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arXiv:1907.12434 · Physical review. B./Physical review. B

Quantum state tomography on a plaquette in the two-dimensional Hubbard model

Stephan Humeniuk

Published Sep 11, 2019• 5 citations
arXivHTMLPDF

Abstract

Motivated by recent quantum gas microscope experiments for fermions in optical lattices, we present proof-of-principle calculations showing that it is possible to obtain the complete information about the quantum state on a small subsystem from equilibrium determinantal quantum Monte Carlo simulations. Both diagonal (in the occupation number basis) and off-diagonal elements of the reduced density matrix are calculated for a square plaquette, which is embedded in a much larger system of the two-dimensional Hubbard model, both at half filling and in the doped case. The diagonalization of the reduced density matrix is done by exploiting the point group symmetry and particle number conservation, which allows one to attach symmetry labels to its eigenvalues. Knowledge of the probabilities of plaquette occupation number configurations is useful for meticulous benchmarking of quantum gas microscope experiments. As the quantum state on the plaquette is exact and self-consistently embedded in an exact, correlated bath, the present approach connects to various cluster approximation techniques.

DOI
10.1103/physrevb.100.115121

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Motivated by recent quantum gas microscope experiments for fermions in optical lattices, we present proof-of-principle calculations showing that it is possible to obtain the complete information about the quantum state on a small subsystem from equilibrium determinantal quantum Monte Carlo simulations. Both diagonal (in the occupation number basis) and off-diagonal elements of the reduced density matrix are calculated for a square plaquette, which is embedded in a much larger system of the two-dimensional Hubbard model, both at half filling and in the doped case. The diagonalization of the reduced density matrix is done by exploiting the point group symmetry and particle number conservation, which allows one to attach symmetry labels to its eigenvalues. Knowledge of the probabilities of plaquette occupation number configurations is useful for meticulous benchmarking of quantum gas microscope experiments. As the quantum state on the plaquette is exact and self-consistently embedded in an exact, correlated bath, the present approach connects to various cluster approximation techniques.
A numerical method is presented for reproducing fermionic quantum gas microscope experiments in equilibrium. By employing nested componentwise direct sampling of fermion pseudodensity matrices, as they arise naturally in determinantal quantum Monte Carlo (QMC) simulations, a stream of pseudosnapshots of occupation numbers on large systems can be produced. There is a sign problem even when the conventional determinantal QMC algorithm can be made sign-problem free, and every pseudosnapshot comes with a sign and a reweighting factor. Nonetheless, this ``sampling sign problem'' turns out to be weak and manageable in a large, relevant parameter regime. The method allows one to compute distribution functions of arbitrary quantities defined in occupation number space and, from a practical point of view, facilitates the computation of complicated conditional correlation functions. While the projective measurements in quantum gas microscope experiments achieve direct sampling of occupation number states from the density matrix, the presented numerical method requires a Markov chain as an intermediate step and thus achieves only indirect sampling, but the full distribution of pseudosnapshots after (signed) reweighting is identical to the distribution of snapshots from projective measurements.
An exact-diagonalization technique on small clusters is used to study the ground states and single-particle excitations of the Hubbard model with on-site (U) and nearest-neighbor (V) Coulombic repulsive interactions. It is shown that the long-range charge-density-wave state realized in a half-filled two-dimensional square lattice for 4V\ensuremath{\gtrsim}U persists up to quarter filling, and that the doped carriers form a small Fermi surface that encloses an area corresponding to the number of doped carriers rather than the total number of electrons involved in the system. The present result offers a well-defined counterexample in the discussion of the Fermi-surface dimensions of strongly correlated electron systems, and may provide a complementary understanding of the large Fermi surface believed to exist in the ordinary Hubbard model.
