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Open Quantum Cluster Embedding Theory

Petar Brinić, Hugo U. R. Strand, Jakša Vučičević

Abstract

The simulation of strongly correlated electron systems remains a formidable challenge. Certain experimentally relevant dynamical response functions are especially difficult to calculate, due to issues of finite-size effects and the ill posed analytic continuation. To address this we propose the quantum cluster embedding theory, an embedded cluster method aimed at computing the response of the system following an external perturbation; the frequency dependent dynamical susceptibility is obtained subsequently by means of inverse linear response theory. The embedded clusters, used within the method as representative of short range correlations, are open quantum systems governed by the Lindblad equation. The short-range correlations extracted from the clusters are used to close the equations of motion for the fermionic bilinear and the local double occupancy on the lattice. In turn, the clusters' Markovian baths are tuned to keep the bilinear and the double occupancy expectation values on the clusters and the lattice identical, throughout the concomitant evolution of the two sets of equations. The theory becomes numerically exact in the non-interacting, atomic and infinite cluster size limits, and it respects the total charge and energy conservation laws. We show that our approach can treat very large lattices while avoiding analytic continuation through the explicit time evolution. Finally we compute the charge-charge correlation function in the square lattice Hubbard model and compare with a recent cold atom experiment, finding good qualitative agreement.

Open Quantum Cluster Embedding Theory

Abstract

The simulation of strongly correlated electron systems remains a formidable challenge. Certain experimentally relevant dynamical response functions are especially difficult to calculate, due to issues of finite-size effects and the ill posed analytic continuation. To address this we propose the quantum cluster embedding theory, an embedded cluster method aimed at computing the response of the system following an external perturbation; the frequency dependent dynamical susceptibility is obtained subsequently by means of inverse linear response theory. The embedded clusters, used within the method as representative of short range correlations, are open quantum systems governed by the Lindblad equation. The short-range correlations extracted from the clusters are used to close the equations of motion for the fermionic bilinear and the local double occupancy on the lattice. In turn, the clusters' Markovian baths are tuned to keep the bilinear and the double occupancy expectation values on the clusters and the lattice identical, throughout the concomitant evolution of the two sets of equations. The theory becomes numerically exact in the non-interacting, atomic and infinite cluster size limits, and it respects the total charge and energy conservation laws. We show that our approach can treat very large lattices while avoiding analytic continuation through the explicit time evolution. Finally we compute the charge-charge correlation function in the square lattice Hubbard model and compare with a recent cold atom experiment, finding good qualitative agreement.
Paper Structure (32 sections, 128 equations, 27 figures, 1 table)

This paper contains 32 sections, 128 equations, 27 figures, 1 table.

Table of Contents

  1. Introduction
  2. Methods
  3. Embedded cluster methods
  4. Open Quantum Cluster Embedding Theory
  5. Formulation - equations of motion
  6. Initial conditions
  7. Numerical implementation
  8. Setting up clusters
  9. OQCET for the square lattice Hubbard model
  10. Inverse linear response theory
  11. Preparing the initial state
  12. Thermal equilibrium lattice initial state
  13. Thermal equilibrium cluster initial state
  14. Cluster dynamics
  15. Choice of jump operators
  16. ...and 17 more sections

Figures (27)

  • Figure 1: The self-consistency relation in embedded cluster theories. Coupling of the representative model to an effective environment is represented by the field $\phi$.
  • Figure 2: Schematic representation of the Anderson impurity model and a Lindbladian open quantum cluster
  • Figure 3: Schematic comparison of the DMFT and OQCET loops. Quantities in the top boxes are imposed onto the representative model by tuning the effective field (i.e. $\Delta$ and $\Gamma_i$); quantities at the bottom are extracted from the representative model and passed onto the lattice equations.
  • Figure 4: OQCET algorithm in time
  • Figure 5: Two overlapping 2$\times$2 clusters, shaded blue and red. Their overlap forms a 2$\times$1 cluster, containing bond $M$.
  • ...and 22 more figures