Entanglement in quantum impurity problems is nonperturbative

The continuous coupling function in quantum impurity problems is exactly partitioned into a part represented by a finite size Wilson chain and a part represented by a set of additional reservoirs, each coupled to one Wilson chain site. These additional reservoirs represent high-energy modes of the environment neglected by the numerical renormalization group and are required to restore the continuum limit of the original problem. We present a hybrid time-dependent numerical renormalization group approach which combines an accurate numerical renormalization group treatment of the non-equilibrium dynamics on the finite size Wilson chain with a Bloch-Redfield formalism to include the effect of these additional reservoirs. Our approach overcomes the intrinsic shortcoming of the time-dependent numerical renormalization group approach induced by the bath discretization with a Wilson parameter $\Lambda > 1$. We analytically prove that for a system with a single chemical potential, the thermal equilibrium reduced density operator is the steady-state solution of the Bloch-Redfield master equation. For the numerical solution of this master equation a Lanczos method is employed which couples all energy shells of the numerical renormalization group. The presented hybrid approach is applied to the real-time dynamics in correlated fermionic quantum-impurity systems. An analytical solution of the resonant-level model serves as a benchmark for the accuracy of the method which is then applied to non-trivial models, such as the interacting resonant-level model and the single impurity Anderson model.

Using detailed balance and scaling properties of integrals that appear in the Coulomb gas reformulation of quantum impurity problems, we establish exact relations between the nonequilibrium transfer rates of the boundary sine-Gordon and the anisotropic Kondo model at zero temperature. Combining these results with findings from the thermodynamic Bethe ansatz, we derive exact closed form expressions for the transfer rate in the biased spin-boson model in the scaling limit. They illustrate how the crossover from weak to strong tunneling takes place. Using a conjectured correspondence between the transfer and the decoherence rate, we also determine the exact lower bound for damping of the coherent oscillation as a function of bias and dissipation strength in this paradigmic model for NMR and superposition of macroscopically distinct states (qubits).

We summarize in this work the relations of the spin-boson Hamiltonian, which was recently studied in connection with the phenomenon of quantum coherence in the presence of dissipation, to three different fermionic Hamiltonians. These relations were obtained through well-known equivalences between Fermi and Bose operators in one dimension. The fermionic Hamiltonians correspond to (a) a two-level system coupled linearly with a fermionic bath, (b) the resonant-level model, and (c) the anisotropic Kondo model. For the first Hamiltonian we reobtain and discuss the relationship between the dimensionless dissipation coefficient \ensuremath{\alpha} and the phase shift of the fermions. The resonant-level model allows us to study the properties of the two-level system for values of \ensuremath{\alpha} around (1/2). At \ensuremath{\alpha}=(1/2) the model reduces to the Toulouse limit, where the Hamiltonian is exactly soluble. A pure exponential decay is obtained for the relaxation of P(t)=〈${\ensuremath{\sigma}}_{z}$(t)〉 for t>0, given that for t<0 the system is known to be localized in one of the two states. The comparison with the Kondo model gives the long-time-limit behavior of the system for (1/2)1 and provides a connection between universal numbers obtained previously for the spin-boson Hamiltonian and the Kondo model.

All-dielectric photonic crystals or metasurfaces supporting optical bound states in the continuum (BIC) are emerging as a promising platform for the study of light-matter interactions in the strong coupling regime. However, coupling of BIC to other optical excitations is generally weak, with the coupling strength limited in the range of several tens of milli electronvolts with short decoherence time. This greatly hinders their applications in solid-base quantum information devices that require a fast information exchange rate and long coherence time. Herein, a cavity-coupling strategy is used to extend the BIC-based coupling platforms. We propose a hybrid system containing an all-dielectric metasurface embedded in an optical microcavity, demonstrating the formation of BIC-cavity polaritons in the strong coupling regime with a surprisingly large Rabi splitting over 220 meV, far beyond the strong coupling regime. A flexible tuning of BIC-cavity coupling from weak to strong coupling regime is realized by controlling the shape of dielectric element of the metasurface. Importantly, a full quantum model based on Heisenberg-Langevin formalism reveals the temporal population dynamics of the coupling system, demonstrating the ultrafast Rabi socillation within 19 fs with a long decoherence time over 200 fs. Utilizing the multipolar expansion approach combined with full wave simulations, we also investigate the coupling-induced polariton characteristics in both the weak and strong coupling regimes. We find that the electric dipole component of the BIC mode dominates in the polariton states in weak coupling regime, which dramatically changes as the system is tuned into strong coupling regime. Such a cavity-coupling strategy in the present hybrid system is expected to be of great use for enhancing BIC-based light-matter interactions for more complex hybrid nanostructures such as three-mode systems with mixing BIC-cavity-exciton interactions.

