Density matrix renormalization group approach to the spin-boson model

Wong, Hang; Chen, Zhide

doi:10.1103/physrevb.77.174305

Cited by 30 publications

(42 citation statements)

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“…Indeed, the standard one-bath spin-boson model (SBM1), obtained for α y = h y = 0, exhibits an interesting and much-studied [1][2][3][4][5][6][7] quantum phase transition (QPT) from a delocalized to a localized phase, with σ x = 0 or = 0, respectively, as α x is increased past a critical coupling α x,c . According to statisticalmechanics arguments, this transition is in the same universality class as the thermal phase transition of the one-dimensional (1D) Ising model with 1/r 1+s interactions.…”

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confidence: 99%

Critical and Strong-Coupling Phases in One- and Two-Bath Spin-Boson Models

et al. 2012

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For phase transitions in dissipative quantum impurity models, the existence of a quantum-toclassical 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.

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Critical and Strong-Coupling Phases in One- and Two-Bath Spin-Boson Models

et al. 2012

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“…17 In the sub-Ohmic case, all the variational calculations predicted a discontinuous transition, 18,19 but nonperturbative numerical calculations, such as numerical renormalization group ͑NRG͒ and densitymatrix renormalization group ͑DMRG͒, found a continuous transition instead. [22][23][24] The situation in the case of T 0 is also confusing. Variational calculations predicted a discontinuous transition at T 0 even in the super-Ohmic case, 25 a result which is in confliction with the known conclusion that the crossover only happens in the Ohmic and sub-Ohmic cases.…”

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Artifact of phonon-induced localization by the existing variational calculations used in the spin-boson model

Chen

Wong

2008

Phys. Rev. B

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We present energy and free-energy analyses on all variational schemes used so far in the spin-boson model at both T = 0 and T 0. It is found that all the variational schemes have fail points, at where the variational schemes fail to provide a lower energy ͑or a lower free energy at T 0͒ than the displaced-oscillator ground state and therefore the variational ground state becomes unstable, which results in a transition from a variational ground state to a displaced-oscillator ground state when the fail point is reached. Such transitions are always misidentified as crossover from a delocalized to localized phases in variational calculations, leading to an artifact of phonon-induced localization. Physics origin of the fail points and explanations for different transition behaviors with different spectral functions are found by studying the fail points of the variational schemes in the single-mode case.

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“…Besides the incorrect employment of the state as the ground state [22], the main reason for obtaining the QPT in previous literatures [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] is the unreasonable treatment of the infrared divergence existing in the spectral functions for Ohmic and sub-Ohmic dissipations, which induced the degeneracy of the ground states in the low frequency domain. To clarify this point, we demonstrate below the potential infrared divergence in the treatments using the bath mode with continuous and discretized spectra, respectively.…”

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“…In contrast, for large α, the dissipation leads to a localization of the particle in one of the two σ z eigenstates, implying a doubly degenerate ground state. On the other hand, intensive studies have recently been paid on the sub-Ohmic dissipation, which also demonstrated the QPT between localized and delocalized phases [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. In particular, the QPT could also happen in the absence of the local field (ǫ = 0) [7][8][9] from a non-degenerate ground state with zero magnetization below a critical coupling to a twofold-degenerate ground state with finite magnetization for a coupling larger than the critical value.…”

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“…In particular, the QPT could also happen in the absence of the local field (ǫ = 0) [7][8][9] from a non-degenerate ground state with zero magnetization below a critical coupling to a twofold-degenerate ground state with finite magnetization for a coupling larger than the critical value. The QPT has been found so far by different numerical studies, such as numerical renormalization group (NRG) technique [4,[6][7][8][10][11][12][13][14], quantum Monte Carlo [9], the exact diagonalization [15], the density matrix renormalization group approach [18], and variational matrix product state approach [19]. Analytical studies, such as unitary transformation method [20,21], were also employed.…”

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No Spin-Localization Phase Transition in the Spin-Boson Model without Local Field

Liu¹,

Miao²,

Li³

et al. 2013

Commun. Theor. Phys.

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We explore the spin-boson model in a special case, i.e., with zero local field. In contrast to previous studies, we find no possibility for quantum phase transition (QPT) happening between the localized and delocalized phases, and the behavior of the model can be fully characterized by the even or odd parity as well as the parity breaking, instead of the QPT, owned by the ground state of the system. Our analytical treatment about the eigensolution of the ground state of the model presents for the first time a rigorous proof of no-degeneracy for the ground state of the model, which is independent of the bath type, the degrees of freedom of the bath and the calculation precision. We argue that the QPT mentioned previously appears due to incorrect employment of the ground state of the model and/or unreasonable treatment of the infrared divergence existing in the spectral functions for Ohmic and sub-Ohmic dissipations. A two-level system coupled to an environment provides a unique test-bed for fundamental interests of quantum physics. Denoting the environment by a multimode harmonic oscillator, the spin-boson model (SBM) [1, 2] presents a phenomenological description of the open quantum system, which plays an important role in quantum information science and condensed matter physics. Particularly, for the SBM at zero temperature, it has attracted intensive interests for the quantum phase transition (QPT) happening between localized and delocalized phases regarding the spin.The standard SBM Hamiltonian in units of = 1 is given by [1],where σ z and σ x are usual Pauli operators, ǫ and ∆ are, respectively, the local field (also called c-number bias [1]) and tunneling regarding the two levels of the spin. a † k and a k are creation and annihilation operators of the bath modes with frequencies ω k , and λ k is the coupling between the spin and the bath modes. The effect of the harmonic oscillator environment is reflected by the spectral function J(ω) = π k λ 2 k δ(ω − ω k ) for 0 < ω < ω c with the cutoff energy ω c . In the infrared limit, i.e., ω →0, the power laws regarding J(ω) are of particular importance. Considering the low-energy details of the spectrum, we have J(ω) = 2παω 1−s c ω s with 0 < ω < ω c and the dissipation strength α. The exponent s is responsible for different bath with super-Ohmic bath s >1, Ohmic bath s =1 and sub-Ohmic bath s <1. * Electronic address: liutao849@163.com † Electronic address: mangfeng@wipm.ac.cnThe local field ǫ introduces asymmetry in the model, which was considered to be less important than the tunneling ∆ and thereby sometimes neglected for convenience of treatments. For the Ohmic dissipation, it was mentioned [3,4] that the SBM has a delocalized and a localized zero temperature phase, separated by a Kosterlitz-Thouless (KT) transition in the case of ǫ = 0. In the delocalized phase, realized at small dissipation strength α, the non-degenerate ground state behaves like a damped tunneling particle. In contrast, for large α, the dissipation leads to a localization of the particle ...

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Density matrix renormalization group approach to the spin-boson model

Cited by 30 publications

References 32 publications

Critical and Strong-Coupling Phases in One- and Two-Bath Spin-Boson Models

Critical and Strong-Coupling Phases in One- and Two-Bath Spin-Boson Models

Artifact of phonon-induced localization by the existing variational calculations used in the spin-boson model

No Spin-Localization Phase Transition in the Spin-Boson Model without Local Field

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