1 Introuction In the past few years there has been renewe interest in the nerson impurity moel. This moel was originally propose by nerson [2], for a

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1 Theory of the nerson impurity moel: The Schrieer{Wol transformation re{examine Stefan K. Kehrein 1 an nreas Mielke 2 Institut fur Theoretische Physik, uprecht{karls{universitat, D{69120 Heielberg, F.. Germany October 5, 1995 bstract We apply the metho of innitesimal unitary transformations recently introuce by Wegner [1] to the nerson single impurity moel. It is emonstrate that this metho provies a goo approximation scheme for all values of the on-site interaction U, it becomes exact for U = 0. We are able to treat an arbitrary ensity of states, the only restriction being that the hybriization shoul not be the largest parameter in the system. Our approach constitutes a consistent framework to erive various results usually obtaine by either perturbative renormalization in an expansion in the hybriization, nerson's \poor man's" scaling approach or the Schrieer{Wol unitary transformation. In contrast to the Schrieer{Wol result we n the correct high{energy cuto an avoi singularities in the inuce couplings. n important characteristic of our metho as compare to the \poor man's" scaling approach is that we continuously ecouple moes from the impurity that have a large energy ierence from the impurity orbital energies. In the usual scaling approach this criterion is provie by the energy ierence from the Fermi surface. 1 E{mail: kehrein@marvin.tphys.uni-heielberg.e 2 E{mail: mielke@hybri.tphys.uni-heielberg.e

2 1 Introuction In the past few years there has been renewe interest in the nerson impurity moel. This moel was originally propose by nerson [2], for a recent review see [3]. It has been introuce to stuy the well{known Kono problem, the behaviour of a single magnetic impurity couple to a conuction ban of electrons. The Hamiltonian contains the electron ban, the energy of the impurity orbital together with a repulsive interaction on the impurity site, an a hybriization between the ban states an the impurity state, H = X k; k c y k; c k; + X y + X k; V k (c y k; + y c k; ) + U y + y + : (1.1) In the case of a linear ispersion relation for the ban an V k = V = const: the moel was solve using a Bethe{ansatz [4]. But if one wants to stuy the nerson impurity moel in a more general situation, one nees a ierent approach. There are several methos available, most of them are reviewe in [3]. The most prominent metho amongst them is probably the numerical renormalization group evelope by Wilson [5] for the original Kono Hamiltonian an applie to the nerson impurity moel by Krishna-murthy et al [6]. ecently we applie a new technique to this moel [7]. We use continuous unitary transformations in a form introuce by Wegner [1] to iagonalize the Hamiltonian approximately. continuous unitary transformation yiels ow equations for the Hamiltonian, or equivalently ow equations for the coupling constants. The approximation we use neglecte some aitional couplings generate by the ow that were not present in the initial moel. Unfortunately we were not able to obtain quantitative results. The reason is that the continuous transformations yiels couple, nonlinear ierential equations for the ierent parameters in the Hamiltonian, which we were not able to treat analytically. In a more recent work we applie continuous unitary transformation to the well known spin{ boson moel [8]. For this moel a unitary transformation exists, the polaron transformation, which has been use to treat the Hamiltonian of the spin{boson moel approximately. In ef. [8] we use a simple moication of the previous ansatz by Wegner [1] for the generator of the continuous unitary transformation. Thereby we were able to construct a continuous polaron transformation. This new transformation oes not have the isavantages of the usual polaron transformation as it treats the slow bosonic moes in a satisfactory way. In aition, we were able to reuce the set of ierential equations to a single, non{linear ierential equation. This nally allowe us to obtain quantitative results which are in goo agreement with results obtaine by other methos. In the case of the nerson impurity moel, a unitary transformation similar to the polaron transformation is known, the Schrieer{Wol transformation [9]. It has been introuce to eliminate the hybriization between the electronic states in the ban an in the impurity orbital present in the nerson moel. Thereby it renormalizes the impurity energy an the repulsive interaction. Furthermore it generates a spin{spin interaction between the impurity electron an the conuction ban electrons. This interaction is responsible for the Kono eect. In this treatment one usually takes only the terms into account which are of secon orer in the hybriization. Other interactions are generate as well, but they are neglecte. The Schrieer-Wol transformation has some isavantages. First it is equivalent to a secon orer perturbational treatment of the hybriization term in the Hamiltonian. Therefore the general valiity of the result is unclear. In particular the nerson impurity moel in the Kono regime is mappe onto a Kono problem with an eective ban with of orer of the conuction ban with. This is known to be wrong since the high{energy cuto in the nerson impurity 1

