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Impact of nonequilibrium fluctuations on prethermal dynamical phase transitions in long-range interacting spin chains

Alessio Lerose, Bojan Žunkovič, Jamir Marino, Andrea Gambassi, and Alessandro Silva
Phys. Rev. B 99, 045128 – Published 16 January 2019
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Abstract
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Article Text
  • INTRODUCTION
  • DYNAMICAL PHASE TRANSITION IN THE…
  • STATIC AND DYNAMICAL SPIN-WAVE…
  • IMPACT OF THE SPIN WAVES ON THE…
  • STRONG INTERACTIONS
  • FINITE-SIZE EFFECTS
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
    • References

    Abstract

    We study the nonequilibrium phase diagram and the dynamical phase transitions occurring during the prethermalization of nonintegrable quantum spin chains, subject to either quantum quenches or linear ramps of a relevant control parameter. We consider spin systems in which long-range ferromagnetic interactions compete with short-range, integrability-breaking terms. We capture the prethermal stages of the nonequilibrium evolution via a time-dependent spin-wave expansion at leading order in the spin-wave density. In order to access regimes with strong integrability breaking, instead, we perform numerical simulations based on the time-dependent variational principle with matrix product states. By investigating a large class of quantum spin models, we demonstrate that nonequilibrium fluctuations can significantly affect the dynamics near critical points of the phase diagram, resulting in a chaotic evolution of the collective order parameter, akin to the dynamics of a classical particle in a multiple-well potential subject to quantum friction. We also elucidate the signature of this novel dynamical phase on the time-dependent correlation functions of the local order parameter. We finally establish a connection with the notion of dynamical quantum phase transition associated with a possible nonanalytic behavior of the return probability amplitude, or Loschmidt echo, showing that the latter displays cusps whenever the order parameter vanishes during its real-time evolution.

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    • Received 25 July 2018
    • Revised 7 December 2018

    DOI:https://doi.org/10.1103/PhysRevB.99.045128

    ©2019 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Dynamical phase transitionsNonequilibrium statistical mechanicsQuantum quench
    1. Physical Systems
    1-dimensional spin chains
    1. Techniques
    Holstein-Primakoff methodQuantum spin chains
    Statistical Physics & ThermodynamicsCondensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Alessio Lerose1,2, Bojan Žunkovič3, Jamir Marino4,5, Andrea Gambassi1,2, and Alessandro Silva1

    • 1SISSA–International School for Advanced Studies, via Bonomea 265, I-34136 Trieste, Italy
    • 2INFN–Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, I-34136 Trieste, Italy
    • 3Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia
    • 4Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
    • 5Department of Quantum Matter Physics, University of Geneva, 1211, Genève, Switzerland

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    Vol. 99, Iss. 4 — 15 January 2019

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    • Figure 1
      Figure 1

      Classical energy landscape (5) of the collective spin σ⃗ of the LMG model along the plane σy≡0 as a function of the magnetization σx, in the ferromagnetic phase 0<g<gcr≡2λρ. The location of the two symmetric minima is determined by Eq. (7). In the thermodynamic limit, the degenerate ground-state wave functions of the collective spin are localized at the two classical minima, respectively, and σ⃗ behaves like a classical particle at rest at the bottom of one of the two wells (e.g., black dot in the figure). At finite size, however, quantum tunneling induced by the presence of the other well occurs over an exponentially long time scale (see Sec. 2c).

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    • Figure 2
      Figure 2

      Left panel: equilibrium order parameter σx of the infinite-range Ising model at zero temperature as a function of the external field g, determined by Eq. (7). Right panel: frequency ω<,> of small oscillations of the collective spin around the minimum [see Eqs. (11) and (14)], equal to the energy gap above the ground state. In both cases, the critical behavior is characterized by a square-root singularity.

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    • Figure 3
      Figure 3

      Classical energy landscapes (5) of the collective spin σ⃗ of the LMG model in the plane σy≡0 as a function of the magnetization σx in the ferromagnetic phase, with a postquench value g such that 0<g<gcr (black solid line) and several possible prequench values g0 such that 0<g0<g (blue, red, and green dashed lines) of the transverse magnetic field. If the system is prepared in a ground state, e.g., with positive magnetization as illustrated by the blue, red, and green dots for decreasing values of g0, and the magnetic field is suddenly quenched to a larger value g0<g<gcr, then depending on the strength g−g0 of the quench, the resulting nonequilibrium evolution may display dynamical ferromagnetic or paramagnetic behavior, exemplified by the blue and green lines, respectively, separated by a critical trajectory with a diverging period, corresponding to the red line and associated with the dynamical critical point g=gdyn. In contrast to Fig. 4, here the various resulting evolutions correspond to varying the prequench parameter g0, with a fixed postquench value g.

