Non-Fermi-liquid Kondo screening under Rabi driving
NNon-Fermi-liquid Kondo screening under Rabi driving
Seung-Sup B. Lee, Jan von Delft, and Moshe Goldstein Faculty of Physics, Arnold Sommerfeld Center for Theoretical Physics,Center for NanoScience, and Munich Center for Quantum Science and Technology,Ludwig-Maximilians-Universit¨at M¨unchen, Theresienstraße 37, 80333 M¨unchen, Germany Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel (Dated: February 11, 2020)We investigate a Rabi-Kondo model describing an optically driven two-channel quantum dotdevice featuring a non-Fermi-liquid Kondo effect. Optically induced Rabi oscillation between thevalence and conduction levels of the dot gives rise to a two-stage Kondo effect: Primary screeningof the local spin is followed by secondary nonequilibrium screening of the local orbital degree offreedom. Using bosonization arguments and the numerical renormalization group, we compute thedot emission spectrum and residual entropy. Remarkably, both exhibit two-stage Kondo screeningwith non-Fermi-liquid properties at both stages.
I. INTRODUCTION
The Kondo effect, involving a local spin entangled witha bath of delocalized electrons, has been studied exten-sively in bulk systems and in transport through quan-tum dots. Some years ago, a landmark experiment [1]showed that it can also be probed optically: A weaklydriven optical transition between the valence and con-duction levels of the dot was used to abruptly switch theKondo effect on or off, leaving telltale power-law signa-tures [2] in the dot emission spectrum. The case of strongspin-selective optical driving was subsequently studiedtheoretically within the context of a single-channel Rabi-Kondo (1CRK) model [3], involving Rabi oscillations be-tween the dot valence and conduction levels. This waspredicted to lead to a novel nonequilibrium quantum-correlated state featuring two-stage Kondo screening:The local spin is screened by a primary screening cloudvia the single-channel Kondo (1CK) effect, then the Rabi-driven levels by a larger, secondary screening cloud. De-spite its nonequilibrium nature, this state has a simpleFermi-liquid (FL) description in terms of scattering phaseshifts, since only a single screening channel is involved.This raises an intriguing question: What type ofnonequilibrium state will arise when the Rabi-driven dotcouples to two spinful channels, described by a two-channel Rabi-Kondo (2CRK) model? Without Rabi driv-ing, it reduces to the standard two-channel Kondo (2CK)model, known to have a non-Fermi liquid (NFL) groundstate [4], describable by Bethe Ansatz [5–7], conformalfield theory (CFT) [8–10] or bosonization [11–14]. How-ever, NFL physics is known to be very sensitive to pertur-bations such as channel asymmetry or a magnetic field.Do the NFL properties survive under Rabi driving? Ifso, what are their fingerprints? In this paper, we answerthese questions. We use a combination of bosonizationarguments and numerical renormalization group (NRG)[15–17] calculations to compute the 2CRK emission spec-trum and impurity entropy. We find that NFL behaviorsurvives, and, remarkably, leaves clear fingerprints in theemission in both the primary and secondary screening regimes.The rest of this paper is organized as follows. In Sec. II,we introduce our system, the 2CRK model. In Sec. III,we provide a qualitative description of the screening pro-cesses in the 2CRK model. In Secs. IV and V, we studythe impurity contribution to the entropy and the Kondocloud, respectively. In Sec. VI, the main points of ourbosonization approach are outlined. In Sec. VII, we an-alyze the emission spectrum. We conclude in Sec. VIII.App. A offers the details of our bosonization approach.
II. TWO-CHANNEL RABI-KONDO MODEL
In this section, we first introduce the system in the labframe, and then derive the effective Hamiltonian in therotating frame to be treated by NRG and bosonization.We consider a small quantum dot ( d ) with a conduction( c ) and a valence ( v ) level as the impurity, and two largedots as the bath [Fig. 1(a)]. The small dot is modelledby the Hamiltonian H d = (cid:88) x = c,v (cid:20) U xx n x ( n x −
1) + (cid:15) x n x (cid:21) + U cv n c n v , (1)where n x = (cid:80) σ d † xσ d xσ denotes the particle number op-erator for the x level ( x = c, v ), and d xσ annihilates spin- σ electron at the x level of energy (cid:15) x . U cc , U vv , and U cv are the Coulomb interaction strengths. The level sepa-ration (cid:15) c − (cid:15) v is of the order of the semiconductor bandgap ∼ n c , n v ) = (1 , c and v levels. Acircularly polarized laser would have coupled to one spinspecies due to an optical selection rule [3, 18]. In thefollowing we will consider the case of linearly polarizedlight, which symmetrically couples to both spin states.The laser frequency ω L is chosen to be close to thebare dot transition between ( n c , n v ) = (1 ,
2) states and( n c , n v ) = (2 ,
1) states, i.e., ω L (cid:39) U cc + (cid:15) c − U vv − (cid:15) v . (We a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b (a) (b) FIG. 1. (a) Schematic depiction of the 2CRK model in thelab frame. A small dot with two levels (conduction c andvalence v ) has its c level coupled to two large dots via spinexchange J . The analogous 1CRK model has only one largedot. Linearly polarized laser induces Rabi oscillation of fre-quency Ω in which electrons transition between the c and v levels, accompanied by the absorption and emission of light.(b) The states of the small dot having n d = n c + n v = 3electrons. The states of ( n c , n v ) = (2 ,
1) are connected tothe states of ( n c , n v ) = (1 ,
2) via the Rabi oscillation. In therotating frame, the c - v coupling becomes time-independentwith amplitude Ω. set (cid:126) = k B = 1.) Hence the ( n c , n v ) = (2 ,
