SO(5) non-Fermi liquid in a Coulomb box device
SSO(5) non-Fermi liquid in a Coulomb box device
Andrew K. Mitchell, Alon Liberman, Eran Sela, and Ian Affleck School of Physics, University College Dublin, Belfield, Dublin 4, Ireland School of Physics and Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel Department of Physics and Astronomy and Stewart Blusson Quantum Matter Institute,University of British Columbia, Vancouver, B.C., Canada, V6T 1Z1
Non-Fermi liquid (NFL) physics can be realized in quantum dot devices where competing interactionsfrustrate the exact screening of dot spin or charge degrees of freedom. We show that a standardnanodevice architecture, involving a dot coupled to both a quantum box and metallic leads, can hostan exotic SO(5) symmetry Kondo effect, with entangled dot and box charge and spin. This NFLstate is surprisingly robust to breaking channel and spin symmetry, but destabilized by particle-holeasymmetry. By tuning gate voltages, the SO(5) state evolves continuously to a spin and then “flavor”two-channel Kondo state. The expected experimental conductance signatures are highlighted.
Nanoelectronic circuit realizations of fundamentalquantum impurity models allow the nontrivial physicsassociated with strong electron correlations to be probedvia quantum transport measurements [1]. Quantum dotdevices, in particular, can exhibit the Kondo effect atlow temperatures [2]: a localized magnetic moment onthe dot is dynamically screened by conduction electronsin the metallic leads. Single-dot devices can behave assingle-electron transistors, with Kondo-enhanced spin-flip scattering strongly boosting the conductance betweensource and drain leads measured in experiments [3–5].The conventional Kondo effect [6] involves a localized“impurity” spin- degree of freedom, coupled to a singleeffective channel of conduction electrons, and has SU(2)spin symmetry. However, the Kondo effect is also ob-served in more complex systems, such as coupled quan-tum dot devices [7, 8] and single-molecule transistors[9, 10], involving spin and orbital degrees of freedom.In such systems, it is possible to realize variants of theclassic spin- single-channel Kondo paradigm; e.g. orbital[11], spin-1 [12, 13], and ferromagnetic [14] Kondo effects.In particular, the symmetry of the effective model is im-portant in determining the low-energy physics. Kondoeffects with SU(4) symmetry can be realized in doublequantum dots [15, 16] and carbon nanotube dots [17, 18],and also have Fermi liquid (FL) ground states.More exotic non-Fermi liquid (NFL) states can be re-alized in multi-channel systems, where competing inter-actions frustrate exact screening of the dot spin or chargedegrees of freedom at special high-symmetry points [19].This results in a residual dot entropy characteristic offractionalized excitations, and anomalous conductancesignatures [20, 21]. However, this kind of NFL physicsis typically delicate, being found at the quantum criticalpoint between more standard FL phases, and is unstableto relevant symmetry-breaking perturbations.Experimentally, the major challenge to realize NFLKondo physics in quantum dot devices is to prevent mix-ing between multiple conduction electron channels. Twoprominent scenarios to achieve this utilize an interactingquantum box (“Coulomb box”) [22, 23]. The quantum box is a large quantum dot, hosting a macroscopicallylarge number of electrons, but due to quantum confine-ment has a discrete level spacing δ and finite chargingenergy E c . For δ < T < E C the box effectively providesa continuum reservoir of conduction electrons, but alsodisplays charge quantization [24].Spin-two channel Kondo (s-2CK) physics can be real-ized in a device involving a small quantum dot coupledto a quantum box as well as metallic leads [22]. The low-energy effective model consists of a dot spin- exchangecoupled to two conduction electron channels (leads andbox), with mixing between the channels suppressed bythe large box charging energy. Both channels competeto Kondo-screen the dot spin, resulting in an NFL state.Breaking channel or spin symmetry relieves the frustra-tion and results in a standard FL state. This physics wasrealized experimentally in Refs. [25, 26].By contrast, a charge-2CK (c-2CK) effect can be real-ized when a quantum box tuned to its charge degeneracypoint is coupled to two leads, as proposed in Ref. [23] andrealized experimentally in Refs. [27, 28]. In this case, the BdLs Ld V d V B Figure 1.
Right:
Schematic of the device: a quantum dotcoupled to a quantum box and source/drain leads.
