Quantum phase transition dynamics in the two-dimensional transverse-field Ising model

The quantum Kibble-Zurek mechanism (QKZM) predicts universal dynamical behavior near the quantum phase transitions (QPTs). It is now well understood for the one-dimensional quantum matter. Higher-dimensional systems, however, remain a challenge, complicated by the fundamentally different character of the associated QPTs and their underlying conformal field theories. In this work, we take the first steps toward theoretical exploration of the QKZM in two dimensions for interacting quantum matter. We study the dynamical crossing of the QPT in the paradigmatic Ising model by a joint effort of modern state-of-the-art numerical methods, including artificial neural networks and tensor networks. As a central result, we quantify universal QKZM behavior close to the QPT. We also note that, upon traversing further into the ferromagnetic regime, deviations from the QKZM prediction appear. We explain the observed behavior by proposing an extended QKZM taking into account spectral information as well as phase ordering. Our work provides a testing platform for higher-dimensional quantum simulators.

[1]  J. Dziarmaga.  Time evolution of an infinite projected entangled pair state: Neighborhood tensor update , 2021, Physical Review B.

[2]  M. Lukin,et al.  Probing topological spin liquids on a programmable quantum simulator , 2021, Science.

[3]  H. Neven,et al.  Realizing topologically ordered states on a quantum processor , 2021, Science.

[4]  M. Lukin,et al.  Quantum phases of matter on a 256-atom programmable quantum simulator , 2020, Nature.

[5]  D. Barredo,et al.  Quantum simulation of 2D antiferromagnets with hundreds of Rydberg atoms , 2020, Nature.

[6]  G. Volovik,et al.  Suppressing the Kibble-Zurek Mechanism by a Symmetry-Violating Bias. , 2019, Physical review letters.

[7]  Daniel A. Lidar,et al.  Probing the universality of topological defect formation in a quantum annealer: Kibble-Zurek mechanism and beyond , 2020, 2001.11637.

[8]  M. Heyl,et al.  Quantum Many-Body Dynamics in Two Dimensions with Artificial Neural Networks. , 2019, Physical review letters.

[9]  W. Zurek,et al.  Sonic horizons and causality in phase transition dynamics , 2019, Physical Review B.

[10]  A. Sandvik,et al.  Scaling and Diabatic Effects in Quantum Annealing with a D-Wave Device. , 2019, Physical review letters.

[11]  W. Zurek,et al.  Symmetry Breaking Bias and the Dynamics of a Quantum Phase Transition. , 2019, Physical review letters.

[12]  M. Plenio,et al.  Quantum Kibble-Zurek physics in long-range transverse-field Ising models , 2019, Physical Review A.

[13]  Dorian Krause,et al.  JUWELS: Modular Tier-0/1 Supercomputer at Jülich Supercomputing Centre , 2019, Journal of large-scale research facilities JLSRF.

[14]  P. Corboz,et al.  Time evolution of an infinite projected entangled pair state: An efficient algorithm , 2018, Physical Review B.

[15]  Sylvain Schwartz,et al.  Quantum Kibble–Zurek mechanism and critical dynamics on a programmable Rydberg simulator , 2018, Nature.

[16]  R. Sarpong,et al.  Bio-inspired synthesis of xishacorenes A, B, and C, and a new congener from fuscol† †Electronic supplementary information (ESI) available. See DOI: 10.1039/c9sc02572c , 2019, Chemical science.

[17]  G. Carleo,et al.  Symmetries and Many-Body Excitations with Neural-Network Quantum States. , 2018, Physical review letters.

[18]  Wojciech H. Zurek,et al.  Defects in Quantum Computers , 2017, Scientific Reports.

[19]  I. Bloch,et al.  Coherent many-body spin dynamics in a long-range interacting Ising chain , 2017, 1705.08372.

[20]  Matthias Troyer,et al.  Solving the quantum many-body problem with artificial neural networks , 2016, Science.

[21]  Logan W. Clark,et al.  Universal space-time scaling symmetry in the dynamics of bosons across a quantum phase transition , 2016, Science.

