Thermodynamics and screening in the Ising-Kondo model
aa r X i v : . [ c ond - m a t . s t r- e l ] J u l physica status solidi Thermodynamics and screening inthe Ising-Kondo model
Kevin Bauerbach , Zakaria M.M. Mahmoud , Florian Gebhard *,1 Fachbereich Physik, Philipps-Universit¨at Marburg, 35032 Marburg, Germany Physics Department, Faculty of Science, King Khalid University, P.O. Box 960, 61421 Asir-Abha, Saudi Arabia Physics Department, Faculty of Science at New Valley, Assiut University, 71515 Assiut, EgyptJuly 6, 2020Received XXXX, revised XXXX, accepted XXXXPublished online XXXX
Key words:
Impurity scattering, screening at finite temperatures, Fermi systems ∗ Corresponding author: e-mail fl[email protected] , Phone: +49-6421-2821318, Fax: +49-6421-2824511
We introduce and study a simplification of the symmet-ric single-impurity Kondo model. In the Ising-Kondomodel, host electrons scatter off a single magnetic im-purity at the origin whose spin orientation is dynami-cally conserved. This reduces the problem to potentialscattering of spinless fermions that can be solved ex-actly using the equation-of-motion technique. The Ising-Kondo model provides an example for static screening.At low temperatures, the thermodynamics at finite mag-netic fields resembles that of a free spin-1/2 in a reduced external field. Alternatively, the Curie law can be inter-preted in terms of an antiferromagnetically screened ef-fective spin. The spin correlations decay algebraicallyto zero in the ground state and display commensurateFriedel oscillations. In contrast to the symmetric Kondomodel, the impurity spin is not completely screened, i.e.,the screening cloud contains less than a spin-1/2 elec-tron. At finite temperatures and weak interactions, thespin correlations decay to zero exponentially with corre-lation length ξ ( T ) = 1 / (2 πT ) . Copyright line will be provided by the publisher
Dilute magnetic spin-1/2 impuritiesstrongly influence the physical properties of a metallichost at low temperatures. The most celebrated exampleis the Kondo resistivity minimum [1] that results fromspin-flip scattering of conduction electrons off magneticimpurities. The magnetic response of these systems isalso peculiar: the zero-field magnetic susceptibility of theimpurities does not obey Curie’s law [2] down to lowesttemperatures but remains finite in the ground state; forfinite temperatures, characteristic logarithmic correctionsare discernible, see Ref. [3] for a review.The finite zero-field susceptibility shows that the impu-rity spin is screened by the conduction electrons. At zerotemperature they form a non-magnetic ‘Kondo singlet’ thatis separated from the triplet by a finite energy gap. Thescreening cloud around the impurity spreads over a sizabledistance and serves as a scattering center for the conduc-tion electrons [4]. Since the size, and thus the scatteringphase shift, of the screening cloud decreases as a function of temperature, the resistivity decreases from its value atzero temperature before it eventually increases again dueto electron-phonon scattering. In this way, the occurrenceof the Kondo resistance minimum is qualitatively under-stood.The Kondo physics is properly incorporated in Zener’s s - d model [3,5], also known as ‘Kondo model’. Unfor-tunately, the Kondo model poses a true many-body prob-lem and its solution requires sophisticated analytical ap-proaches such as the Bethe Ansatz [6,7], or advanced nu-merical techniques such as the Numerical RenormalizationGroup technique [8,9]. Therefore, it is advisable to analyzesimpler models to study the thermodynamics and screen-ing in interacting many-particle problems. Examples arethe non-interacting single-impurity and two-impurity An-derson models [10,11,12].In this work, we address the Ising-Kondo model thatdisregards the spin-flip scattering in the s - d model. There-fore, it only contains the effects of static screening because Copyright line will be provided by the publisher
K. Bauerbach, Z.M.M. Mahmoud, and F. Gebhard: Thermodynamics and screening in the Ising-Kondo model the impurity spin is dynamically conserved, i.e., there is noterm in the Ising-Kondo Hamiltonian that changes the im-purity spin orientation. This has the advantage that its exactsolution requires only the solution of a single-particle scat-tering problem off an impurity at the origin. Therefore, thefree energy and the spin correlation function can be calcu-lated exactly. The lack of dynamical screening in the Ising-Kondo model has the drawback that the Kondo singlet doesnot form at low temperatures. Therefore, the zero-field sus-ceptibility displays the Curie behavior of a free spin downto zero temperature with a reduced Curie constant that re-flects the static screening by the host electrons.Our work is organized as follows. In Sect. 2 we in-troduce the model Hamiltonian, define the free energy,thermodynamic potentials (internal energy, entropy, mag-netization), and response functions (specific heat, mag-netic susceptibilities), and introduce the spin correlationfunction and the amount of unscreened spin at some dis-tance from the impurity to visualize the screening cloud.In Sect. 3 we calculate the free energy and discuss thethermodynamics of the Ising-Kondo model at zero and fi-nite magnetic field. While the formulae apply for arbitrarymagnetic fields
B < , we focus on small fields, B ≪ . InSect. 4 we restrict ourselves to the case of zero magneticfield and one spatial dimension. We discuss the spin cor-relation function and the unscreened spin as a function ofdistance from the impurity at zero and finite temperatures.In particular, we analytically determine the asymptotic be-havior at large distances. Short conclusions, Sect. 5, closeour presentation. Technical details of the calculations forspinless fermions are deferred to appendix A. The extrac-tion of correlation lengths is discussed in appendix B. We start ouranalysis with the definition of the Kondo and Ising-Kondomodels. Next, we consider the thermodynamic quantitiesof interest (free energy, chemical potential at half band-filling, thermodynamic potentials, susceptibilities). At last,to analyze the screening cloud, we define the spin correla-tion function and the unscreened spin as a function of thedistance from the impurity.
The Hamiltonian for theKondo model reads [2,3,13] ˆ H K = ˆ T + ˆ V + ˆ H m , (1)where ˆ T is the kinetic energy of the host electrons, ˆ V istheir interaction with the impurity spin at the lattice ori-gin, and ˆ H m describes the electrons’ interaction with theexternal magnetic field. The kinetic energy of the hostelectrons is given by ˆ T = X σ ˆ T σ , ˆ T σ = X i,j t i,j ˆ c + i,σ ˆ c j,σ . (2)Here, ˆ c + i,σ ( ˆ c i,σ ) creates (annihilates) an electron with spin σ = ↑ , ↓ on lattice site i , and t i,j = t ∗ j,i are the matrix elements for the tunneling of an electron from site j to site i on a lattice with L sites. Assuming translational invariance, t i,j = t ( i − j ) , the kinetic energy is diagonal in momentumspace, ˆ T σ = X k ǫ ( k )ˆ a + k,σ ˆ a k,σ , (3)where ˆ a + k,σ = 1 √ L X r e i kr ˆ c + r,σ , ˆ c + r,σ = 1 √ L X k e − i kr ˆ a + kσ . (4)The corresponding density of states of the host electrons isgiven by ρ ( ω ) = 1 L X k δ ( ω − ǫ ( k )) . (5)We assume particle-hole symmetry. It requires that thereexists half a reciprocal lattice vector Q for which ǫ ( Q − k ) = − ǫ ( k ) for all k . Consequently, ρ ( − ω ) = ρ ( ω ) . Inthe following, we set half the bandwidth W as our energyunit, i.e., W = 2 , so that ρ ( | ω | >
1) = 0 .The Hilbert transform of the density of states ρ ( ω ) provides the real part Λ ( ω ) of the local host-electronGreen function g ( ω ) , g ( ω ) = 1 L X k ω − ǫ ( k ) + i η ≡ Λ ( ω ) − i πρ ( ω ) (6)with Λ ( ω ) = Z − d ǫ ρ ( ǫ ) ω − ǫ . (7)For | ω | < , this is a principal-value integral. To be defi-nite, we frequently choose to work with a one-dimensionaldensity of states for electrons with nearest-neighbor elec-tron transfer on a ring, ρ ( | ω | ≤
1) = 1 π √ − ω , (8)with its Hilbert transform Λ ( | ω | >
1) = sgn( ω ) √ ω − ,Λ ( | ω | <
1) = 0 , (9)where sgn( x ) = x/ | x | is the sign function.Alternatively, we shall employ the semi-elliptic den-sity of states that corresponds to electrons with nearest-neighbor electron transfer on a Bethe lattice with infinitecoordination number, ρ se0 ( | ω | ≤
1) = 2 π p − ω , (10) Copyright line will be provided by the publisher ss header will be provided by the publisher 3 with its Hilbert transform Λ se0 ( | ω | ≤
1) = 2 ω ,Λ se0 ( ω >
1) = 2 (cid:16) ω − p ω − (cid:17) ,Λ se0 ( ω < −
1) = 2 (cid:16) ω + p ω − (cid:17) . (11)In the following we shall consider the case where the hostelectron system is filled on average with ¯ N = ¯ N ↑ + ¯ N ↓ electrons; the thermodynamic limit, ¯ N, L → ∞ with n =¯ N/L fixed, is implicit.
In the (anisotropic) Kondomodel, the host electrons interact locally with the impurityspin at the origin, ˆ V = ˆ V ⊥ + ˆ V z , ˆ V ⊥ = J ⊥ (cid:16) ˆ s x ˆ S x + ˆ s y ˆ S y (cid:17) = J ⊥ (cid:16) ˆ c +0 , ↑ ˆ c , ↓ ˆ d + ⇓ ˆ d ⇑ + ˆ c +0 , ↓ ˆ c , ↑ ˆ d + ⇑ ˆ d ⇓ (cid:17) , ˆ V z = J z ˆ s z ˆ S z = J z (cid:16) ˆ c +0 , ↑ ˆ c , ↑ − ˆ c +0 , ↓ ˆ c , ↓ (cid:17) (cid:16) ˆ d + ⇑ ˆ d ⇑ − ˆ d + ⇓ ˆ d ⇓ (cid:17) . (12)The operators ˆ d + s ( ˆ d s ) create (annihilate) an impurity elec-tron with spin s = ⇑ , ⇓ . In eq. (12) it is implicitly under-stood that the impurity is always filled with an electronwith spin ⇑ or ⇓ . For the isotropic Kondo model we have J ⊥ = J z . We couple the elec-trons to a global external magnetic field ˆ H m = − B (cid:0) ˆ n d ⇑ − ˆ n d ⇓ (cid:1) − B X i (ˆ n i, ↑ − ˆ n i, ↓ ) (13)with the local density operators ˆ n ds = ˆ d + s ˆ d s and ˆ n i,σ =ˆ c + i,σ ˆ c i,σ to investigate the magnetic properties of the Ising-Kondo model. Here, the magnetic energy reads B = g e µ B H / > , (14)where H is the external field, g e ≈ is the electrons’ gy-romagnetic factor, and µ B is Bohr’s magneton. In the definitionof the particle-hole transformation we include a spin-flipoperation, e τ S : ˆ a k,σ ˆ a + Q − k, ¯ σ , ˆ d s ˆ d +¯ s , ˆ a + k,σ ˆ a Q − k, ¯ σ , ˆ d + s ˆ d ¯ s , (15)where ¯ ↑ = ↓ ( ¯ ⇑ = ⇓ ) and ¯ ↓ = ↑ ( ¯ ⇓ = ⇑ ) denotes the flippedspin. The particle-hole transformation implies ˆ c +0 ,σ ˆ c , ¯ σ .The Hamiltonian is invariant under the transformation, e τ S :ˆ H K ˆ H K . In this work, we investi-gate the Ising-Kondo model where we disregard the spin-flip terms in eq. (12), J ⊥ = 0 , ˆ H IK = ˆ T + ˆ V z + ˆ H m . (16)The anisotropic Kondo model reduces to the Ising-Kondomodel for an infinitely strong anisotropy in z -direction.Thus, the Ising-Kondo model and the anisotropic Kondomodel share the same relationship as the Ising model andthe anisotropic Heisenberg model [2]. For the thermodynamics weneed to calculate the free energy from which we obtain thethermodynamic potentials and the response functions.
