Abstract
Quantum quenches in continuum field theory across critical points are known to display different scaling behaviours in different regimes of the quench rate. We extend these results to integrable lattice models such as the transverse field Ising model on a one-dimensional chain and the Kitaev model on a two-dimensional honeycomb lattice using a nonlinear quench protocol which allows for exact analytical solutions of the dynamics. Our quench protocol starts with a finite mass gap at early times and crosses a critical point or a critical region, and we study the behaviour of one point functions of the quenched operator at the critical point or in the critical region as a function of the quench rate. For quench rates slow compared to the initial mass gap, we find the expected Kibble-Zurek scaling. In contrast, for rates fast compared to the mass gap, but slow compared to the inverse lattice spacing, we find scaling behaviour similar to smooth fast continuum quenches. For quench rates of the same order of the lattice scale, the one point function saturates as a function of the rate, approaching the results of an abrupt quench. The presence of an extended critical surface in the Kitaev model leads to a variety of scaling exponents depending on the starting point and on the time where the operator is measured. We discuss the role of the amplitude of the quench in determining the extent of the slow (Kibble-Zurek) and fast quench regimes, and the onset of the saturation.
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References
T.W.B. Kibble, Topology of cosmic domains and strings, J. Phys. A 9 (1976) 1387 [INSPIRE].
W.H. Zurek, Cosmological experiments in superfluid helium?, Nature 317 (1985) 505 [INSPIRE].
S. Mondal, D. Sen and K. Sengupta, Non-equilibrium dynamics of quantum systems: order parameter evolution, defect generation, and qubit transfer, Lect. Notes Phys. 802 (2010) 21 [arXiv:0908.2922].
V. Gritsev and A. Polkovnikov, Universal dynamics near quantum critical points, arXiv:0910.3692 [INSPIRE].
J Dziarmaga, Dynamics of a quantum phase transition and relaxation to a steady state, Adv. Phys. 59 (2010) 1063 [arXiv:0912.4034].
A. Polkovnikov, K. Sengupta, A. Silva and M. Vengalattore, Nonequilibrium dynamics of closed interacting quantum systems, Rev. Mod. Phys. 83 (2011) 863 [arXiv:1007.5331] [INSPIRE].
A. Lamacraft and J.E. Moore, Potential insights into non-equilibrium behaviour from atomic physics, in Ultracold bosonic and fermionic gases, A. Fletcher et al. eds., Elsevier, Germany (2013), arXiv:1106.3567.
A. Chandran et al., Kibble-Zurek problem: universality and the scaling limit, Phys. Rev. B 86 (2012) 064304 [arXiv:1202.5277].
A. Buchel, L. Lehner and R.C. Myers, Thermal quenches in N = 2* plasmas, JHEP 08 (2012) 049 [arXiv:1206.6785] [INSPIRE].
A. Buchel, L. Lehner, R.C. Myers and A. van Niekerk, Quantum quenches of holographic plasmas, JHEP 05 (2013) 067 [arXiv:1302.2924] [INSPIRE].
A. Buchel, R.C. Myers and A. van Niekerk, Universality of abrupt holographic quenches, Phys. Rev. Lett. 111 (2013) 201602 [arXiv:1307.4740] [INSPIRE].
S.R. Das, D.A. Galante and R.C. Myers, Universal scaling in fast quantum quenches in conformal field theories, Phys. Rev. Lett. 112 (2014) 171601 [arXiv:1401.0560] [INSPIRE].
S.R. Das, D.A. Galante and R.C. Myers, Universality in fast quantum quenches, JHEP 02 (2015) 167 [arXiv:1411.7710] [INSPIRE].
S.R. Das, D.A. Galante and R.C. Myers, Smooth and fast versus instantaneous quenches in quantum field theory, JHEP 08 (2015) 073 [arXiv:1505.05224] [INSPIRE].
S.R. Das, D.A. Galante and R.C. Myers, Quantum quenches in free field theory: universal scaling at any rate, JHEP 05 (2016) 164 [arXiv:1602.08547] [INSPIRE].
A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys. 321 (2006) 2 [cond-mat/0506438].
J. Dziarmaga, Dynamics of a quantum phase transition: exact solution of the quantum Ising model, Phys. Rev. Lett. 95 (2005) 245701 [cond-mat/0509490].