We calculate the entanglement and the universal boundary entropy (BE) in the critical quantum spin chains, such as the transverse field Ising chain and the XXZ chain, with arbitrary direction of the boundary magnetic field (ADBMF). We determine the boundary universality class that an ADBMF induces. In particular, we show that the induced boundary conformal field theory (BCFT) depends on the point on the Bloch sphere where the boundary magnetic field directs. We show that the classification of the directions boils down to the simple fact that the boundary field breaks the bulk symmetry or does not. We present a procedure to estimate the universal BE, based on the finite-size corrections of the entanglement entropy, that apply to the ADBMF. To calculate the universal BE in the XXZ chain, we use the density matrix renormalization group (DMRG). The transverse field XY chain with ADBMF after Jordan-Wigner (JW) transformation is not a quadratic free fermion Hamiltonian. We map this model to a quadratic free fermion chain by introducing two extra ancillary spins coupled to the main chain at the boundaries, which makes the problem {\it{integrable}}. The eigenstates of the transverse field XY chain can be obtained by proper projection in the enlarged chain. Using this mapping, we are able to calculate the entanglement entropy of the transverse field XY chain using the usual correlation matrix technique up to relatively large sizes.

We study Monte Carlo calculations of the effective potential for a scalar field theory using three techniques. One of these is a new method proposed and tested for the first time. In each case we extract the renormalised quantities of the theory. The system studied in our calculations is a one component $\phi^4$ model in two dimensions. We apply these methods to both the weak and strong coupling regime. In the weak coupling regime we compare our results for the renormalised quantities with those obtained from two-loop lattice perturbation theory. Our results are verified in the strong coupling regime through comparison with the strong coupling expansion. We conclude that effective potential methods, when suitably chosen, can be accurate tools in calculations of the renormalised parameters of scalar field theories.

We have studied dispersion of surface plasmon polaritons (SPPs) in the Kretcshmann’s geometry (prism/Ag/dye-doped polymer) in weak and strong exciton-plasmon coupling regimes. In reflection, the dispersion curves were in a good agreement with simple model predictions at small concentrations of the dye (R6G) in the polymer (PMMA). At the same time, highly unusual multi-segment ladder-like dispersion curves were observed at large dye concentrations. To describe this behavior, we upgraded the theoretical model taking into account strong exciton-plasmon coupling. The dispersion curves obtained in emission experiments were strongly different from those in reflection and did not seem to be affected by the strong coupling.

$J$-coupled nuclear magnetic resonance (NMR) spectroscopy in the strong coupling regime at low magnetic field ($10{}^{\ensuremath{-}7}$ T $lBl{10}^{\ensuremath{-}3}$ T) is more complex than at high field ($Bg{10}^{\ensuremath{-}3}$ T) and at ultralow field ($Bl{10}^{\ensuremath{-}7}$ T). We show that several upper and lower boundaries ${B}_{i}^{\mathrm{up}}$ and ${B}_{i}^{\mathrm{low}}$ of the magnetic field $B$ exist, where the complexity of $J$-coupled NMR spectra changes in terms of the number of lines. The index $i=1,2,\dots{}$ for ${B}_{i}^{\mathrm{up}}$ at high field specifies the perturbation order of the dominating Zeeman interaction and for ${B}_{i}^{\mathrm{low}}$ at ultralow field the perturbation order of the dominating $J$-coupling interaction. Mathematical expressions for these boundaries are derived for the case of a $J$-coupled $S$-$I$${}_{N}$ group where $S$ and $I$ are rare and abundant spins $\frac{1}{2}$ and $N$ counts the abundant spins $I$. The entire $B$-field range can further be delineated into two weak coupling regimes, one at high field with ${B}_{2}^{\mathrm{up}}lBl{B}_{1}^{\mathrm{up}}$ ($10{}^{\ensuremath{-}3}$ T $lBl{10}^{2}$ T), one at low field with ${B}_{1}^{\mathrm{low}}lBl{B}_{2}^{\mathrm{low}}({10}^{\ensuremath{-}8}$ T $lBl{10}^{\ensuremath{-}7}$ T), and a strong coupling regime with ${B}_{2}^{\mathrm{low}}lBl{B}_{2}^{\mathrm{up}}$ ($10{}^{\ensuremath{-}7}$ T $lBl{10}^{\ensuremath{-}3}$ T). The corresponding NMR spectra for the $S\ensuremath{-}{I}_{N}$ group are investigated by experiment and by simulation. In the strong coupling regime, the maximum number of lines is $(N+1){}^{2}$. In the weak coupling regime ${B}_{1}^{\mathrm{low}}lBl{B}_{2}^{\mathrm{low}}$ at low field, symmetric multiplet structures group around the frequencies $0$, $J$, ($3/2$)$J$, $2J$, ($5/2$)$J$, etc. These spectra determine the structure of the $S\ensuremath{-}{I}_{N}$ group unambiguously and are in dual correspondence to the weakly coupled spectra at high field. High-resolution NMR spectroscopy at ultralow field may open up new ways for chemical analysis by small and mobile instruments with many applications in science and technology.