The free energy of a quantum-me-chanical system is given by F ( T, µ ) = − T ln ( Z ( β, µ )) , Z ( β, µ ) = Tr (cid:16) e − β ( ˆ H − µ ˆ N ) (cid:17) ≡ X N,n exp (cid:2) − β ( E Nn − µN ) (cid:3) , (17)where Z ( β, µ ) is the grand-canonical partition function, T is the temperature, β = 1 /T ( k B ≡ ), and µ is thechemical potential. Here, E Nn denote the eigenenergies ofthe Hamiltonian ˆ H for a system with N particles.The chemical potential µ is fixed by the requirementthat the system contains ¯ N particles on average ¯ N = h ˆ N i , (18)where ˆ N counts all electrons; for the (Ising-)Kondo modelwe have ˆ N = X i,σ ˆ n i,σ + X s ˆ n ds . (19)The thermal average of an operator ˆ A is defined by h ˆ A i = 1 Z Tr (cid:16) e − β ( ˆ H − µ ˆ N )) ˆ A (cid:17) ≡ Z X N,n e − β ( E Nn − µN ) h Ψ Nn | ˆ A | Ψ Nn i . (20)Here, | Ψ Nn i denote the eigenstates of the Hamiltonian ˆ H for a system with N particles. In general, eq. (18) has asolution that depends on the temperature T and the averageparticle number ¯ N , i.e., µ ≡ µ ( T, ¯ N ) . We consider thecase of half band-filling, ¯ N = L +1 . For the (Ising-)Kondomodel we have for all interactions µ ( T, L + 1) = 0 . (21)This relation is readily proven using particle-hole sym-metry. The particle-hole transformation (15) leaves the Copyright line will be provided by the publisher
K. Bauerbach, Z.M.M. Mahmoud, and F. Gebhard: Thermodynamics and screening in the Ising-Kondo model anisotropic Kondo Hamiltonian invariant but it affects theparticle number operator, e τ S : ˆ N L + 2 − ˆ N . (22)Therefore, ¯ N ( µ ) = 1 Z Tr (cid:16) e − β ( ˆ H − µ ˆ N ) ˆ N (cid:17) = Tr (cid:16) e − β ( ˆ H − µ (2 L +2 − ˆ N )) (2 L + 2 − ˆ N ) (cid:17) Tr (cid:16) e − β ( ˆ H − µ (2 L +2 − ˆ N )) (cid:17) = 2 L + 2 − ¯ N ( − µ ) , (23)or ¯ N ( µ ) + ¯ N ( − µ ) = 2 L + 2 . (24)This relation holds for the anisotropic Kondo model at alltemperatures. It readily proves eq. (21) when we demandhalf band-filling, ¯ N = L + 1 . Note that this relation holdsfor all values of J ⊥ and J z . In particular, it also applies forthe Ising-Kondo model, J ⊥ = 0 . Thermodynamicpotentials are first derivatives of the free energy. The inter-nal energy is the thermal expectation value of the Hamilto-nian. At fixed chemical potential µ ( T ) = 0 we have U ( T ) = h ˆ H i = − ∂ ln ( Z ( β )) ∂β = − T ∂∂T (cid:18) F ( T ) T (cid:19) . (25)The entropy follows from the general relation F ( T ) = U ( T ) − T S ( T ) as S ( T ) = U ( T ) − F ( T ) T = − ∂F ( T ) ∂T . (26)In the presence of a finite external field, we calculate themagnetization M ( B, T ) = − ∂F∂ H = g e µ B m ( B, T ) ,m ( B, T ) = − ∂F∂B (27) = 12 h ˆ n d ⇑ − ˆ n d ⇓ + X i (cid:0) ˆ n i, ↑ − ˆ n i, ↓ (cid:1) i . For the Kondo and Ising-Kondo models we are interestedin the impurity-induced contributions of order unity. Wedenote these quantities with the an upper index ‘i’, e.g., F i ( T ) and m i ( T ) [7,14,15].The impurity-induced contribution to the magnetiza-tion m i ( B, T ) is a thermodynamic potential. It must bedistinguished from the impurity spin polarization, S z ( B, T ) = 12 h ˆ n d ⇑ − ˆ n d ⇓ i . (28)For a thorough discussion of the difference between theimpurity-induced magnetization, m i ( T ) , and the impurityspin polarization, S z ( T, B ) , see Refs. [14,15]. Response functions (suscepti-bilities) are first derivatives of the thermodynamic poten-tials. For example, the impurity-induced contribution to thespecific heat is defined by c i V ( T ) = ∂U i ( T ) ∂T (29)and the impurity-induced magnetic susceptibility reads χ i ( B, T ) = ∂M i ( B, T ) ∂ H = (cid:16) g e µ B (cid:17) ∂ [2 m i ( B, T )] ∂B . (30)Likewise, we are also interested in the impurity spin-polarization susceptibility, χ i , S ( B, T ) = g e µ B ∂S z ( B, T ) ∂ H = (cid:16) g e µ B (cid:17) ∂ [2 S z ( B, T )] ∂B . (31)Below, in Sect. 3.3, we shall focus on the zero-field sus-ceptibilities, χ i(S)0 ( T ) = χ i(S) (0 , T ) . The Kondo impurity distortsthe charge and spin distribution of the host electronsaround the origin, known as screeningclouds. To describethese clouds, two-point correlation functions at some dis-tance r between impurity and the bath electrons need to beinvestigated.A well-known textbook example is the screening ofan extra charge in an electron gas [16,17]. Apart fromsome Friedel oscillations at large distances from the im-purity, the additional charge is screened on the scale ofthe inverse Thomas-Fermi wave number k − . i.e., the cor-responding charge distribution function decays essentiallyproportional to exp( − k TF r ) as a function of the distance r from the extra charge.To visualize the spin screening cloud for the Ising-Kondo model, we calculate the spin correlation functionbetween the impurity and bath electrons. We work at finitetemperatures, T ≥ , and zero magnetic field, B = 0 , andfocus on the spin correlation function along the spin quan-tization axis. The local correlation function is defined by C Sdd = h ˆ S z ˆ S z i = 14 h (cid:0) ˆ n d ⇑ − ˆ n d ⇓ (cid:1) i = 14 − h ˆ n d ⇑ ˆ n d ⇓ i = 14 , (32)where we used the fact that the impurity is singly occupied.The correlation function between the impurity site andthe bath site r is defined by C Sdc ( r ) = h ˆ S z ˆ s zr i = 14 h (cid:0) ˆ n d ⇑ − ˆ n d ⇓ (cid:1) (cid:0) ˆ n r, ↑ − ˆ n r, ↓ (cid:1) i . (33)To visualize the screening of the impurity spin, we define S (0 , T, V ) = C Sdd + C Sdc (0) and, for R ≥ , S ( R, T, V ) = C Sdd + C Sdc (0) + R X || r || =1 C Sdc ( r ) , (34) Copyright line will be provided by the publisher ss header will be provided by the publisher 5 where || r || denotes a suitable measure for the length ofa lattice vector. The function S ( R, T, V ) describes theamount of unscreened spin at distance R from the impu-rity site.As we shall show below, for the one-dimensional Ising-Kondo model the screening is incomplete at all tempera-tures, S ( R → ∞ , T ≥ , V ) = S ∞ ( T, V ) > . Moreover, S ( R, T ≥ , V ) shows an oscillating convergence to itslimiting value, i.e., it displays Friedel oscillations. Inthis section we derive closed formulae for the free energyof the Ising-Kondo model. We give expressions for somethermodynamic potentials (internal energy, entropy, mag-netization) and for two response functions (specific heat,zero-field magnetic susceptibilities).
First, we express the partition func-tion of the Ising-Kondo model in terms of an (incomplete)partition function for spinless fermions. Next, we provideexplicit expressions for the free energy of spinless fermionsin terms of the single-particle density of states; the deriva-tion is deferred to appendix A.4. Lastly, we express theimpurity-induced contribution to the free energy in termsof the corresponding expressions for spinless fermions.
The trace over the eigenstates in the partition function (17)contains the sum over the two impurity orientations ( V ≡ J z / > ), Z IK ( β, V ) = h⇑ | Tr c e − β ˆ C | ⇑i + h⇓ | Tr c e − β ˆ C | ⇓i , ˆ C = X σ ˆ T σ + V (ˆ n d ⇑ − ˆ n d ⇓ )(ˆ c +0 , ↑ ˆ c , ↑ − ˆ c +0 , ↓ ˆ c , ↓ ) − B (ˆ n d ⇑ − ˆ n d ⇓ ) − B X i (ˆ n i, ↑ − ˆ n i, ↓ ) , (35)where we used µ ( T, V ) = 0 at half band-filling for all tem-peratures T and interaction strengths V , see eq. (21), and Z IK ( β, V ) ≡ Z IK ( β, µ = 0 , V ) henceforth. Since the spinorientation of the impurity spin is dynamically conservedin the Ising-Kondo model, we can evaluate the expectationvalues with respect to the impurity spins. The remainingterms describe potential scattering for spinless fermions inthe presence of an energy shift due to an external field, Z IK ( β, V ) = e βB ¯ Z sf ( β, B, V ) ¯ Z sf ( β, − B, − V )+ e − βB ¯ Z sf ( β, B, − V ) ¯ Z sf ( β, − B, V ) , ¯ Z sf ( β, B, V ) = Tr sf e − β ˆ H sf ( B,V ) , (36) ˆ H sf ( B, V ) = X k ( ǫ ( k ) − B ) ˆ a + k ˆ a k + VL X k,p ˆ a + k ˆ a p . Here, the creation and annihilation operators ˆ a + k and ˆ a k carry no spin index but still obey the Fermionic algebra, ˆ a + k ˆ a p +ˆ a p ˆ a + k = δ k,p , and all other anticommutators vanish.Note that ¯ Z sf is an incomplete grand-canonical parti-tion function because it lacks the chemical potential term, see appendix A.4. The chemical potential is not zero evenat half band-filling because ˆ H sf is not particle-hole sym-metric. As is de-rived in appendix A.4, the chemical potential correction isof the order /L , as had to be expected for a single impurityproblem. Eventually, it drops out of the problem and wefind for the (incomplete) free energy of spinless fermions ¯ Z sf = e − β ¯ F sf , ¯ F sf = F (0)sf ( B, T ) + F isf ( B, T, V ) , (37)where F (0)sf ( B, T ) = − T Z ∞−∞ d ωρ ( ω ) ln h e − β ( ω − B ) i (38)is the free energy of non-interacting spinless fermions with V = 0 in eq. (36), and F isf ( B, T, V ) = − T Z ∞−∞ d ωD ( ω, V ) ln h e − β ( ω − B ) i (39)is the contribution due to the impurity; the impurity-contri-bution D ( ω, V ) to the single-particle density of states iscalculated in appendix A.1. In one dimension we find with ω p ≡ ω p ( V ) = √ V D ( ω, V ) = δ ( ω + ω p ) θ H ( − V ) + δ ( ω − ω p ) θ H ( V ) − δ ( ω + 1) − δ ( ω − (40) − θ H (1 − − | ω | ) 1 π ∂∂ω arctan (cid:20) V √ − ω (cid:21) , and, for V < / , D se0 ( ω, V ) = − θ H (1 − | ω | ) 1 π ∂∂ω arctan (cid:20) V √ − ω − ωV (cid:21) (41)for the semi-elliptic density of states, where θ H ( x ) is theHeaviside step function. For V > / , poles appear alsofor the semi-elliptic host-electron density of states [10]; wedo not analyze this case in our present work. Weinsert eq. (37) into eq. (36) and find that the free energyof the Ising-Kondo model is given by the sum of the freeenergy of the host electrons and of the free energy from theimpurity, F IK ( B, T, V ) = − T ln [ Z IK ( β, V )]= F h ( B, T ) + F i ( B, T, V ) , (42)with the free energy of the non-interacting host electrons F h ( B, T ) = − T X s = ± Z ∞−∞ d ωρ ( ω ) ln h e − β ( ω − sB ) i . (43) Copyright line will be provided by the publisher
K. Bauerbach, Z.M.M. Mahmoud, and F. Gebhard: Thermodynamics and screening in the Ising-Kondo model
The impurity contribution reads F i ( B, T, V ) = − T ln h e − β [ − B + F isf ( B,T,V )+ F isf ( − B,T, − V )] + e − β [ B + F isf ( B,T, − V )+ F isf ( − B,T,V )] i . (44)For the derivation, see appendix A.4.In the following we discuss the impurity contribu-tion (44) that is of order unity, and ignore the host-electroncontribution because the thermodynamic properties of thehost electrons are well understood [2,18]. Note that phys-ical constraints that apply to the total free energy do notnecessarily apply to the impurity contribution alone, e.g.,the condition that the specific heat must be strictly positiveis not necessarily guaranteed when solely F i is considered,see Sect. 3.2.4. Inthis section, we discuss the thermodynamics of the Ising-Kondo model at zero external field.