3 moel cannot be larger than of orer the on-site interaction U [10]. The secon problem with the Schrieer{Wol transformation is that energy enominators occur, which become zero if the energy of the impurity level lies in the conuction ban. If this is the case, the Schrieer{Wol transformation is ill{ene. This problem is similar to the problem of the treatment of the slow bosonic moes in the polaron transformation. Therefore we expect that a moication of our ol treatment of the nerson impurity moel shoul be useful. In the present paper we construct a continuous, or innitesimal Schrieer{Wol transformation in orer to eliminate the hybriization terms in the Hamiltonian. We will show how such a continuous Schrieer{Wol transformation can be constructe systematically. Our ol approach can be moie so that the number of aitional interactions generate by the unitary transformation is reuce. It is clear that applying innitesimal unitary transformations to a given Hamiltonian is a non{perturbative metho. The avantage of our present approach is that our metho treats the nerson impurity moel consistently within one framework inepenent of whether the on{site interaction U or the hybriization is the larger parameter. We are able to reuce the problem to two couple non{linear ierential equations for the impurity orbital energy an the on{site interaction. These are solve approximately yieling self{consistency conitions for these two quantities. Finally the antiferromagetic spin{spin interaction can be calculate. It is emonstrate how stanar results from e.g. renormalization theory can be obtaine in this conceptually new framework. Whereas other methos are not applicable in the whole parameter space or nee aitional assumptions, continuous unitary transformations are conceptually simple an no physically relevant restrictions or aitional assumptions are neee. One can hope that the ow equations approach will be useful too for other problems with less well{establishe results. In so far it is important to stuy the metho using a well{known problem. Inepenent of Wegner [1], Glazek an Wilson [11, 12] have recently also propose to use continuous unitary transformations to construct renormalization group equations for eective Hamiltonians in quantum el theory. Wilson et al. [13] applie this metho to quantum chromoynamics. lthough the general iea is similar, there are some ierences between the approach of Glazek an Wilson an ours. Their goal is to eliminate "far{o{iagonal" matrix elements in a given Hamiltonian, which means o{iagonal matrix elements connecting states that are energetically far from each other. This means that the nal Hamiltonian has a bane structure. In contrast our goal is to eliminate matrix elements such that the nal Hamiltonian is iagonalize approximately or block{iagonal. lthough it is possible to construct ow equations such that the Hamiltonian becomes iagonal, it is not possible to solve these equations an to calculate the eigenenergies. Therefore we use the continuous unitary transformation in orer to eliminate some of the matrix elements. In our case the hybriization is eliminate an the nal Hamiltonian is block{iagonal. The nal Hamiltonian oes not contain matrix elements connecting states with a singly occupie impurity orbital to states for which the impurity orbital is either not occupie or oubly occupie. If now the impurity site is occupie with no or two electrons, the spin on the impurity site is zero an the aitional antiferromagnetic interaction vanishes. In these cases the problem is essentially solve with respect to static properties. In the regime where the impurity site is singly occupie, the problem is reuce to a usual Kono problem, which may be solve by various methos. Our paper is organize as follows. In the following section we erive the general ow equations for the coupling constants of the nerson impurity moel. In section 3 we illustrate the metho in the case of a vanishing interaction of the electrons on the impurity. It is shown that in this case the ow equations yiel the exact solution. Furthermore we introuce an approximation which still gives the exact result for the case of vanishing interaction an which is applie in section 4 to the case of a non{vanishing interaction. We calculate implicit equations for the renormalize 2

4 interaction an the renormalize impurity energy. These equations can in principle be solve for any given ensity of states an hybriization. Our results are compare with the results obtaine by Schrieer an Wol [9]. In section 5 the antiferromagnetic spin{spin interaction is calculate. It can be use to etermine e.g. the Kono temperature. In section 6 we iscuss the metho of continuous unitary transformations as compare to a single unitary transformation in the Schrieer{Wol paper. Section 7 contains a iscussion of our metho as compare to the well{known \poor man's" scaling approach. The last section contains the conclusions an an outlook on other problems. 2 The ow equations for the nerson impurity moel Our starting point is the Hamiltonian for the single impurity nerson moel (1.1), which we write in a normal orere form H = X k; k : c y k; c k; : + X y + X k; V k (c y k; + y c k;) + U y + y + + E 0 : (2.1) We introuce a normal orering for the ban electron operators c y an k; c k;. It is ene by : c y k; c k; := c y k; c k; n k where n k = (exp( k ) + 1) 1 is the occupation number of the ban state with wave vector k. We let the Fermi energy equal to zero. The reason for the normal orering is that aitional interactions, which will be generate by our proceure an which we neglect, shoul be written own in a normal orere form as well since otherwise the groun state expectation value of such aitional contributions oes not vanish. etaile iscussion has been given in [7]. In contrast to [7] we o not introuce a normal orering for the impurity electron operators y an. The reason is that in our approximation contributions containing such operators are not neglecte if they have a non{vanishing expectation value in the groun state. It is possible to introuce a normal orering on the impurity site as well, but the results are not change. Due to the normal orering a constant E 0 = 2 P k kn k has been introuce in the Hamiltonian. We now want to apply a general continuous unitary transformation to the Hamiltonian. Such a transformation is ene by a generator that epens on a continuous variable, which we call `. The continuous unitary transformation is ene by H ` = [(`); H(`)] with the initial conition H(0) = H in (2.1). In orer to simplify notation we will enote the initial values of an U by I ef = (` = 0); U I ef = U(` = 0) (2.2) an the asymptotic (\renormalize") values for `! 1 by ef = (` = 1); U ef = U(` = 1): (2.3) If these parameters appear without an argument this will imply that they are to be consiere as functions of `. We choose to be of the form = X k; + X k; k (c y k; y c k;) + X k;q; k;q (c y k; c q; c y q; c k;) (2) k (cy k; y y y c k; ) (2.4) 3

5 The commutator X of an H is easily calculate, we obtain [; H] = k ( k )(c y k; + y c k;) k; X + k V q (: c y k; c q; : + : c y q;c k; :) k;q; 2 X k; k V k y + 2 X k; k V k n k + U X k; k ( y y c k; + c y k; y ) X k;q; k;q ( k q )(: c y k; c q; : + : c y q; c k; :) + 2 X k;q; k;q V q ( y c k; + c y k; ) X k; (2) k ( k )( y y c k; + c y k; y ) 2 X k; (2) k V k y y + X k;q; (2) k V q(: c y k; y c q; : + : c y q; y c k; : : c y k; y c q; : : c y q; y c k; : c y k; cy q; y y c q; c k; ) + 2 X k; + U X k; (2) k V kn k y (2) k (y y c k; + c y k; y ): (2.5) Many aitional couplings are generate which i not occur in the original Hamiltonian in (2.1). But some of these terms can be eliminate by a suitable choice of. Let us rst consier terms containing : c y k; c q; + c y q; c k; :. Such terms o not occur if we choose k;q ( k q ) = 1 2 ( kv q + q V k ): (2.6) Similarly, terms containing operators of the type y y c k; + c y k; y o not occur if we choose U k + ( k ) (2) k + U (2) k = 0 (2.7) These equations may be use to etermine k;q an (2) k. The only contribution that oes occur aitionally is the term X k;q; (2) k V q(: c y k; y c q; : + : c y q; y c k; : : c y k; y c q; : : c y q; y c k; : c y k; cy q; y y c q; c k; ): (2.8) Notice that the contributions containing : c y k; c q; + c y q;c k; : or : y y c k; + c y k; y : have been classie irrelevant in our former approach [7], whereas the term in (2.8) is marginal in some of the xe points [7]. In principle it has to be inclue in the Hamilonian in (2.1). But the commutator of this term with oes not yiel contributions to the other terms in the 4

