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
aa r X i v : . [ c ond - m a t . s t r- e l ] J un Quantum phase transitions in a resonant-level model with dissipation:Renormalization-group studies
Chung-Hou Chung, Matthew T. Glossop, Lars Fritz,
3, 4
Marijana Kir´can, Kevin Ingersent, and Matthias Vojta Electrophysics Department, National Chiao-Tung University, Hsinchu, Taiwan, R.O.C. Department of Physics, University of Florida, Gainesville, FL 32611-8440, USA Institut f¨ur Theoretische Physik, Universit¨at K¨oln, Z¨ulpicher Straße 77, 50937 K¨oln, Germany Department of Physics, Harvard University, Cambridge MA 02138, USA Max-Planck-Institut f¨ur Festk¨orperforschung, Heisenbergstraße 1, 70569 Stuttgart, Germany (Dated: October 31, 2018)We study a spinless level that hybridizes with a fermionic band and is also coupled via its chargeto a dissipative bosonic bath. We consider the general case of a power-law hybridization functionΓ( ω ) ∝ | ω | r with r ≥
0, and a bosonic bath spectral function B ( ω ) ∝ ω s with s ≥ −
1. For r < , r − < s <
1, this Bose-Fermi quantum impurity model features a continuouszero-temperature transition between a delocalized phase, with tunneling between the impurity leveland the band, and a localized phase, in which dissipation suppresses tunneling in the low-energylimit. The phase diagram and the critical behavior of the model are elucidated using perturbativeand numerical renormalization-group techniques, between which there is excellent agreement in theappropriate regimes. For r = 0 this model’s critical properties coincide with those of the spin-bosonand Ising Bose-Fermi Kondo models, as expected from bosonization. I. INTRODUCTION
Quantum phase transitions in mesoscopic systemsform a growing area of condensed matter research. Froma theoretical perspective, it is known that models ofa finite system (the “impurity”) coupled to infinitebaths may exhibit boundary quantum phase transitions(QPTs), at which only a subset of the degrees of free-dom becomes critical. Such models help to advance ourunderstanding of quantum criticality in strongly corre-lated systems: Concepts and solution techniques devel-oped in the impurity context may be applied to latticemodels, e.g., within the framework of dynamical mean-field theory (DMFT) and its extensions. This approachhas been followed in connection with the “local critical-ity” proposed to underlie the anomalous non-Fermi-liquidbehavior of several heavy-fermion systems. On the ex-perimental side, QPTs in mesoscopic few-level systemsare of great interest, both for the unprecedented oppor-tunity to probe quantum criticality in a direct and highlycontrolled fashion, and for their numerous potentialtechnological applications, e.g., in nanoelectronics andquantum information processing.
In recent years, QPTs have been identified and studiedin a number of quantum impurity models. Such mod-els can contain both fermionic bands (e.g., conduction-electron quasiparticles) and bosonic baths (e.g., phonons,spin fluctuations, or electromagnetic noise). Analyticaland numerical techniques have been refined to analyzethe critical behavior of these models. Analytical ap-proaches based on bosonization or conformal field theoryhave been used extensively, although their applicability islimited, e.g., to certain forms of the bath spectrum. Forother situations, powerful epsilon-expansion techniqueshave been developed. As such expansions are asymp-totic in character, a comparison with numerical results is mandatory to assess their reliability.An example with especially rich behavior is thefermionic pseudogap Kondo model, which featuresQPTs between Kondo-screened and local-momentground states. Essentially perfect agree-ment between the results of various epsilon expan-sions (around different critical dimensions) and numer-ical renormalization-group (NRG) calculations has beenfound in critical exponents as well as universal ampli-tudes such as the residual impurity entropy.
Impurity models that include bosons are harder totackle numerically than pure-fermionic problems due tothe large Hilbert space, and fewer results are available.The development of a bosonic version of Wilson’sNRG approach has made possible a detailed nonper-turbative study of the spin-boson model, where tunnel-ing in a two-state system competes with dissipation. For the case of Ohmic dissipation, the spin-boson modelhas long been known to display a QPT of the Kosterlitz-Thouless type. In the sub-Ohmic case, the model insteadexhibits a line of continuous QPTs governed by interact-ing quantum critical points (QCPs). (The latterlie in a different universality class than the QCP of thepseudogap Kondo model.)Of particular interest, both for mesoscopics and inthe context of extended DMFT for correlated lattice-systems, are impurity models with fermionic and bosonic baths. The best-studied member of this classis the Bose-Fermi Kondo model, with a spin- local moment coupled to fermionic quasiparticles (theregular Kondo model) as well as to a bosonic bath. Thelatter may describe spin or charge fluctuations of the bulksystem in which the impurity is embedded. The scope ofNRG applications has recently been widened to providea comprehensive treatment of an Ising-symmetric versionof the Bose-Fermi Kondo model. The purpose of this paper is to investigate a some-what simpler quantum impurity model containing bothfermionic and bosonic baths, namely a resonant-levelmodel of spinless electrons, with the impurity charge cou-pled to a dissipative reservoir. In standard notation, itsHamiltonian is H = ε f f † f + X k v k (cid:16) f † c k + H.c. (cid:17) + X k ε k c † k c k + ( f † f − ) X q g q ( b q + b †− q ) + X q ω q b † q b q , (1)with v k characterizing the hybridization between conduc-tion electrons of energy ε k and the impurity level at en-ergy ε f , and g q coupling bosons of energy ω q to the im-purity occupancy. Without loss of generality, v k and g q are taken to be real and non-negative. Equation (1) rep-resents perhaps the simplest nontrivial Bose-Fermi quan-tum impurity model, making it a paradigm for this classand an ideal problem for detailed comparison betweenanalytical and numerical results.The model is completely specified by the impurity levelenergy ε f , the hybridization functionΓ( ω ) ≡ π X k v k δ ( ω − ε k ) = Γ (cid:12)(cid:12)(cid:12) ωD (cid:12)(cid:12)(cid:12) r for | ω | < D, (2)and the bosonic bath spectral function B ( ω ) ≡ π X q g q δ ( ω − ω q ) = B (cid:18) ωω c (cid:19) s for 0 < ω < ω c , (3)with D and ω c acting as fermionic and bosonic cutoffs, re-spectively. Thus, in