1) states are ac-cessed via the Rabi oscillation from the ( n c , n v ) = (1 , n d = n c + n v (cid:54) = 3can be accessed only via virtual processes due to the en-ergy cost of the Coulomb interaction.Since the optical transition is close to the material’sbandgap, that is, of order 1 eV, and much larger thanall the other energy scales (which are typically not morethan a few tens of meV), one could make the rotatingwave approximation, under which a transfer of electronfrom the v to the c level involves the absorption of aphoton and vice versa. We will further assume that thelaser can be described as a classical field, and hence thatspontaneous emission could be neglected. Then the light-induced Hamiltonian term in the lab frame is given by H (lab) L = Ω (cid:88) σ (cid:0) d † cσ d vσ e − iω L t + h . c . (cid:1) , (2)where Ω is the Rabi frequency.In addition, the c level of the small dot is symmetri-cally tunnel-coupled to two identical large dots (channels (cid:96) = 1 , n d + N and n d + N do not fluctuate,where N (cid:96) means the particle number at the large dot (cid:96) .Under these conditions, the whole system Hamiltonianin the lab frame can be approximated, via the Schrieffer-Wolff transformation [19] and up to an overall constant,by H (lab) = (cid:88) (cid:96) J (cid:126)S c · (cid:126)s (cid:96) + δ L n v + H bath + Ω (cid:88) σ (cid:0) d † cσ d vσ e − iω L t + h . c . (cid:1) , (3)where the Hilbert space for the small dot is re-stricted to the four-dimensional subspace of n d = 3shown in Fig. 1(b). Here (cid:126)S c = (cid:80) σσ (cid:48) d † cσ (cid:126)σ σσ (cid:48) d cσ (cid:48) and (cid:126)s (cid:96) = (cid:80) σσ (cid:48) (cid:82) D − D d (cid:15) d (cid:15) (cid:48) D c † (cid:15)(cid:96)σ (cid:126)σ σσ (cid:48) c (cid:15) (cid:48) (cid:96)σ (cid:48) are c -leveland (cid:96) -channel spin operators, respectively. H bath = (cid:80) (cid:96)σ (cid:82) D − D d (cid:15) (cid:15) c † (cid:15)(cid:96)σ c (cid:15)(cid:96)σ describes the large dots with half-bandwidth D , and c (cid:15)(cid:96)σ annihilates channel- (cid:96) electron ofenergy (cid:15) and spin σ . The coupling strength J is propor-tional to 1 / ( U cc + 2 U cv + (cid:15) c ) − / (2 U cv + (cid:15) c ).We will now go to the the rotating frame with respectto the laser-mode Hamiltonian, via the transformation U = e iω L n v t . The rotating-frame Hamiltonian H (rot) = U † H (lab) U + i (d U † / d t ) U will become time-independent, H (rot) = (cid:88) (cid:96) J (cid:126)S c · (cid:126)s (cid:96) + δ L n v + H bath + Ω (cid:88) σ (cid:0) d † cσ d vσ + h . c . (cid:1) , (4)where δ L = ω L − ( U cc + (cid:15) c − U vv − (cid:15) v ) is the detuningof laser frequency from the bare dot transition. This isthe 2CRK Hamiltonian to be studied in the rest of thispaper. For reference, we also include some results for theanalogous 1CRK model ( (cid:96) = 1 only), and the standard2CK and 1CK models (without v level).Since the coupling to the fermionic bath is assumed tobe the main relaxation mechanism and dominates overspontaneous emission, the system would relax to an elec-tronic equilibrium state in the rotating frame, which cor-responds to a time-dependent state in the lab frame.Thus we can analyze the system in the rotating frameemploying equilibrium concepts such as entropy.Note that our setup, which is driven optically, isdifferent from previous setups driven by ac magneticfield [20, 21], in two key aspects. First, the laser canbe focused within the length scale of optical wavelength,so one can selectively drive the small dot only. This selec-tivity has been demonstrated in experiments [1]. Second,the rotating wave approximation works very well for oursystem, since the energy scale of the laser frequency islarger than the other energy scales in the system by atleast two orders of magnitudes. The selectivity and therotating wave approximation are, however, unlikely forthe systems driven by ac magnetic field that are in themicrowave or rf regime. III. QUALITATIVE CONSIDERATIONS
Without Rabi driving, Ω = 0, the “trion” and “Kondo”sectors, with c and v level occupancies ( n c , n v ) = (2 , , v level isinert. The trion sector is a trivial FL, with the doublyoccupied c level forming a local spin singlet. The Kondosector constitutes a standard Kondo model, involving thespin of the singly-occupied c level. Below a characteristicKondo temperature T K , it will be screened by bath elec-trons. For the 2CRK model, it is overscreened, leadingto NFL behavior characteristic of the 2CK model. Forthe 1CRK model, it is fully screened, showing standard1CK FL behavior. FIG. 2. Impurity contribution to the entropy, S imp , for the (a)2CRK and (b) 1CRK models, for four Ω-values (solid lines).Arrows indicate the corresponding values of ω max /T K , the en-ergy scale associated with the peak in the emission spectrumshown in Fig. 4. For comparison, dashed lines show S imp forthe standard 2CK and 1CK models, respectively, for the samevalue of J . For weak driving , 0 < Ω (cid:28) T K , Rabi oscillations be-tween the c and v levels couple the Kondo and trionsectors. Then primary screening of the c -level spin, oc-curring at energies (cid:46) T K , will be followed by secondaryscreening of c - v transitions at the renormalized Rabi cou-pling Ω ∗ (as in Ref. [3]), provided that the ground stateenergies of the two (decoupled) sectors differ by less thanΩ ∗ . (A precise definition of Ω ∗ will be given later.) Wethus fine-tune δ L such that for Ω = 0 the Kondo andtrion ground states are degenerate, following a strategydiscussed in the Supplemental Fig. S2 of Ref. [3].Finally, for strong driving , Ω (cid:38) T K , the Rabi couplinggenerates a strong splitting of bonding and anti-bondingstates built from the c and v levels. The local spin of thebonding state will then undergo single-stage screening,as for the standard 2CK or 1CK models.These qualitative arguments will be substantiatedquantitatively below by NRG calculations and bosoniza-tion arguments. For the former, we use J = 0 . D throughout, leading to T K (cid:39) × − D and T K (cid:39) × − D when Ω = 0. The bath discretization grid isset by Λ = 4 and Λ = 2 .
