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arXiv:1912.13448 · Physical review. B./Physical review. B

Quantum phase transitions in the spin-boson model: Monte Carlo method versus variational approach à la Feynman

G. De Filippis, ANTONIO DE CANDIA, Loris Maria Cangemi, Maura Sassetti, Rosario Fazio, V. Cataudella

Published May 26, 2020• 22 citations
arXivHTMLPDF

Abstract

The effectiveness of the variational approach \`a la Feynman is proved in the spin-boson model, i.e., the simplest realization of the Caldeira-Leggett model able to reveal the quantum phase transition from delocalized to localized states and the quantum dissipation and decoherence effects induced by a heat bath. After exactly eliminating the bath degrees of freedom, we propose a trial, nonlocal in time, interaction between the spin and itself simulating the coupling of the two-level system with the bosonic bath. It stems from a Hamiltonian where the spin is linearly coupled to a finite number of harmonic oscillators whose frequencies and coupling strengths are variationally determined. We show that a very limited number of these fictitious modes is enough to get a remarkable agreement, up to very low temperatures, with the data obtained by using an approximation-free Monte Carlo approach, predicting (1) in the Ohmic regime, a Berezinski-Thouless-Kosterlitz quantum phase transition exhibiting the typical universal jump at the critical value; and (2) in the sub-Ohmic regime ($s\ensuremath{\le}0.5$), mean-field quantum phase transitions, with logarithmic corrections for $s=0.5$.

DOI
10.1103/physrevb.101.180408

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The effectiveness of the variational approach `a la Feynman is proved in the spin-boson model, i.e., the simplest realization of the Caldeira-Leggett model able to reveal the quantum phase transition from delocalized to localized states and the quantum dissipation and decoherence effects induced by a heat bath. After exactly eliminating the bath degrees of freedom, we propose a trial, nonlocal in time, interaction between the spin and itself simulating the coupling of the two-level system with the bosonic bath. It stems from a Hamiltonian where the spin is linearly coupled to a finite number of harmonic oscillators whose frequencies and coupling strengths are variationally determined. We show that a very limited number of these fictitious modes is enough to get a remarkable agreement, up to very low temperatures, with the data obtained by using an approximation-free Monte Carlo approach, predicting (1) in the Ohmic regime, a Berezinski-Thouless-Kosterlitz quantum phase transition exhibiting the typical universal jump at the critical value; and (2) in the sub-Ohmic regime (s\ensuremath≤0.5s\ensuremath{\le}0.5s\ensuremath≤0.5), mean-field quantum phase transitions, with logarithmic corrections for s=0.5s=0.5s=0.5.
By using worldline and diagrammatic quantum Monte Carlo techniques, matrix product state, and a variational approach \`a la Feynman, we investigate the equilibrium properties and relaxation features of a quantum system of NNN spins antiferromagnetically interacting with each other, with strength JJJ, and coupled to a common bath of bosonic oscillators, with strength $\ensuremath{\alpha}$. We show that, in the Ohmic regime, a Beretzinski-Thouless-Kosterlitz quantum phase transition occurs. While for J=0J=0J=0 the critical value of $\ensuremath{\alpha}$ decreases asymptotically with 1/N1/N1/N by increasing NNN, for nonvanishing JJJ it turns out to be practically independent on NNN, allowing to identify a finite range of values of $\ensuremath{\alpha}$ where spin phase coherence is preserved also for large NNN. Then, by using matrix product state simulations, and the Mori formalism and the variational approach \`a la Feynman jointly, we unveil the features of the relaxation, that, in particular, exhibits a nonmonotonic dependence on the temperature reminiscent of the Kondo effect. For the observed quantum phase transition we also establish a criterion analogous to that of the metal-insulator transition in solids.
We consider multiple noninteracting quantum mechanical two-level systems coupled to a common bosonic bath and study its quantum phase transition with Monte Carlo simulations using a continuous imaginary time cluster algorithm. The common bath induces an effective ferromagnetic interaction between the otherwise independent two-level systems, which can be quantified by an effective interaction strength. For degenerate energy levels above a critical value of the bath coupling strength $\ensuremath{\alpha}$ all two-level systems freeze into the same state and the critical value ${\ensuremath{\alpha}}_{c}$ decreases asymptotically as 1/N1/N1/N with increasing NNN. For a finite number NNN of two-level systems the quantum phase transition (at zero temperature) is in the same universality class as the single spin-boson model; in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$ the system shows mean-field critical behavior independent of the power of the spectral function of the bosonic bath. We also study the influence of a spatial separation of the spins in a bath of bosonic modes with linear dispersion relation on the location and characteristics of the phase transition as well as on correlations between the two-level systems.
The spin-boson model is a paradigm for studying decoherence, relaxation, entanglement and other effects that arise in a quantum system coupled to environmental degrees of freedom. At zero temperature, a localization-delocalization phase transition is known to exist in the sub-Ohmic regime, where the standard density matrix renormalization group algorithm is inadequate due to the divergence in the number of low-frequency modes. This limitation is circumvented in this work by symmetrically optimizing the phonon basis and introducing an order parameter accounting for the U(1) symmetry for a two-bath spin-boson model, by which we are able to determine the classification and criticality of the phase transition explicitly. Compared with variational results, the critical phase is characterized by spontaneous vanishing of boson displacements in both the baths, resulting in an accurate phase diagram with three model parameters.
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By employing a variational approach, the density matrix renormalization group (DMRG), the exact diagonalization, and symmetry and mean-field analyses, the ground-state properties of the two-bath spin-boson model with simultaneous diagonal and off-diagonal coupling are systematically studied in the sub-Ohmic regime. A quantum phase transition from a doubly degenerate ``localized phase'' to the other doubly degenerate ``delocalized phase'' is uncovered. Via the multi-D1{\mathrm{D}}_{1}D1​ Ansatz as the variational wave function, transition points are determined accurately, consistent with the results from DMRG and exact diagonalization. An effective spatial dimension deff=2.37(6){d}_{\mathrm{eff}}=2.37(6)deff​=2.37(6) is then estimated, which is found to be compatible with the mean-field prediction. Furthermore, the quantum phase transition is inferred to be of first order for the baths described by a continuous spectral density function. In the single-mode case, however, the transition is softened.
Abstract By means of a trial wave function, the multi‐D1 ansatz, extensive variational calculations with more than 10 000 parameters are carried out to study quantum phase transitions in the ground states of a two‐impurity system embedded in a common Ohmic bath of bosons. Quantum criticality in both the impurity system and the Ohmic bosonic bath is investigated with relevant transition points and critical exponents determined accurately. With the linear grid of the Ohmic spectral density, numerical calculations herein produce a much better description of the ground states with lower energies than other calculations employing a logarithmic grid with a discretization factor far greater than unity. This offers a possible resolution to the considerable controversy on the critical coupling in the literature. Moreover, the ground‐state phase transition is inferred to be of first order in the presence of strong antiferromagnetic spin–spin coupling, at variance with that in the ferromagnetic regime or in the absence of spin–spin coupling where the transition belongs to the Kosterlitz–Thouless universality class.