2010 **Abstract:** In this paper we give a general introduction to quantum critical phenomena, which we practically illustrate by a detailed study of the low energy properties of the spin boson model (SBM), describing the dynamics of a spin 1/2 impurity (or more generically a two-level system) coupled to a bath of independent harmonic oscillators. We show that the behavior of the model is very sensitive to the bath spectrum, in particular how the properties of the quantum critical point in the SBM are affected by the functional …

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“…2(d) where we show our estimate for its 68% and 95% confidence intervals. These suggest that α c 1.25, consistent with the known analytic results [30,40,42]. We note that identifying α c precisely from the time dependence of S z is particularly challenging: since the localisation transition is in the BKT class [40], the order parameter approaches zero continuously.…”

confidence: 82%

“…2(d) where we show our estimate for its 68% and 95% confidence intervals. These suggest that α c 1.25, consistent with the known analytic results [30,40,42]. We note that identifying α c precisely from the time dependence of S z is particularly challenging: since the localisation transition is in the BKT class [40], the order parameter approaches zero continuously.…”

confidence: 82%

“…This model is known to show a rich variety of physics depending on the particular form of spectral density and system parameters chosen. When the spectral density is Ohmic, J(ω) = 2αω exp(−ω/ω c ), the model is known to exhibit a quantum phase transition in the BKT universality class [40], at a critical value of the systemenvironment coupling α = α c [30,41]. The transition takes the system from a delocalised phase below α c , where any spin excitation decays ( S z = 0 in the steady state), to a localised phase above α c ( S z = 0 in the steady state).…”

confidence: 99%

“…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.…”

mentioning

confidence: 99%

“…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.…”

mentioning

confidence: 99%