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

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

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.

► BibTeX data

► References

[1] Sheldon L. Glashow. Partial-symmetries of weak interactions. Nuclear Physics, 22(4):579 – 588, 1961. doi:10.1016/​0029-5582(61)90469-2.
https:/​/​doi.org/​10.1016/​0029-5582(61)90469-2

[2] Steven Weinberg. A model of leptons. Phys. Rev. Lett., 19:1264–1266, Nov 1967. doi:10.1103/​PhysRevLett.19.1264.
https:/​/​doi.org/​10.1103/​PhysRevLett.19.1264

[3] Ian Affleck, Z. Zou, T. Hsu, and P. W. Anderson. Su(2) gauge symmetry of the large-$u$ limit of the hubbard model. Phys. Rev. B, 38:745–747, Jul 1988. doi:10.1103/​PhysRevB.38.745.
https:/​/​doi.org/​10.1103/​PhysRevB.38.745

[4] Qiu-Hong Huo, Yunguo Jiang, Ru-Zhi Wang, and Hui Yan. Non-abelian vortices in the emergent u(2) gauge theory of the hubbard model. EPL (Europhysics Letters), 101(2):27001, 2013. doi:10.1209/​0295-5075/​101/​27001.
https:/​/​doi.org/​10.1209/​0295-5075/​101/​27001

[5] Owe Philipsen. Status of the qcd phase diagram from lattice calculations. [hep-ph], 11 2011. arXiv:https:/​/​arxiv.org/​abs/​1111.5370.
arXiv:1111.5370

[6] Philippe de Forcrand. Simulating qcd at finite density. PoS LAT2010, 05 2010. arXiv:https:/​/​arxiv.org/​abs/​1005.0539.
arXiv:1005.0539

[7] I. Montvay and G. Muenster. Quantum Fields on a lattice. Cambridge Univ. Press, Cambridge, 1994.

[8] M. Creutz. Quarks, gluons and lattices. Cambridge University Press, Cambridge, 1997.

[9] C. Gattringer and C. B. Lang. Quantum Chromodynamics on the Lattice. Springer-Verlag, 2010.

[10] L. Tagliacozzo, A Celi, P. Orland, M. W. Mitchell, and M. Lewenstein. Simulation of non-Abelian gauge theories with optical lattices. Nat. Commun., 4:1–8, 2013. doi:10.1038/​ncomms3615.
https:/​/​doi.org/​10.1038/​ncomms3615

[11] D. Banerjee, M. Bögli, M. Dalmonte, E. Rico, P. Stebler, U.-J. Wiese, and P. Zoller. Atomic Quantum Simulation of U(N) and SU(N) Non-Abelian Lattice Gauge Theories. Phys. Rev. Lett., 110(12):125303, mar 2013. doi:10.1103/​PhysRevLett.110.125303.
https:/​/​doi.org/​10.1103/​PhysRevLett.110.125303

[12] Erez Zohar, J. Ignacio Cirac, and Benni Reznik. Quantum simulations of gauge theories with ultracold atoms: Local gauge invariance from angular-momentum conservation. Phys. Rev. A, 88:023617, Aug 2013. doi:10.1103/​PhysRevA.88.023617.
https:/​/​doi.org/​10.1103/​PhysRevA.88.023617

[13] K. Stannigel, P. Hauke, D. Marcos, M. Hafezi, S. Diehl, M. Dalmonte, and P. Zoller. Constrained dynamics via the zeno effect in quantum simulation: Implementing non-abelian lattice gauge theories with cold atoms. Phys. Rev. Lett., 112:120406, Mar 2014. doi:10.1103/​PhysRevLett.112.120406.
https:/​/​doi.org/​10.1103/​PhysRevLett.112.120406

[14] Uwe-Jens Wiese. Towards quantum simulating QCD. Nucl. Phys. A, 931:246–256, 2014. doi:10.1016/​j.nuclphysa.2014.09.102.
https:/​/​doi.org/​10.1016/​j.nuclphysa.2014.09.102

[15] A. Mezzacapo, E. Rico, C. Sabín, I. L. Egusquiza, L. Lamata, and E. Solano. Non-abelian su(2) lattice gauge theories in superconducting circuits. Phys. Rev. Lett., 115:240502, Dec 2015. doi:10.1103/​PhysRevLett.115.240502.
https:/​/​doi.org/​10.1103/​PhysRevLett.115.240502

[16] Erez Zohar, J Ignacio Cirac, and Benni Reznik. Quantum simulations of lattice gauge theories using ultracold atoms in optical lattices. Reports on Progress in Physics, 79(1):014401, 2016. doi:10.1088/​0034-4885/​79/​1/​014401.
https:/​/​doi.org/​10.1088/​0034-4885/​79/​1/​014401

