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

We provide an extensive study of the subohmic spin-boson model with power-law density of states $J\left(\omega \right)\propto {\omega }^s$ (with $0<s\le 1$), 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.

Figure 1

Sketch of the generic phase diagram of the spin-boson model, in the ohmic ($s=1$) and subohmic ($0<s<1$) cases, respectively, as a function of the dissipation parameter $\alpha$ and the dimensionless magnetic field $\Delta /{\omega }_c$. The superohmic ($s>1$) regime shows only the crossover from coherent to incoherent dynamics, without a localization transition. In all cases, the coherent/incoherent crossover happens at comparable coupling strength, namely $\alpha \simeq 0.5$.

Figure 2

The transverse (along the bath) spin susceptibility $C_z\left(\omega \right)$ of the subohmic spin-boson model at $s=0.1$, $\Delta =0.1{\omega }_c$, and $\alpha =0.000125$. Raw data $|\langle 0,z|{\sigma }_z{|k,z\rangle |}^2/\left[2N_z({\epsilon }_{k,z}-{\epsilon }_{0,z})\right]$ are shown as circles for $N_z=20$ combined NRG calculations, the solid line is the perturbative result, and the three dashed lines are the various NRG broadenings discussed in the text. All frequency-dependent plots are given in units of ${\omega }_c=1$.

Figure 3

Spin susceptibilities $C_z\left(\omega \right)$, $C_y\left(\omega \right)$, and $C_x\left(\omega \right)$ (top to bottom panels) for $\Delta /{\omega }_c=0.1$, $s=0.1$, and increasing dissipation strength $\alpha =0.001,0.004,0.007$ (bottom to top). The critical point is located at ${\alpha }_c\simeq 0.007$. The NRG plots were generated with $N_z=20$ interleaved $z$ averages and used the $b$ trick discussed in Sec. 2c, while the perturbative expressions ($G^0{W}^0$ scheme) are given in Eqs. (36) and (37). Note that the upper panel has a range in $\omega$ that differs from the two other panels.

Figure 4

Spin susceptibilities $C_z\left(\omega \right)$, $C_y\left(\omega \right)$, and $C_x\left(\omega \right)$ (top to bottom panels) for $\Delta /{\omega }_c=0.1$, $s=0.5$, and increasing dissipation strength $\alpha =0.02,0.06,0.1$ (bottom to top). The critical point is located at ${\alpha }_c\simeq 0.105$. The NRG plots were generated with $N_z=10$ interleaved $z$ averages and used the $b$ trick discussed in Sec. 2c, while the perturbative expressions ($G^0{W}^0$ scheme) are given in Eqs. (36) and (37). Note that the upper panel has a range in $\omega$ that differs from the two other panels.

Figure 5

Spin susceptibilities $C_z\left(\omega \right)$, $C_y\left(\omega \right)$, and $C_x\left(\omega \right)$ (top to bottom panels) for $\Delta /{\omega }_c=0.1$, $s=1$, and increasing dissipation strength $\alpha =0.1,0.5,0.9$ (bottom to top). The critical point is located at ${\alpha }_c\simeq 1$. The NRG plots were generated with $N_z=3$ interleaved $z$ averages and used the $b$ trick discussed in Sec. 2c, while the perturbative expressions ($G^0{W}^0$ scheme) are given in Eqs. (36) and (37).

Figure 6

Bare vertex of the spin-boson model in the Majorana language.

Figure 7

Exact formal diagrammatic expression for the fermionic and bosonic self-energies in terms of the full (exact) fermionic and bosonic Green’s functions ($G_i$ and $G_{\Phi }$, respectively) and the full (exact) three-particle vertex $\Gamma$.

Figure 8

Lowest-order perturbative Majorana self-energies, obtained from bare fermion and bare bosonic propagators ($G^0{W}^0$ scheme).

Figure 9

Self-consistent fermionic self-energies (analogous to the $G{W}^0$ approximation), allowing improvement of the calculation in frequency ranges where lowest-order perturbation incorrectly predicts the absence of spin excitations. This scheme recovers multiparticle excitations that fill in the spurious gaps of bare perturbation theory.

Figure 10

Longitudinal spin susceptibility $C_x\left(\omega \right)$ for $\Delta /{\omega }_c=0.1$, $s=0.1$, and $\alpha =0.001,0.004,0.007$ (bottom to top), comparing the NRG result to the self-consistent $G{W}^0$ scheme and allowing recovery of the low-energy tail below the threshold $\omega =\Delta$ (results are quantitatively correct only for the smallest $\alpha$ value; see Sec. 4e for the improvements that are required near the quantum critical point).

Figure 11

Transverse spin susceptibility $C_z\left(\omega \right)$ for $\Delta /{\omega }_c=0.1$, $s=0.1$, and $\alpha =0.002$, comparing the raw NRG data to the self-consistent $G{W}^0$ scheme. In contrast to bare perturbation theory, which leads to a gap for $\omega >{\omega }_c$, the self-consistent resummation correctly describes the tail arising from multiparticle excitations above the threshold ${\omega }_c$ (note the linear frequency scale).

Figure 12

Fermionic self-energies (without self-consistency), including a dressed bosonic propagator (this is analogous to the $G^{0}W$ approximation). This scheme is necessary to recover the critical modes near the quantum critical point but asks for a careful evaluation of the bosonic self-energy (see Fig. 13 and the related discussion in Secs. 4b and 4d).

Figure 13

Diagrammatic representation for the bosonic self-energy computed up to two-loop order, leading to Eq. (53).

Figure 14

Phase diagram in the ($s$,$\alpha$) plane for $h\equiv \Delta /{\omega }_c=0.1$, comparing the NRG data to the Kosterlitz flow obtained from the resolution of Eq. (59) (giving reliable results for $0.9<s<1$), and to the one-loop and two-loop formulas (52) and (55) (the latter is accurate on the wide range $0<s<0.8$).

Figure 15

Phase diagram in the ($s$,$\alpha$) plane for $h\equiv \Delta /{\omega }_c=0.00001,0.0001,0.001,0.01,0.1$ (bottom to top) comparing the NRG simulations versus the two-loop weak dissipation result (valid for $s\ll 1$).

Figure 16

(Upper panel) Ladder series for the bosonic self-energy that is needed to obtain the correct phase boundary within the diagrammatic calculation. (Lower panel) Single frequency approximation used at the two-rung level (and similarly for higher-order contributions); see text.

Figure 17

Renormalization factor ${\alpha }^R/\alpha$ of the dissipation strength from Eq. (63) computed for $\Delta /{\omega }_c=0.1$ as a function of $\alpha$ up to the critical point (${\alpha }_c=0.007,0.105$, respectively, for the two values $s=0.1,0.5$).

Figure 18

Spin susceptibilities $C_z\left(\omega \right)$, $C_y\left(\omega \right)$, and $C_x\left(\omega \right)$ (top to bottom panels) for $\Delta /{\omega }_c=0.1$, $s=0.1$, and increasing dissipation strength $\alpha =0.001,0.004,0.007$ (bottom to top) compared to the full Majorana diagrammatics ($GW$$\Gamma$ scheme).

Figure 19

Spin susceptibilities $C_z\left(\omega \right)$, $C_y\left(\omega \right)$, and $C_x\left(\omega \right)$ (top to bottom panels) for $\Delta /{\omega }_c=0.1$, $s=0.5$, and increasing dissipation strength $\alpha =0.02,0.06,0.1$ (bottom to top) compared to the full Majorana diagrammatics ($GW$$\Gamma$ scheme).