Real Time Dynamics and Confinement in the $\mathbb{Z}_{n}$ Schwinger-Weyl lattice model for 1+1 QED

Giuseppe Magnifico1,2, Marcello Dalmonte3,4, Paolo Facchi5,6, Saverio Pascazio5,6,7, Francesco V. Pepe5,6, and Elisa Ercolessi1,2

1Dipartimento di Fisica e Astronomia dell'Università di Bologna, I-40127 Bologna, Italy
2INFN, Sezione di Bologna, I-40127 Bologna, Italy
3Abdus Salam ICTP, Strada Costiera 11, I-34151 Trieste, Italy
4SISSA, Via Bonomea 265, I-34136 Trieste, Italy
5Dipartimento di Fisica and MECENAS, Università di Bari, I-70126 Bari, Italy
6INFN, Sezione di Bari, I-70126 Bari, Italy
7Istituto Nazionale di Ottica (INO-CNR), I-50125 Firenze, Italy

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We study the out-of-equilibrium properties of $1+1$ dimensional quantum electrodynamics (QED), discretized via the staggered-fermion Schwinger model with an Abelian $\mathbb{Z}_{n}$ gauge group. We look at two relevant phenomena: first, we analyze the stability of the Dirac vacuum with respect to particle/antiparticle pair production, both spontaneous and induced by an external electric field; then, we examine the string breaking mechanism. We observe a strong effect of confinement, which acts by suppressing both spontaneous pair production and string breaking into quark/antiquark pairs, indicating that the system dynamics displays a number of out-of-equilibrium features.

Gauge theories represent a fundamental building block of our understanding of physical laws. Despite their elegance, their real-time dynamics is often difficult to solve on classical computers, due to their complexity, that grows exponentially with the system size.

In recent years, several approaches leveraging on quantum information have enabled controlled simulations of such theories in real-time. These novel tools, based on a class of wave-functions known as tensor networks, offer unique opportunities to understand the spreading of quantum correlations, and in particular entanglement, in gauge theories.

We study here the Schwinger-Weyl lattice model, in which both space and the gauge degree of freedom are discretized, in such a way that standard quantum electrodynamics in 1+1 dimensions is reproduced in the continuum limit.

We analyze the stability of the Dirac vacuum, the mechanism of pair (particle/antiparticle) production, as well as the intriguing string breaking mechanism. Strings behave very differently, depending on the particle mass and charge (coupling), and their dynamics leaves characteristics footprints in the entanglement evolution. A number of out-of-equilibrium features are unhearted and scrutinized.

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[1] H. Kleinert, Gauge Fields in Condensed Matter (World Scientific, Singapore 1989). DOI: 10.1142/​0356.

[2] E. Fradkin, Field Theories of Condensed Matter Physics (Cambridge University Press, Cambridge 2013). DOI: 10.1017/​CBO9781139015509.

[3] M. Creutz, L. Jacobs, C. Rebbi, Monte Carlo computations in lattice gauge theories, Phys. Rep. 95, 203 (1983). DOI: 10.1016/​0370-1573(83)90016-9.

[4] H. J. Rothe, Lattice gauge theories (World Scientific, Singapore, 1992). DOI: 10.1142/​1268.

[5] I. Montvay and G. Münster, Quantum Fields on a Lattice (Cambridge University Press, Cambridge, 1994). DOI: 10.1017/​CBO9780511470783.

[6] K. G. Wilson, Confinement of quarks, Phys. Rev. D 10, 2445 (1974). DOI: 10.1103/​PhysRevD.10.2445.

[7] J. B. Kogut and L. Susskind, Hamiltonian formulation of Wilson's lattice gauge theories, Phys. Rev. D 11, 395 (1975). DOI: 10.1103/​PhysRevD.11.395.

[8] L. Susskind, Lattice fermions, Phys. Rev. D 16, 3031 (1977). DOI: 10.1103/​PhysRevD.16.3031.

[9] J. B. Kogut, An introduction to lattice gauge theory and spin systems, Rev. Mod. Phys. 51, 659 (1979). DOI: 10.1103/​RevModPhys.51.659.

[10] U. Schollwöck, The density-matrix renormalization group, Rev. Mod. Phys. 77, 259 (2005). DOI: 10.1103/​RevModPhys.77.259.

[11] R. Orus, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann. Phys. (N.Y.) 349, 117 (2014). DOI: 10.1016/​j.aop.2014.06.013.

[12] R. P. Feynman, Simulating Physics with Computers, Int. J. Theor. Phys. 21, 467 (1982). DOI: 10.1007/​BF02650179.

[13] I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80, 885 (2008). DOI: 10.1103/​RevModPhys.80.885.

[14] M. Lewenstein, A. Sanpera and V. Ahufinger, Ultracold Atoms in Optical Lattices: Simulating Quantum Many-Body Systems (Oxford University Press, New York, 2012). DOI: 10.1093/​acprof:oso/​9780199573127.001.0001.

[15] J. I. Cirac and P. Zoller, Goals and opportunities in quantum simulation, Nat. Phys. 8, 264 (2012). DOI: 10.1038/​nphys2275.

[16] I. Bloch, J. Dalibard, and S. Nascimbène, Quantum simulations with ultracold quantum gases, Nat. Phys. 8, 267 (2012). DOI: 10.1038/​nphys2259.