Left:
NRGphase diagram spanned by dot and box gate voltages, V d ∝ η/U d and V B ∝ n B , showing the NFL line for various channelasymmetries t L /t B . SO(5) point located at n B = ± and η = 0 . Plotted for constant U d = 0 . , E C = 0 . and t B = 0 . . a r X i v : . [ c ond - m a t . s t r- e l ] S e p macroscopic box charge states play the role of a pseu-dospin impurity. Distinctive signatures of the resultingNFL state are observable in quantum transport [27–30].In this Letter, we revisit the device of Refs. [22, 25, 26]but now examine the full phase diagram as function ofdot and box gate voltages, which in turn control the dotand box occupancies – see Fig. 1. We show that the emer-gent SU(4) symmetry of the system arising when the dothosts a local moment and the box is at its charge degen-eracy point, is reduced to SO(5) at particle-hole symme-try. Although the SU(4) state is an FL [31], a novel NFLKondo effect arises at the SO(5) point, in which bothdot and box charge and spin are maximally entangled.We achieve a detailed understanding of this state usinga combination of conformal field theory [32–34] (CFT),bosonization [35], and numerical renormalization group[36, 37] (NRG) techniques. Remarkably, the NFL physicsat this point is robust to breaking channel and/or spinsymmetry. Furthermore, we show that by tuning gatevoltages, the SO(5) state evolves continuously into themore familiar s-2CK state of Refs. [22, 25, 26], and theninto a “flavor”-2CK (f-2CK) effect when the dot local mo-ment is lost but box charge fluctuations persist. The dis-tinctive transport signatures associated with this physicsare accessible in existing experimental setups. Models and mappings.–
The device illustrated in Fig. 1is described by the Hamiltonian H = H + H B + H d + P γ H γ hyb , with γ = Ls , Ld , B for the source/drainleads and box, respectively. H = P γ,k,σ (cid:15) γk c † γkσ c γkσ describes the three conduction electron reservoirs, while H B = E C (cid:16) ˆ N B − N − n B (cid:17) , (1) H d = X σ (cid:15) d d † σ d σ + U d d †↑ d ↑ d †↓ d ↓ , (2)describe the box Coulomb interaction and the dot. Thedot is tunnel-coupled to the leads and box via H γ hyb = P k,σ ( t γk d † σ c γkσ +H . c . ) . Here, σ = ↑ , ↓ denotes (real) spin,and d σ or c αkσ are operators for the dot or conductionelectrons, respectively. ˆ N B = P k,σ c † B kσ c B kσ is the totalnumber operator for the box electrons. The dot and boxoccupations are controllable by gate voltages V d ∝ η = (cid:15) d + U d and V B ∝ n B , respectively. For simplicity wenow take equivalent conduction electron baths (cid:15) γk ≡ (cid:15) k with a constant density of states ν defined inside a bandof halfwidth D = 1 , such that (cid:15) k ∼ k at low energies. Wedefine t γ = P k | t γk | and t = t + t .Following Ref. [38], we incorporate the box interactionterm, Eq. 1, into the hybridization, H B + H Bhyb → E C (cid:16) ˆ T z − n B (cid:17) + X k,σ ( t B k d † σ c B kσ ˆ T − +H . c . ) , where ˆ T ± = P N B | N B ± ih N B | are ladder operators forthe box charge, and ˆ T z = P N B ( N B − N ) | N B ih N B | . Note that the model possesses the symmetry n B → n B ± .Particle-hole (ph) asymmetry is controlled by n B and η ;the model is invariant to replacing n B → − n B and η →− η , related by a ph transformation. Exact ph symmetryarises at η = 0 for any integer or half-integer n B .For the NRG calculations presented here, only a finitenumber of charge states around the reference N are re-quired to obtain converged results [36, 37, 39]. Spin-2CK regime.–
For large box charging energy E C and deep in the dot and box Coulomb blockade regime(near the point η = 0 and n B = 0 ), the dot hosts aneffective spin- local moment, and the box has a well-defined number of electrons N . At low temperatures T (cid:28) E C , U d virtual charge fluctuations on the dot andbox due to H hyb generate the spin-flip scattering respon-sible for the Kondo effect. However, finite E c blockscharge transfer between the leads and box, giving rise to afrustration of Kondo screening and the possibility of NFLphysics [22]. In this regime, a standard Schrieffer-Wolfftransformation (SWT) yields the s-2CK model [19, 22], H s − CK = H + ~S d · (cid:16) J L ~S L + J B ~S B (cid:17) , (3)where ~S d is a spin- operator for the dot, while ~S α =L , B = P σ,σ c † ασ ~ σ σσ c ασ , with c B σ = t B P k t B k c B kσ and c L σ = t L P k ( t Ls k c Ls kσ + t Ld k c Ld kσ ) the local conductionelectron orbitals at the dot position, and where J L = 8 t U d (cid:20) − (cid:16) ηU d (cid:17) (cid:21) − ; J B = 8 t U d (cid:20) − (cid:16) η U d (cid:17) (cid:21) − (4)with U d = U d + 2 E C and η = η + 2 E C n B . Deep inthe s-2CK regime, NFL physics arises when J L = J B .For given physical device parameters U d , E C , t L , t B , Eq. 4implies the existence of two NFL lines in the ( n B , η ) planerelated by the symmetry η → − η and n B → − n B , seeFig. 1. NFL physics can therefore be accessed by tuningthe gate voltages V d ∝ η and V B ∝ n B , as demonstratedexperimentally in this regime in Refs. [25, 26]. At the phsymmetric point η = n B = 0 , s-2CK arises for t B = ζt L with ζ ’ E C /U d . Although this NFL state is robustto ph asymmetry, it is destabilized by channel asymmetry J L = J B or spin asymmetry B = 0 [40]. SO(5) Kondo.