addition to a power-law spectrum forthe bosonic bath density of states (DOS) characterizedby an exponent s , we consider a nonconstant particle-hole(p-h) symmetric hybridization function characterized byan exponent r . Increasing r (and hence depleting the hy-bridization function around the Fermi level ω = 0) andincreasing B both act to suppress tunneling betweenthe local level and the conduction band. For most of thenumerical work presented in Sec. III, we fix r , s , andthe hybridization strength Γ , then tune the dissipationstrength B to the vicinity of a QPT.Although the bath densities of states and v k , g q donot require separate specification, it will facilitate com-parison between numerical and perturbative results toassume that v k = v , g q = g for all k , q . In thiscase, Γ( ω ) = πv ρ c ( ω ) and B ( ω ) = πg ρ b ( ω ), with thefermionic and bosonic DOS given, respectively, by ρ c ( ω ) = N | ω/D | r for | ω | < D , (4) ρ b ( ω ) = ( K /π ) ( ω/ω c ) s for 0 < ω < ω c , (5)where N and K are normalization factors. Thus, Γ = πN v and B = ( K g ) . The metallic case is recoveredfor r = 0, and Ohmic dissipation corresponds to taking s = 1. rs DelocalizedQPTLocalized S imp = ln 2 imp = r ln 2 S ( )( ) r+s= FIG. 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 fermionicand bosonic baths, respectively. (A finite coupling to bothbaths is assumed.) For max(0 , r − < s ≤
1, the modelshows a boundary quantum phase transition (as the couplings v and g are varied) between a delocalized phase and a lo-calized phase. (The physics along the line r = 0 is identicalto that of the spin-boson model.) In contrast, for s > r < S imp is discussed in the text. Two pertur-bative RG expansions are employed: around the free-impurityfixed point, where the expansion is controlled about r = s = 1(Sec. II B) and around the resonant-level fixed point, wherethe expansion is controlled in 1 − r + s (Sec. II C). It is convenient to identify a pseudospin—making clearthe close relationship between model (1) and the spin-boson model and its variants—by writing f † ≡ S + , f ≡ S − , f † f − ≡ S z . (6)In the model described by Eq. (1), the friction causedby the bosonic bath competes with the resonant tunnel-ing of electrons. In contrast to the simpler spin-bosonmodel, the tunneling properties are determined by thehybridization function Γ( ω ).For ε f = 0 the model features a Z symmetry ofparticle-hole type [assuming ρ c ( ω ) = ρ c ( − ω ) as notedabove], namely c k → c † k , f → − f † , and S z → − S z .Then, we expect that the competition between resonanttunneling and dissipation yields a QPT between a “de-localized” phase ( h S z i = 0), in which the principal effectof dissipation is to renormalize the tunneling amplitude,and a “localized” phase ( h S z i 6 = 0) with a doubly degen-erate ground state, where the tunneling amplitude renor-malizes to zero in the low-energy limit. We note that forthe case of a metallic fermionic bath [ r = 0 in Eq. (4)],bosonization techniques can be used to map the model(1) to the spin-boson model. (The same applies to theIsing-symmetric Bose-Fermi Kondo model with r = 0,and this equivalence has been verified using NRG. )In this paper, we employ renormalization-group (RG)techniques to map out the phase diagram of the Hamil-tonian (1) and to establish over what range of bath ex-ponents r and s the model can be tuned to a delocalized-to-localized QPT, akin to that of the spin-boson model.We do so using both perturbative RG methods, basedon epsilon-expansion techniques developed in the con-text of the pseudogap Kondo and Anderson models, and the Bose-Fermi extension of the NRG approach,which allows us to access the entire parameter range ofthe model.Our main result is summarized in Fig. 1, which illus-trates the qualitative behavior of the model in the planespanned by the bath exponents r and s . A delocalized-to-localized transition—which for r = 0 is identically thatof the spin-boson model—is present at r > r = 0 are es-tablished via the mapping to the spin-boson model, weconfirm the equivalence by direct calculation. II. PERTURBATIVE RENORMALIZATIONGROUPA. Zero-temperature phases
We begin by discussing the trivial fixed points of themodel (1) in the presence of p-h symmetry, ε f = 0. As acharacterization, we will refer to the residual impurity en-tropy S imp , which is defined as the impurity contributionto the total entropy in the limit temperature T → For v = g = 0, the impurity is decoupled from bothbaths. We denote this free-impurity fixed point by FImp.The ground state is doubly degenerate: S imp = ln 2.For v = 0 and g = 0 one has a resonant-level modelwith a power-law conduction-band DOS given by Eq.(4). The hybridization is relevant in the RG sense (w.r.t.FImp) for r <
1, and hence the impurity charge stronglyfluctuates.
We refer to this as the delocalized fixedpoint (Deloc), which, as discussed in Ref. 15, is locatedat intermediate RG coupling, ( g, v ) = (0 , v ∗ ). Somewhatsurprisingly, the impurity entropy is S imp = r ln 2, andvanishes only in the metallic case r = 0. For r > The dissipative coupling g turns out to be RG-relevant at the FImp fixed point for s < S imp = ln 2, i.e., a phase with broken Z symmetry. Thislocalized fixed point (Loc) corresponds to coupling values( g, v ) = ( ∞ , s > r < s <
1, a QPT separates a delocalized (small-dissipation)phase from a localized (large-dissipation) phase. Clearly,this applies only to the case of p-h symmetry, ε f = 0.Otherwise the Z symmetry of the Hamiltonian is bro-ken from the outset, and the phase transition upon vari-ation of the dissipation strength will be smeared into acrossover; this is analogous to the behavior of the spin-boson model in the presence of a finite bias. Furthermore,in situations where the system is localized at ε f = 0,there will be a first-order transition upon tuning ε f frompositive to negative values (as in an ordered magnet sub-ject to a field).We now proceed with an RG treatment of the model(1), carried out without recourse to bosonization. Wecan access quantum-critical properties via two distinctexpansions: (i) an expansion around the free-impurityfixed point (Sec. II B), which is formally valid providedthat the couplings to both baths are small, and (ii) an ex-pansion around the resonant-level fixed point (Sec. II C),performed after exactly integrating out the c fermions.The second approach proves to have the wider range ofapplicability. B. RG expansion around the free-impurity limit
In this subsection, we apply an RG epsilon expan-sion for weak couplings near the free-impurity fixed pointwhere v = g = 0.