[17] Esteban A. Martinez, Christine A. Muschik, Philipp Schindler, Daniel Nigg, Alexander Erhard, Markus Heyl, Philipp Hauke, Marcello Dalmonte, Thomas Monz, Peter Zoller, and Rainer Blatt. Real-time dynamics of lattice gauge theories with a few-qubit quantum computer. Nature, 534(7608):516–519, 06 2016. doi:10.1038/​nature18318.
https:/​/​doi.org/​10.1038/​nature18318

[18] M Dalmonte and S Montangero. Lattice gauge theory simulations in the quantum information era. Contemporary Physics, pages 1–25, 2016. doi:10.1080/​00107514.2016.1151199.
https:/​/​doi.org/​10.1080/​00107514.2016.1151199

[19] Stellan Östlund and Stefan Rommer. Thermodynamic Limit of Density Matrix Renormalization. Phys. Rev. Lett., 75(19):3537–3540, nov 1995. doi:10.1103/​PhysRevLett.75.3537.
https:/​/​doi.org/​10.1103/​PhysRevLett.75.3537

[20] F. Verstraete, D. Porras, and J. I. Cirac. Density matrix renormalization group and periodic boundary conditions: A quantum information perspective. Phys. Rev. Lett., 93:227205, Nov 2004. doi:10.1103/​PhysRevLett.93.227205.
https:/​/​doi.org/​10.1103/​PhysRevLett.93.227205

[21] G. Vidal. Entanglement renormalization. Phys. Rev. Lett., 99:220405, Nov 2007. doi:10.1103/​PhysRevLett.99.220405.
https:/​/​doi.org/​10.1103/​PhysRevLett.99.220405

[22] Ulrich Schollwoeck. The density-matrix renormalization group in the age of matrix product states. Ann. Phys., 326(1):96, 2011. doi:10.1016/​j.aop.2010.09.012.
https:/​/​doi.org/​10.1016/​j.aop.2010.09.012

[23] Román Orús. A practical introduction to tensor networks: Matrix product states and projected entangled pair states. Annals of Physics, 349:117–158, 2014. doi:10.1016/​j.aop.2014.06.013.
https:/​/​doi.org/​10.1016/​j.aop.2014.06.013

[24] X.-G. Wen. Quantum Field Theory of Many-Body Systems. Oxford University Press, 2004.

[25] Mark Alford, Krishna Rajagopal, and Frank Wilczek. Qcd at finite baryon density: Nucleon droplets and color superconductivity. Physics Letters B, 422(1):247–256, 1998. doi:10.1016/​S0370-2693(98)00051-3.
https:/​/​doi.org/​10.1016/​S0370-2693(98)00051-3

[26] Ralf Rapp, T Schäfer, E Shuryak, and Momchil Velkovsky. Diquark bose condensates in high density matter and instantons. Physical Review Letters, 81(1):53, 1998. doi:10.1103/​PhysRevLett.81.53.
https:/​/​doi.org/​10.1103/​PhysRevLett.81.53

[27] D. Horn. Finite matrix models with continuous local gauge invariance. Physics Letters B, 100:149, 1981. doi:10.1016/​0370-2693(81)90763-2.
https:/​/​doi.org/​10.1016/​0370-2693(81)90763-2

[28] P. Orland and D. Rohrlich. Lattice gauge magnets: local isospin from spin. Nucl. Phys. B, 338:647, 1990. doi:10.1016/​0550-3213(90)90646-U.
https:/​/​doi.org/​10.1016/​0550-3213(90)90646-U

[29] S. Chandrasekharan and U. J. Wiese. Quantum link models : A discrete approach to gauge theories. Nucl. Phys. B, 492(1-2):455–471, 1997. arXiv:9609042, doi:10.1016/​S0550-3213(97)80041-7.
https:/​/​doi.org/​10.1016/​S0550-3213(97)80041-7
arXiv:9609042

[30] R. Brower, S. Chandrasekharan, and U.-J. Wiese. Qcd as a quantum link model. Phys. Rev. D, 60:094502, 1999. doi:10.1103/​PhysRevD.60.094502.
https:/​/​doi.org/​10.1103/​PhysRevD.60.094502

[31] John Kogut and Leonard Susskind. Hamiltonian formulation of Wilson's lattice guage theoreis. Phys. Rev. D, 11, 1975. doi:10.1103/​PhysRevD.11.395.
https:/​/​doi.org/​10.1103/​PhysRevD.11.395

[32] Julius Wess and Bruno Zumino. Consequences of anomalous ward identities. Physics Letters B, 37(1):95–97, 1971. doi:10.1016/​0370-2693(71)90582-X.
https:/​/​doi.org/​10.1016/​0370-2693(71)90582-X

[33] Edward Witten. Global aspects of current algebra. Nuclear Physics B, 223(2):422–432, 1983. doi:10.1016/​0550-3213(83)90063-9.
https:/​/​doi.org/​10.1016/​0550-3213(83)90063-9