[17] R. Blatt and C. F. Roos, Quantum simulations with trapped ions, Nat. Phys. 8, 277 (2012). DOI: 10.1038/​nphys2252.

[18] E. Kapit and E. Mueller, Optical-lattice Hamiltonians for relativistic quantum electrodynamics, Phys. Rev. A 83, 033625 (2011). DOI: 10.1103/​PhysRevA.83.033625.

[19] E. Zohar, J. I. Cirac, and B. Reznik, Simulating Compact Quantum Electrodynamics with Ultracold Atoms: Probing Confinement and Nonperturbative Effects, Phys. Rev. Lett. 109, 125302 (2012). DOI: 10.1103/​PhysRevLett.109.125302.

[20] L. Tagliacozzo, A. Celi, P. Orland, and M. Lewenstein, Simulation of non-Abelian gauge theories with optical lattices, Nat. Commun. 4, 2615 (2013). DOI: 10.1038/​ncomms3615.

[21] K. Kasamatsu, I. Ichinose, and T. Matsui, Atomic Quantum Simulation of the Lattice Gauge-Higgs Model: Higgs Couplings and Emergence of Exact Local Gauge Symmetry, Phys. Rev. Lett. 111, 115303 (2013). DOI: 10.1103/​PhysRevLett.111.115303.

[22] 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, 125303 (2013). DOI: 10.1103/​PhysRevLett.110.125303.

[23] L. Tagliacozzo, A. Celi, A. Zamora, and M. Lewenstein, Optical Abelian lattice gauge theories, Ann. Phys. (N.Y.) 330, 160 (2013). DOI: 10.1016/​j.aop.2012.11.009.

[24] E. Zohar, J. I. Cirac, and B. Reznik, Quantum simulations of gauge theories with ultracold atoms: Local gauge invariance from angular-momentum conservation, Phys. Rev. A 88, 023617 (2013). DOI: 10.1103/​PhysRevA.88.023617.

[25] 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 (2014). DOI: 10.1103/​PhysRevLett.112.120406.

[26] E. Zohar, J. I. Cirac, and B. Reznik, Quantum simulations of lattice gauge theories using ultracold atoms in optical lattices., Rep. Prog. Phys. 79, 014401 (2016). DOI: 10.1088/​0034-4885/​79/​1/​014401.

[27] E. A. Martinez, C. A. Muschik, P. Schindler, D. Nigg, A. Erhard, M. Heyl, P. Hauke, M. Dalmonte, T. Monz, P. Zoller, and R. Blatt, Real-time dynamics of lattice gauge theories with a few-qubit quantum computer, Nature 534, 516-519 (2016). DOI: 10.1038/​nature18318.

[28] A. Celi, P. Massignan, J. Ruseckas, N. Goldman, I. B. Spielman, G. Juzeliūnas, and M. Lewenstein, Synthetic Gauge Fields in Synthetic Dimensions, Phys. Rev. Lett. 112, 043001 (2014). DOI: 10.1103/​PhysRevLett.112.043001.

[29] G. Pagano, M. Mancini, G. Cappellini, P. Lombardi, F. Schäfer, H. Hu, X.-J. Liu, J. Catani, C. Sias, M. Inguscio, and L. Fallani, A one-dimensional liquid of fermions with tunable spin, Nat. Phys. 10, 198 (2014). DOI: 10.1038/​nphys2878.

[30] F. Scazza, C. Hofrichter, M. Höfer, P. C. De Groot, I. Bloch, and S. Fölling, Observation of two-orbital spin-exchange interactions with ultracold SU(N)-symmetric fermions, Nat. Phys. 10, 779 (2014). DOI: 10.1038/​nphys3061.

[31] M. Mancini, G. Pagano, G. Cappellini, L. Livi, M. Rider, J. Catani, C. Sias, P. Zoller, M. Inguscio, M. Dalmonte, L. Fallani, Observation of chiral edge states with neutral fermions in synthetic Hall ribbons., Science 349, 1510 (2015). DOI: 10.1126/​science.aaa8736.

[32] L. F. Livi, G. Cappellini, M. Diem, L. Franchi, C. Clivati, M. Frittelli, F. Levi, D. Calonico, J. Catani, M. Inguscio, L. Fallani, Synthetic Dimensions and Spin-Orbit Coupling with an Optical Clock Transition, Phys. Rev. Lett. 117, 220401 (2016). DOI: 10.1103/​PhysRevLett.117.220401.

[33] J. Schwinger, On Gauge Invariance and Vacuum Polarization, Phys. Rev. 82, 664 (1951). DOI: 10.1103/​PhysRev.82.664.

[34] F. A. Wilczek, Nobel Lecture: Asymptotic freedom: From paradox to paradigm, Rev. Mod. Phys. 77, 857 (2005). DOI: 10.1103/​RevModPhys.77.857.

[35] R. Nandkishore and D. A. Huse, Many-Body Localization and Thermalization in Quantum Statistical Mechanics, Annu. Rev. Condens. Matter Phys. 6, 15 (2015). DOI: 10.1146/​annurev-conmatphys-031214-014726.

[36] F. Alet and N. Laflorencie, Many-body localization: An introduction and selected topics, C. R. Phys. 19, 498 (2018). DOI: 10.1016/​j.crhy.2018.03.003.

