Analytical calculation of the finite-size crossover spectrum of the anisotropic two-channel Kondo model

Analytical calculation of the finite-size crossover spectrum of the anisotropic two-channel Kondo model

Gergely Zara´nd

Institute of Physics, Technical University of Budapest, H-1521 Budafoki u´t 8, Budapest, Hungary and International School for Advanced Studies, I-34014 Trieste, Italy

Jan von Delft

Institut fu¨r Theoretische Festko¨rperphysik, Universita¨t Karlsruhe, D-76128 Karlsruhe, Germany 共Received 10 December 1998; revised manuscript received 23 September 1999兲

We present a conceptually simple, analytical calculation of the finite-size crossover spectrum of the aniso- tropic two-channel Kondo共2CK兲model at its Toulouse point. We use Emery and Kivelson’s method, gener- alized in two ways. First, we construct all boson fields and Klein factors explicitly in terms of the model’s original fermion operators and, secondly, we clarify explicitly how the Klein factors needed when refermion- izing act on the original Fock space. This enables us to follow the evolution of the 2CK model’s free-fermion states to its exact eigenstates for arbitrary magnetic fields and spin-flip coupling strengths. We thus obtain an analytic description of the crossover of the finite-size spectrum to the non-Fermi-liquid fixed point, where we recover the conformal field theory results共implying a direct proof of Affleck and Ludwig’s fusion hypothesis兲. From the finite-size spectrum we extract the operator content of the 2CK fixed point and the dimension of various relevant and irrelevant perturbations. Our method can easily be generalized to include various symmetry-breaking perturbations, and to study the crossover to other fixed points produced by these. Further- more, it establishes instructive connections between different renormalization group schemes. We also apply our method to the single-channel Kondo model.

I. INTRODUCTION

One of the most intriguing aspects of a non-Fermi liquid 共NFL兲 is that its elementary excitations are not simply re- lated to the bare excitations of the non-interacting Fermi liq- uid; gaining an understanding of the nature of the elementary excitations of a NFL is thus an important conceptual chal- lenge. The two-channel Kondo 共2CK兲 model, introduced in 1980 by Nozie`res and Blandin,¹ is one of the simplest and most-studied quantum impurity models with NFL behavior, and offers the rare opportunity to address this question di- rectly: it has both a free and a NFL fixed point, and the crossover between the two, including the change in the na- ture of the elementary excitations, can be analyzed exactly using the bosonization approach of Emery and Kivelson² 共EK兲.

In the 2CK model two channels of spinful conduction electrons interact with a single spin 1/2 impurity via a local antiferromagnetic exchange interaction. In contrast to the single-channel Kondo 共1CK兲 model, which has a stable infinite-coupling fixed point at which the conduction elec- trons screen the impurity spin completely, in the two-channel case the impurity spin is overscreened at infinite coupling, and the 2CK model’s infinite-coupling fixed point is un- stable. A stable NFL fixed point exists at intermediate cou- pling, and is characterized by a nonzero residual entropy and nonanalytical behavior for various physical quantities. The relevance of this model to physical systems is extensively reviewed in Ref. 3.

In this paper, we use EK’s method to perform a concep- tually simple, analytic calculation of the finite-size crossover spectrum of the 2CK model between the free and the NFL

fixed points, a result first reported in Ref. 4. The calculation enables us to elucidate the nature of the NFL excitations at the fixed point in great and instructive detail, and to see explicitly how the symmetries of the NFL fixed point emerge as it is approached from the crossover region. Furthermore it establishes instructive connections between various popular renormalization group 共RG兲 schemes, since it allows one to analytically illustrate their main ideas.

The two-channel Kondo model has of course already been studied theoretically by an impressive number of different methods, which are comprehensively reviewed in Ref. 3.

They include approximate methods such as multiplicative^1,5,6 and path-integral^7,8 RG approaches and slave-boson methods;^9–11effective models such as the so-called compac- tified model,^12–15 which is partially equivalent to the 2CK model; the numerical RG 共NRG兲;^16–18 and exact methods, such as the Bethe ansatz,^19–21 conformal field theory 共CFT兲,^18,22–24and Abelian bosonization.^2,4,25–30

Among the several exact approaches to solving the 2CK model, the one that in our opinion is the most simple and straightforward, is that introduced by Emery and Kivelson 共EK兲,² who employ one-dimensional Abelian bosonization 共pedagogically reviewed in Ref. 31兲and refermionization to show that along the so-called Emery-Kivelson line共Toulouse point兲 the anisotropic 2CK model maps onto a quadratic resonant-level model. Since spin anisotropy is irrelevant for the 2CK Kondo model¹⁸ 共as also shown below兲, their work also yielded new insight into the generic behavior of the isotropic 2CK model.

Though the approach is constrained to the vicinity of the EK line, the latter is stable³² and connects the Fermi-liquid and non-Fermi-liquid regimes, so that EK’s method captures PRB 61

0163-1829/2000/61共10兲/6918共16兲/$15.00 6918 ©2000 The American Physical Society

both the model’s NFL behavior and the crossover from the free to the NFL fixed point. EK calculated a number of ther- modynamic and impurity properties and also some electron correlation functions, and explained the NFL behavior by the observation that only ‘‘one half’’ of the impurity’s Majorana degrees of freedom couples to the electrons. Although at the EK line the properties of the model are somewhat specific since the leading irrelevant operator vanishes along it, the generic behavior can easily be derived by perturbation theory in its vicinity. The EK method has since been fruitfully ap- plied and generalized to several related quantum impurity problems.^25–29Ye in particular showed how to use the EK method and simple scaling arguments²⁷to identify easily the fixed points of various bosonizable quantum impurity mod- els, including the k-channel spin anisotropic Kondo model,^27共a兲 and how to calculate electronic correlation func- tions at these fixed points.