We start from eq. (44) that simpli-fies to F i ( T, V ) = F spin ( T ) + ∆F i ( T, V ) ,F spin ( T ) = − T ln(2) ,∆F i ( T, V ) = F isf ( T, V ) + F isf ( T, − V ) ,F isf ( T, V ) = − T Z ∞−∞ d ωD ( ω, V ) ln (cid:2) e − βω (cid:3) (45)in the absence of an external field, where we abbreviate F i ( T, V ) ≡ F i ( B = 0 , T, V ) , etc. The free energy ofthe isolated spin-1/2 is given by the entropy term alone, F spin ( T ) = − T S spin with S spin = ln(2) . The inter-action contribution to the impurity-induced free energy ∆F i ( T, V ) obeys ∆F i ( T, V = 0) = 0 for all tempera-tures.For large temperatures, the entropy contribution fromthe free spin dominates the interaction term, ∆F i ( T ≫ , V ) ≈ − V T , (46)both for the one-dimensional density of states and for thesemi-elliptic density of states.The one-dimensional density of states is very specialbecause the total density of states consists of isolated peaksonly [ ω p ≡ ω p ( V ) = √ V ] D ( ω, V ) + D ( ω, − V ) = δ ( ω + ω p ) − δ ( ω + 1) (47) + δ ( ω − ω p ) − δ ( ω − , as seen from eq. (40) due to the antisymmetry of the arc-tan function. In contrast to our expectation, only the (anti-)bound states and the band edges matter for the free energy,the states near the Fermi edge drop out in F i ( T, V ) .Therefore, the free energy becomes particularly simple, ∆F i1d ( T, V ) = − T ln (cid:20) ω p ( V ) /T ]1 + cosh[1 /T ] (cid:21) (48) Δ F i ( T , V ) J z = 0.4,ΔB = 01d se T ≫ 1 Figure 1
Interaction contribution to the impurity-inducedfree energy at zero field ∆F i ( T, V ) as a function of tem-perature for the one-dimensional and semi-elliptic densityof states for J z = 4 V = 0 . . Also included is the large-temperature asymptote (46).with ω p ( V ) = √ V . For the semi-elliptic density ofstates, eq. (45) can be simplified to ∆F ise ( T, V ) = − Z − d ωπ tanh h ω T i × arctan (cid:20) V √ − ω − ωV (cid:21) (49)for V = J K / < / . In general, the integral must beevaluated numerically.In Fig. 1 we show the interaction contribution to theimpurity-induced free energy for zero magnetic field as afunction of temperature for the one-dimensional and semi-elliptic density of states. For high temperatures, the freeenergy becomes independent of the choice of the densityof states. It is seen from Fig. 1 that the high-temperatureformula (46) becomes applicable for T & .At T = 0 , the free energy is identical to the ground-state energy, F ( T = 0 , V ) = e ( V ) , e ( V ) = Z −∞ d ω ω ( D ( ω, V ) + D ( ω, − V )) . (50)From appendix A.2 or, alternatively, from eq. (48) we findthe ground-state energy of the Ising-Kondo model e ( V ) = 1 − ω p ( V ) = 1 − p V ≈ − V for V ≪ (51)for the one-dimensional density of states, and e se0 ( V ) = 1 π − V πV arctan (cid:20) V − V (cid:21) ≈ − π V for V ≪ (52) Copyright line will be provided by the publisher ss header will be provided by the publisher 7 for the semi-elliptic density of states. The ground-state en-ergy for the semi-elliptic density of states is lower thanthe ground-state energy for the one-dimensional density ofstates.As we mentioned earlier, in one dimension only thebound state and the lower band edge contribute to the freeenergy for low temperatures. Therefore, as a function oftemperature, the changes in the interaction contribution tothe impurity-induced free energy are exponentially smallin ∆F i1d ( T, V ) .In contrast, ∆F ise ( T, V ) displays the generic quadraticdependence in T for the semi-elliptic density of states. Tomake this dependence explicit, we note that the tempera-ture dependence of the free energy in eq. (49) results fromthe region | ω | . T . We readily find ∆F ise ( T, V ) ≈ e se0 ( V ) + F se2 ( V ) T + O ( T ) ,F se2 ( V ) = 4 πV V ) . (53)Note that F se2 ( V ) is positive for all interaction strengths V .This leads to a negative contribution to the specific heat forlow temperatures, see Sect. 3.2.4. We use eq. (25) to calculatethe impurity-induced internal energy from the impuritycontribution to the free energy. For the one-dimensionaldensity of states we find from eq. (48) U i1d ( T, V ) = − ω p ( V ) tanh (cid:20) ω p ( V )2 T (cid:21) + tanh (cid:20) T (cid:21) (54)with ω p ( V ) = √ V for the impurity-induced contri-bution to the internal energy. For the semi-elliptic densityof states, eq. (49) yields U ise ( T, V ) = − Z − d ωπ (cid:18) ω/ (2 T )cosh [ ω/ (2 T )] + tanh h ω T i(cid:19) × arctan (cid:20) V √ − ω − ωV (cid:21) . (55)For high temperatures, eq. (46) gives U i ( T ≫ , V ) = − V T . (56)This result is independent of the choice of the density ofstates.For zero temperature, the internal energy reduces to theground-state energy, U i (0 , V ) = e ( V ) , where the ground-state energy is given in eq. (51) for the one-dimensionaldensity of states (8) and in eq. (52) for the semi-ellipticdensity of states (10). In one dimension, eq. (54) showsthat finite-temperature corrections to the ground-state en-ergy are exponentially small at low temperatures. Forthe generic semi-elliptic density of states, we find fromeq. (53) and eq. (25) that U ise ( T, V ) ≈ e se0 ( V ) − F se2 ( V ) T + O ( T ) . (57) U i ( T , V ) J z = 0.4, B = 01d se T ≫ 1 Figure 2
Impurity-induced internal energy at zero field U i ( T, V ) as a function of temperature for the one-dimensional and semi-elliptic density of states for J z =4 V = 0 . . Also included is the large-temperature asymp-tote (56).The impurity contribution to the internal energy decreases as a function of temperature. This again implies that theimpurity contribution to the specific heat is negative at lowtemperatures, see Sect. 3.2.4.In Fig. 2 we show the internal energy as a function oftemperature for J z = 4 V = 0 . for the one-dimensionaland semi-elliptic density of states. Both curves are quali-tatively similar. The common high-temperature asymptoticis reached for T & . at J z = 0 . . For small temperaturesand in one dimension, the gap for thermal excitations leadsto exponentially small changes of the internal energy fromthe ground-state energy. The semi-elliptic density of statesleads to the generic quadratic dependence of the internalenergy as a function of temperature for small T . The entropy consists of the free impu-rity contribution S spin = ln(2) and the interaction-inducedimpurity terms. Using eq. (26) and eq. (45) we can write S i ( T, V ) = S spin + U i ( T, V ) − ∆F i ( T, V ) T . (58)Explicit expressions for U i ( T, V ) and ∆F i ( T, V ) for theone-dimensional density of states are given in eqs. (48)and (54), their counterparts for the semi-elliptic density ofstates are found in eq. (49) and (55).For large temperatures, we use the high-temperaturelimit (56) for U i ( T, V ) and (46) for ∆F i ( T, V ) to deter-mine the limiting behavior of the entropy, S i ( T ≫ , V ) ≈ ln(2) − V T ; (59)again, the result is independent of the choice of the densityof states. For small temperatures, the interaction-induced Copyright line will be provided by the publisher
K. Bauerbach, Z.M.M. Mahmoud, and F. Gebhard: Thermodynamics and screening in the Ising-Kondo model S i ( T , V ) J z = 0.4, B = 0 Figure 3
Impurity-induced entropy at zero field S i ( T, V ) as a function of temperature for the one-dimensional andsemi-elliptic density of states for J z = 4 V = 0 . . Alsoincluded is the large-temperature asymptote (59).contribution to the impurity entropy is exponentially smallfor the one-dimensional density of states. For the semi-elliptic density of states, we obtain from eq. (53) in eq. (26) S ise ( T ≪ , V ) ≈ ln(2) − F se2 ( V ) T , (60)which displays a linear dependence of the entropy on tem-perature that is generic for fermionic systems. The negativeprefactor shows that the interaction tends to reduce the en-tropy of the free spin.In Fig. 3 we show the impurity entropy. It is seen thatthe Ising-Kondo interaction decreases the impurity-spinentropy S spin = ln(2) . Note, however, that for small in-teractions the reduction is small for all temperatures andvanishes for both small and large temperatures. This indi-cates that screening is not very effective in the Ising-Kondomodel. The high-temperature asymptote (59) is reached for T & . As last point in this subsection,we discuss the specific heat in the absence of a magneticfield. In one dimension, it explicitly reads c i , V ( T, V ) = [ ω p ( V )] T cosh [ ω p ( V ) / (2 T )] − T cosh [1 / (2 T )] (61)with ω p ( V ) = √ V . For the semi-elliptic density ofstates eq. (55) leads to c i , se V ( T, V ) = Z − d ω ω (2 T − ω tanh[ ω/ (2 T )])2 πT cosh [ ω/ (2 T )] × arctan (cid:20) V √ − ω − ωV (cid:21) (62) c i V ( T , V ) J z = 0.4, B = 01d se T ≫ 1 Figure 4
Impurity contribution to the specific heat at zerofield c i V ( T, V ) as a function of temperature for the one-dimensional and semi-elliptic density of states for J z =4 V = 0 . . Also included is the large-temperature asymp-tote (63).for V < / . The limit of high temperatures is independentof the choice of the density of states, c i V ( T ≫ , V ) ≈ V T , (63)using eq. (56) in eq. (29).For small temperatures, the specific heat is exponen-tially small for the one-dimensional density of states. Us-ing eq. (57) in eq. (29) the impurity-induced contributionto the specific heat for the semi-elliptic density of statesshows the generic linear dependence on T but with a neg-ative coefficient, c i , se V ( T ≪ , V ) ≈ − F se2 ( V ) T (64)with F se2 ( V ) from eq. (53). Note that the total specific heatof the system remains positive as required for thermody-namic stability since the impurity provides only a smallnegative contribution.In Fig. 4 we show the impurity contribution to the spe-cific heat at zero field as a function of temperature forthe one-dimensional and semi-elliptic density of states for J z = 4 V = 0 . . The specific heat is negative for smalltemperatures and displays a minimum around T . J z / ( T . J z / ) and a broad maximum around T ≈ J z ( T ≈ J z ) for the one-dimensional (semi-elliptic) densityof states. The high-temperature asymptote (63) becomesapplicable for T & . . Asour last subsection, we discuss the thermodynamics at fi-nite magnetic field. While the results are applicable forgeneral ≤ B < , we restrict the discussion to the exper-imentally realistic region B ≪ V ≪ . Copyright line will be provided by the publisher ss header will be provided by the publisher 9
To address the impurity-inducedcontribution to the free energy, we abbreviate ¯ F ( B, T, V ) = F isf ( B, T, V ) + F isf ( − B, T, − V ) , (65)where F isf ( B, T, V ) is calculated in appendix A.4, so thatin eq. (44) we can write F i ( B, T, V ) = − T ln h e − β [ − B + ¯ F ( B,T,V )] + e − β [ B + ¯ F ( − B,T,V )] i . (66)We split ¯ F ( B, T, V ) = ¯ F s ( B, T, V ) + ¯ F a ( B, T, V ) intotwo parts that are symmetric and antisymmetric in B , ¯ F s ( − B, T, V ) = ¯ F s ( B, T, V ) , ¯ F a ( − B, T, V ) = − ¯ F a ( B, T, V ) , (67)and find in eq. (66) F i ( B, T, V ) = ¯ F s ( B, T, V ) − T ln (cid:2) B eff ( B, T, V ) /T ] (cid:3) , (68) B eff ( B, T, V ) = B − ¯ F a ( B, T, V ) . (69)For small fields we have ¯ F s ( B ≪ , T, V ) = F i ( T, V ) + O ( B ) , ¯ F a ( B ≪ , T, V ) = α ( T, V ) B + O ( B ) ,α ( T, V ) = ∂ ¯ F ( B, T, V ) ∂B (cid:12)(cid:12)(cid:12)(cid:12) B =0 (70)so that, in the small-field limit, B eff ( B ≪ , T, V ) = (1 − α ( T, V )) B ,F i ( B ≪ , T, V ) ≈ F i ( T, V ) − T ln (cid:20) h (1 − α ( T, V )) BT i(cid:21) . (71)For a free spin we obtain F spin ( B, T ) = − T ln (cid:20) h BT i(cid:21) . (72)A comparison with eq. (71) shows that, for small externalfields, the impurity-contribution to the free energy consistsof the field-free term discussed in Sect. 3.2 and the contri-bution of a free spin in the effective field B eff ( B, T, V ) =(1 − α ( T, V )) B .To present tangible results, we use the one-dimensionalhost-electron density of states in eq. (40) and the semi-elliptic host-electron density of states in eq. (41) when V < / to evaluate the free energy for spinless fermionsfrom eq. (39). Performing a partial integration we can write( F , ses , a ≡ F , ses , a ( B, T, V ) , ω p ( V ) = √ V ) ¯ F = − T ln (cid:20) cosh( B/T ) + cosh( ω p ( V ) /T )cosh( B/T ) + cosh(1 /T ) (cid:21) , (73) compare eq. (48), and ¯ F = Z − d ωπ arctan (cid:20) V √ − ω (cid:21)(cid:18)
11 + e ( ω − B ) /T −
11 + e ( ω + B ) /T (cid:19) . (74)Moreover, ¯ F ses = − Z − d ω π (cid:18) tanh (cid:20) ω − B T (cid:21) + tanh (cid:20) ω + B T (cid:21)(cid:19) × arctan (cid:20) V √ − ω − ωV (cid:21) , (75)compare eq. (49), and ¯ F sea = − Z − d ω π (cid:18) tanh (cid:20) ω − B T (cid:21) − tanh (cid:20) ω + B T (cid:21)(cid:19) × arctan (cid:20) V √ − ω − ωV (cid:21) . (76)We again split the impurity free energy into the interactioncontributions and that of the free spin, ∆F i ( B, T, V ) = F i ( B, T, V ) − F spin ( B, T ) (77)with F spin ( B, T ) from eq. (72) so that ∆F i ( B, T, V =0) = 0 for all fields and temperatures.Simplifications of the above expressions are only pos-sible in limiting cases. For high temperatures, T ≫ , weexpand − βF isf ( B, T ≫ , V ) ≈ − T ω ( V )+ 18 T ( ω ( V ) − Bω ( V )) ,ω n = Z ∞−∞ d ω ω n D ( ω, V ) . (78)Note that ω n ( − V ) = ( − n ω n ( V ) due to the symmetry D ( ω, − V ) = D ( − ω, V ) . Then, we obtain F spin ( B, T ≫ ≈ − T ln(2) − B T ,∆F i ( B, T ≫ , V ) ≈ − ω ( V )4 T + ω ( V ) B T , (79)with corrections of the order /T . Using M ATHEMAT - ICA [19] we find ω ( V ) = V and ω ( V ) = V so thatfor T ≫ ∆F i ( B, T ≫ , V ) ≈ − V T + V B T , (80)up to and including second order in /T . Using pertur-bation theory in J z /T [7,15], it is readily shown that ∆F i ( B, T, V ) is indeed independent of the host-electrondensity of states up to second order in J z /T . Eqs. (79)and (80) show that small magnetic fields induce smallcorrections, of the order B . Copyright line will be provided by the publisher