J. Dziarmaga, Dynamics of a quantum phase transition in the random Ising model, Phys. Rev. B 74 (2006) 064416 [cond-mat/0603814] [INSPIRE].
K. Sengupta, D. Sen and S. Mondal, Exact results for quench dynamics and defect production in a two-dimensional model, Phys. Rev. Lett. 100 (2008) 077204 [arXiv:0710.1712].
S. Mondal, D. Sen and K. Sengupta, Quench dynamics and defect production in the Kitaev and extended Kitaev models, Phys. Rev. B 78 (2008) 045101 [arXiv:0802.3986].
T. Hikichi, S. Suzuki, and K. Sengupta, Slow quench dynamics of the Kitaev model: Anisotropic critical point and effect of disorder, Phys. Rev. B 82 (2010) 174305 [arXiv:1009.0323].
K. Sengupta, S. Powell and S. Sachdev, Quench dynamics across quantum critical points, Phys. Rev. A 69 (2004) 053616 [cond-mat/0311355].
P. Calabrese and J.L. Cardy, Time-dependence of correlation functions following a quantum quench, Phys. Rev. Lett. 96 (2006) 136801 [cond-mat/0601225] [INSPIRE].
P. Calabrese and J. Cardy, Quantum quenches in extended systems, J. Stat. Mech. 0706 (2007) P06008 [arXiv:0704.1880] [INSPIRE].
A.A. Patel and A. Dutta, Sudden quenching in the Kitaev honeycomb model: study of defect and heat generation, Phys. Rev. B 86 (2012) 174306 [arXiv:1209.0072].
M. Schmitt and S. Kehrein, Dynamical quantum phase transitions in the Kitaev honeycomb model, Phys. Rev. B 92 (2015) 075114 [arXiv:1505.03401].
S. Sotiriadis and J. Cardy, Quantum quench in interacting field theory: a self-consistent approximation, Phys. Rev. B 81 (2010) 134305 [arXiv:1002.0167] [INSPIRE].
G. Mandal, R. Sinha and N. Sorokhaibam, Thermalization with chemical potentials and higher spin black holes, JHEP 08 (2015) 013 [arXiv:1501.04580] [INSPIRE].
G. Mandal, S. Paranjape and N. Sorokhaibam, Thermalization in 2D critical quench and UV/IR mixing, arXiv:1512.02187 [INSPIRE].
J.S. Cotler, M.P. Hertzberg, M. Mezei and M.T. Mueller, Entanglement growth after a global quench in free scalar field theory, JHEP 11 (2016) 166 [arXiv:1609.00872] [INSPIRE].
J.B. Kogut, An introduction to lattice gauge theory and spin systems, Rev. Mod. Phys. 51 (1979) 659 [INSPIRE].
P. Smacchia et al., Statistical mechanics of the Cluster-Ising model, Phys. Rev. A 84 (2011) 022304 [arXiv:1105.0853] [INSPIRE].
H.D. Chen and Z. Nussinov, Exact results on the Kitaev model on a hexagonal lattice: spin states, string and brane correlators, and anyonic excitations, J. Phys. A 41 (2008) 075001 [cond-mat/0703633].
Z. Nussinov and G. Ortiz, Autocorrelations and thermal fragility of anyonic loops in topologically quantum ordered systems, Phys. Rev. B 77 (2008) 064302 [arXiv:0709.2717].
X.-Y. Feng, G.-M. Zhang and T. Xiang, Topological characterization of quantum phase transitions in a spin-1/2 model, Phys. Rev. Lett. 98 (2007) 087204 [cond-mat/0610626] [INSPIRE].
A. Duncan, Explicit dimensional renormalization of quantum field theory in curved space-time, Phys. Rev. D 17 (1978) 964 [INSPIRE].
E. Barouch and B. McCoy, Statistical mechanics of the XY model. I, Phys. Rev. A 2 (1970) 1075.
T. Hartman, S. Jain and S. Kundu, Causality constraints in conformal field theory, JHEP 05 (2016) 099 [arXiv:1509.00014] [INSPIRE].
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Das, D., Das, S.R., Galante, D.A. et al. An exactly solvable quench protocol for integrable spin models. J. High Energ. Phys. 2017, 157 (2017). https://doi.org/10.1007/JHEP11(2017)157
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DOI: https://doi.org/10.1007/JHEP11(2017)157