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Many-body-based DFT treatment of fermions in optical lattices

Jérémie Westergren

Published Jan 1, 2017• 0 citations
OpenAlex

Abstract

With recent advances in the field of ultra cold atoms one can, by trapping atoms at low temperatures by laser beams, simulate systems which can be adequately described by single-band lattice Hamiltonians. Also, due to the high parameter tunability of the experimental setups, lattice disorder can be introduced in a con- trolled fashion in these systems. This thesis considers disordered/ordered interacting fermion lattice systems in equilibrium in one and two dimensions, subject to trap- ping parabolic potentials, where different levels of description, ranging from exact where possible (in 1D, Density Matrix Renormalization Group, DMRG), to approxi- mate (based on several local-density approximations within the framework of Lattice Density Functional Theory) are used. The exchange-correlation potentials consid- ered come from Many-Body Approximations obtained using Green’s functions, as well as from an exact local-density approximation (LDA) based on the Bethe-Ansatz (BALDA) in 1D. Both one- and two-dimensional systems were studied in equilib- rium, essentially looking at the ground state density profiles. Furthermore, in 2D, a pseudo-dynamics representing the trap-opening in the complete adiabatic limit was also studied. For one-dimensional systems, it is found that, in general, BALDA yields very good results compared to DMRG, except for the low density limit, but DMRG can describe features that none of the LDA:s considered can reproduce. It is also found that the strength of the external potential affects the impact that the exchange-correlation potential has on the system. Many of these features translate to the 2D case. However, a new aspect emerges in two dimensions, related to the competition of disorder and interaction. Here an important outcome is that, on opening the trap at an ideally adiabatic rate, different MBA:s (and thus different LDA:s) provide different minimal vs maximal expansion radii of the particle cloud, as a result of the interplay of disorder and interaction, and the underlying square lattice structure. For the 2D results, exact benchmarks were not available, and our findings may thus need further validation, by e.g. considering several disorder configurations or, ideally, by performing full Green’s function calculations. These considerations are summarised in our conclusions and outlook remarks, where possible directions for future investigation are highlighted.

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Interacting two-component Fermi gases loaded in a one-dimensional (1D) lattice and subjected to a harmonic trapping potential exhibit interesting compound phases in which fluid regions coexist with local Mott-insulator and/or band-insulator regions. Motivated by experiments on cold atoms inside disordered optical lattices, we present a theoretical study of the effects of a correlated random potential on these ground-state phases. We employ a lattice version of density-functional theory within the local-density approximation to determine the density distribution of fermions in these phases. The exchange-correlation potential is obtained from the Lieb-Wu exact solution of Fermi-Hubbard model. On-site disorder (with and without Gaussian correlations) and harmonic trap are treated as external potentials. We find that disorder has two main effects: (i) it destroys the local insulating regions if it is sufficiently strong compared with the on-site atom-atom repulsion, and (ii) it induces an anomaly in the inverse compressibility at low density from quenching of percolation. For sufficiently large disorder correlation length the enhancement in the inverse compressibility diminishes.

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Interacting two-component Fermi gases loaded in a one-dimensional (1D) lattice and subjected to an harmonic trapping potential exhibit interesting compound phases in which fluid regions coexist with local Mott-insulator and/or band-insulator regions. Motivated by experiments on cold atoms inside disordered optical lattices, we present a theoretical study of the effects of a correlated random potential on these ground-state phases. We employ a lattice version of density-functional theory within the local-density approximation to determine the density distribution of fermions in these phases. The exchange-correlation potential is obtained from the Lieb-Wu exact solution of Fermi-Hubbard model. On-site disorder (with and without Gaussian correlations) and harmonic trap are treated as external potentials. We find that disorder has two main effects: (i) it destroys the local insulating regions if it is sufficiently strong compared with the on-site atom-atom repulsion, and (ii) it induces an anomaly in the inverse compressibility at low density from quenching of percolation. For sufficiently large disorder correlation length the enhancement in the inverse compressibility diminishes.

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With recent advances in the field of ultra cold atoms one can, by trapping atoms at low temperatures by laser beams, simulate systems which can be adequately described by single-band lattice Hamiltonians. Also, due to the high parameter tunability of the experimental setups, lattice disorder can be introduced in a con- trolled fashion in these systems. This thesis considers disordered/ordered interacting fermion lattice systems in equilibrium in one and two dimensions, subject to trap- ping parabolic potentials, where different levels of description, ranging from exact where possible (in 1D, Density Matrix Renormalization Group, DMRG), to approxi- mate (based on several local-density approximations within the framework of Lattice Density Functional Theory) are used. The exchange-correlation potentials consid- ered come from Many-Body Approximations obtained using Green’s functions, as well as from an exact local-density approximation (LDA) based on the Bethe-Ansatz (BALDA) in 1D. Both one- and two-dimensional systems were studied in equilib- rium, essentially looking at the ground state density profiles. Furthermore, in 2D, a pseudo-dynamics representing the trap-opening in the complete adiabatic limit was also studied. For one-dimensional systems, it is found that, in general, BALDA yields very good results compared to DMRG, except for the low density limit, but DMRG can describe features that none of the LDA:s considered can reproduce. It is also found that the strength of the external potential affects the impact that the exchange-correlation potential has on the system. Many of these features translate to the 2D case. However, a new aspect emerges in two dimensions, related to the competition of disorder and interaction. Here an important outcome is that, on opening the trap at an ideally adiabatic rate, different MBA:s (and thus different LDA:s) provide different minimal vs maximal expansion radii of the particle cloud, as a result of the interplay of disorder and interaction, and the underlying square lattice structure. For the 2D results, exact benchmarks were not available, and our findings may thus need further validation, by e.g. considering several disorder configurations or, ideally, by performing full Green’s function calculations. These considerations are summarised in our conclusions and outlook remarks, where possible directions for future investigation are highlighted.
This thesis investigates different methods for treating strongly correlated systems, and discusses their respective strengths and weaknesses. Many of the results presented in this thesis come from comparing the different methods and approximations to exact results. Paper I: We studied the real-time dynamics of a trapped fermion gas, as it expands after removal of a trapping potential. Paper II: We constructed a new exchange-correlation potential, to be used in three dimensions. The potential is non-perturbative in the interaction, meaning that we could use it both for weakly interacting (metallic) systems, as well as for strongly interacting (Mott insulating) systems. Paper III: We studied transport of electrons through small disordered wires contacted to leads. Using TDDFT, we have studied how the inclusion of both disorder and large electron-electron interactions affect conduction. At finite bias, we saw that the effects where competitive, and that interactions could increase the current through disordered samples. Paper IV: We studied fermions in 3D, using our newly developed exchange-correlation potentials, presented in Paper II. We studied how a cloud of fermions expands when released from a trapping potential. The simulated systems were large enough to be relevant for actual experimental conditions. Paper V: We studied transport of electrons through small disordered wires contacted to leads. The setup was similar to the one in Paper III, but another method was used - NEGF. Many of the observed trends were similar to those seen in our previous investigations. The differences were attributed to non-local effects.
Interacting two-component Fermi gases loaded in a one-dimensional (1D) lattice and subjected to a harmonic trapping potential exhibit interesting compound phases in which fluid regions coexist with local Mott-insulator and/or band-insulator regions. Motivated by experiments on cold atoms inside disordered optical lattices, we present a theoretical study of the effects of a correlated random potential on these ground-state phases. We employ a lattice version of density-functional theory within the local-density approximation to determine the density distribution of fermions in these phases. The exchange-correlation potential is obtained from the Lieb-Wu exact solution of Fermi-Hubbard model. On-site disorder (with and without Gaussian correlations) and harmonic trap are treated as external potentials. We find that disorder has two main effects: (i) it destroys the local insulating regions if it is sufficiently strong compared with the on-site atom-atom repulsion, and (ii) it induces an anomaly in the inverse compressibility at low density from quenching of percolation. For sufficiently large disorder correlation length the enhancement in the inverse compressibility diminishes.
Interacting two-component Fermi gases loaded in a one-dimensional (1D) lattice and subjected to an harmonic trapping potential exhibit interesting compound phases in which fluid regions coexist with local Mott-insulator and/or band-insulator regions. Motivated by experiments on cold atoms inside disordered optical lattices, we present a theoretical study of the effects of a correlated random potential on these ground-state phases. We employ a lattice version of density-functional theory within the local-density approximation to determine the density distribution of fermions in these phases. The exchange-correlation potential is obtained from the Lieb-Wu exact solution of Fermi-Hubbard model. On-site disorder (with and without Gaussian correlations) and harmonic trap are treated as external potentials. We find that disorder has two main effects: (i) it destroys the local insulating regions if it is sufficiently strong compared with the on-site atom-atom repulsion, and (ii) it induces an anomaly in the inverse compressibility at low density from quenching of percolation. For sufficiently large disorder correlation length the enhancement in the inverse compressibility diminishes.
Beautiful experiments have shown that bosons in optical lattices provide a physical system with an unprecedented degree of control; that they can generate strongly correlated states; and that, as long-term goals, they may be useful for simulating other many-body systems, or realising a quantum computer. Recently a practical scheme for measuring temperature in an optical lattice has been demonstrated, which marks an important milestone necessary for the detailed study of the thermodynamics of this system. In this thesis, we consider both the translationally-invariant lattice, and the combined harmonic trap and optical lattice. We use an extended Bose-Hubbard Hamiltonian which goes beyond the usual Bose-Hubbard approach, and is valid for shallower lattices and higher temperatures, by allowing for beyond nearest-neighbour hopping and excited bands, and we have developed an approximation scheme for off-site interactions. We derive the equations of the Popov approximation to the Hartree-Fock-Bogoliubov method for our Hamiltonian, and diagonalise these equations in the local density approximation (LDA). We examine the density of states of the optical lattice in detail and in various limits. We derive new results on the structure of the density of states, and, in the ideal case, we compare the density of states from the full diagonalisation with our LDA calculation. We make an efficient numerical implementation of our theory and compare the results with the full diagonalisation (for the non-interacting case) and with the limited experimental results currently available. We consider the significance of beyond nearest-neighbour hopping and excited bands and illustrate the properties of our model. In contrast to the trapped gas with no lattice, few thermodynamic results for cold bosons in an optical lattice have been calculated. We analytically derive the first practical formula for the critical temperature in an optical lattice by using simplified shapes for the density of states. We derive corrections for the influence of excited bands, the finite-size effect and mean-field interactions. In all of these cases, we compare our results to full numerical calculations and show that the validity range of our method is complementary to that of the effective-mass approximation, so that the simple descriptions extend over a wide range.
We use numerical simulation to study ultracold, quantum degenerate, atomic gases. In the first part of the thesis we study the effects of disorder, introduced via a bichromatic optical lattice, in one and two dimensional systems. We employ the Aubry-Andr\'{e} model and use time-dependent numerical simulations to investigate the disorder dependent transition to strong localisation present in the model. Weak s-wave interactions are added to the model and we observe the interaction between localisation and interaction induced self-trapping. We then add a tilted lattice potential to the model. In the homogeneous model this induces Bloch oscillations. While one might expect that a strong enough force will break the strong localisation or self-trapping, within the bounds of the single-band model, the trapping effect of the Bloch oscillations reinforces both of the other effects leading to increased confinement, albeit lacking the clear single frequency oscillation signature of pure Bloch oscillations. Along with the two dimensional bichromatic optical lattice we add a term to the Hamiltonian equivalent to that of a uniform external magnetic field on charged particles. Since the experimental realisation of this model would employ neutral atoms, the magnetic field is synthetic, the equivalent effect being produced by an appropriate set of lasers and magnetic fields. We show that in the ballistic regime (weak bichromatic disorder) the system displays positive magnetoresistance. Conversely in the strong localisation regime the system exhibits negative magnetoresistance. In the latter part of the thesis we use density functional theory to calculate the ground-state density of a harmonically trapped dipolar Fermi gas. We then use these to calculate the lowest energy collective mode oscillation frequencies under the hydrodynamic approximation. We find that increasing the strength of the dipoles has the effect of increasing the mode frequencies. The increase saturates for large dipole strengths. We verify this analytically and show that such is due to the local nature of the two dimensional energy functional and not dependent on the specific equation of state. We employ an average density approximation to construct an energy functional for the inhomogeneous, 2D degenerate Fermi gas. The ground-state densities for a cylindrically symmetric harmonic trap are compared to the Kohn-Sham results, showing extremely good agreement in the tail region and good agreement with the exact ground-state energy. We then do the same for higher order polynomial traps and obtain improved agreement for higher degree.
This dissertation serves as a summary of my Ph.D. work numerically studying equilibrium and non-equilibrium properties of strongly-interacting one-dimensional (1D) boson systems. This work is motivated by the fact that 1D systems are realizable and highly controllable with ultracold atoms in optical lattice and atom chip experiments. We apply a recent worm algorithmic Monte Carlo approach developed for 1D continuous systems to study their equilibrium properties, both with and without an underlying lattice. We also apply an exact lattice approach based on the Bose-Fermi mapping to check our Monte Carlo results in the Tonks-Girardeau limit, and more importantly, to study far-from-equilibrium expansion dynamics of the systems. We first study the scaling of one-particle correlations of the harmonically trapped Lieb-Liniger gas with changing temperature and interaction strength. Based on the universal behaviors of the density and momentum profiles, we are able to determine the effective parameters needed to fully characterize the system. We also find that the Tonks-Girardeau limit at low temperatures is the ideal regime for the experimental observation of the 1/k41/k^41/k4 momentum tail. An extra periodic lattice can drive the transition from superfluid to Mott insulator states. Exact and complete phase diagrams for such transitions are available only in the weak interacting and deep lattice limit, in which the system can be described using one-band Bose-Hubbard model. Beyond this limit, we use the worm algorithm in continuous space to map out the phase diagrams at various interaction strengths. We compare our phase diagrams with one-band Bose-Hubbard predictions and identify the regime where the one-band description breaks down. We introduce an inverse confined scattering solution to obtain effective Hubbard parameters, with which the Bose-Hubbard model provides correct results for strong interactions and deep lattices at unit filling. In addition to the equilibrium properties, we also study the expansion dynamics of ultracold atoms in the hard-core limit. Experimentally, this is usually achieved by turning off confining potentials and letting atoms expand in optical lattices. Theoretical studies from initial ground states predicted the occurrence of fermionization of the momentum distribution after long expansion times. In addition, quasicondensation at finite momenta emerges when expanding from Mott insulating domains. Here, we develop a finite-temperature extension of the lattice approach for dynamics. We find the dynamical ferminoization of the momentum distributions at all temperatures. For expansion from initial Mott domains, we observe enhanced correlations reminiscent of dynamical quasicondensation. Surprisingly, we find the systems appear to cool down during the melting of the Mott domains. We use an emergent local Hamiltonian to understand these emergent phenomena.