A numerical method is presented for reproducing fermionic quantum gas microscope experiments in equilibrium. By employing nested componentwise direct of fermion pseudo-density matrices, as they arise naturally in determinantal quantum Monte Carlo (QMC) simulations, a stream of pseudo-snapshots of occupation numbers on large systems can be produced. There is a sign even when the conventional determinantal QMC algorithm can be made sign-problem free, and every pseudo-snapshot comes with a sign and a reweighting factor. Nonetheless, this sampling sign problem turns out to be weak and manageable in a large, relevant parameter regime. The method allows to compute arbitrary distribution functions in Fock space and, from a practical point of view, facilitates the computation of complicated conditional correlation functions.
We investigate the quench of half-filled one-dimensional and two-dimensional fermionic Hubbard models to models without Coulomb interaction. Since the time propagation is Gaussian, we can use a variety of time-dependent quantum Monte Carlo methods to tackle this problem without generating a dynamical sign problem. Using a continuous time quantum Monte Carlo method, we achieve a system size of 128 sites in one dimension, and using a Blankenbecler-Scalapino-Sugar type algorithm, we were able to simulate $20\ifmmode\times\else\texttimes\fi{}20$ square lattices. Applying these methods to study the dynamics after the quench, we observe that the final state of the system can be reasonably well described by a thermal single-particle density matrix that takes the initial single-particle conservation laws into account. The characteristic decay toward this limit is found to be oscillatory with an additional power law decay that depends on the dimensionality. This numerically exact result is shown to compare favorably to mean-field approximations as well as to perturbation theory. Furthermore, we observe the information propagation in the one-dimensional case in the charge-charge and spin-spin correlations and find that it is linear with a velocity of roughly $v\ensuremath{\approx}4$ in units of the hopping amplitude.
Numerical exact diagonalisation (ED) of the Hubbard model is traditionally constrained by the exponential growth of the Hilbert space. In this study, we implemented a symmetry-based ED framework to investigate the one-dimensional Hubbard model at half-filling for system sizes up to L=16. By exploiting particle number conservation, spin projection, and translation invariance, we significantly reduced the Hamiltonian dimension, enabling the resolution of ground state energies and dispersion relations without additional approximations. Our results confirmed the transition from a metallic to a correlated insulating regime with increasing on-site repulsion U. Furthermore, extending calculations from ten to sixteen sites revealed scaling trends that asymptotically approached the thermodynamic limit predicted by the Bethe Ansatz (𝐿= ∞). These results served as precise, reproducible benchmarks for validating approximate many-body techniques such as the Density Matrix Renormalisation Group (DMRG).
In order to understand various local phenomena related to electronic phase separation discovered recently in strongly correlated materials, quantum cluster methods are applied to a theoretical model, i.e., Hubbard model, which has successfully predicted basic properties of strongly correlated systems. Our calculations are performed at two levels: exact calculations based on the isolated small clusters via exact diagonalization and approximate calculations applied to large lattices using the variational cluster approximation (VCA).\nWe first study phase separation instabilities with the formation of a pairing gap in the two-dimensional Hubbard model for 8-site Betts clusters. The exact diagonalization method is applied to extract exact ground and excited states of the Hubbard model. The results show that the electronic states with one hole off half filling are unstable and they energetically prefer creating the spatial phase separation. The effect of next nearest hopping is also discussed in this part.\nWe studied the variation of the pairing when the out-of-plane correlation is present by adding an interacting apical site above the two-dimensional cluster plane. The calculations indicate that the out-of-plane correlation can be detrimental to the in-plane pairing effect. When it is not too strong to destroy electron pairs on the plane, the modulation of the apical site can drive the pairing gap to show a sinusoidal variation which is consistent with recent experimental discoveries.\nVariational cluster approximation is introduced to explore local properties of larger lattices. The local nematic state is studied in the square lattice using this method, which indicates that it is possible that the electronic states can locally break the C4 symmetry (90o$ rotational symmetry) to form a pattern where the electronic properties, such as charge and spin correlations, are different in x and y directions.\nThe spatial phase separation in the square and honeycomb lattices is also investigated by the variational cluster approximation. The study shows that Coulomb interaction and lattice geometry are both crucial for electronic phase separation. We propose that electronic phase separation is the result of an instability of the Fermi surface at the boundary of the first Brillouin zone of antiferromagnetic states.