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Correlation functions in Schwarzian liquid

Yong-Hui Qi, Yunseok Seo, Sang-Jin Sin, and Geunho Song
Phys. Rev. D 99, 066001 – Published 5 March 2019
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  • INTRODUCTION
  • CORRELATION FUNCTIONS FROM SCHWARZIAN
  • QUANTUM LIQUID WITH SCHWARZIAN…
  • HIGH-POINT CORRELATION FUNCTIONS
  • DISCUSSIONS AND CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We analytically study low-temperature universal properties of a class of SYK-type models in the large N limit from the AdS2 gravity dual side in terms of SL(2,R)-invariant Schwarzian action. The quantum correction to the conformal field theory CFT1 two-point correlation function due to the Schwarzian action produces a transfer of degree of freedom from the quasiparticle peak to the Hubbard band in density of states (DOS), a signature strong correlation. In Schwinger-Keldysh (SK) formalism, we calculate higher-point thermal out-of-time order correlation (OTOC) functions, which indicate quantum chaos by having a Lyapunov exponent. Higher-order local spin-spin correlations are also calculated, which can be related to the dynamical local susceptibility of quantum liquids such as the spin liquid in disordered metals, marginal Fermi liquid, non-Fermi liquid, etc.

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    • Received 24 April 2018

    DOI:https://doi.org/10.1103/PhysRevD.99.066001

    Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

    Published by the American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Quantum field theoryQuantum gravitySpontaneous symmetry breaking
    1. Physical Systems
    Quantum chaotic systemsSpin glassesStrongly correlated systems
    1. Techniques
    Finite temperature field theoryNon-Fermi-liquid theorySachdev-Ye-Kitaev model
    Particles & FieldsGravitation, Cosmology & AstrophysicsCondensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Yong-Hui Qi1,2,*, Yunseok Seo1,3,†, Sang-Jin Sin1,‡, and Geunho Song1,§

    • 1Department of Physics, Hanyang University, Seoul, 04763, Korea
    • 2Center for High Energy Physics, Peking University, Beijing, 100871, Peoples Republic of China
    • 3GIST College, Gwangju Institute of Science and Technology, Gwangju 500-712, Korea

    • *yhqi@pku.edu.cn
    • †yseo@gist.ac.kr
    • ‡sjsin@hanyang.ac.kr
    • §sgh8774@gmail.com

    See Also

    Topological non-Fermi liquid

    Rong-Gen Cai, Yong-Hui Qi, Yue-Liang Wu, and Yun-Long Zhang
    Phys. Rev. D 95, 124026 (2017)

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    Vol. 99, Iss. 6 — 15 March 2019

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    • Figure 1
      Figure 1

      (a) Feynman rules for two-point functions of scalars (dashed line) G(t1,t2) and the corrections from gravitational soft mode (double wave lines) for B1(t12) and B2(t12) as shown in Eq. (2.42). (b) Feynman diagrams for two-point correlation functions G2(t1,t2) of scalar fields with loop corrections from soft modes as shown in Eq. (2.17).

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    • Figure 2
      Figure 2

      Typical Feynman diagrams for odd-point matter correlation functions of matter fields (dashed line) with loop corrections from soft modes (wave lines) as shown in Eq. (2.17): (a) Three-point correlation functions G3(t1,t2,t3) with the corrections from gravitational soft mode to C1(t123) and C2(t123) as shown in Eq. (2.51); a six-point function G6(t123456)∝λ32ε2⟨C1(t123)C1(t456)⟩ diagram is also shown on the right; (b) Five-point correlation functions G5(t12345)∝λ3ε2⟨C1(t123)B1(t45)⟩; (c) Seven-point correlation functions G7(t1234567)∝λ3ε4⟨C2(t123)B1(t45)B1(t67)⟩; (d) Ninepoint correlation functions G9(t123456789)∝λ3ε5⟨C2(t123)B1(t45)B1(t67)B1(t89)⟩.

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    • Figure 3
      Figure 3

      Dynamical susceptibility or retarded Green’s functions of Schwarzian MFL with Δ=1/4: χ¯(ω)=−GR(ω)/π given in Eq. (3.39): (a) Evolution with coupling strength (2πC)−1: C=1/3π (blue/purple solid line), C=1/2π (cyan/magenta dashed line); C=1/π (green/orange dashed line) and C=+∞ (black/red dotted line). (b) Evolution with temperature T=β−1: In front of χ¯(ω), we have multiplying a temperature depending factor π/β. For different β: β=2π (blue/purple solid line), β=20π/3 (cyan/magenta dashed line), β=20π (green/orange dashed line); β=200π (black/red dotted line). We have chosen input parameters as β=2π.

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    • Figure 4
      Figure 4

      Dynamical susceptibility or thermal retarded Green’s functions χ(ω)=GR(ω) given in Eq. (3.39) for quantum liquid with (solid line) or without (dashed line) Schwarzian correction. For Δ=1/4 cases: (a) -Re GR(ω) (blue line); (b) -Im GR(ω) (orange line). We have chosen β=2π and C=1/(2π). Local dynamical spin-spin correlation functions of quantum liquid χloc(2)(ω) as given in Eq. (3.51) in high temperature case with β=2π (solid green/pink line) and low temperature case with β=20π (dashed cyan/magenta line): (c) Re χloc(2)(ω), (d) Im χloc(2)(ω)∼tanh(ωβ/2). To avoid singularity of χloc(2)(ω) at Δ=1/4, we have chosen Δ=1/4−ε with ε=10−3.