7, and no z -averaging isused. We use the QSpace tensor library [22] to exploit theSU(2) symmetries of spin and channel where applicable. IV. ENTROPY
Figure 2(a,b) shows our NRG results for the impuritycontribution to the entropy [16], S imp , which quantifiesthe effective degrees of freedom of the dot at differenttemperatures. At high temperatures, T (cid:29) T K , Ω, theentropy S imp = ln 4 simply counts all four configurationsof the dot [Fig. 2(b)] for both the 2CRK and 1CRK mod- els. At lower temperatures, the behavior of the entropydepends on the relation of Ω and T K .For strong driving Ω (cid:38) T K , only two bonding stateswith different spins are accessible for T <
Ω. Hence S imp ( T (cid:46) Ω) shows a plateau at ln(2), followed by asingle crossover to T = 0 value of ln(2) = ln( √
2) orln(1) = 0 for the 2CRK or 1CRK models, respectively.These values are the same as in the standard 2CK or 1CKmodels [6–8] (shown as dashed lines), respectively. Theyreflect overscreening of a local spin by two spinful chan-nels (resulting in a decoupled local Majorana mode [11–14]), or its complete screening by a single spinful chan-nel [23] (resulting in a spin singlet), respectively.In contrast, for weak driving 0 < Ω (cid:28) T K , two-stage screening occurs. For intermediate temperatures S imp (Ω ∗ (cid:28) T (cid:28) T K ) shows a primary-screening plateauat ln(2 + √
2) or ln(2 + 1) for the 2CRK or 1CRK models:the NFL- or FL-screened local spin contributes √ v ) states. At the lowest temperatures, T (cid:28) Ω ∗ ,the c - v transitions lead to a secondary-screening limit-ing value of S imp = ln √ ω max = 0), the primary-screening plateau in S imp persists down to T = 0. V. KONDO CLOUDS
To further study the nature of the screening clouds in-volved in primary and secondary screening, we have com-puted spin-spin correlation functions between the impu-rity and bath spin operators, see Fig. 3. As described inthe caption thereof, for weak driving we find a nested,two-stage cloud, screening the c -level spin at energies (cid:38) T K , and c - v transitions at energies (cid:39) ω max . In con-trast, for strong driving we find just a single screeningcloud. VI. BOSONIZATION
We proceed to a more detailed analysis the weak driv-ing case, 0 < Ω (cid:28) T K , using bosonization (since themethods of Ref. [3] do not easily generalize to the 2CRKmodel). Here we outline the main points, relegating fur-ther details to App. A. With uniaxial anisotropy, thebosonized form [11–14] of 2CRK Hamiltonian H bath + H d is: H = (cid:88) (cid:96) =1 , (cid:26) u π (cid:90) ∞−∞ d x [ ∂ x φ (cid:96) ( x )] + J z π √ P K S z ∂ x φ (cid:96) (0)+ J xy πa P K (cid:0) S + e i √ φ (cid:96) (0) + h . c . (cid:1)(cid:27) + 2Ω τ x , (5)where S ± = S x ± iS y , while τ + = (cid:80) σ d † cσ d vσ , τ − = τ † + ,and τ z = n c − n v are Pauli matrices in the orbital c - v FIG. 3. Spin-spin correlators between the impurity and bathspin operators, revealing the structure of the screening cloudsfor the (a) 2CRK and (b) 1CRK models. We display χ vm = − (cid:104) P T S vz S mz (cid:105) / (cid:104) P T (cid:105) (solid) and χ cm = − (cid:104) P K S cz S mz (cid:105) / (cid:104) P K (cid:105) (dashed), where S vz , S cz and S mz are z -component spin oper-ators for the v level, c level and the Wilson chain site m ≥ m = 0 is directly coupled to the c level. P T = (cid:80) σ n vσ (1 − n v ¯ σ ) n c ↑ n c ↓ and P K = (cid:80) σ n v ↑ n v ↓ n cσ (1 − n c ¯ σ ) are projectors onto the trion and Kondo sectors, involv-ing a singly-occupied v or c level, respectively. Both χ vm and χ cm are obtained by averaging two lines, interpolatingodd and even m ’s, respectively. We choose the abscissa asΛ − m/ D/T K , where Λ − m/ D is the energy scale (and also theinverse length scale [15, 24]) associated with the chain site m .For strong driving (red), χ cm and χ vm have coinciding peaks,reflecting single-stage screening of the bonding-level spin. Incontrast, for intermediate (yellow) and weak (blue) driving,we observe two-stage screening: the peaks of χ cm , reflectingthe screening of the c -level spin, occur at higher energies thanthose of χ vm , reflecting the screening of the c - v transitions.The area under each peak is (cid:39)
1. Arrows indicate the corre-sponding values of ω max /T K . (Kondo-trion) pseudo-spin space, and P K = (1 + τ z ) / u and a = D/u are the Fermi velocity and lattice spacing (inversemomentum cutoff), and φ (cid:96) ( x ) is the chiral (unfolded)bosonic spin field (the charge sector decouples). It obeysthe commutation relation [ φ (cid:96) ( x ) , φ (cid:96) ( x (cid:48) )] = iπ sgn( x − x (cid:48) ),where ∂ x φ (cid:96) (0) / ( π √
2) is the density of the z -componentof the channel- (cid:96) electron spin density at the dot site. A. 1CRK
