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Unveiling the Kondo cloud: unitary RG study of the Kondo model
Anirban Mukherjee,1, ∗ Abhirup Mukherjee,1, † N. S. Vidhyadhiraja,2, ‡ A. Taraphder,3, § and Siddhartha Lal1, ¶
1Department of Physical Sciences, Indian Institute of Science Education and Research-Kolkata, W.B. 741246, India2Theoretical Sciences Unit, Jawaharlal Nehru Center for Advanced Scientific Research, Jakkur, Bengaluru 560064, India
3Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India(Dated: January 5, 2022)
We analyze the single-channel Kondo model using the recently developed unitary renormalisationgroup (URG) method, and obtain a comprehensive understanding of the Kondo screening cloud.The fixed-point low-energy Hamiltonian enables the computation of a plethora of thermodynamicquantities (specific heat, susceptibility, Wilson ratio, etc.) as well as spectral functions, all of whichare found to be in good agreement with known results. By integrating out the impurity, we obtain aneffective Hamiltonian for the excitations of the electrons comprising the Kondo cloud. This is foundto contain both k−space number-diagonal (Fermi liquid) as well off-diagonal four-fermion scatteringterms. Our conclusions are reinforced by a URG study of the two-particle entanglement and many-body correlations among members of the Kondo cloud and impurity. The entanglement between theimpurity and a cloud electron, as well as between any two cloud electrons, is found to increase underflow towards the singlet ground state at the strong-coupling fixed point. Both the number-diagonaland off-diagonal correlations within the conduction cloud are also found to increase as the impurity isscreened under the flow, and the latter are found to be responsible for the macroscopic entanglementof the Kondo-singlet ground state. The unitary RG flow enables an analytic computation of thephase shifts suffered by the conduction electrons at the strong-coupling fixed point. This revealsan orthogonality catastrophe between the local moment and strong-coupling ground states, and isrelated to a change in the Luttinger volume of the conduction bath under the crossover to strongcoupling. Our results offer fresh insight on the nature of the emergent many-particle entanglementwithin the Kondo cloud, and pave the way for further investigations in more exotic contexts suchas the fixed point of the over-screened multi-channel Kondo problem.
I. INTRODUCTION
In this work, we present a unitary renormalisationgroup (URG) analysis of the single channel Kondo model.The model involves a quantum impurity interactingwith a conduction bath. It can be derived from theparticle-hole symmetric single-impurity Anderson modelusing a Schrieffer-Wolff transformation that integratesout single-particle interactions [1]. The model was ini-tially developed to study metals with magnetic impuri-ties, and was used by Jun Kondo [2, 3] to explain theresistivity minimum that appears in these materials atlow temperatures. Since then, various methods havebeen applied to determine the low energy behaviour ofthis model. Notable attempts include the Coulomb gasapproach by Anderson and Yuval [4, 5] and perturba-tive renormalisation group approach (poor man’s scaling)by Anderson [6]. The latter method showed that theKondo exchange coupling J flows to larger values as wego to lower energy scales, at least for small J . The fullcrossover from the local moment phase at small J to thescreened moment strong-coupling phase at large J waslater obtained by the numerical renormalisation group(NRG) technique developed by Wilson [7, 8], the Bethe
∗ [email protected]† [email protected]‡ [email protected]§ [email protected]¶ [email protected]
ansatz solution by Andrei and Wiegmann [9, 10] and theconformal field theory (CFT) approach [11, 12]. Anotherimportant strong-coupling approach based on argumentsof scattering phase shifts is that of the local Fermi liquidtheory [13, 14] by Nozieres. The low-temperature proper-ties were found to be universal functions of a single energyscale, the Kondo temperature TK [7, 15–18]. All the pre-dicted aspects of the Kondo effect, including the existenceof the Kondo cloud [19–21], were observed experimen-tally in quantum dot systems [22–26]. Scanning tunnel-ing spectroscopy (STS) measurements have revealed thatthe Kondo effect often depends on the neighborhood ofthe impurities, in Cu and Co atoms [27, 28]. It was alsoshown [29], using quantum dots, that the out of equilib-rium Kondo effect also displays universality, the physicsat low temperatures being decided by only two energyscales, the frequency and amplitude of the perturbation.The impurity spectral function has been calculated usingNRG[30], both at T = 0 [31, 32] as well as T > 0 [33], aswell as using diagrammatic methods [34]. The electricalresistivity was found to obey single-parameter scaling be-havior in T/TK [35]. Nozieres went further and analyzeda more realistic model, the multi-channel Kondo prob-lem in which multiple conduction bath channels interactwith quantum impurity at the center [36]. Such a modelwas found, through methods like the Bethe ansatz, CFTand bosonization among others, to host a non-Fermi liq-uid low energy phase [36–47]. Kondo effect also occursin light quark matter which interact with heavy quarkimpurities through gluon-exchange interactions; the scat-tering amplitude goes through a similar logarithmic di-
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vergence and renormalisation group calculations reveal aKondo scale in such systems [48, 49]. Kondo effect canalso be realized for other fermionic systems like graphene[50], Dirac/Weyl semi-metals [51, 52] and dense nuclearmatter [48, 53, 54].
Obtaining the form of the effective Hamiltonian gov-erning the low-energy physics of the Kondo cloud and themany-body singlet wavefunction remained a challenge.In this work, we have analyzed the Kondo model on atight-binding lattice using the recently developed URGmethod [55, 56]. The method has already been appliedto a variety of problems such as the 2D Hubbard model[57, 58], the quantum kagome antiferromagnet [59], a gen-eralized model of electrons in two spatial dimensions withattractive U(1)-symmetric interactions [60], the 1D Hub-bard model [61], as well as other generalized models ofelectrons with repulsive interactions with and withouttranslation invariance [56]. The method involves apply-ing unitary transformations (see eq. 2) on the Hamilto-nian that decouple high energy electrons, creating a fam-ily of unitarily-connected Hamiltonians at progressivelysmaller energy scales. The decoupled electrons lose theirquantum fluctuations in the process, becoming integralsof motion (IOMs). This program leads to the one of themain results of this work - a unitarily-transformed strong-coupling fixed-point Hamiltonian. It is from this fixed-point Hamiltonian that we obtain the effective Hamilto-nian for the electrons that comprise the Kondo cloud,by integrating out the impurity. This effective Hamil-tonian is found to contain both Fermi liquid (diagonal)and off-diagonal four-fermion interaction terms, sheddingnew light on the nature of the Kondo cloud the interac-tions within. It is seen that the flow towards the strong-coupling fixed point is concomitant with the growth ofthese off-diagonal terms, and this picture is made clearerby the evolution of the spectral function calculated usingthe effective Hamiltonians. For the purposes of bench-marking, we have also computed several thermodynamicquantities (e.g., the impurity susceptibility, specific heat,Wilson ratio etc.) from this effective theory. These aredescribed in the appendices, and are found to be in goodagreement with the results known from the literature.
Another important result of this work involves studiesof the evolution of the macroscopic entanglement of theKondo cloud with the impurity spin and the many-bodycorrelations inside the Kondo cloud under renormalisa-tion towards strong-coupling. Experimental studies ofentanglement and correlations have been performed indouble quantum dots [62]. For the purpose of studyingthe entanglement and many-body correlations, we em-ploy the entanglement RG method developed recentlyby some of us in Refs.[60, 61, 63]. These calculations areenabled by the fact that the RG transformations are uni-tary and preserve the total spectral weight. Thus, itera-tive applications of the inverse of these unitary transfor-mations on the IR fixed-point singlet wavefunction gen-erates a family of wavefunctions under a reversal of theRG flow towards the UV. The entanglement and corre-
lations measures are then calculated from this family ofwavefunctions, and represents their RG evolution. Thecorrelations (diagonal and off-diagonal) as well as the mu-tual information (among members of the cloud as well asbetween the impurity and the cloud) are found to in-crease towards the strong-coupling fixed point, and thisis consistent with the presence of the off-diagonal scat-tering term which generates the correlation among themembers of the cloud. Our quantification of the many-particle entanglement through the mutual informationof the Kondo screening cloud and the impurity and thestudy of its renormalisation is complementary to methodsthat have been used previously in the literature. This in-cludes the impurity contribution to the entanglement ofa region in the conduction bath (dubbed as the impurityentanglement entropy in Refs.[64–66]), the entanglemententropy obtained by tracing out the impurity [67–69], andthe concurrence between the impurity with the rest of theconduction bath [70]. Recent works have also shown, us-ing negativity as a measure of entanglement [71], that theimpurity is maximally entangled with the cloud [72].
The rest of the paper is outlined as follows. In Sec.II,we introduce the Hamiltonian for the single channelKondo model. We perform the URG analysis on theHamiltonian in Sec.III. Sec.IV constitutes results on thescaling of the Kondo coupling, description of the low-energy stable fixed point in terms of the effective Hamil-tonian and ground state wavefunction, and the redistri-bution of spectral weight under the RG flow. In Sec.V,we obtain the effective Hamiltonian for the Kondo cloudexcitations and describe the salient features. In Sec.VI,we study the entanglement features and many-body cor-relations of the Kondo cloud. We end in Sec.VII with adiscussion of the results and some avenues for future in-vestigations. Appendix A shows the derivation of the RGequation using the URG method. In appendices B andC, we obtain a real-space effective Hamiltonian for thelow-energy excitations of the strong-coupling fixed pointHamiltonian and calculate the conduction electron scat-tering phase shifts close to the ground state. AppendicesD 1 and D 2 focus on calculating some thermodynamicquantities (impurity susceptibility, impurity specific heatand impurity thermal entropy) in order to benchmarkwith existing results.