In a recent publication,⁴we have shown that the power of EK-bosonization can actually be increased even more 关see points共i兲–共vi兲and共x兲–共xii兲below兴by generalizing it to fi- nite system size L. Though retaining terms of order 1/L natu- rally requires some additional technical effort, none of the conceptual simplicity of the EK approach is thereby lost. The present paper is devoted to presenting the calculations by which the results of Ref. 4 were obtained in explicit detail, and includes discussions of a number of subtleties and results not mentioned there.

The generalization to finite system size necessitates two important modifications relative to the work of EK. 共1兲 While they use the field-theoretical approach to bosonization in which the bosonization relation ␺␣j⯝F_␣je^⫺i␾␣j is used merely as a formal correspondence, we use the more careful constructive bosonization procedure of Haldane,^33,31 where both the boson fields ␾␣j and Klein factors F_␣j are con- structed explicitly from the original␺_␣joperators, so that the bosonization formula becomes an operator identity in Fock space.共2兲Since EK were interested mainly in impurity prop- erties, they did not need to discuss at all the Klein factors F_␣j 关which lower the number of ␣j electrons by one and ensure proper anticommutation relations for the ␺␣j’s兴. However, as has been pointed out by several authors recently,28,31,34,35

these Klein may be extremely important in some situations, and they are essential for quantities like the finite-size spectrum or various electron correlation functions.^33,4 Therefore it is crucial to specify how the new Klein factors of the refermionized operators act on the Fock space. As we shall see, these new Klein factors are only well defined on a suitably enlarged Fock space that also contains unphysical states, which must be discarded at the end using certain gluing conditions.

With these modifications, EK’s bosonization approach en- ables us by straightforward diagonalization of the quadratic resonant-level model 共i兲 to analytically calculate the cross- over of the 2CK model’s finite-size spectrum from the FL to the NFL fixed point, at which we reproduce the fixed-point spectrum previously found by CFT using a certain fusion hypothesis 共which we thereby prove directly兲; 共ii兲 to con- struct the eigenstates of the 2CK model corresponding to this crossover spectrum explicitly, thereby elucidating the nature of the NFL excitations; and共iii兲to extract the operator con- tent of the NFL fixed point and determine the dimensions of

different relevant and irrelevant operators. We also prove that the leading irrelevant operator is missing along the EK line but is present away from it. Since our method works also in the presence of an arbitrary magnetic field 共unlike CFT兲, we can also 共iv兲 investigate how a finite magnetic field de- stroys the NFL spectrum for the low-energy excitations of the model and restores the FL properties. 共v兲 Furthermore, our finite-size bosonization approach can easily be related to various popular RG methods; it therefore not only provides a useful bridge between them, but can potentially be used as a pedagogical tool for analytically illustrating their main ideas.

共vi兲For completeness, we also construct the analytical finite size spectrum of the single channel Kondo model, and cal- culate the crossover between its weak and strong coupling Fermi liquid fixed points.

In a future publication³⁶we shall show that EK’s method furthermore allows one共vii兲to construct very easily the scat- tering states of the model; 共viii兲 to verify explicitly the va- lidity of the bosonic description of the NFL fixed point worked out in Refs. 30 and 27; 共ix兲to determine the fixed point boundary conditions at the impurity site for the differ- ent currents and fields in a very straightforward way, 共x兲as well as the leading corrections to these;共xi兲to calculate all correlation functions at and around the NFL fixed point; and 共xii兲 to clarify the role of the dynamics of Klein factors in correlation functions. 关Although共vii兲to共ix兲can also be ob- tained in a system of infinite size, 共x兲 to 共xii兲 turn out to depend crucially on the finite-size results of the present pa- per.兴 This implies that all CFT results can be checked from first principles using bosonization.

The paper is organized as follows. In Sec. II we define the 2CK model to be studied. For completeness, and since the proper use of Klein factors is essential, Sec. III briefly re- views the ‘‘constructive’’共operator identity-based兲approach to finite-size bosonization used throughout this paper. The Emery-Kivelson mapping onto a resonant-level model is dis- cussed in Sec. IV, using our novel, more explicit formulation of refermionization within a suitably extended Fock space.

The solution of the resonant level model and the construction of the NFL spectrum using generalized gluing conditions is presented in Sec. V. In Sec. VI the results of our finite-size calculations are compared with and interpreted in terms of various RG procedures. In Sec. VII we show the finite-size spectrum for the 1CK model. Finally, in Sec. VIII we sum- marize our conclusions.

The centerpiece of the main text is our uncommonly care- ful and detailed finite-size formulation of the EK mapping.

Technicalities not related to this mapping are relegated to four Appendixes 共see Ref. 37兲. Appendix A discusses in some detail matters related to the choice of an ultraviolet cutoff, and also gives the often-used position-space defini- tion of the 2CK model, to facilitate comparison with our momentum-space version. The construction of the extended Fock space needed for refermionization is discussed in Ap- pendix B, and the technical details used to diagonalize the resonant-level model and to calculate several of its properties are given in Appendix C. Finally, in Appendix D we present our finite-size bosonization calculation for the one-channel Kondo model as well.