1. RG equations
We model the bosonic bath by a relativistic scalar field, φ = b + b † , in d = 2 + s dimensions, with the action S φ = Z β dτ Z Λ q d d q (2 π ) d φ − q ( τ ) (cid:0) − ∂ τ + q (cid:1) φ q ( τ ) , (7)Λ q being a momentum-space cutoff (related to the energycutoff ω c via ω c = c Λ q with c = 1 being a velocity). Thisproduces a DOS of the form ρ φ ( ω ) = sgn( ω ) S s | ω | s = sgn( ω ) K π (cid:12)(cid:12)(cid:12)(cid:12) ωω c (cid:12)(cid:12)(cid:12)(cid:12) s , (8)for | ω | < ω c , with S d = 2 / [(4 π ) d/ Γ( d/ ρ φ is just a symmetrized version of ρ b defined in Eq. (a) (b) (c) FIG. 2: Diagrams appearing in the perturbative expansionfor the dissipative resonant-level model. Dashed, solid, andwiggly lines denote respectively f , c , and φ propagators. Thegray (black) circles are the interaction vertices v ( g ). (a) and(b): f fermion self-energy diagrams to one-loop order. (c)One-loop vertex renormalization of g . (5).] Similarly, we represent the fermionic bath by Diracfermions in (1 + r ) dimensions: S c = Z β dτ Z Λ k − Λ k dk | k | r (2 π ) r ¯ c k ( ∂ τ + k ) c k , (9)with Λ k = D/v F and v F = 1 being the (Fermi) velocity,which reproduces the DOS defined in Eq. (4). A path-integral representation of Eq. (1) reads S = S c + S φ + Z β dτ ¯ f ∂ τ f + g Z β dτ ( ¯ f f − ) φ ( τ, v Z β dτ (cid:2) ¯ f c ( τ,
0) + c.c. (cid:3) . (10)Power counting yields the bare scaling dimensions offields and couplings with respect to v = g = 0: [ f ] = 0,[ φ q ] = − (1 + s ) /
2, [ c k ] = − (1 + r ) /
2, [ v ] = (1 − r ) / g ] = (1 − s ) /
2. Thus, we can carry out an RGexpansion around r = 1 and s = 1, where both v and g become marginal, defining ǫ = 12 (1 − s ) , ǫ ′ = 12 (1 − r ) . (11)In order to proceed with the RG analysis, we define arenormalized field f R and couplings v and g according to f = p Z f f R ,v = µ ǫ ′ s D r N Z f Z v v ,g = µ ǫ √ ω sc πZ g K Z f g , (12)where µ is an arbitrary renormalization energy scale and Z f , Z v , and Z g are renormalization factors. As is usualfor impurity problems, there is no renormalization of thebosonic and fermionic bulk propagators, since the im-purity only provides a one-over-volume correction to thebulk properties. The relevant diagrams for obtaining theone-loop RG beta functions are shown in Fig. 2.Following standard procedures, the one-loop RGbeta functions of the dissipative resonant-level model aregiven by β ( v ) = − ǫ ′ v + v + g v ,β ( g ) = − ǫg + 2 v g , (13) where the calculation parallels that of Ref. 15. The cor-responding Z factors, to one-loop accuracy, are Z f =1 − v /ǫ ′ − g / ǫ , Z v = 1, and Z g = 1 − g / ǫ .The RG flows arising from Eqs. (13) are plotted in Fig.3. In this subsection, we consider the case 0 < s <
1; theregime s < g ∗ , v ∗ ) = (0 , ǫ ′ ) and ( g ∗ , v ∗ ) = ( ∞ ,
0) describe thedelocalized (Deloc) and localized (Loc) phases, respec-tively. For r < r + , where r + = (1 + s ) / , (14)both these fixed points are stable: For small g andlarge v , the ground state is delocalized, characterizedby strong local charge fluctuations due to resonant tun-neling between the impurity and the conduction electronbath ( h S z i = 0). In the opposite limit of small v andlarge g , we find a localized ground state where chargetunneling renormalizes to zero in the low-energy limit( h S z i 6 = 0). An unstable critical fixed point [Cr], locatedat ( g ∗ , v ∗ ) = (2 ǫ ′ − ǫ, ǫ/ g - v planebetween the delocalized and localized phases.As r approaches r + from below, the critical fixedpoint merges with the delocalized fixed point (which it-self merges with FImp as r → r ≥ r + : Deloc and FImp are unsta-ble w.r.t. infinitesimal bosonic coupling, such that theground state is always localized for g = 0.
2. Correlation-length exponent
In the following, we discuss the properties of theboundary QPT, controlled by the critical fixed point Cr.We start with the correlation-length exponent ν , describ-ing the flow away from criticality: The characteristic en-ergy scale T ∗ above which quantum-critical behavior isobserved vanishes as T ∗ ∝ | t | ν , (15)where t is a dimensionless measure of the distance tocriticality, defined such that t > t <
0) correspondsto the localized (delocalized) phase. Upon linearizationof the RG beta functions around the Cr fixed point, weobtain1 ν = r ǫ ǫ (cid:16) ǫ ′ − ǫ (cid:17) − ǫ O (cid:0) ǫ , ǫ ′ (cid:1) . (16)Clearly, ν diverges as s → r → ν − =1 − r + s . The same result, valid for small 1 − r + s , isalso obtained in Sec. II C following an RG expansion validnear the strong-coupling fixed point. The divergence of ν as 1 − r + s → v g v v g g (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) LocLocFImpDeloc Loc(a)(b)(c)Deloc Cr
FIG. 3: Schematic RG flow diagrams for the dissipativeresonant-level model with p-h symmetry. Although these di-agrams are obtained by expansion about the free-impurityfixed point and hence are formally valid as r, s →
1, theyare confirmed by expansion about the delocalized fixed point(Sec. II C) and NRG calculations (Sec. III) to capture thecorrect physics for all r ≥ < s <
1. The hori-zontal axis denotes the renormalized bosonic coupling g ; thevertical axis denotes the renormalized hybridization v . (a) r < r + = (1 + s ) /
2: Stable fixed points at Deloc and Loc de-scribe the delocalized and localized phases, respectively. Thecontinuous impurity QPT is controlled by the critical fixedpoint Cr. (b) r + < r <
1: The delocalized (Deloc) fixedpoint is unstable against finite g . As r → − , the Deloc fixedpoint merges with the free-impurity fixed point (FImp). (c) r ≥ v is irrelevant. In both (b) and (c), the flow is towardLoc for any finite g .
3. Response to a local field
The local impurity susceptibility χ loc ( T ) is the impu-rity response to a field applied only to the impurity. Here, for the spinless resonant-level model under con-sideration, the level energy ǫ f plays the role of a lo-cal electric field. Defining the impurity “magnetization” m imp = h S z i , with the pseudospin S z as specified in Eq.(6), it follows that χ loc = − ∂m imp ∂ε f (17)is nothing other than the impurity capacitance. Near criticality, χ loc ( T ) is expected to follow a power-law form χ loc ( T ) ∝ T − η χ for T ∗ ≪ T ≪ T , (18)up to a nonuniversal cutoff scale T . This relation definesthe anomalous exponent η χ , which governs the anoma-lous decay of the impurity “spin-spin” correlation func-tion and is calculated via η χ = µ ∂ ln Z χ ∂µ (cid:12)(cid:12)(cid:12)(cid:12) v ∗ ,g ∗ . (19)The renormalization factor Z χ obeys the exactrelation Z − χ = ( Z g /Z f ) , (20)which is graphically represented in Fig. 4(a). This allowsus to derive the exact result η χ = 2 ǫ = 1 − s (21)at the Cr fixed point, a relation that is borne out by thenumerical results presented in Sec. III.
