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arXiv:0804.1457 · New Journal of Physics

Dynamics of large anisotropic spin in a sub-ohmic dissipative environment close to a quantum-phase transition

Frithjof B. Anders

Published Nov 20, 2008• 6 citations
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Abstract

We investigate the dynamics of a large anisotropic spin whose easy-axis component is coupled to a bosonic bath with a spectral function . Such a spin complex might be realized in a single-molecular magnet. Using the non-perturbative renormalization group, we calculate the line of quantum-phase transitions (QPTs) in the sub-ohmic regime ( s <1). These QPTs only occur for integer spin J . For half-integer J , the low temperature fixed-point is identical to the fixed-point of the spin-boson model without quantum tunneling between the two levels. Short-time coherent oscillations in the spin decay prevail even into the localized phase in the sub-ohmic regime. The influence of the reorganization energy and the recurrence time on the decoherence in the absence of quantum tunneling is discussed.

DOI
10.1088/1367-2630/10/11/115007

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We investigate the dynamics of a large anisotropic spin whose easy-axis component is coupled to a bosonic bath with a spectral function . Such a spin complex might be realized in a single-molecular magnet. Using the non-perturbative renormalization group, we calculate the line of quantum-phase transitions (QPTs) in the sub-ohmic regime ( s <1). These QPTs only occur for integer spin J . For half-integer J , the low temperature fixed-point is identical to the fixed-point of the spin-boson model without quantum tunneling between the two levels. Short-time coherent oscillations in the spin decay prevail even into the localized phase in the sub-ohmic regime. The influence of the reorganization energy and the recurrence time on the decoherence in the absence of quantum tunneling is discussed.
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 investigate the behavior of an NNN-component quantum rotor coupled to a bosonic dissipative bath having a sub-Ohmic spectral density $J(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{s}$ with s<1s<1s<1. With increasing dissipation strength, this system undergoes a quantum phase transition from a delocalized phase to a localized phase. We determine the exact critical behavior of this transition in the large-NNN limit. For 1>s>1/21>s>1/21>s>1/2, we find nontrivial critical behavior corresponding to an interacting renormalization group fixed point, while we find mean-field behavior for s<1/2s<1/2s<1/2. The results agree with those of the corresponding long-range interacting classical model. The quantum-to-classical mapping is therefore valid for the sub-Ohmic rotor model.
Open quantum spin systems offer a previously unexplored route to realizing deconfined quantum criticality. We consider a spin-1/2 JJJ-Q3Q_3Q3​ chain, consisting of an antiferromagnetic (AFM) Heisenberg exchange and a competing multi-spin interaction favoring a valence-bond solid (VBS) state, with each spin component coupled to a bosonic bath. Using non-Abelian bosonization and renormalization-group (RG) analysis, combined with large-scale quantum Monte Carlo (QMC) simulations, we determine the phase diagram and the associated phase transitions of the model. We show that strong dissipation stabilizes an AFM phase for sub-Ohmic, Ohmic, and super-Ohmic baths. Continuous AFM-VBS transitions at finite dissipation are found upon increasing the multi-spin interaction in both the sub-Ohmic and Ohmic regimes. Critical properties are obtained through perturbative RG analysis and QMC simulations. In the Ohmic case, the critical point features spinon deconfinement and emergent O(4) symmetry. In the sub-Ohmic regime, the transition may also involve spinon deconfinement, provided that spinons remain deconfined in the dissipative VBS phase. In addition, in the super-Ohmic regime, we propose a transition from AFM phase to a quasi-long-range ordered phase.
We investigate the real-time dynamics of the sub-Ohmic spin-boson model across a broad range of coupling strengths, using the numerically exact inchworm quantum Monte Carlo algorithm. From short- and intermediate-time dynamics starting from an initially decoupled state, we extract signatures of the zero-temperature quantum phase transition between localized and delocalized states. We show that the dynamical phase diagram thus obtained differs from the equilibrium phase diagram in both the values of critical couplings and the associated values of the critical exponent. We also identify and quantitatively analyze two competing mechanisms for the crossover between coherent oscillations and incoherent decay. Deep in the sub-Ohmic regime, the crossover is driven by the damping of the oscillation amplitude, while closer to the Ohmic regime the oscillation frequency itself drops sharply to zero at large coupling.
The effective theories for many quantum phase transitions can be mapped onto those of classical transitions. Here we show that the naive mapping fails for the sub-Ohmic spin-boson model which describes a two-level system coupled to a bosonic bath with power-law spectral density, J(omega) proportional, variantomega(s). Using an epsilon expansion we prove that this model has a quantum transition controlled by an interacting fixed point at small s, and support this by numerical calculations. In contrast, the corresponding classical long-range Ising model is known to display mean-field transition behavior for 0 < s < 1/2, controlled by a noninteracting fixed point. The failure of the quantum-classical mapping is argued to arise from the long-ranged interaction in imaginary time in the quantum model.
We show that the low-frequency modes of a sub-Ohmic bosonic heat bath generate an effective dynamical asymmetry for an intrinsically symmetric quantum spin-1/2. An initially fully polarized spin first decays toward a quasiequilibrium determined by the dynamical asymmetry, thereby showing coherent damped oscillations on the (fast) time scale of the spin splitting. On top of this, the dynamical asymmetry itself decays on an ultraslow time scale and vanishes asymptotically since the global equilibrium phase is symmetric. We quantitatively study the nature of the initial fast decay to the quasiequilibrium and discuss the features of ultraslow dynamics of the quasiequilibrium itself. The dynamical asymmetry is more pronounced for smaller values of the sub-Ohmic exponent and for lower temperatures, which emphasizes the quantum many-body nature of the effect. The symmetry breaking is related to the dynamic crossover between coherent and overdamped relaxation of the spin polarization and is not connected to the localization quantum phase transition. In addition to this delocalized phase, we identify a phase which is characterized by damped coherent oscillations in the localized phase. This allows for a sketch of the zero-temperature phase diagram of the sub-Ohmic spin-boson model with four distinct phases.