6 Hamiltonian. Therefore we o not take it into account in our rst analysis of the problem. But it is clear that this aitional term is important. part of it yiels the antiferromagnetic interaction between the impurity spin an the spins of the ban electrons that is responsible for the Kono eect. We will come back to this interaction later. Our rst goal is to calculate the ow equations for the parameters in the Hamiltonian (2.1). From (2.5) we obtain V k ` = k( k ) + 2 X p k;p V p (2.9) k ` = 2 kv k (2.10) ` = 2X k U ` = 4X k k V k + 2 X k (2) k V k (2) k V kn k (2.11) (2.12) E 0 ` = 4X k k V k n k (2.13) The last equation yiels irectly E 0 = 2 P k kn k, which is the energy of the lle Fermi sea. In the following we are intereste in the thermoynamic limit. For large N, the number of states in the ban, one has V k / N 1=2. Thus, k must as well be of the orer N 1=2, an the erivative of k with respect to ` is of orer N 1. For large values of N the ban energies o not epen on `. This shoul have been expecte. The thermoynamic bath of electrons is not aecte by the single impurity. This means that the global ensity of states in the ban is xe. But this oes not mean that the local ensity of states is xe as well. In contrary, one shoul expect that the local ensity of states near the impurity site is aecte by the impurity. We will come back to this point in the iscussion. 3 Vanishing interaction U = 0 To illustrate the avantages of our metho, let us rst stuy the case U = 0. Then we have a quaratic Hamiltonian that can be solve exactly, see for example ef. [14]. We will show that our metho yiels the exact solution in this case. For U = 0 the ow equations simplify to V k ` = k( k ) + 2 X p k;p V p (3.1) ` = 2X k k V k (3.2) The ow equations are exact in the case U = 0 since the neglecte terms in (2.5) vanish in this limit. We let k = V k f( k ; `) (3.3) an introuce J(; `) = X k V 2 k ( k ): (3.4) 5

7 In the literature one often introuces the parameter = ( F ) V kf (0) 2 = J( F ; 0); (3.5) where ( F ) is the ensity of states at the Fermi surface. The ow equations for an J(; `) are ` = 2 f(; `)J(; `) (3.6) Z = 2J(; `)f(; `)( ) J(; `)J(0 ; `)(f(; `) + f( 0 ; `)) 0 : (3.7) For the integral in the last equation one has to take its principal value. The set of ow equations may be solve if one introuces a function G(; `) Z 0 J( 0 ; `) 0 + G(; `) : (3.8) Taking the erivative with respect to `, we obtain an implicit equation for this erivative, which can be solve. The nal result is Z = J( 0 ; `) ( 0 + G(; `)) 2 1 Z 0 f( 0 ; `)J( 0 ; `) 0 + G(; `) 0 + G(; `) Calculating the erivative of G(; `) with respect to we obtain similarly Z 1 + = Z = 2 0 J( 0 ; `) ( 0 + G(; `)) 2 1 : (3.10) 0 f( 0 ; `)J( 0 ; `) 0 + G(; `) 0 + G(; `) Comparing the right han sie with the erivative of with respect to `, we obtain = (`) (3.11) : (3.12) This equation can be integrate an the nal result is (`) + G( (`); `) = : (3.13) In the last step we use G(; 1) = 0, which follows irectly from J(; 1) = 0 an hols for an appropriate choice of f(; `). Solving for (`), we obtain (`) = Z J(; `) (3.14) gain, for the integral on the right han sie we have to take its principal value. This means that we have to choose J(; `) an therefore f(; `) such that the principal value exists for all `. We obtain the value of, if we let ` = 0 an solve for. s a simple example we take a semi-circle J(; 0) = 2V 2 D 2 p D2 2 6 (3.15)

8 where 2D is the ban with an V = q P k V 2 k. Here we have = 2V 2 =D. The main reason for this choice of the hybriization is that all the integrals can be worke out in close form in the sequel. But it shoul be note that it is a main avantage of our approach that it can be use for arbitrary functions J(; 0), in particular for any istribution of the ensity of states in the conuction ban. However, if one chooses a linear ispersion relation an constant hybriization V k, that is J(; 0) = V 2 (D jj) as usually one in renormalization group treatment of the 2D nerson impurity moel, one must be careful ue to the iscontinuous behaviour close to the ban ege. The self{consistency equations in the ow equations approach will then generally have more than one solution, however, the actual solution of the ierential equations chooses the correct one. Close to the ban ege one expects unphysical behaviour anyway ue to the unphysical choice of J(; 0) an neither approach shoul be truste. Therefore it is natural in our approach to choose a function J(; 0) that is continuous at the ban ege as shoul be expecte on physical grouns anyway. Let us come back to our example introuce in Eq. (3.15). We have to istinguish between the case where lies in the ban an the case where it lies outsie the ban. We rst consier the latter case. The integral can be calculate an we obtain q sign( ) = 0 (3.16) I + D ( )2 D 2 if > D. This equation yiels a simple quaratic equation for, which has always two solutions. If < D at most one of these solutions lies outsie the ban. If > D an I > D, there is a single solution for outsie the ban, but if I < D we obtain two solutions outsie the conuction ban. The situation is much simpler when lies insie the ban. The integral in (3.14) has to be interprete as its principal value an we obtain I D = 0: (3.17) The only solution is = I : 1 D (3.18) lies insie the ban if < D I. The various cases are shown in Fig. 1. The fact that for a suciently large value of V ( > D in our example) two solutions for exist, is generic. It hols for any J(; 0) with a connecte support of length 2D. It is clear that with the present approach of ow equations only a single solution can be obtaine. Nevertheless, the secon solution is of physical importance. It is possible that a localize state evelops from the original ban states that has an energy which lies outsie the ban. Such a state cannot be obtaine within the present formulation of the ow equations. In the case U = 0 one can introuce a ierent representation of the Hamiltonian an of that inclues a localize ban state explicitly. Since we are at present not able to eal with similar problems in the case U > 0, we restrict ourselves to the parameter regime < D (in fact later we will nee < D=2). This is reasonable from a physical point of view since we o not expect that the hybriization is the largest parameter in the system. Eq. (3.14) is obtaine as well if one uses two simple approximations to the ow equation. The rst approximation neglects the terms proportional to c y k; c q; in the Hamiltonian that are generate by the transformation an consequently one neglects such terms in as well. This 7