II. DEFINITION OF THE MODEL

Throughout the main part of this paper we shall use the standard 2CK Hamiltonian in momentum space. We con- sider a magnetic impurity with spin 1/2 placed at the origin of a sphere of radius R⫽L/2, filled with two species of free, spinful conduction electrons, labeled by a spin index ␣

⫽(↑,↓)⫽(⫹,⫺) and a channel or flavor index j⫽(1,2)⫽ (⫹,⫺). We assume that the interaction between the impurity and the conduction electron is sufficiently short-ranged that it involves only s-wave conduction electrons, whose kinetic energy can be written as

H₀⫽

兺

k␣j

k:c_k^†_␣jc_k␣j: 共vF⫽ប⫽1兲. 共1兲 The operator c_k^†_␣j creates an s-wave conduction electron of species (␣j ) with radial momentum k⬅p⫺p_Frelative to the Fermi momentum p_F, and the dispersion has been linearized around the Fermi energy ␧F: ␧k⫺␧F⬇k. The colons in Eq.

共1兲denote normal ordering with respect to the free Fermi sea or ‘‘vacuum state’’兩0ជ典0:

c_k␣j兩0ជ典^{0⬅0 for k⬎0, 共2a兲} c_k^†_␣j兩0ជ典^{0⬅0 for k⭐0. 共2b兲} The c_k␣j’s obey standard anticommutation relations 兵^ck␣j,c_k

⬘^␣⬘^j⬘

† 其^⫽␦kk⬘␦␣␣⬘␦j j⬘, where due to radial momen- tum quantization the values of k are quantized:

k⫽2␲

L 共n_k⫺P₀/2兲, n_k苸Z. 共3兲 Here P₀⫽0 or 1, since at zero temperature the chemical potential 共and hence p_F) must either coincide with a degen- erate level ( P₀⫽0) or lie midway between two of them ( P₀⫽1). The level spacing in both cases is

⌬L⫽2␲^/L. 共4兲

The s-wave conduction electrons can also be described by a one-dimensional chiral field^23共b兲

␺␣j共x兲⬅

冑

²L^␲_n

兺

k苸Ze^⫺ikxc_k␣j,

冉

^x苸

冋

^⫺L2,L2

册冊

^{, 共5兲}

兵␺␣j共x兲,␺_␣^†_⬘j⬘共x

⬘

兲其⫽␦␣␣⬘␦j j⬘2␲␦共x⫺x

⬘

兲. 共6兲 In the continuum limit L→⬁, the x⬎0 and x⬍0 portions of

␺_␣j(x) can be associated with the incoming and outgoing scattering states, respectively. Note that for P₀⫽0 or 1 the fields ␺_␣j(x) have periodic or antiperiodic boundary condi- tions at x⫽⫾L/2, respectively, hence P₀ will be called the

‘‘periodicity parameter.’’

We assume a short-ranged anisotropic exchange interac- tion between the impurity spin and the s-wave conduction electron spin density at the origin of the form

H_int⫽⌬L_␮,k,k

兺

_⬘

␣,␣⬘^{, j}

␭_␮S_␮:c_k^†_␣j

冉

^{12␴␣␣␮ ⬘}

冊

^{ck⬘␣⬘j: . 共7兲}

Here the S_␮(␮⫽x,y ,z) are the impurity spin operators, with S_z eigenvalues (⇑,⇓)⫽(¹₂,⫺¹2), and the ␭_␮’s denote dimen- sionless couplings: ␭z generates different phase shifts for spin-up and spin-down conduction electrons, while ␭x⬅␭y

⬅␭_⬜ describe spin-flip scattering off the impurity. Finally, we add a magnetic term

H_h⫽h_iS_z⫹h_eNˆ

s, 共8兲

where h_i and h_e denote the magnetic fields acting on the impurity and conduction electron spins, respectively, andNˆ_s denotes the total spin of the conduction electrons.

Since the constructive bosonization method requires an unbounded spectrum, the fermion bandwidth cutoff is re- moved共i.e., taken to be infinite兲in the equations above. This ultraviolet cutoff will only be restored when we define the new Bose fields in Eq. 共13兲below.

III. BOSONIZATION BASICS

The key to diagonalizing the Hamiltonian is to find the relevant quantum numbers of the problem and to bosonize the Hamiltonian carefully. While bosonization is a widely used technique, the so-called Klein factors mentioned in the Introduction are often neglected or not treated with sufficient care. In the present section we therefore discuss our bosonization approach in somewhat more detail than usual, formulating it as a set of operator identities in Fock space, and emphasizing in particular the proper use of Klein factors to ladder between states with different particle numbers in Fock space.

A. Bosonization ingredients As a first step we introduce the operators

Nˆ

␣j⬅

兺

_k ^{:ck†␣jck␣j:, 共9兲}

which count the number of electrons in channel (␣^{j ) with} respect to the free electron reference ground state 兩0ជ典^{0. The} nonunique eigenstates of Nˆ_␣j will generically be denoted by 兩Nជ典⬅兩N_↑1典^丢兩N_↓1典^丢兩N_↑2典^丢兩N_↓2典, where the N_␣j’s can be arbitrary integers, i.e., Nជ_苸Z⁴.