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    • Figure 4
      Figure 4

      Nonequilibrium dynamics of the LMG model (2) in the thermodynamic limit, after a sudden quench g0→g of the transverse magnetic field starting from a ferromagnetic ground state of H(g0). The first row shows the semiclassical phase portrait of the prequench Hamiltonian Hcl(g0), where the initial state is represented by one of the two minima. The second row shows the semiclassical phase portrait of the postquench Hamiltonian Hcl(g), where the initial state is no longer a stationary point but moves along a nontrivial nonequilibrium trajectory, in the three qualitatively different cases corresponding to g<gdyn,g=gdyn and g>gdyn in the first, second, and third columns, respectively. The third row shows the dynamics of the order parameter as a function of time for the three cases. First column: for a weak quench, the dynamics remains trapped within the starting ferromagnetic sector; second column: for the critical quench, the initial state lies on a separatrix of the postquench Hamiltonian and its subsequent evolution approaches the unstable equilibrium point at infinite time; third column: for a strong quench, the semiclassical orbit encircles both ferromagnetic minima, hence, the symmetry is dynamically restored and the time-averaged order parameter is zero. In contrast to Fig. 3, here the different trajectories correspond to a varying postquench parameter g, with a fixed prequench value g0.

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    • Figure 5
      Figure 5

      Left panel: nonequilibrium order parameter σx¯, defined in Eq. (18), of the infinite-range Ising model (2) after a quench of the external magnetic field starting from a ferromagnetic ground state with g0=0 and positive magnetization, as a function of the postquench field g. Right panel: classical frequency Ωcl of the mean-field dynamical trajectory, which represents the characteristic timescale of the nonequilibrium evolution, as a function of the postquench field g. For both quantities, the nature of the singular behavior at the dynamical critical point g=gdyn is logarithmic, as explained in the text. These plots can be compared with the analogous ones in equilibrium conditions in Fig. 2.

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    • Figure 6
      Figure 6

      Convergence to the classical behavior in the thermodynamic limit N→∞ of the quantum dynamics governed by the Hamiltonian (2) with finite size N and with s=12. This is studied via exact diagonalization in the maximal spin sector. Left panels: evolution of the dynamical order parameter with g/λ=0.9 (top), g/λ=1.1 (center), g/λ=1.7 (bottom), and increasing system size N=16,64,256, starting from a fully polarized state along the x̂ direction, i.e., from a ground state with g0=0. The classical limit is shown by the black dashed curve. Right panels: corresponding infinite-time average distribution of the order parameter, as obtained from the diagonal ensemble pde(m)=|Ψ(t)|m|2¯, where m is the state with magnetization m and the overline stands for infinite-time average. The classical “microcanonical” distributions, obtained by averaging over the trajectory of Hcl with energy E=〈ψ0|H|ψ0〉/N=−λ, are shown by the black dashed curve. Note that the quantum evolution agrees with its classical limit over a time window that increases with N. After this time, quantum phenomena emerge. In all cases, damping of the classical oscillations takes place as a consequence of the quantum spreading of the wave packet. Furthermore, for system sizes N as small as 16, additional quantum effects become observable. In the top left panel, quantum tunneling to the opposite well can be observed in the dynamical ferromagnetic phase at relatively small time, which scales as Ttun=O(ecN); note that the corresponding infinite-time distribution of the magnetization is suppressed in the classically forbidden region m≈0 as N→∞. In the center left panel, a remnant of ferromagnetic behavior can be observed in the dynamical paramagnetic phase, due to contributions to the wave packet coming from ferromagnetic initial conditions (in order to visualize this, one should replace the small black dot in Fig. 4 with an extended circle of radius 1/N). In the bottom left panel, recurrences in the evolution of the order parameter emerge at relatively small time Trec=O(N), due to wave-packet refocusing after spreading. All these three effects occur at larger times for N=64,256, and thus do not appear in the relative plots.

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    • Figure 7
      Figure 7

      Dynamical phase diagram of the model in Eq. (20) after a quench of the magnetic field g0=0→g starting from the fully polarized ground state with positive magnetization, as a function of g and J. Here, N=100. As energy scale we choose λ¯≡λ+J. The color of each point of the diagram is determined by the sign of the long-time average σx¯ of σx(t): light yellow for σx¯>0, orange for σx¯=0, and blue for σx¯<0. Regions A and B are perturbative extensions of the dynamical ferromagnetic and paramagnetic phases of the LMG model with J=0, corresponding to the horizontal axis (see Fig. 8 for an illustration of the dynamics within A and B). Upon increasing J at fixed g, in a neighborhood of the mean-field critical point g=λ¯, a new chaotic dynamical ferromagnetic phase C emerges, within which the magnetization σx(t), after an initial dynamical paramagnetic behavior, gets trapped in one of the two symmetry-broken sectors with opposite signs of the collective magnetization [process (a) in the inset], in some cases followed by hopping between them [process (b) in the inset] (see Fig. 9 for an illustration of the dynamics within C). The extent and features of the three phases A, B, C are stable as N is increased.

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    • Figure 8
      Figure 8

      Dynamical behavior of the order parameter σx(t) (first row) and of the spin-wave density ε(t) (second row) in the presence of a short-range interaction J/λ¯=0.1 (solid red line) and 0.2 (dashed blue line), after a quench from a fully polarized ferromagnetic state (g0=0). Left panels: dynamical ferromagnetic phase with g/λ¯=0.9. Right panels: dynamical paramagnetic phase with g/λ¯=1.5. These dynamical phases are characterized by the sign of the time average of σx(t), shown in the top panels. The quantity ε(t) shown in the bottom panels represents the total amount of spin-wave excitations generated during the nonequilibrium evolution. This is the control parameter for the validity of the low-density expansion, which is consistent if ε≪1, i.e., if the length of the total spin |σ⃗(t)|=1−ε(t) remains close to its maximal value. The presence of a short-range interaction, even of sizable strength J/λ¯=0.2, produces a perturbative modification of the mean-field evolution and, correspondingly, a small amount of spin waves. In particular, the mean-field persistent oscillations are not damped by the self-generated “bath.” In the plots, N=100 and the mean-field dynamical critical point is gdyn/λ¯=1.

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    • Figure 9
      Figure 9

      Evolution of the order parameter σx(t) within the chaotic dynamical ferromagnetic phase C in Fig. 7, for g/λ¯=1.03 (solid red line) and 1.031 (dashed blue line), and with J/λ¯=0.1. Here, N=100. The two lines are practically indistinguishable during the initial paramagnetic transient, but they have markedly distinct fates at the onset of the critical process denoted by (a) in the inset of Fig. 7 and they eventually end up into distinct wells. In both cases, ε(t) grows from ε(t=0)=0 to values around 0.04. This extreme sensitivity on the value of g (and of J) is at the origin of the “mosaic” structure of region C in Fig. 7.

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    • Figure 10
      Figure 10

      Evolution of the order parameter σx(t) after a quench from a pure ferromagnetic state (g0=0) in four different generalizations of the Ising Hamiltonian (20). Top left: XY spin chain with an infinite-range and a nearest-neighbor interaction, defined by Eq. (63) with αy=0.25,αz=0,g/λ¯=1.03 (solid red line) and 1.032 (dashed blue line), with J/λ¯=0.4. Top right: XYZ spin chain with an infinite-range and a nearest-neighbor interaction, defined by Eq. (63) with αy=0.25,αz=0.125,g/λ¯=0.9 (solid red line) and 0.902 (dashed blue line), with J/λ¯=0.4. Bottom left: Ising spin chain with an infinite-range and a next-to-nearest-neighbor interaction, defined by Eq. (64) with v(r)=δr,1+0.5δr,2,g/λ¯=1.03 (solid red line) and 1.031 (dashed blue line), with J/λ¯=0.2. Bottom right: Ising spin chain with an infinite-range and a power-law decaying interaction, defined by Eq. (64) with v(r)=1/r2,g/λ¯=1.03 (solid red line) and 1.031 (dashed blue line), with J/λ¯=0.2. In all simulations, N=100. These trajectories have been obtained by numerically integrating the evolution equations given by the time-dependent spin-wave theory, analogous to Eqs. (50) and (54), derived for the generalized spin chains above through the same procedure as that explained in details in Sec. 3b for the Ising model.

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    • Figure 11
      Figure 11

      Space-time density plots of the dynamical correlation function Cxx(r,t) [see Eq. (65)] after a quench of the magnetic field g from a fully polarized ferromagnetic state (g0=0) to g/λ¯=0.7,0.9,1.025,3, in clockwise order from top left. In all plots, J/λ¯=0.25,N=240. For a small quench occurring deep in the ferromagnetic phase (top left), the overall amplitude of the correlation function is weak (few excitations are produced) and the light cone is narrow due to an almost constant spin-wave dispersion relation. The amplitude and the width become larger as the dynamical critical region is approached (top right). In the chaotic dynamical phase (bottom right), a “knee” is visible, marked by the black arrow, witnessing a change of the maximal velocity of propagation due to the trapping of the orbit, after a paramagnetic transient, into a ferromagnetic sector (notice the change of scale, highlighting a larger amplitude of the correlations). Finally, deep in the paramagnetic phase (bottom left), the maximal velocity approaches the value analytically predicted in Eq. (67), indicated by the black line. An approximately periodic modulation of the amplitude of Cxx(r,t) is visible in all cases, which reflects the approximately periodically driven nature of the spin waves, induced by the precession of the collective spin.

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    • Figure 12
      Figure 12

      Long-time behavior of the correlation function C(ξt,t) along two close space-time rays with fixed ξ=0.5λ¯ (left) and 0.55λ¯ (right) in a log-log scale, after a quench of the magnetic field from a pure ferromagnetic state (g0=0) to g/λ¯=0.9 with J/λ¯=0.25,N=240, corresponding to the data of the top right panel of Fig. 11. Apart from an (approximately) periodic modulation, C(ξt,t) decays, in the large space-time limit, as a power law (left) or as an exponential (right) as a function of time. The red line highlights the t−1/2 decay suggested by the argument in the text [see Eq. (69)]. Data are consistent with a maximal velocity of propagation of the effective free quasiparticles between 0.25λ¯ and 0.275λ¯ in this specific numerical instance.

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    • Figure 13
      Figure 13

      Dynamical phase diagrams for linear ramps [see Eq. (70)] of the transverse magnetic field for the model described by Eq. (20) starting from g0=0 (fully polarized ferromagnetic state), in the plane of the dimensionless final magnetic field g/λ¯ and short-range interaction strength J/λ¯, analogous to Fig. 7. Here, N=100. The color of each point in the diagrams indicates the asymptotic sign of the time-averaged order parameter, with the same graphical conventions as in Fig. 7. The dimensionless duration λ¯τ of the ramp is 0.7 (left), 1.00 (middle), 1.15 (right). As the driving becomes slower, the mean-field dynamical critical point for J→0 shifts from the sudden quench value gdyn/λ=1 towards that in the adiabatic limit, i.e., the equilibrium critical point gcr/λ=2 which is witnessed by the progressive shift rightwards along the horizontal axis of the border between the yellow and orange regions in the plot. Simultaneously, the chaotic dynamical ferromagnetic phase shrinks, due to the progressively smaller amount of nonequilibrium excitations produced by the increasingly slower ramp.

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    • Figure 14
      Figure 14

      Left panel: dynamical phase diagram for linear ramps of the magnetic field g of the same model as in Fig. 13, in the plane of the dimensionless final magnetic field g/λ¯, and of the dimensionless ramp duration λ¯τ, but with fixed J/λ¯=0.2. The color of each point of the diagram is assigned as in Figs. 13 or 7, and the diagram corresponds to taking a horizontal cut of those in Fig. 13 at fixed J/λ¯=0.2 and varying τ continuously. As the ramp becomes slower, we notice two features: first, the two boundaries of the chaotic phase shift from the sudden quench position around gdyn/λ¯=1 towards the equilibrium critical point gcr/λ¯=2−(5/8)(J/λ¯)2+O(J/λ¯)3≈1.975 [see Eq. (43)] in the adiabatic limit, marked by the black vertical line. Second, the chaotic dynamical phase shrinks and practically disappears as τ is increased. Both these features are clearly visible in the picture. The “oscillatory” dependence of the phase boundary on τ is already present at the mean-field level. Right panel: long-time average of the density ε(t) of spin-wave excitations generated in the nonequilibrium dynamics.

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    • Figure 15
      Figure 15

      Time-averaged order parameter σx¯ of the model of Eq. (61) as a function of the postquench values g of the transverse field, for various values of the coupling J around 0.5, with λ¯=λ+J=1. The black crosses (J=0.58) correspond to a finer grid of values of g, with δg=0.008, are shown in order to display the high sensitivity of the chaotic phase to postquench parameters. These data show that the dynamically ferromagnetic and chaotic region persist also at large nearest-neighbor interactions. The data are calculated for system size N=200.

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    • Figure 16
      Figure 16

      Comparison between the evolution of the order parameter σx at large J (solid lines) with those of the mean-field model with J=0 (dashed lines) far from the critical region and for the same model as in Fig. 15. We observe that the evolution corresponding to both the ferromagnetic and paramagnetic phases is not altered qualitatively by the effects of quantum fluctuations. The decay of the oscillations' amplitude upon increasing time is a finite-size effect. In these simulations, N=400 and D=300.

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    • Figure 17
      Figure 17

      Stability of the flipped ferromagnetic region. We show several trajectories within a wide range of different quench parameters as a part of the same region with a flipped final magnetization. This demonstrates stability of the flipped ferromagnetic region at large nearest-neighbor interactions. Simulations were performed with N=200,D=200.

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    • Figure 18
      Figure 18

      Evolution of the order parameter σx(t) for a quench occurring close to the mean-field dynamical critical point within the chaotic phase. These curves at fixed J/λ¯=0.583 show a sensitive dependence on the value of g, and they may oscillate for a long time before settling eventually in a sector with definite positive or negative order parameter. By changing the quench parameter g/λ¯ only slightly (approximately by 0.08) we observe a large change in the final magnetization which jumps from the positive to the negative sector and finally back to the positive sector. The curves in this plot correspond to the data points indicated by black crosses in Fig. 15. Simulations were performed with N=200, and with D=600 (full lines) and D=500 (dashed lines).

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    • Figure 19
      Figure 19

      Relationship between the vanishing of the order parameter σx and the change of sign of the difference between the probability P1 to return to the initial state and the probability P2 to reach the state with the opposite magnetization. Upper panels: comparison between the order parameter (orange) and the sign of (P1−P2) (blue). Lower panels: evolution of the return probabilities P1 (blue) and P2 (red). In the ferromagnetic region (left) we observe that the order parameter remains close to one, which corresponds to a large difference in the probabilities P1 and P2. The return probability to the initial state P1 remains at all times much larger than the return probability to a state with the opposite magnetization P2. On the other hand, in the paramagnetic region (right panels) the order parameter periodically changes the sign. These changes correspond well with the cusps in the return probability which appear at points where P1−P2 changes its sign. Parameters: D=300,N=400,J/λ¯=0.5,g/λ¯=0.5 (top), 1.5 (bottom).

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    • Figure 20
      Figure 20

      Same plots as in Fig. 19 with different values of the parameters corresponding to the flipped and chaotic ferromagnetic regions. Initially, the zeros of the order parameter correspond precisely to the cusps in the return probability, i.e., to P1=P2. At intermediate times, the quality of this correspondence decreases, due to finite-size effects (and is improved by increasing the system size). (Left) At late times the return probability P2 to the state with the opposite magnetization becomes larger than the return probability P1 to the initial state. This corresponds well to the flipped time-dependent order parameter at late times. (Right) By changing the parameters only slightly, the final magnetization changes its sign, but the correspondence between the zeros of the order parameter and the cusps in the return probability remains valid. Parameters: D=600,N=200,J/λ¯=0.583,g/λ¯=1.15 (top), 1.158 (bottom).

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    • Figure 21
      Figure 21

      Nonequilibrium phase diagram for quenches with fixed postquench parameters g/λ¯=1.03,J/λ¯=0.25 as a function of the direction on the Bloch sphere of the prequench fully polarized spin-coherent initial state, parametrized by the canonically conjugated phase-space coordinates ϕ0 and cosθ0 (cf. Sec. 3c). As in Fig. 7, this plot is obtained via numerically integrating the evolution equations of the time-dependent spin-wave theory in the thermodynamic limit, with the same graphical conventions thereof. The lowest-order finite-size correction consists in replacing a classical, uncertainty-free initial condition, specified by (ϕ0,cosθ0), with a Gaussian wave packet in phase space centered around it, with linear extension ℏeff=1/Ns, which takes into account the quantum uncertainty at the lowest order in the semiclassical expansion, as discussed in Sec. 2c. The circles superimposed to the diagram indicate the width of these Gaussian distributions centered around (0,0) for various values of N, corresponding to quenches from a ground state of the prequench Hamiltonian with g0=0 considered in all simulations reported in this work. We see that for N≲102 the corresponding wave packet encompasses initial conditions eventually belonging to all possible phases of the model. Accordingly, one expects the chaotic dynamical phase to be blurred by these quantum fluctuations when N is sufficiently small. This effect is more severe when N is in the range ≲16 accessible to full exact diagonalization of the Hamiltonian (61), which makes it hard to observe signatures of the chaotic dynamical phase via this exact method. The latter is observed in MPS-TDVP simulations with N in the range 102–103 and stronger perturbation J/λ¯≈0.5, as reported in Sec. 5, in correspondence of which the extension of the regions with a uniform sign of the asymptotic magnetization becomes sufficiently large compared to the coarse-graining scale 1/N.

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