[22]  H. M. Bharath,et al.  Quantum Kibble-Zurek Mechanism in a Spin-1 Bose-Einstein Condensate. , 2015, Physical review letters.

[23]  W. Zurek,et al.  Space and time renormalization in phase transition dynamics , 2015, 1510.06132.

[24]  D. Ceperley,et al.  Probing the Bose glass–superfluid transition using quantum quenches of disorder , 2015, Nature Physics.

[25]  Ivan Oseledets,et al.  Unifying time evolution and optimization with matrix product states , 2014, 1408.5056.

[26]  G. Maret,et al.  Kibble–Zurek mechanism in colloidal monolayers , 2015, Proceedings of the National Academy of Sciences.

[27]  J. Dalibard,et al.  Emergence of coherence via transverse condensation in a uniform quasi-two-dimensional Bose gas , 2014, Nature Communications.

[28]  Alexander L. Gaunt,et al.  Critical dynamics of spontaneous symmetry breaking in a homogeneous Bose gas , 2014, Science.

[29]  Immanuel Bloch,et al.  Emergence of coherence and the dynamics of quantum phase transitions , 2014, Proceedings of the National Academy of Sciences.

[30]  S. Cheong,et al.  Topological defects as relics of emergent continuous symmetry and Higgs condensation of disorder in ferroelectrics , 2014, Nature Physics.

[31]  P. Chesler,et al.  Defect formation beyond Kibble-Zurek mechanism and holography , 2014, 1407.1862.

[32]  F. Dalfovo,et al.  Observation of solitonic vortices in Bose-Einstein condensates. , 2014, Physical review letters.

[33]  Wojciech H. Zurek,et al.  Universality of Phase Transition Dynamics: Topological Defects from Symmetry Breaking , 2013, 1310.1600.

[34]  M. Plenio,et al.  Topological defect formation and spontaneous symmetry breaking in ion Coulomb crystals , 2013, Nature Communications.

[35]  J. Rossnagel,et al.  Observation of the Kibble–Zurek scaling law for defect formation in ion crystals , 2013, Nature Communications.

[36]  T. Schaetz,et al.  Trapping of topological-structural defects in Coulomb crystals. , 2012, Physical review letters.

[37]  M. Fiebig,et al.  Scaling Behavior and Beyond Equilibrium in the Hexagonal Manganites , 2012 .

[38]  S. Cheong,et al.  Direct observation of the proliferation of ferroelectric loop domains and vortex-antivortex pairs. , 2012, Physical review letters.

[39]  S. Gubser,et al.  Kibble-Zurek problem: Universality and the scaling limit , 2012, 1202.5277.

[40]  D. Huse,et al.  Nonequilibrium dynamic critical scaling of the quantum Ising chain. , 2011, Physical review letters.

[41]  B. Clark,et al.  Nonequilibrium dynamics of bosonic Mott insulators in an electric field , 2011, 1106.4031.

[42]  F. Brennecke,et al.  Exploring symmetry breaking at the Dicke quantum phase transition. , 2011, Physical review letters.

[43]  B. Demarco,et al.  Quantum quench of an atomic Mott insulator. , 2011, Physical review letters.

[44]  K. Rzążewski,et al.  Solitons as the early stage of quasicondensate formation during evaporative cooling. , 2011, Physical review letters.

[45]  Alessandro Silva,et al.  Colloquium: Nonequilibrium dynamics of closed interacting quantum systems , 2010, 1007.5331.

[46]  A. Campo,et al.  Spontaneous nucleation of structural defects in inhomogeneous ion chains , 2010, 1006.5937.

[47]  Jacek Dziarmaga,et al.  Dynamics of a quantum phase transition and relaxation to a steady state , 2009, 0912.4034.

[48]  Wojciech H. Zurek,et al.  Critical dynamics of decoherence , 2009, 0911.5729.

[49]  A. Polkovnikov,et al.  Adiabatic Perturbation Theory: From Landau–Zener Problem to Quenching Through a Quantum Critical Point , 2009, 0910.2236.