– At n B = , the box states with N and ( N + 1) electrons are exactly degenerate. Neglect-ing other box charge states (which are at least E C higher in energy), we may define charge pseudospin- operators ˆ T +B = | N + 1 ih N | , ˆ T − B = ( ˆ T +B ) † , and ˆ T z B = ( | N + 1 ih N + 1 | − | N ih N | ) . The charge pseu-dospin is flipped by electronic tunneling between the dotand box. The effective low-energy effective model is ob-tained by projecting onto the dot spin and box pseu-dospin sectors using a generalized SWT. We now considerexplicitly the special point with ph symmetry ( n B = and η = 0 ) and channel symmetry ( J L = J B ≡ J , whichimplies t B = ξt L with ξ ’ E C /U d ), whence [41, 42] H eff = H + J ~S d · (cid:18) c † ~ σ c (cid:19) + V z ˆ T z B (cid:18) c † τ z c (cid:19) + Q ⊥ ~S d · (cid:16) c † ~ σ ( τ + ˆ T − B + τ − ˆ T +B ) c (cid:17) , (5)where we have suppressed spin σ = ↑ , ↓ and channel α =L , B labels for clarity, and with Pauli matrices σ a or τ b acting in spin or channel space, respectively.Although initially the coupling constants in Eq. 5 takedifferent values, perturbative scaling [43] shows that themodel develops an emergent symmetry J = V z = Q ⊥ atan isotropic low-temperature fixed point. Then the RGequations reduce to dJ/dl = 3 J , and we have a Kondoscale T SO(5)K ∼ D exp( − / νJ ) .The fixed point has an unusual SO(5) symmetry, whichcan be seen by writing Eq. 5 in the symmetric form, H SO(5) = H + J X A =1 J A M A , (6)where J A = c † T A c (with c ≡ c ασ ( x = 0) the con-duction electron operators at the dot position as be-fore) and M A = f † T A f in terms of a fermionic ‘im-purity’ operator f ≡ f ασ which carries both ‘flavor’and spin labels subject to the constraint f † ασ f ασ = 1 such that ˆ S ad = f † σ a f and ˆ T b B = f † τ b f . Here, { T A } are the ten non-zero generators of SO(5) [44], T ab = − T ba (with a, b = 1 ... ) satisfying the algebra [ T ab , T cd ] = − i ( δ bc T ad − δ ac T bd − δ bd T ac + δ ad T bc ) . Theequivalence between Eqs. 5 and 6 is then established by, σ = T , σ = T , σ = T , σ a =1 , , τ = T a , σ a =1 , , τ = T a , τ = T . We applied the machinery of CFT [32–34] to analyzethe fixed point properties using the symmetry decompo-sition U(1) c × Z × SO(5) . Here, U(1) corresponds to thecharge sector, Z is an Ising model. The primary fieldsof the SO(5) theory consist of the singlet ( ) of scalingdimension 0, the spinor ( ) of scaling dimension andthe vector ( ) of scaling dimension [45]. The strongcoupling fixed point describing the SO(5) fixed point canbe obtained by fusion with the spinor ( ) , under whichthe impurity transforms.The energies ( E ) and degeneracies ( ) of the result-ing finite size spectrum characterize the fixed point. Wefind [39] ( E, , , , , , ... ,consistent with our NRG results, and establishing thenew SO(5) fixed point as NFL. Interestingly, this spec-trum is identical to that of the standard s-2CK model[33]. The entropy at the fixed point S = log g is given interms of the modular S-matrix [32–34], yielding a ln(2) entropy, consistent with NRG, see Fig. 2; again reminis-cent of s-2CK. Figure 2. Physical properties of the SO(5) Kondo effect, ob-tained by NRG for n B = , η = 0 , U d = 0 . , E C = 0 . , t L = 0 . , and t B ’ ξt L = 0 . . (a) Impurity contributionto entropy S imp ( T ) , showing overscreening of the entangledspin and flavor degrees of freedom on the scale of the Kondotemperature T K ∼ − . For T K (cid:28) T (cid:28) E C , free impurityspin and flavor give a ln(4) entropy, while S imp ( T ) = ln(2) for T (cid:28) T K , characteristic of the free Majorana fermion atthe SO(5) fixed point. (b) T = 0 local spin and flavor dynam-ical susceptibilities, both showing apparent FL-like behavior χ loc ( ω ) ∼ ω for ω (cid:28) T K . (c) Linear response conductancethrough the dot G ( T ) /G (blue line), with G = G at T = 0 ,and leading linear behavior G ( T ) − G (0) ∼ +( T /T K ) G (inset,dashed line). The standard spin-2CK conductance lineshapeis given for comparison as the dotted line. However, differences from the standard s-2CK picturecan be seen in dynamical quantities such as the local sus-ceptibilities and conductance – see middle and bottompanels in Fig. 2. Since the impurity spin ~S and pseu-dospin ~T operators are absorbed into the conduction elec-trons at the strong coupling fixed point, they must trans-form among the 10 generators of SO(5). But such fieldsoccur only as descendants in SO(5) , and so spin-spincorrelation functions appear FL-like, χ s loc ( ω ) ∼ ω (simi-larly for flavor susceptibility). This contrasts to the reg-ular k -channel Kondo effect: the spin SU(2) k theory con-tains a vector field which transforms as the 3 componentsof the impurity spin, with scaling dimension k , whichleads to anomalous NFL properties in the spin suscepti-bility χ s loc ∼ ω − k − k +2 . At the SO(5) point, we find leadinglinear behavior of the conductance G ( T ) − G (0) ∼ T fromNRG. This contrasts to standard s-2CK conductancewhich approaches its fixed point value as √ T [22, 46–48], or the T FL conductance of 1CK [2].To gain further insight, we expand on the bosoniza-tion and refermionization techniques [49] developed byEmery and Kivelson (EK) for the s-2CK model [35], andinclude the coupling to the flavour degree of freedom.This method allows us to express an anisotropic versionof Eq. 5 in terms of local fermions d ∝ S − and a ∝ T − ,relating to impurity spin and flavor degrees of freedom,with corresponding Majorana operators d + = √ ( d † + d ) and d − = √ i ( d † − d ) , and similarly for a , as well as a1D bulk fermionic field denoted ψ sf ( x ) , with Majoranacomponents χ + = ψ † sf (0)+ ψ sf (0) √ , χ − = ψ † sf (0) − ψ sf (0) √ i . Theresulting EK Hamiltonian takes the form, H EK = H + iJ ⊥ d − χ + − iQ ⊥ d + a − . (7)Details of the derivation are given in the SupplementalMaterial [39]. The J ⊥ term is the usual EK form of thes-2CK interaction. The spin-flavour coupling term Q ⊥⊥ couples and gaps out the pair d + and a − . Unlike in thes-2CK model where d + remains decoupled, here we seethat it is a + that is free at the SO(5) fixed point, and isresponsible for the ln(2) residual entropy. Stability of SO(5) Kondo.–