4. Conduction electron T -matrix The conduction electron T -matrix, describing the scat-tering of the c electrons off the impurity, is another im-portant observable, being central to the calculation oftransport properties. For a resonant-level model, the T -matrix is given by T ( ω ) = v G f ( ω ) where G f is the fullimpurity ( f -electron) Green’s function, graphically rep-resented in Fig. 4(b). As with the local susceptibility,we expect a power-law behavior of the T -matrix spectraldensity near criticality: T ( ω ) ∝ | ω | − η T for T ∗ ≪ | ω | ≪ T . (22)It has been shown that all critical fixed points for 0 0, which behavior has beenobserved in a number of separate studies. Using the exact relation Z T = Z f /Z v , we can derivean exact result for the critical point of the dissipativeresonant level model: η T = 1 − r . (23)Thus, even though the multiplicative prefactor of the be-havior (22) is expected to exhibit both r and s depen-dence, the power law followed at criticality is identical tothat of the pseudogap Kondo and Anderson models. χ loc = T = (a) (b) FIG. 4: (a) Exact relation for the local susceptibility. Theblack triangle denotes the full vertex function and the dasheddouble line denotes the full impurity level propagator. (b)The large dot denotes the full hybridization vertex. 5. Hyperscaling and other critical exponents The QCP is expected to satisfy hyperscaling relationscharacteristic of an interacting fixed point, including ω/T scaling in dynamical quantities. It follows that thecorrelation-length exponent ν and the anomalous expo-nent η χ are sufficient to determine all critical exponentsassociated with the application of a local field. Forexample, one can define exponents γ and γ ′ through the T → χ loc ( t < T = 0) ∝ ( − t ) − γ , γ = ν (1 − η χ ) ,T χ loc ( t > T = 0) ∝ t γ ′ , γ ′ = νη χ . (24)One can also determine critical exponents β and δ asso-ciated with the local magnetization m imp : m imp ( t > T = 0 , ε f → ∝ t β , β = νη χ / ,m imp ( ε f ; t = 0 , T = 0) ∝ | ε f | /δ , δ = 2 /η χ − . (25)Thus, near criticality β = ǫ p ǫ / ǫ ( ǫ ′ − ǫ/ − ǫ/ O (cid:0) ǫ , ǫ ′ (cid:1) (26)and δ = 1 ǫ − O (cid:0) ǫ , ǫ ′ (cid:1) , (27)where, in contrast to Eqs. (21) and (23), the higher-ordercorrections do not cancel. Section III reports NRG re-sults for several of these critical exponents that demon-strably obey the hyperscaling relations. C. RG expansion around the delocalized fixedpoint In addition to the RG expansion for r → s → 1. RG equations To begin, we integrate out the conduction electrons,which is an exact operation for the present model. The resulting action is S = X ω n ¯ f ( ω n ) (cid:2) iA sgn( ω n ) | ω n /D | r + iA ω n (cid:3) f ( ω n )+ S φ + g Z β dτ (cid:0) ¯ f f − (cid:1) φ ( τ, , (28)where the local f fermions are now “dressed” by the con-duction lines, A = πN v sec (cid:16) πr (cid:17) = Γ sec (cid:16) πr (cid:17) (29)is a nonuniversal energy scale, and A = 1 + O ( v ). For r < 1, the | ω n | r term dominates the f propagator atlow energies. Then, dimensional analysis of the bosoniccoupling (here w.r.t. the Deloc fixed point) yields[ g ] = 2 r − − s , (30)which implies that an RG expansion can be controlled inthe smallness of 2˜ ǫ = 1 − r + s. (31)We introduce a dimensionless coupling according g = µ − ˜ ǫ A √ ω sc πZ g K Z f g , (32)and, following the procedure described in Sec. II B, wefind that the only contribution to Z g is that shown inFig. 2(c), which reads (note that Z f = 1 to this order) Z g = 1 + csc (cid:16) πs (cid:17) g ˜ ǫ . (33)The RG beta function for g is β ( g ) = ˜ ǫg − (cid:16) πs (cid:17) g . (34)It is clear from Eq. (34) that for s > ǫ > 0, thereexists a critical fixed point at g ∗ = ˜ ǫ (cid:16) πs (cid:17) , (35)which controls the delocalized-to-localized transition.The RG flow diagram is sketched in Fig. 5.Note that the critical coupling g ∗ approaches zero as˜ ǫ → + and/or as s → + , suggesting that beyondthese limiting cases the delocalized fixed point is un-stable towards the localized fixed point. The same in-stability has already been deduced for ˜ ǫ < r > r + = (1 + s ) / s ≤ g ∗ ∞ g LocCrDeloc FIG. 5: RG flow diagram of the dissipative resonant-levelmodel near the delocalized (Deloc) fixed point for s , ˜ ǫ > g = 0)and localized ( g = ∞ ) fixed points, separated by the criticalfixed point [ g = g ∗ specified in Eq. (35)]. 2. The regime s ≤ For s ≤ 0, the perturbation theory described in Sec.II C is singular due to the divergent DOS in the bosonicpropagator. In this range of s , the delocalized fixed pointis always unstable against any infinitesimal bosonic cou-pling g , which favors the localized fixed point.We can gain a better understanding of this instabilityby considering the local bosonic propagator G φ ( iω n ) = P q G φ ( q , iω n ) in the presence of the impurity. Includingimpurity effects via the boson self-energy, the local bosonpropagator is given by G − φ ( iω n ) = ( ω sn + s Λ s − g for s > ,ω − sn − g for s ≤ , (36)where Λ is a momentum cutoff energy scale. Let us dis-cuss s > s Λ s > g > 0, the local bosonpropagator is massive, meaning that the ground state forthe bulk is just the empty state. For g > s Λ s > 0, bycontrast, the local boson propagator has “negative mass”,as a consequence of which the local boson condenses atzero temperature with an expectation value h φ i 6 = 0.This drives the system to the localized phase where thepseudospin operator S z also assumes a nonzero expecta-tion value. This reasoning supports the existence of aQPT for s > 0, with criticality reached at g ∗ = s Λ s .For s ≤ 0, the local boson propagator G φ always hasa negative mass, i.e., the impurity is localized. (Techni-cally, the impurity induces a bound state in G φ .) Theobservation that the ground state is always localized for s ≤ and the Bose-Fermi Kondo model, whichbelong to the same universality class as the dissipativeresonant-level model in the metallic limit r = 0. 3. Phase diagram The RG flow allows us to deduce that the qualitativephase diagram of the dissipative resonant-level model inthe parameter space specified by r and s is as shown inFig. 