At low temperatures, eq. (71) shows that the temper-ature dependence of the impurity contribution to the freeenergy is dominated by the logarithm. Therefore, the Som-merfeld expansion of ¯ F ( T, V ) [2,18] can be restricted tothe leading-order term, i.e., we use ¯ F ( B, T ≪ , V ) ≈ ¯ F ( B, T = 0 , V ) ≡ ¯ E ( B, V ) . Thus, we find in eq. (68) F i ( B, T ≪ , V ) ≈ ¯ E s ( B, V ) − T ln (cid:2) B eff ( B, V ) /T ] (cid:3) , (81) B eff ( B, V ) = B − ¯ E a ( B, V ) (82)with B eff ( B, V ) ≡ B eff ( B, T = 0 , V ) . Eq. (81) showsthat the low-temperature thermodynamics of the model atfinite fields is described by a free spin in an effective field B eff ( B, V ) . Eq. (81) is actually applicable in the temper-ature region T . B eff ( B, V ) . At low temperatures, theinteraction-induced impurity contribution to the free en-ergy becomes ∆F i ( T ≪ ≈ ¯ E s ( B, V ) − T ln (cid:20) cosh[( B − ¯ E a ( B, V )) /T ]cosh[ B/T ] (cid:21) , (83)where we abbreviate ∆F i ( T ≪ ≡ ∆F i ( B, T ≪ , V ) .The above expressions can be worked out further whenthe density of states is specified. For the one-dimensionaldensity of states we have ¯ E ( B, V ) = e ( V ) = 1 − ω p ( V ) = 1 − p V , ¯ E ( B, V ) = Z B − B d ωπ arctan (cid:20) V √ − ω (cid:21) (84)with e ( V ) from eq. (51), and, with ω p ( V ) = √ V , ¯ E ( B, V ) = 2 Vπ arcsin( B ) + 2 Bπ arctan (cid:20) V √ − B (cid:21) − ω p ( V ) π arctan (cid:20) BVω p ( V ) √ − B (cid:21) , (85)where we used M ATHEMATICA [19] to carry out the inte-gral.For the semi-elliptic density of states we find ¯ E ses ( B, V ) = Z − B − d ωπ arctan " V √ − ω − ωV − Z B d ωπ arctan " V √ − ω − ωV (86)and ¯ E sea ( B, V ) = Z B − B d ωπ arctan " V √ − ω − ωV . (87) With ¯ ω ( V ) = √ V we explicitly have ¯ E ses ( B, V ) = √ − B π + Bπ arctan " V √ − B − BV − Bπ arctan " V √ − B BV (88) + ¯ ω ( V )8 πV arctan (cid:20) B ¯ ω ( V ) − V √ − B (1 − V ) (cid:21) − ¯ ω ( V )8 πV arctan (cid:20) B ¯ ω ( V ) + 4 V √ − B (1 − V ) (cid:21) and ¯ E sea ( B, V ) = − (1 − V ) arcsin( B )4 πV + Bπ arctan " V √ − B − BV + Bπ arctan " V √ − B BV (89) + ¯ ω ( V )8 πV arctan (cid:20) B ¯ ω ( V ) − V √ − B (1 − V ) (cid:21) + ¯ ω ( V )8 πV arctan (cid:20) B ¯ ω ( V ) + 4 V √ − B (1 − V ) (cid:21) , where we used M ATHEMATICA [19] to carry out the inte-grals.At T = 0 , the impurity-contribution to the ground-stateenergy is obtained from eq. (81) as e ( B, V ) = ¯ E s ( B, V ) − | B − ¯ E a ( B, V ) | . (90)The absolute value can be ignored because the argument isalways positive for B > . Thus, we have e ( B, V ) ≡ F i ( B, T = 0 , V )= ¯ E s ( B, V ) + ¯ E a ( B, V ) − B (91)for the interaction contribution to the impurity-inducedchange in the ground-state energy at finite fields B .In Fig. 5 we show the interaction contribution to theimpurity-induced free energy ∆F i ( B, T, V ) as a functionof temperature for the one-dimensional and semi-ellipticdensity of states for B = 0 . and J z = 4 V = 0 . .The low-temperature expression (83) works very well un-til ∆F i ( B, T, V ) reaches its limiting value ∆F i ( B, T ≫ B, V ) = ¯ E s ( B, V ) for T & B eff . Thus, it is clearly seenthat, at very low temperatures T . B ≪ , the thermo-dynamics of the Ising-Kondo model can be described bya spin-1/2 in an effective field, see eq. (82). In the region T & V , a small magnetic field becomes irrelevant and wemay approximate ∆F i ( B ≪ , T & B, V ) ≈ ∆F i ( B =0 , T, V ) . Copyright line will be provided by the publisher ss header will be provided by the publisher 11 Δ F i ( B , T , V ) J z = 0.4,ΔB = 0.01 ΔF i,1d ΔF i,1d (T ≪ 1)ΔF i,1d (B = 0) ΔF i,s≪ ΔF i,s≪ (T ≪ 1)ΔF i,s≪ (B = 0) Figure 5
Interaction contribution to the impurity-inducedfree energy ∆F i ( B, T, V ) as a function of temperature forthe one-dimensional and semi-elliptic density of states for B = 0 . and J z = 4 V = 0 . . Also included is thefree energy at zero field, eqs. (48) and (49), and the low-temperature approximation (83).For further reference, we list the results in the limit ofsmall fields. We have ¯ E ( B, V ) ≈ Bπ arctan( V ) + B π V V + O ( B ) , ¯ E ses ( B, V ) ≈ e se0 ( V ) + 4 B π V V + O ( B ) , ¯ E sea ( B, V ) ≈ Bπ arctan(2 V ) − B π − V (1 + 4 V ) + O ( B ) (92)for B ≪ with e se0 ( V ) from eq. (52). Moreover, for theground-state energy we find e ( B ≪ , V ) ≈ e ( V ) − B (cid:18) − π arctan( V ) (cid:19) ,e se0 ( B ≪ , V ) ≈ e se0 ( V ) + 4 B π V V − B (cid:18) − π arctan(2 V ) (cid:19) , (93)up to and including second order in B > with e ( V ) from eq. (51) and e se0 ( V ) from eq. (52).For small fields, the effective field in eq. (82) is scaledlinearly, see eqs. (70) and (71) with α ( V ) ≡ α ( T = 0 , V ) , B eff ( B, V ) ≈ (1 − α ( V )) B ,α ( V ) = 2 π arctan( V ) ,α se ( V ) = 2 π arctan(2 V ) . (94) For small interactions, the effect is small, of the order V .Due to the interaction, the effective magnetic field is some-what smaller than the external field. This is readily under-stood from the fact that the conduction electrons screen theimpurity and thus weaken the externally applied field. Next, we brieflydiscuss the impurity-induced internal energy and entropyfor the case of small fields.For all temperatures, couplings, and fields, the internalenergy and the entropy are obtained from the free energyby differentiation with respect to T , see eq. (25) for theinternal energy and eq. (26) for the entropy. Since we as-sume a small magnetic field, typically B ≪ J z ≪ , theimpurity-induced internal energy and entropy follow thecurves shown in Fig. 2 and Fig. 3 when T & V , with smallcorrections of the order B .For small temperatures, T . B , we start from eq. (83)and find for the internal energy of a spin in an effectivefield U i ( B, T . B, V ) = ¯ E s ( B, V ) (95) − B eff ( B, V ) tanh (cid:20) B eff ( B, V ) T (cid:21) with B eff ( B, V ) from eq. (82). The impurity-contributionto the entropy reads S i ( B, T . B, V ) = ln (cid:2) B eff ( B, V ) /T ) (cid:3) (96) − B eff ( B, V ) T tanh (cid:20) B eff ( B, V ) T (cid:21) for low temperatures; for a free spin, replace B eff by B ineqs. (95) and (96).In Fig. 6 we show the impurity-induced internal en-ergy U i ( B, T, V ) as a function of temperature for theone-dimensional and semi-elliptic density of states for J z = 4 V = 0 . , a small external field B = 0 . , andlow temperatures. The internal energy increases from itsvalue e ( B, V ) , eq. (91), only exponentially slowly be-cause the magnetic field induces an energy gap of theorder B eff ( B, V ) between the two spin orientations.When the temperature becomes of the order of theeffective magnetic field B eff ( B, V ) , the impurity contri-bution to the internal energy U i ( B, T, V ) approaches thevalue ¯ E s ( B, V ) ≈ e ( V ) , with corrections of the or-der B , and the approximate low-temperature internal en-ergy (95) starts to deviate from the exact result. At tem-peratures T & V , the internal energy becomes essentiallyidentical to its zero-field value shown in Fig. 2 on a largertemperature scale.In Fig. 7 we show the impurity contribution to the en-tropy S i ( B, T, V ) as a function of temperature for the one-dimensional and semi-elliptic density of states for J z =4 V = 0 . , and a small external field B = 0 . . In contrastto the zero-field case, the entropy is zero at zero tempera-ture because the impurity spin is oriented along the effec-tive external field. Due to the excitation gap, the entropy is Copyright line will be provided by the publisher U i ( B , T , V ) J z = 0.4, B = 0.01U i,1d U i,1d (T ≪ 1)U i,1d (B = 0) U i,s≪ U i,s≪ (T ≪ 1)U i,s≪ (B = 0) Figure 6
Impurity-induced internal energy U i ( B, T, V ) as a function of temperature for the one-dimensional andsemi-elliptic density of states for J z = 4 V = 0 . and external field B = 0 . . Also included is the low-temperature approximation (95) and the zero-field approx-imation shown in Fig. 2. S i ( B , T , V ) J z = 0.4, B = 0.01S i,1d S i,1d (T ≪ 1)S i,1d (B = 0) S i,s≪ S i,s≪ (T ≪ 1)S i,s≪ (B = 0) Figure 7
Impurity-induced entropy S i ( B, T, V ) as a func-tion of temperature for the one-dimensional and semi-elliptic density of states for J z = 4 V = 0 . and externalfield B = 0 . . Also included is the low-temperature ap-proximation (96) and the zero-field approximation shownin Fig. 3.exponentially small for T ≪ B eff ( B, V ) . When the tem-perature becomes of the order of B eff ( B, V ) , the impurityentropy approaches S ≈ ln(2) . For T & V , it becomesessentially identical to its zero-field value shown in Fig. 3on a larger temperature scale. For the calculation of the impurity-induced magne- m i ( B , V ) J z = 0.4, T = 0m i,1d m i,se S z Figure 8
Impurity-induced magnetization at zero temper-ature m i ( B, T = 0 , V ) as a function of magnetic field forthe one-dimensional and semi-elliptic density of states for J z = 4 V = 0 . from eq. (99). Also included is the valuefor the impurity spin polarization, S z ( B > , T = 0 , V ) =1 / from eq. (104).tization m i ( B, T, V ) ≡ m i , see eq. (27), we start fromeq. (68) and find m i = − (cid:20) ∂ ¯ F s ∂B − (cid:18) − ∂ ¯ F a ∂B (cid:19) tanh (cid:20) B − ¯ F a T (cid:21)(cid:21) (97)with ¯ F s / a ≡ ¯ F s / a ( B, T, V ) . We numerically perform thederivatives with respect to B for all temperatures.At temperature T = 0 , eq. (91) gives for B > m i ( B, V ) ≡ m i ( B, , V ) ,m i ( B, V ) = 12 (cid:18) − ∂ ¯ E s ( B, V ) ∂B − ∂ ¯ E a ( B, V ) ∂B (cid:19) . (98)For the model density of states we obtain m i , ( B, V ) = 12 − π arctan (cid:20) V √ − B (cid:21) ,m i , se ( B, V ) = 12 − π arctan " V √ − B − BV . (99)At V = 0 , we recover the value for a free spin in a finitefield at zero temperature, m i ( B > , T = 0 , V = 0) = m spin ( B > , T = 0) = 1 / . For finite antiferromagneticinteractions, V > , the impurity-induced magnetization issmaller than the free-spin value because the impurity spinis screened by the band electrons in its surrounding, seeSect. 4.In Fig. 8 we show the impurity-induced magnetizationat zero temperature m i ( B, V ) ≡ m i ( B, T = 0 , V ) asa function of magnetic field for the one-dimensional andsemi-elliptic density of states for J z = 4 V = 0 . . Due tothe larger density of states at the Fermi energy for small B , Copyright line will be provided by the publisher ss header will be provided by the publisher 13 the screening is more effective for the semi-elliptic den-sity of states than for the one-dimensional density of states.Since the one-dimensional density of states diverges at theband edges, the two curves cross at some (very large) mag-netic fields, B ≈ . . At B = 1 , the screening vanishesfor the semi-elliptic density of states, m i , se ( B = 1 , T =0 , V ) = 1 / , and becomes perfect for the one-dimensionaldensity of states, m i , ( B = 1 , T = 0 , V ) = 0 , reflectingthe behavior of the density of states at the band edges.For low temperatures, T . B , the asymptotic expres-sion for the magnetization is obtained from eq. (97) as m i ( B, T . B, V ) ≈ − ∂ ¯ E s ∂B + 12 (cid:16) − ∂ ¯ E a ∂B (cid:17) tanh (cid:20) B eff ( B, V ) T (cid:21) (100)with B eff ( B, V ) from eq. (82). For small fields this can befurther simplified to give m i ( B ≪ , T ≪ , V ) ≈ B eff ( B, V )2 B tanh (cid:20) B eff ( B, V ) T (cid:21) (101)with B eff ( B, V ) /B = 1 − α ( V ) from eq. (94). This showsthat, for small fields and temperatures, the magnetizationis a universal function of B/T , as for a free spin [2,18].For high temperatures, we can derive the asymptotefrom eqs. (79) and (80) as m i ( B, T ≫ ≈ B T (cid:18) − VT (cid:19) , (102)with corrections of the order /T .In Fig. 9 we show the impurity-induced magnetiza-tion m i ( B, T, V ) as a function of temperature for the one-dimensional of states for J z = 4 V = 0 . and B = 0 . ;the curves for the semi-elliptic density of states are qualita-tively very similar. For small interactions, the temperature-dependence of the magnetization follows that of a free spinin an effective field, i.e., the temperature dependence isvery small as long as T . B eff ( B, V ) , see eq. (100). When T exceeds B eff the magnetization rapidly declines and ap-proaches zero for high temperatures. Indeed, as seen fromeq. (102), at large temperatures the magnetization vanishesproportional to /T , as for a free spin; interaction correc-tions are smaller, of the order V /T .To demonstrate the screening of the host electrons, wecompare the impurity-induced magnetization with the im-purity spin polarization. We have from eq. (28) S z = 12 e − β [ − B + ¯ F ( B,T,V )] − e − β [ B − ¯ F ( − B,T,V )] e − β [ − B + ¯ F ( B,T,V )] + e − β [ B − ¯ F ( − B,T,V )] = 12 tanh (cid:20) B eff ( B, T, V ) T (cid:21) (103)with S z ≡ S z ( B, T, V ) and B eff ( B, T, V ) from eq. (71). m i ( B , T , V ) m i,1d m i,1d (T ≪ 1)m i (T ≫ 1) S i,1dz S i,1dz (T ≪ 1)S iz (T ≫ 1)0 0.01 0.02 0.03 0.04T00.10.20.30.40.5 m i ( B , T , V ) J z ≫ 0.4≪≪B ≫ 0.01 Figure 9