9 is an approximation that can be justie from a physical point of view, since these terms are irrelevant in all xe points [7]. Then the secon term in (2.9) vanishes an the equation for V k is linear in V k. Similarly, the secon term in (3.7) vanishes, whereas (3.6) remains unchange. Both equations together yiel ` : (3.19) Furthermore we assume that (`) converges rapily to, so that we can replace (`) with on the right han sie (3.19). This yiels (3.14). We will use similar approximations for U > 0 as well. lthough it is possible to choose in such a way that only very few new terms are generate, the ow equations become very complicate. In orer to be able to analyse the ow equations, one has to neglect higher interactions. This is often possible ue to physical reasons. One woul like to unerstan why the self{consistency conition obtaine by replacing by on the right han sie of (3.19) yiels a goo approximation to the exact solution of (3.19). To iscuss this point let us introuce a special choice of f(; `). Since we want J(; `) to vanish in the limit `! 1, a natural choice woul be f(; `) = ( ). But for nite U we will have to make a ierent choice for f(; `) in the next section. For consistency we therefore take f(; `) = ( ) 3 =(4 2 ). This obviously works as well an it is easy to see that in the present case both choices are essentially equivalent. The following argument hols in both cases. Unless =, J(; `) ecays exponentially on a scale set by ` / 2 =( ) 4. If lies outsie the ban it will ten to exponentially an the approximation on the right han sie of (3.19) is justie. On the other han, if lies insie the ban, we can estimate the relevant `{scale on which changes by calculating the ratio of the total change of to its erivative with respect to ` for small `. This shows that changes on a scale set by ` / 2 =D4, i.e. much faster than J(; `) for values of near the Fermi energy. Therefore can be replace by its renormalize value on the right han sie of (3.19). We will use the same approximation in the next section to iscuss the case U > 0, it can be justie in the same manner. 4 Non{vanishing interaction U > 0 With the approximations introuce at the en of the last section, the ow equations for U > 0 may be written in the form V k ` = k( k ); (4.1) ` = 2X k U ` = 4X k k V k + 2 X k (2) k V k: (2) k V kn k ; (4.2) (4.3) ccoring to (2.7) we take (2) k = U k ( k + U) 1 an as above k = V k f( k ; `). We assume that P k V k 2 < D2 so that the renormalize is unique for U = 0. We expect that it is unique for U > 0 as well. With these assumptions we procee as in the previous section. We introuce J(; `) as in (3.4) an obtain the ow = 2f(; `)J(; `)( ); (4.4) 8

10 ` = Z U ` = 2 `) + (1 + ( )( + U) ; `) ( )( + U) : (4.6) n() is the Fermi istribution. The rst equation may be use to parametrize J(; `). suitable parametrization is Z `! ( ) 2 ( + U) 2 J(; `) = J(; 0) exp ` ( : (4.7) + U) 2 We will see that with this choice the hybriization ows to zero for all, in particular also for = or = + U. The reason is that ecays like ` 1=2 as we will see below. J( ; `) ecays algebraically to zero. Furthermore, Eq. (4.7) correspons to the following function f(; `) f(; `) = ( )( + U) ( + U) 2 : (4.8) The reason for this choice of f(; `) or J(; `) is that now no pole terms appear in the integrals on the right han sies of (4.5) an (4.6). The enominator in (4.8) is just introuce for convenience so that limits like lim U!1 can be performe in all the equations without iculties. Later we will come back to the question of other parametrizations f(; `). In fact Eq. (4.8) belongs to a class of parametrizations that all give the same physical results, whereas other parametrizations lea to ivergencies or J(; `) oes not ow to zero everywhere. Let us consier for a moment the simplie case lim U!1. The equation for takes the form ` = Z J(; 0)(1 + n())( ) exp Z ` 0 ( ) 2 `0! : (4.9) This equation is very similar to the ow equation for the renormalize tunneling frequency in the spin-boson problem [8]. The asymptotic behaviour can be obtaine as in [8], one ns (`) / ` 1 2 for large ` if lies insie the ban. Otherwise it ecays exponentially. (4.9) shows that 1= p` plays the role of an eective ban with if 1= p` becomes smaller than the original ban with of J(; 0). The eective ban with is reuce with increasing `. This is similar to a renormalization group proceure, where moes with high energies are integrate out. In our case, these moes are ecouple from the system. The analogies with renormalization theory will be worke out in more etail in section 7. For nite U the situation is somewhat more complicate, but the results are similar. The eective ban with is 1= p` an leas to an asymptotic behaviour U U(`) = C 1` 1 2 an (`) = C 2` 1 2 for large ` with some constants C 1 an C 2. This again hols if an + U lie insie the ban. C 1 an C 2 are positive if lies below an + U lies above the Fermi energy. One possibility to obtain these results is to make the ansatz U(`) = U + C 1` an (`) = C 2`. Inserting these expressions in the ow equations one shows easily that = = 1 is the only possible solution. We now replace U an 2 by their asymptotic values on the right han sie of (4.5) an (4.6). Both equations can be integrate an we obtain Z (`) = J(; `) + (1 + n())u ( )( + U ) ; (4.10) Z U(`) = U U + 2 J(; `) ( )( + U ) : (4.11) 9

