Quantum phase transitions in a resonant-level model with dissipation: Renormalization-group studies

We study a spinless level that hybridizes with a fermionic band and is also coupled via its charge to a dissipative bosonic bath. We consider the general case of a power-law hybridization function $\Gamma \left(\omega \right)\propto {\mid \omega \mid }^r$, with $r⩾0$, and a bosonic-bath spectral function $B\left(\omega \right)\propto {\omega }^s$, with $s⩾-1$. For $r<1$ and $\max (0,2r-1)<s<1$, this Bose-Fermi quantum impurity model features a continuous zero-temperature transition between a delocalized phase, with tunneling between the impurity level and the band, and a localized phase, in which dissipation suppresses tunneling in the low-energy limit. The phase diagram and the critical behavior of the model are elucidated using perturbative and numerical renormalization-group techniques, between which there is excellent agreement in the appropriate regimes. For $r=0$, this model’s critical properties coincide with those of the spin-boson and Ising Bose-Fermi Kondo models, as expected from bosonization.

Figure 1

Schematic phase diagram of the dissipative resonant-level model (1), in the parameter space spanned by exponents $r$ and $s$ characterizing the low-energy behavior of fermionic and bosonic baths, respectively. (A nonzero coupling to both baths is assumed.) For $\max (0,2r-1)<s⩽1$, the model shows a boundary quantum phase transition (as the couplings $v_0$ and $g_0$ are varied) between a delocalized phase and a localized phase. (The physics along the line $r=0$ is identical to that of the spin-boson model.) In contrast, for $s>1$ and $r<1$, the system is generically delocalized, whereas it is localized in the rest of the parameter regime (shaded). The impurity entropy $S_{\mathrm{imp}}$ is discussed in the text. Two perturbative RG expansions are employed: around the free-impurity fixed point, where the expansion is controlled about $r=s=1$ (Sec. 2B), and around the resonant-level fixed point, where the expansion is controlled in $1-2r+s$ (Sec. 2C).

Figure 2

Diagrams appearing in the perturbative expansion for the dissipative resonant-level model. Dashed, solid, and wiggly lines denote, respectively $f$, $c$, and $\varphi$ propagators. The gray (black) circles are the interaction vertices $v$ $\left(g\right)$. [(a) and (b)] $f$ fermion self-energy diagrams to one-loop order. (c) One-loop vertex renormalization of $g$.

Figure 3

Schematic RG flow diagrams for the dissipative resonant-level model with p-h symmetry. Although these diagrams are obtained by expansion about the free-impurity fixed point and hence are formally valid as $r,s\to 1$, they are confirmed by expansion about the delocalized fixed point (Sec. 2C) and NRG calculations (Sec. 3) to capture the correct physics for all $r⩾0$ and $0<s<1$. The horizontal axis denotes the renormalized bosonic coupling $g^2$; the vertical axis denotes the renormalized hybridization $v^2$. (a) $r<r_{+}=(1+s)∕2$: Stable fixed points at Deloc and Loc describe the delocalized and localized phases, respectively. The continuous impurity QPT is controlled by the critical fixed point Cr. (b) $r_{+}<r<1$: The delocalized (Deloc) fixed point is unstable against finite $g$. As $r\to 1^{-}$, the Deloc fixed point merges with the free-impurity fixed point (FImp). (c) $r⩾1$: $v$ is irrelevant. In both (b) and (c), the flow is toward Loc for any nonzero $g$.

Figure 4

(a) Exact relation for the local susceptibility. The black triangle denotes the full vertex function and the dashed double line denotes the full impurity level propagator. (b) Definition of the conduction-electron $T$ matrix. The large dot denotes the full hybridization vertex.

Figure 5

RG flow diagram of the dissipative resonant-level model near the delocalized (Deloc) fixed point for $s$, $\widetilde{ϵ}>0$. The two stable phases are governed by the delocalized $(g=0)$ and localized $(g=\infty )$ fixed points, separated by the critical fixed point [$g=g^{*}$ specified in Eq. (35)].

Figure 6

(Color online) (a) The lowest NRG eigenstates $E_N$ vs even iteration number $N$ for $(r,s)=(0.85,0.9)$, hybridization strength ${\Gamma }_0=0.1$, and a range of dissipation strengths $B_0-B_{0,c}=0,\pm {10}^{-3},\pm {10}^{-2}$. The flows are typical of those for $\max (0,2r-1)<s<1$ with $0⩽r<1$. The levels at the critical coupling $B_0=B_{0,c}\approx 0.3731$ are shown as bold dotted lines, while those nearby in the delocalized $(B_0<B_{0,c})$ [localized $(B_0>B_{0,c})$] phase are shown as solid [dashed] lines. As $B_0$ approaches $B_{0,c}$ in either phase, the levels follow those of the unstable critical fixed point down to progressively lower temperatures, before crossing over to the levels characteristic of the delocalized or localized stable fixed point. (b) NRG level flows for $(r,s)=(0.975,0.9)$. In this case, and more generally for $s<\max (0,2r-1)$ with $0⩽r<1$, the flow is toward the localized fixed point for any $B_0>0$, but follows the delocalized fixed point down to progressively lower temperatures as $B_0$ is reduced toward zero. The solid lines show the flow for $B_0=0$.

Figure 7

(Color online) Phase diagram in the $r\text{−}{B}_0$ plane, obtained using NRG for the fixed bosonic-bath exponent $s$ and the hybridization strength ${\Gamma }_0$ shown in the legend. For $0<r<r_{+}=(1+s)∕2$, we find a continuous QPT between delocalized (Deloc) and localized (Loc) phases. The critical dissipation strength $B_{0,c}$ is found to vanish continuously at $r=r_{+}$. For $r⩾r_{+}$, only the localized phase can be accessed for $B_0>0$. The inset shows the vanishing of $B_{0,c}$ with decreasing $1-2r+s$ in each case, compared to the results obtained from the perturbative analysis.