We consider the effect ofsymmetry-breaking perturbations at the SO(5) point.Channel asymmetry, corresponding to t B = ξt L in thebare model, generates an extra term in Eq. 5 given by δH ch = J − ~S d · ( c † ~ σ τ z c ) with J − ∝ J B − J L . As in theEK mapping of the s-2CK model, this becomes δH ch = − iJ − d + χ − . In contrast to the s-2CK model, the SO(5)point is robust to detuning away from channel symmetry.This is because the d + Majorana involved in J − is alreadygapped out by the spin-flavor coupling Q ⊥ in Eq. 7. Thisis confirmed by NRG [39].Breaking spin symmetry by applying a dot magneticfield δH s = B ˆ S zd = − iBd + d − is similarly irrelevant atthe SO(5) fixed point. NFL physics is therefore robustto B , also confirmed by NRG [39].Keeping n B = , ph symmetry is broken by η = 0 .Performing the SWT yields an additional contribution toEq. 5 of the form [42] δH ph = V ⊥ P b =1 , ˆ T b B ( c † τ b c ) + Q z P a =1 , , ˆ S ad ˆ T ( c † σ a τ c ) where V ⊥ , Q z ∝ η . This per-turbation contains an additional 5 generators, which to-gether with the 10 from SO(5) form the defining repre-sentation of SU(4). Indeed, under RG the system flowsto a fully isotropic SU(4) FL fixed point, as discussed inRefs. [16, 31, 41], with zero residual entropy. Breakingph symmetry therefore destabilizes the NFL SO(5) fixedpoint, with an emergent FL crossover scale T ∗ ∼ η [39].Unusually then, lowering the symmetry of the bare modelby introducing finite η leads to a low-energy SU(4) fixedpoint with higher symmetry than the SO(5) fixed pointobtained at η = 0 . Applying the EK mapping, we ob-tain [39] δH ph = − iV ⊥ a + χ − . This is an RG relevantterm with scaling dimension : the previously free a + Majorana is now coupled to the χ − field, quenching the ln(2) entropy and leading to an FL state, with χ s,f loc ∼ ω and G ( T ) − G (0) ∼ T [50]. Evolution of NFL line.–
We now explore the evolutionof the NFL state in the ( n B , η ) plane. For n B = , phsymmetry is broken, generating δH ph , and also flavordegeneracy is removed, generating an effective flavor field δH f = B f ˆ T z B = − iB f a + a − , with B f = E C (1 − n B ) .Both effects might be expected to destabilize the NFL.However, we shall see that an NFL line extends awayfrom the SO(5) point at n B = and η = 0 .Combining the flavour field − iB f a + a − with the spin-flavour coupling − iQ ⊥ d + a − , we define a new localMajorana basis d + = d + cos θ + a + sin θ and a + = a + cos θ − d + sin θ with tan θ = B f Q ⊥ . The Hamilto-nian with additional channel- and ph-asymmetry per-turbations, H EK + δH f + δH ch + δH ph remains of thesame form as with B f = 0 in the new Majorana basis,only taking modified coupling constants ˜ Q ⊥ , ˜ J − , ˜ V ⊥ . Forthe relevant perturbation, δH ph = − i ˜ V ⊥ a + χ − , we find ˜ V ⊥ = V ⊥ cos θ − J − sin θ . Importantly, we see that a + can be completely decoupled to yield the NFL fixed pointwhen ˜ V ⊥ = 0 . Near the point η = 0 and n B = , thishappens along the line Q ⊥ V ⊥ = J − B f , a result we haveconfirmed using NRG. Although ph asymmetry destabi-lizes the SO(5) NFL fixed point, the two sources of phasymmetry from η and n B can “cancel out”. Along theresulting NFL line, the free Majorana contributing the ln(2) residual entropy is the rotated a + . Moving awayfrom n B = , the free Majorana smoothly transformsfrom being flavor-like to spin-like. NRG phase diagram.–
Finally, we examine numericallythe full phase diagram in the plane ( n B , η ) using NRG.Returning to Fig. 1 we see that the s-2CK effect at n B = 0 and the SO(5) Kondo effect at n B = are continuouslyconnected for all t B /t L > ζ (red and black lines). In-terestingly, the NFL lines continue into an unexpectedregion of the phase diagram with | η | /U d > , where thedot no longer hosts a local moment and f-2CK pertains.These NFL lines diverge with η → ±∞ as n B → ± .Moving along an entire NFL line from n B = + to − ,the spin and flavour susceptibilities χ s,f (see [39]) showa continuous crossover from spin-flavour SO(5) Kondo tos-2CK and ultimately f-2CK. While both spin and flavorfluctuations are important at SO(5), flavor (spin) fluctu-ations are suppressed in the s-2CK (f-2CK) regimes.For t B /t L < ζ , the NFL line instead terminates at apoint ( n ∗ B , η ∗ ) , beyond which the condition ˜ V ⊥ = 0 canno longer be satisfied (green and pink lines, Fig. 1). Spinand flavor fluctuations are both important; no pure s-2CK or f-2CK effect can be realized.The NFL line with t B /t L = ζ (blue line) is specialsince it is invariant to the ph transformation n B → − n B and η → − η . It therefore smoothly connects SO(5) NFLpoints at all half-odd-integer n B . Conclusions.–
We revisit a classic model describingquantum dot/box experiments used to probe NFLphysics, uncovering a rich range of new physics, includ-ing a novel spin-flavor SO(5) Kondo effect. We studythe evolution of the NFL line as a function of dot andbox gate voltages using a combination of analyticaland numerical techniques, showing that the well-knowns-2CK effect can continuously transform into the f-2CKor SO(5) Kondo effects. Distinctive experimentalsignatures of this new physics should be observable inconductance [50].