1. The solid line denotes the locus of points satis-fying 1 − r + s = 0. In the unshaded region to the leftof the line [i.e., for max(0 , r − < s < 1, or equiva-lently < r + < r < r + defined in Eq. (14)], theRG expansion predicts a continuous QPT as v and g are varied. For s < max(0 , r − 1) (shaded area), theground state of the model is always localized for any fi-nite bosonic coupling g . This is consistent with the RGflow diagrams presented in Fig. 3, where the RG expan-sion is carried out for r, s → 1. The phase diagram isconfirmed by NRG results in Sec. III. 4. Critical exponents By linearizing the RG equation around the fixed point,the correlation-length exponent at the critical point g ∗ isfound to satisfy 1 ν = 2˜ ǫ + O (cid:0) ˜ ǫ (cid:1) . (37)For the anomalous exponent η χ associated with thelocal susceptibility [Eq. (18)], we again have the exactproperty Eq. (20) [see also Fig. 4(a)], from which it fol-lows that η χ = 1 − s . (38)The exponents β and δ can be obtained from the hyper-scaling relations (25): β = 1 − s ǫ + O (cid:0) ˜ ǫ (cid:1) , (39)and δ = 1 + s − s + O (cid:0) ˜ ǫ (cid:1) . (40)The exponent η T , associated with conduction-electron T -matrix, is also found to obey η T = 1 − r [see Eq. (23)]. Ofcourse, all critical exponents for the two RG expansions(one for r, s → − r + s → 0) are expectedto be compatible since the expansions describe the sameQPT. In the limit r, s → 1, the square root of Eq. (16)may be expanded to yield Eq. (37). The equivalences ofEqs. (26) and (39) for β and of Eqs. (27) and (40) for δ are also readily verified. III. NUMERICAL RENORMALIZATIONGROUP The NRG method has recently been extended to pro-vide nonperturbative results for the Bose-Fermi Kondomodel. In the following, we implement the same ap-proach for the spinless resonant-level model (1), whichalso involves both fermionic and bosonic baths.There are three essential features of the NRG: (i) Theenergy axis is logarithmically discretized, introducing adiscretization parameter Λ. (ii) The Hamiltonian is thenmapped to a chain form, with the impurity degrees offreedom coupled to the first site only of one or moretight-binding chains. (iii) Owing to the discretization,the tight-binding coefficients decay exponentially with N E N N E N (a)(b) FIG. 6: (Color online) (a) The lowest NRG eigenstates E N vs even iteration number N for ( r, s ) = (0 . , . = 0 . 1, and a range of dissipation strengths B − B ,c = 0 , ± − , ± − . The flows are typical of thosefor max(0 , r − < s < ≤ r < 1. The levels at thecritical coupling B = B ,c ≈ . B < B ,c ) [local-ized ( B > B ,c )] phase are shown as solid [dashed] lines. As B approaches B ,c in either phase, the levels follow those ofthe unstable critical fixed point down to progressively lowertemperatures, before crossing over to the levels characteristicof the delocalized or localized stable fixed point. (b) NRGlevel flows for ( r, s ) = (0 . , . s < max(0 , r − 1) with 0 ≤ r < 1, the flow istowards the localized fixed point for any B > 0, but followsthe delocalized fixed point down to progressively lower tem-peratures as B is reduced towards zero. The solid lines showthe flow for B = 0. increasing chain length. This allows the problem to besolved in an iterative fashion, diagonalizing progressivelylonger finite-length chains and thereby including expo-nentially smaller energy scales, T N ≈ D Λ − N/ , at eachiterative step N = 0, 1, 2, . . . . The RG transformationrelating the effective Hamiltonians at consecutive itera-tions eventually reaches a scale-invariant fixed point thatdetermines the low-temperature properties of the system.In all applications of the NRG, the maximum number N s of many-body eigenstates retained from iteration N toform basis states for iteration N +1 must be truncated forsufficiently large N due to the limitations of finite com-putational power. The presence of one or more bosonicchains introduces additional considerations. First, thebosonic Hilbert space must be truncated even at itera-tion N = 0, allowing a maximum of N b bosons per siteof a bosonic chain. Second, for problems involving bothfermionic and bosonic chains, the fact that the bosonictight-binding coefficients decay as the square of those forfermionic chains must be reflected in the specific iterativescheme employed. That is, only (bosonic and fermionic)excitations of the same energy scale should be consideredat the same iterative step. Thus, while the fermionicchain is extended at each iteration, the bosonic chain isextended only at every second iteration. These issues, to- gether with further details of the implementation of theBose-Fermi NRG, are discussed in detail in Ref. 29.The NRG method has provided a comprehensive nu-merical account of the quantum-critical properties of anumber of impurity problems, e.g., the fermionic pseudo-gap Kondo and Anderson models, the spin-boson model,and the Bose-Fermi Kondo model. In all cases it isfound that the critical properties (such as exponents) areinsensitive to the discretization parameter Λ and con-verge rapidly with the number of retained states N s . Formodels involving bosonic baths, critical exponents alsorapidly converge with increasing bosonic truncation pa-rameter N b . In the following we take Λ = 3, with all datasuitably converged for the choice N s = 500 and N b = 8.For convenience we set D = ω = 1. A. Phase diagram Figure 6 shows the flow of the lowest NRG eigenstates E N of the effective Hamiltonian H N at even iterationnumbers N for two representative cases for s > 0: (a)1 − r + s > − r + s < 0. Figure 6(a) showsdata obtained for ( r, s ) = (0 . , . 9) and Γ = 0 . 1. Here,and for any 1 − r + s > 0, the flow is schematized byFig. 3(a), which follows from the perturbative analysis.For B < B ,c , the NRG flow is towards the delocalizedfixed point, where the spectrum coincides with that forcoupling B = 0 to the bosonic bath. For B > B ,c the NRG flow is towards the localized fixed point, wherethe spectrum coincides with that for coupling Γ = 0 tothe fermionic band. For B close to B ,c , as consideredin Fig. 6(a), the flow in either case is first towards thecritical spectrum. The departure from the critical flow,at a crossover scale T ∗ that vanishes at B = B ,c , isgoverned by the correlation-length exponent discussed inSec. III B 1.Figure 6(b) shows NRG level flows for ( r, s ) =(0 . , . 9) and Γ = 0 . 