Impurity-induced magnetization m i ( B, T, V ) and impurity spin polarization S z ( B, T, V ) as a functionof temperature for the one-dimensional density of states for J z = 4 V = 0 . and external field B = 0 . . Also includedare the approximate results for low temperatures and smallfields from eq. (101) and eq. (104), respectively, and thelarge-temperature asymptotes, eq. (102) and eq. (105). In-set: region of small temperatures.For low temperatures and B > , this expression sim-plifies to S z ( B, T . B, V ) ≈
12 tanh (cid:20) B eff ( B, V ) T (cid:21) (104)with B eff ( B, V ) from eq. (82). For T = 0 and B > ,the impurity spin is aligned with the external field. Acomparison with eq. (101) shows that, at low temper-atures and small fields, the impurity spin polarizationand the impurity-induced magnetization differ by a fac-tor B eff ( B, V ) /B = 1 − α ( V ) . The impurity-inducedmagnetization is smaller because it is more sensitive to thescreening by the host electrons.For large temperatures, we find from eq. (78) S z ( B, T ≫ , V ) ≈ B T (cid:18) − V T (cid:19) (105)with corrections of the order T − . The impurity-inducedmagnetization and the impurity spin polarization agree tofirst order in /T (free spin) but slightly differ already insecond order, compare eq. (102) and eq. (105). Again, theimpurity-induced magnetization is smaller than the impu-rity spin polarization because of the larger screening con-tribution from the host electrons. The results for the im-purity spin polarization are visualized in Fig. 9 in compari-son with those for the impurity-induced magnetization. Forsmall fields and interactions, the differences between m i and S z are small but discernible. Copyright line will be provided by the publisher c i V ( B , T , V ) J z = 0.4, B = 0.01c i,1dV c i,1dV (T ≪ 1)c i,1dV (B = 0) Figure 10
Impurity contribution to the specific heat c i V ( B, T, V ) as a function of temperature for the one-dimensional density of states for J z = 4 V = 0 . and B = 0 . . Also included is the specific heat for zero fieldfrom eq. (61) as shown in Fig. 4 on a larger temperaturescale. Lastly, we discuss thespecific heat in the presence of a small magnetic fieldand the zero-field susceptibilities for the impurity-inducedmagnetization and the impurity spin polarization.In general, we calculate the specific heat from the in-ternal energy using eq. (29), see eqs. (61) and (62) for thezero-field case and the two model density of states. Forsmall B , we thus focus on small temperatures where wecan use eq. (95) to find c i V ( B, T . B, V ) ≈ (cid:18) B eff ( B, V ) T cosh[ B eff ( B, V ) /T ] (cid:19) (106)with B eff ( B, V ) from eq. (82). It is seen that the specificheat displays a peak around B eff ( B, V ) . Due to the screen-ing by the host electrons, the impurity spin in the Ising-Kondo model behaves like a free spin in an effective field.We show the specific heat in Fig. 10 for J z = 4 V = 0 . and B = 0 . as a function of temperature for the one-dimensional density of states; the curves for the semi-elliptic density of states differ only slightly. For smallfields, the specific heat approaches the zero-field valuearound T & V , with small corrections of the order B .Lastly, we consider the zero-field susceptibilities at fi-nite temperature T > B = 0 . Since we keep T finiteand let B go to zero first, none of the approximate ex-pressions is applicable that were derived for the impurity-induced magnetization or the impurity spin polarization inSect. 3.3.3. χ i ( T , V ) / ( g e μ B ) J z = 0.4, μ = 0 χ S,1d0 χ i,1d0 χ i,1d0 (T ≪ 1)χ i0 (T ≫ 1) Figure 11
Impurity contribution to the zero-field magneticsusceptibilities χ i(S)0 ( T, V ) as a function of temperature forthe one-dimensional density of states for J z = 4 V = 0 . .Also included are the asymptotic expressions for χ i0 ( T, V ) ,eq. (110) for high temperatures and eq. (111) for low tem-peratures. Note the logarithmic scale on the ordinate.Using the impurity-induced free energy (71) we can de-rive the zero-field susceptibility as χ i0 ( T, V )( g e µ B ) = 14 T (1 − α ( T, V )) , (107)see eq. (70). Explicitly, for the one-dimensional density ofstates we have α ( T, V ) = tanh " √ V T − tanh (cid:20) T (cid:21) + 2 T Z − d ωπ (cid:18)
12 cosh[ ω/ (2 T )] (cid:19) × arctan (cid:20) V √ − ω (cid:21) , (108)and for the semi-elliptic density of states we find α se ( T, V ) = 2 T Z − d ωπ (cid:18)
12 cosh[ ω/ (2 T )] (cid:19) × arctan " V √ − ω − ωV . (109)For high temperatures, this gives χ i0 ( T ≫ , V )( g e µ B ) = 14 T (cid:18) − V T (cid:19) + O ( T − ) . (110)for a general density of states. For small temperatures wefind χ i0 ( T ≪ , V )( g e µ B ) = 14 T (1 − α ( V )) ≡ ˜ CT , (111)
Copyright line will be provided by the publisher ss header will be provided by the publisher 15 where − α ( V ) is the reduction factor for the magneticfield for small fields at zero temperature, see eq. (94). Thishad to be expected because, for small fields and low tem-peratures, the system describes an impurity spin in an ef-fective field. Therefore, we obtain the Curie law (111) witha modified Curie constant ˜ C [2,18].In Fig. 11 we show the impurity-induced zero-fieldmagnetic susceptibility χ i0 ( T, V ) as a function of tempera-ture for the one-dimensional density of states for V = 0 . ( J z = 0 . ); the curves for the semi-elliptic density ofstates are almost identical. The high-temperature asymp-tote (110) and the low-temperature asymptote (111) to-gether provide a very good description of the zero-fieldimpurity-induced magnetic susceptibility.Since the Curie constant is proportional to S ( S + 1) / with S = 1 / in our case, we can argue that the Ising-Kondo interaction with the host electrons reduces the ef-fective spin on the impurity, S eff ( V ) = 12 − − r − α ( V )2 + 3 α ( V )4 ! ≈ − α ( V )4 for α ≪ . (112)It is only for V = ∞ that α ( V ) = 1 , i.e., there alwaysremains an unscreened spin on the impurity.Finally, we address the spin impurity susceptibility, χ i , S0 ( T, V )( g e µ B ) = 14 T (1 − α ( T, V )) , (113)where we took the derivative of S z ( B, T, V ) in eq. (103)with respect to B and put B = 0 afterwards. The impurityspin susceptibility is also reduced from its free-spin valuebut the reduction factor is only linear in (1 − α ( T, V )) in-stead of quadratic as for the impurity-induced magneticsusceptibility, see eq. (107). Fig. 11 also shows the im-purity spin susceptibility as a function of temperature for J z = 4 V = 0 . . In this section we first calculatethe matrix element for the spin correlation between impu-rity and bath electrons. Next, we focus on the spin correla-tion function on a chain. Lastly, we discuss the screeningcloud in one spatial dimension.
We evaluate the spincorrelation function (33). At B = 0 , the partition functionis given by Z IK ( β, V ) = 2 ¯ Z sf ( β, V ) ¯ Z sf ( β, − V ) (114)with ¯ Z sf ( β, V ) ≡ ¯ Z sf ( β, B = 0 , V ) and V = J z / > because, in the absence of a magnetic field, the two impu-rity orientations contribute equally and the bath electronsexperience either a repulsive or an attractive potential at the origin. Then, C Sdc ( r ) = 12 Z h⇑ | Tr c (cid:20) e − β [ ˆ T + V (ˆ n d ⇑ − ˆ n d ⇓ )(ˆ c +0 , ↑ ˆ c , ↑ − ˆ c +0 , ↓ ˆ c , ↓ )] × (cid:0) ˆ n d ⇑ − ˆ n d ⇓ (cid:1) (cid:16) ˆ n r, ↑ − ˆ n r, ↓ (cid:17)(cid:21) | ⇑i (115)because the impurity spin configuration | ⇓i gives thesame contribution due to spin symmetry. At half band-filling, µ ( T, V ) = 0 for all temperatures T and interactionstrengths V , see eq. (21).Since ˆ n d ⇑ | ⇑i = | ⇑i and ˆ n d ⇓ | ⇑i = 0 , we find C Sdc ( r ) = 12 Z Tr c, ↑ (cid:20) e − β [ ˆ T ↑ + V ˆ c +0 , ↑ ˆ c , ↑ ] ˆ c + r, ↑ ˆ c r, ↑ (cid:21) × Tr c, ↓ (cid:20) e − β [ ˆ T ↓ − V ˆ c +0 , ↓ ˆ c , ↓ ] (cid:21) − Z Tr c, ↑ (cid:20) e − β [ ˆ T ↑ + V ˆ c +0 , ↑ ˆ c , ↑ ] (cid:21) × Tr c, ↓ (cid:20) e − β [ ˆ T ↓ − V ˆ c +0 , ↓ ˆ c , ↓ ] ˆ c + r, ↓ ˆ c r, ↓ (cid:21) = 14 (cid:16) h ˆ c + r ˆ c r i sf ( V ) − h ˆ c + r ˆ c r i sf ( − V ) (cid:17) , (116)where h ˆ A sf i sf ( V ) = 1¯ Z sf ( β, V ) Tr (cid:16) e β [ ˆ T + V ˆ c +0 ˆ c ] ˆ A sf (cid:17) (117)is the thermal expectation value for an operator ˆ A sf forspinless fermions with impurity scattering of strength V at the origin. The ex-pressions (116) for spinless fermions in one dimensionare evaluated in Appendix A.5. In the following, we use r ≥ because the spin correlation function is inversionsymmetric. We analytically derive explicit expressions forthe long-range asymptotics of the spin correlation functionat zero and finite temperatures. The correlation func-tion contains contributions from the poles and from theband part ( V = J z / > ), see appendix A.5, C Sdc ( r ) = 14 (cid:0) N ( r, T, V ) − N ( r, T, − V ) (cid:1) ≡ C S, p dc ( r ) + C S, b dc ( r ) (118)with C S, p dc ( r ) = 14 (cid:0) N p0 ( r, T, V ) − N p0 ( r, T, − V ) (cid:1) = − V (cid:0) V + √ V (cid:1) − r √ V tanh (cid:20) √ V T (cid:21) (119) Copyright line will be provided by the publisher C Sd c ( r ) T = 0, J z = 0.4asymptoticexact Figure 12
Spin correlation function as a function of dis-tance r from the impurity for J z = 4 V = 0 . at tem-perature T = 0 in one dimension. The numerical data arecompared with the asymptotic expressions (125).and C S, b dc ( r ) = 14 (cid:0) N b0 ( r, T, V ) − N b0 ( r, T, − V ) (cid:1) = ( − r V π Z π/ − π/ d pf (sin( p ) , T ) sin(2 pr ) cos( p ) V + cos ( p ) (120)with the Fermi function f ( ω, T ) = 1 / (1 + exp( ω/T )) . Ingeneral, the integral must be evaluated numerically.In Fig. 12 we show the spin correlation function as afunction of distance in the ground state for J z = 4 V =0 . . It is seen that the asymptotic expression (125) as de-rived in Sect. 4.2.2 becomes applicable for r & r with r ( J z / ≈ or r ≈ for J z = 0 . .In Fig. 13 we show the logarithm of the absolute valueof the spin correlation function as a function of distance for J z = 4 V = 0 . for various small temperatures. It is seenthat the correlation function decays to zero exponentially.The correlation length agrees with the analytically deter-mined value ( ξ ) − = 2 πT from eq. (127), as derived inSect. 4.2.3. The polecontribution to the spin correlation function decays expo-nentially for all temperatures, as has to be expected forbound and anti-bound states that are localized around theimpurity. Therefore, the long-range asymptotic is governedby the Friedel oscillations of the band contribution. We fo-cus on the limit of small interactions, V = J z / ≪ .For the band contribution to the correlation function weconsider at T = 0 C S, b dc ( r, T = 0) = V π c S ( r, V ) (121) l n | C Sd c ( r ) | J z = 0.4asymptoticasymptoticasymptotic T=0.02T=0.04T=0.1 Figure 13
Logarithm of the absolute value of the spin cor-relation function as a function of distance r from the im-purity for J z = 4 V = 0 . at temperatures T = 0 . , T = 0 . , T = 0 . in one dimension. The numerical dataare compared with the analytically determined exponentialdecay with exponent ( ξ ) − = 2 πT , see eq. (127).with c S ( r, V ) = ( − r Z − π/ d p sin(2 pr ) cos( p ) V + cos ( p ) (122) = Z π/ d u sin(2 ur )sin( u ) (cid:18) − V sin ( u ) + V (cid:19) . The first term in the brackets can be integrated analyticallyusing M
ATHEMATICA [19], c S ( r, V ) = 14 (cid:20) ψ (cid:18)
14 + r (cid:19) − ψ (cid:18) − r (cid:19) − ψ (cid:18)
34 + r (cid:19) + ψ (cid:18) − r (cid:19)(cid:21) ≈ π − ( − r r for r ≫ , (123)where ψ ( x ) = Γ ′ ( x ) /Γ ( x ) is the digamma function.The integrand in the second term of eq. (122) is of theorder V when sin( u ) is of order unity. Therefore, onlysmall arguments are of interest, c S ( r, V ) ≈ − Z γV d u sin(2 ru ) u V u + V ≈ − Z ∞ d x (2 rV ) x + (2 rV ) sin( x ) x = − π (cid:0) − e − rV (cid:1) ≈ − π for r ≫ (124)with γ smaller than one but of the order unity so that rγV ≫ for r ≫ . The integral was evaluated usingM ATHEMATICA [19].
Copyright line will be provided by the publisher ss header will be provided by the publisher 17
Summing the two terms from eq. (123) and (124) givesthe long-range asymptotics of the spin correlation functionat zero temperature for small interaction strengths, V ≪ , C S, b dc ( r ≫ , T = 0) = − ( − r V πr . (125)The correlation function decays to zero algebraically, anddisplays Friedel oscillations [2] that are commensuratewith the lattice at half band-filling. At finitetemperature and small interactions V = J z / ≪ , thecorrelation function decays to zero exponentially as a func-tion of distance r from the impurity, C Sdc ( r ≫ , T > ∼ ( − r e − r/ξ ( T ) (126)with ξ ( T ) = 12 πT . (127)A detailed derivation is given in Appendix B.3. Note thatthe same correlation length was obtained earlier for thenon-interacting single-impurity Anderson model [11]. Lastly, wediscuss the screening cloud. We analytically derive thelong-range asymptotics of the unscreened spin.
