We study interaction quenches of the Fermi-Hubbard model initiated from various high-temperature and high-energy states, motivated by cold atom experiments, which currently operate above the ordering temperature(s). We analytically calculate the dynamics for quenches from these initial states, which are often strongly interacting, to the noninteracting limit. Even for high-temperature uncorrelated initial states, transient connected correlations develop. These correlations share many features for all considered initial states. We observe light-cone spreading of intertwined spin and density correlations. The character of these correlations is quite different from their low-temperature equilibrium counterparts: for example, the spin correlations can be ferromagnetic. We also show that an initially localized hole defect affects spin correlations near the hole, suppressing their magnitude and changing their sign.
We study interaction quenches of the Fermi-Hubbard model initiated from various high-temperature and high-energy states, motivated by cold atom experiments, which currently operate above the ordering temperature(s). We analytically calculate the dynamics for quenches from these initial states, which are often strongly interacting, to the noninteracting limit. Even for high-temperature uncorrelated initial states, transient connected correlations develop. These correlations share many features for all considered initial states. We observe light-cone spreading of intertwined spin and density correlations. The character of these correlations is quite different from their low-temperature equilibrium counterparts: for example, the spin correlations can be ferromagnetic. We also show that an initially localized hole defect affects spin correlations near the hole, suppressing their magnitude and changing their sign.
We study a nonequilibrium dynamics of a one-dimensional spin-imbalanced Fermi-Hubbard model following a quantum quench of on-site interaction, realizable, for example, in Feshbach-resonant atomic Fermi gases. We focus on the post-quench evolution starting from the initial BCS and Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) ground states and analyze the corresponding spin-singlet, spin-triplet, density-density, and magnetization-magnetization correlation functions. We find that beyond a light-cone crossover time, rich post-quench dynamics leads to thermalized and pre-thermalized stationary states that display strong dependence on the initial ground state. For initially gapped BCS state, the long-time stationary state resembles thermalization with the effective temperature set by the initial value of the Hubbard interaction. In contrast, while the initial gapless FFLO state reaches a stationary pre-thermalized form, it remains far from equilibrium. We suggest that such post-quench dynamics can be used as a fingerprint for identification and study of the FFLO phase.
The study of quantum dynamics is an increasingly important aspect of atomic, molecular and optical physics, as well as condensed-matter physics and other disciplines. It is essential in order to understand non-equilibrium phenomena which do not fit within the framework of equilibrium thermodynamics, such as many-body localization, dynamical quantum phase transitions, and non-equilibrium steady states in driven-dissipative systems. However, the exact calculation of dynamical observables is extremely difficult in general for several reasons, which makes it challenging for theoretical predictions and analyses to keep pace with rapidly advancing experiments. In this thesis I use and develop several computational methods to investigate the dynamics of quantum many-body systems, with the two goals of better understanding the evolution of local observables and correlation functions in spin and Fermion systems as well as creating and testing more effective numerical methods for solving quantum dynamics problems. I explore quantum dynamics by using Hamiltonian parameter quenches in several commonly-used models in AMO physics and elsewhere: the Fermi-Hubbard model, the XXZ model, and the transverse-Ising model. The first represents a system of short-range interacting Fermions, while the latter two represent spin systems with different spin-spin couplings and magnetic field terms. All of these models can be realized in current experiments. First, I derive an exact solution for an extreme interaction quench in the Fermi-Hubbard model, from a strongly-interacting initial state to a noninteracting final Hamiltonian. From this result I show that there are nontrivial transient connected correlations, despite the fact that the system has very high temperature and a noninteracting Hamiltonian. Those features would suppress such correlations in equilibrium, but the dynamical correlations are still significant. Then, I develop and examine two numerical methods for solving dynamics of the spin expectation values and correlation functions in the transverse-Ising and XXZ models. Both methods rely on approximating a thermodynamically large system with a much smaller and more computationally tractable cluster of lattice sites, or a series of clusters. The first method is a dynamical numerical linked cluster expansion (NLCE). NLCEs have already been used successfully on systems in thermodynamic equilibrium, and I extended the technique to directly compute the dynamics of an observable after a quantum quench. I review the general NLCE method, describe my dynamical variant of it, and compute the quench dynamics of spin and correlation observables in the transverse-Ising and XXZ models. The second method, which I term the adaptive-boundary cluster method, relies on recently refined bounds on information propagation in quantum systems to choose a cluster of lattice sites that minimizes the finite size error bound of a dynamical computation. I briefly review these bounds, and describe a new algorithm which uses them to construct a cluster whose boundary is chosen according to a related error estimate. Such an adaptive-boundary cluster has the property that there exists a computable upper bound on the finite size error of observables measured at the cluster center, and this error bound is nearly minimal when compared to that of other clusters of the same size. I then compute the quench dynamics of spin and correlation observables in the transverse-Ising model using the adaptive boundary clusters. I apply this method to both uniform and strongly disordered systems. To evaluate the effectiveness of both dynamical NLCE and adaptive-boundary clusters, I compare the results obtained using them to results from a standard and general purpose numerical method: exact solvers applied a rectangular cluster with periodic boundary conditions. Both of the new methods usually converge faster and are more accurate than the standard method for comparable cluster sizes. This thesis illustrates that choosing numerical methods which take advantage of the geometric and entanglement structure of the initial state and Hamiltonian can substantially enhance our ability to study quantum dynamics, by reducing the computational cost of numerically exact solutions. I conclude with a discussion of various avenues for future research to build on this work.