Let us start from the single-channel case, where (cid:96) =1 [ φ ( x ) does not exist]. The unitary transformation U α = e − iαS z P K φ (0) with α = J z / ( π √ u ) eliminates the J z term at the cost of modifying the J xy term by a shiftto the coefficient of φ (0) in the exponent.At energies (cid:29) T K (cid:29) Ω we may ignore the Rabi term,and follow the usual perturbative renormalization group(RG) flow of the 1CK problem. J xy flows since it hasa nontrivial scaling dimension, set by the correspondingbosonic exponent (after the above-mentioned transfor-mation). In addition, second-order spin-flip ( J xy ) pro- cesses revive the non-spin-flip J z term, which may thenbe transformed away as above. J z thus flows to a fixedpoint value, J z = 2 πu , corresponding to the Kondo fixed-point π/ J xy grows until it becomesof the order of the reduced cutoff, which could serve todefine the primary c -spin Kondo scale T K . The U α -typetransformations applied throughout the RG flow modifythe Rabi term. Thus, below T K we obtain the followingintermediate-scale effective Hamiltonian: H int1CRK = u π (cid:90) ∞−∞ d x [ ∂ x φ ( x )] + J ren xy πa P K S x + Ω τ + (cid:104) P ↑ e − iφ (0) / √ + P ↓ e iφ (0) / √ (cid:105) + h . c ., (6)where P ↑ , ↓ = 1 / ± S z is a projector into the subspace S z = ± /
2, and J ren xy ∼ T K (cid:29) Ω. The latter largecoupling fixes the dot spin to S x = 1 /
2, which corre-sponds, in the original basis, to an entangled state ofthe impurity and bath spins, i.e., the primary Kondosinglet. Thus P ↑ , ↓ are replaced by their expectation val-ues (cid:104) P ↑ , ↓ (cid:105) = 1 /
2. The resulting model describes the hy-bridization between the pseudo-spin ( c - v or Kondo-trion)degree of freedom and the channel, which is equivalent(up to a transformation similar to U α but involving τ z instead of S z ) to an anisotropic Kondo model for thepseudo-spin space. The Rabi coupling Ω is relevant,with scaling dimension η = 1 /
4, determined by thecorresponding bosonic exponent in Eq. (6), or, withinCFT, from its role as boundary condition changing op-erator, turning on and off 1CK screening [25]. Hence, Ωflows to strong coupling, creating a new scale, the renor-malized Rabi frequency (secondary Kondo temperature),Ω ∗ ∼ T K (Ω /T K ) / (1 − η ) = T K (Ω /T K ) / (cid:28) Ω, where oneexpects a peak in the dot emission spectrum to occur,instead of the more usual peak at Ω for strong drivingΩ (cid:29) T K . Below this scale, the pseudo-spin is screenedby the creation of a secondary “Kondo singlet”. B. 2CRK
Let us now perform a similar analysis of the 2CRKmodel. Defining the fields φ ± ( x ) = [ φ ( x ) ± φ ( x )] / √ J z , and could be eliminatedby a transformation similar to U α defined with √ φ + (0)instead of φ (0). For Ω (cid:28) T K one may proceed withthe primary 2CK RG flow, which drives J z to πu , corre-sponding to a π/ J xy to J ren xy ∝ T K (cid:29) Ω.At the same time, the Rabi coupling gets modified. Onthe scale of T K we thus arrive at: H int2CRK = (cid:88) p = ± u π (cid:90) ∞−∞ d x [ ∂ x φ p ( x )] + J ren xy πa P K S x cos φ − (0)+ Ω τ + (cid:104) P ↑ e − iφ + (0) / + P ↓ e iφ + (0) / (cid:105) + h . c . (7)The first line describes the 2CK fixed point, at whichthe φ − remains coupled: S x assumes a definite value FIG. 4. (a) Log-log plot of the emission spectrum S ( ω ), and(b) its finite-frequency peak position ω max and zero-frequencyspectral weight S as functions of Rabi driving Ω /T K , forthe 2CRK (solid) and 1CRK (dashed) models at T = 0.Guide-to-the-eye grey lines depict the power laws predictedby bosonization arguments (see text). S x = ± /
2, and correspondingly φ − (0) is locked to aminimum or maximum of the cosine function. Refermion-izing the local spin- φ − system, the J xy term couples a lo-cal Majorana fermion ( ∝ S x ) to the lead, leaving anotherlocal Majorana ( ∝ S y ) unscreened [11, 12].We now turn to the second line. Since J ren xy ∼ T K (cid:29) Ω,we may again set P ↑ , ↓ → /
2. The remaining term is aproduct of τ ± with bosonic exponents. The exponentscontribute 1 / τ ± turns on or off the J ren xy term, which is equivalent toturning on or off a local backscattering impurity in a Lut-tinger liquid, with scaling dimension 1/16 [26, 27]. Thus,the overall scaling dimension of Ω is η = 3 /
16. Thismatches the corresponding CFT analysis of its role as aboundary-condition changing operator [25]. Thus, Ω isrelevant, flowing to strong coupling and generating a newscale Ω ∗ ∼ T K (Ω /T K ) / (1 − η ) = T K (Ω /T K ) / (cid:28) Ω,below which secondary screening of the c - v (Kondo-trion)fluctuations is achieved. Importantly, since the Rabiterm is spin symmetric, it does not interfere with theprimary NFL 2CK screening, and leaves the decoupledMajorana ( S y ) unscreened: While the Rabi term contains S z ∝ S x S y , the corresponding processes are suppressedby the dominant J ren xy term, and all higher order (in Ω)processes which leave the system within the low-energymanifold of J ren xy term do not couple to S y . VII. EMISSION SPECTRUM
Having established the general picture of the two-stageNFL screening, we can now analyze its effect on the mainexperimental observable, the dot emission spectrum. Theemission spectrum of linear polarization at detuning ω from the driving laser frequency is proportional to thespectral function [3], S ( ω ) = (cid:88) jj (cid:48) ρ j (cid:12)(cid:12) (cid:104) j (cid:48) | (cid:80) σ d † vσ d cσ | j (cid:105) (cid:12)(cid:12) δ ( ω + E j (cid:48) − E j ) , (8)where | j (cid:105) and E j are energy eigenstates and eigenvaluesof the Rabi-Kondo Hamiltonian, and ρ j = e − E j /T /Z .This is the spectral function of the Rabi term with itself.At temperature T = 0, the emission spectrum has weightonly for ω ≤