II. THE MODEL
The Kondo model [2, 6] describes the coupling betweena magnetic quantum impurity localized in real space witha bath of conduction electrons
H =∑kσ
εknkσ +J
2
∑k,k′
S · c†kασαβck′β . (1)
We will consider a 2D electronic bath with a disper-sion εk = −2t(cos kx + cos ky) and with the Fermi en-ergy set to EF = µ. Here J is the Kondo scattering
3
FIG. 1. The Kondo model is composed of a two-dimensionalconduction electron bath (Fermi liquid) coupled to a mag-netic impurity via a spin-flip (solid) / non spin-flip (dashed)scattering coupling.
coupling between the impurity and the conduction elec-trons. An important feature of the Kondo coupling is thetwo different classes of scattering processes: one involv-ing spin-flip scattering processes for the bath electrons
(c†k↑ck′↓ + h.c.), and the other, non spin-flip (i.e., simple
potential scattering processes). In the antiferromagneticregime J > 0, the spin-flip scattering processes generatequantum entanglement between the impurity spin and amacroscopic number of bath electrons (called the “Kondocloud”), resulting in the complete screening of the impu-rity via the formation of a singlet spin state. It is clearthat the dynamical Kondo cloud corresponds to an effec-tive single spin-1/2, such that the screening is an exampleof macroscopic quantum entanglement arising from elec-tronic correlations. It is the nature of this entanglement,and the underlying quantum liquid that forms the Kondocloud, that we seek to learn more of.
III. URG THEORY FOR THE KONDO MODEL
In constructing an effective low-energy theory for theKondo singlet, we employ the unitary RG formalism [55–61] to the Kondo model such that electronic states fromthe non-interacting conduction bath are step-wise disen-tangled, starting with the highest energy electrons at thebandwidth and eventually scaling towards the FS. Whilethis aspect is similar to Anderson’s poor man’s scaling [6],we shall see that several other aspects of the unitary RGformalism are different from those adopted in the poorman’s scaling approach.
The electronic states are labeled in terms of the normaldistance Λ from the FS and the orientation unit vectors(Fig.2) s: kΛs = kF (s)+Λs, where s = ∇εk
|∇εk| |εk=EF . The
various s represent the normal directions of the Fermisurface. The states are labeled as |j, l, σ〉 = |kΛj s, σ〉, l :=(sm, σ). The Λ’s are arranged as follows: ΛN > ΛN−1 >. . . > 0, where the electronic states farthest from FS ΛNare disentangled first, eventually scaling towards the FS.This leads to the Hamiltonian flow equation [55]
H(j−1) = U(j)H(j)U†(j) , (2)
FIG. 2. Fermi surface geometry for a circular Fermi volumeof non-interacting electrons in 2 spatial dimensions.
where the unitary operation U(j) is the unitary mapat RG step j. U(j) disentangles all the electronicstates |kΛj sm , σ〉 on the isogeometric curve and has theform [55, 57]
U(j) =∏l
Uj,l, Uj,l =1√2
[1 + ηj,l − η†j,l] , (3)
where ηj,l are electron-hole transition operators followingthe algebra
{ηj,l, η†j,l} = 1 ,[ηj,l, η
†j,l
]= 1 . (4)
The transition operator can be represented in terms ofthe diagonal (HD) and off-diagonal (HX) parts of theHamiltonian as follows
ηj,l = Trj,l(c†j,lHj,l)cj,l
1
ωj,l − Trj,l(HDj,lnj,l)nj,l
. (5)
We note that in the numerator of the expression for ηj,l,
the operator Trj,l(c†j,lHj,l)cj,l + h.c. is composed of all
possible scattering vertices that modify the configurationof the electronic state |j, l〉 [55]. The generic forms of HD
j,l
and HXj,l are as follows
HDj,l =
∑Λs,σ
εj,lnkΛs,σ +∑α
Γ4,(j,l)α nkσnk′σ′
+∑β
Γ8,(j,l)β nkσnk′σ′(1− nk′′σ′′) + . . . ,
HXj,l =
∑α
Γ2αc†kσck′σ′ +
∑β
Γ2βc†kσc†k′σ′ck′1σ′1ck1σ1
+ . . . .
(6)The indices α and β are strings that denote the quantumnumbers of the incoming and outgoing electronic statesat a particular interaction vertex Γnα or Γmβ . The op-erator ωj,l accounts for the quantum fluctuations arisingfrom the non-commutation between different parts of therenormalised Hamiltonian and has the following form [55]
ωj,l = HDj,l +HX
j,l −HXj,l−1 . (7)
Upon disentangling electronic states s, σ along a isogeo-metric curve at distance Λj , the following effective Hamil-
4
tonian Hj,l is generated
Hj,l =
l∏m=1
Uj,mH(j)[
l∏m=1
Uj,m]† . (8)
Here, τj,l = nj,l − 12 . We note that Hj,2nj+1 = H(j−1)
is the Hamiltonian obtained after disentangling all 2njnumber of electronic states on the isogeometric curvej at distance Λj . Below, we depict the different termsgenerated upon successive disentanglement of the states|kΛj sl , σ〉 on a given curve,
Hj,l+1 = Trj,l(H(j,l)) + {c†j,lTrj,l(H(j,l)cj,l), ηj,l}τj,l, τj,l
= nj,l −1
2Hj,l+2 = Trj,l+1(Trj,l(H(j,l)))
+ Trj,l+1({c†j,lTrj,l(H(j,l)cj,l), ηj,l}τj,l)+ {c†j,l+1Trj,l+1(Trj,l(H(j,l))cj,l+1), ηj,l+1}τj,l+1
+ {c†j,l+1Trj,l+1({c†j,lTrj,l(H(j,l)cj,l), ηj,l}cj,l+1),
ηj,l+1}τj,lτj,l+1 .
Hj,l+3 = . . . terms with τj,l, τj,l+1, τj,l+2 . . .
+ . . . terms with τj,lτj,l+1, τj,l+1τj,l+2, τj,lτj,l+2 . . .
+ . . . terms with τj,lτj,l+1τj,l+2. (9)
Upon disentangling multiple electronic states placedin the tangential direction on a given momentum shell atdistance Λj generates RG contribution from leading orderscattering processes (terms multiplied with τj,l, τj,l+1,etc.) that goes as Area/V ol = 1/L and higher orderprocesses like terms multiplied with τj,lτj,l+1 that goesas (Area)2/V ol2 = 1/L2 , τj,lτj,l+1τj,l+2 that goes as(Area)3/V ol3 = 1/L3 [55]. Here, each factor of areaarises from decoupling an entire shell of single-particlestates (τj,l) at every RG step, and every factor of volumearises from the Kondo coupling. Accounting for only theleading tangential scattering processes, as well as othermomentum transfer processes along the normal directions, the renormalised Hamiltonian H(j−1) has the form [55]
Trj,(1,...,2nj)(H(j))+
2nj∑l=1
{c†j,lTrj,l(H(j)cj,l), ηj,l}τj,l . (10)
From the effective Hamiltonian obtained at the stablefixed point H∗, we can compute the (unnormalised) den-
sity matrix operator (ρ = e−βH∗) and thence the finite-
temperature partition function as
Z = Tr [ρ] = Tr[e−βH
∗]
= Tr[U†e−βH
∗U]
= Tr[e−βH
],
(11)
where β = 1/kBT , U =∏j∗
1 U(j), H is the bare Hamil-tonian and j∗ is the RG step at which the IR stablefixed point is reached. This indicates that the partitionfunction is preserved along the RG flow as the unitarytransformations preserve the eigenspectrum.
IV. CROSSOVER FROM LOCAL MOMENT TOSTRONG-COUPLING
A. IR fixed point Hamiltonian and wavefunction
The unitary RG process generates the effective Hamil-tonian’s H(j)(ω)’s for various eigendirections |Φ(ω)〉 ofthe ω operator. Note the associated eigenvalue ω iden-tifies a sub-spectrum in the interacting many bodyeigenspace. The form of H(j)(ω) is given by
∑j,l,σ
εj,lnj,l + J (j)(ω)S · s< +
j,nj∑a=N,m=1
J (a)Szsza,s,m , (12)
where the label a indexes the isoenergetic shells that havealready been decoupled and hence run from the furthestshell a = N to the most recently decoupled shell a = j.