Strong electron correlations lie at the origin of high-temperature superconductivity. Its essence is believed to be captured by the Fermi-Hubbard model of repulsively interacting fermions on a lattice. Here we report on the site-resolved observation of charge and spin correlations in the two-dimensional (2D) Fermi-Hubbard model realized with ultracold atoms. Antiferromagnetic spin correlations are maximal at half-filling and weaken monotonically upon doping. At large doping, nearest-neighbor correlations between singly charged sites are negative, revealing the formation of a correlation hole, the suppressed probability of finding two fermions near each other. As the doping is reduced, the correlations become positive, signaling strong bunching of doublons and holes, in agreement with numerical calculations. The dynamics of the doublon-hole correlations should play an important role for transport in the Fermi-Hubbard model.
The study of the interplay between interaction-induced correlations and nonequilibrium initial conditions in many-body systems has recently attracted a lot of attention. New experimental techniques provide high control over many-body systems and systematic access to their nonequilibrium regime. In many systems the nonequilibrium properties of a Fermi liquid can be relevant. A first approach to their understanding is the main content of this thesis. At the beginning, nonequilibrium phenomena, Landau's theory of a Fermi liquid, the Hubbard model, time evolution in quantum mechanics, related experiments, and the flow equation method are reviewed. The key observation of this thesis, namely a characteristic mismatch of expectation values in equilibrium and nonequilibrium, is first illustrated for the squeezed oscillator. For this one-particle model Hamiltonian the perturbative approach can be compared with the exact solution and it is shown that the mismatch holds even beyond perturbation theory. Afterwards these observations are generalized to a larger class of one-particle models. Then the nonequilibrium behavior of a Fermi liquid is examined by analyzing the Fermi liquid phase of the Hubbard model in more than one dimension. After a sudden switch-on of a weak two-particle interaction to the noninteracting Fermi gas the relaxation of the many-body system is observed. For this purpose, the flow equation transformation is implemented for the Hubbard Hamiltonian. Then the discussion of the momentum distribution function and of the kinetic energy displays a three-step relaxation behavior of the Fermi liquid from the initial perturbation until thermalization is reached. Firstly, the sudden switch inserts excitation energy into the system which drives the following dynamics. Then a rapid initial build-up of correlations is caused by initial dephasing and leads to the establishment of a quasiparticle picture. By that the system enters into a quasi-stationary state which can be long-lasting for weak interaction. This state shows prethermalization of the kinetic energy which already has relaxed to its final values. However, the momentum distribution still resembles a zero temperature Fermi liquid. Its later relaxation on a second time scale is caused by a residual two-particle interaction which allows for scattering processes and can be described by a quantum Boltzmann equation. The physical origin of this delayed relaxation can be traced back to an interplay of translational invariance of the Hubbard model and the Pauli principle for fermions; together they restrict the phase space for scattering events. Comparing with similar work I conjecture on the generic nature of the findings made for the quenched Fermi liquid in other many-body systems. Finally I point out to the potential relevance of the delayed relaxation for the observation of further nonequilibrium phenomena, for instance in BCS systems. In order to extend the study of sudden switching to arbitrary switching processes the calculation is repeated using the Keldysh perturbation theory. First evaluations for a linear increase of the interaction strength are given. Extensions of this work are suggested and may motivate further research.
In this work, we study the non-equilibrium dynamics of low-dimensional strongly correlated quantum systems, which in our case are generated by sudden and time-dependent quantum quenches. Such non-equilibrium dynamics are a particularly difficult problem, because the time evolution of the initial many-body quantum state, which is governed by the post-quench Hamiltonian, is highly non-trivial. On top of that, applying an operator to this many-body quantum state in order to calculate its expectation values, typically creates an intricate set of excitations. Throughout the thesis, we furthermore contrast the results that we have obtained using exact numerical methods for two microscopic models, namely the Heisenberg and Fermi-Hubbard chains, with results derived from the Luttinger liquid theory, which, in equilibrium, describes the universal behavior of the two models.In the second part of the thesis, we reduce the dimensionality even further and employ complementary numerical and analytical tools to examine the unusual non-equilibrium behavior of a ring-shaped quantum impurity hosting interacting spin-less fermions.
Using the time-dependent density matrix renormalization-group method and exact diagonalization, we study the nonequilibrium dynamics of the one-dimensional Fermi-Hubbard model following a quantum quench or a ramp of the on-site interaction strength. We are particularly interested in the nonequilibrium evolution of Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) correlations, which, at finite spin polarizations and for attractive interactions, are the dominant two-body correlations in the ground state. For quenches from the noninteracting to the attractive regime, we investigate the dynamical emergence of FFLO correlations and their signatures in the pair quasi--momentum distribution function. We observe that the postquench double occupancy exhibits a maximum as the interaction strength becomes of the order of the bandwidth. Finally, we study quenches and ramps from attractive to repulsive interactions, which imprint FFLO correlations onto repulsively bound pairs. We show that a quite short ramp time is sufficient to wipe out the characteristic FFLO features in the postquench pair momentum distribution functions.