11 These equations are goo approximations to the solution of the ow equations (4.5) an (4.6). They give the correct asymptotic behaviour an a numerical integration of the ow equations shows that the true solution iers only slightly from the approximate value. It shoul be note that this is a ierent level of approximation than the previous restriction to a certain set of interactions inclue in the ow equations. This restriction was a physical approximation whereas the approximate solutions (4.10) an (4.11) can be controlle by solving the original ierential equations numerically. We have one that too an always foun very goo agreement. For U = 0 we showe that (4.10) yiels the exact result. For U = 1, (4.10) is correct up to terms quaratic in J(; `). This can be seen if one notices that (4.10) is the exact solution of a set of equations similar to (3.6) an (3.7) but with J(; `) replace by J(; `)(1 + n()). n aitional argument to justify this approach is similar to the one given at the en of the previous section. The relevant `{scale for changes of an U is smaller than the scale on which J(; `) varies. The crossover to the asymptotic behaviour occurs for ` > D 2, whereas J(; `) oes not change too much on this scale. Taking ` = 0, (4.10) an (4.11) yiel the self{consistency conitions for an U. Let us mention that the results for an U obtaine from (4.10) an (4.11) o not epen on the special choice of f(; `) in (4.8). Nevertheless (4.10) an (4.11) are only goo approximations to the ow equations (4.5) an (4.6) for special choices of f(; `) like the one in (4.8). The important point is that J(; `) has to be chosen so that the principal value of the integrals in (4.10) an (4.11) is well{ene for all values of `. This is clearly true for J(; `) given in (4.7). Our results here o not epen on the etails of the continuous unitary transformation. But the transformation has to be chosen such that the ow for all the parameters in the Hamiltonian is well{ene. In section 7 we will see an example of a ierent parametrization of J(; `) where this is not the case. p Let us again consier the case J(; 0) = 2V 2 D 2 D2 2. The equation for U oes not contain a factor n() an the integral is easily evaluate. The result is U I = U 2 D U ( +( + U D)sign( D)sign( ) q + U ) ( + U ) 2 D 2 q( )2 D 2 : (4.12) This equation shows that if lies below the Fermi energy an + U lies above the Fermi energy, then U is larger than the initial value U I. This is also true if both, an + U lie in the ban. It this case we simply obtain U I U = : 1 2 D On the other han, if an + U lie above the energy ban, U is smaller than U I. (4.13) Symmetric nerson moel In the symmetric case U I = 2 I, the ow equations yiel U(`) = 2 (`) an the above conitions give U = 2 as it shoul be. Therefore we have q ( : (4.14) I = + 2 D D)sign( ) ( )2 D 2 This equation is very similar to the one obtaine for U = 0. If value is I < D 2 the renormalize = I =(1 2 =D) an lies insie the ban. If I > D 2 we obtain a quaratic 10

12 equation for. For 2 < D this equation has a single solution outsie the ban, whereas for 2 > D we can obtain two solutions of the self{consistency equations outsie the conuction ban similar to the case U = 0. symmetric nerson moel In the general case (U 6= 2 ), we have to calculate the integral in (4.10). It contains a factor n() ue to the normal orering we introuce in the Hamiltonian. Therefore the renormalize impurity energy epens on the temperature an the chemical potential. We let T = 0 an F = 0, so that n() = 1 (). In this case the integrals can easily be evaluate explicitly. We have to istinguish various cases of whether the impurity orbital energies lie insie the conuction ban or outsie. ; We obtain + U > D: I = 2D! 2 U + sign( + U ) q( + U ) 2 D arcsin D + U sign( ) q ( )2 D arcsin D! : (4.15) In the limit U I = 0 we have U = 0 an the conition for is the same as in the previous section. In the limit U = 1, we obtain a single equation for, I = + q sign( ) ( 2D )2 D arcsin D! + 2 D 3 : (4.16) In both expressions, > I. The impurity orbital energy is pushe away from the ban as it shoul have been expecte. < D; We obtain + U > D: I = 2 U + 2 q 2D D 2 ( )2 ln 0q 1 2 ( )2 + D! +sign( + U ) q( + U ) 2 D arcsin D + U : (4.17) If we let U = 1 in this case, this expression simplies to I = 3 2 2D D + 2 q D 2 ( )2 ln 0q 1 2 ( )2 + D : (4.18) 11

13 Finally for D U = I + ln 2D e + O( D ): (4.19) U This is shown in Fig. 2. Eq. (4.19) contains a logarithmic singularity on the right han sie for! 0. If I is negative an suciently far away from the Fermi energy, is negative as well an increases with increasing I. t some (still negative) value of I the renormalize impurity energy jumps iscontinuously from a given value below the Fermi energy to a value above the Fermi energy an then increases further with increasing I. This behaviour can be euce from the numerical solution of the ierential equations an is epicte by the full line in Fig. 2. However, it is icult to obtain reliable numerical results in this regime. never reaches the Fermi level except for the trivial case where V = 0 an I = 0. ; We obtain + U < D: I = 2 U + 2 q 2D 2 q D 2 ( + U ) 2 ln D 2 ( )2 ln D 2 ( )2 + D D 2 ( + U ) 2 + D + U 1 1 : (4.20) This expression contains two logarithmic singularities, it iverges if either! 0 or + U! 0. s a consequence, both an + U cannot approach the Fermi energy as long as V 6= 0. For U D one ns in particular = I + ln U + O(U ): (4.21) D gain the solution of Eq. (4.21) is non{unique for some range of the initial parameter I. In fact Eq. (4.21) is well{known from renormalization theory [6, 10, 16]: In the valence uctuation regime of the asymmetric nerson moel one has to replace I by an eective impurity orbital energy E (we use the notation from [6]). E can be obtaine as the solution of the following equation T 2 = E (T 2 ) ef = E (4.22) with E (T ) = I ln U T : (4.23) One easily checks E = with from the ow equations approach. This result for will play an important role for the calculation of the Kono temperature in this regime in 12