[37] C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papić, Weak ergodicity breaking from quantum many-body scars, Nat. Phys. 14, 745 (2018) DOI: 10.1038/​s41567-018-0137-5; Quantum scarred eigenstates in a Rydberg atom chain: Entanglement, breakdown of thermalization, and stability to perturbations, Phys. Rev. B 98, 155134 (2018). DOI: 10.1103/​PhysRevB.98.155134.

[38] V. Khemani, C. R. Laumann, and A. Chandran, Signatures of integrability in the dynamics of Rydberg-blockaded chains, Phys. Rev. B 99, 161101 (2019). DOI: 10.1103/​PhysRevB.99.161101.

[39] P. Hauke, D. Marcos, M. Dalmonte, and P. Zoller, Quantum Simulation of a Lattice Schwinger Model in a Chain of Trapped Ions, Phys. Rev. X 3, 041018 (2013). DOI: 10.1103/​PhysRevX.3.041018.

[40] S. Kühn, J. I. Cirac, and M.C. Bañuls, Quantum simulation of the Schwinger model: A study of feasibility, Phys. Rev. A 90, 042305 (2014). DOI: 10.1103/​PhysRevA.90.042305.

[41] S. Notarnicola, E. Ercolessi, P. Facchi, G. Marmo, S. Pascazio and F. V. Pepe, Discrete Abelian gauge theories for quantum simulations of QED, J. Phys. A: Math. Theor. 48, 30FT01 (2015). DOI: 10.1088/​1751-8113/​48/​30/​30FT01.

[42] V. Kasper, F. Hebenstreit, F. Jendrzejewski, M K Oberthaler, and J. Berges, Implementing quantum electrodynamics with ultracold atomic systems, New J. Phys. 19, 023030 (2017). DOI: 10.1088/​1367-2630/​aa54e0.

[43] S. Notarnicola, M. Collura, and S. Montangero, Real time dynamics quantum simulation of (1+1)-D lattice QED with Rydberg atoms, Phys. Rev. Research 2, 013288 (2020). DOI: 10.1103/​PhysRevResearch.2.013288.

[44] F. M. Surace, P. P. Mazza, G. Giudici, A. Lerose, A. Gambassi, and Marcello Dalmonte, Lattice gauge theories and string dynamics in Rydberg atom quantum simulators, Phys. Rev. X 10, 021041 (2020). DOI: 10.1103/​PhysRevX.10.021041.

[45] D. Banerjee, M. Dalmonte, M. Müller, E. Rico, P. Stebler, U. J. Wiese, and P. Zoller, Atomic Quantum Simulation of Dynamical Gauge Fields Coupled to Fermionic Matter: From String Breaking to Evolution after a Quench, Phys. Rev. Lett. 109, 175302 (2012). DOI: 10.1103/​PhysRevLett.109.175302.

[46] 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, 201601 (2014). DOI: 10.1103/​PhysRevLett.112.201601.

[47] D. Horn, Finite Matrix Models With Continuous Local Gauge Invariance, Phys. Lett. 100B, 149 (1981). DOI: 10.1016/​0370-2693(81)90763-2.

[48] 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.

[49] S. Chandrasekharan and U. J. Wiese, Quantum Link Models: A Discrete Approach to Gauge Theories, Nucl. Phys. B 492, 455 (1997). DOI: 10.1016/​S0550-3213(97)80041-7.

[50] U. J. Wiese, Ultracold quantum gases and lattice systems: quantum simulation of lattice gauge theories, Ann. Phys. (Berl.) 525, 777 (2013). DOI: 10.1002/​andp.201300104.

[51] B. Buyens, S. Montangero, J. Haegeman, F. Verstraete and K. Van Acoleyen, Finite-representation approximation of lattice gauge theories at the continuum limit with tensor networks, Phys. Rev. D 95, 094509 (2017). DOI: 10.1103/​PhysRevD.95.094509.

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

[53] T. Pichler, M. Dalmonte, E. Rico, P. Zoller, and S. Montangero, Real-Time Dynamics in U(1) Lattice Gauge Theories with Tensor Networks, Phys. Rev. X 6, 011023 (2016). DOI: 10.1103/​PhysRevX.6.011023.

[54] B. Buyens, J. Haegeman, F. Hebenstreit, F. Verstraete and K. Van Acoleyen, Real-time simulation of the Schwinger effect with matrix product states, Phys. Rev. D 96, 114501 (2017). DOI: 10.1103/​PhysRevD.96.114501.

[55] B. Buyens, J. Haegeman, H. Verschelde, F. Verstraete, K. Van Acoleyen, Confinement and String Breaking for $QED_2$ in the Hamiltonian Picture, Phys. Rev. X 6, 041040 (2016). DOI: 10.1103/​PhysRevX.6.041040.

[56] Y. Kuno, S. Sakane, K. Kasamatsu, I. Ichinose, and Tetsuo Matsui, Quantum simulation of (1+1)-dimensional U(1) gauge-Higgs model on a lattice by cold Bose gases, Phys. Rev. D 95, 094507 (2017). DOI: 10.1103/​PhysRevD.95.094507.

[57] J. Park, Y. Kuno, and I. Ichinose, Glassy dynamics from quark confinement: Atomic quantum simulation of the gauge-Higgs model on a lattice, Phys. Rev. A 100, 013629 (2019). DOI: 10.1103/​PhysRevA.100.013629.