Acknowledgments.–
AKM thanks the Stewart Blus-son Quantum Matter Institute (UBC) for travel sup-port and acknowledges funding from the Irish Re-search Council Laureate Awards 2017/2018 throughgrant IRCLA/2017/169. ES acknowledges support fromARO (W911NF-20-1-0013), the Israel Science Founda-tion grant number 154/19 and US-Israel Binational Sci-ence Foundation (Grant No. 2016255). IA acknowledgessupport from NSERC Discovery Grant 04033-2016. [1] L. L. Sohn, L. P. Kouwenhoven, and G. Schön,
Meso-scopic electron transport , Vol. 345 (Springer Science &Business Media, 2013).[2] M. Pustilnik and L. Glazman, Journal of Physics: Con-densed Matter , R513 (2004).[3] D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu,D. Abusch-Magder, U. Meirav, and M. Kastner, Nature , 156 (1998).[4] S. M. Cronenwett, T. H. Oosterkamp, and L. P. Kouwen-hoven, Science , 540 (1998).[5] W. Van der Wiel, S. De Franceschi, T. Fujisawa, J. Elz-erman, S. Tarucha, and L. Kouwenhoven, science ,2105 (2000).[6] A. C. Hewson, The Kondo Problem to Heavy Fermions (Cambridge University Press, 1997).[7] H. Jeong, A. M. Chang, and M. R. Melloch, Science ,2221 (2001).[8] J. Malecki, E. Sela, and I. Affleck, Physical Review B , 205327 (2010).[9] W. Liang, M. P. Shores, M. Bockrath, J. R. Long, andH. Park, Nature , 725 (2002).[10] A. K. Mitchell, K. G. Pedersen, P. Hedegård, andJ. Paaske, Nature communications , 1 (2017).[11] R. López, D. Sánchez, M. Lee, M.-S. Choi, P. Simon, andK. Le Hur, Physical Review B , 115312 (2005).[12] S. Sasaki, S. De Franceschi, J. Elzerman, W. Van derWiel, M. Eto, S. Tarucha, and L. Kouwenhoven, Nature , 764 (2000).[13] J. Paaske, A. Rosch, P. Wölfle, N. Mason, C. Marcus,and J. Nygård, Nature Physics , 460 (2006).[14] A. K. Mitchell, T. F. Jarrold, and D. E. Logan, PhysicalReview B , 085124 (2009).[15] A. Keller, S. Amasha, I. Weymann, C. Moca, I. Rau,J. Katine, H. Shtrikman, G. Zaránd, and D. Goldhaber-Gordon, Nature Physics , 145 (2014).[16] L. Borda, G. Zaránd, W. Hofstetter, B. Halperin, andJ. Von Delft, Physical review letters , 026602 (2003); M. R. Galpin, D. E. Logan, and H. Krishnamurthy, ibid . , 186406 (2005).[17] M.-S. Choi, R. López, and R. Aguado, Physical reviewletters , 067204 (2005).[18] F. B. Anders, D. E. Logan, M. R. Galpin, and G. Finkel-stein, Physical review letters , 086809 (2008).[19] P. Nozieres and A. Blandin, Journal de Physique , 193(1980).[20] I. Affleck and A. W. Ludwig, Physical Review Letters ,161 (1991).[21] I. Affleck and A. W. Ludwig, Physical Review B , 7297(1993).[22] Y. Oreg and D. Goldhaber-Gordon, Physical review let-ters , 136602 (2003).[23] A. Furusaki and K. Matveev, Physical Review B ,16676 (1995).[24] K. Matveev, Physical Review B , 1743 (1995).[25] R. Potok, I. Rau, H. Shtrikman, Y. Oreg, andD. Goldhaber-Gordon, Nature , 167 (2007).[26] A. Keller, L. Peeters, C. Moca, I. Weymann, D. Mahalu,V. Umansky, G. Zaránd, and D. Goldhaber-Gordon, Na-ture , 237 (2015).[27] Z. Iftikhar, S. Jezouin, A. Anthore, U. Gennser, F. Par-mentier, A. Cavanna, and F. Pierre, Nature , 233(2015).[28] Z. Iftikhar, A. Anthore, A. Mitchell, F. Parmentier,U. Gennser, A. Ouerghi, A. Cavanna, C. Mora, P. Si-mon, and F. Pierre, Science , 1315 (2018).[29] A. K. Mitchell, L. Landau, L. Fritz, and E. Sela, Physicalreview letters , 157202 (2016).[30] G. A. van Dalum, A. K. Mitchell, and L. Fritz,Physical Review B , 041111 (2020); G. vanDalum, A. Mitchell, and L. Fritz, arXiv preprintarXiv:2007.07239 (2020).[31] K. Le Hur and P. Simon, Physical Review B , 201308(2003).[32] I. Affleck, Nuclear Physics B , 517 (1990).[33] I. Affleck and A. W. Ludwig, Nuclear Physics B , 641(1991).[34] I. Affleck, arXiv preprint cond-mat/9512099 (1995).[35] V. Emery and S. Kivelson, Physical Review B , 10812(1992).[36] K. G. Wilson, Reviews of modern physics , 773 (1975);R. Bulla, T. A. Costi, and T. Pruschke, Reviews of Mod-ern Physics , 395 (2008).[37] A. K. Mitchell, M. R. Galpin, S. Wilson-Fletcher, D. E.Logan, and R. Bulla, Physical Review B , 121105(2014); K. Stadler, A. Mitchell, J. von Delft, and A. We-ichselbaum, ibid . , 235101 (2016).[38] F. B. Anders, E. Lebanon, and A. Schiller, Physical Re-view B , 201306 (2004).[39] See Supplemental Material for further details.[40] I. Affleck, A. W. Ludwig, H.-B. Pang, and D. Cox, Phys-ical Review B , 7918 (1992).[41] K. Le Hur, P. Simon, and L. Borda, Physical Review B , 045326 (2004).[42] See Eq. 17 in Ref. 41.[43] See Eq. 20 in Ref. 41 with V ⊥ = Q z = 0 .[44] H. Georgi, Lie algebras in particle physics: from isospinto unified theories (CRC Press, 2018).[45] P. Francesco, P. Mathieu, and D. Sénéchal,