[50]  A. Polkovnikov,et al.  Quench dynamics near a quantum critical point , 2009, 0909.5181.

[51]  Bogdan Damski,et al.  Soliton creation during a Bose-Einstein condensation. , 2009, Physical review letters.

[52]  Brian P. Anderson,et al.  Spontaneous vortices in the formation of Bose–Einstein condensates , 2008, Nature.

[53]  K. Sengupta,et al.  Defect production in nonlinear quench across a quantum critical point. , 2008, Physical review letters.

[54]  K. Sengupta,et al.  Quench dynamics and defect production in the Kitaev and extended Kitaev models , 2008, 0802.3986.

[55]  K. Sengupta,et al.  Exact results for quench dynamics and defect production in a two-dimensional model. , 2007, Physical review letters.

[56]  A. Polkovnikov,et al.  Breakdown of the adiabatic limit in low-dimensional gapless systems , 2007, 0803.3967.

[57]  T. Kibble.  Phase-transition dynamics in the lab and the universe , 2007 .

[58]  Masahito Ueda,et al.  Kibble-Zurek mechanism in a quenched ferromagnetic Bose-Einstein condensate , 2007, 0704.1377.

[59]  W. Zurek,et al.  Entropy of entanglement and correlations induced by a quench: Dynamics of a quantum phase transition in the quantum Ising model , 2007, cond-mat/0701768.

[60]  U. R. Fischer,et al.  Vortex quantum creation and winding number scaling in a quenched spinor Bose gas. , 2006, Physical review letters.

[61]  D. Stamper-Kurn,et al.  Spontaneous symmetry breaking in a quenched ferromagnetic spinor Bose–Einstein condensate , 2006, Nature.

[62]  Jacek Dziarmaga,et al.  Dynamics of a quantum phase transition: exact solution of the quantum Ising model. , 2005, Physical review letters.

[63]  P. Zoller,et al.  Dynamics of a quantum phase transition. , 2005, Physical review letters.

[64]  B. Damski,et al.  The simplest quantum model supporting the Kibble-Zurek mechanism of topological defect production: Landau-Zener transitions from a new perspective. , 2004, Physical review letters.

[65]  A. Polkovnikov.  Universal adiabatic dynamics in the vicinity of a quantum critical point , 2003, cond-mat/0312144.

[66]  Joakim Nivre.  AN EFFICIENT ALGORITHM , 2003 .

[67]  Youjin Deng,et al.  Cluster Monte Carlo simulation of the transverse Ising model. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[68]  R. Monaco,et al.  Zurek-Kibble domain structures: the dynamics of spontaneous vortex formation in annular Josephson tunnel junctions. , 2001, Physical review letters.

[69]  Carmi,et al.  Observation of spontaneous flux generation in a multi-josephson-junction loop , 2000, Physical review letters.

[70]  L. Bettencourt,et al.  Vortex string formation in a 3D U(1) temperature quench , 1998, hep-ph/9811426.

[71]  W. Zurek,et al.  Vortex formation in two dimensions: when symmetry breaks, how big are the pieces? , 1998, hep-ph/9801223.

[72]  P. Laguna,et al.  Density of Kinks after a Quench: When Symmetry Breaks, How Big are the Pieces? , 1996, gr-qc/9607041.

[73]  Wen Xu,et al.  Vortex formation in neutron-irradiated superfluid 3He as an analogue of cosmological defect formation , 1996, Nature.

[74]  G. R. Pickett,et al.  Laboratory simulation of cosmic string formation in the early Universe using superfluid 3He , 1996, Nature.

[75]  W. Zurek.  Cosmological experiments in condensed matter systems , 1996, cond-mat/9607135.

[76]  W. Zurek.  Cosmic strings in laboratory superfluids and the topological remnants of other phase transitions , 1993 .

[77]  W. H. Zurek,et al.  Cosmological experiments in superfluid helium? , 1985, Nature.

[78]  T. Kibble,et al.  Some Implications of a Cosmological Phase Transition , 1980 .

[79]  T W B Kibble,et al.  Topology of cosmic domains and strings , 1976 .

[80]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.