[58] E. Ercolessi, P. Facchi, G. Magnifico, S. Pascazio, and F. V. Pepe, Phase transitions in $Z_n$ gauge models: Towards quantum simulations of the Schwinger-Weyl QED, Phys. Rev. D 98, 074503 (2018). DOI: 10.1103/​PhysRevD.98.074503.

[59] J. Schwinger, Gauge Invariance and Mass. II, Phys. Rev. 128, 2425 (1962). DOI: 10.1103/​PhysRev.128.2425.

[60] G. Magnifico, D. Vodola, E. Ercolessi, S. P. Kumar, M. Müller, and A. Bermudez, Symmetry-protected topological phases in lattice gauge theories: Topological $QED_2$, Phys. Rev. D 99, 014503 (2019). DOI: 10.1103/​PhysRevD.99.014503.

[61] G. Magnifico, D. Vodola, E. Ercolessi, S. P. Kumar, M. Müller, and A. Bermudez, $\mathbb{Z}_N$ gauge theories coupled to topological fermions: $QED_2$ with a quantum-mechanical $\theta$ angle, Phys. Rev. B 100, 115152 (2019). DOI: 10.1103/​PhysRevB.100.115152.

[62] M. Kormos, M. Collura, G. Takacs, and P. Calabrese, Real time confinement following a quantum quench to a non-integrable model, Nat. Phys. 13 246, (2017). DOI: 10.1038/​nphys3934.

[63] H. Weyl, The Theory of Groups and Quantum Mechanics (Dover Publications, New York, Dover Publications, 1950).

[64] J. Schwinger and B. G. Englert, Quantum Mechanics: Symbolism of Atomic Measurements (Springer, Berlin, 2001). DOI: 10.1007/​978-3-662-04589-3.

[65] M. Dalmonte and S. Montangero, Contemp. Lattice gauge theories simulations in the quantum information era, Phys. 57, 388 (2016). DOI: 10.1080/​00107514.2016.1151199.

[66] E. Zohar and J.I. Cirac, Removing staggered fermionic matter in U(N) and SU(N) lattice gauge theories, Phys. Rev. D 99, 114511 (2019). DOI: 10.1103/​PhysRevD.99.114511.

[67] B. M. McCoy and T. T. Wu, Two-dimensional Ising field theory in a magnetic field: Breakup of the cut in the two-point function, Phys. Rev. D 18, 1259 (1978). DOI: 10.1103/​PhysRevD.18.1259.

[68] P. Calabrese, J. cardy and B. Doyon, Entanglement Entropy in Extended Quantum Systems Special Issue, Journal of Physics A 42 (2009). DOI: 10.1088/​1751-8121/​42/​50/​500301.

[69] S. Rachel, M. Haque, A. Bernevug, A. Laeuchli and E. Fradkin (Guest Editors), Quantum Entanglement in Condensed Matter Physics, Special Issue, J. Stat. Mech. (2015). Link: https:/​/​​journal/​1742-5468/​page/​extra.special4.

[70] P. Calabrese and J. L. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. 2005, P04010 (2005). DOI: 10.1088/​1742-5468/​2005/​04/​P04010.

[71] E. H. Lieb and D. W. Robinson, The finite group velocity of quantum spin systems, Commun. Math. Phys. 28, 251 (1972). DOI: 10.1007/​BF01645779.

[72] A. Laeuchli and C. Kollath, Spreading of correlations and entanglement after a quench in the one-dimensional Bose-Hubbard model, J. Stat. Mech. 2008, P05018 (2008). DOI: 10.1088/​1742-5468/​2008/​05/​P05018.

[73] S. R. Manmana, S. Wessel, R. M. Noack, and A. Muramatsu, Time evolution of correlations in strongly interacting fermions after a quantum quench, Phys. Rev. B 79, 155104 (2009). DOI: 10.1103/​PhysRevB.79.155104.

[74] H. Kim and D. A. Huse, Ballistic Spreading of Entanglement in a Diffusive Nonintegrable System, Phys. Rev. Lett. 111, 127205 (2013). DOI: 10.1103/​PhysRevLett.111.127205.

[75] P. Barmettler, D. Poletti, M. Cheneau, and C. Kollath, Propagation front of correlations in an interacting Bose gas, Phys. Rev. A 85, 053625 (2012). DOI: 10.1103/​PhysRevA.85.053625.

[76] G. Carleo, F. Becca, L. Sanchez-Palencia, S. Sorella, and M. Fabrizio, Light-cone effect and supersonic correlations in one- and two-dimensional bosonic superfluids, Phys. Rev. A 89, 031602 (2014). DOI: 10.1103/​PhysRevA.89.031602.

[77] L. Bonnes, F. H. L. Essler, and A. M. Lauchli, ``Light-Cone'' Dynamics After Quantum Quenches in Spin Chains, Phys. Rev. Lett. 113, 187203 (2014). DOI: 10.1103/​PhysRevLett.113.187203.

[78] R. Geiger, T. Langen, I. E. Mazets, and J. Schmiedmayer, Local relaxation and light-cone-like propagation of correlations in a trapped one-dimensional Bose gas, New J. Phys. 16, 053034 (2014). DOI: 10.1088/​1367-2630/​16/​5/​053034.

