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Superluminal moving defects in the Ising spin chain

Alvise Bastianello and Andrea De Luca
Phys. Rev. B 98, 064304 – Published 14 August 2018
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Article Text
  • INTRODUCTION
  • GENERALITIES OF THE ISING MODEL
  • MOVING DEFECT AND THE SCATTERING PROBLEM
  • AN EXACT SOLUTION: THE δ DEFECT
  • SEMICLASSICAL APPROXIMATION
  • FRICTION
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    Quantum excitations in lattice systems always propagate at a finite maximum velocity. We probe this mechanism by considering a defect traveling at a constant velocity in the quantum Ising spin chain in transverse field. Independently of the microscopic details of the defect, we characterize the expectation value of local observables at large times and large distances from the impurity, where a local quasistationary state (LQSS) emerges. The LQSS is strongly affected by the defect velocity: for superluminal defects, it exhibits a growing region where translational invariance is spontaneously restored. We also analyze the behavior of the friction force exerted by the many-body system on the moving defect, which reflects the energy required by the LQSS formation. Exact results are provided in the two limits of extremely narrow and very smooth impurity. Possible extensions to more general free-fermion models and interacting systems are discussed.

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    • Received 9 May 2018

    DOI:https://doi.org/10.1103/PhysRevB.98.064304

    ©2018 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Ballistic transportNonequilibrium statistical mechanicsQuantum quenchQuantum transportTransport phenomena
    1. Physical Systems
    Lattice models in statistical physicsNonequilibrium lattice modelsNonequilibrium systems
    1. Techniques
    Exact solutions for many-body systemsIntegrable systemsIsing model
    Statistical Physics & Thermodynamics

    Authors & Affiliations

    Alvise Bastianello1 and Andrea De Luca2

    • 1SISSA & INFN, via Bonomea 265, 34136 Trieste, Italy
    • 2The Rudolf Peierls Centre for Theoretical Physics, Oxford University, Oxford, OX1 3NP, United Kingdom

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    Vol. 98, Iss. 6 — 1 August 2018

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    Images

    • Figure 1
      Figure 1

      Spreading of the LQSS due to a moving defect (continuous red line), for subluminal (v<vM) and superluminal (v>vM) defects. In both cases, the local magnetization density 〈σ̂z〉 is considered. In the subluminal case, the perturbation is entirely contained within the light cone spreading from the initial position of the defect (dashed lines), while in the superluminal case the system is affected beyond such a light one, following the defect. In both cases, vM=1 and a δ-like defect is considered (see Sec. 4 for notation). Parameters used for v<vM: h=1.15, c=0.5, v=0.5. Parameters used for v>vM: h=1.15, c=2, v=2.

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    • Figure 2
      Figure 2

      Dynamically accessible scattering channels 1→1 and 1→2 (see Sec. 3b for notation). The scattering channels are identified as the solutions of Eq. (31) which are ultimately pulled back to the first Brillouin zone. The magnetic field is set h=1.15, thus vM=1.

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    • Figure 3
      Figure 3

      The analytic LQSS generated by a δ defect is compared with the numeric, in particular the local magnetization 〈σ̂z〉, trivially connected with the fermionic density σ̂jz=1−2d̂j†d̂j, is considered in the subluminal (a) and superluminal (b) cases. Notice the presence of the plateau in the superluminal case, accordingly with the discussion of Sec. 3c. In both cases, the light velocity is vM=1 and the ray ζ is measured with respect to the defect, which is thus located at ζ=0. Parameters (a): h=1.15, c=0.6, v=0.6, 1800 sites. Parameters (b): h=1.15, c=2, v=2, 1100 sites.

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    • Figure 4
      Figure 4

      Energy levels ω(k,x)=const for a transmitted (a) and a reflected (b) classical particle. The classical scattering is fully determined by the energy levels: assuming the particle has initial momentum kIN infinitely far from the defect, the lines at constant level ω(k,x)=ω(kIN,∞) starting from kIN and x=±∞ (depending on the direction of the incoming particle) determined whether the particle is transmitted or reflected (dashed red line). Above, the case of a Gaussian potential V(x)=0.5e−Ax2/2 moving at v=0.5 is considered, with A=0.04 and constant magnetic field h=1.3. The initial momentum kIN has been taken kIN=2 and kIN=1 in (a) and (b), respectively.

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    • Figure 5
      Figure 5

      (a) The analytic semiclassical prediction for the LQSS is compared with the numerics, in the case of the local magnetization 〈σ̂z〉 and for a smooth defect. (b) Within the semiclassical regime, a superluminal defect does not produce a LQSS. Above, the magnetization profile as a function of the lattice sites at a given time t=150. At late time, the system is affected only in the neighborhood of the defect (rightmost peak) where the scattering theory is no longer valid. At finite time, a transient included within the lightcone propagating from the initial position of the defect is displayed. Semiclassically, this transient is due to the quasiparticle initially sat on the defect that experience a sudden change of their energy and therefore of their velocity: a bunch of excitation starts traveling across the system leaving behind a hole. Parameters (a): V(x)=0.5e−Ax2/2 (A=0.04), v=0.5, and h=1.3. At time t=0 the system is initialized in a thermal ensemble with inverse temperature β=0.5, 1600 sites are used. Parameters (b): V(x)=e−Ax2/2 (A=0.04), v=3, and h=1.3. At time t=0 the system is initialized in a thermal ensemble with inverse temperature β=0.5, 800 sites are used.

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    • Figure 6
      Figure 6

      (a) The mean friction force F¯ is plotted as a function of the defect's velocity v for a δ-like defect, showing excellent agreement between analytic and numerics. We considered velocities v>0.2 since the increasing number of singularities in Eq. (39) makes difficult its practical evaluation for v→0. Parameters: h=1.15, c=1, the initial state is the ground state. (b) The numerical mean friction force F¯ for large velocities in the case of a δ-like and a smooth defect, their normalization has been chosen in such a way they share the same asymptotic decay (47) (dashed black line). Parameters: h=1.15, the strength of the δ defect is set c=1. The smooth defect is chosen as δh(x)=0.56e−0.98x2. The system has been initialized in the ground state.

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    • Figure 7
      Figure 7

      Oscillations of local observables can be present in proximity of the defect due to finite-time/-size effects. Above, we numerically compute the local magnetization 〈σ̂z〉. The same parameters of Fig. 3 have been used, but at finite time t≃850. The defect, activated in j=0 at time t=0, is now placed nearby site j≃500 where the discontinuity occurs.

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