[34] Sergei Petrovich Novikov. The hamiltonian formalism and a many-valued analogue of morse theory. Russian mathematical surveys, 37(5):1–56, 1982. URL: http:/​/​stacks.iop.org/​0036-0279/​37/​i=5/​a=R01.
http:/​/​stacks.iop.org/​0036-0279/​37/​i=5/​a=R01

[35] RM Konik, T Pálmai, G Takács, and AM Tsvelik. Studying the perturbed wess–zumino–novikov–witten su (2) k theory using the truncated conformal spectrum approach. Nuclear Physics B, 899:547–569, 2015. doi:10.1016/​j.nuclphysb.2015.08.016.
https:/​/​doi.org/​10.1016/​j.nuclphysb.2015.08.016

[36] Steven R. White. Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett., 69(19):2863–2866, nov 1992. doi:10.1103/​PhysRevLett.69.2863.
https:/​/​doi.org/​10.1103/​PhysRevLett.69.2863

[37] Steven R. White. Density-matrix algorithms for quantum renormalization groups. Phys. Rev. B, 48(14):10345–10356, oct 1993. doi:10.1103/​PhysRevB.48.10345.
https:/​/​doi.org/​10.1103/​PhysRevB.48.10345

[38] Guifré Vidal. Efficient simulation of one-dimensional quantum many-body systems. Phys. Rev. Lett., 93:040502, Jul 2004. doi:10.1103/​PhysRevLett.93.040502.
https:/​/​doi.org/​10.1103/​PhysRevLett.93.040502

[39] Horacio Casini, Marina Huerta, and José Alejandro Rosabal. Remarks on entanglement entropy for gauge fields. Physical Review D, 89(8):085012, 2014. doi:10.1103/​PhysRevD.89.085012.
https:/​/​doi.org/​10.1103/​PhysRevD.89.085012

[40] William Donnelly. Decomposition of entanglement entropy in lattice gauge theory. Physical Review D, 85(8):085004, 2012. doi:10.1103/​PhysRevD.85.085004.
https:/​/​doi.org/​10.1103/​PhysRevD.85.085004

[41] Karel Van Acoleyen, Nick Bultinck, Jutho Haegeman, Michael Marien, Volkher B. Scholz, and Frank Verstraete. Entanglement of distillation for lattice gauge theories. Phys. Rev. Lett., 117:131602, Sep 2016. doi:10.1103/​PhysRevLett.117.131602.
https:/​/​doi.org/​10.1103/​PhysRevLett.117.131602

[42] T. M. R. Byrnes, P. Sriganesh, R. J. Bursill, and C. J. Hamer. Density matrix renormalization group approach to the massive schwinger model. Phys. Rev. D, 66:013002, Jul 2002. doi:10.1103/​PhysRevD.66.013002.
https:/​/​doi.org/​10.1103/​PhysRevD.66.013002

[43] Takanori Sugihara. Matrix product representation of gauge invariant states in a $z_2$ lattice gauge theory. Journal of High Energy Physics, 2005(07):022, 2005. doi:10.1088/​1126-6708/​2005/​07/​022.
https:/​/​doi.org/​10.1088/​1126-6708/​2005/​07/​022

[44] E. Rico, T. Pichler, M. Dalmonte, P. Zoller, and S. Montangero. Tensor networks for lattice gauge theories and atomic quantum simulation. Phys. Rev. Lett., 112(20):1–5, 2014. doi:10.1103/​PhysRevLett.112.201601.
https:/​/​doi.org/​10.1103/​PhysRevLett.112.201601

[45] Pietro Silvi, Enrique Rico, Tommaso Calarco, and Simone Montangero. Lattice gauge tensor networks. New Journal of Physics, 16(10):103015, 2014. doi:10.1088/​1367-2630/​16/​10/​103015.
https:/​/​doi.org/​10.1088/​1367-2630/​16/​10/​103015

[46] Luca Tagliacozzo and Guifre Vidal. Entanglement renormalization and gauge symmetry. Phys. Rev. B, 83(11):115127, mar 2011. doi:10.1103/​PhysRevB.83.115127.
https:/​/​doi.org/​10.1103/​PhysRevB.83.115127

[47] L. Tagliacozzo, A. Celi, and M. Lewenstein. Tensor Networks for Lattice Gauge Theories with Continuous Groups. Phys. Rev. X, 4(4):041024, nov 2014. doi:10.1103/​PhysRevX.4.041024.
https:/​/​doi.org/​10.1103/​PhysRevX.4.041024

[48] M.C. Bañuls, K Cichy, J.I. Cirac, and K Jansen. The mass spectrum of the Schwinger model with matrix product states. J. High Energy Phys., 2013(11):158, nov 2013. doi:10.1007/​JHEP11(2013)158.
https:/​/​doi.org/​10.1007/​JHEP11(2013)158

[49] Stefan Kühn, J. Ignacio Cirac, and Mari-Carmen Bañuls. Quantum simulation of the Schwinger model: A study of feasibility. Phys. Rev. A, 90(4):042305, oct 2014. arXiv:1407.4995, doi:10.1103/​PhysRevA.90.042305.
https:/​/​doi.org/​10.1103/​PhysRevA.90.042305
arXiv:1407.4995