We also defined s< = 12
∑j1,j2<j
∗,m,m′
c†j1,sm,ασαβcj2,sm′ ,β
as the total spin operator for the Kondo cloud andszl,s,m = 1
2 (nl,sm,↑− nl,sm,↓). The derivation of the aboveequation is presented in Appendix A. Here, the secondterm is the effective Hamiltonian for the coupling of theKondo cloud to the impurity spin, while the third en-codes the interaction between the impurity spin momentand the decoupled electronic degrees of freedom that donot belong to the Kondo cloud (lying on radial shells inmomentum-space indexed by the RG step j). Note thatthe third term is an Ising coupling and involves only po-tential scattering of the decoupled electrons and the im-purity, and hence does not cause any spin-flip scatteringof the impurity spin. Importantly, we will see later thateq.(12) encodes the entire T = 0 RG crossover. Employ-ing eq.(11), this enables a computation of the entire RGcrossover at finite T as well. The Kondo coupling RGequation for the RG steps (Appendix A) has the form
∆J (j)(ω)
∆ log ΛjΛ0
=nj(J
(j))2[(ω − ~vFΛj
2 )]
(ω − ~vFΛj2 )2 − (J(j))
2
16
, (13)
where nj =∑s 1 is the number of states on the isogeo-
metric shell at distance Λj from the Fermi surface. Wehave assumed that the conduction bath dispersion is lin-ear near the Fermi surface: εj = ~vFΛj , vF being the
Fermi velocity. Note that the denominator ∆ log ΛjΛ0
= 1
for the RG scale parametrisation Λj = Λ0 exp(−j). Wenow redefine Kondo coupling as a dimensionless param-eter
K(j) =J (j)
ω − ~vF2 Λj
, (14)
and operate in the regime ω > ~vF2 Λj . With the above
parametrisation of eq.(14), we can convert the differenceRG relation (eq.(13)) into a continuum RG equation
dK
d log ΛΛ0
=
(1− ω
ω − 12~vFΛ
)K +
n(Λ)K2
1− K2
16
, (15)
5
FIG. 3. Schematic RG phase diagram for the Kondo problem.The red dot represents intermediate coupling fixed point atK∗ = 4 for the case of the antiferromagnetic Kondo coupling.The blue dot represents the critical fixed point at K∗ = 0 forthe case of the ferromagnetic Kondo coupling.
where n(Λ) is the continuum counterpart of nj and repre-sents the number of electronic states on the isogeometricshell at momentum Λ.
Upon approaching the Fermi surface Λj → 0, hence(1− ω
ω−~vFΛ
)→ 0 and n(Λ) can be replaced by density
of states on the Fermi surface n(0):
dK
d log ΛΛ0
=n(0)K2
1− K2
16
(16)
At this point, we observe an important aspect of theRG equation: for K << 1, the RG equation reduces tothe one loop form: dK
d log ΛΛ0
= K2 [6]. On the other hand,
the non-perturbative form of the flow equation obtainedfrom the URG formalism shows the presence of interme-diate coupling fixed point at K∗ = 4 in the antiferromag-netic regime K > 0. Upon integrating the RG equationand using the fixed point value K∗ = 4 we obtain theKondo energy scale (TK) and thence the effective lengthof the Kondo cloud (ξK)
1
K0− 1
2+K0
16= −n(0) log
Λ∗
Λ0, (17)
Λ∗ = Λ0 exp
(1
2n(0)− 1
n(0)K0− K0
n(0)16
), (18)
TK =~vFΛ0
kBexp
(1
2n(0)− 1
n(0)K0− K0
n(0)16
), (19)
ξK =2π
Λ∗. (20)
At the IR fixed point in the antiferromagnetic regimethe effective Hamiltonian is given by,
H∗ =∑|Λ|<Λ∗
~vFΛnΛ,s,σ + J∗S · s< +
j∗,nj′∑j′=N,m=1
Jj′Szszj′,m ,
(21)In the equation, m refers to the various normal directionssm of the Fermi surface. In the above equation, the sec-ond term is the effective Hamiltonian for the coupling ofthe Kondo cloud to the impurity spin. The third encodesthe Ising interaction between the impurity spin momentand the magnetic moment of the decoupled electronic de-grees of freedom that do not belong to the Kondo cloud
0 20 40 60 80 100←− RG steps
0
1
2
3
4
K
K > 0
K < 0
−0.025
−0.020
−0.015
−0.010
−0.005
K
FIG. 4. Red: Renormalisation of the dimensionless Kondocoupling K towards the strong-coupling fixed point (K > 0)with RG steps (log Λj/Λ0). The growth of the Kondo couplingto a finite value of the intermediate coupling fixed point isevident . Blue: Decay of the dimensionless Kondo coupling Kto zero towards the local moment critical fixed point (K < 0)with RG steps (log Λj/Λ0). For these plots, we chose ω =~vF Λj .
(lying on radial shells in momentum-space indexed bythe RG step j, and corresponding to the integrals of mo-tion (IOMs) generated during the RG flow). As reportedin the appendices, the fixed point Hamiltonian has beenused to compute various thermodynamic quantities likethe impurity susceptibility, impurity specific heat, ther-mal entropy, Wilson number and the Wilson ratio. Thevalues obtained are found to be in good agreement withthose from other methods like the NRG or the Betheansatz [8, 73–75].
We can now extract a zero mode from the above Hamil-tonian that captures the low energy theory near theFermi surface,
Hcoll =1
N
∑|Λ|<Λ∗
~vFΛ∑|Λ|<Λ∗
nΛ,s,σ + J∗S · s<
+
j,nj′∑j′=N,m=1
Jj′Szszj′,m
= J∗S · s< +
j,nj′∑j′=N,m=1
Jj′Szszj′,m .
(22)
In the first step, the first term vanishes as the sum overwavevector Λ within the symmetric window Λ∗ aroundthe Fermi surface itself vanishes. Indeed, by taking theIsing coupling between the impurity spin and the IOMs(szj′,m) to be a constant, we observe that the Kondo sin-glet state
|Ψ∗〉 =1√2
(|↑d,⇓〉 ⊗ |↓〉 − |↓d,⇑〉 ⊗ |↑〉) (23)
corresponds to the ground state of the zero mode Hamil-tonian at the IR fixed point (eq.(22)). In the singlet statewritten above, |↑d〉 and |↓d〉 represent the configuration
6
of the impurity spin Sz, |⇑〉 and |⇓〉 represent the config-uration of the spin of the Kondo cloud sz< and |↑〉 and |↓〉represent the spin of the IOMs sz =
∑j1>j
∗,m
nj1,sm,ασzαβ .
Signatures of such a singlet ground state have beenexperimentally probed using NMR experiments in Cu-nuclei [76], using conductance measurements in carbonnanotubes [77–79] and using STS measurements in quan-tum corrals [80–83].
B. Kondo cloud size and effective Kondo couplingas functions of bare coupling J0
In figures 5, we show the variation of the Kondo cloudsize ξK and effective Kondo coupling J∗ as function ofbare coupling J (in units of t) in the range 5.7× 10−5 <J < 5.4. All plots below are obtained for momentum-space grid 100×100 and with RG scale factor Λj = bΛj+1
(b = 0.9999 = 1− 1/N2 , N = 100). The EF for the 2dtight binding band −W < Ek = −2t(cos kx + cos ky) <W (W = 4t) is chosen at EF = −3.9t, and the bare k-space cutoff is set at Λ0 ' 2.83. The variation of the
10−4 10−3 10−2 10−1 100
J(bare coupling in units of t)
2.5
3.0
3.5
4.0
4.5
ξ K
6
8
10
12
14
16
J∗ (
ren.
Kon
doC
oupl
ing)
FIG. 5. Red: Kondo cloud length ξK vs. bare Kondo couplingJ . Blue: Renormalised Kondo coupling J∗ vs. J (x-axis inlog scale). The bare Kondo coupling J is chosen to lie withinthe range 5.7× 10−5 < J < 5.4.
renormalised Kondo coupling J∗ with the bare J shownin Fig.5 clearly indicates the flow under RG towards sat-uration at a strong-coupling value of J∗sat ∼ 16. Sim-ilarly, the variation of the Kondo screening length ξKwith J in Fig. 5 shows a fall to an asymptotic value ofξK ∼ 3 lattice sites at the strong-coupling fixed point.We recall that Wilson’s NRG calculation for the Kondoproblem (for a bath of conduction electrons in the con-tinuum with linear dispersion and a very large UV cutoffD) shows that the renormalised Kondo coupling J∗ →∞under flow to strong coupling. This can be reconciledfrom our URG results by noting that the value of J∗satincreases upon rescaling the conduction bath bandwidthD to larger values (by rescaling the nearest neighbor hop-ping strength t). This is shown in Fig. 6, where we see
that J∗sat increases from a value of O(1) (in units of t)to O(109) as t is increased from O(1) to O(104). Thus,taking the limit of D →∞ will lead to J∗sat →∞.
100 101 102 103 104
t
102
103
104
105
J∗ sat
FIG. 6. Variation of the renormalised Kondo coupling J∗ withthe hopping parameter O(1) < t < O(104) of the electronicbath (and hence the bandwidth).
FIG. 7. Emergence of the Kondo length scale under URG.The red circle represents the impurity site, while the wavelikestructure represents the shortest wavelength interacting withthe impurity at any given point of the RG. The leftmost imageis closest to the local moment fixed point while the rightmostimage is closest to the strong-coupling one.
There is another aspect of this method that is worthpointing out - the size of the Kondo cloud emerges as anatural length scale of the low energy theory. The URGmoves forward by decoupling high momentum states.Each momentum state wavefunction Ψk is associatedwith a de Broglie wavelength λk ∼ 1
k . At high energies(temperatures), all possible wavelengths are interactingwith the impurity. Under RG, the shortest of these wave-lengths get decoupled and the impurity interacts withonly longer wavelengths. The shortest such wavelengththat is still interacting with the impurity at the fixedpoint defines the Kondo length scale ξK .