14 Sect. 5. It will then become apparent why it is important to n the eective impurity orbital energy from renormalization theory to be ientical to our asymptotic value. Finally it shoul be emphasize that the ow equations immeiately give the correct high{ energy cuto in Eqs. (4.19) an (4.21). The smaller of the two parameters U an D appears in the logarithm an this comes about here very naturally. These results, especially the self{consistency conitions in the general form Z = I + J(; 0) + (1 + n())u ( )( + U ) Z U = U I U 2 J(; 0) ( )( + U ) (4.24) (4.25) can be compare with the result of the Schrieer{Wol transformation. If one applies a usual Schrieer{Wol transformation to the Hamiltonian, the impurity energy an the interaction U are renormalize as well. But the result is a simple result of a secon orer perturbational treatment. It may be obtaine from our self{consistency conitions if they are solve recursively to rst orer in J(; 0). This simply means that on the right han sies of (4.24) an (4.25) the renormalize values are approximate by the initial values. s alreay mentione our result is exact if U = 0. In this case the self{consistency conition for can be solve iteratively. This correspons to summing up the whole perturbational series. Similarly, the self{consistency conitions (4.24) an (4.25) can be solve recursively though this will in general not give the exact result. But it is seems that a large part of the perturbational series is summe up when one solves these equations ue to the same reasons that were alreay mentione at the en of section 3. p One of the main results that we have obtaine for J(; 0) = 2V 2 D 2 D2 2 was that the renormalize values an + U behave iscontinuously at the Fermi energy as a function of the initial values I an I + U I. This is a consequence of the fact that ue to the normal orering a factor n() appears in (4.24). This result is generic an hols for a general function J(; 0). similar eect occurs at the ban ege if J(; 0) is not a continuous function at the ban ege. Then we obtain singularities in the integral in (4.24) if or + U approach the ban ege. Consequently these quantities behave iscontinuously at the ban ege as a function of the initial values. There is another interesting point that we woul like to mention. In the symmetric case U = 2 the renormalize value of U oes not epen on the temperature. If the system eviates only a bit from the symmetric case, (4.24) an (4.25) show that the system is pushe in the irection of the symmetric case. This can be seen if one takes = U=2+ an expans the right han sie of (4.24). The renormalize value of is smaller than the initial value of. In this sense the symmetric situation is stable. Near the symmetric point the temperature epenence of the renormalize values will be weak. But in the general case the renormalize values of the impurity energy an the interaction will epen on the temperature. For small enough temperature the integral in (4.24) can be evaluate using the usual Sommerfel expansion. This yiels = I + Z J(; 0) This shows that for + (1 + ( ))U ( )( + U ) + 6 (k B T ) 2 ( ) 2 ( + U ) 2 +O(T 4 )(4.26) + U > we obtain (T ) > (T = 0), whereas for + U < we obtain (T ) < (T = 0). Generally Eq. (4.26) is a goo approximation only for T < : 13

15 When T becomes larger one can show that (T ) will ecrease as a function of T for + U >. The temperature epenence of leas to a weak temperature epenence of U. In the special case J(; 0) / p D 2 2 we obtain a temperature epenence of U only if + U lie outsie the ban. If ; the temperature. 5 The inuce spin-spin interaction + U < D, (4.12) shows that U oes not epen on So far the contribution (2.8), which is generate by the unitary transformation, an which therefore has to be inclue in the Hamiltonian, was not taken into account. This interaction gives rise to a spin{spin coupling term 2 X k;q V (2) ( 1 y k;q k 2 ~ 1 q) ( y 2 ~ ) (5.1) or with k =!! c k;+ ; c = + ; (5.2) k; the so calle potential scattering term 1 2 X k;q V (2) k;q ( y k q) ( y ); an a term 1 2 X k;q (5.3) V (2) k;q (cy k; cy q; + y y c q; c k; ) : (5.4) The nal Hamiltonian contains no couplings between states that have a singly occupie impurity orbital an states for which the impurity orbital is either empty or oubly occupie. Whereas the spin{spin coupling (5.1) acts only on the part of the Hilbert space of states with a singly occupie impurity orbital, the term (5.4) vanishes on this part of the Hilbert space. It is important if the impurity orbital is either empty or oubly occupie. In this sense, these two terms are conjugate to each other. In fact one has a simple interpretation for these couplings in the symmetric nerson moel. Whereas the asymmetric nerson moel has only the usual SU(2){ spin symmetry, the symmetric nerson moel has an aitional SU(2){pseuo{spin symmetry. Introucing the wave vector, which has all components equal to, the symmetric energy ban has the symmetry k = k, V k = V k. For = U=2 the Hamiltonian also commutes with the operators ^S z = y + + y + X k (1 c y k;+ c k;+ c y k; c k; )! ^S + = + + X k ^S = y + y + X k c k;+ c k; (5.5) c y k; cy k;+ : These operators form the secon SU(2) symmetry mentione above. The potential scattering term (5.3) an the term (5.4) together can be written as a pseuo{spin interaction. It is clear 14

