Time-dependent Matrix Product States Research Articles

Matrix product study of spin fractionalization in the one-dimensional Kondo insulator

The Kondo lattice is one of the classic examples of strongly correlated electronic systems. We conduct a controlled study of the Kondo lattice in one dimension, highlighting the role of excitations created by the composite fermion operator. Using time-dependent matrix product state methods, we compute various correlation functions and contrast them with both large-N mean-field theory and the strong-coupling expansion. We show that the composite fermion operator creates long-lived, charge-e and spin-1/2 excitations, which cover the low-lying single-particle excitation spectrum of the system. Furthermore, spin excitations can be thought to be composed of such fractionalized quasiparticles with a residual interaction which tend to disappear at weak Kondo coupling. Published by the American Physical Society 2024

Unified Definition of Exciton Coherence Length for Exciton–Phonon Coupled Molecular Aggregates

Exciton coherence length (ECL) characterizes the spatial extent of coherently delocalized excited states of molecular aggregates. Constructive/destructive superpositions of coherent molecular dipoles lead to superradiance/subradiance, where the radiative rate is enhanced/suppressed compared to that of a single molecule. Longer ECLs correspond to faster/slower radiative rates for superradiant/subradiant aggregates. However, previous ECL definitions fail to produce monotonic relationships when exciton–phonon coupling is considered, even for simple 1D exciton–phonon systems. This problem is exacerbated for 2D aggregates with both constructive and destructive superpositions. In this Letter, we propose a novel ECL definition by virtue of sum rule for oscillator strengths, ensuring a bijective and monotonic relationship between ECL and radiative rate for both 1D/2D superradiant and subradiant aggregates. Using numerically accurate time-dependent matrix product states, we study large-scale, exciton–phonon coupled 2D aggregates and predict the existence of maximum superradiance at finite temperature, in contrast to the previously believed 1/T law. Our results provide new insights into the design and optimization of efficient light emitting materials.

Qubit-oscillator relationships in the open quantum Rabi model: the role of dissipation

Using a dissipative quantum Rabi model, we study the dynamics of a slow qubit coupled to a fast quantum harmonic oscillator interacting with a bosonic bath from weak to strong and ultra-strong coupling regimes. Solving the quantum Heisenberg equations of motion, perturbative in the internal coupling between qubit and oscillator, we derive functional relationships directly linking the qubit coordinates in the Bloch sphere to oscillator observables. We then perform accurate time-dependent Matrix Product State simulations and compare our results both with the analytical solutions of the Heisenberg equations of motion, and with numerical solutions of a Lindblad master equation, perturbative in the external coupling between oscillator and environment. Indeed, we show that, up to the strong coupling regime, the qubit state accurately fulfils the derived functional relationships. We analyse in detail the case of a qubit starting with generic coordinates on the Bloch sphere of which we evaluate the three components of the Bloch vector through the averages of oscillator observables. Interestingly, a weak to intermediate oscillator coupling to the bath is able to simplify the Bloch vector evaluation since qubit-oscillator relationships are more immediate. Moreover, by monitoring the qubit fidelity with respect to free limit, we find the parameter regime where the combined effect of internal and external couplings is able to hinder the reliable evaluation of the qubit Bloch vector. Finally, in the ultra-strong coupling regime, non-Markovian effects become robust and the dynamics of qubit and oscillator are inextricably entangled making the qubit Bloch vector evaluation difficult.Graphical

Radio-frequency driving of an attractive Fermi gas in a one-dimensional optical lattice

We investigate the response to radio-frequency driving of an ultracold gas of attractively interacting fermions in a one-dimensional optical lattice. We study the system dynamics by monitoring the driving-induced population transfer to a third state, and the evolution of the momentum density and pair distributions. Depending on the frequency of the radio-frequency field, two different dynamical regimes emerge when considering the evolution of the third level population. One regime exhibits (off)resonant many-body oscillations reminiscent of Rabi oscillations in a discrete two-level system, while the other displays a strong linear rise. Within this second regime, we connect, via linear response theory, the extracted transfer rate to the system single-particle spectral function, and infer the nature of the excitations from Bethe ansatz calculations. In addition, we show that this radio-frequency technique can be employed to gain insights into this many-body system coupling mechanism away from equilibrium. This is done by monitoring the momentum density redistributions and the evolution of the pair correlations during the drive. Capturing such non-equilibrium physics goes beyond a linear response treatment, and is achieved here by conducting time-dependent matrix product state simulations.

Preparation of Low Entropy Correlated Many-Body States via Conformal Cooling Quenches.