0. Without Rabi driving, S ( ω → − ) showsa power-law divergence. For weak driving, the divergenceis cut off, giving way to a power-law decrease. Accord-ingly a wide peak at | ω | = ω max and a delta-functionpeak S δ ( ω ) of weight S at ω = 0 emerge. We identify ω max with the renormalized Rabi frequency Ω ∗ .Figure 4(a) shows a log-log plot of the emission spec-trum, revealing its various power laws. For weak driv-ing, there are two distinct regimes: (i) The intermediate-detuning regime, ω max (cid:46) | ω | (cid:46) T K , is dominated by theKondo exchange coupling and reflects primary screening.Here the correlations of the Rabi term with itself are gov-erned by its scaling dimension η , giving S ( ω ) ∝ | ω | η − with η = η = 1 / η = η = 3 /
16 for the 1CRKor 2CRK models, respectively. Thus, this part of thespectrum reveals the scaling of 1CK vs. 2CK bound-ary condition changing operators [25]. (ii) The small-detuning regime, | ω | (cid:46) ω max , is dominated by the Rabicoupling and reflects secondary c - v screening. In thisregime, S ( ω ) corresponds to the correlation function ofthe exchange interaction (cid:80) (cid:96) (cid:126)S · (cid:126)s (cid:96) with itself in the stan-dard Kondo models, which yields S ( ω ) ∝ | ω | and ∝ | ω | for the 1CRK and 2CRK models, respectively. The poweris reduced in the 2CRK case, as the unscreened Majo-rana S y appearing in the Rabi term in Eq. (7) (through S z ∝ S x S y ) reduces the corresponding scaling dimensionby 1 /
2. Thus, the | ω | -behavior is a clear fingerprint ofthe NFL nature of the nonequilibrium secondary screen-ing nature in the 2CRK system.Figure 4(b) shows that ω max and S increase as powerlaws in Ω. For weak driving, our previous analysis showsthat, in accordance with the numerical data, ω max ∼ Ω ∗ ∝ Ω / (1 − η ) ∼ Ω / or Ω / , and moreover (as wewill momentarily explain), S ∼ Ω η/ (1 − η ) ∼ Ω / orΩ / for the 2CRK or 1CRK models, respectively. In-deed, S takes up the spectral weight missing at smalldetuning due to Ω ∗ cutting off the intermediate detuning S ( ω ) ∝ | ω | η − behavior. Hence, S ∝ (cid:82) Ω ∗ d ω | ω | η − ∼ Ω η/ (1 − η ) . Alternatively, by Eq. (8) S is the squareof the expectation value of τ x (before transformations)in the ground state. But τ x is the Rabi term dividedby Ω, and the Rabi term should scale as Ω ∗ , leadingto S ∼ (Ω ∗ / Ω) ∼ Ω η/ (1 − η ) , as before. Thus, theKondo boundary condition changing operators governsboth ω max and S . For strong driving, ω max ∼ Ω corre-sponds to the transition energy between the bonding andanti-bonding states of c and v levels. VIII. CONCLUSIONS
We have identified a two-stage NFL screening pro-cess in a Rabi driven quantum dot. The NFL naturesurvives in nonequilibrium as the Rabi driving respectsboth spin and channel symmetries. We have developeda new bosonization approach that explains the power-law exponents obtained numerically. The distinct powerlaws in the emission spectra should motivate opticalspectroscopy studies on the multi-channel quantum dotdevices. The case of non-negligible spontaneous emis-sion, which goes beyond the description of the time-independent Hamiltonian in the rotating frame, wouldbe an interesting question for future study. We envisionour findings to also be relevant for higher-dimensionaldriven strongly-correlated materials.
ACKNOWLEDGMENTS
We thank E. Sela for useful discussions. This jointwork was supported by German Israeli Foundation(Grant No. I-1259-303.10). S.-S.B.L. and J.v.D. are sup-ported by the Deutsche Forschungsgemeinschaft (DFG,German Research Foundation) under Germanys Excel-lence Strategy – EXC-2111 – 390814868; S.-S.B.L. fur-ther by Grant No. LE 3883/2-1. M.G. acknowl-edges support by the Israel Science Foundation (GrantNo. 227/15), the US-Israel Binational Science Founda-tion (Grant No. 2016224), and the Israel Ministry of Sci-ence and Technology (Contract No. 3-12419).
Appendix A: Bosonization details
In this Appendix, we develop in details the theoryof the multichannel Kondo effect, by first reviewing thebosonization description of the ordinary single- and two-channel Kondo (1CK, 2CK), then going on to their Rabi-Kondo versions, without and with spin rotation symme-try (the latter being the case considered in the maintext).We note that the Yuval-Anderson (YA) Coulomb gasapproach [28–33] is known to give equivalent results tothe bosonization approach for all universal (i.e., cutoff-independent) quantities, such as critical dimensions.Meanwhile, the Coulomb gas approach provides more ac-curate microscopic expressions for the phase shifts thatare cutoff-dependent. We have verified that the same istrue for the systems discussed in this work. However, in this paper we employ the bosonization approach, since itis more succinct than the Coulomb gas approach.