The members of the Kondo cloud then involve all stateswith wavelengths starting from ξK and extending up to∞. From existing results as well as our own investiga-tions (inset of fig.8), it is clear that the width of thespectral function increases with temperature. This widthdefines the relevant energy scale ω(T ) (and hence a rel-evant length scale ξ(T ) ∼ ω−1(T )) for interactions withthe impurity. Higher temperatures therefore lead to re-
7
duced ξ(T ). This means that at sufficiently large tem-peratures, the relevant length scale ξ(T ) is shorter thanthe Kondo length scale ξK : ξ(T ) � ξK , which meansthat the Kondo effect will not manifest at temperaturesT � TK : the RG flow is not able to filter out the low en-ergy physics because of the thermal fluctuations. Similarinsights were obtained in [84] using a real space renor-malisation group flow.
C. Redistribution of spectral weight under RG flow
The presence of a unitarily-transformed effectiveHamiltonian at each stage of the RG allows one to mapthe journey in terms of changes in the distribution of thetotal spectral weight. In this section, we will computethe impurity spectral function at various values of therunning coupling J , following the expression [85, 86]
Aσd = − 1
πIm
⟨{Oσ, O†σ}
⟩, Oσ =
J
2
(S−σc0,−σ + Szc0,σ
)(24)
where 〈...〉 indicates a thermal average, {, } indicates ananti-commutator and c0,σ =
∑k ckσ.
In order to plot the spectral function (fig. 8), we chosevarious values of J to mimic the crossover from the localmoment to the strong-coupling fixed points. The spec-tral function is normalised by dividing with the total areaunder the curve. The evolution of the spectral functionsshows the increase in the spectral weight at the zero en-ergy resonance at the cost of that at higher frequencies.This transfer of spectral weight occurs because the high-frequency excitations are slowly getting integrated outand consumed into the IOMS, while the Kondo cloudgets distilled into purely low-energy modes proximate tothe Fermi surface. This should be contrasted with thesharpening of the Abrikosov-Suhl resonance in the spec-tral function of the bare Hamiltonian as obtained fromother methods (e.g., NRG), and which corresponds to theexcitations of the local Fermi liquid.
The inset of Fig. 8 also shows the change in the spectralfunction when temperature is introduced. The broaden-ing of the central peak implies a decrease in the rele-vant length scale and hence a destruction of screeningfor T � TK. This was also shown in [85].
V. EFFECTIVE HAMILTONIAN FOREXCITATIONS OF THE KONDO CLOUD
In this section we will study the effect of the renor-malised Kondo coupling on the self energy of the elec-trons comprising the Kondo cloud. We integrate out thedecoupled electronic states to obtain the effective Hamil-tonian H∗ of the impurity + electronic cloud system. Inthis Hamiltonian we have additionally kept the electronicdispersion to study the effect of electronic density fluctu-ation due to the inter-electronic interaction mediated by
−4 −3 −2 −1 0 1 2 3 4ω/D ×102
0
1
2
3
Impu
rity
spec
tral
func
tion
×10−3
ω/D
T/D = 0.00
T/D = 1.26
T/D = 3.15J/D = 0.01
J/D = 0.10
J/D = 1.00
J/D = 10.00
FIG. 8. Impurity spectral function as a function of ω/D atvarious values of J , starting from weak-coupling and extend-ing up to strong-coupling. D is the half-bandwidth, and isset to 1. Inset shows the spectral function at various temper-atures. The width increases with increasing temperature (seefinal paragraph of subsection IV B).
the impurity spin,
H∗ = H∗0 +J∗
2
∑j1,j2<j
∗,m,m′
S · c†j1,sm,ασαβcj2,sm′ ,β , (25)
where H∗0 =∑|Λj |<Λ∗,σ εj nj,s,σ. In order to study the
inter-electronic interaction we need to isolate the quan-tum impurity from the Kondo cloud. This can be done byfirst recasting the many body eigenstate |Ψ〉 of H∗ in the|↑d〉 , |↓d〉 basis of the impurity and the associated config-uration of the rest of the electronic states. We will thenperform a perturbative expansion in the small parametert2
J from which we obtain the effective k-space Hamilto-nian for the Kondo cloud. This is justified by the factthat the Kondo coupling can be made arbitrarily large byperforming the expansion as close to the strong-couplingfixed point as desired.At the ground state, the wavefunction will be a sin-glet composed of the impurity spin and the compos-ite spin formed by the conduction electrons at the ori-gin (|⇑〉 , |⇓〉): |Ψ〉 = a0 |↑d〉 |⇓〉 + a1 |↓d〉 |⇑〉. With thisground state in mind, we can rewrite the eigenvalue rela-tion H∗K |Ψ〉 = EGS |Ψ〉 as a set of two coupled equations:
a0(H∗0 +J∗
2sz) |⇓〉+ a1
J∗
2s− |⇑〉 = a0EGS |⇓〉 ,
a0J∗
2s+ |⇓〉+ a1(H∗0 −
J∗
2sz) |⇑〉 = a1EGS |⇑〉 .
(26)
By combining these two equations and eliminating |⇑〉,we obtain the effective Hamiltonian for excitations in thesubspace of states |⇓〉:[H∗0 +
J∗
2sz + s−
(J∗)2
4
EGS + J∗
2 sz −H∗0s+
]|⇓〉 = EGS |⇓〉
(27)
8
From here we obtain the form of the effective Hamilto-nian accounting for the leading order density-density andoff-diagonal four Fermi interaction terms, by expandingthe denominator in powers of H∗0/J .
H∗0 +J∗
2sz + s−
(J∗)2
4
(EGS + J∗
2 sz)(1− 1EGS+ J∗
2 szH∗0 )
s+
≈ H∗0 +J∗
2sz + s−
(J∗)2
4
EGS + J∗
2 sz(1 +
1
EGS + J∗
2 szH∗0
+1
EGS + J∗
2 szH∗0
1
EGS + J∗
2 szH∗0 )s+ +O((H∗0 )3)
≈ H∗0 +J∗
2sz + s−
(J∗)2
4
EGS + J∗
2 szs+ + s−
(J∗)2
4
(EGS + J∗
2 sz)2H∗0 s+
+ s−
(J∗)2
4
(EGS + J∗
2 sz)2H∗0
1
EGS + J∗
2 szH∗0 s+,
≈ H∗0 +J∗
2sz + (
1
2− sz)
(J∗)2
4
EGS + J∗
4
+ s+
(J∗)2
4
(EGS + J∗
2 sz)2
(s−H∗0 + [H∗0 , s−])+
+ s+
(J∗)2
4
(EGS + J∗
2 sz)2H∗0
1
EGS + J∗
2 sz(s−H
∗0 + [H∗0 , s−]).
(28)
In the second and third term, we can substitute sz = − 12 ,
because sz |⇓〉 = − 12 . We will also substitute the singlet
binding energy − 3J∗
4 as the ground state energy EGS
because J � εk at the strong-coupling fixed point, andH∗0 = E∗0 as the kinetic energy part of the ground state|⇓〉:
Heff = −3J∗
4+ E∗0 + s+
(J∗)2
4
(EGS + J∗
2 sz)2
(s−H∗0 + [H∗0 , s−]
+H∗01
EGS + J∗
2 sz(s−H
∗0 + [H∗0 , s−])
).
(29)This effective Hamiltonian has the expected structure;there are constant pieces that represent the diagonal partof the Hamiltonian, and then there are terms that scat-ter within the subspace. We will henceforth drop theconstant terms, and keep only the excitations within thesubspace. After simplifying the terms and retaining atmost four-fermion interactions, we obtain the followingeffective Hamiltonian:
Heff = 2(H∗0 +1
J∗H∗0
2) +∑1234
V1234c†k4↑c
†k3↓ck2↓ck1↑ (30)
where V1234 = (εk1− εk3
)[1− 2
J∗ (εk3− εk1
+ εk2+ εk4
)].
The first term, second and third terms represent a ki-netic energy (eq.(V)), a density-density correlation
(see eq.(D8) below), and a spin-fluctuation mediatedelectron-electron scattering process respectively. Thepresence of the off-diagonal scattering term is expectedbecause the ground state singlet is highly-entangled andintegrating out the impurity from this singlet shouldlead to scattering among the conduction electron states.In other words, it is this off-diagonal interaction that isa signal of the strongly screened local moment.
VI. MANY BODY CORRELATIONS ANDENTANGLEMENT PROPERTIES OF THE
KONDO CLOUD
In order to study the effect of the off-diagonal termseq.(30) on the constituents of the Kondo cloud we per-form a reverse URG treatment (shown in Fig.10) start-ing from the Kondo model ground state |Ψ∗〉 at the IRfixed point eq.(23). For this, we employ the entangle-ment RG method developed recently by some of us inRefs.[60, 61, 63]. For the present study we take a toy
FIG. 9. Figure describes the electronic Kondo cloud (dashedrectangle) coupled to the impurity spin(red circle and arrow).The blue dots represent momentum+spin states inside theKondo cloud, while the black dots represent states outside theKondo cloud (that is, among the IOMS). The ground statesinglet is formed by the impurity (red circle) and the Kondocloud members (blue dots).