1. These flows are typical of thosefor any 1 − r + s < B > 0. As B is reduced towards zero, the levelsfollow those of the delocalized fixed point (obtained for B = 0) down to progressively lower energy scales.Figure 7 shows the phase diagram of the model on the r - B plane for three different combinations of the bosonicbath exponent 0 < s < . For all s and Γ pairs considered, the phase-boundaryvalue of B decreases monotonically with increasing r from that found for a metallic conduction band ( r = 0).This is particularly clear from the data set obtained for s = 0 . = 10 − (circles in Fig. 7), where themetallic system undergoes a continuous QPT at a critical B ,c ( r = 0) ≈ . r , and hence grow-ing depletion of the conduction electron density of statesaround the Fermi level, the critical dissipation strength B ,c required to localize the system is reduced, as ex-pected on physical grounds. B ,c ( r ) is found to vanish r B -2 -1 -3 r + s -6 -4 -2 DelocLoc s Γ FIG. 7: (Color online) Phase diagram in the r - B plane,obtained using NRG for the fixed bosonic bath exponent s and the hybridization strength Γ shown in the legend. For0 < r < r + = (1 + s ) / 2, we find a continuous QPT betweendelocalized (Deloc) and localized (Loc) phases. The criticaldissipation strength B ,c is found to vanish continuously at r = r + . For r ≥ r + only the localized phase can be ac-cessed for B > 0. The inset shows the vanishing of B ,c withdecreasing 1 − r + s in each case, compared to the resultsobtained from the perturbative analysis. continuously at r = r + , with r + as defined in Eq. (14).This vanishing is illustrated in the inset to Fig. 7, whichshows B ,c vs 1 − r + s on a logarithmic scale.For r > r + , localized solutions are found for arbitrarilysmall dissipation strength B > 0. The symbols at thelargest r (= r + ) in each case, which lie at B = 0, markthe point at and above which no delocalized solutionscan be found with B > 0. Thus, we find that we cantune the system to a QPT if, and only if, 0 < s ≤ ≤ r ≤ r + , in complete agreement with the scenariodeduced via the perturbative analyses and illustrated inFig. 1.For 0 ≤ r < s = 1, we find a line of Kosterlitz-Thouless-like transitions between delocalized and local-ized ground states, and for s > > r > s > 1, the essential physics is controlled by the free-impurity fixed point, regardless of the couplings Γ and B .For a given ( r, s ) pair that exhibits a continuous QPT,the critical dissipation strength B ,c varies with the hy-bridization strength Γ as B ,c ∝ Γ (1 − s ) / (1 − r )0 (41)provided that all scales are small compared to the cutoffs.This result, which follows from dimensional arguments[Eq. (41) can readily be obtained using Eq. (13)] and is −5 −4 −3 −2 −1 Γ −5 −4 −3 −2 −1 B , c (0.7, 0.8)(0.9, 0.9) FIG. 8: Critical dissipation strength B ,c vs hybridizationstrength Γ for the ( r, s ) pairs specified in the legend. Wefind that B ,c ∝ Γ x , with x = (1 − s ) / (1 − r ). confirmed numerically in Fig. 8, identifies Γ / (1 − r )0 as thetunneling amplitude analogous to ∆ of the spin-bosonmodel, where the critical dissipation strength is α c ∝ ∆ − s . A similar result for the Bose-Fermi Kondo modelfinds B ,c ∝ T − sK , with T K the bare Kondo temperatureserving as a tunneling amplitude between impurity spinstates. It is interesting to compare the location of the phaseboundary obtained using NRG with that inferred fromanalytical expansion. We have in mind fixing the hy-bridization strength Γ and the bosonic-bath exponent s (as in Fig. 7), and finding the critical coupling B as afunction of the conduction-band exponent r . However,an analysis of the expansion around the free-impurityfixed point (Sec. II B) reveals no simple analytical ex-pression for the phase boundary, due to the fact thatthe problem is described by a two-parameter flow, whichcannot be linearized in general. We have therefore ana-lyzed the coupled differential flow equations numerically.The phase boundary can be obtained by determining theeigenvalues and eigenvectors of the linearized RG equa-tions near the critical point and then following the RGflow backwards along the separatrix.The inset of Fig. 7 compares phase boundaries deter-mined via NRG (symbols) with those obtained via theperturbative RG equations (13) (dashed lines). For therange of 1 − r + s considered by NRG, B ,c appears tovanish as a power law, with an exponent that depends onboth the bosonic bath exponent s and the hybridizationΓ . This apparent power law does not reflect the asymp-totic behavior, revealed by the perturbative calculationsto be B ,c ∝ ˜ ǫ as ˜ ǫ → 0. (This regime is inaccessibleto NRG because the merging of the critical and delocal-ized fixed points with decreasing ˜ ǫ make it impossible toreliably determine the critical coupling B ,c .) Neverthe-less, we find the level of agreement remarkable and stressthat there is no fitting procedure involved in making thiscomparison.From the expansion around the delocalized fixed point0 −7 −6 −5 −4 −3 −2 −1 | t | −60 −40 −20 T * (0.65, 0.8), Deloc(0.65, 0.8), Loc(0.65, 0.9), Deloc(0.8, 0.9), Deloc FIG. 9: Crossover scale T ∗ vs | t | = | B − B ,c | /B ,c for the( r, s ) pairs specified in the legend. In the vicinity of the tran-sition ( | t | ≪ T ∗ ∝ | t | ν . The correlation-length exponent ν ( r, s ) is independent both of the hybridization Γ and of thephase from which the QCP is approached. (Sec. II C), where we have a one-parameter flow, it seemspossible to obtain an analytical expression for the phaseboundary. However, we have to keep in mind that thedressed f propagator in Eq. (28) contains terms withdifferent frequency dependencies, and is dominated by | ω n | r in the low-energy limit only. (The coefficient A is nonzero in general, except right at the Deloc fixedpoint.) The interplay of the | ω n | r and ω n terms intro-duces a nonuniversal crossover scale into the problem,and a proper treatment including elevated energies wouldrequire a multistage RG scheme, which is beyond thescope of this paper. B. Critical exponents 1. Correlation-length exponent The correlation-length exponent ν defined in Eq. (15)is readily extracted from the crossover scale T ∗ ∝ Λ − N ∗ / in the NRG level flows between the unstable and either ofthe stable fixed points. Here, N ∗ denotes the NRG iter-ation number at which crossover is observed in a chosenNRG eigenvalue E N . (See Refs. 28 and 29 for further de-tails.) Figure 9 shows