Session V34: Precision Many Body Physics V

Floquet Supersymmetry

We introduce the notion of time-reflection symmetry in periodically driven (Floquet) quantum systems, and show that it enables a Floquet variant of quantum-mechanical supersymmetry. In particular, we find Floquet analogues of the Witten index that place lower bounds on the degeneracies of states with quasienergies 0 and π. We provide a simple class of disordered, interacting, and ergodic Floquet models with an exponentially large number of states at quasienergies 0,π, which are robust as long as the time-reflection symmetry is preserved. Floquet supersymmetry manifests itself in the evolution of certain local observables as a period-doubling effect with dramatic finite-size scaling, providing a clear signature for experiments.   

Spin transport of weakly disordered Heisenberg chain at infinite temperature

We study the disordered Heisenberg spin chain, which exhibits many body localization at strong disorder, in the weak to moderate disorder regime. A continued fraction calculation of dynamical correlations is devised, using a variational extrapolation of recurrents. Good convergence for the infinite chain limit is shown. We find that the local spin correlations decay at long times as C ∼ t^-β, while the conductivity exhibits a low frequency power law σ ∼ ω^α. The exponents depict sub-diffusive behavior β<1/2, α>0 at all finite disorders, and convergence to the scaling result, α+2β=1, at large disorders.   

Time crystals in doped semiconductors

Spin ensembles in semiconductors provide a unique experimental platform for the observation of emergent many-body dynamics like many-body localization and quantum chaos. High precision quantum control of electron and nuclear spins in phosphorus doped silicon have been utilized to uncover the many-body correlations in decoherence. We present here the experimental observation of a many-body time crystal, a novel dynamical phase of matter with spontaneously broken time translation symmetry, in periodically driven nanoscale electron spins in phosphorus doped silicon. The signatures of time crystalline correlations were studied as a function of spin concentration. These observations can be theoretically captured by the paradigmatic central spin model previously used for describing decoherence in this system.   

Quantum quenches and relaxation dynamics in the thermodynamic limit

We implement numerical linked cluster expansion (NLCEs) to study dynamics of one-dimensional systems of hard-core bosons close to an integrable point in the thermodynamic limit. We find that local observables exhibit exponential relaxation in the time scales accessible to us. We calculate the relaxation rates and find that they scale quadratically with the strength of the integrability breaking perturbation.
  

Non-Equilibrium Transport in the Quantum Dot: Quench Dynamics and Non-Equilibrium Steady State

We present an exact method of calculating the non-equilibrium current driven by a voltage drop across a quantum dot. The system is described by the two lead Kondo model with non-interacting Fermi-liquid leads at zero temperature. We prepare the system in an initial state consisting of a free Fermi sea in each lead with the voltage drop given as the difference between the two Fermi levels. We quench the system by coupling the dot to the leads at $t=0$ and following the time evolution of the wavefunction. In the long time limit a steady state emerges provided that the size of the system is large compared to the time of evolution (open system limit). A new type of Bethe Ansatz wavefunction describes the system - it satisfies the Lippmann-Schwinger equation with the two Fermi seas serving as the boundary conditions. We present this wavefunction and give an infinite series expression for the current evaluated in this state to obtain the nonequilibrium steady state current as a function of the voltage. Evaluation of this series is discussed.   

One-dimensional Bose gas dynamics: breather fragmentation and quantum correlations

Both exact conservation laws and dynamical invariants exist in the topical case of the one-dimensional ultra-cold atomic Bose gas, providing a validation of methods for precision quantum dynamics. The first four quantum conservation laws are exactly conserved in the approximate truncated Wigner approach to many-body quantum dynamics, which is a 1/N expansion for N atoms. Center-of-mass position variance is exactly calculable, and is nearly exact in the truncated Wigner approximation, apart from small terms that vanish as N^-3/2 as N→∞.

As an example, we calculate the dynamics of a higher-order soliton in the mesoscopic case of N=10³-10⁴ particles, giving predictions for quantum soliton breather fragmentation. Our results show evidence for gradual fragmentation with formation of multiple single particle condensate states and nonlocal quantum correlations. These results disagree with variational calculations that violate exact dynamical results, due to the use of too few natural modes in variational theories. Methods for going beyond the 1/N expansion will be treated as well.

Such dynamical predictions are testable in BEC experiments using attractive Bose gases with either ⁷Li or ⁸⁵Rb condensates.   

Quantum spin chains with multiple dynamics

Many-body systems with multiple emergent time scales arise in various contexts, including correlated quantum materials and ultra-cold atoms. We investigate such non-trivial quantum dynamics in a new setting: a spin-1 bilinear-biquadratic chain. It has a solvable entangled groundstate, but a gapless excitation spectrum that is poorly understood. By using large-scale DMRG simulations, we find that the lowest excitation has a dynamical exponent z that varies from 2 to 3.2 as we vary a coupling in the Hamiltonian. We find an additional gapless mode with a continuously varying exponent, which establishes the presence of multiple dynamics. In order to explain these striking properties, we construct a field theory and determine various correlation functions and entanglement properties, as well as an exact mapping to the non-equilibrium dynamics of a classical spin chain. Finally, we discuss the connections to other spin chains and to a family of quantum critical models in 2d.   