FIG. 10. The upper line represent the Hamiltonian RG flowvia the unitary maps, while the lower line represents the re-verse RG flow on the ground state wavefunction obtained atthe Kondo IR fixed point. The reverse RG re-entangles decou-pled electronic states with the Kondo singlet. This will resultin generation of the many-body eigenstates at UV scales.
model construction of the ground state wavefunction |Ψ∗〉eq.(23) at IR, the Kondo impurity couples to 12 elec-tronic states |kσ〉 of which three are occupied and 9 areunoccupied. The net spin of the electrons comprisingthe Kondo cloud is oppositely aligned to that the Kondoimpurity. The Kondo cloud system is in tensor product
9
with 14 separable electronic states. This constructionis represented in Fig.9 At each URG step Uj,↑Uj,↓ twoelectronic states |kj , ↑〉, |kj↓〉 are disentangled in reach-ing the IR fixed point. Upon performing reverse RG ateach step two electrons are re-entangled into the eigen-
states via the inverse unitary maps U†j,↑U†j,↓, Fig.10, all
total we perform seven reverse RG steps. This reverseRG program is numerically implemented using python.Since the forward RG is driven by the unitary transfor-mation U = 1√
2
(1 + η − η†
), the inverse transformation
is 1√2
(1− η + η†
). The η have already been described in
appendix-A. The inverse transformation for re-entanglingan electron of spin σ and energy εq can therefore be writ-ten as
U−1qσ =
1√2
[1− J2
2
1
2ωτqσ − εqτqσ − JSzszq
(O + O†
)](31)
where O =∑k<Λ∗
∑α=↑,↓
∑a=x,y,z S
aσaασc†kαcqσ. The
wavefunctions for the reverse RG are generated by re-peat application of this inverse unitary operator on thefixed point ground state |Ψ∗〉. The total operator to re-entangle the energy states εq1 , εq2 , ..., εqN is
|Ψj〉 =
qN∏q=q1
U−1q↑ U
−1q↓ |Ψ∗〉 (32)
We use the wavefunctions generated under the reverseRG to compute the mutual information between (a) anelectron in the Kondo cloud and the impurity electron,and (b) two electrons within the cloud. MI measuresthe total amount of quantum and classical correlationsin a system [87]. The mutual information between twoelectrons is given by
I(i : j) = −Tr(ρi ln ρi)− Tr(ρj ln ρj) + Tr(ρij ln ρij) ,(33)
where ρi or ρj and ρij are the 1- and 2-electron reduceddensity matrices respectively obtained from the wave-functions obtained at each step of the reverse RG simula-tion. In Fig.11, we present the RG flow of both types ofmutual information mentioned above. The orange curvein Fig.11(a) represents the plot for the maximum mutualinformation I(e : e) (eq.(33)) between any two of the elec-trons comprising the Kondo cloud, and shows that themaximum entanglement content/quantum correlation in-creases under RG flow from UV to IR. This implies thatthe electrons within the Kondo cloud is not simply aseparable state in momentum space expected of a localFermi liquid. This is a strong indication of the fact that
the two-particle off-diagonal (c†k4↑c†k↓ck2↓ck1↑) scattering
term in eq.(30) is playing a role in the electronic entangle-ment within the Kondo cloud. Further, the blue curve inFig.11(a) shows that the maximum mutual informationbetween the Kondo impurity and any one member of theelectronic cloud also increases under RG and to a highervalue compared to that between electrons. This origi-nates from the maximally entangled singlet state that is
formed between the impurity and the electronic cloud (ascan also be observed from the ln 2 entanglement entropyobtained by tracing out the impurity spin from the sin-glet state). In this manner, the Kondo impurity mediatesthe entanglement between electrons (orange curve) in theKondo cloud.
In order to understand this further, we alsostudy (i) the maximum density-density correlationsmaxk,k1
〈nk↑nk1↓〉 (blue curve in Fig.11(b)), and (ii)the maximum two-particle off-diagonal correlations
maxk,k1〈c†k↑c
†k1↓ck2↓ck3↑〉 (orange curve in Fig.11(b)) be-
tween electrons within the Kondo cloud. The plots showclearly that both the correlations grow under RG fromUV to IR, finally reach the same value at the IR fixedpoint. On the other hand, the large values of the off-diagonal correlations reinforce our observation of a non-zero mutual information content between the cloud elec-trons. This implies that the electronic cloud contains,in general, interaction terms beyond the Fermi liquiddensity-density interaction leading to non-zero entangle-ment content.
0 2 4 6RG Steps →
0.0
0.1
0.2
0.3
0.4M
utua
lIn
form
atio
n(I
)Imax(e:e)
Imax(imp:e)
0 1 2 3 4 5 6 7RG Steps →
0.05
0.10
0.15
0.20
0.25
Cor
rela
tion
s
〈n1↑n2↑〉〈c†1↑c
†2↓c3↓c4↑〉
FIG. 11. Upper panel shows the RG flows for the maximummutual information between the impurity and electron (blackcurve) and diagonal correlations (red curve). Lower panelshows the RG flows for the off-diagonal (black curve) anddiagonal correlations (red curve).
10
VII. CONCLUSIONS
The Kondo problem [2] is one of the oldest and most well-studied problems of electronic correlations in condensedmatter physics [6, 7]. The unitary RG analysis of theKondo Hamiltonian leads to a zero temperature phasediagram revealing a strong coupling fixed point for an an-tiferromagnetic Kondo coupling. At the IR fixed point,we obtained the effective Hamiltonian, the ground statewavefunction and the energy eigenspectrum. This en-abled the computation of various thermodynamic quan-tities such as the impurity susceptibility, specific heat co-efficient, Wilson ratio, Wilson number and thermal en-tropy, all of which are found to be in good agreementwith that obtained from the NRG studies [8]. It is alsonoteworthy that we were able to capture the entire RG(crossover) flow at finite temperatures from the completeeffective Hamiltonian (eq.(D1)) obtained upon reachingthe IR stable fixed point (as seen, for instance, for the sus-ceptibility χ(T ) in Fig.13). As the URG relies purely onunitary transformations, the eigenspectrum is preservedunder the RG flow, and thence so is the partition func-tion. Thus, at each RG fixed point, the effective Hamil-tonian eq.(D1) enables the construction of the densitymatrix for a given temperature scale kBT , such that onecan compute the finite temperature partition function(eq.(11)).
Furthermore, we found that the effective Hamiltonianfor the Kondo cloud (eq. 30), obtained by integratingout the impurity spin, contains a density-density repul-sion (corresponding to a Fermi liquid) as well as a four-fermion interaction term . In order to better understandthe roles of the two types of electronic correlations, weperformed a comparative study of the RG evolution offour-point number diagonal and number off-diagonal cor-relators. By using the singlet IR ground state wavefunc-tion obtained from the URG analysis, we also studiedthe RG evolution of the mutual information (an entan-glement based measure) between (a) the impurity andan electron in the cloud and (b) two electrons in thecloud. The results show strong inter-electronic as wellas electron-impurity entanglement upon approaching theIR fixed point. This is in agreement with the presence ofboth types of two-particle correlators at the IR groundstate. We find that both the number diagonal and num-ber off-diagonal correlators reach same value at the IRfixed point, indicating that the electronic configurationwithin the Kondo cloud is not simply a local Fermi liq-uid. The large entanglement within the Kondo cloud isan indication of the spin singlet it forms together with theimpurity spin. This is an example of quantum mechan-ics at the macro-scale: impurity-bath correlations drivemany-particle entanglement between bath electrons, sig-naling the emergence of a collective spin that binds withthe impurity. Given that the Kondo problem representsa simple model for a single qubit coupled to a fermionicenvironment, our insights can spur further investigationsinto more complex qubit-bath interactions relevant to the
realization and workings of present-day quantum com-puters in noisy environments.
The calculation of impurity susceptibility in appendixD 1 defines the Kondo temperature as that energy scalebelow which only the singlet state contributes to thesusceptibility and the other states in the entangled 4-dimensional Hilbert space of the impurity spin and thetotal conduction electron spin drop out. The subsequentcalculation of the wavefunction renormalisation Z = 1 inappendix backs up the presence of a local Fermi liquidphase at the fixed point. In this way, the low energyphysics of the Kondo model can be captured in greatdepth purely from the URG fixed point Hamiltonian. Animportant point can now be highlighted. The physics ofthe emergent Kondo cloud Hamiltonian (Sec.V) is foundto be in good agreement with that of the local Fermiliquid. This includes various thermodynamic quantities(e.g., the impurity susceptibility, specific heat, thermalentropy etc.) and the Wilson ratio (see appendices D 1and D 2). We have found that the k-space Kondo cloudeffective Hamiltonian describes an extended real-spaceobject comprised of electron waves with extent greaterthan ξK . On the other hand, the local Fermi liquid isfound to reside at a distance ξK from the impurity. Thus,it appears tempting to conclude that there exists a holo-graphic bulk-boundary relationship between the Kondocloud system and the local Fermi liquid. We specu-late that the emergent change in the Luttinger’s volume(see eq.C11 in Appendix C) of the conduction bath atthe strong-coupling fixed point corresponds to the wind-ing number topological quantity [88] that signals such aholography. We do not know at present whether the ef-fective theory for the Kondo cloud found by us can berelated to a bulk gravity theory obtained from an AdS-CFT treatment [89]. We leave this for a future work.
Future studies need to be performed for investigatingthe nature of the correlated metal comprising the Kondocloud. This work also opens up the prospect of perform-ing a similar study on other variants of the Kondo prob-lem, such as the multi-channel Kondo model [36–47]. Thecurrent method can allow one to investigate the criti-cal properties of the over-screened multi-channel Kondointermediate-coupling fixed point. The low-energy ef-fective Hamiltonian of such models should be helpful inidentifying the microscopic origins of the non-Fermi liq-uid phase of such a fixed point. Entanglement studiesof the zero mode will also be helpful in identifying thebreakdown of screening, and such studies will lay barethe distinction between various variants of the Kondoimpurity models.