Next, we define bosonic electron-hole creators by b_q␣^† _j⬅ i

冑

ⁿq _n

兺

k苸Z

c_k^†_⫹q␣jc_k␣j, 共10兲 where q⫽2␲ⁿq/L⬎0 and the n_q are positive integers. The operators b_q^†_␣j create ‘‘density excitations’’ with momentum q in channel ␣j, satisfy standard bosonic commutation rela- tions, and commute with the Nˆ_␣j’s:

关b_q␣j,b_q

⬘^␣⬘^j⬘

† 兴⫽␦qq⬘␦␣␣⬘␦j j⬘, 关b_q␣j,Nˆ_␣

⬘^j⬘兴⫽0.

共11兲 Among all states兩Nជ典with given Nជ, there is a unique state, to be denoted by 兩Nជ典0, that contains no holes and thus has the defining property

b_q␣j兩Nជ典^0⫽0 共for any q⬎0, ␣^{, j}兲. 共12兲 We shall call it the ‘‘Nជ-particle ground state,’’ since in the absence of interactions no兩Nជ典has a lower energy than兩Nជ典0; likewise, no 兩Nជ典0 has a lower energy than the ‘‘vacuum state’’ 兩0ជ典0 defined in Eq. 共2兲. Note, though, that if P₀⫽0, the states c_0␣j兩0ជ典⁰ are degenerate with 兩0ជ典⁰, because then c_0␣j removes a zero-energy electron. Any Nជ-electron state 兩Nជ典 can be written as兩Nជ典⫽f (b^†)兩Nជ典0, i.e., by acting on the Nជ-electron ground state with an appropriate function of electron-hole creation operators.^33,31

Next, we define bosonic fields by

␾␣j共x兲⬅_q

兺

_⬎0

_冑

^⫺_n¹

q

共e^⫺iqxb_q␣j⫹e^iqxb_q^†_␣j兲e^⫺aq/2. 共13兲 Here a⬃1/p_F is a short-distance cutoff; it is introduced to cure any ultraviolet divergences the theory may have ac- quired by taking the fermion bandwidth to be infinite. It is well known, however, that within this bosonization cutoff scheme the coupling constants have different meanings than for other standard regularization schemes using a finite fer- mion bandwidth, and that the relations between coupling constants in different regularization schemes can be found by requiring that they yield the same phase shifts. This and other cutoff related matters are discussed in Appendix A.³⁷ The fields ⳵x␾_␣j(x) are canonically conjugate to the

␾␣j(x)’s

关␾_␣j共x兲,⳵x⬘␾_␣⬘j⬘共x

⬘

兲兴⫽2␲ⁱ„␦a共x⫺x

⬘

兲⫺1/L…␦␣␣⬘␦j j⬘, 共14兲 where␦a(x)⫽a/␲^(x2⫹a²) is the smeared delta function.

As final bosonization ingredient, we need the so-called Klein factors F_␣j, which ladder between states with differ- ent N_␣j’s. By definition, the F_␣j’s are required to satisfy the following relations:

关F_␣j,Nˆ_␣

⬘^j⬘兴⫽␦␣␣⬘␦j j⬘F_␣j, 共15a兲 关F_␣j,b_q␣⬘j⬘兴⫽关F_␣j,b_q␣

⬘^j⬘

† 兴⫽0, 共15b兲

F_␣jF_␣^†_j⫽F_␣^†_jF_␣j⫽1, 共15c兲 兵^F␣j,F_␣^†_⬘j⬘其^⫽2␦␣␣⬘␦j j⬘ 共15d兲 兵^F␣j,F_␣⬘j⬘其^{⫽0 for 共}␣^j兲⫽共␣

⬘

^j

⬘

兲. 共15e兲 These relations imply that F_␣j (F_␣^†_j) decreases 共increases兲 the electron number in channel兵␣^j其by one without creating particle-hole excitations. As shown in Refs. 33 or 31, the construction F_␣j⫽a^1/2␺_␣j(0)e^i␾␣j(0), which explicitly ex- presses F_␣j in terms of the fermion operators c_k␣j, has all the desired properties.

B. Bosonization identities

Any expression involving the fermion operators c_k␣j can be rewritten in terms of the the Klein factors F_␣jand boson fields ␾␣j defined above. In our notation, the standard

bosonization identities³³ for the fermion field, density and kinetic energy take the following forms:³¹

␺_␣j共x兲⫽F_␣ja^⫺1/2e^{⫺i(Nˆ␣j⫺P0/2)2␲x/L}e^{⫺i␾␣j(x)}, 共16兲 1

2␲^:␺_␣j

† 共x兲␺_␣j共x兲:⫽ 1

2␲ ⳵^x␾_␣j共x兲⫹Nˆ

␣j/L, 共17兲

H₀⫽

兺

_␣j

⌬L

2 Nˆ_␣j共Nˆ_␣j⫹1⫺P₀兲⫹

兺

_␣j q⬎0

qb_q^†_␣jb_q␣j. 共18兲

Several comments are in order:共i兲in the limit a→0 Eqs.

共16兲 to 共18兲 are not mere formal correspondences between the fermionic and bosonic expressions, but hold as rigorous operator identities in Fock space. For a⫽0, they should be viewed as conveniently regularized redefinitions of the fer- mion fields and densities 共see³⁷ Appendix A 2兲. 共ii兲 The Klein factors F_␣j in Eq.共16兲 play a twofold role: First, by Eq. 共15a兲 they ensure that the right-hand side of Eq. 共16兲 acting on any state indeed does lower the number of ␣^j electrons by one, just as ␺_␣j does; and secondly, by Eqs.

