Kondo physics in the algebraic spin liquid

# Kondo physics in the algebraic spin liquid

## Abstract

We study Kondo physics in the algebraic spin liquid, recently proposed to describe [Phys. Rev. Lett. 98, 117205 (2007)]. Although spin dynamics of the algebraic spin liquid is described by massless Dirac fermions, this problem differs from the Pseudogap Kondo model, because the bulk physics in the algebraic spin liquid is governed by an interacting fixed point where well-defined quasiparticle excitations are not allowed. Considering an effective bulk model characterized by an anomalous critical exponent, we derive an effective impurity action in the slave-boson context. Performing the large- analysis with a spin index , we find an impurity quantum phase transition from a decoupled local-moment state to a Kondo-screened phase. We evaluate the impurity spin susceptibility and specific heat coefficient at zero temperature, and find that such responses follow power-law dependencies due to the anomalous exponent of the algebraic spin liquid. Our main finding is that the Wilson’s ratio for the magnetic impurity depends strongly on the critical exponent in the zero temperature limit. We propose that the Wilson’s ratio for the magnetic impurity may be one possible probe to reveal criticality of the bulk system.

###### pacs:
71.10.-w, 71.10.Hf, 71.27.+a

## I Introduction

Recent experiments have claimed the emergence of spin liquid (SL) phases in materials of geometrically frustrated lattices such as ,Cs_2CuCl_4 () ,kappa-(ET)_2Cu_2(CN)_3 () and ,ZnCu_3(OH)_6Cl_2 (), where no symmetries associated with spin rotations (magnetic ordering) and lattice translations (valance bond ordering) are broken at low temperatures while charge fluctuations are frozen due to strong electron-electron interactions (Mott insulator). An important issue is the nature of such SL phases. Although spin susceptibility, specific heat, and thermal transport measurements can determine possible spin liquids, there still remains uncertainty.

Consider with an anisotropic triangular lattice.Cs_2CuCl_4 () Although this material exhibits magnetic long-range spiral ordering below with an incommensurate wave vector, the spin-fluctuation spectrum in inelastic neutron scattering experiments has shown large high-energy continuum beyond the spin-wave description. In addition, this continuum spectrum survives above the Neel temperature. More detailed analysis revealed that the continuum follows with an anomalous exponent , suggesting the presence of deconfined critical spinons. Such spin-fluctuation measurements suggest several candidates of SL scenarios, for example, decoupled one-dimensional chains,Tsvelik () proximate gapped SLs,YBKim () algebraic spin liquid (ASL),Wen () algebraic vortex liquid,Fisher () and so on.Senthil ()

Recently, Florens et al. studied the role of magnetic impurities in both the Z SL phase and the O(4) QCP separating the spiral magnetic order from the Z SL.Florens_SL_KE () Although impurity moments coupled to spin- bosons (spin singlet-triplet excitations) in conventional paramagnets are only partially screened even at the bulk O(3) QCP,Sachdev_Vojta () they have shown that the presence of deconfined bosonic spinons can display a bosonic version of the Kondo effect. Furthermore, they found a weak-coupling impurity quantum phase transition (I-QPT) from a local-moment state to a fully-screened phase. This study implies that the magnetic impurity can be utilized as a probe for elementary excitations, thus identifying the nature of SLs.

In this paper we investigate the Kondo effect in the ASL, recently proposed to be realized in the Kagome antiferromagnet ,PALee_ASL () where no magnetic order is observed down to very low temperature compared to the Curie-Weiss temperature , and there is no sign of spin gap in dynamical neutron scattering.ZnCu_3(OH)_6Cl_2 () However, it is not perfectly clear whether all experiments are consistent with the ASL conjecture. The ASL picture is not consistent with the temperature-linear specific heat below and saturation of the spin susceptibility to a finite value below ,PALee_ASL () because these measurements indicate the existence of a finite density of states at the Fermi energy. This discrepancy may result from the presence of disorder in real materials. To examine the role of magnetic impurities in the ASL can be an important test in revealing the genuine nature of the SL phase of this compound.

