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Universal equilibrium scaling functions at short times after a quench

Markus Karl, Halil Cakir, Jad C. Halimeh, Markus K. Oberthaler, Michael Kastner, and Thomas Gasenzer
Phys. Rev. E 96, 022110 – Published 7 August 2017
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  • ACKNOWLEDGMENTS
  • APPENDICES
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    Abstract

    By analyzing spin-spin correlation functions at relatively short distances, we show that equilibrium near-critical properties can be extracted at short times after quenches into the vicinity of a quantum critical point. The time scales after which equilibrium properties can be extracted are sufficiently short so that the proposed scheme should be viable for quantum simulators of spin models based on ultracold atoms or trapped ions. Our results, analytic as well as numeric, are for one-dimensional spin models, either integrable or nonintegrable, but we expect our conclusions to be valid in higher dimensions as well.

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    • Received 5 May 2017
    • Revised 9 July 2017

    DOI:https://doi.org/10.1103/PhysRevE.96.022110

    ©2017 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Magnetic phase transitionsNonequilibrium statistical mechanicsQuantum phase transitionsQuantum statistical mechanics
    1. Physical Systems
    Quantum spin models
    1. Techniques
    Density matrix renormalization groupIntegrable systemsIsing model
    Statistical Physics & Thermodynamics

    Authors & Affiliations

    Markus Karl1, Halil Cakir1,*, Jad C. Halimeh2, Markus K. Oberthaler1, Michael Kastner3,4, and Thomas Gasenzer1,†

    • 1Kirchhoff-Institut für Physik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany
    • 2Physics Department and Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität München, D-80333 München, Germany
    • 3National Institute for Theoretical Physics (NITheP), Stellenbosch 7600, South Africa
    • 4Department of Physics, Institute of Theoretical Physics, University of Stellenbosch, Stellenbosch 7600, South Africa

    • *Present address: Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany.
    • †t.gasenzer@uni-heidelberg.de

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    Vol. 96, Iss. 2 — August 2017

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    • Figure 1
      Figure 1

      Order-parameter correlations Czz(t,ℓ) of the one-dimensional transverse-field Ising model as a function of the spin-spin distance ℓ, for various fixed times t after a quench from h0=∞ to ɛ=h−1=0.001 (left) and ɛ=0.1 (right). Time is given in units of 1/J=1. A quasistationary short-distance falloff of the correlations is reached within a few times 1/J.

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    • Figure 2
      Figure 2

      Order-parameter correlation functions and correlation lengths after a quench from h0≫1 into the vicinity of the quantum phase transition of the one-dimensional transverse-field Ising model. (a) Correlation function |Czz| as a function of the spin-spin separation ℓ at two different times t after the quench from h0=103 to ɛ=h−1=0.1. Circles mark the exact solution obtained by diagonalisation in terms of Bogoliubov fermions while the solid line shows the approximate function (3). Note the logarithmic scale. The envelope of the correlations shows approximately exponential decay, |Czz|∼exp(−ℓ/ξi), with different correlation lengths ξi, i=1,2, below and above a characteristic scale ℓ1(t), see main text. (b) As in (a) but for a quench to ɛ=0.001. (c) Inset: Time evolution of ξ1, obtained by fitting exponential functions to the short-distance falloff of |Czz| for finite spin chains. Data for four different quenches are shown, with final quench parameters ɛ=1.0 (blue), 0.1 (green), 0.01 (red), and 0.001 (yellow). Main graph: Correlation length ξ1 at t=10, as a function of the postquench parameter ɛ. Red circles are obtained from exponential fits to the data, the blue and green lines are the analytical expressions (10) for ξ1 and (A2) for ξ2, respectively. ξT marks the thermal correlation length (13) for an effective temperature T=1.58. (d) Analytic results for the transverse-field Ising model in the scaling limit. Main graph: Asymptotic postquench correlation lengths ξ1,2(t→∞;Δ) (blue and green lines, respectively) as given in Eqs. (18) and (A3), after a quench from Δ0=1 to Δ. The black dashed line shows the thermal correlation length ξT at an effective temperature T=Teff=2Δ0/π. This choice of temperature ensures that ξTeff and the postquench correlation lengths ξ1,2(t) coincide at the critical point, i.e., ξTeff=limt→∞ξ1,2(t) for Δ=0. The inset shows that, for Δ/Δ0≪1, the same linear corrections apply to ξ1 and ξTeff.

