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arXiv:1106.2655 · Physical Review B

Dissipative spin dynamics near a quantum critical point: Numerical renormalization group and Majorana diagrammatics

Serge Florens, Axel Freyn, D. Venturelli, Rajesh Narayanan

Published Oct 13, 2011• 25 citations
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Abstract

We provide an extensive study of the subohmic spin-boson model with power-law density of states $J(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{s}$ (with $0<s\ensuremath{\le}1$), focusing on the equilibrium dynamics of the three possible spin components, from very weak dissipation to the quantum critical regime. Two complementary methods, the bosonic numerical renormalization group and Majorana diagrammatics, are used to explore the physical properties in a wide range of parameters. We show that the bosonic self-energy is the crucial ingredient for the description of critical fluctuations, but that many-body vertex corrections need to be incorporated as well in order to obtain quantitative agreement of the diagrammatics with the numerical simulations. Our results suggest that the out-of-equilibrium dynamics in dissipative models beyond the Bloch-Redfield regime should be reconsidered in the long-time limit. Regarding also the spin-boson Hamiltonian as a toy model of quantum criticality, some of the insights gained here may be relevant for field theories of electrons coupled to bosons in higher dimensions.

DOI
10.1103/physrevb.84.155110

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We provide an extensive study of the subohmic spin-boson model with power-law density of states J(\ensuremathω)\ensuremath∝\ensuremathωsJ(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{s}J(\ensuremathω)\ensuremath∝\ensuremathωs (with 0<s\ensuremath≤10<s\ensuremath{\le}10<s\ensuremath≤1), focusing on the equilibrium dynamics of the three possible spin components, from very weak dissipation to the quantum critical regime. Two complementary methods, the bosonic numerical renormalization group and Majorana diagrammatics, are used to explore the physical properties in a wide range of parameters. We show that the bosonic self-energy is the crucial ingredient for the description of critical fluctuations, but that many-body vertex corrections need to be incorporated as well in order to obtain quantitative agreement of the diagrammatics with the numerical simulations. Our results suggest that the out-of-equilibrium dynamics in dissipative models beyond the Bloch-Redfield regime should be reconsidered in the long-time limit. Regarding also the spin-boson Hamiltonian as a toy model of quantum criticality, some of the insights gained here may be relevant for field theories of electrons coupled to bosons in higher dimensions.
Employing the nonperturbative numerical renormalization group method, we study the dynamics of the spin-boson model, which describes a two-level system coupled to a bosonic bath with a spectral density J(omega) proportional to omega(s). We show that, in contrast with the case of Ohmic damping, the delocalized phase of the sub-Ohmic model cannot be characterized by a single energy scale only, due to the presence of a nontrivial quantum phase transition. In the strongly sub-Ohmic regime, s<<1, weakly damped coherent oscillations on short time scales are possible even in the localized phase--this is of crucial relevance, e.g., for qubits subject to electromagnetic noise.
We analyze the static and dynamic properties of two Ising-coupled quantum spins embedded in a common bosonic bath as an archetype of dissipative quantum mechanics. First, we elucidate the ground-state phase diagram for an Ohmic and a sub-Ohmic bath using a combination of bosonic numerical renormalization group (NRG), analytical techniques, and intuitive arguments. Second, by employing the time-dependent NRG we investigate the system's rich dynamical behavior arising from the complex interplay between spin-spin and spin-bath interactions. Interestingly, spin oscillations can synchronize due to the proximity of the common non-Markovian bath and the system displays highly entangled steady states for certain nonequilibrium initial preparations. We complement our nonperturbative numerical results by exact analytical solutions when available and provide quantitative limits on the applicability of the perturbative Bloch-Redfield approach at weak coupling.
This thesis investigates the physics of strongly interacting driven–dissipative bosonic systems in one dimension, combining field-theoretical approaches with various numerical simulation techniques. While weakly interacting condensates are accurately described by a generalized Gross–Pitaevskii equation, the strongly correlated regime of the same microscopic models remains far less understood, particularly out of equilibrium. This thesis aims to bridge this gap by developing controlled quantum-many-body simulations for open systems governed by Lindblad dynamics, as well as investigating the corresponding action under the functional renormalization group.In the first part, we formulate a field-theoretical description of driven–dissipative one-dimen-sional bosons using the Schwinger–Keldysh formalism and the functional renormalization group (fRG).Two models are analyzed: (i) a driven-dissipative condensate whose phase dynamics is expected to exhibit emergent Kardar–Parisi–Zhang (KPZ) behavior, for which we search the emergent KPZ fixed point starting from the bosonic field theory; and (ii) a two-photon–driven system with two-body losses realizing dissipative cat qubits. For the latter, we determine the phase diagram and identify a continuous phase transition between symmetry-broken and symmetric phases, which motivates the construction of an fRG Ansatz capturing this critical behavior.The second part of the thesis focuses on the open system dynamics using exact diagonalization and tensor-network methods. For the Bose–Hubbard model with incoherent pump and loss, we characterize the steady state, and study spectral functions across the full parameter space. We show that key ground-state features are absent under dissipation, while the high-temperature unitary spectrum captures aspects of the open-system dynamics. In an extended model with additional two-body and non-local losses, we establish the semi-classical emergence of KPZ scaling and extract the KPZ dynamical exponent from the spectral function. A finite-size analysis exposes the current limitations of the tensor-network methods for reaching the high occupancy regime in the full quantum evolution.Finally, a last project develops a two-loop perturbative RG analysis of a spin-1 chain at its continuous symmetry breaking quantum critical point, revealing critical exponents distinct from the Ising universality class.Together, these results advance the understanding of strongly interacting open quantum systems, elucidating their symmetries, effective field theories, and dynamical properties, and outlining key directions for future theoretical and numerical work.