We have C Sdd = 1 / from eq. (32) and C dc (0) = C p dc (0) . After summing thespin correlation function from | r | = 1 up to | r | = R wefind for the unscreened spin at distance R ≥ S ( R, T, V ) = 14 − V √ V tanh (cid:20) √ V T (cid:21)! + s p R ( T, V ) + s b R ( T, V ) (128)with ( K = V + √ V , − K = − V K ) s p R ( T, V ) = − tanh √ V T ! (cid:0) − K − R (cid:1) K √ V , (129)and s b R ( T, V ) = V π Z π/ − π/ d pf (sin( p ) , T ) 1 V + cos ( p ) × (cid:0) ( − R sin[(2 R + 1) p ] − sin( p ) (cid:1) = − V π Z π/ d p tanh (cid:20) sin( p )2 T (cid:21) V + cos ( p ) × (cid:0) ( − R sin[(2 R + 1) p ] − sin( p ) (cid:1) . (130)In Fig. 14 we show the unscreened spin as a func-tion of distance in the ground state for J z = 4 V = 0 . .Even at zero temperature, the impurity spin is not com-pletely screened at infinite distance from the impurity butreaches the limiting value given in eq. (131), as derived in
10 20 30 40 50R0.2300.2320.2340.2360.2380.240 S ( R , T = , V ) T = 0, J z = 0.4asymptoticexactlimiting value Figure 14
Unscreened spin as a function of distance R from the impurity for J z = 4 V = 0 . at temperature T =0 in one dimension. The numerical data are compared withthe limiting value (131) and the asymptotic behavior (132)for small couplings. l n | Δ S Δ R , T , V ) | J z = 0.4analyticanalyticanalytic T=0.02T=0.04T=0.1 Figure 15
Logarithm of the decaying part of theunscreened spin, ∆ S ( R, T, V ) = |S ( R, T, V ) − S ∞ ( T, V ) | , as a function of distance R from the impurityfor J z = 4 V = 0 . at temperatures T = 0 . , T = 0 . , T = 0 . in one dimension. The numerical data are com-pared with the analytically determined exponential decaywith exponent ( ξ ) − = 2 πT , see eq. (136).Sect. 4.3.2. The unscreened spin displays Friedel oscilla-tions around its limiting value that decay algebraically tozero, see eq. (132).In Fig. 15 we show the decaying part of the unscreenedspin, ∆ S ( R, T, V ) = |S ( R, T, V ) − S ∞ ( T, V ) | , as afunction of distance for J z = 4 V = 0 . for various smalltemperatures. It is seen that the correlation function de-cays to zero exponentially. The correlation length agrees Copyright line will be provided by the publisher with the analytically determined value ( ξ ) − = 2 πT fromeq. (136), as derived in Sect. 4.3.3. First, wedetermine the unscreened spin in the ground state for R →∞ . As shown in Appendix A.5, the Friedel sum rule [2]gives S ∞ ( V ) = lim R →∞ S ( R, T = 0 , V )= 14 + 12 ∆N ( T = 0 , V )= 14 − π arctan( V ) (131)because ∆N ( T, − V ) = − ∆N ( T, V ) , see eq. (209) inappendix A.5.3. For all finite interactions, the impurity spinis not completely screened, S ∞ ( T, V ) > , even at zerotemperature. In fact, for small interactions, V ≪ , wehave S ∞ ( V ≪ ≈ / − V / (2 π ) , i.e., the screening isvery small, of the order V .Next, we use eq. (125) to determine the approach of theunscreened spin to its limiting value for small V , S ( R, T = 0 , V ) −S ∞ ( V ) ≈ − ( − R πR V + O (cid:0) V /R (cid:1) . (132)The Friedel oscillations seen in the correlation functionalso show up in the unscreened spin. As dis-cussed in Appendix A.5, the Friedel sum rule is slightlymodified at finite temperatures ( β = 1 /T ), S ∞ ( T, V ) = lim R →∞ S ( R, T, V )= 14 − (cid:2) exp (cid:0) β √ V (cid:1) − exp( β ) (cid:3) / (cid:0) (cid:0) β √ V (cid:1)(cid:1) (1 + exp( β )) − Z β/ − β/ d x arctan " V p − (2 x/β ) × π cosh ( x ) . (133)At low temperatures, we find S ∞ ( T ≪ , V ) ≈ − π arctan( V ) − π V V T , (134)with corrections of the order V T . In one dimension, thedensity of states increases around the Fermi energy. There-fore, at finite temperatures, more electrons are available toscreen the impurity spin so that the screening becomes a lit-tle bit more effective for small but finite temperatures thanin the ground state. Note, however, that the corrections aresmall, of the order V T for small V and T .At finite temperature and small interactions V = J z / ≪ , as seen from Fig. 15, the amount of the un-screened spin decays exponentially to its limiting value S ∞ ( T, V ) as a function of distance r from the impurity, S ( R ≫ ξ , T, V ) = S ∞ ( T, V )+˜ s ( T, V )( − R e − R/ξ ( T ) (135)with an unspecified proportionality constant ˜ s ( T, V ) andthe correlation length ξ ( T ) = 12 πT (136)in one dimension. A detailed derivation is given in Ap-pendix B.2. In this work, we calculated and dis-cussed the thermodynamics and spin correlations in theexactly solvable Ising-Kondo model. In this problem, theimpurity spin orientation is dynamically conserved so thatthe partition function and thermal expectation values canbe expressed in terms of the single-particle density of statesof spinless fermions with a local scattering potential.As examples, we studied in detail the case of nearest-neighbor hopping on a chain and on a Bethe lattice withinfinite coordination number at half band-filling. The lat-ter condition considerably simplifies the analysis because itfixes the chemical potential to zero for all temperatures andinteraction strengths. We gave explicit equations for thefree energy, several thermodynamic potentials such as theinternal energy, entropy, and magnetization, and responsefunctions such as the specific heat and magnetic suscepti-bilities.The Ising-Kondo model provides an instructive exam-ple for static screening. At zero external magnetic field, theimpurity scattering is attractive for one spin species of thehost electrons and repulsive for the other. For an antifer-romagnetic Ising-Kondo coupling, host electrons with spinopposite to the impurity accumulate near the impurity siteand partially screen the localized spin.Since there is no dynamic coupling of the two im-purity spin orientations in the Ising-Kondo model, theground state remains doubly degenerate. This is seen in theimpurity-induced entropy that remains S spin ( T = 0) =ln(2) at zero temperature. Due to the thermally activatedscreening, the impurity-induced entropy is reduced fromits limiting value for all temperatures, as seen in Fig. 3.As a consequence, the impurity contribution to the specificheat is negative at low temperatures, see Fig. 4.The static screening also shows up for small externalmagnetic fields B . At low temperatures, T . B , the ther-modynamics of the Ising-Kondo model becomes identicalto that of a free spin in an effective magnetic field, seeFigs. 8 and 9, e.g., the impurity-induced zero-field suscep-tibility obeys a Curie law. The effective field B eff ( B, J z ) issmaller than the external field B due to the antiferromag-netic screening by the host electrons. Alternatively, onemay argue that the host electrons reduce the size of thelocal spin to S eff < / . This effective spin remains finiteas long as the Ising-Kondo interaction does not diverge. In Copyright line will be provided by the publisher ss header will be provided by the publisher 19 our work, we provide explicit results for the effective fieldas a function of temperature T , external magnetic field B ,and Ising-Kondo interaction J z .The incomplete static screening is also seen in the spincorrelation function. In the ground state, the spin correla-tion function displays an algebraic decay to zero with com-mensurate Friedel oscillations, see Fig. 12. At finite tem-peratures and in one dimension, the spin correlations decayexponentially with correlation length ξ ( T ) = 1 / (2 πT ) forweak interactions, see Fig. 13. For J z ≪ , the correla-tion length is independent of the Ising-Kondo interactionand identical to that for the non-interacting single-impurityAnderson model [11]. The amount of unscreened spin re-mains finite at zero temperature even at infinite distancefrom the impurity, see Fig. 14.The extensive analysis in our work provides tangibleresults for a non-trivial many-particle problem. The ex-plicit formulae may be used as benchmark tests for so-phisticated numerical methods that can be applied to gen-eral many-body problems such as the (anisotropic) Kondomodel; recall that the Ising-Kondo model is the limitingcase of the anisotropic Kondo model where the spin-flipterms are completely suppressed. Thus, the Ising-Kondomodel may also serve as a starting point for the analysisof the Kondo model for large anisotropy. However, whenpursuing the goal of an analytical approach to the Kondomodel, a systematic treatment of spin-flip excitations forthe description of dynamical screening in the Kondo modelcontinues to pose an intricate many-particle problem. Acknowledgements
Z.M.M. Mahmoud thanks the Fach-bereich Physik at the Philipps Universit¨at Marburg for its hospi-tality during his research stay. The authors extend their appreci-ation to the Deanship of Scientific Research at King Khalid Uni-versity for funding this work through research groups programunder grant number G.R.P-36-41.
A Spinless fermions
We treat spinless fermions thatinteract with a scattering potential at the lattice origin ˆ H ps = X i,j t i,j ˆ c + i ˆ c j + V ˆ c +0 ˆ c = X k ǫ ( k )ˆ a + k ˆ a k + VL X k,p ˆ a + k ˆ a p (137)for a L -site system with periodic boundary conditions. Thistextbook problem is addressed, e.g., in Ref. [20] for aquadratic dispersion relation.In the main text, we encounter the case where an exter-nal field of strength B couples to each fermion mode, ˆ H sf ( B ) = ˆ H ps − ˆ H ext , ˆ H ext = B X k ˆ a + k ˆ a k . (138)The external field acts like an external chemical potentialbecause it couples to the operator for the particle number ˆ N = X k ˆ a + k ˆ a k . (139) Therefore, we first consider ˆ H ps alone, and later absorb B in the chemical potential when we focus on ˆ H sf . A.1 Calculation of the Green function
We need tocalculate the retarded Green function G ret k,p ( t ) = ( − i) θ H ( t ) h (cid:2) ˆ a k ( t ) , ˆ a + p (cid:3) + i , (140)where ˆ A ( t ) = exp(i ˆ H ps t ) ˆ A exp( − i ˆ H ps t ) is the Heisen-berg operator assigned to the Schr¨odinger operator ˆ A ,and θ H ( x ) is the Heaviside step function. The discussionclosely follows Ref. [21]. A.1.1 Equation-of-motion method
The derivativeof the retarded Green function obeys i ˙ G ret k,p ( t ) = δ k,p δ ( t ) + ǫ ( k ) G ret k,p ( t ) + VL X p ′ G ret p ′ ,p ( t ) . (141)A Fourier transformation leads to the result ( η = 0 + ) ˜ G ret k,p ( ω ) = δ k,p + V H p ( ω ) ω − ǫ ( k ) + i η (142)with the abbreviation H p ( ω ) = 1 L X p ′ ˜ G ret p ′ ,p ( ω ) . (143)We insert eq. (142) into eq. (143) to find H p ( ω ) = 1 L X p ′ δ p ′ ,p + V H p ( ω ) ω − ǫ ( p ′ ) + i η = 1 L ω − ǫ ( p ) + i η + V g ( ω ) H p ( ω ) ,H p ( ω ) = 1 L − V g ( ω ) 1 ω − ǫ ( p ) + i η , (144)where g ( ω ) = 1 L X p ω − ǫ ( p ) + i η = Λ ( ω ) − i πρ ( ω ) (145)is the local Green function of the non-interacting hostfermions; eqs. (8) and (9) [eqs. (10) and (11)] give explicitexpressions for nearest-neighbor transfers on a chain [on aBethe lattice with infinite coordination number].Then, the solution of eq. (142) can be cast into the form ˜ G ret k,p ( ω ) = ˜ G ret , h k,p ( ω ) + ˜ G ret , i k,p ( ω ) , ˜ G ret , h k,p ( ω ) = δ k,p ω − ǫ ( k ) + i η , ˜ G ret , i k,p ( ω ) = V /L − V g ( ω ) 1 ω − ǫ ( k ) + i η ω − ǫ ( p ) + i η . (146)The host-electron part ˜ G ret , h k,p ( ω ) provides a bulk contribu-tion that is independent of V . Only the impurity-inducedpart ˜ G ret , i k,p ( ω ) of order unity is relevant for our further con-siderations. Copyright line will be provided by the publisher
A.1.2 Density of states
The impurity-induced con-tribution to the single-particle density of states is given by D ( ω ) = − π Im X k ˜ G ret k,k ( ω ) ! − Lρ ( ω )= − Im (cid:20) πL X k V − V g ( ω ) (cid:16) ω − ǫ ( k ) + i η (cid:17) (cid:21) = − π ∂∂ω Im [ln (1 − V g ( ω ))] . (147)For general g ( ω ) we note the useful relation D ( − ω, − V ) = D ( ω, V ) , (148)where we made explicit the V -dependence of the impurity-induced contribution to the density of states. Moreover, Z ∞−∞ d ωD ( ω, V ) = 0 (149)where we use eq. (147) and the fact g ( | ω | → ∞ ) = 0 . One-dimensional host-electron density of states
Let | ω | > . We obtain the (anti-)bound state from − V Λ ( ω b , ab ) = 0 . (150)For the one-dimensional case we thus find ω b , ab = ± p V . (151)There is a bound state at ω b = −√ V for V < and an anti-bound state at ω ab = √ V for V > . Tocalculate the contribution to the density of states from thebound states outside the band where we have ρ ( ω ) = η ≡ + , we expand R( ω ) ≡ − V Λ ( ω ) ≈ R ′ ( ω )( ω − ω ) (152)in the vicinity of ω ≡ ω b , ab . Then, D b , ab0 ( ω ) = − π ∂∂ω (cid:20) cot − (cid:18) R ′ ( ω )( ω − ω ) πV η (cid:19)(cid:21) = 1 π ˜ η ˜ η + ( ω − ω ) = δ ( ω − ω ) (153)with ˜ η = πV η/ R ′ ( ω ) → + . Thus, the bound and anti-bound states contribute D b , ab0 ( ω ) = δ ( ω − ω b ) θ H ( − V )+ δ ( ω − ω ab ) θ H ( V ) (154)to the impurity part of the density of states.For the band contribution we consider the region thatincludes the band edges, | ω | ≤ + . In general, we obtain D band0 ( ω ) = − π sgn( V ) ∂∂ω Cot − [ ϕ ( ω )] ,ϕ ( ω ) = 1 − V Λ ( ω ) π | V | ρ ( ω ) , (155) where Cot − ( x ) = πθ H ( − x ) + cot − ( x ) is continuousand differentiable across x = 0 , and sgn( x ) = x/ | x | is thesign function.In one dimension and for V > , the phase ϕ ( ω ) jumpsby π/ when going from ω = ( − − to ω = ( − + .The same jump appears at ω = 1 . For V < , we obtainthe same discontinuities. Inside the band we have instead Λ ( | ω | <
1) = 0 so that we find altogether ( ω p ( V ) ≡ ω p = √ V ) D ( ω ) = δ ( ω + ω p ) θ H ( − V ) + δ ( ω − ω p ) θ H ( V ) − δ ( ω + 1) − δ ( ω − (156) − θ H (1 − − | ω | ) 1 π ∂∂ω arctan (cid:20) V √ − ω (cid:21) . Semi-elliptic host-electron density of states
For the semi-elliptic density of states, see eqs. (10)and (11), there are no bound states for
V < / [10] and nosingularities at the band edges. Therefore, the semi-ellipticdensity of states displays only a band contribution, D se0 ( ω ) = − θ H (1 − | ω | ) 1 π ∂∂ω arctan (cid:20) V √ − ω − ωV (cid:21) . (157) A.2 Ground-state energy
When we calculate theground-state energy, we need to know the Fermi energy.At half band-filling, the interaction-induced changes in theFermi energy vanish in the thermodynamic limit and thusare irrelevant for the calculation of the interaction-inducedchange in the ground-state energy.