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    • Figure 5
      Figure 5

      Susceptibility or thermal retarded Green’s functions χ(ω)=GR(ω) of the quantum liquid with (solid line) or without (dashed line) the Schwarzian correction as given in Eq. (3.39). For Δ=1/3 case: (a)-Re GR(ω) (black line); (b)-Im GR(ω) (red line). We have chosen β=20π and C=1/(2π). Local dynamical spin-spin correlation functions of quantum liquid χloc(2)(ω) as given in Eq. (3.51): (c) Re χloc(2)(ω), (d) Im χloc(2)(ω) at high temperature with β=2π (solid green/pink line) or at low temperature with β=20π (dashed cyan/magenta line).

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    • Figure 6
      Figure 6

      Dynamical susceptibility or retarded Green’s functions of Schwarzian FL with Δ=1/2: χ(ω)=−G¯R(ω)/π is given in Eq. (3.39). (a) Evolution with different coupling strength (2πC)−1: C=1/3π (blue/purple solid lines), C=1/2π (cyan/magenta dashed line); C=1/π (green/orange dashed line) and C=+∞ (black/red dotted line). We have chosen input parameters as β=2π. (b) Evolution with different temperature T: In front of Eq. (3.39), we have multiplying a temperature depending factor π/β. For different T=β−1: β=2π (blue/purple solid lines), β=20π/3 (cyan/magenta dashed line); β=20π (green/orange dashed line); β=200π (black/red dotted line).

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    • Figure 7
      Figure 7

      Retarded Green’s functions GR(ω) of quantum liquid with (solid line) or without (dashed line) Schwarzian correction as given in Eq. (3.39). Re/Im GR(ω) for (a) Δ=1/8 (black or red line); (b) Δ=1/16 (blue or orange line); (c) Δ=1/32 (cyan or magenta line); (d) Δ=1/64 (green or purple line). For all six cases, there are clean signatures of Hubbard band in DOS. Local dynamical susceptibility χ(2)(ω) with different p≡1/Δ (p=8, 16, 32, 64 corresponds to solid, dashed, dot-dashed and dotted lines, respectively): (e) Reχ(2)(ω); (f) Imχ(2)(ω). We have chosen input parameters as β=2π and C=1/(2π). With the increase of p, the density spectral function becomes more central localized at low frequency region, i.e., ω∼0.

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    • Figure 8
      Figure 8

      Feynman diagrams for four-point correlation functions G4(t1,t2,t3,t4) of scalar fields with loop corrections from soft modes, as shown in Eq. (2.44).

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    • Figure 9
      Figure 9

      Schwinger-Keldysh four contour for four-point OTOCs, with time chosen as in Eq. (4.24).

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    • Figure 10
      Figure 10

      The four-point OTOCs in frequency space with Lyapnov exponent as in Eq. (4.26) −GVWVW(4)(ω,β)/π of Schwarzian liquid: maximal chaotic behavior with λL=1 (β=2π) (cyan/purple thick lines) or nonmaximal chaotic behavior with λL=1/2 (β=4π) (green/magnet dashed lines). We have chosen a set of input parameters as Δ=1/4, C=1/(2π).

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    • Figure 11
      Figure 11

      Typical Feynman diagrams for six-point correlation functions G6(t1,…,t6) of scalar fields with loop corrections from soft modes as shown in Eq. (2.44), in TOCs and OTOCs, respectively.

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    • Figure 12
      Figure 12

      Schwinger-Keldysh six contour for six-point OTOCs, with time chosen as in Eq. (4.28). One may entail an infinitesimal Imt52=δ>0 to entail that t2 is earlier than t5 along imaginary time line, and in the end set it to be zero, which does not affect the results.

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    • Figure 13
      Figure 13

      Typical Feynman diagrams for eight-point functions G8(t1,…,t8) of scalar fields with loop corrections from soft modes as shown in Eq. (2.44), in TOCs and OTOCs, respectively.

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    • Figure 14
      Figure 14

      Schwinger-Keldysh eight contour for eight-point OTOCs, with time chosen as in Eq. (4.33). One may entail infinitesimals Imt52=δ>0 and Imt74=δ′>0 to entail that t2,4 is earlier than t5,7, respectively, along the imaginary time axis.

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    • Figure 15
      Figure 15

      Spectral asymmetric angle θ and number density n vs effective IR charge ed and conformal scaling dimension Δ: (a) θ(ed,Δ); (b) n(n,Δ). Where θ is given in Eq. (c134) and n is given in (c136). For Δ=1/2, the number density n has a typical Fermi liquid like behavior at zero temperature, since it becomes as a step function of ed. The spectral asymmetric angle also has a similar behavior at Δ=1/2 but is vanishing when Δ=0.

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    • Figure 16
      Figure 16

      Residual entropy s0 at zero temperature: (a) s1(ed,Δ); (b) s1(n,Δ); (c) s1(θ,Δ); (d) s0(ed,Δ); (e) s0(n,Δ); (f) s0(θ,Δ). s1 is in Eq. (c200) and s0 is in Eq. (c202) with the relation s1=s0 in the density symmetric case (ed=0 or θ=0, n=0). ed is the IR effective gauge coupling, n is the number density as in Eq. (c136) or Eq. (c204), θ is the spectral asymmetry angle in Eq. (c134). We have chosen the parameter q=1, ed∈[−1/2,1/2], n∈(0,1), θ∈[−π/2,π/2] and Δ∈[0,1/2].

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