[50] Stefan Kühn, Erez Zohar, J. Ignacio Cirac, and Mari Carmen Bañuls. Non-Abelian string breaking phenomena with matrix product states. J. High Energy Phys., 2015(7):130, jul 2015. arXiv:1505.04441, doi:10.1007/​JHEP07(2015)130.
https:/​/​doi.org/​10.1007/​JHEP07(2015)130
arXiv:1505.04441

[51] Boye Buyens, Jutho Haegeman, Karel Van Acoleyen, Henri Verschelde, and Frank Verstraete. Matrix product states for gauge field theories. Phys. Rev. Lett., 113:091601, Aug 2014. doi:10.1103/​PhysRevLett.113.091601.
https:/​/​doi.org/​10.1103/​PhysRevLett.113.091601

[52] Boye Buyens, Karel Van Acoleyen, Jutho Haegeman, and Frank Verstraete. Matrix product states for hamiltonian lattice gauge theories. In PROCEEDINGS OF SCIENCE, page 7, 2014.

[53] Boye Buyens, Jutho Haegeman, Henri Verschelde, Frank Verstraete, and Karel Van Acoleyen. Confinement and string breaking for ${\mathrm{qed}}_{2}$ in the hamiltonian picture. Phys. Rev. X, 6:041040, Nov 2016. doi:10.1103/​PhysRevX.6.041040.
https:/​/​doi.org/​10.1103/​PhysRevX.6.041040

[54] Boye Buyens, Jutho Haegeman, Frank Verstraete, and Karel Van Acoleyen. Tensor networks for gauge field theories. In PROCEEDINGS OF SCIENCE, page 7, 2016.

[55] H. Saito, M. C. Bañuls, K. Cichy, J. I. Cirac, and K. Jansen. Thermal evolution of the one-flavour schwinger model using matrix product states. [hep-lat], 11 2015. arXiv:https:/​/​arxiv.org/​abs/​1511.00794.
arXiv:1511.00794

[56] Mari Carmen Bañuls, K Cichy, J Ignacio Cirac, K Jansen, and H Saito. Thermal evolution of the schwinger model with matrix product operators. Physical Review D, 92(3):034519, 2015. doi:10.1103/​PhysRevD.92.034519.
https:/​/​doi.org/​10.1103/​PhysRevD.92.034519

[57] Ashley Milsted. Matrix product states and the non-abelian rotor model. Phys. Rev. D, 93:085012, Apr 2016. doi:10.1103/​PhysRevD.93.085012.
https:/​/​doi.org/​10.1103/​PhysRevD.93.085012

[58] Yannick Meurice, Alan Denbleyker, Yuzhi Liu, Tao Xiang, Zhiyuan Xie, Ji-Feng Yu, Judah Unmuth-Yockey, and Haiyuan Zou. Comparing Tensor Renormalization Group and Monte Carlo calculations for spin and gauge models. PoS, LATTICE2013:329, 2014. arXiv:1311.4826.
arXiv:1311.4826

[59] Benjamin Bahr, Bianca Dittrich, Frank Hellmann, and Wojciech Kaminski. Holonomy spin foam models: Definition and coarse graining. Phys. Rev. D, 87:044048, Feb 2013. doi:10.1103/​PhysRevD.87.044048.
https:/​/​doi.org/​10.1103/​PhysRevD.87.044048

[60] Mark Alford, Krishna Rajagopal, and Frank Wilczek. Qcd at finite baryon density: Nucleon droplets and color superconductivity. Physics Letters B, 422(1):247–256, 1998. doi:10.1016/​S0370-2693(98)00051-3.
https:/​/​doi.org/​10.1016/​S0370-2693(98)00051-3

[61] Mark G. Alford, Andreas Schmitt, Krishna Rajagopal, and Thomas Schäfer. Color superconductivity in dense quark matter. Rev. Mod. Phys., 80:1455–1515, Nov 2008. doi:10.1103/​RevModPhys.80.1455.
https:/​/​doi.org/​10.1103/​RevModPhys.80.1455

[62] D Banerjee, F-J Jiang, P Widmer, and U-J Wiese. The (2+1)-d u(1) quantum link model masquerading as deconfined criticality. Journal of Statistical Mechanics: Theory and Experiment, 2013(12):P12010, 2013. doi:10.1088/​1742-5468/​2013/​12/​P12010.
https:/​/​doi.org/​10.1088/​1742-5468/​2013/​12/​P12010

[63] Pasquale Calabrese and John Cardy. Entanglement entropy and quantum field theory. Journal of Statistical Mechanics: Theory and Experiment, 2004(06):P06002, 2004. doi:10.1088/​1742-5468/​2004/​06/​P06002.
https:/​/​doi.org/​10.1088/​1742-5468/​2004/​06/​P06002