Conformalfield theory (Springer Science & Business Media, 2012).[46] M. Pustilnik, L. Borda, L. Glazman, and J. Von Delft,Physical Review B , 115316 (2004). [47] E. Sela, A. K. Mitchell, and L. Fritz, Physical reviewletters , 147202 (2011).[48] A. K. Mitchell and E. Sela, Physical Review B , 235127(2012).[49] J. Von Delft and H. Schoeller, Annalen der Physik , 225 (1998).[50] A. Liberman, A. K. Mitchell, I. Affleck and E. Sela, inpreparation. upplemental Material: SO(5) non-Fermi liquid in a Coulomb box device Andrew K. Mitchell, Alon Liberman, Eran Sela, and Ian Affleck School of Physics, University College Dublin, Belfield, Dublin 4, Ireland School of Physics and Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel Department of Physics and Astronomy and Stewart Blusson Quantum Matter Institute,University of British Columbia, Vancouver, B.C., Canada, V6T 1Z1
S-1. SUPPLEMENTARY DATA
Figure S1 shows the impurity contribution to the totalentropy S imp ( T ) vs temperature T obtained by NRG atand in the vicinity of the SO(5) symmetric point. (a)channel asymmetry, t B /t L = ξ ; (b) spin asymmetry dueto a dot magnetic field, B = 0 ; (c) ph asymmetry due to η = 0 . At n B = , NFL physics is robust to channel andspin asymmetry, but unstable to ph symmetry breaking.Figure S2 shows the T = ω = 0 spin and flavor sus-ceptibilities (black and red lines) of a system with fixed t B /t L > ζ as a function of n B along the entire NFL line(black curves in Fig. 1 in the main text). Deep in the boxCoulomb blockade regime, flavor fluctuations are sup-pressed by the box charging energy E C , as seen fromthe small values of χ f loc . However, the spin susceptibil-ity is enhanced, characteristic of the s-2CK effect, with T K χ s loc = const. (panel c) over a wide range of n B . Closeto n B = , the flavor susceptibility χ f loc is strongly en-hanced. However, we find χ s loc (0) ∼ ( − n B ) , consis-tent with χ s loc ( ω ) ∼ ω , while χ f loc remains finite (panelb). Strong dot charge fluctuations between one and twoelectron states at η/U d = − give rise to enhanced spinand flavor fluctuations, χ s loc = χ f loc . This situation arisesat the point with the maximum Kondo temperature T K (blue line, panel a). Decreasing n B further towards − we see enhancement (suppression) of the flavor (spin)susceptibility, with T K χ f loc → const. and T K χ s loc → as n B → − , indicative of a crossover from s-2CK to f-2CK. S-2. DETAILS OF NRG CALCULATIONS
Following Ref. 1 we implement with NRG the model H ALS = P α =L , B ( H α + H α hyb ) + H B + H d , where, H α = X k,σ (cid:15) αk c † αkσ c αkσ , (S1) H B = E C (cid:16) ˆ T z − n B (cid:17) , (S2) H d = X σ (cid:15) d d † σ d σ + U d d †↑ d ↑ d †↓ d ↓ , (S3) H Lhyb = X k,σ ( t L k d † σ c L kσ + H . c . ) (S4) H Bhyb = X k,σ ( t B k d † σ c B kσ ˆ T − + H . c . ) , (S5) where ˆ T z = P mn = − m n | n ih n | and ˆ T + = P m − n = − m | n +1 ih n | with ˆ T − = ( ˆ T + ) † in terms of the box charge states | n i . In practice we take m = 2 , which yields well-converged low-temperature results for E C = 0 . and t α = P k | t αk | ∼ O (10 − ) as used here. For simplic-ity we take equivalent lead and box bands, with a flatconduction electron density ν within bands of halfwidth D ≡ . We define η = (cid:15) d + U d as before.The free conduction electron bands are discretized log-arithmically using Λ = 2 . , and N s ∼ states areretained at every step of the iterative diagonalizationprocedure. We utilized the ‘interleaved NRG’ method onthe generalized Wilson chain, exploiting conserved totalcharge and spin projection. Dynamical quantities arecalculated using the full density matrix NRG method. S-3. CONDUCTANCE
In the ‘split-lead’ geometry utilized in the experimentsof Refs. 5 and 6 and depicted in Fig. 1 of the main pa-per, the free lead Hamiltonian is decomposed as H L0 = P γ =Ls , Ld P k,σ (cid:15) L k c † γkσ c γkσ into source and drain com-ponents, while H Lhyb = P γ =Ls , Ld P k,σ ( t γk d † σ c γkσ +H . c . ) ,and with t γ = P k | t γk | .A voltage bias, V bias , is applied symmetrically tothe leads, H bias = eV bias ( ˆ N Ls − ˆ N Ld ) with ˆ N γ = P k,σ c † γkσ c γkσ , and we measure the current into the drainlead, I Ld = h ddt ˆ N Ld i . The linear-response differentialconductance, G = lim V bias → dI Ld dV bias , (S6)is related to the equilibrium dot Green’s function G dd ( ω, T ) ≡ hh d σ ; d † σ ii via the Meir-Wingreen formula, G ( T ) = G Z ∞−∞ dω df ( ω ) dω (cid:2) πνt Im G dd ( ω, T ) (cid:3) , (S7)where f ( ω ) is the Fermi function, and G = 2 e /h × t t /t . We obtain the dot Green’s function G dd ( ω, T ) as an entire function of (real) frequency ω ata given temperature T using NRG as above.We find πνt Im G dd (0 ,