ACKNOWLEDGMENTS
The authors thank P. Majumdar, A. Mitchell, S. Sen,S. Patra, M. Mahankali and R. K. Singh for several dis-cussions and feedback. Anirban Mukherjee thanks theCSIR, Govt. of India and IISER Kolkata for fund-
11
ing through a research fellowship. Abhirup Mukher-jee thanks IISER Kolkata for funding through a re-search fellowship. AM and SL thank JNCASR, Banga-lore for hospitality at the inception of this work. NSV
acknowledges funding from JNCASR and a SERB grant(EMR/2017/005398). We thank two anonymous refereesfor their helpful comments and suggestions.
Appendix A: Calculation of effective Hamiltonian from URG
Starting from the Kondo Hamiltonian eq.(1) and using the URG based Hamiltonian RG equation eq.(10), we obtainthe renormalised Hamiltonian :
∆H(j) =
nj∑m=1,β=↑/↓
(J (j))2τj,sm,β2(2ωτj,sm,β − εj,lτj,sm,β − J (j)Szszj,sm)
×[SaSbσaαβσ
bβγ
∑(j1,j2<j),
n,o
c†j1,sn,αcj2,so,γ(1− nj,sm,β)
+ SbSaσbβγσaαβ
∑(j1,j2<j),
n,o
cj2,so,γc†j1,sn,α
nj,sm,β
]+
nj∑m=1,β=↑/↓
(J (j))2
2(2ωτj,sm,β − εj,lτj,sm,β − J (j)Szszj,sm)
×[SxSyσxαβσ
yβαc†j,sm,α
cj,sm,βc†j,sm,β
cj,sm,α + SySxσxαβσyβαc†j,sm,β
cj,sm,αc†j,sm,α
cj,sm,β
]. (A1)
The first term in eq.(A1) corresponds to the renormalisation of the Kondo coupling and describes the s-d exchangeinteractions for the entangled degrees of freedom
∆H1(j) =
nj∑m=1,β=↑/↓
(J (j))2τj,sm,β(2ωτj,sm,β − εj,lτj,sm,β − J (j)Szszj,sm)
∑(j1,j2<j),
n,o
S · c†j1,sn,ασαβ
2cj2,so,β
=1
2
nj∑m=1,β=↑/↓
τj,sm,β(J (j))2[(2ωτj,sm,β − εj,lτj,sm,β) + J (j)Szszj,m)
](ω − εj,l
2 )2 − (J(j))2
16
∑(j1,j2<j),
n,o
S · c†j1,sn,ασαβ
2cj2,so,β
=1
2
nj∑m=1,β=↑/↓
(J (j))2[(ω2 −
εj,l4 )]
(ω − εj,l2 )2 − (J(j))
2
16
+1
2
nj∑m=1
(J (j))3Szszj,m(τj,sm,↑ + τj,sm,↓)
(ω − εj,l2 )2 − (J(j))
2
16
∑(j1,j2<j),
n,o
S · c†j1,sn,ασαβ
2cj2,so,β
=nj(J
(j))2[(ω − εj
2 )]
(ω − εj,l2 )2 − (J(j))
2
16
S ·∑
(j1,j2<j),n,o
c†j1,sn,ασαγ
2cj2,so,γ .
(A2)In the second last step of the calculation, we have used the result τ2
j,sm,↑ = 14 , where τj,sm,↑ = nj,sm,↑− 1
2 . In obtainingthe last step of the calculation we have assumed εj,l = εj for a circular Fermi surface geometry. Further, we havereplaced τj,sm,↑ and τj,sm,↓ by their eigenvalues, τj,sm,↑ = −τj,sm,↓ = 1
2 , i.e., the resulting decoupled electronic wavevector |j, sm〉 carries a non-zero spin angular momentum. This configuration promotes the spin scattering betweenthe Kondo impurity and the fermionic bath. The second term in eq.(A1) corresponds to the renormalisation of thenumber diagonal Hamiltonian for the immediately disentangled electronic states |j, sm, σ〉:
∆H2(j) =
nj∑m=1,β=↑/↓
(J (j))2
(2ωτj,sm,β − εj,lτj,sm,β − J (j)Szszj,sm)
[SxSyσxαβσ
yβαc†j,sm,α
cj,sm,βc†j,sm,β
cj,sm,α (A3)
+ SySxσxαβσyβαc†j,sm,β
cj,sm,αc†j,sm,α
cj,sm,β
](A4)
=
nj∑m=1,β=↑/↓
(J (j))2
(2ωτj,sm,β − εj,lτj,sm,β − J (j)Szszj,sm)Szσzαα
2
[nj,sm,α(1− nj,sm,β)− nj,sm,β(1− nj,sm,α)
](A5)
12
=
nj∑m=1
(J (j))2
(2ωτj,sm,β − εj,lτj,sm,β − J (j)Szszj,sm)Szszj,sm , (A6)
where we have used nj,sm,α(1− nj,sm,β)− nj,sm,β(1− nj,sm,α) = nj,sm,α− nj,sm,β in the last step, and the spin densityfor the state |j, sm〉 is given by szj,sm = 1
2 (nj,sm,↑ − nj,sm,↓). In obtaining the above RG equation we have replaced
ω(j) = 2ωτj,sm,β . We set the electronic configuration τj,sm,↑ = −τj,sm,↓ = 12 to account for the spin scattering between
the Kondo impurity and the fermionic bath. The operator ω(j) (eq.(7)) for RG step j is determined by the occupation
number diagonal piece of the Hamiltonian HD(j−1) attained at the next RG step j − 1. This demands a self-consistent
treatment of the RG equation to determine the ω. In this fashion, two-particle and higher order quantum fluctuationsare automatically encoded into the RG dynamics of ω. In the present work, however, we restrict our study by ignoringthe RG contribution in ω. The electron/hole configuration (|1j,sm,β〉/|0j,sm,β〉) of the disentangled electronic state(and associated with energy ±εj,l) is accounted by the fluctuation energy scales ±ω. From the above HamiltonianRG equations eq.(A2), we can obtain the form of the Kondo coupling RG equation (eq.(13))
∆J (j) =nj(J
(j))2[ω − εj,l
2
](εj,l2 − ω)2 − (J(j))
2
16
. (A7)
It is easy to generalize this method to the more general anisotropic Kondo model with distinct transverse (J⊥) andIsing (Jz) couplings:
JzSz∑kk′
1
2
(c†k↑ck′↑ − c
†k↓ck′↓
)+
1
2J⊥S
+∑kk′
c†k↓ck′↑ +1
2J⊥S
−∑kk′
c†k↑ck′↓ (A8)
We now briefly sketch the calculation.
• Since the denominator involves the diagonal Szsz part of the Hamiltonian, only the Ising coupling Jz will enterthe denominator of the RG equation.
• The Ising coupling Jz will be renormalised only by scattering processes that involve one S+ operator and oneS− operator, so that the corresponding scattering vertices that renormalise Jz are of strength J⊥
2.
• On the other hand, the transverse coupling J⊥ will be renormalised only by processes that involve one S±
operator and one Sz operator, so that the corresponding vertices are of strength JzJ⊥.
With these modifications, the final result of eq. A2 becomes
nj(ω − εj2 )
(ω − εj,l2 )2 −
(J
(j)z
)2
16
(J (j)⊥
)2 ∑(j1,j2<j),n,o,α
Szσzααc†j1,sn,α
cj2,so,α + J (j)z J
(j)⊥
∑(j1,j2<j),
n
S+c†j1,sn,↓cj2,so,↑ + h.c.
. (A9)
The RG equations for the Ising and transverse couplingscan now be read off from the renormalised Hamiltonian:
∆J (j)z =
nj(J(j)⊥ )2
[ω − εj,l
2
](ω − εj,l
2 )2 − (J(j))2
16
,
∆J(j)⊥ =
njJ(j)z J
(j)⊥[ω − εj,l
2
](ω − εj,l
2 )2 − (J(j))2
16
.
(A10)
These coupled RG equations have the same RG invariantas the Berezinskii-Kosterlitz-Thouless (BKT) RG equa-tions [90, 91] as well as the 1D Hubbard model RG equa-tions [61]:
∆J(j)z
∆J(j)⊥
=J
(j)⊥
J(j)z
=⇒ J(j)⊥
2− J (j)
z
2= constant (A11)
One can also show that the anisotropy in the couplings isirrelevant by writing the RG equation for the anisotropy
parameter κ = J(j)⊥ − J
(j)z .
∆κ =njJ
(j)t
(J
(j)⊥ − J
(j)z
) [ω − εj,l
2
](ω − εj,l
2 )2 − (J(j))2
16
(A12)
Dividing this equation by the RG equation of J(j)z gives
∆κ
∆J(j)z
= − κ
J(j)⊥
. (A13)
Since both ∆J(j)z and J
(j)⊥ are positive in the strong-
coupling regime, we find that ∆κ/κ < 0. This relation
13
implies that if κ is initially positive (negative), the renor-malisation is negative (positive), and κ flow towards zero.This shows that all strong-coupling flows enforce the ir-relevance of the anisotropy parameter κ.
Appendix B: Effective Hamiltonian for low energyexcitations of the Kondo singlet - the local Fermi
liquid
The central resonance in the impurity spectral func-tion can be attributed to quasiparticle excitations aris-ing from the local Fermi liquid component of the effectiveHamiltonian. Here we will derive an effective Hamilto-nian for low-energy excitations in the singlet subspace ofthe fixed-point theory and hence expose the local Fermiliquid component.
