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arXiv:0706.2085 · Physical Review B

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

Chung‐Hou Chung, Matthew T. Glossop, Lars Fritz, Marijana Kirćan, Kevin Ingersent, Matthias Vojta C

Published Dec 4, 2007• 12 citations
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Abstract

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 $\ensuremath{\Gamma}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\mid}\ensuremath{\omega}\ensuremath{\mid}}^{r}$, with $r\ensuremath{\geqslant}0$, and a bosonic-bath spectral function $B(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{s}$, with $s\ensuremath{\geqslant}\ensuremath{-}1$. For $r<1$ and $\mathrm{max}(0,2r\ensuremath{-}1)<s<1$, 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.

DOI
10.1103/physrevb.76.235103

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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 \ensuremathΓ(\ensuremathω)\ensuremath∝\ensuremath∣\ensuremathω\ensuremath∣r\ensuremath{\Gamma}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\mid}\ensuremath{\omega}\ensuremath{\mid}}^{r}\ensuremathΓ(\ensuremathω)\ensuremath∝\ensuremath∣\ensuremathω\ensuremath∣r, with r\ensuremath⩾0r\ensuremath{\geqslant}0r\ensuremath⩾0, and a bosonic-bath spectral function B(\ensuremathω)\ensuremath∝\ensuremathωsB(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{s}B(\ensuremathω)\ensuremath∝\ensuremathωs, with s\ensuremath⩾\ensuremath−1s\ensuremath{\geqslant}\ensuremath{-}1s\ensuremath⩾\ensuremath−1. For r<1r<1r<1 and max(0,2r\ensuremath−1)<s<1\mathrm{max}(0,2r\ensuremath{-}1)<s<1max(0,2r\ensuremath−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=0r=0r=0, this model's critical properties coincide with those of the spin-boson and Ising Bose-Fermi Kondo models, as expected from bosonization.
We extend the numerical renormalization-group method to treat Bose-Fermi Kondo models (BFKMs) describing a local moment coupled both to a conduction band and to a dissipative bosonic bath representing, e.g., lattice or spin collective excitations of the environment. We apply the method to the Ising-symmetry BFKM with a structureless band and a bath spectral function $\ensuremath{\eta}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{s}$. The method is valid for all bath exponents sss and all temperatures TTT. For 0<s<10<s<10<s<1, the range of interest in the context of heavy-fermion quantum criticality, an interacting critical point, characterized by hyperscaling of exponents and $\ensuremath{\omega}∕T$ scaling, describes a continuous quantum phase transition between Kondo-screened and localized phases. For Ohmic dissipation s=1s=1s=1, where the model is relevant to certain dissipative mesoscopic qubit devices, the transition is found to be Kosterlitz-Thouless-like. In both regimes the impurity spectral function for the corresponding Anderson model shows clearly the collapse of the Kondo resonance at the transition. Connection is made to other recent results for the BFKM and the spin-boson model.
We study the competition between Kondo physics and dissipation within an Anderson model of a magnetic impurity level that hybridizes with a metallic host and is also coupled, via the impurity charge, to the displacement of a bosonic bath having a spectral density proportional to ${\ensuremath{\omega}}^{s}$. As the impurity-bath coupling increases from zero, the effective Coulomb interaction between two electrons in the impurity level is progressively renormalized from its repulsive bare value until it eventually becomes attractive. For weak hybridization, this renormalization in turn produces a crossover from a conventional spin-sector Kondo effect to a charge Kondo effect. At particle-hole symmetry, and for sub-Ohmic bath exponents 0<s<10<s<10<s<1, further increase in the impurity-bath coupling results in a continuous zero-temperature transition to a broken-symmetry phase in which the ground-state impurity occupancy ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}_{d}$ acquires an expectation value ${⟨{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}_{d}⟩}_{0}\ensuremath{\ne}1$. The response of the impurity occupancy to a locally applied electric potential features the hyperscaling of critical exponents and $\ensuremath{\omega}∕T$ scaling that are expected at an interacting critical point. The numerical values of the critical exponents suggest that the transition lies in the same universality class as that of the sub-Ohmic spin-boson model. For the Ohmic case s=1s=1s=1, the transition is instead of Kosterlitz-Thouless type. Away from particle-hole symmetry, the quantum phase transition is replaced by a smooth crossover but signatures of the symmetric quantum critical point remain in the physical properties at elevated temperatures and/or frequencies.