16 that if the original Hamiltonian has these symmetries, the transforme Hamiltonian has these symmetries too. Therefore the term (5.4) is present if the corresponing spin{spin interaction is present. lthough in the asymmetric case the Hamiltonian oes not have the aitional symmetry, the two terms (5.3) an (5.4) have the same interpretation. The only ierence is that now the coupling constant V (2) k;q is not symmetric with respect to a transformation k! k. In the remaining part of this section we want to iscuss the regime where the impurity orbital is singly occupie. Since y = 1 in this regime, the pseuo{spin interaction reuces to a scattering of ban electrons. Such a term has alreay been neglecte an therefore this contribution is not taken into account. In spite of the fact that the aitional couplings V (2) k;q are small compare to the other parameters of the nerson moel, they cannot be ignore. In the regime where the impurity orbital is singly occupie, < F an + U > F, even a small antiferromagnetic spin{spin coupling at the Fermi surface V (2) k F ;k F < 0 gives rise to the Kono eect for low temperatures. In the Kono moel the Kono temperature can be ene as T K = (2 imp (T = 0)) 1 [4] where imp (T = 0) is the impurity contribution to the susceptibility at zero temperature. Let us remark that other enitions of the Kono temperature can be foun in the literature, the enition by Wilson [5] is somewhat ierent. For a etaile iscussion see e.g. [4]. Base on a Bethe-ansatz solution, Tsvelick an Wiegmann argue that the Kono temperature T K is given by k B T K = 2 D exp h (2( F )V (2) k F ;k F ) with the universal function [5] i (y) = 1 jyj 1 ln jyj + O(y): (5.7) 2 For the Kono problem D is the conuction ban with. In the nerson impurity moel one knows that D in (5.6) has to be replace by an eective ban with D e that cannot be larger than U [10]. If one follows the perturbative calculation of e.g. the susceptibility in the Kono problem, one notices that the breakown of the perturbation expansion is ue to the matrix elements V (2) k F ;q: These escribe the scattering of an electron from the Fermi surface with the impurity to some wave vector q an then back to the Fermi surface. For this reason one is not only intereste in the coupling at the Fermi surface V (2) k F ;k F, but also in V (2) k F ;q since this etermines the eective ban with of the associate Kono problem. Let us now calculate the matrix elements V (2) k;q in the ow equations approach. We alreay mentione that the aitional couplings (2.8) o not lea to a contribution in the equations for an U. Therefore we calculate the coupling constant simply by integrating the coecient in front of the interaction term in (5.1). To be consistent with the notation in our previous paper we symmetrize this coecient. We have to calculate Z 1 V (2) = k;q `( (2) V k q + q (2) V k ): (5.8) 0 Using (2.7) an (3.3) to replace (2) an furthermore (4.4), we obtain V (2) k;q = 1 2 Z 1 0 `V k V q U ln J( k ln ( k )( k + U) (5.6) 1 : (5.9) ( q )( q + U) The `-epenence of V k is obtaine from (4.1). Using again (3.3) an (4.4) we obtain s J(k ; `) V k (`) = V k (0) J( k ; 0) : (5.10) 15

17 This yiels V (2) k;q = 1 2 V k(0)v q (0) Z 1 0 `U(J( k ; `)J( k ; 0)J( q ; `)J( q ; 0)) 1 k q ; `) ( k )( k + U) + 1 k ; `) : (5.11) ( q )( q + U) Using the parametrization for J(; `) introuce in (4.7), this expression simplies to V (2) k;q = 1 2 V k(0)v q (0) Z 1 0 exp 1 2 ` U ( k )( k + U) + ( q )( q + U) 2 + ( + U) 2! ( k ) 2 ( k + U) ( + ( q ) 2 ( q + U) 2 `0 + U) ( + U) 2 Z ` 0! : (5.12) Let us replace an U on the right han sie by their renormalize values. The same reasoning applies with respect to this approximation as at the en of section 3, in particular for the important matrix elements at the Fermi surface. One ns V (2) k;q = V k (0)V q (0)U ( k)( k + U ) + ( q)( q + U ) ( k) 2 ( k + U ) 2 + ( q) 2 ( q + U ) 2 : (5.13) This formula is a very goo approximation to (5.12) if k or q are not too close to or +U. If both ban energies become equal to or + U, the approximate result iverges. In this special case the asymptotic behaviour of an U becomes important. One can show that the integral in (5.12) has a logarithmic ivergence, which yiels a logarithmic ivergence of V (2) if k;q k an q approach or + U. Such a ivergence causes no problems since it is integrable as a function of the energy k ; also higher powers of V (2) k;q remain integrable. When calculating the Kono temperature, we only nee V (2) k;q for the case where at least one of the ban energies is equal (or at least very close) to the Fermi energy. Since or + U can only be equal to the Fermi energy if the hybriization vanishes, we can use (5.13) in the following. t the Fermi surface this yiels V (2) k F ;k F = V kf (0) 2 U ( + U ) ; (5.14) an with only one wave vector at the Fermi surface V (2) k F ;q = V k F (0) V q (0) U ( + U ) + ( q)( q + U ) ( )2 ( + U ) 2 + ( q) 2 ( q + U ) 2 : (5.15) Before proceeing with the calculation of the Kono temperature in various cases, it is interesting to compare this result with the coupling obtaine by the Schrieer{Wol unitary transformation in ef. [9]. There one ns V (2) k;q = 1 2 V k(0) V q (0) U I 1 ( I k)( I k + U I ) + 1 ( I q)( I q + U I ) in particular at the Fermi surface! ; (5.16) V (2) k F ;k F = V kf (0) 2 U I I (I + U I ) : (5.17) 16