[64] Pasquale Calabrese, Massimo Campostrini, Fabian Essler, and Bernard Nienhuis. Parity effects in the scaling of block entanglement in gapless spin chains. Phys. Rev. Lett., 104:095701, Mar 2010. doi:10.1103/​PhysRevLett.104.095701.
https:/​/​doi.org/​10.1103/​PhysRevLett.104.095701

[65] Pasquale Calabrese, John Cardy, and Ingo Peschel. Corrections to scaling for block entanglement in massive spin chains. Journal of Statistical Mechanics: Theory and Experiment, 2010(09):P09003, 2010. doi:10.1088/​1742-5468/​2010/​09/​P09003.
https:/​/​doi.org/​10.1088/​1742-5468/​2010/​09/​P09003

[66] Michael E. Fisher and Michael N. Barber. Scaling theory for finite-size effects in the critical region. Phys. Rev. Lett., 28:1516–1519, Jun 1972. doi:10.1103/​PhysRevLett.28.1516.
https:/​/​doi.org/​10.1103/​PhysRevLett.28.1516

[67] Pietro Silvi, Tommaso Calarco, Giovanna Morigi, and Simone Montangero. Ab initio characterization of the quantum linear-zigzag transition using density matrix renormalization group calculations. Phys. Rev. B, 89:094103, Mar 2014. doi:10.1103/​PhysRevB.89.094103.
https:/​/​doi.org/​10.1103/​PhysRevB.89.094103

[68] Erik S Sørensen, Ming-Shyang Chang, Nicolas Laflorencie, and Ian Affleck. Quantum impurity entanglement. Journal of Statistical Mechanics: Theory and Experiment, 2007(08):P08003, 2007. doi:10.1088/​1742-5468/​2007/​08/​P08003.
https:/​/​doi.org/​10.1088/​1742-5468/​2007/​08/​P08003

[69] L. Tagliacozzo, Thiago. R. de Oliveira, S. Iblisdir, and J. I. Latorre. Scaling of entanglement support for matrix product states. Phys. Rev. B, 78:024410, Jul 2008. doi:10.1103/​PhysRevB.78.024410.
https:/​/​doi.org/​10.1103/​PhysRevB.78.024410

[70] Vid Stojevic, Jutho Haegeman, I. P. McCulloch, Luca Tagliacozzo, and Frank Verstraete. Conformal data from finite entanglement scaling. Phys. Rev. B, 91:035120, Jan 2015. doi:10.1103/​PhysRevB.91.035120.
https:/​/​doi.org/​10.1103/​PhysRevB.91.035120

[71] B. Pirvu, G. Vidal, F. Verstraete, and L. Tagliacozzo. Matrix product states for critical spin chains: Finite-size versus finite-entanglement scaling. Phys. Rev. B, 86:075117, Aug 2012. doi:10.1103/​PhysRevB.86.075117.
https:/​/​doi.org/​10.1103/​PhysRevB.86.075117

[72] Alexander O. Gogolin, Alexander A. Nersesyan, and Alexei M. Tsvelik. Bosonization and Strongly Correlated Systems. Cambridge University Press, 2004.

[73] Thierry Giamarchi. Quantum Physics in One Dimension. Oxford University Press, 2003.

[74] Alexander Moreno, Alejandro Muramatsu, and Salvatore R Manmana. Ground-state phase diagram of the one-dimensional t-j model. Physical Review B, 83(20):205113, 2011. doi:10.1103/​PhysRevB.83.205113.
https:/​/​doi.org/​10.1103/​PhysRevB.83.205113

[75] P. C. Hohenberg. Existence of long-range order in 1 and 2 dimensions. Phys. Rev., 158:383–386, 1967. doi:10.1103/​PhysRev.158.383.
https:/​/​doi.org/​10.1103/​PhysRev.158.383

[76] N. D. Mermin and H. Wagner. Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic heisenberg models. Phys. Rev. Lett., 17:1133–1136, 1966. doi:10.1103/​PhysRevLett.17.1133.
https:/​/​doi.org/​10.1103/​PhysRevLett.17.1133

[77] T.D. Kühner, S.R. White, and H. Monien. One-dimensional Bose-Hubbard model with nearest-neighbor interaction. Phys. Rev. B, 61(18):12474, 2000. doi:10.1103/​PhysRevB.61.12474.
https:/​/​doi.org/​10.1103/​PhysRevB.61.12474

[78] Erez Zohar, Michele Burrello, Thorsten B. Wahl, and J. Ignacio Cirac. Fermionic projected entangled pair states and local u(1) gauge theories. Annals of Physics (2015), pp. 385-439, 07 2015. doi:10.1016/​j.aop.2015.10.009.
https:/​/​doi.org/​10.1016/​j.aop.2015.10.009

[79] Erez Zohar and Michele Burrello. Formulation of lattice gauge theories for quantum simulations. Phys. Rev. D, 91(5):054506, mar 2015. arXiv:1409.3085, doi:10.1103/​PhysRevD.91.054506.
https:/​/​doi.org/​10.1103/​PhysRevD.91.054506
arXiv:1409.3085