Recent work is reviewed on continuous quantum phase transitions in impurity models, both with fermionic and bosonic baths – these transitions are interesting realizations of boundary critical phenomena at zero temperature. The models with fermion bath are generalizations of the standard Kondo model, with the common feature that Kondo screening of the localized spin can be suppressed due to competing processes. The models with boson bath are related to the spin–boson model of dissipative two-level systems, where the interplay between tunneling and friction results in multiple phases. The competition inherent to all models can generate unstable fixed points associated with quantum phase transitions, where the impurity properties undergo qualitative changes. Interestingly, certain impurity transitions feature both lower-critical and upper-critical “dimensions” and allow for epsilon-type expansions. Results are presented for a number of observables, obtained by both analytical and numerical renormalization group techniques, and connections made to experiments.
We study a pseudogap Anderson-Holstein model of a magnetic impurity level that hybridizes with a conduction band whose density of states vanishes in power-law fashion at the Fermi energy, and couples, via its charge, to a nondispersive bosonic mode (e.g., an optical phonon). The model, which we treat using poor-man's scaling and the numerical renormalization group, exhibits quantum phase transitions of different types depending on the strength of the impurity-boson coupling. For weak impurity-boson coupling, the suppression of the density of states near the Fermi energy leads to quantum phase transitions between strong-coupling (Kondo) and local-moment phases. For sufficiently strong impurity-boson coupling, however, the bare repulsion between a pair of electrons in the impurity level becomes an effective attraction, leading to quantum phase transitions between strong-coupling (charge Kondo) and local-charge phases. Even though the Hamiltonian exhibits different symmetries in the spin and charge sectors, the thermodynamic properties near the two types of quantum phase transition are closely related under spin-charge interchange. Moreover, the critical responses to a local magnetic field (for small impurity-boson coupling) and to an electric potential (for large impurity-boson coupling) are characterized by the same exponents, whose values place these quantum-critical points in the universality class of the pseudogap Anderson model. One specific case of the pseudogap Anderson-Holstein model may be realized in a double-quantum-dot device, where the quantum phase transitions manifest themselves in the finite-temperature linear electrical conductance.
We extend the numerical renormalization-group method to Bose-Fermi Kondo models (BFKMs), describing a local moment coupled to a conduction band and a dissipative bosonic bath. We apply the method to the Ising-symmetry BFKM with a bosonic bath spectral function eta(omega) proportional omega(s), of interest in connection with heavy-fermion criticality. For 0 < s < 1, an interacting critical point, characterized by hyperscaling of exponents and omega/T scaling, describes a quantum phase transition between Kondo-screened and localized phases. A connection is made to other results for the BFKM and the spin-boson model.
We address the phenomenon of critical Kondo destruction in pseudogap Bose-Fermi Anderson and Kondo quantum impurity models. These models describe a localized level coupled both to a fermionic bath having a density of states that vanishes like ${|\ensuremath{\epsilon}|}^{r}$ at the Fermi energy ($\ensuremath{\epsilon}=0$) and, via one component of the impurity spin, to a bosonic bath having a sub-Ohmic spectral density proportional to ${|\ensuremath{\omega}|}^{s}$. Each bath is capable by itself of suppressing the Kondo effect at a continuous quantum phase transition. We study the interplay between these two mechanisms for Kondo destruction using continuous-time quantum Monte Carlo for the pseudogap Bose-Fermi Anderson model with 0<r<120<r<\frac{1}{2}0<r<21​ and $\frac{1}{2}\ensuremath{\le}s<1$, and applying the numerical renormalization group to the corresponding Kondo model. At particle-hole symmetry, the models exhibit a quantum-critical point between a Kondo (fermionic strong-coupling) phase and a localized (Kondo-destroyed) phase. The two solution methods, which are in good agreement in their domain of overlap, provide access to the many-body spectrum, as well as to correlation functions including, in particular, the single-particle Green's function and the static and dynamical local spin susceptibilities. The quantum-critical regime exhibits the hyperscaling of critical exponents and $\ensuremath{\omega}/T$ scaling in the dynamics that characterize an interacting critical point. The (r,s)(r,s)(r,s) plane can be divided into three regions: one each in which the calculated critical properties are dominated by the bosonic bath alone or by the fermionic bath alone, and between these two regions, a third in which the bosonic bath governs the critical spin response but both baths influence the renormalization-group flow near the quantum-critical point.