1. Ordinary Kondo
First, we review the ordinary (equilibrium) 1CK and2CK effects from the bosonization perspective. a. Single-channel ordinary Kondo
Let us start from the ordinary single-channel Kondoeffect. Using a bosonic description of the channel, thecharge sector decouples, while the spin sector can be writ-ten in terms of a single right-moving chiral boson overthe entire 1D line (instead of a single non-chiral bosonon the 1D half-line), leading to the following Hamilto-nian [11, 34]: H = u π (cid:90) ∞−∞ d x [ ∂ x φ ( x )] + J z π √ S z ∂ x φ (0)+ J xy πa (cid:16) S + e i √ φ (0) + h . c . (cid:17) , (A1)where S ± = S x ± iS y , S x,y,z are the impurity spin-1/2operators, u and a are the Fermi velocity and lattice spac-ing (inverse momentum cutoff), the bosonic field obeysthe commutation relation [ φ ( x ) , φ ( x (cid:48) )] = iπ sgn( x − x (cid:48) ),and ∂ x φ (0) / ( π √
2) is the conduction electron spin den-sity at the dot site. Applying the transformation H → H (cid:48) = U α H U † α where U α = e − iαS z φ (0) with α = J z / ( π √ u ), the J z term is eliminated, at the cost of mod-ifying the exponent in the J xy term: H (cid:48) = u π (cid:90) ∞−∞ d x [ ∂ x φ ( x )] + J xy πa (cid:16) S + e i √ − J z / (2 πv )] φ (0) + h . c . (cid:17) . (A2)We now proceed with perturbative RG, using Cardy’s op-erator product expansion (OPE) version [35]. J xy flowsbecause it has a nontrivial scaling dimension (due to thecorresponding nontrivial bosonic exponent), whereas theOPE of the two J xy terms reintroduces the J z term.This can be transformed again into the bosonic expo-nent. Defining the dimensionless exchange couplings J xy,z = J xy,z / (2 πu ), and denoting the energy cutoff by D = v/a , we thus obtain Anderson’s well-known RGequations: − D d J xy d D = (cid:104) − (1 − J z ) (cid:105) J xy , (A3) − D d J z d D = (1 − J z ) J xy . (A4)Thus, J z flows to the strong-coupling fixed point value J z = 1 ( π/ ∝ J xy S x (where at strong coupling J xy ∝ T K , the Kondo temperature), and seemingly po-larized the impurity spin in the x direction. Recallingthat the S x operator has undergone a succession of trans-formations dressing it with the bosonic field, we recognizethat, in terms of the original fields, this actually signifies(an anisotropic version of) the Kondo singlet. Indeed, thefact that the spin flip terms in the original Hamiltonian, S ± e ± i √ φ (0) , have been renormalized to S ± means thatthe renormalized versions of the original S ± operators are S ± e ∓ i √ φ (0) . The correlation function of these two oper-ators decays in time as 1 /t (due to the bosonic factor), inaccordance with Fermi liquid theory (in which one positsthat at the fixed point the impurity spin “merges” withthe Fermi sea, so its correlator behaves like the correla-tion function of the lead fermion density). Another wayto get this result is to notice that, generically (that is, ina higher order RG than what we considered), S z could get dressed by the lead spin density at the impurity site, ∝ ∂ x φ (0), hence its correlation would decay as 1 /t . Us-ing similar arguments, the connected correlation functionof the exchange terms in the original Hamiltonian turnsinto a connected correlator of two lead spin operatorswith two lead spin operators, decaying as 1 /t , trans-lating into an ω behavior of the corresponding spectralfunction at low frequencies. Finally, since the impurityHamiltonian reduces to ∝ J xy S x at the fixed point, if amagnetic field in the z direction is introduced, the im-purity susceptibility becomes ∝ J − xy ∝ T − . This willalso give a finite expectation value to the lead spin cor-relators, making the leading contribution (at long time)to the exchange-exchange correlation function decay as1 /t , or ω in the frequency domain. Finally, the impurityentropy is ln 2 at T (cid:29) T K , and goes to zero at T (cid:28) T K ,due to the Kondo screening. b. Two-channel ordinary Kondo Now the starting Hamiltonian is: H = (cid:88) (cid:96) =1 , (cid:26) u π (cid:90) ∞−∞ d x [ ∂ x φ (cid:96) ( x )] + J z π √ S z ∂ x φ (cid:96) (0) + J xy πa S + e i √ φ (cid:96) (0) + h . c . (cid:27) , (A5)where (cid:96) = 1 , ∂ x φ (cid:96) (0) / ( π √
2) gives their respective spin densities at thedot site, and we assume channel symmetry. Here it is useful to define the symmetric and antisymmetric combinations, φ ± ( x ) = [ φ L ( x ) ± φ R ( x )] / √
2, which keep the commutation relations the same. We now apply the transformation U α + = e − iαS z φ + (0) with α = J z / ( πu ) to eliminate the J z term and get H = (cid:88) p = ± u π (cid:90) ∞−∞ d x [ ∂ x φ p ( x )] + J xy πa S + cos [ φ − (0)] (cid:0) e i [1 − J z / ( πv )] φ + (0) + h . c . (cid:1) . (A6)The RG equations are similar to the 1CK case, butwith (1 − J z ) → (1 − J z ). Hence, J z flows to a valueof 1/2 ( π/ φ − . J xy continues to flow tostrong coupling, where it becomes ∝ T K (this strong cou-pling bosonic description corresponds to the intermediatecoupling non-Fermi-liquid fixed point in the traditionaldescription in terms of the original fermions). If onerefermionizes the local spin and the bosonic subsystem φ − , the J xy term becomes a coupling of a local Majo-rana operator ( S x ) to a Majorana field density in thelead at the adjacent site, namely cos[ φ − (0)], while S y becomes a local decoupled Majorana. This is the famousEmery-Kivelson point. Therefore, the low-temperatureimpurity entropy is ln √
2. Since S z ∝ iS x S y , its cor-relator with itself is a convolution of the correlators ofa localized Majorana fermion ( ∝ S y ) and a propagatingone ( ∝ S x ), and decays in time as 1 /t , as that of one freefermion times one localized fermion, leading to a loga- rithmic divergence of the susceptibility with the largestcutoff energy (magnetic field, temperature, or frequency).For a similar reason, the original non-spin-flip exchangeterm, ∝ S z ∂ x φ + (0), has correlations decaying as 1 /t ,implying a low-frequency power-law behavior of ω , inthe absence of a magnetic field (a magnetic field sup-presses the non-Fermi-liquid 2CK physics, and restoresthe Fermi-liquid 1CK ω behavior). If we look at cor-relators of S + (with its conjugate), we can use the factthat the series of transformations map it to S + e − iφ + (0) ,leading to a 1 /t behavior, similar to S z .One can recover the behavior of the susceptibility usingpurely bosonic language [36–39]. At the strong J xy fixedpoint, S x picks a value ± /
2, and then φ − (0) is pinned toeither a minimum or a maximum of the cosine function,respectively. With that one can calculate the suscepti-bility, that is, the retarded correlator of S z with itself.Indeed, S z anticommutes with S x , hence with the spin-flip exchange term. Since the spin-flip exchange termmodifies by unity the spin of one of the leads, the oper-ator V = e iπN − , where N − = N − N is the differencebetween the refermionized populations of the two leads(corresponding to S z of the original electrons, since thebosonic fields are all related to the original electronic spindegrees of freedom), also anticommutes with the spin-flipexchange term. Hence, the correlator of S z could be re-placed by a correlator of V . Conservation of the over-all refermionized population, N + = N − N allows oneto write replace V → e i πN = e iφ (0) . Rememberingthat at the fixed point the two leads are effectively well-coupled, N behaves as the population of one half of aninfinite lead. With this, the correlation function of V with itself decays in time as 1 /t , again leading to a log-arithmic divergence of the susceptibility with the largestcutoff energy (magnetic field, temperature, or frequency).