We study the time evolution of the reduced density matrix for the Ohmic spin boson model out of an uncorrelated but otherwise arbitrary initial state. We consider arbitrary bias $\ensuremath{\epsilon}$ and tunneling $\mathrm{\ensuremath{\Delta}}$ at zero temperature for a weak coupling $\ensuremath{\alpha}$ to the bosonic bath. Using the real-time renormalization group method, we present a consistent weak-coupling expansion one order beyond the Bloch-Redfield approximation within a renormalized perturbation theory with analytical results covering the whole crossover regime from small times $\mathrm{\ensuremath{\Omega}}t\ensuremath{\ll}1$ to large times $\mathrm{\ensuremath{\Omega}}t\ensuremath{\gg}1$, where $\mathrm{\ensuremath{\Omega}}=\sqrt{{\ensuremath{\epsilon}}^{2}+{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{\ensuremath{\Delta}}}}^{2}}$ denotes the Rabi frequency in terms of the renormalized tunneling $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{\ensuremath{\Delta}}}$. In addition, for exponentially small or large times, we perform a nonperturbative resummation of all logarithmic terms. We show that standard Born approximation schemes calculating the effective Liouvillian of the kinetic equation up to first order in $\ensuremath{\alpha}$ are not sufficient to account for various important corrections one order beyond the Bloch-Redfield solution. (1) The resummation of all secular terms $\ensuremath{\sim}{(\mathrm{\ensuremath{\Gamma}}t)}^{n}$ is necessary to obtain the correct exponential decay of all terms of the time evolution with decay rate $\mathrm{\ensuremath{\Gamma}}$ or $\mathrm{\ensuremath{\Gamma}}/2$, together with the correct pre-exponential functions. (2) The resummation of all logarithmic terms at high and low energies leads to a renormalized tunneling $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{\ensuremath{\Delta}}}$ and to pre-exponential functions of logarithmic and power-law form. (3) The fact that two eigenvalues of L(E)L(E)L(E) are close to each other by $O(\mathrm{\ensuremath{\Gamma}})$ requires degenerate perturbation theory for times $\mathrm{\ensuremath{\Gamma}}t\ensuremath{\sim}O(1)$, where certain terms of the Liouvillian in $O({\ensuremath{\alpha}}^{2})$ are needed to calculate the stationary state and the time evolution of the nonoscillating purely decaying modes up to $O(\ensuremath{\alpha})$. In contrast to the zero-bias case, we find two further interesting results for the time dynamics of the oscillating modes. (4) The terms of the pre-exponential functions with a strong time dependence show a leading long-time tail $\ensuremath{\sim}\ensuremath{\alpha}/(\mathrm{\ensuremath{\Omega}}t)$, besides other subleading terms $\ensuremath{\sim}\ensuremath{\alpha}/{(\mathrm{\ensuremath{\Omega}}t)}^{2}$ well-known from the zero-bias case. (5) The terms of the pre-exponential functions with a weak (logarithmic) time dependence vary according to a power law ${(\frac{1}{\mathrm{\ensuremath{\Omega}}t})}^{2\ensuremath{\alpha}\frac{{\ensuremath{\epsilon}}^{2}}{{\mathrm{\ensuremath{\Omega}}}^{2}}}$ for exponentially large times. The power-law exponent depends on the bias and has to be contrasted to the one at exponentially small times where it crosses over to the bias-independent result $2\ensuremath{\alpha}$. We discuss that the complexity to calculate one order beyond Bloch-Redfield approximation is rather generic and applies also to other models of dissipative quantum mechanics.
For phase transitions in dissipative quantum impurity models, the existence of a quantum-to-classical correspondence has been discussed extensively. We introduce a variational matrix product state approach involving an optimized boson basis, rendering possible high-accuracy numerical studies across the entire phase diagram. For the sub-Ohmic spin-boson model with a power-law bath spectrum ∝ω(s), we confirm classical mean-field behavior for s<1/2, correcting earlier numerical renormalization-group results. We also provide the first results for an XY-symmetric model of a spin coupled to two competing bosonic baths, where we find a rich phase diagram, including both critical and strong-coupling phases for s<1, different from that of classical spin chains. This illustrates that symmetries are decisive for whether or not a quantum-to-classical correspondence exists.
Numerical renormalization-group (NRG) calculations of quantum impurity models, based on a logarithmic discretization in energy of electronic or bosonic Hamiltonians, provide a powerful tool to describe physics involving widely separated energy scales, as typically encountered in nanostructures and strongly correlated materials. This main advantage of the NRG was however considered a drawback for resolving sharp spectral features at finite energy, such as dissipative atomic peaks. Surprisingly, we find a bunching of many-body levels in NRG spectra near dissipative resonances, and exploit this by combining the widely used Oliveira's zzz trick, using an averaging over few discrete NRG spectra, with an optimized frequency-dependent broadening parameter $b(\ensuremath{\omega})$. This strategy offers a tremendous gain in computational power and extracts all the needed information from the raw NRG data without a priori knowledge of the various energy scales at play. As an application we investigate with high precision the crossover from coherent to incoherent dynamics in the spin boson model.