[80] Alexei Kitaev and John Preskill. Topological entanglement entropy. Phys. Rev. Lett., 96:110404, Mar 2006. doi:10.1103/​PhysRevLett.96.110404.
https:/​/​doi.org/​10.1103/​PhysRevLett.96.110404

[81] Michael Levin and Xiao-Gang Wen. Detecting topological order in a ground state wave function. Phys. Rev. Lett., 96:110405, Mar 2006. doi:10.1103/​PhysRevLett.96.110405.
https:/​/​doi.org/​10.1103/​PhysRevLett.96.110405

[82] Mari Carmen Bañuls, Krzysztof Cichy, J Ignacio Cirac, Karl Jansen, Stefan Kühn, and Hana Saito. The multi-flavor schwinger model with chemical potential-overcoming the sign problem with matrix product states. DESY 16-198, 2016. arXiv:https:/​/​arxiv.org/​abs/​1611.01458.
arXiv:1611.01458

[83] Mari Carmen Bañuls, Krzysztof Cichy, J. Ignacio Cirac, Karl Jansen, and Stefan Kühn. Density induced phase transitions in the schwinger model: A study with matrix product states. Phys. Rev. Lett., 118:071601, Feb 2017. doi:10.1103/​PhysRevLett.118.071601.
https:/​/​doi.org/​10.1103/​PhysRevLett.118.071601

[84] D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac. Matrix product state representations. Quant. Inf. Comput., 7:401, 2007. arXiv:https:/​/​arxiv.org/​abs/​quant-ph/​0608197.
arXiv:quant-ph/0608197

[85] Jorge Dukelsky, Miguel A Martín-Delgado, Tomotoshi Nishino, and Germán Sierra. Equivalence of the variational matrix product method and the density matrix renormalization group applied to spin chains. EPL (Europhysics Letters), 43(4):457, 1998. doi:10.1209/​epl/​i1998-00381-x.
https:/​/​doi.org/​10.1209/​epl/​i1998-00381-x

[86] Sukhwinder Singh, Robert NC Pfeifer, and Guifré Vidal. Tensor network decompositions in the presence of a global symmetry. Physical Review A, 82(5):050301, 2010. doi:10.1103/​PhysRevA.82.050301.
https:/​/​doi.org/​10.1103/​PhysRevA.82.050301

[87] Sukhwinder Singh, Robert N. C. Pfeifer, and Guifre Vidal. Tensor network states and algorithms in the presence of a global u(1) symmetry. Phys. Rev. B, 83:115125, Mar 2011. doi:10.1103/​PhysRevB.83.115125.
https:/​/​doi.org/​10.1103/​PhysRevB.83.115125

[88] Steven R. White and Adrian E. Feiguin. Real-time evolution using the density matrix renormalization group. Phys. Rev. Lett., 93:076401, 2004. doi:10.1103/​PhysRevLett.93.076401.
https:/​/​doi.org/​10.1103/​PhysRevLett.93.076401

Cited by

[1] Giulia Mazzola, Simon V. Mathis, Guglielmo Mazzola, and Ivano Tavernelli, "Gauge-invariant quantum circuits for U (1) and Yang-Mills lattice gauge theories", Physical Review Research 3 4, 043209 (2021).

[2] Marco Rigobello, Simone Notarnicola, Giuseppe Magnifico, and Simone Montangero, "Entanglement generation in(1+1)DQED scattering processes", Physical Review D 104 11, 114501 (2021).

[3] Julian Bender, Patrick Emonts, and J. Ignacio Cirac, "Variational Monte Carlo algorithm for lattice gauge theories with continuous gauge groups: A study of (2+1) -dimensional compact QED with dynamical fermions at finite density", Physical Review Research 5 4, 043128 (2023).

[4] Alessio Celi, "Different models of gravitating Dirac fermions in optical lattices", The European Physical Journal Special Topics 226 12, 2729 (2017).

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

[6] Angus Kan, Lena Funcke, Stefan Kühn, Luca Dellantonio, Jinglei Zhang, Jan F. Haase, Christine A. Muschik, and Karl Jansen, "Investigating a (3+1)D topological θ -term in the Hamiltonian formulation of lattice gauge theories for quantum and classical simulations", Physical Review D 104 3, 034504 (2021).

[7] Mari Carmen Bañuls, Krzysztof Cichy, Ying-Jer Kao, C.-J. David Lin, Yu-Ping Lin, and David T.-L. Tan, "Phase structure of the (1+1)-dimensional massive Thirring model from matrix product states", Physical Review D 100 9, 094504 (2019).

[8] Román Orús, "Tensor networks for complex quantum systems", Nature Reviews Physics 1 9, 538 (2019).

[9] 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).