共15d兲and共15e兲they ensure that fields with different (␣^{j )’s} do have the proper anticommutation relations 共6兲. 共iii兲 In Eqs.共18兲the first⌬Lterm is just ₀具^Nជ兩H0兩Nជ典0, the energy of the Nជ-particle ground state兩Nជ典0 relative to兩0ជ典0. Since the Klein factors do not commute with this term, they evidently cannot be neglected when calculating the full model’s finite- size spectrum, for which all terms of order ⌬L must be re- tained. The second term of Eq.共18兲describes the energy of electron-hole excitations relative to 兩0ជ典0.

IV. EMERY-KIVELSON MAPPING

In this section, we map the 2CK model onto a resonant level model, using a finite-size version of the strategy in- vented by Emery and Kivelson: using bosonization and re- fermionization, we make a unitary transformation to a more convenient basis, in which the Hamiltonian is quadratic for a certain choice of parameters.

A. Conserved quantum numbers

The quantum numbers N_␣jof Eq.共9兲are conserved under the action of H₀, H_h, and H_z 共the ␭z term of H_int⬅H_z

⫹H_⬜), but fluctuate under the action of the spin-flip interac- tion H_⬜ 共the ␭_⬜ term兲. On the other hand, the total charge and flavor of the conduction electrons is obviously conserved by all terms in the Hamiltonian, including H_⬜. Therefore it is natural to introduce the following new quantum numbers:

冉

^{NNNNˆˆˆˆcsxf}

冊

^⬅12

^冉

^{1111 ⫺⫺1111 ⫺⫺1111 ⫺⫺1111}

^{冊 冉}

^{NNNNˆˆˆˆ↑↓↑↓1122}

^冊

^{, 共19兲}

where 2Nˆ

c, Nˆ

s, and Nˆ

f denote the total charge, spin, and flavor of the conduction electrons, andNˆ

xmeasures the spin difference between channels 1 and 2. Clearly, any conduc-

tion electron state 兩Nជ典 can equally well be labeled by the corresponding quantum numbers Nជ_⬅(Nc,Ns,Nf,Nx).

However, whereas the N_␣j’s take arbitrary independent inte- ger values, theNជ ’s generated by Eq.共19兲 共with Nជ_苸Z⁴) can easily be shown to satisfy the following free gluing condi- tions:

Nជ_苸共Z⫹P/2兲⁴, 共20a兲 Nc⫾Nf⫽共Ns⫾Nx兲mod 2, 共20b兲 where the parity index P equals 0 or 1 if the total number of electrons is even or odd, respectively. Equation共20a兲formal- izes the fact that the addition or removal of one ␣^{j electron} to or from the system changes each of theNy’s by⫾1/2, so that they are either all integers or all half-integers. Equation 共20b兲 selects from the set of all Nជ of the form 共20a兲 the physical ones for which Nជ_苸Z⁴, and eliminates the unphysi- cal ones for which Nជ_苸(Z⫹1/2)⁴.

In the new basis,NcandNfare conserved; moreover,Ns

fluctuates only ‘‘mildly’’ between the values S_T⫿1/2, since the total spin

S_T⬅Ns⫹S_z 共21兲

is conserved. In contrast,Nxfluctuates ‘‘wildly,’’ because an appropriate succession of spin flips can produce anyNxthat satisfies Eq. 共20b兲, as illustrated in Fig. 1. This wildly fluc- tuating quantum number will be seen below to be at the heart of the 2CK model’s NFL behavior. In revealing contrast, the 1CK model, which shows no NFL behavior, lacks such a wildly fluctuating quantum number 共see Appendix D兲.

Since S_T,Nc, andNfare conserved, the Fock spaceFphys

of all physical states can evidently be divided as follows into subspaces invariant under the action of H:

Fphys⫽

兺

丣⬘^ST,Nc,Nf

Sphys共S_T,Nc,Nf兲, 共22兲

Sphys共S_T,Nc,Nf兲⫽_丣

兺

_⬘N

x

兵^兩Nc,S_T⫺1/2,Nf,Nx;⇑典

丣兩Nc,S_T⫹1/2,Nf,Nx⫹1;⇓典其^. 共23兲 In both equations the prime on the sum indicates a restriction to thoseNy’s that satisfy the free gluing conditions共20兲. To

diagonalize the Hamiltonian for given S_T, Nc, and Nf, it evidently suffices to restrict one’s attention to the corre- sponding subspaceSphys(S_T,Nc,Nf).

B. Emery-Kivelson transformation

Following Emery and Kivelson, we now introduce, in analogy to Eq. 共19兲, new electron-hole operators and boson fields via the transformations

b_qy⬅

兺

_␣j ^{Ry ,␣j bq␣j}

␸y⬅

兺

_␣j ^{Ry ,␣j ␾␣j}

冎

^{共y⫽c,s, f ,x兲, 共24兲}

where R_{y ,␣j} is the unitary matrix in Eq. 共19兲. These obey relations analogous to Eqs.共11兲and共14兲, with␣^j→y . More- over, we define兩Nជ典0, theNជ-particle vacuum state, to satisfy b_qy兩Nជ典0⫽0, as in Eq. 共12兲. IfNជ and Nជ are related by Eq.