18 s a rst remark we mention that for k = q both results are ientical if one replaces the initial values of I an U I in the result by Schrieer an Wol with the renormalize values an U. For k 6= q our result iers from the Schrieer{Wol result. The rst problem with the inuce spin{spin interaction in the formalism of Schrieer an Wol are the pole terms in V (2) if k F ;q I or I + U I lie in the conuction ban. secon problem is apparent in the following limit V (2) j qj!1 k F ;q! 1 2 V U I k F (0) V q (0) I (I + U 6= 0: (5.18) I ) This immeiately implies that the eective ban with in the corresponing Kono problem is of orer the conuction ban with D e / D; (5.19) which is known to be wrong. It is quite obvious from Eq. (5.15) that both these problems o not show up in the ow equations approach. In orer to obtain some more quantitative results, let us now iscuss two particular regimes of the nerson moel. Symmetric nerson moel with < D In the symmetric case one has coupling are = U =2 an the relevant matrix elements of the spin{spin V (2) = k F ;q V 2 k F (0) V q (0) U q 2 q (U ) 2 4 (U ) (U ) : (5.20) This is epicte in Fig. 3 where it can be compare with the Schrieer{Wol result. For simplicity we have assume V q (0) = V kf (0) for all wave vectors q in the iagram. Furthermore, we have replace U I by U in the Schrieer{Wol result, see also our iscussion in section 6 for this point. The Kono temperature epens mainly on the coupling at the Fermi surface (compare Eq. (5.6)) 2( F ) V (2) k F ;k F = : (5.21) U I D This is consistent with the Schrieer{Wol result except that we n an aitional (usually small) correction term 2 =D. The main ierence to the Schrieer{Wol result shows up in the eective ban with of the associate Kono problem. In the ow equations approach the eective ban with is obviously proportional to the on{site interaction D e / U (5.22) since the spin{spin coupling becomes ferromagnetic for j q j > U = p 2 an ecays to zero even further away from the Fermi surface. Since the spin{spin coupling inuce by our unitary transformation is not constant (there is no physical reason why it shoul be), it is icult to say quantitatively what the proportionality factor in Eq. (5.22) is: The Kono problem is usually only treate for a constant spin{spin coupling with the conuction ban electrons. Nevertheless, 17

19 we woul like to estimate the proportionality constant in (5.22) for the special case of a constant ensity of states an constant V q (0). This value can be compare with known results. To get some rather approximate estimate one can e.g. replace V (2) regare as a function of k F ;q q by its value at the Fermi surface in an interval aroun the Fermi surface that has the same area as the original curve. This shoul give a lower boun on D e. We obtain D e > 0:36 U : (5.23) This result can be compare with the result from the Bethe-ansatz solution. There one obtains k B T K = U p 4 exp( (2( F )V (2) k F ;k F )): (5.24) We ientify the prefactor with the one in (5.6) an obtain p D e = U = 0:443 U: 4 (5.25) This is in goo agreement with (5.23). Estimates similar to (5.23) can be obtaine for an arbitrary ensity of states an arbitrary V q (0). Valence uctuation regime 0 < U D This regime shows characteristic new features of the asymmetric nerson moel. The spin{ spin coupling is given by V (2) = k F ;q V 2 k F (0) V q (0) q ( (5.26) q) 2 + ( )2 with the value at the Fermi surface 2( F ) V (2) k F ;k F = 2 : (5.27) Essentially the same remarks apply as in the previous section. The coupling is epicte in Fig. 4 where it can be compare with the Schrieer{Wol result (again with V q (0) = V kf (0) an with I, U I replace by, U ). s before the ow equations approach yiels the correct scaling behaviour of the eective ban with D e / : (5.28) It is important that the \renormalize" value of the impurity orbital energy enters in Eqs. (5.26) an (5.27). This behaviour is known from perturbative renormalization [10, 16] or numerical renormalization [6] which give the same value of the renormalize impurity orbital energy that we have calculate in Eq. (4.21). In this regime the Schrieer{Wol unitary transformation not only gives the wrong scaling behaviour of the eective ban with D e / D, but also the wrong coupling at the Fermi surface since the initial value of the impurity orbital energy enters. t this point one can woner about the temperature epenence of that has been foun in Eq. (4.26). This eect seems to be unobserve in renormalization treatments. However, the maximum eect of non{zero temperature is to increase (T = 0) by a value of orer. ccoring to Fig. 2 the smallest value of with < 0 is = =, that is of orer. Thus one might expect to see some inuence of the temperature epenence of in this regime with <. But this is just the mixe valence regime (for k B T < ) where scaling breaks own anyway [16]. For this reason there is no contraiction. 18

20 6 Comparison with other methos I (Schrieer{Wol unitary transformation) t this point we woul like to explain in more etail why the Schrieer{Wol transformation an our continuous unitary transformation yiel ierent results. t a rst glance these two transformations are very similar. Our has the same general structure as the generator S in the Schrieer{Wol transformation H! e S He S [9] with S = X k; V k (0) U I ( I k)( I k + U I ) y c y k; + 1 c y k I k;! h:c: (6.1) In both approaches the same more complicate interactions generate ue to higher commutators are neglecte. But as we have seen the two transformations still show important ierences that have to be explaine. First of all, let us mention that it is possible to construct a continuous unitary transformation with the initial values of an U replace by their renormalize values. In fact one can achieve this by choosing the parametrization that reprouces the Schrieer{Wol result (5.16) for the coupling V (2) k;q f(; `) = instea of Eq. (4.8). Then the hybriization J(; `) shows a very simple ow (6.2) J(; `) = J(; 0) exp( l): (6.3) This is epicte in Fig. 5 where it can be compare with the behaviour for our choice of f(; `) from Eq. (4.8). s compare to the original Schrieer{Wol transformation this is an improvement since now the renormalize parameters enter into the expressions for the inuce spin{spin interaction at the Fermi surface V (2) k F ;k F. In the valence uctuation regime this is of importance as iscusse in the previous section. Still the main ierences between the Schrieer{Wol result an the ow equations are not resolve so easily. That is the couplings V (2) k;q show the wrong high{energy cuto an contain pole terms. Obviously our result for the spin{spin coupling iers from the result by Schrieer an Wol alreay in secon orer in the hybriization V k. This is ue to the fact that our transformation iers in this orer from the Schrieer{Wol transformation. In principle it is of course possible to write our transformation in the form exp(s) too. S can be calculate from the expansion S = Z 1 0 ` (`) Z 1 0 ` Z ` 0 `0[(`); (`0)] + : : : : (6.4) In secon orer in V k the term containing the commutator of at two ierent values of the ow parameter becomes important. It is of the form X k;q; S (2) k;q (: cy k; y c q; : : c y q; y c k; : : c y k; y c q; : + : c y q; y c k; : +c y k; cy q; y y c q; c k; ): (6.5) Our choice of f(; `) leas to a controlle expansion without any pole terms here: The worst ivergencies occuring in the generate couplings are only logarithmic pole terms an are therefore 19