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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<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>S<\/mml:mi><mml:mi>U<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mn>2<\/mml:mn><mml:msub><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mn>2<\/mml:mn><\/mml:msub><\/mml:math>Wess-Zumino-Novikov-Witten model. Here we do not perform the continuum limit but we explicitly address the global<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>U<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mn>1<\/mml:mn><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>charge conservation symmetry.<\/jats:p>","DOI":"10.22331\/q-2017-04-25-9","type":"journal-article","created":{"date-parts":[[2017,4,25]],"date-time":"2017-04-25T08:58:44Z","timestamp":1493110724000},"page":"9","source":"Crossref","is-referenced-by-count":58,"title":["Finite-density phase diagram of a<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mn>1<\/mml:mn><mml:mo>+<\/mml:mo><mml:mn>1<\/mml:mn><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>\u2212<\/mml:mo><mml:mi>d<\/mml:mi><\/mml:math>non-abelian lattice gauge theory with tensor networks"],"prefix":"10.22331","volume":"1","author":[{"given":"Pietro","family":"Silvi","sequence":"first","affiliation":[{"name":"Institute for complex quantum systems & Center for Integrated Quantum Science and Technologies (IQST), Universit\u00e4t Ulm, D-89069 Ulm, Germany"},{"name":"Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria"}]},{"given":"Enrique","family":"Rico","sequence":"additional","affiliation":[{"name":"Department 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"}]},{"given":"Marcello","family":"Dalmonte","sequence":"additional","affiliation":[{"name":"Institute 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"},{"name":"Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, Trieste, Italy"}]},{"given":"Ferdinand","family":"Tschirsich","sequence":"additional","affiliation":[{"name":"Institute for complex quantum systems & Center for Integrated Quantum Science and Technologies (IQST), Universit\u00e4t Ulm, D-89069 Ulm, Germany"}]},{"given":"Simone","family":"Montangero","sequence":"additional","affiliation":[{"name":"Institute for complex quantum systems & Center for Integrated Quantum Science and Technologies (IQST), Universit\u00e4t Ulm, D-89069 Ulm, Germany"},{"name":"Institute for Complex Quantum Systems & Center for Integrated Quantum Science and Technologies, Universit\u00e4t Ulm, D- 89069 Ulm, Germany"}]}],"member":"9598","published-online":{"date-parts":[[2017,4,25]]},"reference":[{"key":"1","doi-asserted-by":"publisher","unstructured":"Sheldon L. 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