T ∗ vs | t | = | B − B ,c | /B ,c for the( r, s ) pairs specfied in the legend. The dashed lines arelinear fits to the log-log data, which yield the correlationlength exponent ν ( r, s ), independent of the hybridizationstrength Γ and the phase (Deloc or Loc) from which theQCP is accessed.The r dependence of the correlation-length exponent isdemonstrated in Fig. 10(a) for two values of the bosonicbath exponent s . As anticipated, for r = 0 we find thatwithin our estimated numerical error of about 1%, ν (0 , s )is in essentially exact agreement with ν ( s ) for the spin-boson model (and the Ising-symmetry Bose-FermiKondo model, demonstrated in Ref. 28 to share the same r / ν s = 0.9 s = 0.8 r + s / ν (a) (b) FIG. 10: (Color online) (a) Correlation-length exponent ν vsconduction-band exponent r for two values of the bosonic-bath exponent s , as shown in the legend. The symbolsshow NRG data, while the dashed lines are the correspond-ing perturbative results [Eq. (16)], expanding about the free-impurity fixed point. We find that ν − vanishes at r = r + ,in keeping with the qualitatively distinct behavior for 2˜ ǫ ≡ − r + s ≷ 0. (b) The same data plotted vs 2˜ ǫ . For small˜ ǫ , ν − ≈ ǫ , a result [Eq. (37)] (shown as a dotted line)obtained by a perturbative expansion about the delocalizedfixed point. universality class). By increasing r we find that ν ( r, s )diverges as r → r + from below, i.e., as 1 − r + s → + .The dashed lines are the corresponding perturbative re-sults [Eq. (16)], with which there is excellent agreementfor r approaching r + . Figure 10(b) shows the same dataplotted vs 2˜ ǫ = 1 − r + s . With decreasing ˜ ǫ > 0, thecurves approach the result ν − ≈ ǫ (shown as a dottedline), as obtained in Sec. II.C.3 by an expansion aboutthe delocalized fixed point. 2. Response to a local field As discussed in Sec. II B 3, the response to a field ap-plied only at the impurity provides a useful probe of thelocally critical properties of the model. The inset to Fig.11(a) shows m imp ( t ; T = 0) vs t = ( B − B ,c ) /B ,c for( r, s ) = (0 . , . 9) and hybridization strength Γ = 0 . m imp ( t ; T = 0) is finite in the localized phase ( t > m imp ( t ; T = 0) ≈ for t ≫ t → + . In the delocalized phase( t < m imp ( t ; T = 0) = 0. The main part of Fig.11 shows m imp ( t ; T = 0) vs t > β is found to be β = 0 . t = 0), the dependence of m imp ( t = 0 , T = 0)on the field ε f defines the exponent δ according to Eq.(25). We typically observe such power-law behavior overseveral orders of magnitude of ε f , as shown in Fig. 11(b).For ( r, s ) = (0 . , . /δ = 0 . ≤ r < s = 1, m imp ( t ; T =1 −8 −6 −4 −2 ε f m i m p ( ε f ; t = , T = ) −2 −1 t −3 −2 −1 m i m p ( t > ; T = , ε f = + ) −1 0 100.250.5 (a) (b) FIG. 11: (Color online) Critical exponents β and δ , definedin Eq. (25), for ( r, s ) = (0 . , . 9) and Γ = 0 . 1, where B ,c ≈ . m imp vs t = ( B − B ,c ) /B ,c as t → + with characteristic exponent β (extracted as the limiting slope of the data on a logarithmicscale). The inset shows the data on an absolute scale. (b)Variation of m imp ( T = 0) with local level energy ε f at thecritical point t = 0. The data clearly follow a power law forsmall ε f , defining the exponent δ . −14 −12 −10 −8 −6 −4 −2 T χ l o c ( T ) FIG. 12: Static local susceptibilty χ loc ( T ) vs T for ( r, s ) =(0 . , . = 0 . 1, and B = 0 . . ≈ B ,c (stars), and 0 . η χ = 1 − s , independentof r . See text for further discussion. , ε f = 0 + ) undergoes a jump at the critical point t = 0.Here, the essential behavior has been discussed in Refs.28,29,30 and 37 for the case ( r, s ) = (0 , 1) relevant tocharge fluctuations on a metallic island subject to elec-tromagnetic noise.We calculate the static local susceptibility via χ loc ( T ) = − ∂m imp ∂ε f (cid:12)(cid:12)(cid:12)(cid:12) ε f =0 = lim ε f → − m imp ε f . (42)In the delocalized phase B < B ,c , m imp ( T = 0) van-ishes linearly with ε f and thus χ loc ( T ) ≈ const. for T ≪ T ∗ . In the localized phase B > B ,c , m imp is nonzero as ε f → χ loc ( T ≪ T ∗ ) ∝ /T . In thequantum-critical regime T ∗ ≪ T ≪ T , χ loc ( T ) divergesas a power law with an anomalous exponent η χ definedin Eq. (18). For all ( r, s ) pairs considered (such that1 − r + s > η χ = 1 − s , (43)independent of r . The behavior described above is clearlyillustrated in Fig. 12, which shows three data sets for( r, s ) = (0 . , . η χ = 0 . 3. Hyperscaling As discussed in Sec. II B 5, critical exponents for thepresent model are expected to obey hyperscaling rela-tions derived via a scaling ansatz for the critical partof the free energy that assumes the critical fixed pointis interacting. This expectation is borne out by thenumerical analysis: we find hyperscaling relations tobe obeyed to within the estimated error (typically lessthan 1%) across the range of ( r, s ) displaying critical be-havior. For example, for the case ( r, s ) = (0 . , . /ν = 0 . η χ = 0 . β = 0 . /δ = 0 . C. Spectral function We now turn to the single-particle spectral function A ( ω ), calculated via A ( ω ) = X n,m (cid:12)(cid:12) h n | f † | m i (cid:12)(cid:12) e − βE m + e − βE n Z δ ( ω − E n + E m ) , (44)where | m i is a many-body eigenstate of NRG iteration N , and Z = P n exp( − βE n ) is the partition function; A ( ω ) = A ( − ω ) for the p-h symmetric parameters stud-ied. The discrete delta-functions are Gaussian broadenedon a logarithmic scale: a standard NRG procedure dis-cussed, e.g., in Ref. 18. We set the broadening parame-ter b such that A ( ω ) for the simplest resonant-level model(with r = 0, B = 0, and ε f = 0) is in optimal agreementwith the exact result A ( ω ) = π − Γ / ( ω + Γ ).Figure 13(a) shows A ( ω ) vs | ω | on a logarithmic scalefor r = 0 . s = 0 . 8, Γ = 10 − , and the dissipationstrengths B ≤ B ,c specified in the figure caption. Forthe delocalized phase B < B ,c , we find that the dissi-pation does not alter the asymptotic low-frequency be-havior of A ( ω ) found for B = 0, i.e., A ( ω ) = 1 π Γ cos (cid:16) πr (cid:17) | ω | − r for | ω | ≪ T ∗ . (45)2 -30 -28 -26 -24 -22 | ω | A ( ω ) -30 -20 -10 | ω | F ( ω ) (a)(b) FIG. 13: (Color online) (a) Spectral function A ( ω ) vs | ω | for r = 0 . s = 0 . 