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Dynamical mean field theory extension to the nonequilibrium two-particle self-consistent approach

Olivier Simard and Philipp Werner
Phys. Rev. B 107, 245137 – Published 27 June 2023
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  • INTRODUCTION
  • MODEL AND METHODS
  • RESULTS
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    Nonlocal correlations play an essential role in correlated electron systems, especially in the vicinity of phase transitions and crossovers, where two-particle correlation functions display a distinct momentum dependence. In nonequilibrium settings, the effect of nonlocal correlations on dynamical phase transitions, prethermalization phenomena, and trapping in metastable states is not well understood. In this paper, we introduce a dynamical mean field theory (DMFT) extension to the nonequilibrium two-particle self-consistent (TPSC) approach, which allows to perform nonequilibrium simulations capturing short- and long-ranged nonlocal correlations in the weak-, intermediate-, and strong-correlation regimes. The method self-consistently computes local spin and charge vertices, from which a momentum-dependent self-energy is constructed. Replacing the local part of the self-energy by the DMFT result within this self-consistent scheme provides an improved description of local correlation effects. We explain the details of the formalism and the implementation, and demonstrate the versatility of DMFT+TPSC with interaction quenches and dimensional crossovers in the Hubbard model.

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    • Received 3 March 2023
    • Accepted 5 June 2023

    DOI:https://doi.org/10.1103/PhysRevB.107.245137

    ©2023 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Quasiparticles & collective excitations
    1. Physical Systems
    Strongly correlated systems
    1. Techniques
    Dynamical mean field theoryGreen's function methodsHubbard model
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Olivier Simard and Philipp Werner

    • Department of Physics, University of Fribourg, 1700 Fribourg, Switzerland

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    Vol. 107, Iss. 24 — 15 June 2023

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    • Figure 1
      Figure 1

      Second-order Hartree self-energy diagram. The fermionic propagators represent the Weiss Green's functions G0.

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    • Figure 2
      Figure 2

      Second-order self-energy diagram.

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    • Figure 3
      Figure 3

      Third-order diagrams ΣH3a (top left corner), ΣH3b (top right corner), and ΣH3c (bottom).

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    • Figure 4
      Figure 4

      Third-order diagrams Σ3a (left) and Σ3b (right).

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    • Figure 5
      Figure 5

      Third-order diagrams Σ3d (top left corner), Σ3c (top right corner) and Σ3e (bottom).

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    • Figure 6
      Figure 6

      Flow chart describing the self-consistent determination of D(z), χsp, and Γsp (alternative method). In the actual simulations, we modify the BSE as in Eq. (70) and use the multidimensional root-finding method.

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    • Figure 7
      Figure 7

      Flow chart describing the self-consistent determination of χch and Γch. In the actual simulations, we modify the BSE as in Eq. (70).

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    • Figure 8
      Figure 8

      Flow chart describing the self-consistent DMFT+TPSC procedure. In the actual calculations, the Bethe-Salpeter equations inside the yellow panel are approximated by Eq. (70).

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    • Figure 9
      Figure 9

      Flow graph showing the connections between the two levels of TPSC, namely, the first- (blue boxes) and second-level (green boxes) approximations. The red line shows that second-level irreducible vertices could, in principle, be obtained from the second-level self-energy Σ(1). The α renormalization of the vertices introduced via Eq. (72) modifies the irreducible vertices such that the two levels of the approximation become consistent.

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    • Figure 10
      Figure 10

      Imaginary part of the Matsubara self-energy at the antinode (k=(0,π)) for the half-filled Hubbard model at U=2. Results for T=0.33 (top subplot) and T=0.1 (bottom subplot) are shown for the various methods indicated in the legend. This figure can be compared with the “TPSC” panel in Fig. 10 of Ref. [63].

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    • Figure 11
      Figure 11

      Bandwidth-renormalized spin and charge irreducible vertices as a function of normalized bare interaction for the nearest-neighbor square (2D) and cubic (3D) lattices for TPSC (bold lines) and TPSC+GG (dashed lines). The dimensionless temperature is T/W=0.05 and we consider half-filled systems. These data are taken from Ref. [30].

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    • Figure 12
      Figure 12

      Γch (top panel) and Γsp (bottom panel) as a function of T/W for U=2,3,4,5 in the half-filled 3D Hubbard model, calculated with TPSC+GG. The values of the vertices are normalized by U for presentation reasons.

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    • Figure 13
      Figure 13

      Γch (top panel) and Γsp (bottom panel) as a function of T/W for U=2,3,4,5 in the half-filled 3D Hubbard model, calculated with TPSC. The values of the vertices are normalized by U.

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    • Figure 14
      Figure 14

      Dimensionless spin and charge irreducible vertices as a function of normalized bare interaction for the square and cubic lattices, calculated with DMFT+TPSC. The dimensionless temperature is T/W=0.05 and the systems are half filled.

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    • Figure 15
      Figure 15

      Γch (top panel) and Γsp (bottom panel) as a function of T for U=2,3,4,5 in the 3D half-filled nearest-neighbor Hubbard model. The values of the vertices are normalized by U and were obtained using DMFT+TPSC. At lower temperatures for U=5, the DMFT solution could not be converged.

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    • Figure 16
      Figure 16

      Static spin susceptibility of the 3D (top subplot) and 2D (bottom subplot) models at momentum kπ as a function of temperature for the interactions U=2,3,4,5 and half filling. Results are shown for TPSC (bold lines) and TPSC+GG (dashed lines). The vertical lines coincide with those in Fig. 13.

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    • Figure 17
      Figure 17

      Static spin susceptibility of the half-filled 2D and 3D model at momentum kπ as a function of temperature for interactions U=1,2,3,4 (2D) and U=2,3,4,5 (3D), obtained with DMFT+TPSC. The vertical lines coincide with those in Fig. 15.