Figure 8

Critical dissipation strength $B_{0,c}$ vs hybridization strength ${\Gamma }_0$ for the $(r,s)$ pairs specified in the legend. We find that $B_{0,c}\propto {\Gamma }_0^x$, with $x=(1-s)∕(1-r)$.

Figure 9

Crossover scale $T^{*}$ vs $\mid t\mid =\mid B_0-B_{0,c}\mid ∕B_{0,c}$ for the $(r,s)$ pairs specified in the legend. In the vicinity of the transition $(\mid t\mid ⪡1)$, $T^{*}\propto {\mid t\mid }^{\nu }$. The correlation-length exponent $\nu (r,s)$ is independent both of the hybridization ${\Gamma }_0$ and of the phase from which the QCP is approached.

Figure 10

(Color online) (a) Correlation-length exponent $\nu$ vs conduction-band exponent $r$ for two values of the bosonic-bath exponent $s$, as shown in the legend. The symbols show NRG data, while the dashed lines are the corresponding perturbative results [Eq. (16)], expanding about the free-impurity fixed point. We find that ${\nu }^{-1}$ vanishes at $r=r_{+}$, in keeping with the qualitatively distinct behavior for $2\widetilde{ϵ}\equiv 1-2r+s\gtrless 0$. (b) The same data plotted vs $2\widetilde{ϵ}$. For small $\widetilde{ϵ}$, ${\nu }^{-1}\approx 2\widetilde{ϵ}$, a result [Eq. (37)] (shown as a dotted line) obtained by a perturbative expansion about the delocalized fixed point.

Figure 11

(Color online) Critical exponents $\beta$ and $\delta$, defined in Eq. (25), for $(r,s)=(0.85,0.9)$ and ${\Gamma }_0=0.1$, where $B_{0,c}\approx 0.3731$. (a) Continuous vanishing of order parameter $m_{\mathrm{imp}}$ vs $t=(B_0-B_{0,c})∕B_{0,c}$ as $t\to 0^{+}$ with characteristic exponent $\beta$ (extracted as the limiting slope of the data on a logarithmic scale). The inset shows the data on an absolute scale. (b) Variation of $m_{\mathrm{imp}}(T=0)$ with local level energy ${\epsilon }_f$ at the critical point $t=0$. The data clearly follow a power law for small ${\epsilon }_f$, defining the exponent $\delta$.

Figure 12

Static local susceptibility ${\chi }_{\mathrm{loc}}\left(T\right)$ vs $T$ for $(r,s)=(0.2,0.5)$, ${\Gamma }_0=0.1$, and $B_0=0.4902$ (circles), $0.5002\approx B_{0,c}$ (stars), and 0.5102 (squares). The anomalous exponent in the quantum-critical regime is found to be ${\eta }_{\chi }=1-s$, independent of $r$. See text for further discussion.

Figure 13

(Color online) (a) Spectral function $A\left(\omega \right)$ vs $\mid \omega \mid$ for $r=0.65$, $s=0.8$, ${\Gamma }_0={10}^{-3}$, and three values of the dissipation strength: $B_0=0$ (dotted line), $B_0=B_{0,c}-{10}^{-5}$ (solid line), and $B_0=B_{0,c}=0.03247113$ (dashed line). At the quantum critical point $B_0=B_{0,c}$, $A\left(\omega \right)\propto {\mid \omega \mid }^{-r}$, which behavior is also followed for $B_0$ close to $B_{0,c}$ and $\mid \omega \mid ⪢T^{*}$. In the delocalized phase $B_0<B_{0,c}$, there is a crossover in $A\left(\omega \right)$ to the behavior in Eq. (45) for $\mid \omega \mid ⪡T^{*}$. For the data shown, $T^{*}\sim \mathcal{O}\left({10}^{-26}\right)$. (b) The crossover behavior is more readily seen in the modified spectral function $\mathcal{F}\left(\omega \right)=\pi {\Gamma }_0{\sec }^2\left(\frac{\pi }{2}r\right){\mid \omega \mid }^{r}A\left(\omega \right)$, which shows the ultimate low-$\omega$ behavior $\mathcal{F}(\omega =0)=1$ throughout the delocalized phase $B_0<B_{0,c}$, $0<\mathcal{F}(\omega =0)<1$ for $B_0=B_{0,c}$, and $\mathcal{F}(\omega =0)=0$ throughout the localized phase $B_0>B_{0,c}$. In order of decreasing crossover scale, delocalized-phase spectra are shown for $B_0=0$ (dotted line) and for $B_{0,c}-B_0={10}^{-3}$, ${10}^{-4}$, ${10}^{-5}$, and ${10}^{-6}$ (solid lines); localized-phase spectra (dashed lines) are shown for $B_0=0.05$ and $B_0-B_{0,c}={10}^{-3}$, ${10}^{-4}$, ${10}^{-5}$, and ${10}^{-6}$. The critical spectrum is shown as a thick dashed line.

Figure 14

(Color online) $\pi {\Gamma }_{0}A\left(\omega \right)$ vs $\omega$ for the case of a metallic fermionic density of states $(r=0)$ and Ohmic dissipation $(s=1)$ for ${\Gamma }_0={10}^{-3}$. Spectra are shown for increasing dissipation strength $B_0$ (see legend) in the delocalized phase. The spectrum is a simple Lorentzian for $B_0=0$, and the vanishing width as $B_0\to B_{0,c}^{-}$ indicates a suppression of tunneling between the local level and the conduction band.