A.2.1 Fermi energy
The impurity Hamiltonian (137)is not particle-hole symmetric. Therefore, the Fermi energy ǫ F ( V ) depends on V . Since the scattering only appears ata single site, we have ǫ F ( V ) = ǫ (0)F + ǫ (1)F ( V ) L (158)to leading order in /L . Here, ǫ (0)F is determined from theparticle number, N = L Z ǫ (0)F −∞ d ωρ ( ω ) . (159)At half band-filling, N = L/ , and for a symmetric densityof states, ρ ( − ω ) = ρ ( ω ) , it is readily shown that the bulkFermi energy is zero, ǫ (0)F = 0 .We can calculate ǫ (1)F ( V ) from L Z ǫ (0)F + ǫ (1)F ( V ) /Lǫ (0)F d ωρ ( ω ) + Z ǫ (0)F −∞ d ωD ( ω, V ) , (160)so that ǫ (1)F ( V ) = − ρ ( ǫ (0)F ) Z ǫ (0)F −∞ d ωD ( ω, V ) (161) Copyright line will be provided by the publisher ss header will be provided by the publisher 21 in the thermodynamic limit.At half band-filling, we do not need the correction tocalculate the ground-state energy because ǫ (0)F = 0 and thebulk contribution to the energy is E bulk0 ( V ) = L Z −∞ d ωωρ ( ω ) + L Z ǫ (1)F /L d ωωρ ( ω )= E bulk0 ( V = 0) + Lρ (0) 12 ǫ (1)F L ! = E bulk0 ( V = 0) + O (1 /L ) . (162)Thus, we can calculate the scattering contribution to theground-state energy from the single-particle density ofstates as e ( V ) = E ( V ) − E bulk0 ( V = 0) = Z −∞ d ωωD ( ω, V ) . (163)In Sect. A.1.2 we provide explicit expressions for the im-purity-induced single-particle density of states D ( ω, V ) for nearest-neighbor electron transfer on a chain and ona Bethe lattice with infinite coordination number, seeSect. 2.1.1. A.2.2 One-dimensional density of states
There isno bound state for
V > and the ground-state energy canbe calculated from the band contribution alone, e ( V >
0) = 12 − π (cid:20) ω arctan ( πV ρ ( ω )) (cid:21) − + + 1 π Z − d ω arctan [ πV ρ ( ω )]= 1 π Z − d ω arctan [ πV ρ ( ω )]= 12 (cid:16) V − p V (cid:17) . (164)For the last step we rely on M ATHEMATICA [19].For attractive interactions,
V < , we can investigatethe particle-hole transformed Hamiltonian, τ +ph ˆ H ps ( V )ˆ τ ph = ˆ H ps ( − V ) + V . (165)At half filling, this implies for the scattering contributionto the ground-state energy e ( V ) = V + e ( − V ) . (166)Thus, we find ( V < e ( V ) = V + 12 (cid:16) − V − p V (cid:17) = 12 (cid:16) V − p V (cid:17) . (167)Eq. (167) is formally identical to eq. (164). Alternatively, we can calculate e ( V ) for V < fromthe density of states. We include the bound state and find e ( V <
0) = − p V + 12+ 1 π (cid:20) ω arctan ( π | V | ρ ( ω )) (cid:21) − + − π Z − + d ω arctan [ π | V | ρ ( ω )]= 12 (cid:16) V − p V (cid:17) , (168)which is identical to eq. (167), and e ( V ) = 12 (cid:16) V − p V (cid:17) (169)holds for all V . A.2.3 Semi-elliptic density of states
For the den-sity of states in eq. (10) and < V < / there are no(anti-)bound states [10]. Eq. (163) gives e se0 ( V >
0) = − π Z − d ω ω ∂∂ω arctan " V √ − ω − V ω = 1 π Z − d ω arctan " V √ − ω − V ω = 12 π + V − V πV arctan (cid:20) V − V (cid:21) (170)after a partial integration. In the last step, we used M ATH - EMATICA [19] to carry out the integration. For − / We con-sider the case of potential scattering only. Before we cancalculate the free energy, we must determine the chemicalpotential µ ( N, T, V ) . A.3.1 Chemical potential For finite temperatures T ,eq. (158) generalizes to µ ( N, T, V ) = µ (0) ( N, T ) + µ (1) ( N, T, V ) L (171)to leading order in /L . By definition, µ (0) ( N, T ) is thechemical potential for non-interacting spinless fermions attemperature T with average particle number N , N = L Z ∞−∞ d ω ρ ( ω )1 + exp[ β ( ω − µ (0) ( N, T )] , (172)where β = 1 /T . When we consider the particle number N as a function of µ (0) ( T ) , we can use particle-hole symme- Copyright line will be provided by the publisher try, ρ ( ω ) = ρ ( − ω ) , to write N ( µ (0) ( T )) = L Z ∞−∞ d ω ρ ( ω )1 + exp[ β ( − ω − µ (0) ( T )]= L Z ∞−∞ d ω ρ ( ω ) exp[ β ( ω + µ (0) ( T )]1 + exp[ β ( ω + µ (0) ( T )]= L − N ( − µ (0) ( T )) . (173)which implies µ (0) ( L − N, T ) = − µ (0) ( N, T ) , (174)i.e., when µ (0) ( T ) fixes the average particle number to N ,the chemical potential − µ (0) ( T ) leads to the average par-ticle number to L − N . Thus, for half band-filling, a zerochemical potential µ (0) ( T ) = 0 , (175)implies half band-filling, N = L/ , for all temperatures.In the thermodynamic limit, we can calculate the cor-rection µ (1) ( N, T, V ) in eq. (171) from − L Z ∞−∞ d ω ρ ( ω + µ (0) ( N, T ))1 + exp( βω )+ L Z ∞−∞ d ω ρ ( ω + µ (0) ( N, T ))1 + exp[ β ( ω − µ (1) ( N, T, V ) /L )]+ Z ∞−∞ d ω D ( ω + µ (0) ( N, T ))1 + exp( βω ) , (176)where we used D ( ω ) = Lρ ( ω ) + D ( ω ) so that µ (1) ( N, T, V ) = − A ( N, T, V ) A ( N, T ) ,A ( N, T, V ) = Z ∞−∞ d ω D ( ω + µ (0) ( N, T ) , V )1 + exp( βω ) ,A ( N, T ) = Z ∞−∞ d ω ρ ( ω + µ (0) ( N, T )) β exp( βω )[1 + exp( βω )] (177)in the thermodynamic limit. Note that, for T → , we re-cover µ (1) ( T = 0 , N, V ) = ǫ (1)F ( V ) from eq. (161).With eq. (148) it is readily shown that A ( L − N, T, − V ) = Z ∞−∞ d ω D ( ω + µ (0) ( N, T ) , V )1 + exp( − βω )= Z ∞−∞ d ωD ( ω + µ (0) ( N, T, V )) − Z ∞−∞ d ω D ( ω + µ (0) ( N, T ) , V )1 + exp( βω )= − A ( N, T, V ) , (178)where we used eqs. (149) and (174). Likewise we find A ( L − N, T ) = A ( N, T ) . (179)Thus, µ (1) ( L − N, T, − V ) = − µ (1) ( N, T, V ) . (180) A.3.2 Free energy For non-interacting fermions withsingle-particle density of states D ( ω ) , the free energy canbe written as [16,17] F = − T Z ∞−∞ d ω ln (1 + exp[ − β ( ω − µ ]) D ( ω ) , (181)where F ≡ F ( N, T ) , µ ≡ µ ( N, T ) for notational sim-plicity. For the spinless fermion model in eq. (137) weuse D ( ω, V ) = Lρ ( ω ) + D ( ω, V ) to write ( µ (0) ≡ µ (0) ( N, T ) , µ (1) ≡ µ (1) ( N, T, V ) ) F ps = − T L Z ∞−∞ d ω ln h e − β ( ω − µ (1) /L ) i ρ ( ω + µ (0) ) − T Z ∞−∞ d ω ln (cid:0) e − βω (cid:1) D ( ω + µ (0) , V )= F (0)ps − µ (1) Z ∞−∞ d ω ρ ( ω + µ (0) )1 + exp( βω ) (182) − T Z ∞−∞ d ω ln (cid:0) e − βω (cid:1) D ( ω + µ (0) , V ) , where F (0)ps ≡ F ps ( N, T ) = F ps ( N, T, V = 0) is the freeenergy for free spinless fermions, F (0)ps = − T Z ∞−∞ d ω ln (cid:16) − β ( ω − µ (0) ] (cid:17) ρ ( ω ) . (183)Using the definition of µ (0) in eq. (172), we readily find F ps ( N, T, V ) = F (0)ps ( N, T ) − µ (1) ( N, T, V ) NL + F ips ( N, T, V ) ,F ips ( N, T, V ) = − T Z ∞−∞ d ω ln (cid:0) e − βω (cid:1) (184) × D ( ω + µ (0) ( N, T ) , V ) . A.4 Free energy (external field) In the following,we consider ˆ H sf , see eq. (138), where the spinless fermionsencounter an external field. The external field B can beabsorbed in the chemical potential, i.e., we simply haveto replace µ (0) ( T ) by µ (0) ( T ) + B in all formulae of thepreceding section A.3. A.4.1 Half band-filling We focus on a half-filled sys-tem at B = 0 , i.e., we set µ (0) ( T ) = 0 . Thus, for finite B we have N ≡ N ( B ) = L Z ∞−∞ d ω ρ ( ω )1 + exp[ β ( ω − B )] (185)for the particle number. Note that we choose B smallenough to not completely fill or empty the system. Notethat N ( − B ) + N ( B ) = L (186)which expresses the half-filling condition at B = 0 . Copyright line will be provided by the publisher ss header will be provided by the publisher 23 We proceed analogously to Sect. A.3.1 and find ¯ µ (1) ( B, T, V ) = − ¯ A ( B, T, V )¯ A ( B, T ) , ¯ A ( B, T, V ) = Z ∞−∞ d ω D ( ω + B, V )1 + exp( βω ) , ¯ A ( B, T ) = Z ∞−∞ d ω ρ ( ω + B ) β exp( βω )[1 + exp( βω )] . (187)In analogy to eq. (180) we have ¯ µ (1) ( − B, T, − V ) = − ¯ µ (1) ( B, T, V ) (188)for the impurity-induced correction to the chemical po-tential at half band-filling in the presence of an externalfield B .For the free energy of a half-filled system in the pres-ence of an external field we find F sf ( B, T, V ) = F (0)sf ( B, T ) − ¯ µ (1) ( B, T, V ) N ( B ) L + F isf ( B, T, V ) ,F isf ( B, T, V ) = − T Z ∞−∞ d ω ln (cid:0) e − βω (cid:1) × D ( ω + B, V ) (189)with N ( B ) from eq. (185) and F (0)sf ( B, T ) = − T Z ∞−∞ d ω ln (1 + exp( − βω )) ρ ( ω + B ) (190)for non-interacting spinless fermions at half band-filling inan external field. A.4.2 Incomplete free energy In the main text, weencounter the incomplete partition function ¯ Z sf = Tr e − β ˆ H sf (191)that lacks the chemical potential term µ (1) in the partitionfunction for spinless fermions at half band-filling in thepresence of an external field, Z sf = Tr e − β ( ˆ H sf − µ (1) ˆ N/L ) , (192)where ˆ N is the particle-number operator, see eq. (139).We add the chemical potential term in eq. (191), ¯ Z sf = e − βµ (1) N/L Tr e − β ( ˆ H sf − µ (1) ˆ N/L + µ (1) ( ˆ N − N ) /L ) , (193)where N is the average particle number from eq. (185).Since particle-number fluctuations are small, we may ex-pand ¯ Z sf ≈ Z sf e − βµ (1) N/L (cid:18) − β µ (1) L h ˆ N − N i sf + β [ µ (1) ] L h ( ˆ N − N ) i sf (cid:19) , (194) where h ˆ A sf i sf = 1 Z sf Tr (cid:16) e − β ( ˆ H sf − µ (1) ˆ N/L ) ˆ A sf (cid:17) (195)is the thermal expectation value of an operator ˆ A sf for themodel of spinless fermions, see eq. (117). By construction, h ˆ N − N i sf = 0 . Moreover, L h ( ˆ N − N ) i sf = O ( N ) L = O (1 /L ) (196)so that the second-order term and all higher-order termsin the expansion in eq. (194) vanish in the thermodynamiclimit. Thus, ¯ F sf = − T ln ¯ Z sf = µ (1) NL + F sf . (197)Together with eq. (189) we find ¯ F sf ( B, T, V ) = F (0)sf ( B, T ) + F isf ( B, T, V ) , (198)as used in the main text.Eq. (198) shows that the chemical potential µ (1) isirrelevant for the calculation of the effective free energy ¯ F sf ( B, T, V ) . This can readily be understood from the factthat, in the grand canonical ensemble, the particle num-ber is only fixed on average, with fluctuations of the order / √ N , see eq. (196). Thus, the small changes in the parti-cle number induced by the interaction on a single site canbe ignored from the beginning by putting µ (1) /L ≡ . A.5 Local density For the calculation of the screen-ing cloud, we need the impurity-induced change in the lo-cal density, N ( r, T, V ) = h ˆ c + r ˆ c r i V − h ˆ c + r ˆ c r i V =0 . (199)After a Fourier transformation and using the retardedsingle-particle Green function, this single-particle expec-tation value can be expressed as [16] N ( r, T, V ) = 1 L X k,p e i( k − p ) r Z ∞−∞ d ωf ( ω, T ) D ( k, p ; ω ) (200)with the Fermi function f ( ω, T ) = 11 + exp(( ω − µ ) /T ) (201)and the impurity-induced contribution to the single-particlespectral function D ( k, p ; ω ) = − π Im (cid:16) ˜ G ret , i k,p ( ω ) (cid:17) . (202)For the impurity-induced part of the Green function, seeeq. (146). Copyright line will be provided by the publisher Using inversion symmetry we perform the sum over k and p and arrive at N ( r, T, V ) = Z ∞−∞ d ωf ( ω, T ) (cid:20) − π Im (cid:18) V Q r ( ω )1 − V g ( ω ) (cid:19)(cid:21) (203)where Q r ( ω ) = Z π − π d k π e i kr ω + cos( k ) + i η (204)with ǫ ( k ) = − cos( k ) when W = 2 is the bandwidth.With the help of M ATHEMATICA [19], the integrals can becarried out analytically, Q r ( ω > 1) = (cid:0) ω + √ ω − (cid:1) −| r | √ ω − ,Q r ( ω < − 1) = − (cid:0) ω − √ ω − (cid:1) −| r | √ ω − ,Q r ( | ω | < 1) = (i)i r cos( pr ) + i sin( p | r | )cos( p ) , (205)where ω = sin( p ) for | ω | < [21].We split the frequency integral in eq. (203) into thepole contribution for | ω | > and the band contributionfor | ω | < , and discuss them separately. A.5.1 Pole contribution In eq. (203), the poles at ω b = − ω p for V < and at ω ab = ω p for V > with ω p = √ V contribute ( r ≥ ) N p0 ( r, T, V ) = − V √ V θ ( − V ) f ( − ω p , T ) Q r ( − ω p )+ V √ V θ ( V ) f ( ω p , T ) Q r ( ω p )= V √ V (cid:16) | V | + p V (cid:17) − r × [ θ ( V ) f ( ω p , T ) − θ ( − V ) f ( − ω p , T )] , (206)where we used eq. (205) and − π Im (cid:18) V − V Λ ( ω ) + i V η (cid:19) = 1 | Λ ′ ( ω ) | δ ( ω − ω ) , | Λ ′ ( ω ) | = V sgn( V ) √ V (207)with ω = ± ω p for the bound and anti-bound states.Eq. (206) shows that the pole contribution decays expo-nentially as a function of distance, with exponent /ξ = − K ) , K = | V | + √ V , where ξ is the correla-tion length for the pole contribution. A.5.2 Band contribution We substitute ω = sin( p ) to find the band contribution for | ω | < as ( r ≥ ) N b0 ( r, T, V ) = ( − r Vπ Z π/ − π/ d p f (sin( p ) , T )cos ( p ) × sin(2 pr ) − V cos(2 pr ) πρ (sin( p ))1 + [ V πρ (sin( p ))] = ( − r Vπ Z π/ − π/ d pf (sin( p ) , T ) × sin(2 pr ) cos( p ) − V cos(2 pr ) V + cos ( p ) . (208)In general, the integral can only be evaluated numerically. A.5.3 Sum rule Lastly, we calculate the shift in theparticle number due to the impurity scattering, ∆N ( T, V ) = X r N ( r, T, V ) = Z ∞−∞ d ωf ( ω, T ) D ( ω ) , (209)where we used eq. (200) and D ( ω ) = P k D ( k, k ; ω ) ,see eq. (147). We note in passing that ∆N ( T, − V ) = − ∆N ( T, V ) . This is readily shown using eqs. (148) andeq. (149)At zero temperature, we recover the Friedel sum rulewhich states that the shift in particle number is determinedby the scattering phase shifts at the Fermi energy [2], ∆N ( T = 0 , V ) = Z −∞ d ωD ( ω )= − π Im [ln (1 − V g (0))]= − π arctan ( πV ρ (0)) (210)because g ( ±∞ ) = 0 and Λ (0) = 0 from particle-holesymmetry.For T > and in one dimension, we use the density ofstates (156) and find after a partial integration ( β = 1 /T ) ∆N ( T, V ) = sgn( V ) (cid:2) exp( β ) − exp (cid:0) β √ V (cid:1)(cid:3)(cid:0) (cid:0) β √ V (cid:1)(cid:1) (1 + exp( β )) − Z β/ − β/ d x arctan " V p − (2 x/β ) × π cosh ( x ) . (211)The first term is exponentially small for small tempera-tures. The denominator in the integrand guarantees thatonly values | x | . noticeably contribute to the integral.Consequently, for small temperatures, we may expand thesquare root and perform the integrals over the real axis, ∆N ( T, V ) ≈ − arctan( V ) π − π V V T . (212)Corrections are of the order V T . Copyright line will be provided by the publisher ss header will be provided by the publisher 25 B Extracting correlation lengths Physical quanti-ties often display an exponential decay as a function oftime or distance. We discuss how exponents can be ex-tracted from data or intricate analytic dependencies. B.1 Analytic considerations We start with somebasic analytic considerations. We apply them to the case ofthe Ising-Kondo model in appendix B.2. B.1.1 Constant and exponential dependency Weassume that some quantity decays exponentially to a con-stant value as a function of time, f ( t ) = c + c e − t/τ , (213)and values f i = f ( t i ) are measured at some time t i . Thedecay time τ is of interest. Since the measuring time islimited, and the constant c is unknown or of no interest, itis advisable to fix a time interval ∆ and to consider F ∆ ( t ) = f ( t + ∆ ) − f ( t ) = c (cid:16) e − ∆/τ − (cid:17) e − t/τ . (214)Apparently, the constant c drops out of the problem, andthe slope of the data for ln[ F ∆ ( t i )] versus t i gives ( − /τ ) . B.1.2 Constant and two exponentials Let us nowconsider the case where a correlation function decays withtwo exponentials, f ( x ) = c + c e − x/ξ + c e − x/ξ . (215)We introduce two shifts ∆ and ∆ to write f ( x + ∆ ) − f ( x ) = c (cid:16) e − ∆ /ξ − (cid:17) e − x/ξ + c (cid:16) e − ∆ /ξ − (cid:17) e − x/ξ ,f ( x + ∆ ) − f ( x ) = c (cid:16) e − ∆ /ξ − (cid:17) e − x/ξ + c (cid:16) e − ∆ /ξ − (cid:17) e − x/ξ . (216)We assume that we know the exponent ξ . Then, F ∆ ,∆ ( x ) = (cid:16) − e − ∆ /ξ (cid:17) ( f ( x + ∆ ) − f ( x )) − (cid:16) − e − ∆ /ξ (cid:17) ( f ( x + ∆ ) − f ( x ))= e C e − x/ξ , e C = c (cid:16) − e − ∆ /ξ (cid:17) (cid:16) − e − ∆ /ξ (cid:17) − c (cid:16) − e − ∆ /ξ (cid:17) (cid:16) − e − ∆ /ξ (cid:17) . (217)The slope of ln[ F ∆ ,∆ ( x )] versus x gives ( − /ξ ) . B.2 Application to the screening cloud We nowcalculate the correlation length for the screening cloud. B.2.1 Analytic expressions In the main text, weshowed that ( V = J z / > ) S ( R, T, V ) = const + s p R ( T, V ) + s b R ( T, V ) (218) with s p R ( T, V ) = − tanh (cid:16) ω p T (cid:17) ω p K (cid:16) − e − R ln( K ) (cid:17) ,s b R ( T, V ) = − V π Z π/ d k cos[(2 R + 1) k ]sin ( k ) + V tanh h cos( k )2 T i (219)with ω p = √ V and K = V + √ V . Apparently,we have /ξ = 2 ln( K ) for the exponential decay of thepole contribution s p R ( T, V ) .Since we showed numerically that S R ( T, V ) decays tozero with the screening length ξ , we can conclude that theband contribution s b R ( T, V ) asymptotically behaves like s b R ≫ ( T, V ) ∼ c + c e − R/ξ + c e − R/ξ . (220)It displays the structure that we analyzed in appendix B.1. B.2.2 Identifying the exponent In eq. (217) we set x ≡ R , ∆ = − , and ∆ = 1 , /ξ = 2 ln( K ) and f ( R ) = − V π Z π/ d k cos[(2 R + 1) k ]sin ( k ) + V tanh h cos( k )2 T i . (221)Thus, we find F − , ( R ) = (1 − e − /ξ ) ( f ( R − − f ( R )) − (1 − e /ξ ) ( f ( R + 1) − f ( R )) . (222)Moreover, we are interested in the limit of small couplings, V ≪ , so that we use /ξ ≈ V , − exp( ± /ξ ) ≈∓ V so that we find ( F − , ≡ F − , ( R ) ) F − , ≈ V [ f ( R + 1) + f ( R − − f ( R )]= 8 V Z π/ d k π tanh h cos( k )2 T i cos[(2 R + 1) k ] × sin ( k )sin ( k ) + V ≈ V Z π/ d k π tanh h cos( k )2 T i cos[(2 R + 1) k ] , (223)neglecting terms formally of the order V in the last step.After a substitution we arrive at F − , ( R ) ≈ V ( − R d R h R,T , (224) h R,T = Z d R d u sin( u ) tanh h sin( uπ/ (2 d R ))2 T i , where d R = ( π/ R + 1) . We split the integral and use d R ≫ for R ≫ to approximate h R,T ≈ Z ∞ d u sin( u ) (cid:20) tanh h sin( uπ/ (2 d R ))2 T i − (cid:21) = 2 d R T sinh(2 d R T ) ≈ d R T e − d R T , (225) Copyright line will be provided by the publisher where we used M ATHEMATICA [19] in the next-to-last stepand d R ≫ again in the last step. Altogether we have ineq. (224) F − , ( R ) ≈ V T ( − R e − πT R (226)for R ≫ . Using eq. (217) we can read off the exponent ξ = 12 πT , (227)as claimed in the main text. Note that we also reproducethe numerically observed oscillating convergence. B.3 Application to the correlation function In thislast section, we calculate the correlation length for the spincorrelation function. B.3.1 Analytic expressions In the main text, weshowed that ( V = J z / > ) the band contribution to thespin correlation function reads C S, b dc ( r ) = − ( − r V π Z π/ − π/ d p tanh (cid:20) sin( p )2 T (cid:21) × sin(2 pr ) cos( p ) V + cos ( p ) . (228)As for the screening cloud, we have /ξ = 2 ln( K ) for theexponential decay of the pole contribution C S, p dc ( r ) . More-over, the band part goes to zero for large distances, C S, b dc ( r ≫ 1) = ˜ c e − r/ξ + ˜ c e − R/ξ . (229)It displays the structure that we analyzed in appendix B.1. B.3.2 Identifying the exponent In eq. (217) we set x ≡ R , ∆ = − , and ∆ = 1 , /ξ = 2 ln( K ) and f ( R ) = Z π/ d p tanh h sin( p )2 T i sin(2 rp ) cos( p )cos ( p ) + V . (230)As in the previous section B.2 we find in the limit of smallinteractions ( F − , ≡ F − , ( r ) ) F − , = − V Z π/ d p tanh h sin( p )2 T i sin(2 pr ) cos( p ) × cos ( p )cos ( p ) + V ≈ − V Z π/ d p tanh h sin( p )2 T i sin(2 pr ) cos( p ) , (231)neglecting terms formally of the order V in the last step.After a substitution we arrive at F − , ( r ) ≈ − V " r r − h r,T r , (232) ˜ h r,T = Z ˜ d r d u sin( u ) " tanh h sin( uπ/ (2 ˜ d r ))2 T i − with ˜ d r = πr . Here, we approximated cos( p ) ≈ in theintegrand in eq. (231) because the dominant contribution tothe integral results from the region p ≪ . We use ˜ d r ≫ for r ≫ to extend the integration limit to infinity so that ˜ h r,T ≈ πrT sinh(2 πrT ) − , (233)where we used M ATHEMATICA [19] to evaluate the inte-gral. Altogether we have from eq. (228) C S, b dc ( r ) ≈ ( − r e − πT r (234)for r ≫ . Using eq. (217) we can read off the exponent ξ = 12 πT , (235)as for the screening cloud. This result is not surprisingbecause the sum over an exponentially decaying functiongives an exponentially decaying function with the same ex-ponent. References [1] J. Kondo, Progress in Theoretical Physics , 37 (1964).[2] J. S´olyom, Fundamentals of the Physics of Solids (Sprin-ger, Berlin, 2009), Vols. 1-3.[3] A. Hewson, TheKondoProblemtoHeavyFermions(Cam-bridge University Press, Cambridge, 1993).[4] I. V. Borzenets, J. Shim, J. C. H. Chen, A. Ludwig, A. D.Wieck, S.Tarucha, H. Sim, and M. Yamamoto, Nature ,210 (2020).[5] C. Zener, Phys. Rev. , 440 (1951).[6] A. Tsvelick and P. Wiegmann, Advances in Physics , 453(1983).[7] N. Andrei, K. Furuya, and J. H. Lowenstein, Rev. Mod.Phys. , 331 (1983).[8] K. G. Wilson, Rev. Mod. Phys. , 773 (1975).[9] R. Bulla, T. Costi, and T. Pruschke, Rev. Mod. Phys. ,395 (2008).[10] Z. Mahmoud and F. Gebhard, Ann. Phys. (Berlin) , 794(2015).[11] Z. M. M. Mahmoud and F. Gebhard, physica status solidi(b) , 1800670 (2019).[12] Z. M. M. Mahmoud, J. B¨unemann, and F. Gebhard, physicastatus solidi (b) , 1600842 (2017).[13] J. R. Schrieffer and P. A. Wolff, Phys. Rev. , 491 (1966).[14] M. H¨ock and J. Schnack, Phys. Rev. B , 184408 (2013).[15] G. Barcza, K. Bauerbach, F. Eickhoff, F. B. Anders, F. Geb-hard, and ¨O. Legeza, Phys. Rev. B , 075132 (2020).[16] A. L. Fetter and J. D. Walecka, QuantumTheoryofMany-ParticleSystems (McGraw-Hill, Boston, 1971).[17] G. D. Mahan, Many Particle Physics (Plenum, New York,2000).[18] N. Ashcroft and D. Mermin, Solid State Physics (Holt,Rinehart and Winston, Philadelphia, 1976).[19] Wolfram Research, Inc., M ATHEMATICA , ver. 11 (Wolf-ram Research, Inc., Champaign, IL, 2016). Copyright line will be provided by the publisher ss header will be provided by the publisher 27 [20] S. Doniach and E. H. Sondheimer, Green’s Functions forSolid State Physicists, 3rd edition (Addison-Wesley, Red-wood City, 1982).[21] G. Barcza, F. Gebhard, T. Linneweber, and ¨O. Legeza,Phys. Rev. B , 165130 (2019)., 165130 (2019).