We propose and analyze a method for preparing low entropy many-body states in isolated quantum optical systems of atoms, ions, and molecules. Our approach is based upon shifting entropy between different regions of a system by spatially modulating the magnitude of the effective Hamiltonian. We conduct two case studies, on a topological spin chain and the spinful fermionic Hubbard model, focusing on the key question: can a "conformal cooling quench" remove sufficient entropy within experimentally accessible timescales? Finite-temperature, time-dependent matrix product state calculations reveal that even moderately sized bath regions can remove enough energy and entropy density to expose coherent low-temperature physics. The protocol is particularly natural in systems with long-range interactions, such as lattice-trapped polar molecules and Rydberg-excited atoms, where the magnitude of the Hamiltonian scales directly with the interparticle spacing. To this end, we propose simple, near-term implementations of conformal cooling quenches in systems of atoms or molecules, where signatures of low-temperature phases may be observed.

Numerically Exact Treatment of Many-Body Self-Organization in a Cavity.

We investigate the full quantum evolution of ultracold interacting bosonic atoms on a chain and coupled to an optical cavity. Extending the time-dependent matrix product state techniques and the many-body adiabatic elimination technique to capture the global coupling to the cavity mode and the open nature of the cavity, we examine the long time behavior of the system beyond the mean-field elimination of the cavity field. We investigate the many-body steady states and the self-organization transition for a wide range of parameters. We show that in the self-organized phase the steady state consists in a mixture of the mean-field predicted density wave states and excited states with additional defects. In particular, for large dissipation strengths a steady state with a fully mixed atomic sector is obtained crucially different from the predicted mean-field state.

Lattice modulation spectroscopy of one-dimensional quantum gases: Universal scaling of the absorbed energy

Lattice modulation spectroscopy is a powerful tool for probing low-energy excitations of interacting many-body systems. By means of bosonization we analyze the absorbed power in a one dimensional interacting quantum gas of bosons or fermions, subjected to a periodic drive of the optical lattice. For these Tomonaga Luttinger liquids we find a universal $\omega^3$ scaling of the absorbed power, that at very low-frequency turns into an $\omega^2$ scaling when scattering processes at the boundary of the system are taken into account. We confirm this behavior numerically by simulations based on time-dependent matrix product states. Furthermore, in the presence of impurities, the theory predicts an $\omega^2$ bulk scaling. While typical response functions of Tomonaga Luttinger liquids are characterized by exponents that depend on the interaction strength, modulation spectroscopy of cold atoms leads to a universal power-law exponent of the absorbed power. Our findings can be readily demonstrated in ultracold atoms in optical lattices with current experimental technology.

Parton theory of angle-resolved photoemission spectroscopy spectra in antiferromagnetic Mott insulators

Angle-resolved photoemission spectroscopy (ARPES) has revealed peculiar properties of mobile dopants in correlated anti-ferromagnets (AFMs). But describing them theoretically, even in simplified toy models, remains a challenge. Here we study ARPES spectra of a single mobile hole in the $t-J$ model. Recent progress in the microscopic description of mobile dopants allows us to use a geometric decoupling of spin and charge fluctuations at strong couplings, from which we conjecture a one-to-one relation of the one-dopant spectral function and the spectrum of a constituting spinon in the \emph{undoped} parent AFM. We thoroughly test this hypothesis for a single hole doped into a 2D Heisenberg AFM by comparing our semi-analytical predictions to previous quantum Monte Carlo results and our large-scale time-dependent matrix product state (td-MPS) calculations of the spectral function. Our conclusion is supported by a microscopic trial wavefuntion describing spinon-chargon bound states, which captures the momentum and $t/J$ dependence of the quasiparticle residue. Our conjecture suggests that ARPES measurements in the pseudogap phase of cuprates can directly reveal the Dirac-fermion nature of the constituting spinons. Specifically, we demonstrate that our trial wavefunction provides a microscopic explanation for the sudden drop of spectral weight around the nodal point associated with the formation of Fermi arcs, assuming that additional frustration suppresses long-range AFM ordering. We benchmark our results by studying the cross-over from two to one dimension, where spinons and chargons are confined and deconfined respectively.

Evolution of two-time correlations in dissipative quantum spin systems: Aging and hierarchical dynamics

We consider the evolution of two-time correlations in the quantum XXZ spin-chain in contact with an environment causing dephasing. Extending quasi-exact time-dependent matrix product state techniques to consider the dynamics of two-time correlations within dissipative systems, we uncover the full quantum behavior for these correlations along all spin directions. Together with insights from adiabatic elimination and kinetic Monte Carlo, we identify three dynamical regimes. For initial times, their evolution is dominated by the system unitary dynamics and depends on the initial state and the Hamiltonian parameters. For weak spin-spin interaction anisotropy, after this initial dynamical regime, two-time correlations enter an algebraic scaling regime signaling the breakdown of time-translation invariance and the emergence of aging. For stronger interaction anisotropy, these correlations first go through a stretched exponential regime before entering the algebraic one. Such complex relaxation arises due to the competition between the proliferation dynamics of energetically costly excitations and their motion. As a result, dissipative heating dynamics of spin systems can be used to probe the entire spectrum of the underlying Hamiltonian.

