Quantum criticality in the spin-isotropic pseudogap Bose-Fermi Kondo model: Entropy, scaling, and the $g$-theorem

We study the behavior of the entropy of the pseudogap Bose-Fermi Kondo model within a dynamical large-$N$ limit, where $N$ is related to the symmetry group of the model. This model is a general quantum impurity model that describes a localized level coupled to a fermionic bath having a density of states that vanishes in a power-law fashion near the Fermi energy and to a bosonic bath possessing a power-law spectral density below a cutoff energy. As a function of the couplings to the baths, various quantum phase transitions can occur. We study how the impurity entropy changes across these zero-temperature transitions and compare our results with predictions based on the $g$-theorem. This is accomplished by an analysis of the leading and subleading scaling behaviors. Our analysis shows that the $g$-theorem does not apply to the pseudogap Bose-Fermi Kondo model at the large-$N$ level. This inapplicability originates from an anomalous contribution to the scaling function in the hydrodynamic regime where $k_{B}T>\hslash \omega$ which is absent in the quantum coherent regime, i.e., for $k_{B}T<\hslash \omega$. We also compare our results with those obtained for the Sachdev-Ye-Kitaev model.

Figure 1

(a) Sketch of the BFKM. (b) Flow diagram for different values of $r$ and $0\le \epsilon \le 1$. For the case considered in this paper ($\kappa =M/N=\frac{1}{2}$) one finds $r_c\approx 0.3115$. For details, see main text.

Figure 2

Leading-order contribution. Each vertex contains factor of $\sqrt{\frac{1}{N}}$. Summation is assumed for all $\sigma ,\alpha$ which makes the diagram order $N$.

Figure 3

Numerical results of Green's function with leading power-law fitting at different fixed points. (a) $\mathrm{Im}\left[G_f(\omega ,T={10}^{-9}D)\right]$, (b) $\mathrm{Im}\left[G_B(\omega ,T={10}^{-9}D)\right]$.

Figure 4

(a) Temperature dependency of $\mathrm{Im}\left[G_f(\omega =0)\right]$ and the power-law fitting at different fixed points. (b), (c) $\omega /T$-scaling behavior of $G_f\left(\omega \right)$ and $G_B\left(\omega \right)$ with the exponent obtained from scaling ansatz.

Figure 5

Numerical results of $\omega /T$-scaling at ${\text{C}}^{\prime }$ fixed point for (a) $G_f\left(\omega \right)$, (b) $G_B\left(\omega \right)$, (c) $\chi \left(\omega \right)$, (d) $\mathcal{G}\left(\omega \right)$. The parameters are $r=\frac{1}{4},\epsilon =\frac{2}{5},\kappa =\frac{1}{2}$.

Figure 6

Numerical results of Green's function with leading power-law fitting at different fixed points. (a) $G_f(\tau ,T)$. (b) $G_B(\tau ,T)$ at the intermediate fixed points C, ${\text{C}}^{\prime }$, ${\text{LM}}^{\prime }$, and MCK. The dashed lines indicate the leading power-law behavior. The exponent is in line with ${\alpha }_f$ and ${\alpha }_B$ from Table 1.

Figure 7

The impurity entropy $s\left(T\right)$ for $r=0$ and $g=0$ vs $T/T_K^0$ for $r=0$. The fixed-point values at MCK [$s(T=0)$] and LM [$s(T/T_K^0\to \infty )$] are recovered independently of the value of $J_K$. $s(T=0)$ depends on $\kappa =M/N$ (see Ref. [50]).

Figure 8

(a) Boundary entropy of the pseudogap BFKM model for $r=\frac{1}{4},\epsilon =\frac{2}{5},\kappa =\frac{1}{2}$ and coupling constants $J_K$ and $g$ such that $s(T=0)=s_{\mathrm{MCK}}$ for $J_K=0.7D,g=0$, $s(T=0)=s_{\text{LM}}$ for $J_K=0.325D,g=0$, $s(T=0)=s_{\text{C}}$ for $J_K=0.45D,g=0$, $s(T=0)=s_{{\text{C}}^{\prime }}$ for $J_K=0.8D,g=0.375D$, and $s(T=0)=s_{{\text{LM}}^{\prime }}$ for $J_K=0,g=0.325D$. (b) Same as (a) for values of $s(T\to 0)$ between 0.68 and 0.7 to show the differences between $s_{\text{C}}$ and $s_{\text{LM}}$.

Figure 9

Comparison of different approach of determining the residual boundary entropy, labeled by $s$, $s^{*}$, and $s^{\mathrm{corr}}$. The specifics of each of these approaches are given in the main text. (a) $r$ dependence of $s$, $s^{*}$, and $s^{\mathrm{corr}}$ at the MCK fixed point and (b) $\epsilon$ dependence of $s$, $s^{*}$, and $s^{\mathrm{corr}}$ at the ${\text{LM}}^{\prime }$ fixed point with difference $\epsilon$.

Figure 10

Comparison of $arctan\left[\text{Re}\left\{G_f\right\}/\text{Im}\left\{G_f\right\}\right]$ obtained from Eq. (41) and denoted by black dashed-dotted lines with results obtained from self-consistent solutions of the saddle-point equations. (a) Comparison for $r=0.2$ at the MCK fixed point. (b) Comparison for $\epsilon =0.5$ at the ${\text{LM}}^{\prime }$ fixed point. (c) $r$-dependent comparison at the MCK fixed point at low $T$ and for $\omega \lesssim T$. (d) $\epsilon$-dependent comparison at the ${\text{LM}}^{\prime }$ fixed point at low $T$ and for $\omega \lesssim T$.

Figure 11

The behavior of $G_f\left(\tau \right)$ of ${\text{LM}}^{\prime }$ fixed point with $\epsilon =0.4$. (a) The temperature dependency and (b) near $\beta /2$ regime.

Figure 12

(a) Boundary entropy $s(J_K,g,T_0)$ at $T_0={10}^{-7}D$ as a function of $J_k$ and $g$. (b) $s(J_K^0,g,T)$ across ${\text{C}}^{\prime }$ at $J_k^0=0.8D$ vs $g$ and $T$. (c) $s(J_K,g=0,T)$ across C vs $J_k$ and $T$. (d) $s(J_K=0,g,T)$ near ${\text{LM}}^{\prime }$ vs $g$ and $T$. (e) $s(J_K,g=0,T)$ near the MCK of $r=0$ case vs $J_k$ and $T$. (f) Flow diagram for $0<r<r_c$ case.

Figure 13

The contour $\mathcal{C}=c_1+c_2+c_3+c_4+c_5+c_6$ used to evaluate the integral (D4).

Figure 14

Behavior of $G_f\left(\tau \right)$ for the $r=0$ MCK fixed point case. (a) $G_f\left(\tau \right)$ at various temperatures. The black dashed line is the fitting of $G_f\left(\tau \right)$ with $\frac{1}{A{x}^{-{\alpha }_f}+2}$ at $r=0$ case. (b) Behavior of $G_f\left(\tau \right)$ near $\beta /2$ regime.

Figure 15

Numerical solution of (a) $G\left(\tau \right)$ and (b) $\mathrm{\Sigma }\left(\tau \right)$ at different temperature. $\omega /T$-scaling of $\mathrm{Im}G\left(\omega \right)$ and $\mathrm{Im}\mathrm{\Sigma }\left(\omega \right)$ at different temperature.

Figure 16

Temperature dependence of the entropy of the SYK model. Our result interpolates between the analytical known results for $s_{\mathrm{SYK}}(T=0)$ and $s_{\mathrm{SYK}}(T\gg J)$.