[79] Compare for example our Fig. 8(a) with Fig. 6 of Ref. [62].

[80] H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletić, M. D. Lukin, Probing many-body dynamics on a 51-atom quantum simulator, Nature 551, 579 (2017). DOI: 10.1038/​nature24622.

[81] P. Calabrese and J. Cardy, Time Dependence of Correlation Functions Following a Quantum Quench, Phys. Rev. Lett. 96, 136801 (2006) DOI: 10.1103/​PhysRevLett.96.136801; Quantum quenches in extended systems, J. Stat. Mech. 2007 P06008 (2007). DOI: 10.1088/​1742-5468/​2007/​06/​P06008.

[82] F. Hebenstreit, R. Alkofer, and H. Gies, Schwinger pair production in space- and time-dependent electric fields: Relating the Wigner formalism to quantum kinetic theory, Phys. Rev. D 82, 105026 (2010). DOI: 10.1103/​PhysRevD.82.105026.

[83] F. Hebenstreit, J. Berges, and D. Gelfand, Simulating fermion production in 1+1 dimensional QED, Phys. Rev. D 87, 105006 (2013). DOI: 10.1103/​PhysRevD.87.105006.

[84] F. Liu, R. Lundgren, P. Titum, G. Pagano, J. Zhang, C. Monroe, and A. V. Gorshkov, Confined Quasiparticle Dynamics in Long-Range Interacting Quantum Spin Chains, Phys. Rev. Lett. 122, 150601 (2019). DOI: 10.1103/​PhysRevLett.122.150601.

[85] O. Pomponio, L. Pristyák, and G. Takács, Quasi-particle spectrum and entanglement generation after a quench in the quantum Potts spin chain, J. Stat. Mech. (2019) 013104. DOI: 10.1088/​1742-5468/​aafa80.

[86] U. Schollwöck and S. R. White, Methods for Time Dependence in DMRG, AIP Conf. Proc. 816, 155 (2006). DOI: 10.1063/​1.2178041.

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[24] Wibe A. de Jong, Kyle Lee, James Mulligan, Mateusz Płoskoń, Felix Ringer, and Xiaojun Yao, "Quantum simulation of nonequilibrium dynamics and thermalization in the Schwinger model", Physical Review D 106 5, 054508 (2022).

[25] Adrien Florio, David Frenklakh, Kazuki Ikeda, Dmitri Kharzeev, Vladimir Korepin, Shuzhe Shi, and Kwangmin Yu, "Real-Time Nonperturbative Dynamics of Jet Production in Schwinger Model: Quantum Entanglement and Vacuum Modification", Physical Review Letters 131 2, 021902 (2023).

[26] Johannes Knaute, Matan Feuerstein, and Erez Zohar, "Entanglement and confinement in lattice gauge theory tensor networks", Journal of High Energy Physics 2024 2, 174 (2024).

[27] Roland C. Farrell, Ivan A. Chernyshev, Sarah J. M. Powell, Nikita A. Zemlevskiy, Marc Illa, and Martin J. Savage, "Preparations for quantum simulations of quantum chromodynamics in 1+1 dimensions. I. Axial gauge", Physical Review D 107 5, 054512 (2023).

[28] S. Pradhan, A. Maroncelli, and E. Ercolessi, "Discrete Abelian lattice gauge theories on a ladder and their dualities with quantum clock models", Physical Review B 109 6, 064410 (2024).

[29] W. L. Tan, P. Becker, F. Liu, G. Pagano, K. S. Collins, A. De, L. Feng, H. B. Kaplan, A. Kyprianidis, R. Lundgren, W. Morong, S. Whitsitt, A. V. Gorshkov, and C. Monroe, "Domain-wall confinement and dynamics in a quantum simulator", Nature Physics 17 6, 742 (2021).

[30] 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).

[31] Hiroki Ohata, "Monte Carlo study of Schwinger model without the sign problem", Journal of High Energy Physics 2023 12, 7 (2023).

[32] Titas Chanda, Ruixiao Yao, and Jakub Zakrzewski, "Coexistence of localized and extended phases: Many-body localization in a harmonic trap", Physical Review Research 2 3, 032039 (2020).

[33] Piotr Sierant, Maciej Lewenstein, and Jakub Zakrzewski, "Polynomially Filtered Exact Diagonalization Approach to Many-Body Localization", Physical Review Letters 125 15, 156601 (2020).

[34] Octavio Pomponio, Miklós Antal Werner, Gergely Zaránd, and Gabor Takacs, "Bloch oscillations and the lack of the decay of the false vacuum in a one-dimensional quantum spin chain", SciPost Physics 12 2, 061 (2022).

[35] K. Pakrouski, P. N. Pallegar, F. K. Popov, and I. R. Klebanov, "Many-Body Scars as a Group Invariant Sector of Hilbert Space", Physical Review Letters 125 23, 230602 (2020).

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

[37] Takuya Okuda, "Schwinger model on an interval: Analytic results and DMRG", Physical Review D 107 5, 054506 (2023).

[38] Arata Yamamoto, "Real-time simulation of (2+1)-dimensional lattice gauge theory on qubits", Progress of Theoretical and Experimental Physics 2021 1, 013B06 (2021).