0) = , implying G (0) = G ,everywhere along the NFL line. Deep in the Coulombblockade regime of the box where the standard spin-2CK effect is observed, G ( T ) − G (0) ∼ − p T /T K G . At a r X i v : . [ c ond - m a t . s t r- e l ] S e p (a) Channel symmetry breaking(b) Spin symmetry breaking(c) Particle-hole symmetry breaking Figure S1. Dot contribution to the total entropy S imp ( T ) vs T obtained by NRG at and in the vicinity of the SO(5) sym-metric point. The SO(5) point for n B = , η = 0 , U d = 0 . , E C = 0 . , t L = 0 . , and t B ’ ξt L = 0 . is shown as the boldred lines in each panel: the entropy flows from ln(4) for freeimpurity spin and flavor degrees of freedom, to ln(2) char-acteristic of the NFL SO(5) fixed point, on the scale of theKondo temperature T SO(5)K ∼ − . (a) Channel symmetrybreaking, parametrized by λ = ξ t /t = 1 , , for the red,blue, and black lines respectively, with fixed t (1 + λ ) = 0 . .NFL physics is seen to be robust to this perturbation, with S imp (0) = ln(2) in all cases. Strong channel asymmetry gen-erates a two-stage Kondo effect with first-stage single-channelscreening on the scale of T s K of the spin by the more stronglycoupled channel, followed by two-channel overscreening ofthe flavor degrees of freedom on the scale of T f K . (b) Spinsymmetry breaking, due to a dot magnetic field B/T
SO(5)K =0 , , . , for the red, green, blue and black lines respectively.For magnetic fields B/T
SO(5)K (cid:29) , the dot spin is effectivelyfrozen, suppressing spin-Kondo screening. The free flavor de-gree of freedom (giving a ln(2) entropy contribution) is thentwo-channel overscreened below T f K . (c) Particle-hole symme-try breaking, due to η = (cid:15) d + U d = 0 , − , − , − . , − for the red, pink, green, blue, and black lines respectively. Phasymmetry is seen to destabilize the NFL fixed point, gener-ating a second Fermi liquid (FL) scale T ∗ ∼ η , below whichthe residual impurity entropy is quenched, S imp (0) = 0 , atan SU(4) symmetric fixed point. With T ∗ (cid:28) T SO(5)K , this FLcrossover from SO(5) to SU(4) Kondo takes a universal form. n B = and η = 0 where spin and flavor fluctuationsboth play an important role in driving the Kondo effect,we obtain G ( T ) − G (0) ∼ ± p T /T K G , with the + or Figure S2. Evolution along the NFL line as a function of n B for fixed t B = 0 . , t L = 0 . , U d = 0 . , E C = 0 . (alongthe black curves in Fig. 1 in the main text). (a) T = ω = 0 susceptibilities χ s loc (black), χ f loc (red), and Kondo tempera-ture T K (blue). (b) Same near n B = 0 . on a log-log plot vs − n B . (c) T K χ s loc (black dashed) and T K χ f loc (red dashed). − sign, respectively, for channel asymmetry t B > ξt L or t B < ξt L . This is due to a two-stage Kondo effect in thestrongly channel asymmetric case. However, when pre-cisely at the SO(5) point, where t B = ξt L , the square-rootbehavior vanishes, leaving a leading linear temperaturedependence, G ( T ) − G (0) ∼ +( T /T K ) G (see Fig. 2 ofthe main paper). This is reminiscent of the behavior ofconductance in the two-impurity Kondo model. At the SO(5) point, the full conductance lineshape is auniversal function of
T /T K , and is entirely characteristicof the novel SO(5) Kondo effect. This should be measur-able in experiment (note also that the Kondo tempera-ture itself is enhanced at the SO(5) point). S-4. CFT TREATMENT OF THESO(5) FIXED POINT
In this appendix we derive the finite size spectrum atthe SO(5) symmetric fixed point using CFT. For a reviewof CFT methods for Kondo problems see Ref. 9.We employ a U(1) c × Z × SO(5) conformal embed-ding. Here, U(1) stands for the charge sector, whoseprimary fields are labelled by integer charges Q and havescaling dimension Q ; Z is the Ising model consistingof three primary fields: the identity field of scaling di-mension , the spin field σ of scaling dimension , andthe fermion field (cid:15) with scaling dimension 1/2, which sat-isfy the fusion rules σ × σ = 1 + (cid:15) , σ × (cid:15) = σ ; primaryfields of the SO(5) theory are labelled by SO(5) repre-sentations, consisting of the singlet ( ) of scaling di-mension 0, the spinor ( ) of scaling dimension andthe vector ( ) of scaling dimension / . The SO(5) the-ory can also be thought of as 5 Ising models, and itsfusion rules follow by identifying the fields ( ) , ( ) , ( ) with , σ, (cid:15) . We note that other representations of SO(5)appear as descendants of the above primary fields, forexample J A = c † T A c which transforms under the di-mensional representation, similar to the impurity spinand flavour degrees of freedom, is a (Kac-Moody) de-scendent of the singlet ( ) . Table I. Finite size spectrum.Free spectrum Fusion with ( ) Q Ising SO(5) E Q Ising SO(5) E − ( ) ( ) ± σ ( ) ± σ ( ) 0 ± σ ( ) ± (cid:15) ( ) ± (cid:15) ( ) ± ( ) ± ( ) (cid:15) ( ) (cid:15) ( ) The free fermion spectrum can be recovered by the glu-ing conditions shown in the left hand side part in TableI. [As reviewed in Ref. 9, energy levels are obtained byadding up scaling dimensions in units of πv F /L , where L is the length of the effective 1D system and v F the Fermivelocity.] We obtain the strong coupling fixed point byfusion with the spinor field ( ) of SO(5) . Physically, theimpurity is absorbed into the electrons and changes theirboundary condition. The resulting finite size spectrum-listed in the right side of Table I, is fully consistent withour NRG observations. The first energies and degenera-cies are given by ( E, , / , / , / , , . . . (S8) S-5. EMERY-KIVELSON HAMILTONIANOF SPIN-FLAVOUR MODEL