8, Γ = 10 − , and three values of the dis-sipation strength: B = 0 (dotted line), B = B ,c − − (solid line), and B = B ,c = 0 . B = B ,c , A ( ω ) ∝ | ω | − r ,which behavior is also followed for B close to B ,c and | ω | ≫ T ∗ . In the delocalized phase B < B ,c , there is acrossover in A ( ω ) to the behavior Eq. (45) for | ω | ≪ T ∗ .For the data shown, T ∗ ∼ O (10 − ). (b) The crossoverbehavior is more readily seen in the modified spectral func-tion F ( ω ) = π Γ sec ( π r ) | ω | r A ( ω ), which shows the ultimatelow- ω behavior F ( ω = 0) = 1 throughout the delocalizedphase B < B ,c , 0 < F ( ω = 0) < B = B ,c , and F ( ω = 0) = 0 throughout the localized phase B > B ,c . Inorder of decreasing crossover scale, delocalized-phase spectraare shown for B = 0 (dotted line) and for B ,c − B = 10 − ,10 − , 10 − , and 10 − (solid lines); localized-phase spectra(dashed lines) are shown for B = 0 . 05 and B − B ,c = 10 − ,10 − , 10 − , and 10 − . The critical spectrum is shown as athick dashed line. For B = 0 the spectrum is identical to that obtained forthe noninteracting ( U = 0) limit of the (spinful) pseudo-gap Anderson model at p-h symmetry, where the resultEq. (45) holds for 0 < r < Moreover, it is known that the form Eq. (45) persists throughout the Kondo-screened phase of the pseudogap Anderson model withinteractions present (i.e., for all U < U c ), which in thep-h symmetric case is confined to 0 < r < .In the vicinity of the QCP, B ≈ B ,c , we find A ( ω ) = ˜ c ( r, s ) π Γ cos (cid:16) πr (cid:17) | ω | − r for T ∗ ≪ | ω | ≪ T , (46)where ˜ c ( r, s ) ≤ T is a high-frequency cutoff setby the bare hybridization strength Γ . This behaviorconfirms Eqs. (22) and (23).In the localized phase, by contrast, A ( ω ) vanishes as ω → A ( ω ) ∝ | ω | a for | ω | ≪ T ∗ . (47)The exponent a is positive, and in general depends onboth r and s .The crossover between these behaviors is more read-ily apparent in the modified spectral function F ( ω ) = π Γ sec ( πr/ | ω | r A ( ω ). Any low-frequency divergenceof A ( ω ) is canceled in F ( ω ), and F (0) = 1 is pinnedthroughout the delocalized phase of the model. Asdiscussed in the context of the pseudogap Andersonmodel, this generalizes the well-known pinning π Γ A (0) = 1 of the spectral function for the regular( r = 0, fermionic) Anderson model. In the delocalizedphase, the scale T ∗ , playing the role of a renormalizedtunneling amplitude, is then manifest as the width of thepinned resonance at the Fermi level ω = 0, vanishing as B → B − ,c .Figure 13(b) shows F ( ω ) vs | ω | for r = 0 . s = 0 . = 10 − , and the B values specified in the figure cap-tion. Throughout the delocalized phase (0 ≤ B < B ,c ), F (0) = 1 remains satisfied to within a few percent, asis typical for NRG. Close to the QCP in either phase, F ( ω ) ≈ ˜ c ( r, s ) down to the scale T ∗ .We close by considering the single-particle spectrumfor the case of a metallic fermionic density of states( r = 0) and Ohmic dissipation ( s = 1). Here the modeldescribes charge fluctuations on a quantum dot or res-onant tunneling device close to a degeneracy point andsubject to electromagnetic noise. The essential physics—a Kosterlitz-Thouless-like QPT between delocalized andlocalized states—has been investigated in a number ofearlier studies , e.g., via a Bose-Fermi Kondomodel, and we will not repeat the discussion here. Wesimply show, in Fig. 14, the spectrum for Γ = 0 . 001 anda range of dissipation strengths; for B = 0, A ( ω ) is ofLorentzian form. The vanishing width of the central res-onance as B → B − ,c indicates a suppression of tunnelingbetween dot and leads due to the noisy electromagneticenvironment. IV. CONCLUSIONS In this paper, we have analyzed the phase diagram andthe quantum phase transitions of a paradigmatic quan-tum impurity model with both fermionic and bosonicbaths, namely a dissipative resonant-level model. Forweak dissipation, the resonant tunneling of electrons isrenormalized due to the friction of the bosonic bath, butthe ground state remains delocalized. For strong dissi-pation, by contrast, the tunneling amplitude renormal-izes to zero in the low-energy limit leading to a local-ized ground state. We have employed both analyticaland numerical techniques, utilizing epsilon expansions re-cently developed in the context of the pseudogap Ander-son and Kondo model, and an extension of Wilson’s nu-3 −0.01 −0.005 0 0.005 0.01 ω π Γ A ( ω ) FIG. 14: (Color online) π Γ A ( ω ) vs ω for the case of a metal-lic fermionic density of states ( r = 0) and Ohmic dissipation( s = 1) for Γ = 0 . B (see legend) in the delocalized phase. Thespectrum is a simple Lorentzian for B = 0, and the vanish-ing width as B → B − ,c indicates a suppression of tunnelingbetween the local level and the conduction band. merical renormalization-group approach, generalized totreat both fermionic and bosonic baths.The transition between delocalized and localizedphases exists for a wide range of exponents r and s char-acterizing the conduction-band and bosonic-bath densi-ties of states, respectively. Our epsilon expansions, for-mulated in the original degrees of freedom, are in ex-cellent agreement with numerics in the vicinity of theexpansion points. For the case of a metallic bath, inac-cessible to the analytical techniques used here, we have presented numerical results, making contact with earlierbosonization studies of related models.We finally mention a few applications. In the contextof nanostructures, a resonant-level model may describethe tunneling of electrons between a lead and a smallisland or quantum dot. Taking into account elec-tromagnetic noise of a fluctuating environment directlyleads to a model of type (1), provided that the spin de-gree of freedom of the electrons can be neglected (e.g., ifelectrons are spin-polarized due to a large applied mag-netic field). Related situations, mainly corresponding tobath exponents r = 0 and s = 1, have been discussed inthe literature. Apart from the common situation ofohmic noise ( s = 1), sub-ohmic dissipation ( s < 1) canoccur, e.g., in RLC transmission lines which display a √ ω spectrum in the R-dominant limit. Further, a bath with r = 1 may be realized using Dirac electrons of grapheneor quasiparticles of a d -wave superconductor. Acknowledgments We thank S. Florens and N. Tong for fruitful discus-sions on the present paper and related subjects. 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