simone Montangero, Jutho Haegeman, Frank Verstraete, and Karel Van Acoleyen, "Finite-representation approximation of lattice gauge theories at the continuum limit with tensor networks", Physical Review D 95 9, 094509 (2017).

[8] Boye Buyens, Jutho Haegeman, Florian Hebenstreit, Frank Verstraete, and Karel Van Acoleyen, "Real-time simulation of the Schwinger effect with matrix product states", Physical Review D 96 11, 114501 (2017).

[9] Kai Zapp and Román Orús, "Tensor network simulation of QED on infinite lattices: Learning from (1+1) d , and prospects for (2+1) d", Physical Review D 95 11, 114508 (2017).

[10] Mari Carmen Bañuls, Krzysztof Cichy, Ying-Jer Kao, C. -J. David Lin, Yu-Ping Lin, and David Tao-Lin Tan, "Tensor Network study of the (1+1)-dimensional Thirring Model", arXiv:1710.09993.

[11] M. C. Banuls, K. Cichy, J. I. Cirac, K. Jansen, S. Kühn, and H. Saito, "The multi-flavor Schwinger model with chemical potential - Overcoming the sign problem with Matrix Product States", Proceedings of the 34th annual International Symposium on Lattice Field Theory (LATTICE2016). 24-30 July 2016. University of Southampton 316 (2016).

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[13] Erez Zohar, "Gauss law, Minimal Coupling and Fermionic PEPS for Lattice Gauge Theories", arXiv:1807.01294.

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