[10] Nilin Abrahamsen, Yu Tong, Ning Bao, Yuan Su, and Nathan Wiebe, "Entanglement area law for one-dimensional gauge theories and bosonic systems", Physical Review A 108 4, 042422 (2023).

[11] Mari Carmen Bañuls, Rainer Blatt, Jacopo Catani, Alessio Celi, Juan Ignacio Cirac, Marcello Dalmonte, Leonardo Fallani, Karl Jansen, Maciej Lewenstein, Simone Montangero, Christine A. Muschik, Benni Reznik, Enrique Rico, Luca Tagliacozzo, Karel Van Acoleyen, Frank Verstraete, Uwe-Jens Wiese, Matthew Wingate, Jakub Zakrzewski, and Peter Zoller, "Simulating lattice gauge theories within quantum technologies", The European Physical Journal D 74 8, 165 (2020).

[12] Julian Bender, Patrick Emonts, Erez Zohar, and J. Ignacio Cirac, "Real-time dynamics in 2+1D compact QED using complex periodic Gaussian states", Physical Review Research 2 4, 043145 (2020).

[13] 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).

[14] Yasar Y. Atas, Jinglei Zhang, Randy Lewis, Amin Jahanpour, Jan F. Haase, and Christine A. Muschik, "SU(2) hadrons on a quantum computer via a variational approach", Nature Communications 12 1, 6499 (2021).

[15] Timo Felser, Pietro Silvi, Mario Collura, and Simone Montangero, "Two-Dimensional Quantum-Link Lattice Quantum Electrodynamics at Finite Density", Physical Review X 10 4, 041040 (2020).

[16] Giuseppe Magnifico, Timo Felser, Pietro Silvi, and Simone Montangero, "Lattice quantum electrodynamics in (3+1)-dimensions at finite density with tensor networks", Nature Communications 12 1, 3600 (2021).

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

[18] Gertian Roose, Nick Bultinck, Laurens Vanderstraeten, Frank Verstraete, Karel Van Acoleyen, and Jutho Haegeman, "Lattice regularisation and entanglement structure of the Gross-Neveu model", Journal of High Energy Physics 2021 7, 207 (2021).

[19] Zohreh Davoudi, Alexander F. Shaw, and Jesse R. Stryker, "General quantum algorithms for Hamiltonian simulation with applications to a non-Abelian lattice gauge theory", Quantum 7, 1213 (2023).

[20] 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).

[21] Katsumasa Nakayama, Lena Funcke, Karl Jansen, Ying-Jer Kao, and Stefan Kühn, "Phase structure of the CP(1) model in the presence of a topological θ -term", Physical Review D 105 5, 054507 (2022).

[22] 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).

[23] 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).

[24] Emil Mathew and Indrakshi Raychowdhury, "Protecting local and global symmetries in simulating (1+1)D non-Abelian gauge theories", Physical Review D 106 5, 054510 (2022).

[25] 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).

[26] Adrián Franco-Rubio and Guifré Vidal, "Entanglement renormalization for gauge invariant quantum fields", Physical Review D 103 2, 025013 (2021).

[27] Jan F. Haase, Luca Dellantonio, Alessio Celi, Danny Paulson, Angus Kan, Karl Jansen, and Christine A. Muschik, "A resource efficient approach for quantum and classical simulations of gauge theories in particle physics", Quantum 5, 393 (2021).

[28] Sarmed A Rahman, Randy Lewis, Emanuele Mendicelli, and Sarah Powell, "SU(2) lattice gauge theory on a quantum annealer", Physical Review D 104 3, 034501 (2021).

[29] Mari Carmen Bañuls, Krzysztof Cichy, J. Ignacio Cirac, Karl Jansen, and Stefan Kühn, "Efficient Basis Formulation for (1+1)-Dimensional SU(2) Lattice Gauge Theory: Spectral Calculations with Matrix Product States", Physical Review X 7 4, 041046 (2017).

[30] Yannick Meurice, Ryo Sakai, and Judah Unmuth-Yockey, "Tensor lattice field theory for renormalization and quantum computing", Reviews of Modern Physics 94 2, 025005 (2022).

[31] Timo Jakobs, Marco Garofalo, Tobias Hartung, Karl Jansen, Johann Ostmeyer, Dominik Rolfes, Simone Romiti, and Carsten Urbach, "Canonical momenta in digitized Su(2) lattice gauge theory: definition and free theory", The European Physical Journal C 83 7, 669 (2023).

[32] Erez Zohar, "Wilson loops and area laws in lattice gauge theory tensor networks", Physical Review Research 3 3, 033179 (2021).

[33] Takis Angelides, Lena Funcke, Karl Jansen, and Stefan Kühn, "Computing the mass shift of Wilson and staggered fermions in the lattice Schwinger model with matrix product states", Physical Review D 108 1, 014516 (2023).

[34] Daniel C. Hackett, Kiel Howe, Ciaran Hughes, William Jay, Ethan T. Neil, and James N. Simone, "Digitizing gauge fields: Lattice Monte Carlo results for future quantum computers", Physical Review A 99 6, 062341 (2019).