[39] Chunping Gao, Zheng Tang, Fei Zhu, Yunbo Zhang, Han Pu, and Li Chen, "Nonthermal dynamics in a spin- 12 lattice Schwinger model", Physical Review B 107 10, 104302 (2023).

[40] Bart van Voorden, Jiří Minář, and Kareljan Schoutens, "Quantum many-body scars in transverse field Ising ladders and beyond", Physical Review B 101 22, 220305 (2020).

[41] Tomasz Szołdra, Piotr Sierant, Maciej Lewenstein, and Jakub Zakrzewski, "Unsupervised detection of decoupled subspaces: Many-body scars and beyond", Physical Review B 105 22, 224205 (2022).

[42] Jens Nyhegn, Chia-Min Chung, and Michele Burrello, "ZN lattice gauge theory in a ladder geometry", Physical Review Research 3 1, 013133 (2021).

[43] Roland C. Farrell, Marc Illa, Anthony N. Ciavarella, and Martin J. Savage, "Scalable Circuits for Preparing Ground States on Digital Quantum Computers: The Schwinger Model Vacuum on 100 Qubits", PRX Quantum 5 2, 020315 (2024).

[44] Jared Jeyaretnam, Jonas Richter, and Arijeet Pal, "Quantum scars and bulk coherence in a symmetry-protected topological phase", Physical Review B 104 1, 014424 (2021).

[45] Rocco Maggi, Elisa Ercolessi, Paolo Facchi, Giuseppe Marmo, Saverio Pascazio, and Francesco V. Pepe, "Dimensional reduction of electromagnetism", Journal of Mathematical Physics 63 2, 022902 (2022).

[46] Roberto Verdel, Guo-Yi Zhu, and Markus Heyl, "Dynamical Localization Transition of String Breaking in Quantum Spin Chains", Physical Review Letters 131 23, 230402 (2023).

[47] Qianqian Chen and Zheng Zhu, "Inverting multiple quantum many-body scars via disorder", Physical Review B 109 1, 014212 (2024).

[48] Mari Carmen Bañuls, Michal P. Heller, Karl Jansen, Johannes Knaute, and Viktor Svensson, "Quantum information perspective on meson melting", Physical Review D 108 7, 076016 (2023).

[49] Adith Sai Aramthottil, Utso Bhattacharya, Daniel González-Cuadra, Maciej Lewenstein, Luca Barbiero, and Jakub Zakrzewski, "Scar states in deconfined Z2 lattice gauge theories", Physical Review B 106 4, L041101 (2022).

[50] Pablo Arrighi, Giuseppe Di Molfetta, and Nathanaël Eon, "Gauge-invariance in cellular automata", Natural Computing 22 3, 587 (2023).

[51] Marc Illa and Martin J. Savage, "Basic elements for simulations of standard-model physics with quantum annealers: Multigrid and clock states", Physical Review A 106 5, 052605 (2022).

[52] Masazumi Honda, Etsuko Itou, Yuta Kikuchi, Lento Nagano, and Takuya Okuda, "Classically emulated digital quantum simulation for screening and confinement in the Schwinger model with a topological term", Physical Review D 105 1, 014504 (2022).

[53] Christopher M. Langlett and Shenglong Xu, "Hilbert space fragmentation and exact scars of generalized Fredkin spin chains", Physical Review B 103 22, L220304 (2021).

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

[55] Domenico Pomarico, Leonardo Cosmai, Paolo Facchi, Cosmo Lupo, Saverio Pascazio, and Francesco V. Pepe, "Dynamical Quantum Phase Transitions of the Schwinger Model: Real-Time Dynamics on IBM Quantum", Entropy 25 4, 608 (2023).

[56] A. A. Michailidis, C. J. Turner, Z. Papić, D. A. Abanin, and M. Serbyn, "Stabilizing two-dimensional quantum scars by deformation and synchronization", Physical Review Research 2 2, 022065 (2020).

[57] Cheng-Ju Lin, Vladimir Calvera, and Timothy H. Hsieh, "Quantum many-body scar states in two-dimensional Rydberg atom arrays", Physical Review B 101 22, 220304 (2020).

[58] Ruixiao Yao and Jakub Zakrzewski, "Many-body localization of bosons in an optical lattice: Dynamics in disorder-free potentials", Physical Review B 102 10, 104203 (2020).

[59] Giuliano Angelone, Elisa Ercolessi, Paolo Facchi, Davide Lonigro, Rocco Maggi, Giuseppe Marmo, Saverio Pascazio, and Francesco V Pepe, "Dimensional reduction of the Dirac theory", Journal of Physics A: Mathematical and Theoretical 56 6, 065201 (2023).

[60] Flávia B. Ramos, Máté Lencsés, J. C. Xavier, and Rodrigo G. Pereira, "Confinement and bound states of bound states in a transverse-field two-leg Ising ladder", Physical Review B 102 1, 014426 (2020).

[61] Roberto Verdel, Fangli Liu, Seth Whitsitt, Alexey V. Gorshkov, and Markus Heyl, "Real-time dynamics of string breaking in quantum spin chains", Physical Review B 102 1, 014308 (2020).

[62] Berislav Buča, "Unified Theory of Local Quantum Many-Body Dynamics: Eigenoperator Thermalization Theorems", Physical Review X 13 3, 031013 (2023).