The ASL can be found from the fermion representation of the Heisenberg model via the flux mean-field ansats.Marston_flux () This reminds us of the previous study of the Kondo effect in the flux phase by Cassanello and Fradkin.Fradkin_KE_Flux () More generally, one may regard the present impurity problem as the class of the Pseudogap Kondo model,Fradkin_KE_Flux (); Vojta_PG_KM (); Ingersent_NRG () where the fermion density of states vanishes as near the Fermi energy. The case of corresponds to Fermi liquid while the case coincides with Dirac fermions arising from the flux phase or superconductor. In contrast with the Kondo effect of the Fermi liquid, the Pseudogap Kondo model has shown that Kondo screening of the magnetic impurity can appear beyond some critical value of the Kondo coupling constant. Thus, the I-QPT from a local-moment state to a Kondo-screened phase was found in this model. Furthermore, the exponent in the density of states was shown to play the role of an effective dimension in the problem. The case was found to be its upper critical dimension, thus exhibiting logarithmic corrections to scaling while the case of lies in its lower critical one.

However, there is an important difference between the Pseudogap Kondo problem and ASL Kondo physics. The bulk physics in the Pseudogap Kondo problem is governed by a noninteracting (Gaussian) fixed point, thus allowing well-defined electron-like quasiparticle excitations. On the other hand, the ASL physics is determined by an interacting fixed point (the conformal invariant fixed point of QED),ASL_fixed_point () where well-defined spinon quasiparticle excitations corresponding to electrons do not exist. The absence of quasiparticle excitations prohibits us from applying the conventional picture of the Pseudogap Kondo physics to the ASL Kondo problem. In this respect the Kondo effect at such an interacting fixed point is an interesting problem.

The main difficulty is how to introduce the absence of well-defined spinon excitations in the ASL Kondo problem. Long-range gauge interactions would result in the anomalous critical exponent in the single spinon propagator, destroying the quasiparticle pole in the Green’s function. Unfortunately, such critical physics can be found within the summation of infinite diagrams of gauge interactions, and this procedure prohibits us from analyzing the ASL Kondo problem in a simple mean-field way such as the large- approximation with a spin index , well utilized in the Pseudogap Kondo problem.Fradkin_KE_Flux () Considering the mathematical derivation in the large- context, the main problem is how to derive an effective impurity action from the ASL-Kondo Lagrangian through integrating out bulk degrees of freedom, critical spinon and gauge fluctuations coupled to the magnetic impurity. More precisely speaking, a bulk-spinon propagator appears to govern the impurity dynamics in the effective impurity action, thus how to write its accurate form is an important problem since the presence of gauge interactions makes such a task nontrivial.

In the present paper we assume the expression of the spinon Green’s function as an ansatz, introducing an anomalous critical exponent . In the text we discuss the validity of this ansatz in great detail. This effective representation allows us to analyze the ASL Kondo problem in the large- context. Performing the slave-boson saddle-point analysis for the effective impurity action, we find an I-QPT from a decoupled local-moment state to a Kondo-screened phase. We evaluate the impurity spin susceptibility and specific heat coefficient at zero temperature, and find that such responses follow power-law dependencies due to the ASL anomalous exponent. The main finding of the present study is that the Wilson’s ratio for the magnetic impurity depends strongly on the ASL critical exponent in the zero temperature limit. We propose that the Wilson’s ratio for the magnetic impurity be a probe to reveal criticality of the bulk system.

## Ii Review of the algebraic spin liquid and its Kondo problem

For completeness of this paper, it is necessary to review how the effective Lagrangian so called QED describing the ASL is derived from a microscopic model such as the antiferromagnetic Heisenberg model, with . Inserting the fermion representation of spin into the Heisenberg model, and performing the Hubbard-Stratonovich transformation for an exchange channel, we find an effective one-body Hamiltonian for fermionic spinons () coupled to a hopping parameter (), . Notice that the hopping parameter is a complex number defined on links . Thus, it can be decomposed into , where and are the amplitude and phase of the hopping parameter, respectively. Inserting this representation of into the effective Hamiltonian, we obtain , where the constant contribution for the ground state energy is omitted. Then, we can see that this effective Hamiltonian has an internal U(1) gauge symmetry, under the following U(1) phase transformation, and . This implies that the phase of the hopping parameter plays the same role as the U(1) gauge field .