[35] Christian Schweizer, Fabian Grusdt, Moritz Berngruber, Luca Barbiero, Eugene Demler, Nathan Goldman, Immanuel Bloch, and Monika Aidelsburger, "Floquet approach to ℤ2 lattice gauge theories with ultracold atoms in optical lattices", Nature Physics 15 11, 1168 (2019).

[36] Pietro Silvi, Yannick Sauer, Ferdinand Tschirsich, and Simone Montangero, "Tensor network simulation of an SU(3) lattice gauge theory in 1D", Physical Review D 100 7, 074512 (2019).

[37] Mari Carmen Bañuls and Krzysztof Cichy, "Review on novel methods for lattice gauge theories", Reports on Progress in Physics 83 2, 024401 (2020).

[38] Zi-Yong Ge and Franco Nori, "Confinement-induced enhancement of superconductivity in a spin- 12 fermion chain coupled to a Z2 lattice gauge field", Physical Review B 107 12, 125141 (2023).

[39] Patrick Emonts and Erez Zohar, "Gauss law, minimal coupling and fermionic PEPS for lattice gauge theories", SciPost Physics Lecture Notes 12 (2020).

[40] Simone Montangero, Enrique Rico, and Pietro Silvi, "Loop-free tensor networks for high-energy physics", Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 2216, 20210065 (2022).

[41] Sarmed A Rahman, Randy Lewis, Emanuele Mendicelli, and Sarah Powell, "Self-mitigating Trotter circuits for SU(2) lattice gauge theory on a quantum computer", Physical Review D 106 7, 074502 (2022).

[42] Erez Zohar, Thorsten B. Wahl, Michele Burrello, and J. Ignacio Cirac, "Projected Entangled Pair States with non-Abelian gauge symmetries: An SU(2) study", Annals of Physics 374, 84 (2016).

[43] Zohreh Davoudi, Indrakshi Raychowdhury, and Andrew Shaw, "Search for efficient formulations for Hamiltonian simulation of non-Abelian lattice gauge theories", Physical Review D 104 7, 074505 (2021).

[44] L. Funcke, T. Hartung, K. Jansen, S. Kühn, M. Schneider, P. Stornati, and X. Wang, "Towards quantum simulations in particle physics and beyond on noisy intermediate-scale quantum devices", Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 2216, 20210062 (2022).

[45] Mari Carmen Bañuls, Krzysztof Cichy, J. Ignacio Cirac, Karl Jansen, and Stefan Kühn, "Density Induced Phase Transitions in the Schwinger Model: A Study with Matrix Product States", Physical Review Letters 118 7, 071601 (2017).

[46] Ian C. Cloët, Matthew R. Dietrich, John Arrington, Alexei Bazavov, Michael Bishof, Adam Freese, Alexey V. Gorshkov, Anna Grassellino, Kawtar Hafidi, Zubin Jacob, Michael McGuigan, Yannick Meurice, Zein-Eddine Meziani, Peter Mueller, Christine Muschik, James Osborn, Matthew Otten, Peter Petreczky, Tomas Polakovic, Alan Poon, Raphael Pooser, Alessandro Roggero, Mark Saffman, Brent VanDevender, Jiehang Zhang, and Erez Zohar, "Opportunities for Nuclear Physics & Quantum Information Science", arXiv:1903.05453, (2019).

[47] Sarmed A Rahman, Randy Lewis, Emanuele Mendicelli, and Sarah Powell, "SU(2) lattice gauge theory on a quantum annealer", arXiv:2103.08661, (2021).

[48] Mari Carmen Bañuls, Krzysztof Cichy, J. Ignacio Cirac, Karl Jansen, Stefan Kühn, and Hana Saito, "Towards overcoming the Monte Carlo sign problem with tensor networks", European Physical Journal Web of Conferences 137, 04001 (2017).

[49] 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).

[50] Shinichiro Akiyama, Yannick Meurice, and Ryo Sakai, "Tensor Renormalization Group for fermions", arXiv:2401.08542, (2024).

[51] 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", European Physical Journal Web of Conferences 175, 11017 (2018).

[52] Adrian Franco-Rubio and Guifre Vidal, "Entanglement renormalization for gauge invariant quantum fields", arXiv:1910.11815, (2019).

[53] 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, (2017).

[54] Tomoya Hayata, Yoshimasa Hidaka, and Kentaro Nishimura, "Dense $\textrm{QCD}_2$ with matrix product states", arXiv:2311.11643, (2023).

[55] Ali H. Z. Kavaki and Randy Lewis, "From square plaquettes to triamond lattices for SU(2) gauge theory", arXiv:2401.14570, (2024).

The above citations are from Crossref's cited-by service (last updated successfully 2024-03-19 10:01:13) and SAO/NASA ADS (last updated successfully 2024-03-19 10:01:14). The list may be incomplete as not all publishers provide suitable and complete citation data.