[63] Aritra Das, Umberto Borla, and Sergej Moroz, "Fractionalized holes in one-dimensional Z2 gauge theory coupled to fermion matter: Deconfined dynamics and emergent integrability", Physical Review B 107 6, 064302 (2023).

[64] Titas Chanda, Marcello Dalmonte, Maciej Lewenstein, Jakub Zakrzewski, and Luca Tagliacozzo, "Spectral properties of the critical (1+1)-dimensional Abelian-Higgs model", Physical Review B 109 4, 045103 (2024).

[65] Kazuki Ikeda, Dmitri E. Kharzeev, and Yuta Kikuchi, "Real-time dynamics of Chern-Simons fluctuations near a critical point", Physical Review D 103 7, L071502 (2021).

[66] Kyle Lee, James Mulligan, Felix Ringer, and Xiaojun Yao, "Liouvillian dynamics of the open Schwinger model: String breaking and kinetic dissipation in a thermal medium", Physical Review D 108 9, 094518 (2023).

[67] K. Pakrouski, P. N. Pallegar, F. K. Popov, and I. R. Klebanov, "Group theoretic approach to many-body scar states in fermionic lattice models", Physical Review Research 3 4, 043156 (2021).

[68] Maksym Serbyn, Dmitry A. Abanin, and Zlatko Papić, "Quantum many-body scars and weak breaking of ergodicity", Nature Physics 17 6, 675 (2021).

[69] Arata Yamamoto, "Quantum variational approach to lattice gauge theory at nonzero density", Physical Review D 104 1, 014506 (2021).

[70] Domenico Pomarico, "Multiscale Entanglement Renormalization Ansatz: Causality and Error Correction", Dynamics 3 3, 622 (2023).

[71] Artur Maksymov, Piotr Sierant, and Jakub Zakrzewski, "Many-body localization in a one-dimensional optical lattice with speckle disorder", Physical Review B 102 13, 134205 (2020).

[72] Kazuki Ikeda, Dmitri E. Kharzeev, René Meyer, and Shuzhe Shi, "Detecting the critical point through entanglement in the Schwinger model", Physical Review D 108 9, L091501 (2023).

[73] 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).

[74] Gianluca Lagnese, Federica Maria Surace, Márton Kormos, and Pasquale Calabrese, "Quenches and confinement in a Heisenberg–Ising spin ladder", Journal of Physics A: Mathematical and Theoretical 55 12, 124003 (2022).

[75] Anthony N. Ciavarella, Stephan Caspar, Hersh Singh, and Martin J. Savage, "Preparation for quantum simulation of the (1+1) -dimensional O(3) nonlinear σ model using cold atoms", Physical Review A 107 4, 042404 (2023).

[76] D. González-Cuadra, A. Dauphin, M. Aidelsburger, M. Lewenstein, and A. Bermudez, "Rotor Jackiw-Rebbi Model: A Cold-Atom Approach to Chiral Symmetry Restoration and Charge Confinement", PRX Quantum 1 2, 020321 (2020).

[77] Joseph Vovrosh, Rick Mukherjee, Alvise Bastianello, and Johannes Knolle, "Dynamical Hadron Formation in Long-Range Interacting Quantum Spin Chains", PRX Quantum 3 4, 040309 (2022).

[78] Yannick Meurice, "Discrete aspects of continuous symmetries in the tensorial formulation of Abelian gauge theories", Physical Review D 102 1, 014506 (2020).

[79] Rasmus Berg Jensen, Simon Panyella Pedersen, and Nikolaj Thomas Zinner, "Dynamical quantum phase transitions in a noisy lattice gauge theory", Physical Review B 105 22, 224309 (2022).

[80] Gianluca Lagnese, Federica Maria Surace, Márton Kormos, and Pasquale Calabrese, "Confinement in the spectrum of a Heisenberg–Ising spin ladder", Journal of Statistical Mechanics: Theory and Experiment 2020 9, 093106 (2020).

[81] Natalie Klco, Alessandro Roggero, and Martin J Savage, "Standard model physics and the digital quantum revolution: thoughts about the interface", Reports on Progress in Physics 85 6, 064301 (2022).

[82] Shane Thompson and George Siopsis, "Quantum computation of phase transition in the massive Schwinger model", Quantum Science and Technology 7 3, 035001 (2022).

[83] Thomas Iadecola and Michael Schecter, "Quantum many-body scar states with emergent kinetic constraints and finite-entanglement revivals", Physical Review B 101 2, 024306 (2020).

[84] Zhi-Cheng Yang, Fangli Liu, Alexey V. Gorshkov, and Thomas Iadecola, "Hilbert-Space Fragmentation from Strict Confinement", Physical Review Letters 124 20, 207602 (2020).

[85] Yannick Meurice, "Quantum Field Theory; A quantum computation approach", Quantum Field Theory (2021).

[86] Joseph Vovrosh and Johannes Knolle, "Confinement and entanglement dynamics on a digital quantum computer", Scientific Reports 11, 11577 (2021).

[87] Roland C. Farrell, Marc Illa, Anthony N. Ciavarella, and Martin J. Savage, "Quantum Simulations of Hadron Dynamics in the Schwinger Model using 112 Qubits", arXiv:2401.08044, (2024).