共19兲, then the states兩Nជ典0 and兩Nជ典0 are equal up to an unim- portant phase 共see³⁷ Appendix B兲, because both have the same Nˆ

␣jandNˆ

yeigenvalues and both are annihilated by all b_q␣j’s and b_qy’s.

Using the quantum numbers Nˆ

y and the bosonic fields

␸y(x), the H₀ of Eq.共18兲becomes

H₀⫽⌬L

冋

^{Nˆc共1⫺P0兲⫹}

^兺

^{y Nˆy2/2}

册

^{⫹y , q}

^兺

^{⬎0 qbqy† bqy,}

共25兲 while Eqs. 共17兲and共16兲are used to obtain, respectively,

H_z⫽␭z关⳵x␸s共0兲⫹⌬LNˆ

s兴S_z, 共26兲

H_⬜⫽␭_⬜

2a关e^⫺i␸s(0)S_⫹共F_↓^†₁F_↑1e^⫺i␸x(0)

⫹F_↓^†₂F_↑2e^i␸x(0)兲⫹H.c.兴. 共27兲 Equations共25兲–共27兲and共8兲constitute the bosonized form of the Hamiltonian for the anisotropic 2CK model, up to and including terms of order⌬L.

Next we simplify H_z. It merely causes a phase shift in the spin sector, which can be obtained explicitly using a unitary transformation共due to EK兲parametrized by a real number␥^, to be determined below:

H→H

⬘

⫽UHU^†, U⬅e^{i␥Sz␸s(0)}. 共28兲 The impurity spin, spin-diagonal part of H, spin boson field and fermion fields then transform as follows共using, e.g., the identities in Appendix C of Ref. 31兲:

S_⫾→US_⫾U^†⫽e^{⫾i␥␸s(0)}S_⫾, 共29兲 H₀⫹H_z→H₀⫹共␭z⫺␥兲⳵x␸s共0兲S_z⫹␭z⌬LNˆ_sS_z

⫹␥²关1/共4a兲⫺␲^/共4L兲兴, 共30兲

␸s共x兲→␸s共x兲⫺2␥^Szarctan共x/a兲 共兩x兩ⰆL兲, 共31兲

␺_␣j共x兲→␺_␣j共x兲e^i␣␥Szarctan(x/a) 共兩x兩ⰆL兲. 共32兲 FIG. 1. Under a succession of spin flips, Ns fluctuates mildly

between S_T⫿1/2共here S_T⫽1/2); in contrast,Nx fluctuates wildly, since it can acquire any value consistent with the gluing conditions 共20兲. The dotted line represents the reference energy 0 up to which the free Fermi sea is filled for P₀⫽1, the filled and empty circles represent filled and empty single-particle states with energy k, which increases from left to right.

Equation共30兲is most easily derived in the momentum-space representation, but for Eq.共31兲, the position-space represen- tation is more convenient关first evaluate U⳵x␸s(x)U^⫺1using Eq. 共14兲, then integrate兴. Equation 共32兲 follows from Eq.

共31兲, since ␺_␣j⬀e^{⫺i␣␸s/2}.

Recalling that ⳵x␸s(x)/2␲ contributes to the conduction electron spin density, we note by differentiating Eq.共31兲that the EK transformation produces a change in the spin density of⫺2␥^Sz␲␦a(x)/2␲, and thus ties a spin of⫺␥^Szfrom the conduction band to the impurity spin S_z.

To eliminate the S_z⳵x␸sterm in Eq.共30兲, we now choose

␥⬅␭z; then the spin-flip-independent part of the Hamil- tonian takes the form

H

⬘

共␭_⬜⫽0兲⫽␭z⌬LNˆ

sS_z⫹

兺

_y ^{⌬LNˆy2/2⫹}_{y , q}

兺

_⬎0 ^{qbqy† bqy}

⫹H_h⫹const, 共33兲

and H_⬜

⬘

contains the factors e^{⫾i(1⫺␭z)␸s(0)}. These factors are simply equal to 1 at the Emery-Kivelson line ␭z⫽1, where H_⬜

⬘

simplifies to

H_⬜

⬘

⫽␭_⬜

2a关S_⫹共F_↓^†₁F_↑1e^⫺i␸x(0)⫹F_↓^†₂F_↑2e^i␸x(0)兲⫹H.c.兴. 共34兲 We shall henceforth focus on the case ␭z⫽1, which will enable us to diagonalize the model exactly by refermioniza- tion. Deviations from the EK line will be shown in Sec. VI C to be irrelevant, by taking␥⫽1 but␭z⫽1⫹␦␭z, and doing perturbation theory in

␦^Hz

⬘

⫽␦␭z关⳵x␸s共0兲⫹⌬LNˆ

s兴S_z. 共35兲

The crucial property of the EK line is that it contains the NFL intermediate-coupling fixed point. A Heuristic way to see this it to note that on the EK line, the impurity spin is in fact ‘‘perfectly screened:’’ the spin⫺␥^Sz from the conduc- tion band, that is tied to the impurity by the EK transforma- tion, is equal to ⫺S_z if ␥⫽␭z⫽1. It thus precisely ‘‘can- cels’’ the impurity’s spin S_z, and forms a ‘‘perfectly screened singlet’’ with zero total spin 共without breaking channel symmetry兲, in agreement with the heuristic argu- ments of Nozie`res and Blandin.¹

Of course, there are more rigorous ways of seeing that the NFL fixed point lies on the EK line. First, for␭z⫽1 it fol- lows from Eq. 共32兲 that the phase shift ␦ of the outgoing relative to the incoming fields, defined by ␺␣j(0^⫺)

⬅e^i2␦␺_␣j(0^⫹)共with兩0^⫾兩Ⰷa), is兩␦兩⫽␲/4, which is just the value known for the NFL fixed point from other approaches.^7,18Secondly, we shall deduce in Sec. VI C from an analysis of the finite-size spectrum that the leading irrel- evant operators with dimensions 1/2 vanish exclusively along this line, but not away from it. Since the presence or absence of the leading irrelevant operators strongly influ- ences the low-temperature properties of the model such as its critical exponents,^2,26and since these must stay invariant un- der any RG transformation, one concludes that the Emery- Kivelson line must be stable under RG transformations.