2. Spin-asymmetric Rabi-Kondo
We now add to the Kondo effect a laser, which triesto Rabi-flip the electron constituting the impurity spininto a level decoupled from the leads. We will introducea corresponding two-level degree of freedom, with Paulimatrices τ x,y,z , whose two states τ z = ± c ) level (Kondo)and in the valence ( v ) level (trion), respectively. TheRabi flopping ( τ x ) is induced by a laser with amplitudeΩ. If the laser has a proper circular polarization, it onlycouples to a spin-up electron, S z = 1 / a. Single-channel spin-asymmetric Rabi-Kondo Let us start from the single-channel spin-asymmetricRabi-Kondo (1CARK) case, analyzed in our previouswork [3]. Based on all the above considerations, theHamiltonian is: H = u π (cid:90) ∞−∞ d x [ ∂ x φ ( x )] + J z π √ P K S z ∂ x φ (0)+ J xy πa P K (cid:16) S + e i √ φ (0) + h . c . (cid:17) + 2Ω τ x P ↑ . (A7)Here P K = (1 + τ z ) acts as a local projector onto the c level (i.e., the Kondo sector), and P ↑ = + S z as a localprojector onto the spin-up subspace. We will concentrateon the case where the Kondo temperature is much largerthan the Rabi frequency, T K (cid:29) Ω. Then, at energy scaleslarger than T K , we can ignore the Rabi term. The trans-formations and RG flow are as above, with the only differ-ence that every transformation U α = e − iαS z φ (0) shouldbe replaced by U α = e − iα (1+ τ z ) S z φ (0) / . The series oftransformations on the way to the Kondo fixed point at J z = 1 then modifies the Rabi term, giving H int = u π (cid:90) ∞−∞ d x [ ∂ x φ ( x )] + J ren xy πa P K S x + Ω P ↑ (cid:16) τ + e − iφ (0) / √ + h . c . (cid:17) , (A8)where J ren xy ∝ T K ( (cid:29) Ω), as mentioned above. Thus,the corresponding Kondo term is much larger than theRabi term, and effectively eliminates the S z part of P ↑ = + S z (the eliminated part breaks the symme-try under S z → − S z on the scale Ω ∗ introduced below,as a local magnetic field would do, but this has a negli-gible effect in the current 1CK physics, since Ω ∗ (cid:28) T K ).With this the Rabi term looks exactly like the spin-flipexchange term in the pure Kondo problem, Eq. (A2),demonstrating that the Rabi term leads to a secondaryKondo screening process. The scaling dimension, say η ,of the 1CRK term is dictated by the bosonic exponent,giving η = 1 /
4, reflecting the Anderson orthogonalitycatastrophe with a phase shift change of π/ ∗ (cid:28) | ω | (cid:28) T K (where the new low-energyscale Ω ∗ will be defined shortly), the emission spectrum(imaginary part of the retarded correlator of the Rabiterm with itself) scales as | ω | η − = | ω | − / . Moreover,the RG equation for Ω is [35] − D d(Ω /D )d D = (1 − η ) Ω D , (A9)with solution Ω( D ) /D = Ω /T K ( D/T K ) η − , where wehave taken into account that the RG flow of Ω starts atthe scale of T K . Therefore Ω( D ) flows to strong coupling.The scale at which Ω( D ) /D becomes of order unity de-fines the renormalized Rabi frequency (secondary Kondotemperature), Ω ∗ /T K ∼ (Ω /T K ) / (1 − η ) = (Ω /T K ) / .Thus, the impurity entropy starts with the value ln 3 at T (cid:29) T K (the four possible values of S z and τ z , exceptthe excluded possibility of τ z = − S z = − / ∗ (cid:28) T (cid:28) T K (due to the Kondoscreening of the τ z = 1 sector), and then goes to zero for T (cid:28) Ω ∗ , due to the secondary Kondo screening.Below Ω ∗ , secondary Kondo screening (of the τ de-gree of freedom) sets in. The emission spectrum, whichcorresponds to a correlator of the Rabi term with it-self, becomes the spectral function of the correlator ofthe secondary Kondo exchange term with itself. Ourprevious analysis for the single-channel case shows thatthis leads to an | ω | behavior, or, in the presence of de-tuning (which adds to the Hamiltonian a term propor-tional to τ z , that is, a magnetic field in the secondaryKondo language), to an | ω | scaling. Also, at zero fre-quency a delta function appears in the emission spec-trum. Its amplitude can be calculated in two ways. Oneis to note that the spectral weight missing by the emer-gence of Ω and the corresponding change of the spec-tral function from ∝ | ω | η − to a positive power shouldgo into the delta function, giving it a weight scaling as (cid:82) Ω ∗ dω | ω | η − ∼ Ω ∗ η ∼ Ω η / (1 − η ) = Ω / . The otherargument is that the coefficient of the delta function is |(cid:104) G | τ x | G (cid:105)| , the square of the matrix element of τ x (be- fore the transformations) between the ground state anditself, and this matrix element is the ground-state expec-tation value of the Rabi term divided by Ω. The expec-tation value of the Rabi term scales as Ω ∗ , giving againan (Ω ∗ / Ω) = Ω η / (1 − η ) = Ω / scaling of the weightof the delta function. b. Two-channel spin-asymmetric Rabi-Kondo We will now consider the analogous two-channel spin-asymmetric Rabi-Kondo (2CARK) setup. Now the startingHamiltonian is: H = (cid:88) (cid:96) =1 , (cid:26) u π (cid:90) ∞−∞ d x [ ∂ x φ (cid:96) ( x )] + J z π √ P K S z ∂ x φ (cid:96) (0) + J xy πa P K (cid:0) S + e i √ φ (cid:96) (0) + H . c . (cid:1)(cid:27) + 2Ω P ↑ τ x . (A10)At energies larger than T K , we can use similar steps to the above, and arrive at: H int = (cid:88) p = ± u π (cid:90) ∞−∞ d x [ ∂ x φ (cid:96) ( x )] + J ren xy πa P K S x cos [ φ − (0)] + Ω P ↑ (cid:0) τ + e − iφ + (0) / + h . c . (cid:1) (A11)For the 2CRK model, the scaling dimension, say η ,of the Rabi term, seen as a boundary condition changingoperator, is given by η = 3 /