[88] Dmitri E. Kharzeev and Yuta Kikuchi, "Real-time chiral dynamics from a digital quantum simulation", Physical Review Research 2 2, 023342 (2020).

[89] Umberto Borla, Ruben Verresen, Fabian Grusdt, and Sergej Moroz, "Confined Phases of One-Dimensional Spinless Fermions Coupled to Z<SUB>2</SUB> Gauge Theory", Physical Review Letters 124 12, 120503 (2020).

[90] Titas Chanda, Jakub Zakrzewski, Maciej Lewenstein, and Luca Tagliacozzo, "Confinement and Lack of Thermalization after Quenches in the Bosonic Schwinger Model", Physical Review Letters 124 18, 180602 (2020).

[91] Bipasha Chakraborty, Masazumi Honda, Taku Izubuchi, Yuta Kikuchi, and Akio Tomiya, "Classically Emulated Digital Quantum Simulation of the Schwinger Model with Topological Term via Adiabatic State Preparation", arXiv:2001.00485, (2020).

[92] Ana Hudomal, Ivana Vasić, Nicolas Regnault, and Zlatko Papić, "Quantum scars of bosons with correlated hopping", Communications Physics 3 1, 99 (2020).

[93] Fangli Liu, Seth Whitsitt, Przemyslaw Bienias, Rex Lundgren, and Alexey V. Gorshkov, "Realizing and Probing Baryonic Excitations in Rydberg Atom Arrays", arXiv:2007.07258, (2020).

[94] Olalla A. Castro-Alvaredo, Máté Lencsés, István M. Szécsényi, and Jacopo Viti, "Entanglement Oscillations near a Quantum Critical Point", Physical Review Letters 124 23, 230601 (2020).

[95] A. Krasznai and G. Takács, "Escaping fronts in local quenches of a confining spin chain", arXiv:2401.04193, (2024).

[96] Nouman Butt, Simon Catterall, Yannick Meurice, Ryo Sakai, and Judah Unmuth-Yockey, "Tensor network formulation of the massless Schwinger model with staggered fermions", Physical Review D 101 9, 094509 (2020).

[97] Adrien Florio, Andreas Weichselbaum, Semeon Valgushev, and Robert D. Pisarski, "Mass gaps of a $\mathbb{Z}_3$ gauge theory with three fermion flavors in 1 + 1 dimensions", arXiv:2310.18312, (2023).

[98] Masazumi Honda, Etsuko Itou, Yuta Kikuchi, and Yuya Tanizaki, "Negative string tension of a higher-charge Schwinger model via digital quantum simulation", Progress of Theoretical and Experimental Physics 2022 3, 033B01 (2022).

[99] Ashley Milsted, Junyu Liu, John Preskill, and Guifre Vidal, "Collisions of false-vacuum bubble walls in a quantum spin chain", arXiv:2012.07243, (2020).

[100] Alexander M. Czajka, Zhong-Bo Kang, Yuxuan Tee, and Fanyi Zhao, "Studying chirality imbalance with quantum algorithms", arXiv:2210.03062, (2022).

[101] Jad C. Halimeh and Philipp Hauke, "Origin of staircase prethermalization in lattice gauge theories", arXiv:2004.07254, (2020).

[102] Riccardo Javier Valencia Tortora, Pasquale Calabrese, and Mario Collura, "Relaxation of the order-parameter statistics and dynamical confinement", EPL (Europhysics Letters) 132 5, 50001 (2020).

[103] Kazuki Ikeda and Adam Lowe, "Robustness of quantum correlation in quantum energy teleportation", arXiv:2402.00479, (2024).

[104] Riccardo Javier Valencia Tortora, Pasquale Calabrese, and Mario Collura, "Relaxation of the order-parameter statistics and dynamical confinement", arXiv:2005.01679, (2020).

[105] Dongwook Ghim and Masazumi Honda, "Digital Quantum Simulation for Spectroscopy of Schwinger Model", arXiv:2404.14788, (2024).

[106] Arata Yamamoto, "Real-time simulation of (2+1)-dimensional lattice gauge theory on qubits", arXiv:2008.11395, (2020).

[107] Kazuki Ikeda, "Quantum-classical simulation of quantum field theory by quantum circuit learning", arXiv:2311.16297, (2023).

[108] Fanyi Zhao, "3D Imaging via Polarized Jet Fragmentation Functions and Quantum Simulation of the QCD Phase Diagram", arXiv:2309.10838, (2023).

[109] Xiaopeng Cui, Yu Shi, and Ji-Chong Yang, "Circuit-based digital adiabatic quantum simulation and pseudoquantum simulation as new approaches to lattice gauge theory", Journal of High Energy Physics 2020 8, 160 (2020).

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

[111] Yao-Tai Kang, Chung-Yu Lo, Shuai Yin, and Pochung Chen, "Kibble-Zurek mechanism in a quantum link model", Physical Review A 101 2, 023610 (2020).

[112] Yoshimasa Hidaka, Yuya Tanizaki, and Arata Yamamoto, "$\mathbb{Z}_3$ lattice gauge theory as a toy model for dense QCD", arXiv:2404.07595, (2024).

[113] Joshua Lin, Di Luo, Xiaojun Yao, and Phiala E. Shanahan, "Real-time Dynamics of the Schwinger Model as an Open Quantum System with Neural Density Operators", arXiv:2402.06607, (2024).

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