One can perform a saddle-point analysis of the effective Hamiltonian to find its stable mean-field phases in various lattices such as square,Marston_flux () triangular,Wen () Kagome,PALee_ASL (); Hastings () and etc. In the present paper we consider the square lattice for simplicity, where the antiferromagnetic long-range order can be suppressed via next-nearest-neighbor or ring-exchange interactions causing frustration. It is not so difficult to extend the mean-field analysis on the square lattice into that on the Kagome lattice, proposed to show the SL physics of .PALee_ASL ()

It has been shown that one possible stable mean field phase is a -flux state, where a spinon gains the phase of when it turns around one plaquette. The amplitude of the hopping parameter is frozen to be in the low energy limit. Then, one finds the low-energy effective Lagrangian in terms of massless Dirac fermions interacting via compact U(1) gauge fieldsDonKim_QED ()

 Z=∫DψnσDaμe−∫d3xL, L=∑σ=↑,↓2∑n=1¯ψnσγμ(∂μ−iaμ)ψnσ+12e2|∂×a|2. (1)

Here, is the two-component massless Dirac spinon, where represent the nodal points of and , and , SU(2) spin. They are expressed as and , respectively. In the spinon field represent the nodal points, , even and odd sites, and , its spin, respectively. The Dirac matrices are given by the Pauli matrices , satisfying the Clifford algebra . is the U(1) gauge field whose kinetic energy results from particle-hole excitations of high energy spinons. is an effective internal charge, not a real electric charge.

It has been argued that QED has an infrared stable fixed point showing the conformal symmetry in the large- limit ().ASL_fixed_point () This conformal invariant fixed point is identified with the ASL, displaying algebraically decaying correlation functions with anomalous critical exponents. To confirm the ASL as a genuine stable phase a cautious person may ask the stability of such an interacting fixed point against perturbations. Four-fermion interaction terms are irrelevant at this fixed point owing to their high scaling dimensions. In addition, chiral symmetry breaking due to noncompact gauge fluctuations has been shown not to occur in the Schwinger-Dyson-equation analysis when the flavor number of massless Dirac fermions is sufficiently large.CSB () Furthermore, it has been argued that confinement as an instanton effect arising from compact gauge fluctuations does not seem to appear in the large- limit because the scaling dimension of the monopole insertion operator is proportional to the flavor number , thus expected to be irrelevant in the large- ASL.ASL_fixed_point ()

Criticality of the ASL is characterized by critical exponents of correlation functions. The single particle propagator can be expressed as

 GASL(k)≈−iγμkμk2−ηψ, (2)

where is an anomalous critical exponent. One can find such an anomalous dimension in the large- analysis.Exponent_Large_N () However, the critical exponent obtained in this way is difficult to have a definite physical meaning because it is not gauge invariant. In this respect the critical exponent should be evaluated in a gauge invariant way. The following gauge invariant Green’s function can be considered, . Unfortunately, it is not easy to calculate the critical exponent with such a gauge invariant expression. Its precise value is far from consensus and still under current debates. The crucial point is the sign of the exponent while its absolute value is given by in the approximation.Exponent_Large_N (). Most evaluationsWen_ARPES (); Ye_propagator (); Khveshchenko_propagator (); QED_eta () suggest its negative sign, . However, as argued in Ref. Tesanovic_QED (), its negative sign seems to be unphysical in the sense that the spinon propagator becomes more ”coherent” at long distances than the propagator of the free Dirac theory. This result is in contrast with the usual role of interactions, making elementary excitations less coherent. This is indeed true in such critical field theories with local repulsive interactions, for example, the -vector model, where positive critical exponents are well known.N_Vector_Model () If the critical exponent is positive, long-range gauge interactions destabilize the quasiparticle pole. The quasiparticle weight with momentum vanishes in the long-wave length and low-energy limits. In the present paper we do not determine its sign. Instead, we regard the exponent as a phenomenological parameter. Thus, we consider both cases of and . Furthermore, we assume that the renormalized spinon propagator [Eq. (2)] is obtained in a gauge invariant way,Wen_ARPES (); Tesanovic_QED (); Ye_propagator (); Khveshchenko_propagator (); QED_eta () and the critical exponent is also gauge invariant.