To consider the local states as ground states, we willdrop the kinetic energy part and start with the followingzeroth Hamiltonian:
H∗loc = J∗ ~Sd · ~s︸ ︷︷ ︸H0
−t∗∑σ,〈0,l〉
c†0σcl,σ + h.c.
︸ ︷︷ ︸V
(B1)
The four ground states of H0 are
|φ(0)i 〉 = |χi〉︸︷︷︸
site 1
⊗ 1√2
(|↑, ↓〉 − |↓, ↑〉) , i ∈ [1, 4] ,
{χi} = {0, ↑, ↓, 2} , E = −3J∗
4
(B2)
In the singlet part, the first entry is the configuration ofthe zeroth site while the second entry is the configurationof the impurity site.
Since V and V 3 take |φ(0)2,4〉 to a completely orthogonal
subspace, the first and third order corrections are 0. Thesecond order shift in the ground state energy for a stateφ(0) is given by
E(2) =∑i6=φ(0)
| 〈φ(0)|V |i〉2
E(0)φ − Ei
(B3)
This second order shift for both the up-spin and double-occupied states (and hence the down and empty states,using SU(2) and p-h symmetries) comes out to be
E(2) = − 4t2
3J∗(B4)
so that the effective Hamiltonian at second order is sim-ply a constant. We move on to the fourth order correc-tion. The general formula is quite formidable, so we onlywrite down the terms that aren’t outright zero for thisproblem:
E(4) =∑i(0) 6=φ,j(0) 6=φ,k(0) 6=φ
Vφ,kVk,jVj,iVi,φEφ,kEφ,jEφ,i
− E(2)φ
∑m(0) 6=φ
|Vm,φ|2(Eφ,m)
2
(B5)
where Vx,y = 〈x(0)|V |y(0)〉 and Ex,y = E(0)x − E(0)
y . At
fourth order, |φ(0)2 〉is first excited to |0, 2, ↑〉 or |2, 0, ↑〉,
and then to the spin-triplet subspace. The total effectiveHamiltonian up to fourth order (and up to a constantenergy shift) is
Heffloc = −4α
t4
J∗3
∑σ
n1σ + 8αt4
J∗3n1↑n1↓ (B6)
The presence of the repulsive correlation term means thatelectrons that want to occupy the first site will face arepulsion from an electron already present there and theeffective Hamiltonian behaves like a local Fermi liquid[13] on the first site.
Appendix C: Scattering phase shift of conductionelectrons at strong-coupling and wavefunction
renormalisation
In general, the conduction electrons will suffer a phaseshift as they scatter off the impurity. We will obtainan explicit form for this phase shift in terms of the fixedpoint coupling J∗, by writing down a simple Hamiltonianthat models the low energy theory. We will drop thestates of the triplet subspace and retain only the singletstate and the double and hole states of the zeroth site. Inorder to avoid any two-particle terms, we will drop theimpurity site because it is already frozen in the singlet.Also, since the double and hole states are at zero energy,they will drop out of the Hamiltonian, and we will modelthe zeroth site in the form of a single spin. We define |0σ〉as the state where the zeroth site has a single electronof spin σ and |00〉 and |02〉 as the states with 0 and 2electrons on the zeroth site respectively. Then,
spectrum of zeroth site→
state energy
|0σ〉 −3J∗
4
|00〉 , |02〉 0
(C1)
The total simplified Hamiltonian can therefore be writtencompletely in terms of the single-particle states |0σ〉 and|kσ〉. |kσ〉 is the conduction electron state of momentumk and spin σ. This Hamiltonian preserves the spin of thezeroth site and the conduction electrons separately, andthe Hamiltonian for a particular spin is
Hσ = ε0 |0σ〉 〈0σ|+∑k
[−t (|0σ〉 〈kσ|+ h.c.) + εk |kσ〉 〈kσ|]
(C2)
where ε0 = − 3J∗
4 is the effective onsite energy of thezeroth site. The matrix elements of the retarded singleparticle Greens function for spin σ, Gσ, satisfy the equa-tion
∑β (ω −Hσ)αβ Gβγ = δαγ , where α, β and γ repre-
sent matrix elements between any pair of states among0σ and {kσ}, and δ is the Kronecker delta [92]. Solvingthe equations for Gσ00 gives
Gσ00(ω) =1
ω − ε0 − Σ(ω)(C3)
14
Σ(ω) =∑k
t2
ω−εk =∫dε t
2ρ(ε)ω−ε is the self-energy of the
zeroth site of the bath because of hybridisation with therest of the bath through the hopping term t. The imagi-nary part is given by ΣI(ω) = −πρ(ω)t2. Since ε0 � t2,we can drop the real part of the self energy in the de-nominator. The T−matrix of the conduction electronscan now be calculated using this impurity Greens func-tion [93]:
Tσkk′(ω) = t2G00(ω) = t2ω − ε0 + iΣI
(ω − ε0)2
+ Σ2I
(C4)
From scattering theory, it can be shown that the phaseshift δFL
kσ of the conduction electron state |kσ〉 is given bythe phase of Tkk [94]:
tan δFLkσ (ω) =
ΣIω − ε0
=−πρ(ω)t2
ω + 3J∗
4
(C5)
Note that there is an additional phase shift that we havenot accounted for; when we removed the zeroth site fromthe lattice, all the wavefunctions got shifted by a distanceof a: ψk(x) ∼ sin (kx) → sin (kx− ka). This is equiva-lent to a phase shift of δ0
kσ = ka. The total phase shift istherefore
δtotkσ (ω) ≡ δ0
kσ + δFLkσ (ω) = ka+ tan−1 −πρ(ω)t2
ω + 3J∗
4
(C6)
Further, following the arguments involving the Friedelsum rule given in Ref.[95], we know that the total scatter-ing phase shift in the ground state of the single impurityAnderson model (SIAM) at the Fermi surface is equal toπ times the impurity occupation nd. The Kondo modelcorresponds to the local moment regime of the SIAM,and possesses a value of nd = 1. It then follows that
δtotkF (0) = π × nd = π . (C7)
Further, we know that since kFσ = 1/2a for a half-filledone-dimensional Fermi volume (for a s-wave Kondo ex-change coupling), the phase shift δ0
kFis given by
δ0kF =
∑σ
δ0kFσ =
∑σ
kFσa = π = δtotkF (0) . (C8)
Substituting δ0kF
= δtotkF
(0) into eq. C6 then indicates thatthe scattering phase shift for the Landau quasiparticleδFL vanishes identically at the Fermi surface, and thetotal phase shift is determined purely by δ0
kF.
The phase shift can also be connected to the overlapintegral between the ground states |Ψ0〉 and |Ψ〉 beforeand after adding the local interaction respectively. Thelocal interaction can either take the form of the Kondocoupling between the impurity and the Kondo cloud, or ahopping between the zeroth site and the rest of the con-duction bath. In the former case, the ground state |Ψ0〉is that of a decoupled conduction bath, and |Ψ〉 corre-sponds to the ground state of the conduction electrons
in the presence of a strong Kondo coupling. In the lattercase, |Ψ0〉 is the ground state of a conduction bath fromwhich the zeroth site (i.e., the site strongly coupled withthe impurity) is removed, and |Ψ〉 is that of the samesystem but with a small hopping between the zeroth andfirst sites of the conduction gas.
Following Anderson’s orthogonality theorem [96], it’sextension by Yamada and Yosida [97] and the generalizedFriedel sum rule due to Langer and Ambegaokar [98], thesquare of the overlap integral (often called the wavefunc-tion renormalisation) is given by
Z ≡ | 〈Ψ|Ψ0〉 |2 = N−1π2 δkF (0)2
, (C9)
where N is the total number of conduction electrons andδkF (0) is the phase shift produced by the local perturba-tion at the Fermi surface. In the thermodynamic limit,this expression simplifies to
Z(N →∞) = 1 if δkF (0) = 0, and 0 otherwise. (C10)
For the first case of the local interaction mentioned above(i.e., for the Kondo coupling), we obtain the wavefunc-tion overlap (Zimp) in terms of the total phase shift:
Zimp = N−1π2 (δtot
kF(0))
2
→ 0 in the limit N → ∞. Thisshows the orthogonality catastrophe between the groundstates of the conduction bath in the local-moment andstrong-coupling phases. Further, following [99], the or-thogonality catastrophe reflects a change in the Luttingervolume (∆NL) of the conduction bath:
∆NL =δkF (0)
π= 1 . (C11)
This increase in the number of electrons inside the Fermivolume is nothing but the “large Fermi surface” effectseen in heavy fermionic systems [100–103]. Here the ef-fect is infinitesimal because of the local nature of theimpurity.
In the second case (i.e., with the zeroth site as the localperturbation), the relevant phase shift is δFL
kF(0) = 0, and
the overlap integral then pertains to the quasiparticle
residue of the local Fermi liquid: ZFL = N−1π2 (δFL
kF(0))
2
=1 . Further, from Ref.[94], the Wilson ratio (R) of thelocal Fermi liquid is given by
R = 1 + sin2(δkF (0)
2) = 1 + sin2(
π
2) = 2 . (C12)
Appendix D: Calculation of thermodynamicquantities
1. Impurity contribution to the magneticsusceptibility
The complete effective Hamiltonian for the impurityspin (S), Kondo cloud spin (s) and the electrons that
15
comprise the local Fermi liquid has the form,
H2 = ε∗∑m,σ
n∗,m,σ + J∗S · s + J∗Sz∑m
sz∗,m . (D1)
The Hamiltonian H2 has several conserved quantitieswhich we depict below,
[H2, Sz + sz] = 0 , [H2, s
z∗,m] = 0 ∀ 1 ≤ m ≤ nj , (D2)
such that [H∗, Sztot] = 0 where Sztot = Sz +sz +∑m s
z∗,m.