We propose a scheme to achieve CPA not only in the strong-coupling regime but also in the weak-coupling regime. The system under consideration consists of an atomic ensemble coupled to an optical cavity containing an optical parametric amplifier (OPA). We show that when the OPA introduces an effective loss, CPA can occur only in the strong-coupling regime. In contrast, when the OPA provides an effective gain, CPA can emerge in both the weak- and strong-coupling regimes. We further demonstrate that in the weak-coupling regime, CPA cannot occur within the bistable region, whereas in the strong-coupling regime, CPA can indeed appear in the bistable region. Moreover, the output intensity can be flexibly controlled by tuning the effective strength and the phase of the OPA. Our work opens a potential way to design a coherent perfect absorber based on weak coupling mechanism.

By using two independent and complementary approaches, we compute exactly the shot noise in an out-of-equilibrium interacting impurity model, the interacting resonant level model at its self-dual point. An analytical approach based on the thermodynamical Bethe ansatz allows us to obtain the density matrix in the presence of a bias voltage, which in turn allows for the computation of any observable. A time-dependent density matrix renormalization group technique that has proven to yield the correct result for a free model (the resonant level model) is shown to be in perfect agreement with the former method.

A semiclassical approach is developed for nonequilibrium quantum transport in molecular junctions. Following the early work of Miller and White [J. Chem. Phys. 84, 5059 (1986)], the many-electron Hamiltonian in second quantization is mapped onto a classical model that preserves the fermionic character of electrons. The resulting classical electronic Hamiltonian allows for real-time molecular dynamics simulations of the many-body problem from an uncorrelated initial state to the steady state. Comparisons with exact results generated for the resonant level model reveal that a semiclassical treatment of transport provides a quantitative description of the dynamics at all relevant timescales for a wide range of bias and gate potentials, and for different temperatures. The approach opens a door to treating nontrivial quantum transport problems that remain far from the reach of fully quantum methodologies.

The analytical continuation average spectrum method (ASM) and maximum entropy (MaxEnt) method are applied to the dynamic response of a noninteracting resonant level model within the framework of the Kubo formula for electric conductivity. The frequency dependent conductivity is inferred from the imaginary time current-current correlation function for a wide range of temperatures, gate voltages, and spectral densities representing the leads, and compared with exact results. We find that the MaxEnt provides more accurate results compared to the ASM over the full spectral range.

We prove that for various impurity models, in both classical and quantum settings, the self-energy matrix is a sparse matrix with a sparsity pattern determined by the impurity sites. In the quantum setting, such a sparsity pattern has been known since Feynman. Indeed, it underlies several numerical methods for solving impurity problems, as well as many approaches to more general quantum many-body problems, such as the dynamical mean field theory. The sparsity pattern is easily motivated by a formal perturbative expansion using Feynman diagrams. However, to the extent of our knowledge, a rigorous proof has not appeared in the literature. In the classical setting, analogous considerations lead to a perhaps less-known result, i.e., that the precision matrix of a Gibbs measure of a certain kind differs only by a sparse matrix from the precision matrix of a corresponding Gaussian measure. Our argument for this result mainly involves elementary algebraic manipulations and is in particular non-perturbative. Nonetheless, the proof can be robustly adapted to various settings of interest in physics, including quantum systems (both fermionic and bosonic) at zero and finite temperature, non-equilibrium systems, and superconducting systems.

The stochastic mean-field (SMF) approach allows to treat correlations beyond mean-field using a set of independent mean-field trajectories with appropriate choice of fluctuating initial conditions. We show here, that this approach is equivalent to a simplified version of the Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy between one-, two-, ..., N-body degrees of freedom. In this simplified version, one-body degrees of freedom are coupled to fluctuations to all orders while retaining only specific terms of the general BBGKY hierarchy. The use of the simplified BBGKY is illustrated with the Lipkin-Meshkov-Glick (LMG) model. We show that a truncated version of this hierarchy can be useful, as an alternative to the SMF, especially in the weak coupling regime to get physical insight in the effect beyond mean-field. In particular, it leads to approximate analytical expressions for the quantum fluctuations both in the weak and strong coupling regime. In the strong coupling regime, it can only be used for short time evolution. In that case, it gives information on the evolution time-scale close to a saddle point associated to a quantum phase-transition. For long time evolution and strong coupling, we observed that the simplified BBGKY hierarchy cannot be truncated and only the full SMF with initial sampling leads to reasonable results.

By applying the island rule proposed recently, we compute the entanglement entropy of Hawking radiation and study the Page curve for the eternal black holes in massive gravity. We investigate for both the neutral and charged black holes which the corresponding results of Schwarzschild and Reissner–Nordström black holes are restored in the limit of massless graviton. We show for the neutral and non-extremal charged black holes that the island is not formed at the early times of the evaporation and hence the entanglement entropy increases linearly in time. However, for the extremal charged black hole, the calculation of the entanglement entropy at the early times without the island is ill-defined because the metric is divergent at the curvature singularity. This implies that new physics in the UV region must be taken into account to make the metric behaving smoothly at the very short distances. At the late times, with the emergence of one island near the event horizon, the entanglement entropy is saturated by the Bekenstein–Hawking entropy of black holes. In addition, we analyze the impact of massive gravity parameters on the size of island, the entanglement entropy, the Page time, and the scrambling time in detail.