C. Refermionization 1. Definition of new Klein factors

The most nontrivial step in the solution of the model is the proper treatment of Klein factors when refermionizing the transformed Hamiltonian. In their original treatment EK did not discuss Klein factors at all and simply identified e^⫺i␸x(x)/

冑

a as a new pseudofermion field ␺x(x). Though this was adequate for their purposes, the proper consideration of the Klein factors and gluing conditions is essential for solving the model rigorously and obtaining the finite-size spectrum. Other authors tried to improve the Emery- Kivelson procedure by representing the Klein factors by F_␣j⬃e^⫺i⌰␣j, where ⌰_␣j is a ‘‘phase operator conjugate to Nˆ_␣j,’’ and added these to the bosonic fields␾_␣jbefore mak- ing the linear transformation 共24兲. This procedure is prob- lematic, however, since then e^⫺i␸y(0)contains factors such as e^{⫺i⌰␣j/2}, which are ill defined共see Appendix D 2 of Ref. 31兲. A rigorous way of dealing with Klein factors when refer- mionizing was presented in Ref. 4共and adapted in Ref. 31 to treat an impurity in a Luttinger liquid兲: We introduce a set of ladder operators Fy

† and Fy (y⫽c,s, f ,x) to raise or lower the quantum numbers Ny by ⫾1, with, by definition, the following properties:

关Fy,Nˆ

y⬘兴⫽␦y y⬘Fy, 共36a兲

关Fy,b_qy⬘兴⫽关Fy,b^†_qy⬘兴⫽0, 共36b兲 FyFy

†⫽Fy

†Fy⫽1, 共36c兲

兵Fy,F^†y⬘其⫽2␦y y⬘, 共36d兲 兵^Fy,Fy_⬘其^{⫽0 for y⫽y}

⬘

^. 共36e兲 Now, note that the action of any one of the new Klein factors Fy or Fy

† respects the first of the free gluing conditions 共20a兲, but not the second, Eq. 共20b兲. More generally, Eq.

共20b兲 is respected only by products of an even number of new Klein factors, but violated by products of an odd num- ber of them. This implies that the physical Fock spaceFphys

of all 兩Nជ典 satisfying both Eqs. 共20a兲 and 共20b兲 is closed under the action of even but not of odd products of new Klein factors. The action of arbitrary combinations of new Klein factors thus generates an extended Fock space Fext, which containsFphysas a subspace and is spanned by the set of all 兩Nជ典 satisfying Eq. 共20a兲, including unphysical states violating Eq. 共20b兲. In Appendix B we show that Fphyscan indeed be embedded in Fextby explicitly constructing a set of basis states for Fext.³⁷

Since odd products ofFy’s lead out ofFphys, they cannot be expressed in terms of the original Klein factors F_␣j, which leaveFphysinvariant. However, the Hamiltonian con- tains only even products of old Klein factors. Now, any com- bination F_␣^†_jF_␣⬘j⬘or F_␣^†_jF_␣

⬘^j⬘

† of Klein factors just changes two of the N_␣jquantum numbers. Using Eq.共19兲to read off the corresponding changes in Ny, we can thus make the following identifications between pairs of the old and new Klein factors:

Fx

†Fs

†⬅F_↑^†₁F_↓1, FxFs

†⬅F_↑^†₂F_↓2, 共37a兲 Fx

†Ff

†⬅F_↑^†₁F_↑2, Fc

†Fs

†⫽F_↑^†₁F_↑^†₂. 共37b兲 These relations, which each involve an arbitrary choice of sign, can be used to express any product of two old Klein factors in terms of two new ones, e.g., Fs

†Ff

†

⫽⫺(FxFs

†)(Fx

†Ff

†)⫽F_↑^†₁F_↓2. Since relations共37兲 by con- struction respect Eq. 共19兲 共as can be checked by acting on any兩Nជ典), they, and all similar bilinear relations derived from them, also respect both free gluing conditions共20兲.

We can thus replace the Klein factor pairs occurring in Eq. 共34兲by the ones in Eq.共37a兲:

H_⬜

⬘

⫽␭_⬜

2a关S_⫹F_s共F_xe^⫺i␸x(0)⫹F_x^†e^i␸x(0)兲⫹H.c.兴. 共38兲 The only consequence of this change is that we now work in the extended Fock spaceFext, and will diagonalize H

⬘

^{not in} the physical invariant subspace Sphys(S_T,Nc,Nf) of Eq.

共23兲, but in the corresponding extended subspace Sext(S_T,Nc,Nf), given by an equation similar to Eq. 共23兲, but with the 丣

⬘

Nx sum now restricted only to satisfy Eq.