Another important character of the ASL is that the conformally invariant fixed point has an enlarged global symmetry beyond the original lattice model, here the Heisenberg Hamiltonian. Such an emergent symmetry corresponds to Sp(4) in the case of SU(2) gauge interactionsWen_Symmetry () and SU(4) in the case of U(1) onesHermele_Symmetry (). This enlarged symmetry gives rise to an important effect on correlation functions, that is, resulting in the same behaviors between different correlation functions when the operators in the correlators are related with symmetry transformations. For example, staggered spin correlations have the same functional dependency (power-law decay) as the valance bond fluctuations since they are symmetry-equivalent. An interesting point is that such correlations are most susceptible in the ASL.Hermele_Symmetry () This implies that the ASL resides near the antiferromagnetic and valance bond solid phases. Actually, Tanaka and Hu have derived an effective Wess-Zumino-Witten (WZW) Lagrangian from the ASL, describing competition between antiferromagnetic spin correlations and valance bond fluctuations.Tanaka ()

To study the role of magnetic impurities in the ASL bulk, we consider the Kondo coupling term, , where is a spin-fluctuation operator of bulk spinons with momentum and represents an impurity spin. The bulk-spin operator has two contributions in the continuum,

 →S(q)≈→Su(q)+→Ss(q)=∑k∑nσσ′¯ψnσ(k−q)γ0→τσσ′2ψnσ′(k)+∑k∑nσσ′¯ψnσ(k−q)→τσσ′2ψnσ′(k), (3)

where represents the uniform component and denotes the staggered one.DonKim_QED () Then, the ASL Kondo problem is described by the following action

 S=∫dτ{∫d2r(∑σ=↑,↓2∑n=1¯ψnσγμ(∂μ−iaμ)ψnσ+12e2|∂×a|2)+JK2∑q(→Su(q)+→Ss(q))⋅→s}. (4)

The next work is to obtain an effective impurity action, integrating out bulk degrees of freedom, spinon and gauge excitations coupled to the magnetic impurity. One can write down its schematic expression in the following way

 Simp≈−J2K4∫dτdτ′sa(τ)(∑q⟨Sau(q,τ)Sbu(−q,τ′)⟩+∑q⟨Sas(q,τ)Sbs(−q,τ′)⟩)sb(τ′)+⋯, (5)

where is the renormalized correlation function of staggered (uniform) spin fluctuations, and are higher moment contributions. As clearly shown in this expression, dynamics of impurity spin fluctuations is governed by spin correlations of the bulk at the impurity site. An important point is that only staggered spin correlations exhibit an anomalous scaling behavior with a nontrivial critical exponent.AFL_spin_correlation (); Wen_spin_correlation () Uniform spin correlations have no anomalous scaling dimension since they correspond to conserved currents.DonKim_QED (); AFL_spin_correlation () Correlations of conserved currents do not have any anomalous scaling dimensions. This means that the contribution of uniform spin fluctuations is basically the same as the Kondo effect of the Pseudogap Kondo model while that of staggered spin excitations will give rise to new effects on the Pseudogap Kondo physics. Furthermore, staggered spin fluctuations are most singular in the large- ASL,Hermele_Symmetry () thus expected to contribute to the Kondo effect dominantly. In this respect we take into account staggered spin fluctuations only, which is an important assumption in the present paper.

## Iii Kondo physics in the algebraic spin liquid: large-Nσ analysis

Our objective is to construct a mean-field theory for the present Kondo problem. Using the slave-boson representation, the impurity spin is expressed as , and such fermions satisfy the constraint with , where is spin. Inserting this expression into Eq. (4) with Eq. (3), the Kondo coupling term becomes

 HK=−JK2Nσ∑qkNn∑n=1Nσ∑σσ′=1¯ψnσ(k−q)χσχ†σ′γ0ψnσ′(q)−JK2Nσ∑qkNn∑n=1Nσ∑σσ′=1¯ψnσ(k−q)χσχ†σ′ψnσ′(q) (6)

in the large- treatment, where the first and second terms are associated with uniform and staggered spin-fluctuation contributions, respectively. Since staggered spin fluctuations will give main contributions to the ASL Kondo effect, effects of uniform spin fluctuations are neglected in the following.

Performing the Hubbard-Stratonovich transformation for the Kondo-exchange channel, we find an effective ASL-Kondo action as our starting point

 Seff=∫d3x[(Nσ∑σ=1Nn∑n=1¯ψnσγμ(∂μ−iaμ)ψnσ+12e2|∂×a|2)−Nσ∑σ=1Nn∑n=1(bs†nχ†σψnσ(0)+¯ψnσ(0)γ0χσbsn)] +∫dτ[Nσ∑σ=1χ†σ(∂τ−hσ)χσ+iλ(Nσ∑σ=1χ†σχσ−Qχ)+Nσ2JKNn∑n=1bs†nγ0bsn]. (7)

The first part represents the ASL bulk. The second part arises from the Hubbard-Stratonovich decoupling of the Kondo interaction term, where is a two-component hybridization order parameter associated with staggered bulk-spin fluctuations. Such a hybridization order parameter is determined self-consistently in the saddle-point analysis

 Nσ2JKγ0bsn=⟨∫d2k(2π)2Nσ∑σ=1χ†σψnσ(k)⟩. (8)

The third part describes impurity-spinon dynamics, where is an external magnetic field and is a Lagrange multiplier field to impose the pseudo-fermion constraint.

When the bulk system is in the non-interacting fixed point corresponding to the absence of gauge interactions, the effective Kondo model becomes the multi-channel Pseudogap Kondo model, where the channels come from Dirac nodes . This model was argued to show an I-QPT from a decoupled local-moment state to an over-screened phase in the large- approximation although this analysis does not capture the over-screened Kondo physics quite well.Fradkin_KE_Flux () On the other hand, the present bulk system lies at the interacting fixed point characterized by the anomalous critical exponent , where quasiparticle excitations do not exist. In this case it is not clear whether the conventional Kondo screening picture is applicable.

Integrating out bulk-spinon and gauge excitations, we obtain an effective impurity action in energy-momentum space

 Simpeff=∫dk02π[Nσ∑σ=1χ†σ(ik0−hσ+ϵχ)χσ−Nσ∑σ=1Nn∑n=1bs†nχ†σ(∫d2k(2π)2⟨ψnσ(k)¯ψnσ(k)⟩)γ0χσbsn +Nσ2JKNn∑n=1bs†nγ0bsn−ϵχQχ], (9)

where is replaced with to clarify its physical meaning. The main question in this impurity action is how to evaluate the spinon Green’s function. As discussed intensively in the previous section, the single particle propagator has an anomalous scaling exponent, given by . This expression seems to be consistent with Eq. (5) if ”sub-leading” uniform spin-correlation contributions are not taken into account. This is because the critical exponent of the staggered spin-spin correlation function is found to be twice the exponent of the single particle propagator, i.e., in the case of .Wen_spin_correlation (); AFL_spin_correlation () Such correspondence occurs when both critical exponents are calculated in a gauge invariant way. This correspondence was also pointed out in Ref. QED_eta, .

Such spinon excitations with an anomalous scaling exponent result in anomalous energy-dependent (nonlocal in time) interactions for impurity fermions, as reflected in the kernel of . The vector function is obtained to be

 Fμ(k0)=∫d2k(2π)2kμ(k20+k2)1−ηψ/2=k0(k20+Λ2)ηψ/2−|k0|ηψ2πηψδμ0, (10)

thus with , where is a momentum cutoff.

Inserting Eq. (10) into Eq. (9) and integrating over impurity fermions in Eq. (9), we find the following expression for the impurity free energy

 Fimp=−∫∞−∞dk02πNσ∑σ=1ln[(ik0−hσ+ϵχ)+ik0F(k0)Nn∑n=1bs†nbsn]+Nσ2JKNn∑n=1bs†nγ0bsn−ϵχQχ. (11)

Expressing the hybridization order parameter as a two-component spinor , one can find in the saddle-point analysis. Representing the above impurity free energy with , we obtain

 Fimp=−Nσ2∫∞−∞dk02π[ln(ik0−H+ϵχ+ik02Nnπηψ|b|2{(k20+Λ2)ηψ/2−|k0|ηψ}) +ln(ik0+H+ϵχ+ik02Nnπηψ|b|2{(k20+Λ2)ηψ/2−|k0|ηψ})]+2NσNnJK|b|2−ϵχQχ. (12)

Minimizing the impurity free energy with respect to and , we find the saddle-point equations giving the self-consistency

 b(1JK−∫∞0dk02πk202πηψ{(k20+Λ2)ηψ/2−kηψ0}[1+2Nnπηψ|b|2{(k20+Λ2)ηψ/2−kηψ0}]k20[1+2Nnπηψ|b|2{(k20+Λ2)ηψ/2−kηψ0}]2+ϵ2χ)=0, QχNσ=−∫∞0dk02π2ϵχk20[1+2Nnπηψ|b|2{(k20+Λ2)ηψ/2−kηψ0}]2+ϵ2χ. (13)

Since the impurity free energy is momentum-cutoff-dependent, it is necessary to make it cutoff-independent, taking appropriate scaling transformations for all variables. Considering the scaling dimension of given by , one can find and , where represents the scaling dimension of an operator . Then, the scale-free impurity free energy is obtained to be

 fimp≡FimpΛ=−Nσ4π∫∞−∞dx[ln(ix−h+ϵr+ix2Nnπηψ|br|2{(x2+1)ηψ/2−|x|ηψ}) +ln(ix+h+ϵr+ix2Nnπηψ|br|2{(x2+1)ηψ/2−|x|ηψ})]+2NσNnJr|br|2−ϵrQχ, (14)

where such rescaled variables are given by

 br=bΛ−ηψ/2,     Jr=JKΛ−(1+ηψ),     ϵr=ϵχΛ,     x=k0Λ,     h=HΛ.

Notice that these scaled variables are dimensionless. Accordingly, the self-consistent saddle-point equations read

 QχNσ=−1π∫∞0dxϵrx2[1+2Nnπηψ|br|2{(x2+1)ηψ/2−xηψ}]2+ϵ2r. (15)

The QCP of the I-QPT can be found with and in the particle-hole symmetric case, . Then, the critical renormalized Kondo coupling constant is obtained from Eq. (15),

 (16)

as far as the ASL exponent lies in . In the large limit the ASL exponent may also satisfy this condition as discussed in Section II. In addition, this critical value is continuously defined in the limit of , where the impurity critical pointPGKM () is given by

 1Jrc=12π2∫∞0dxln(1+1x2)=12π, (17)

consistent with the previous study.Fradkin_KE_Flux ()

It is interesting to notice that the I-QPT occurs as long as the ASL exponent . Remember that in the regime of critical spinon excitations are less coherent than those in the Pseudogap Kondo model () while in the regime of such spinon excitations become more coherent than quasiparticle excitations in the Fermi liquid with pseudogap. To screen the magnetic impurity, stronger Kondo couplings would be required when quasiparticle excitations are less coherent. Actually, we find such an asymmetric behavior for the ASL exponent in Fig. 1, obtained from Eq. (16).

It might seem mysterious that the critical Kondo coupling vanishes as . As the ASL exponent approaches , critical spinon excitations are not only less coherent but also localized. Considering the spinon propagator Eq. (2), makes it energy-momentum-independent. Such localized spinons are expected to form a Kondo singlet with an impurity spin immediately. When the ASL exponent goes to , it is important that the bare scaling dimension of the Kondo coupling () vanishes, implying that Kondo interactions are marginal perturbations similar to the conventional Kondo effect in the Fermi liquid. In this respect the critical Kondo coupling would go to zero as .

Solving Eq. (15) numerically, one can find the hybridization amplitude as a function of the Kondo coupling . We show the I-QPT in Fig. 2, where both and vanish as . It is important to notice that the -axis is instead of . This means that the impurity QCP matches the origin of the -axis. The absolute value of the impurity chemical potential increases rapidly as the ASL exponent increases from to [Fig. 2(a)]. Accordingly, the increasing ratio of the hybridization order parameter is largest for and smallest for . This may be associated with localization tendency emerging from a positive exponent. A further analysis finds a scaling behavior of the hybridization amplitude not only near the impurity QCP, but also rather away from the QCP, i.e., in the Kondo-screened phase. Such a scaling behavior even in the Kondo phase seems to arise from the criticality of the bulk system. From the log-log plot of Fig. 2(b), we find the scaling relation

 |br|2∼(Jr−Jrc)f(ηψ) (18)

with , confirming that the slope of the positive ASL exponent is larger than that of the negative one.

The I-QPT can be also found in the impurity-spin susceptibility,

 χimp=−∂2fimp(h)∂h2=−Nσπ∫∞0dxϵ2r−x2[1+2Nnπηψ|br|2{(x2+1)ηψ/2−xηψ}]2(x2[1+2Nnπηψ|br|2{(x2+1)ηψ/2−xηψ}]2+ϵ2r)2. (19)

In the decoupled phase () the impurity susceptibility diverges in the zero temperature limit (following the Curie law) while it vanishes in the screened phase. Since for the hybridization amplitude is smallest, the impurity-spin susceptibility becomes largest. Approaching the impurity QCP (), it shows a power-law divergence with an anomalous critical exponent of the ASL bulk. As shown in Fig. 3, such curves are well fitted with

 χimp∼(Jr−Jrc)−g(ηψ), (20)

where the scaling function is . It is valuable to consider how the behavior of the impurity susceptibility differs from that of the Pseudogap Kondo modelPGKM_Susceptibility () which corresponds to the case of .

Next, we evaluate the impurity specific heat. The zero temperature formulation [Eq. (14)] for the impurity free energy can be transformed to the finite temperature version through the Wick rotation. Following Refs. Fradkin_KE_Flux (); Doniach (), we find the impurity free energy at finite temperatures,

 fimp=Nσ∫∞−∞dξπ1eβrξ+1Θ(ξ)+2NσNnJr|br|2−ϵrQχ

with a rescaled temperature , where the ”angle” function is given by

 Θ(ξ) =tan−1(2Nnπηψ|br|2sin(πηψ2)|ξ|1+ηψξ[1+2Nnπηψ|br|2{(−ξ2+1)ηψ/2−cos(πηψ2)|ξ|ηψ}]+ϵr)+π2(1−sign(ξ))   % for  |ξ|<1, (21) =tan−1(2Nnπηψ|br|2sin(πηψ2)ξ{(ξ2−1)ηψ/2−|ξ|ηψ}ξ[1+2Nnπηψ|br|2cos(πηψ2){(ξ2−1)ηψ/2−|ξ|ηψ}]+ϵr)+π2(1−sign(ξ))    for  |ξ|≥1.

Here, the denominator and numerator in the angle function correspond to the real and imaginary parts of the kernel for the impurity free energy [Eq. (14)], respectively.

We find the impurity entropy

 (22)

and specific heat coefficient

 γimp=CimpTr=∂Simp∂Tr=NσT2r∫∞−∞dξπξ2∂∂ξ(1eβrξ+1)∂Θ(ξ)∂ξ. (23)

Taking the zero temperature limit, we find the self-consistent results in Fig. 4, using the solutions of Eq. (15). The latter terms in Eq. (21) ensure that the impurity entropy is in the decoupled phase, consistent with our expectation. In the Kondo phase the impurity entropy becomes vanished. But, we note that more elaborate calculations result in small nonzero entropy contributions in the Kondo phase.NCA (); Entropy () The coefficient shows a behavior similar to the impurity susceptibility , diverging as . It exhibits the scaling behavior ,

 γimp∼(Jr−Jrc)−h(ηψ) (24)

with in our numerical analysis.

Using the impurity susceptibility and specific heat coefficient, one can find the Wilson’s ratio in the zero temperature limit

 (25)

In Fig. 5 we plot this value as a function of the rescaled Kondo coupling . Remember in the Kondo effect of the Fermi liquid. Here, we also obtain a similar value for the Pseudogap Kondo model (). An important observation is that the Wilson’s ratio is strongly dependent on the ASL exponent. For the negative exponent the Wilson’s ratio becomes enhanced while it gets suppressed for the positive one. This implies that the Wilson’s ratio can be utilized as a probe for revealing the nature of SLs, more generally, criticality of the bulk system. This is the main message of the present study.

## Iv Summary and discussion

It is valuable to remind several assumptions for solving the ASL Kondo problem. First of all, we have considered effects of staggered spin fluctuations on dynamics of a magnetic impurity, ignoring those of uniform spin correlations, since antiferromagnetic spin fluctuations are most susceptible in the ASL bulk, thus expected to give dominant contributions on this problem. In addition, ferromagnetic spin correlations do not show anomalous scaling, implying that such contributions would coincide with the Pseudogap Kondo effect, thus not so interesting. Our second assumption is in writing a spinon Green’s function, where effects of gauge fluctuations are introduced in an anomalous critical exponent. In the present paper we have used the scaling exponent as a phenomenological parameter. Both assumptions are compatible since the critical exponent of the staggered spin-spin correlation function is consistent with that of the single particle propagator when both critical exponents are evaluated in a gauge invariant way.

The third one is rather an approximation than an assumption for solving the effective impurity action while the above two are basic assumptions for deriving the impurity action. In the slave-boson representation of this effective impurity action we have performed the large- analysis introducing the hybridization order parameter. Although well-defined quasiparticle excitations do not exist in the case of , it was shown that the I-QPT occurs between the local-moment state and the Kondo-screened phase. Evaluating the impurity spin susceptibility and specific heat coefficient, we found that the Wilson’s ratio depends strongly on the ASL exponent. This result has an important physical meaning because the Wilson’s ratio for the magnetic impurity reflects criticality of the bulk system. This conclusion will be available to general critical systems with exact screening particulary, where the expression of the Kondo vertex is the same as that of the present paper.Criticality_Kondo () In this respect the Wilson’s ratio for the magnetic impurity may be one possible probe for measuring bulk criticality.