Twofold correlation spreading in a strongly correlated lattice Bose gas

We study the spreading of correlations in the Bose-Hubbard chain, using the time-dependent matrix-product state approach. In both the superfluid and the Mott-insulator phases, we find that the time-dependent correlation functions generally display a universal twofold cone structure characterized by two distinct velocities. The latter are related to different microscopic properties of the system and provide useful information on the excitation spectrum. The twofold spreading of correlations has profound implications on experimental observations that are discussed.

Thermal states as convex combinations of matrix product states

We study thermal states of strongly interacting quantum spin chains and prove that those can be represented in terms of convex combinations of matrix product states. Apart from revealing new features of the entanglement structure of Gibbs states our results provide a theoretical justification for the use of White's algorithm of minimally entangled typical thermal states. Furthermore, we shed new light on time dependent matrix product state algorithms which yield hydrodynamical descriptions of the underlying dynamics.

Light-Cone and Diffusive Propagation of Correlations in a Many-Body Dissipative System.

We analyze the propagation of correlations after a sudden interaction change in a strongly interacting quantum system in contact with an environment. In particular, we consider an interaction quench in the Bose-Hubbard model, deep within the Mott-insulating phase, under the effect of dephasing. We observe that dissipation effectively speeds up the propagation of single-particle correlations while reducing their coherence. In contrast, for two-point density correlations, the initial ballistic propagation regime gives way to diffusion at intermediate times. Numerical simulations, based on a time-dependent matrix product state algorithm, are supplemented by a quantitatively accurate fermionic quasiparticle approach providing an intuitive description of the initial dynamics in terms of holon and doublon excitations.

Topological Fractional Pumping with Alkaline-Earth-Like Atoms in Synthetic Lattices.

Alkaline-earth(-like) atoms, trapped in optical lattices and in the presence of an external gauge field, can form insulating states at given fractional fillings. We will show that, by exploiting these properties, it is possible to realize a topological fractional pump. Our analysis is based on a many-body adiabatic expansion, on simulations with time-dependent matrix product states, and, for a specific form of atom-atom interaction, on an exactly solvable model of fractional pump. The numerical simulations allow us to consider a realistic setup amenable of an experimental realization. As a further consequence, the measure of the center-of-mass shift of the atomic cloud would constitute the first measurement of a many-body Chern number in a cold-atom experiment.

Temperature dependence of the NMR spin-lattice relaxation rate for spin-12chains

We use recent developments in the framework of time dependent matrix product state method (t-MPS) to compute the NMR relaxation rate $1/T_1$ for spin-1/2 chains under magnetic field and for different Hamiltonians (XXX, XXZ, isotropically dimerized). We compute numerically the temperature dependence of the $1/T_1$. We consider both gapped and gapless phases, and also the proximity of quantum critical points. At temperatures much lower than the typical exchange energy scale our results are in excellent agreement with analytical results, such as the ones derived from the Tomonaga-Luttinger liquid (TLL) theory and bosonization which are valid in this regime. We also cover the regime for which the temperature $T$ is comparable to the exchange coupling. In this case analytical theories are not appropriate but this regime is relevant for various new compounds with exchange couplings in the range of tens of Kelvin. For the gapped phases, either the fully polarized phase for spins chains or the low magnetic field phase for the dimerized systems, we find an exponential decrease in $\Delta/(k_BT)$ of the relaxation time and can compute the gap $\Delta$. Close to the quantum critical point our results are in good agreement with the a scaling behavior based on the existence of free excitations.

A multiprecision C++ library for matrix-product-state simulation of quantum computing: Evaluation of numerical errors

The time-dependent matrix-product-state (TDMPS) simulation method has been used for numerically simulating quantum computing for a decade. We introduce our C++ library ZKCM_QC developed for multiprecision TDMPS simulations of quantum circuits. Besides its practical usability, the library is useful for evaluation of the method itself. With the library, we can capture two types of numerical errors in the TDMPS simulations: one due to rounding errors caused by the shortage in mantissa portions of floating-point numbers; the other due to truncations of nonnegligible Schmidt coefficients and their corresponding Schmidt vectors. We numerically analyze these errors in TDMPS simulations of quantum computing.