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    • Figure 3
      Figure 3

      Occupation numbers of the Bogoliubov fermions as functions of the quasimomentum k after a quench from h0=103 to three different h=1+ɛ close to the critical value at ɛ=0, marked by solid lines. The black dashed line marks the thermal occupation number for h=1 and an effective temperature T=2, cf. Eq. (12), while the dotted line is the same distribution at T=1.58.

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    • Figure 4
      Figure 4

      Sketch of the ground-state phase diagram of the 1D Heisenberg XXZ chain, after Ref. [69]. The anisotropy γ and the dimensionless transverse field strength h parametrize the antiferromagnetic XXZ Hamiltonian (24). At γ=1 one recovers the Heisenberg XXX chain in a transverse field. Lines and tricritical points separate the equilibrium phases as indicated. We study quenches at fixed γ=0, 2, from ground states at large h into the vicinity of the quantum phase transitions at h≈1.5 and h≈2.5, respectively, as marked by the red arrows.

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    • Figure 5
      Figure 5

      Left and center: Order parameter correlations of the XXZ chain at various times t after quenches from h0≫1 into the vicinity of a quantum critical point. The dashed lines are fits to the exponential decays at short distances ℓ. Data are for anisotropy γ=0 and postquench parameter h=1.54 (left) and for anisotropy γ=2 and postquench parameter h=2.55 (center). Right: Correlation lengths ξ extracted by means of exponential fits to the correlation functions. The distances ɛ=h/hc−1 from the critical point are calculated with hc=1.5 for γ=0 and hc=2.53 for γ=2. The overall shape of ξ(ɛ) is consistent with the findings for the one-dimensional TFIM [blue line in Fig. 2].

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    • Figure 6
      Figure 6

      Time dependence of the limiting scale ℓ1(t) for three different quenches, from a large value of h0=103 to ɛ=0.1 (red), ɛ=0.01 (green), and ɛ=0.001 (blue), on a semilogarithmic scale. ℓ1(t) is extracted as the distance ℓ where the length scales defined by the two terms in Eq. (3), the asymptotic exponential ∼exp{−ℓ/ξ1} and the transient integral contribution, intersect; see inset. The dashed line marks the analytically determined growth ℓ1(t)∼(3ξ1/2)ln(t/t0) (cf. Eq. (37) of Ref. [64]) for quenches to ɛ≳0.1. After an initial period where the exponential correlations build up, the transition scale ℓ1 oscillates, with envelope increasing logarithmically in time, the slower, the smaller ξ1 is.

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    • Figure 7
      Figure 7

      Order-parameter correlation functions and correlation lengths after a quench from h0=1.1 into the vicinity of the quantum phase transition of the one-dimensional transverse-field Ising model. (a) Correlation function |Czz| on a log scale as a function of the spin–spin separation ℓ at two different times t after the quench from h0=1.1 to ɛ=h−1=0.01. Circles mark the exact solution obtained by diagonalization in terms of Bogoliubov fermions, while the solid line shows the approximate function (3). (b) As in (a) but for a quench to ɛ=0.001. (c) Inset: Time evolution of the correlation length ξ1, obtained by fitting exponential functions to the short-distance (ℓ∈{21,⋯,30}) falloff of |Czz| for finite spin chains. Data for four different quenches are shown, with final quench parameters ɛ=0.01 (blue), 0.001 (green), 0.0001 (red), and 0.00001 (yellow). Main graph: Correlation length ξ1 as a function of the postquench parameter ɛ. Circles are obtained from exponential fits to the data for ℓ∈{21,⋯,30} at times t=40 (red) and t=140 (blue circles). The blue line is the analytical expression (10) for ξ1 in the thermodynamic limit. ξT marks the thermal correlation length (13) for an effective temperature T=0.12. The total number of spins in the periodic chain in the numerical computations is N=800, which is large enough to avoid finite-size effects. (d) Correlation length ξ1 for quenches from different h0 to h=1 for a chain of N=800 spins at time t=140 after the quench (red circles). The blue solid line depicts the asymptotic behavior (t→∞) in the thermodynamic limit.

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    • Figure 8
      Figure 8

      Occupation numbers of the Bogoliubov fermions as functions of the quasimomentum k after a quench from h0=1.1 to three different h=1+ɛ close to the critical value at ɛ=0, marked by solid lines. The black dashed line marks the thermal occupation number for h=1 and an effective temperature T=0.1≃2(h0−1)/(h0+1), while the dotted line is the same distribution at T=0.12, leading to the same ɛ→0 limit of the postquench and thermal correlation length, as seen in Fig. 7.

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