16 [25]. One could arrive atthis value using also our abelian bosonization language:The φ + exponent contributes 1/8 to the scaling dimen-sion of the Rabi term. Beyond that, the τ ± operatorsturn on or off the transformed Kondo exchange term in-volving cos[ φ − (0)]. Now, turning on and off such a co-sine appears in the problem of the Fermi edge singular-ity, that is, turning on and off backscattering by impu-rity, in a Luttinger liquid. This problem was analyzed inRefs. [26, 27]. They showed that, at long times, the cosinecan be replaced by a quadratic term (since it is relevant),which allows one to find its contribution to the long timebehavior of the correlation function of τ x . This contribu-tion scales as t − / , corresponding to a scaling dimensionof 1/16. Adding this to the 1 / φ + , we recover the CFT result η = 3 / ∗ (cid:28) | ω | (cid:28) T K the emission spectrum be-haves as | ω | η − = | ω | − / . The RG equation for Ω isthe same as above, with η taking the place of η , re-flecting the different scaling dimension of the Rabi term.Then we get a low-enegry scale Ω ∗ ∝ Ω / (1 − η ) = Ω / ,and the weight of the delta peak at zero frequency scalesas (Ω ∗ ) η ∝ Ω η / (1 − η ) = Ω / . The impurity entropywill be ln 3 for T (cid:29) T K , ln(1 + √
2) (2CK partial screen-ing + exciton state) for Ω ∗ (cid:28) T (cid:28) T K , and zero for T (cid:28) T K .As for the behavior of the emission spectrum at | ω | (cid:28) Ω ∗ , one could argue that the Rabi term, with its explicit S z dependence, breaks the symmetry for flipping S z andhas similar effects to a local magnetic field on the phys-ical spin. Thus, below Ω ∗ the 2CK physics should be suppressed, and one should recover the 1CK behavior of | ω | or | ω | in the absence or presence of detuning, respec-tively.
3. Spin-symmetric Rabi-Kondo
Finally we arrive at the spin-symmetric version of theRabi-Kondo problem, where the applied laser featuresthe two circular polarizations with the same amplitude(i.e., a linear polarization), and thus couples equally toboth spin states. a. Single-channel spin-symmetric Rabi-Kondo
We start from the single-channel spin-symmetric Rabi-Kondo (1CSRK) problem. Now the Hamiltonian is: H = u π (cid:90) ∞−∞ d x [ ∂ x φ ( x )] + J z π √ P K S z ∂ x φ (0)+ J xy πa P K (cid:16) S + e i √ φ (0) + h . c . (cid:17) + 2Ω τ x . (A12)Here on the scale of T K we obtain: H int = u π (cid:90) ∞−∞ d x [ ∂ x φ ( x )] + J ren xy πa P K S x + Ω τ + (cid:104) P ↑ e − iφ (0) / √ + P ↓ e iφ (0) / √ (cid:105) + h . c ., (A13)which is invariant under flipping of S z , together with thelead (integrated) spin density φ ( x ). However, this sym-metry is not essential in the 1CK case, and the analysis0goes basically the same as in the spin-asymmetric case (atleast as long as one considers the spin-symmetric emis-sion spectrum).Let us note that one could formally map the secondaryscreening problem to an anisotropic spin-1 Kondo prob-lem. Indeed, since J ren xy / ( πa ) ∼ T K is large, we can dis-card the S x = − / τ z = 1and S x = − /
2) and the two spin states of the exciton( τ z = − J z / ( πu )is of order 1, i.e., the spin-1 problem is strongly spin-anisotropic. Now, any spin exchange anisotropy wouldcause the creation of impurity-spin terms proportionalto the square of the z component of the effective spin1, which amounts to detuning the exciton and primary-Kondo states. For weak anisotropy, it is sufficient to adda corresponding compensating term to restore the degen-eracy and hence the spin-1 Kondo physics. However, inour case, where the bare secondary exchange anistropy isvery large, the physics never reaches the underscreenedspin-1 Kondo regime. Thus, the impurity entropy goesfrom ln 4 to ln 3 and then to zero as T is lowered through T K and Ω ∗ . b. Two-channel spin-symmetric Rabi-Kondo The last case is the two-channel symmetric Rabi-Kondo (2CSRK) model, with Hamiltonian H = (cid:88) (cid:96) =1 , (cid:26) u π (cid:90) ∞−∞ d x [ ∂ x φ (cid:96) ( x )] + J z π √ P K S z ∂ x φ (cid:96) (0) + J xy πa P K (cid:0) S + e i √ φ (cid:96) (0) + h . c . (cid:1)(cid:27) + 2Ω τ x , (A14)which becomes on the scale of T K : H int = (cid:88) p = ± u π (cid:90) ∞−∞ d x [ ∂ x φ (cid:96) ( x )] + J ren xy πa P K S x cos [ φ − (0)] + Ω τ + (cid:104) P ↑ e − iφ + (0) / + P ↓ e + iφ + (0) / (cid:105) + h . c . (A15)Again the analysis parallels the spin-asymmetric case,except that now flipping S z together with φ + ( x ) remainsa symmetry, so the 2CK physics is not destroyed at lowenergies, and a decoupled Majorana zero mode remains.Indeed, while the Rabi term contains S z ∝ S x S y , thecorresponding processes are suppressed by the dominant J ren xy term, and all higher order processes (in terms of Ω)which leave the system within the low-energy manifold of J ren xy term do not couple to S y . It should show up in the correlation function of the Rabi term with itself, whichdepends on S z ∝ S x S y , and reduce one power of ω fromthe power-law dependence of the emission spectrum on ω for | ω | (cid:28) T K , that is, make it go as | ω | instead of | ω | (in the absence of detuning). Correspondingly, theimpurity entropy goes from ln 4 to ln(2 + √
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