共20a兲, not also Eq. 共20b兲. At the end of the calculation we shall then use the gluing condition共20b兲to discard unphysi- cal states. This approach is completely analogous to the use of gluing conditions in AL’s CFT solution of the 2CK model. It is also somewhat analogous to Abrikosov’s pseudofermion technique³⁸ of representing a spin operator via pseudofermions acting in an enlarged Hilbert space, and projecting out unphysical states at the end.

2. Pseudofermions and refermionized Hamiltonian We now note that H_⬜

⬘

^{of Eq.}共38兲can be written in a form quadratic in fermionic variables

H_⬜

⬘

⫽ ␭_⬜

2

冑

^{a关␺x共0兲⫹␺x†共0兲兴共cd⫺cd†兲, 共39兲}

by defining a local pseudofermion c_d and a pseudofermion field␺x(x) by the following refermionization relations:

c_d⬅F_s^†S_⫺, c_d^†c_d⫽S_z⫹1/2, 共40兲

␺x共x兲 ⬅F_xa^⫺1/2e^{⫺i(Nˆx⫺1/2)2␲x/L}e^⫺i␸x(x) 共41a兲

⬅

冑

²L^␲

兺

¯_k

e^{⫺ik¯ x}c_{k¯ x}, 共41b兲 where Eq.共41b兲defines the c_{¯ xk} as Fourier coefficients of the field␺x(x). For reasons discussed below, the field␺xin Eq.

共41a兲has been defined in such a way that its boundary con- dition at⫾L/2 is P dependent, sinceNx苸Z⫹P/2 and␸x(x) is a periodic function. Thus the quantized k¯ momenta in the Fourier expansion共41b兲must have the form

¯k⫽⌬L关n¯_k⫺共1⫺P兲/2兴 共n¯_k苸Z兲. 共42兲 The new pseudofermions were constructed in such a way that they satisfy the following commutation-anticommutation relations:

兵^c¯ x^k ,c¯_k⬘^x

† 其^⫽␦¯ k¯k ⬘, 兵^cd,cd

†其^{⫽1, 共43兲}

兵^cd,c_{¯ xk}^† 其⫽兵^cd,c_{¯ xk} 其⫽0, 共44兲 关c_d,Nˆ_s兴⫽c_d, 共45兲 which follow directly from the properties of ␸x and Eqs.

共36兲. Note that c_d lowers the impurity spin, raises the total electron spin Nˆ

s and hence conserves the total spin S_T, whereas␺xconserves each of the impurity, electron and total spins.

To relate the number operator for the new x-pseudofermions to the quantum number Nx, we must de- fine a free reference ground state, say兩0典Sext, in the extended subspaceSext, with respect to which the number of pseudo- fermions are counted. In analogy to Eq. 共2兲, we define it by c_{¯ xk} 兩0典S_ext⬅0 for k¯⬎0, 共46a兲 c_{¯ xk}^† 兩0典Sext⬅0 for k¯⭐0, 共46b兲 c_d兩0典Sext⬅0 for ␧d⬎0, i.e., n_d⁽⁰⁾⬅0, 共46c兲 c_d^†兩0典S_ext⬅0 for ␧d⭐0, i.e., n_d⁽⁰⁾⬅1. 共46d兲 Here␧d, whose value will be derived below关see Eq.共52兲兴, is the energy of the c_d pseudofermion, and n_d⁽⁰⁾ denotes its occupation number in the reference ground state 兩0典S_ext. Us- ing colons to henceforth denote normal ordering of the pseudofermions with respect to 兩0典S_ext, we have :c_d^†c_d:

⫽c_d^†c_d⫺n_d⁽⁰⁾. Furthermore, we define the number operator for the x pseudofermions by N¯ˆ

x⬅兺¯k:c_{¯ xk}^† c_{¯ xk} :. Then N¯ˆ

x⫽Nˆ

x⫺P/2 共47兲

holds as an operator identity. This can be seen intuitively by noting that␺x⬃Fx⬃c_{¯ xk} 关by Eq.共41兲兴, hence the application of␺x共or␺x

†) to a state decreases共or increases兲bothNxand N¯_x by one. These two numbers can thus differ only by a constant, which must ensure that N¯_xis an integer. Our defi- nition共46兲of兩0典Sexteffectively fixes this constant to be P/2, by setting N¯

x⫽0 for Nx⫽P/2共see Appendix A 3 for a rig- orous argument³⁷兲.

We are now ready to refermionize the Hamiltonian H

⬘

^. The kinetic energy of the k¯ pseudofermions obeys

兺

_¯k ^{¯ :ck ¯ xk† c¯ xk :⫽⌬}2^LN¯ˆ_x共N¯ˆ_x⫹P兲⫹

兺

_q ^{q bqx† bqx, 共48兲}

an operator identity which follows by analogy with Eqs.共1兲 and 共18兲 共also see³⁷ Appendix A 3兲. Now note that N¯ˆ

x(N¯ˆ

x

⫹P)⫽Nˆ

x

2⫺P/4, i.e., Eq.共48兲does not contain a term linear inNˆ

x. Actually, the choice of the phase e^{⫺i(Nˆx⫺1/2)2␲x/L} in our refermionization ansatz 共41a兲 for ␺x(x) was made spe- cifically to achieve this. Hence Eq.共48兲can be directly used to represent the kinetic energy of the x sector in Eq.共25兲in terms of c_{¯ xk} fermions: