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Low-energy effective theory of nonthermal fixed points in a multicomponent Bose gas

Aleksandr N. Mikheev, Christian-Marcel Schmied, and Thomas Gasenzer
Phys. Rev. A 99, 063622 – Published 24 June 2019
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  • INTRODUCTION
  • LOW-ENERGY EFFECTIVE THEORY
  • KINETIC THEORY AND SCALING ANALYSIS
  • NUMERICAL SIMULATIONS FOR N=3 and d=3
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    Nonthermal fixed points in the evolution of a quantum many-body system quenched far out of equilibrium manifest themselves in a scaling evolution of correlations in space and time. We develop a low-energy effective theory of nonthermal fixed points in a bosonic quantum many-body system by integrating out long-wavelength density fluctuations. The system consists of N distinguishable spatially uniform Bose gases with U(N)-symmetric interactions. The effective theory describes interacting Goldstone modes of the total and relative-phase excitations. It is similar in character to the nonlinear Luttinger-liquid description of low-energy phonons in a single dilute Bose gas, with the markable difference of a universal nonlocal coupling function depending, in the large-N limit, only on momentum, single-particle mass, and density of the gas. Our theory provides a perturbative description of the nonthermal fixed point, technically easy to apply to experimentally relevant cases with a small number of fields N. Numerical results for N=3 allow us to characterize the analytical form of the scaling function and confirm the analytically predicted scaling exponents. The predicted and observed exponentially suppressed coherence at short distances takes the form of that of a quasicondensate in low-dimensional equilibrium systems. The fixed point which is dominated by the relative phases is found to be Gaussian, while a non-Gaussian fixed point is anticipated to require scaling evolution with a distinctly lower power of time.

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    • Received 12 March 2019
    • Revised 15 May 2019

    DOI:https://doi.org/10.1103/PhysRevA.99.063622

    ©2019 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Bose gasesCold atoms & matter wavesCritical exponentsEffective field theoryNonequilibrium statistical mechanicsQuantum kinetic theoryQuantum quench
    1. Physical Systems
    Nonequilibrium systems
    1. Techniques
    Nonequilibrium Green's functionPath-integral methods
    Atomic, Molecular & OpticalParticles & FieldsCondensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Aleksandr N. Mikheev1,*, Christian-Marcel Schmied1,2,†, and Thomas Gasenzer1,‡

    • 1Kirchhoff-Institut für Physik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany
    • 2Dodd-Walls Centre for Photonic and Quantum Technologies, Department of Physics, University of Otago, Dunedin 9016, New Zealand

    • *aleksandr.mikheev@kip.uni-heidelberg.de
    • †christian-marcel.schmied@kip.uni-heidelberg.de
    • ‡t.gasenzer@uni-heidelberg.de

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    Issue

    Vol. 99, Iss. 6 — June 2019

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    • Figure 1
      Figure 1

      (a) Universal scaling dynamics of the single-component occupation number n1(k)≡n1(k,t) according to (2), for a (d=3)-dimensional gas with N=3 components. For details of the truncated Wigner simulations, see Sec. 4. The time evolution is starting from the initial distribution n1(k,t0)=n0Θ(kq−|k|), identical in all three components a (gray solid line), with n0=(4πkq3)−1ρ(0), kq=1.4kΞ (kΞ=Ξ−1=[2mρ(0)g]1/2). Colored dots show n1 at five different times. (b) The collapse of the data to the universal scaling function fS,1(k)=n1(k,tref), with reference time tref/tΞ=31 (units of tΞ−1=gρ(0)/2π), shows the scaling (2) in space and time. For the time window tref=200tΞ<t<350tΞ, we extract exponents α=1.62±0.37, β=0.53±0.09. (c) Universal scaling dynamics of the single-component first-order coherence function g1(1)(r)=g1(1)(r,t) (colored dots), for the same system and the same color encoding of t as in (a) and (b). At short distances r≳Ξ the first-order coherence function takes an exponential form, reminiscent of a quasicondensate, which is characterized by a correlation length rescaling in time. Note the semilog scale.

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    • Figure 2
      Figure 2

      Scaling exponents α/3 (orange stars) and β (blue dots) obtained from least-square rescaling fits of the occupancy spectra n1(k)≡n1(k,t) shown in Fig. 1. The exponents correspond to the mean required to collapse the spectra within the time window [tref,tref+Δt] with Δt=146tΞ. Error bars denote the least-square-fit error.

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    • Figure 3
      Figure 3

      Diagrammatic representation of the interaction terms.

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    • Figure 4
      Figure 4

      Zero-momentum mode of the single-component occupation number n1(k=0,t) (blue dots). According to (2) the universal time evolution is given by n1(k=0,t)=(t/tref)αfS,1(k=0). At late times, t≳200tΞ, we find α≃d/2=32 (black solid line) consistent with the analytical prediction within our low-energy EFT.

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    • Figure 5
      Figure 5

      (a) Enlarged representation of the infrared scaling evolution of the single-component occupation number n1(k)≡n1(k,t), for the same evolution times as shown in Fig. 1 (same color coding). The solid black and dashed gray lines show the results obtained by fitting the corresponding scaling functions to the IR part of the distribution. At late times (cyan and purple), the data are close to the scaling function n1(k,t), defined in (130), which corresponds to a first-order coherence function with exponential times cardinal-sine form (131). For all evolution times, the data differ from the scaling function n1G(k,t) defined in (117) which corresponds to the purely exponential first-order coherence function (116). This shows that, although we observe an exponential decay at short distances in the first-order coherence function, an additional oscillatory contribution is present (see [75] for more details). Note that we do not claim (130) to be the precise scaling form. (b) n1(k)−1−n1(0)−1, with the respective extrapolated fit value inserted for n1(k=0) in order to be independent of possible deviations due to the buildup of a condensate in the zero mode. This representation clearly shows that the data are better described by the scaling function n1 at late times. The solid black line corresponds to the fit of Eq. (130) to the data for the latest time shown (purple dots).

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    • Figure 6
      Figure 6

      (a) Universal scaling dynamics of the relative phases C12(k,t)=〈|(Φ1†Φ2)(k,t)|2〉 for a (d=3)-dimensional gas with N=3 components. The time evolution is starting from the initial box distribution. Colored dots show C12 at five different times. (b) The collapse of the data to the universal scaling function fS,C(k)=C12(k,tref), with reference time tref/tΞ=31 (units of tΞ−1=gρ(0)/2π), shows the scaling (87) in space and time. The universal scaling of C12(k,t) confirms our hypothesis that the scaling behavior also affects the relative phases of the components. Note that C12(k,t) does not show a plateau in the IR but goes over to follow a scaling function with similar falloff at higher momenta as the one for na(k,t). The power law C12∼k−4 in the scaling regime is only slightly modified as compared to that of na. (c) Corresponding first-order coherence function of the relative phases g12(1)(r,t) (colored dots), for the same system and the same color encoding of time as in (a) and (b). Similar to the single-component first-order coherence functions we an exponential form of g12(1)(r,t) at short distances r≳Ξ. Note the semilog scale.

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    • Figure 7
      Figure 7

      Scaling exponents α/3 (orange stars) and β (blue dots) obtained from least-square rescaling fits of the occupancy spectra C12(k)≡C12(k,t) shown in Fig. 6. The exponents correspond to the mean required to collapse the spectra within the time window [tref,tref+Δt] with Δt=146tΞ. Error bars denote the least-square-fit error.

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    • Figure 8
      Figure 8

      Hydrodynamic decomposition of the flow pattern encoded in the phase-angle field θa, at an early time as well as for the four evolution times in the universal scaling regime. Shown are momentum distributions n(δ)(k), derived from the decomposition of the kinetic energy density into ɛ(δ)(k)=k2n(δ)(k). The total occupation number ntot (blue dots) is compared to the distributions representing the quantum pressure part n(q) (orange diamonds), the nematic n(n) (green squares), compressible n(c) (red triangles), incompressible n(i) (purple pentagons), and spin n(s) (brown arrows) parts, as well as their sum nkin (pink thin diamonds). The incompressible part arising from vorticity in the system exhibits a power-law decay with n(i)(k)∼k−14/3 (see dotted line). The total occupation number is dominated by the nematic and the spin parts except for momenta deep in the IR regime where the decomposition is expected to fail. The interplay of the dominant parts with the steep power law in the incompressible part leads to an overall power-law decay of ntot(k)∼k−4 (see dashed line) at intermediate momenta.

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    • Figure 9
      Figure 9

      Universal scaling dynamics of the occupation numbers representing the (a) incompressible n(i), (b) nematic n(n), and (c) spin n(s) parts of the hydrodynamic decomposition (see Fig. 8) according to (2). The collapse of the data to the universal scaling functions fS(δ)(k)=n(δ)(k,tref), with reference time tref=31tΞ, shows, in each case δ=i,n,s, the scaling (2) in space and time. The universal scaling exponents α/3 (orange stars) and β (blue dots) obtained by means of a least-square fit to the data within the time window [tref,tref+Δt] with Δt=146tΞ are depicted in the insets of each panel. For late times we find that the scaling exponent β extracted for the nematic and spin parts approaches β≃0.5 while for the incompressible part it settles into β≃0.58. The scaling exponents corresponding to the time evolution of the nematic and spin parts corroborate our hypothesis that the scaling behavior is dominated by the relative phases of the components.

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    • Figure 10
      Figure 10

      Contour summarizing (B9) and (B10).

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    • Figure 11
      Figure 11

      First nontrivial diagrams that contribute to the proper self-energy Σ(x,y). The lines represent full propagators G(x,y), the vertices are introduced in Fig. 3.

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