Therefore, the eigenvalues of |sz∗,m =↑ / ↓〉 are goodquantum numbers; this is simply an outcome of the URGmethod. For the purposes of computing the impuritymagnetization and susceptibility, we keep only the effec-tive impurity - Kondo electron cloud Hamiltonian fromH2 above
H∗K = J∗S · s +BSz , (D3)
and where we have introduced a local magnetic field Bthat couples to the impurity magnetic moment througha Zeeman coupling.
We can now obtain the impurity magnetization andsusceptibility from the effective Hamiltonian H∗K . Thefour state eigenspectrum of H∗K is given by
E1,2 =1
2(−J
∗
2±√B2 + J∗2), E3,4 =
1
4J∗ − 1
2B,
(D4)The partition function for this Hamiltonian (with β =
1kBT
) is given by
Z(B) = 2e−βJ∗4 cosh(β
B
2) + 2eβ
J∗4 cosh(
β
2(√B2 + J∗2)) .
(D5)The susceptibility is then given by,
χ = limB→0
d
dB
(kBT
Z(B)
dZ(B)
dB
)=
β4 + 1
2J∗ eβ J∗2 sinh(β2J
∗)
1 + eβJ∗2 cosh(β2J
∗).
(D6)In the plot of 4χTK (fig. 12), the blue and orange
lines represent the contributions to the total suscepti-bility coming from the Sz = 0 and Sz 6= 0 sectors respec-tively. The green line is the total susceptibility itself. Theinset shows a further resolution of the Sz = 0 sector intothe S = 0 and S = 1 contributions. We can now makeseveral important observations based on eq.D6 and theplot. Firstly, at low temperatures T � TK , almost theentirety of the contribution comes from the singlet state.Near TK , the singlet contribution starts dropping whilethe Sz 6= 0 contributions start to pick up. In this way, theKondo temperature scale TK arises out of the interplaybetween symmetry-preserved and the symmetry-brokensectors. Coming down from high temperatures, TK canbe thought of as that temperature beyond which the sus-ceptibility contribution from the symmetry-broken statesvanishes.
Secondly, the saturation value of 4TKχ as β →∞ (i.e.,T → 0) is given by
χsat = χ(T = 0) =1
2J∗. (D7)
100 101 102
T/TK
0.0
0.1
0.2
0.3
0.4
4χTK
ofSz
=0
andSz6=
0
100 101 102
T/TK
0.0
0.2
0.4
4χTK
Sz = S = 0
Sz = 0, S = 1
Sz = 0
Sz 6= 0
total
max(χ)
W = 0.413
FIG. 12. Variation of 4TKχ with T/TK , for individual spinsubspaces as well as the total. Inset shows the difference inthe contributions to χ coming from the singlet and the tripletzero states. Dashed line shows the maximum in the χ, andpasses through TK .
We find that the Wilson number W = 4Tkχsat takes theform 2Tk
J∗ . For values of J∗ ' 16.612t and Tk ' 3.433t weobtain W = 0.413. This is in good agreement with thevalue for W obtained from NRG [8] and Bethe Ansatzsolution of the Kondo problem [9, 10]. This is shownin Fig.12. Further, we show the variation of W withthe bare Kondo coupling J in Fig.13. The figure clearlyshows the saturation of W to the value mentioned aboveas J flows to the strong-coupling fixed point.
0.00 0.05 0.10 0.15 0.20 0.25 0.30J0
0.4120
0.4122
0.4124
0.4126
0.4128
0.4130
0.4132
0.4134
4χsatTk
FIG. 13. Variation of the Wilson number W = 4TKχsat withbare Kondo coupling J0. See discussion in text.
Thirdly, the 4Tkχ vs temperature curve shown inFig.12 has a non-monotonic behaviour, i.e., we find amaximum obtained from the transcendental equationdχdβ = 0. We confirm from our numerical studies that
the temperature corresponding to the maximum valueof TKχ tends to Tk as J flows to strong coupling. Fur-ther, we find that the maximum value of TKχ does not
16
vary with the bare J in the range O(10−5) < J < O(1).As discussed before, this maximum emerges as we go tolower temperatures and indicates that temperature atwhich the contributions from the |↑, ↑〉 and |↓, ↓〉 dropout. Thus, the non-monotonic behaviour in χ can bedirectly attributed to the corresponding non-monotonicbehaviour in the contribution coming from these states.We note that the impurity susceptibility obtained fromNRG treatments of the Kondo problem appear to re-semble that obtained by us for the Sz = 0 sector (redcurve in Fig.12) rather than the total susceptibility. Wedo not presently understand the reason for this discrep-ancy. However, it is also interesting to note that a similarnon-monotonic behaviour of the impurity susceptibilityis obtained from a Schwinger boson large-N mean-fieldapproach to the fully screened Kondo model [104].
100 101 102
T/TK
0.00
0.05
0.10
0.15
0.20
0.25
Tχ
ofSz
=0
andSz6=
0
Sz = 0
Sz 6= 0
total
T == TK
FIG. 14. Variation of Tχ with T/TK . See discussion in text.
Finally, we find that the saturation value of T × χ forβ → 0 is given by the universal value kBTχ(T = ∞) =14 , as shown in Fig.14. The saturation value at high-Tarises from the two-fold degenerate impurity and reflectsthe physics of the (almost isolated) local impurity spinmoment, while the vanishing value of Tχ originates fromthe formation of the singlet between the Kondo cloud andthe impurity spin (second term in eq.(D1)).
2. Impurity contribution to specific heat andthermal entropy
In order to study the thermodynamic properties of theFermi liquid we restrict our attention to the density-density terms only. From the density terms we obtainthe low excitation energy functional accounting for thequasiparticle interaction
E = E0 +∑
kΛs,Λ<Λ∗
εkδnkσ+∑k,k′
εkεk′
J∗δnkσδnk′σ′ . (D8)
10−3 10−2 10−1
T/TK
0.00
0.01
0.02
0.03
0.04
0.05
TKγ
imp
10−3 10−2 10−1
T/TK
Cim
p
FIG. 15. Variation of γimpTK with T/TK . The saturation toa constant value at low temperatures shows the Fermi liquid-like behaviour. Inset shows the impurity specific heat Cimp.
−3.0 −2.8 −2.6 −2.4 −2.2 −2.0 −1.8
log (T/TK)
2.0
6.5
11.0
15.5
20.0
R=χ/γ
FIG. 16. Variation of the Wilson ratio R with T/TK . Rsaturates to a value of 2.009 as T → 0.
This leads to the renormalised one-particle dispersion,where the self energy term has the following form
εΛ = εΛ + ΣΛ,ΣΛ = (∑
Λ′,s,σ′
εΛεΛ′
J∗δnΛ′,s,σ′) for Λ < Λ∗ .
(D9)Note that as Λ → 0 , ΣΛ → 0. Next, we compute thespecific heat Cimp (≡ C(J∗)− C(0)) of the impurity fromthe Fermi distribution of the renormalised quasiparticles
Cimp =∑Λ,s,σ
1
T 2
[(εΛ)2eβεΛ
(eβεΛ + 1)2− (εΛ)2eβεΛ
(eβεΛ + 1)2
], (D10)
where C(J∗) is the specific heat for the electronic systemwith the Kondo impurity, and C(0) is the specific heatfor the free electronic system without coupling to Kondoimpurity. The specific heat coefficient is given by γimp =Cimp
kBT.
We computed the impurity specific heat using the sameset of parameters as used in the computation of the sus-ceptibility, and plotted γimpTK in Fig.15. It is seen thatγimpTk rises from 0 at high temperatures T > 102TKand saturates at a value γimp(0) = 1
4J∗ for T < 10−2TK .
17
10−1 100 101
T/TK
0.0
0.2
0.4
0.6
0.8
1.0Simp/
ln2
FIG. 17. Impurity contribution to thermal entropy in unitsof ln 2. The high temperature value of ln 2 shows is indicativeof a doubly-degenerate local moment, while the zero temper-ature value of 0 shows the ground state is a non-degeneratesinglet.
The Wilson ratio R is defined as the ratio of the suscep-tibility χimp and the specific heat coefficient γimp. It isfound that R saturates to a value R = χ/γimp = 2.009for T � TK.
We also calculated the impurity thermal entropy fromthe zero mode Hamiltonian using the expression
Simp(T ) = −1
2
∂F
∂T=
1
2
[lnZ +
β
Z
∑i
εie−βεi
](D11)
Z and εi are the partition function and energy eigen-values of the zero mode Hamiltonian, and the factor ofhalf comes from the fact that since the impurity and theconduction bath site are symmetrical in the zero modeHamiltonian, the impurity contribution to the entropywill be half of the total contribution. This is shown inFig.17. At high temperatures, the impurity behaves likea doubly-degenerate free spin , contributing an entropyof ln 2. At lower temperatures, the impurity is screenedwithin the unique singlet ground state, leading to a van-ishing residual entropy.
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