Incomplete thermalization from trap-induced integrability breaking: lessons from classical hard rods

The familiar Newton’s cradle can be modeled as a one-dimensional gas of hard rods trapped in a harmonic potential, which breaks integrability of the hard-rod interaction in a non-uniform way. We explore the consequences of such broken integrability for the dynamics of a large number of particles and find three distinct regimes: initial, chaotic, and stationary. The initial regime is captured by an evolution equation for the phase-space distribution function. However, for any finite number of particles, this hydrodynamics breaks down due to a “complexity crisis” and the dynamics become chaotic after a characteristic time scale determined by the inter-particle distance and scattering length. At long times, the system fails to thermalize and the time-averaged ensemble of individual trajectories is not micro-canonical, but it is a stationary state of the hydrodynamic evolution. We close by discussing logical extensions of the results to similar systems of quantum particles.   

Behaviour of l-bits Near the Many-Body Localization Transition

Eigenstates of fully many-body localized (FMBL) systems are described by quasilocal operators τ_i^z (l-bits), which are conserved exactly under Hamiltonian time evolution. The algebra of the operators τ_i^z and τ_i^x associated with l-bits τ_i completely define the eigenstates and the matrix elements of local operators between eigenstates at all energies. We develop a non-perturbative construction of the full set of l-bit algebras in the many-body localized phase for the canonical model of MBL. Our algorithm to construct the Pauli-algebra of l-bits combines exact diagonalization and a tensor network algorithm developed for efficient diagonalization of large FMBL Hamiltonians. The distribution of localization lengths of the l-bits is evaluated in the MBL phase and used to characterize the MBL-to-thermal transition.   

Single-particle mobility edges in a one-dimensional bichromatic incommensurate potential

In this talk I will focus on a 1D mutually incommensurate bichromatic lattice system which has been implemented in ultracold atoms to study quantum localization. It has been universally believed that the tight-binding version of this bichromatic incommensurate system is represented by the well-known Aubry-Andre (AA) model. Here we establish that this belief is incorrect and that the AA model description generically breaks down near the localization transition due to the unavoidable appearance of single-particle mobility edges (SPME). As a result, for the full lattice system, an intermediate phase between completely localized and completely delocalized regions appears due to the existence of the SPME, making the system qualitatively distinct from the AA prediction. Our theoretical prediction is subsequently verified in an experiment using a one-dimensional quasi-periodic optical lattice. In particular, a regime is identified where extended and localized single-particle states coexist, in good agreement with our theoretical results. Our work thus presents the first realization of a system with an SPME in one dimension, and it opens up more research prospects in the context of many-body localization, including the question of many-body mobility edges.   

Many-body localization in spin chain systems with quasiperiodic fields

We study the many-body localization of spin chain systems with quasiperiodic fields. We identify the lower bound for the critical disorder necessary to drive the transition between the thermal and many-body localized phase to be Wc>1.85, based on finite-size scaling of entanglement entropy and fluctuations of the bipartite magnetization. We also examine the time evolution of the entanglement entropy of an initial product state where we find power-law and logarithmic growth for the thermal and many-body localized phases, respectively, with a transition point Wc∼2.5. For larger disorder strength, both imbalance and spin-glass order are preserved at long times, while spin-glass order shows dependence on system size. We also explore density matrix renormalization group studies and explore a two-legged ladder model. Quasiperiodic fields have been applied in different experimental systems, and our study finds that such fields are very efficient at driving the many-body localized phase transition.   

Development of time dependent DMRG method for higher dimensional systems and its application to quantum annealing

In order to investigate quantum dynamics on higher dimensional strongly correlated systems, we have developed new kind of the time dependent DMRG (tDMRG) method using the kernel polynomial method (KPM). In our tDMRG method, a time depending state is directly calculated by the KPM. By a three-term recursive formula of special functions, we can perform effective tDMRG calculations. Also, our tDMRG calculations give accurate results as long as we employ large enough DMRG truncation number. In the present study, we have applied our tDMRG method to quantum annealing which is a kind of quantum computing for optimization problems. In the quantum annealing, the Hamiltonian is constructed by a two-dimensional Ising model with transverse magnetic field. Our tDMRG calculations of the quantum annealing give correct answer of optimization problems by using relatively small DMRG truncation number. Thus, our tDMRG method is suitable for numerical studies of the quantum annealing.   

Spectral Functions of the Bilinear-Biquadratic Spin-1 Chain

The bilinear-biquadratic spin-1 chain exhibits a rich phase diagram. Depending on the two coupling strengths, the system can be in a ferromagnetic phase, a gapped dimer phase, the gapped Haldane phase with hidden topological long-range order, or an extended critical phase with period-three structure and dominant quadrupolar correlations. Here, we use density matrix renormalization group (DMRG) algorithms to compute precise static structure factors and spectral functions in the different phases. We relate our results to insights from exact solutions at special points in the phase diagram and field-theoretical approaches.