It is interesting to compare the ASL Kondo physics with the Kondo effect in the Luttinger liquid,Furusaki (); MunDaeKim () since the ASL can be considered as the high dimensional realization of the Luttinger liquid. In the Luttinger liquid the Kondo interaction term can be decomposed to the forward and backward scattering channels, analogous to the uniform and staggered ones in the ASL. It was shown that the forward scattering channel is irrelevant in the renormalization group analysis up to two-loop order.MunDaeKim () Similarly, in this paper we take into account only the antiferromagnetic correlation channel for the ASL Kondo effect, although it is not proven that the ferromagnetic channel is irrelevant. The backward impurity scattering in the Luttinger liquid was shown to cause anomalous scaling, in particular, power-law behavior of the Kondo temperature owing to the presence of the anomalous critical exponent in the Luttinger liquid.MunDaeKim () This is basically the same as the ASL-Kondo effect that the ASL criticality results in anomalous scaling on the impurity physics, although there is no phase transition in the Luttinger liquid owing to one dimensionality.

It should be noted that the present mean-field analysis is difficult to describe correct scaling behaviors in the over-screened phase since the hybridization order parameters are not regarded as dynamic variables but static ones. This approximation scheme seems to be more appropriate when quasiparticle excitations are well defined, thus the conventional Kondo screening picture is applicable. The slave-boson mean-field scheme can be improved using the non-crossing approximation,NCA () where such hybridization parameters are taken to be dynamic variables, thus quantum fluctuations are more involved. Performing the Hubbard-Stratonovich transformation for the nonlocal (for time) hopping term in Eq. (9), we find an effective impurity action

 SNCA=∫dk02π[Nσ∑σ=1χ†σ(ik0−hσ+ϵχ)χσ+Nσ2JKNn∑n=1bs†nγ0bsn−ϵχQχ] +∫dk02π[∫dk′02πΣχ(k0)Σb(k′0)i(k0−k′0)F(k0−k′0)−Nσ∑σ=1χ†σ(k0)Σb(k0)χσ(k0)+Nn∑n=1bs†n(k0)Σχ(−k0)bsn(k0)],

where and are fermion and boson self-energies, respectively, determined by the following self-consistent NCA-type equations

 Σχ(τ′−τ)=F(τ−τ′)⟨Nσ∑σ=1χ†σ(τ)χσ(τ′)⟩, Σb(τ−τ′)=−F(τ−τ′)⟨Nn∑n=1bs†n(τ)bsn(τ′)⟩

with . This kind of approximation is well known to catch non-Fermi liquid physics in the multi-channel Kondo model.NCA () Scaling behaviors of both bosonic and fermionic self-energies are expected in the low energy limit, causing anomalous critical physics to this system even in the case of . Inserting expected scaling forms for both the self-energies and renormalized Green’s functions to the NCA equations, we would obtain the total anomalous scaling exponents which are expected to be sum of the critical exponents of the multichannel Pseudogap Kondo model and the ASL scaling dimension approximately, considering the presence of the ASL scaling exponent in . It will be interesting to examine how the scaling exponents in the conventional bulk are affected by the presence of the ASL exponent.

Applying magnetic fields to the ASL, the impurity QPT is expected to disappear. Because external magnetic fields would result in finite density of states at the Fermi energy, the conventional Kondo physics may appear, where only the over-screened Kondo phase would occur, independent of the Kondo interaction. Furthermore, gauge fluctuations would be dissipative due to the finite density of states, and the bulk system becomes more ”Fermi liquid”-like, supporting the above expectation.

In the present analysis we did not consider scattering due to randomly distributed disorder potentials. One of the present authors has studied the role of random potentials in the ASL, and found that such a spin liquid phase remains stable against weak disorders because massless Dirac spinons at the interacting fixed point live in higher spatial dimensions than two owing to the presence of the anomalous critical exponent.Kim_Disorder () Remember the presence of the delocalization transition above two spatial dimensions. However, it is not clear whether the diffusive nature appears or not in the ASL. If so, the presence of finite density of states due to random potential scattering may destroy the I-QPT as the case of magnetic fields.

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