Emery-Kivelson’s (EK) bosonization and refermioniza-tion technique, originally used to solve the 2CK prob-lem , allows to express the Hamiltonian in a free fermionform. In this appendix we derive the EK form of theHamiltonian of our spin-flavour Kondo model Eq. 5 andits various perturbations. We will outline the rigoroustreatment of the s-2CK case and highlight the requiredadditions involving the flavour impurity degree of free-dom. (A similar derivation for the spin-flavour Kondomodel appears in Ref. 13.)Our starting point is the ph symmetric and channelsymmetric Hamiltonian Eq. 5 in the main text withanisotropic bare coupling constants: (i) In the flavoursector the unperturbed model contains only a V z cou-pling and no a flavour-flip ( V ⊥ ), since the latter breaksph symmetry (treated below as a perturbation). (ii) Inthe spin sector we separate the SU(2) symmetric Kondocoupling J into J z and J ⊥ components. (iii) Also the spin-flavour operator Q ⊥ in Eq. 5 in the main text issplit into spin-flip part Q ⊥⊥ and a non-spin-flip part Q z ⊥ .The EK technique is applied to the hamiltonianas follows: (1) Fermionic fields are transformed intobosonic fields φ σα ( x ) , according to the relation ψ σα ( x ) = F σα e − iφ σα ( F σα being fermionic Klein operators). (2)The new bosonic fields are re-expressed in a basis ofcharge, spin, flavour and spin-flavor degrees of freedomabbreviated as φ A ( x ) , ( A = c , s , f , sf ). (3) A unitary op-erator U = exp[ i ( S z φ s + T z φ f )] acting both in spin andflavor spaces is applied on the Hamiltonian. With a par-ticular choice of the z -part of the spin and flavor Kondocouplings J z and V z , (i) terms of the form ∂ x φ s (0) S z and ∂ x φ f (0) T z effectively drop out of the Hamiltonian, and(ii) vertex parts e − iφ A of both spin and flavour sectorscancel. To achieve both cancellations we must have spe-cific bare couplings J z = V z = v F . (4) Klein operatorsare expressed in the basis of A = c , s , f , sf . (5) The Hamil-tonian is expressed in terms of new fermionic operators: ψ sf ( x = 0) = F sf e − iφ sf (0) , d = F † s S − , a = F † f T − . Thesecan be used to define Majorana fermion fields evaluatedat x = 0 : χ A + ≡ ψ † A (0) + ψ A (0) √ , χ A − ≡ ψ † A (0) − ψ A (0) √ i . (S9)Similarly we define local Majoranas a ± , d ± using the a and d operators for the flavour and spin impurity degreesof freedom, respectively. The presence of 4 local Majo-rana fermions, simply accounts for the entropy ofthe free fermion fixed point, due to both the spin andflavour impurity degrees of freedom S and T . In theseterms, the resulting Hamiltonian is H EK = H + iJ ⊥ d − χ + − Q z ⊥ a − χ + d − d + − iQ ⊥⊥ d + a − . (S10)The term ∝ J ⊥ appears in the usual s-2CK model, whichis a relevant perturbation to the free fermion fixed point H , leading to the absorption of the local Majorana d − into the Majorana field χ + . The last term ∝ Q ⊥⊥ gapsout the sector spanned by d + and a − , meaning that wecan exchange every appearance of the product id + a − byits expectation value. This way, the term ∝ Q z ⊥ becomesthe same as J ⊥ , merely renormalizing it. Eq. 7 in themain text follows (with simplified notation Q ⊥⊥ → Q ⊥ ).Symmetry breaking perturbations include channelasymmetry, magnetic field, and ph symmetry breaking.Channel asymmetry and magnetic field have the sameEK form as in the s-2CK model, as discussed extensivelyin the literature . We focus on ph breaking terms whichare unique to the spin-flavour model.Moving away from the point η = 0 , n B = in thephase diagram, ph symmetry is broken. As a result, theterms ∝ V ⊥ and Q z (missing by symmetry from Eq. 5 inthe main text, and featuring in δH ph ), appear. Focusingon V ⊥ T + c † τ − c + h.c. , and performing the EK transforma-tion, we obtain the relevant operator δH ph = iV ⊥ a + χ − .The operator Q z generates the same relevant operator. F. B. Anders, E. Lebanon, and A. Schiller, Physical Re-view B , 201306 (2004). K. G. Wilson, Reviews of modern physics , 773 (1975);R. Bulla, T. A. Costi, and T. Pruschke, Reviews of ModernPhysics , 395 (2008). A. K. Mitchell, M. R. Galpin, S. Wilson-Fletcher, D. E. Lo-gan, and R. Bulla, Physical Review B , 121105 (2014);K. Stadler, A. Mitchell, J. von Delft, and A. Weichsel-baum, ibid . , 235101 (2016). A. Weichselbaum and J. von Delft, Physical review letters , 076402 (2007). R. Potok, I. Rau, H. Shtrikman, Y. Oreg, andD. Goldhaber-Gordon, Nature , 167 (2007). A. Keller, L. Peeters, C. Moca, I. Weymann, D. Mahalu,V. Umansky, G. Zaránd, and D. Goldhaber-Gordon, Na-ture , 237 (2015). Y. Meir and N. S. Wingreen, Physical review letters ,2512 (1992). A. K. Mitchell, E. Sela, and D. E. Logan, Physical reviewletters , 086405 (2012). I. Affleck, arXiv preprint cond-mat/9512099 (1995). P. Francesco, P. Mathieu, and D. Sénéchal,
Conformalfield theory (Springer Science & Business Media, 2012). V. Emery and S. Kivelson, Physical Review B , 10812(1992). J. Von Delft and H. Schoeller, Annalen der Physik , 225(1998). J. Ye, Physical Review B , R489 (1997). A. K. Mitchell and E. Sela, Physical Review B85