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    • Figure 18
      Figure 18

      ξsp as a function of β=1/T for U=2 in the half-filled 2D Hubbard model. The y axis uses a logarithmic scale. The methods compared are OG TPSC (green circles, called TPSC in Refs. [38, 63]), TPSC+GG (orange diamonds), DMFT+TPSC (cyan crosses), DΓA (blue circles), DiagMC (black triangles), TRILEX (red circles), and PA (green triangles). The data calculated using TRILEX, DiagMC, OG TPSC, DΓA, and PA were taken from Ref. [63]. The third-order IPT impurity solver is used in DMFT+TPSC (see Sec. II B 3 2).

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    • Figure 19
      Figure 19

      Double occupancies Dimp [Eq. (73)] and DTPSC [Eq. (78)] as a function of temperature for several interactions U in the half-filled 3D Hubbard model. The annotated percentages denote the largest absolute variation relative to Dimp.

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    • Figure 20
      Figure 20

      Double occupancy of the 2D Hubbard model calculated from the lattice quantities, Eq. (78), for Σ(2), DMFT (bare and bold), OG TPSC, DMFT+TPSC, and TPSC+GG. The interaction is ramped from U=0 to U=1 in the time interval indicated by the grey shading and the initial temperature is T=0.2.

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    • Figure 21
      Figure 21

      Double occupancies calculated using the impurity quantities Eq. (73) in the cases of DMFT (bare and bold) and DMFT+TPSC. In the case of TPSC and TPSC+GG, the double occupancy taken from Eq. (63) is shown. The result for Σ(2) as well as the parameters are the same as in Fig. 20.

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    • Figure 22
      Figure 22

      Local DMFT+TPSC (dashed lines) and TPSC+GG (solid lines) quantities in the 2D Hubbard model for the ramp from U=1 to U=3 at initial temperature T=0.33. The charge irreducible vertex (top panel), spin irreducible vertex (second panel from top), Dimp (third panel from top), and DTPSC (bottom panel) are plotted for a time window of Δt=8.

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    • Figure 23
      Figure 23

      Color plot illustrating the relation between the potential energy Ep (x axis), the kinetic energy Ek (y axis), and the corresponding equilibrium temperature for DMFT+TPSC and U=1,2,3,4 (see annotations). The 2D square lattice Hubbard model is used. The red (green) cross marks the postramp state, obtained from the interaction ramp shown in Fig. 22 (Fig. 20).

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    • Figure 24
      Figure 24

      Color plot analogous to Fig. 23, but for TPSC+GG. The red (green) cross marks the postramp state obtained from the interaction ramp shown in Fig. 22 (Fig. 20).

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    • Figure 25
      Figure 25

      Local TPSC (solid lines) and TPSC+GG (dashed lines) quantities in a dimensional ramp from a square lattice to a cubic lattice corresponding to a ramp from tzhop=0 to tzhop=1 in the dispersion relation [Eq. (2)]. The initial temperature is T=0.2 and the constant interaction is U=2.5. The charge irreducible vertex (top panel), spin irreducible vertex (second panel from top), Dimp (third panel from top), and α (bottom panel) are plotted for a time window of Δt=7.

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    • Figure 26
      Figure 26

      Local DMFT+TPSC quantities in the dimensional ramp from tzhop=0 to tzhop=1 in the single-band nearest-neighbor Hubbard model for U=2.5 at initial temperature T=0.2. The charge irreducible vertex (top panel), spin irreducible vertex (second panel from top), Dimp (third panel from top) and DTPSC (bottom panel) are plotted for a time window of Δt=8.

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    • Figure 27
      Figure 27

      Imaginary parts of the lesser component of the spin (top subplot) and charge (bottom subplot) susceptibilities for momentum kπ and TPSC. The initial temperature is T=0.2 and the interaction is U=2.5. The inset shows the profile of the perpendicular hopping ramps tzhop with the vertical bars representing the times for which the spectra are calculated. The time window for the Fourier transformation is Δt=2.5.

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    • Figure 28
      Figure 28

      Top (bottom) panels: Difference spectra of the lesser component of the charge (spin) susceptibility after the interaction ramp shown in the inset. The inset black triangle illustrates the path in reciprocal space—within the kz=π plane – along which the spectra are displayed. The times ti and tf used in the calculation of the difference spectra are annotated in each panel. The time window used in the Fourier transformation is Δt=2.5. Each row of panels uses the same color scale. The method used here is TPSC.

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    • Figure 29
      Figure 29

      Top (bottom) panels: Difference spectra of the DMFT+TPSC lesser component of the charge (spin) susceptibility after the perpendicular lattice hopping ramp from tzhop=0 to tzhop=1 shown in the inset. The time window employed in the Fourier transformation is Δt=2.5. Each row of panels uses the same color scale. The initial temperature is T=0.2.

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    • Figure 30
      Figure 30

      Top (bottom) panels: Difference spectra of the DMFT+TPSC lesser component of the charge (spin) susceptibility after the interaction ramp from U=1 to U=3 shown in the inset. The time window employed in the Fourier transformation is Δt=5. Each row of panels uses the same color scale. The initial temperature is T=0.33.

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    • Figure 31
      Figure 31

      Top (bottom) panels: Difference spectra of the lesser component of the charge (spin) susceptibility after the interaction ramp from U=1 to U=3 shown in the inset. The top (bottom) subplot shows the results obtained using TPSC+GG (TPSC). The time window employed in the Fourier transformation is Δt=5. Each row of panels uses the same color scale. The initial temperature is T=0.33.

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