K 6= 0

The study of the system in which Kondo effect between dot1 and terminals of the ring and AF coupling between dot1 and dot2 coexist (K 6= 0) is the main point we discuss in this thesis. Three subjects we are concerned: what happened when changing the Kondo coupling constant K, the size effect in this system, and the phase transition between Kondo state and singlet state in thermodynamic limit. In the following, we discuss the three subjects by changing L and K independently and use these observations to piece up the phase diagram which is sketched in Fig(4.1).

A.

0 0.005 0.01 0.015 0.02 0.025 0.03 0

Figure 4.3: T_k as functions of K. (a)Case I: L=4n (b)Case II: L=4n+2 Other parameters of data: ²d = -0.8, tR = tL = 0.4, φ = 0 in units of t.

We monitor how the Kondo resonance is destroyed by varying the AF coupling strength K at a fixed finite size L. Fig.(4.3) shows T_k as a function of AF coupling strength K at a fixed size L = 300. In both case (I) and (II), T_k vanishes as K → K_c2, indicating the suppression of Kondo resonance by the AF interaction. Moreover, in both cases the decay of Tk close to K = Kc2 follows an exponential form: ln Tk ∝ −1/|K − Kc2|, the characteristic of the KT transition. However, there are minor differences in how fast T_k vanishes as K close to K_c2. In case (II), T_k remains a constant over a wider range of K compared to that in case (I) before it decays. This suggests that the Kondo effect at a finite size seems more robust in case (II) than in case (I) so that the crossover region in case (II) is much narrower than in that in case (I).

B.

Here, we present the finite-size dependence of the Kondo resonance at a fixed AF cou-pling strength K. In analogous to the Numerical Renormation Group (NRG) method, we

decrease the energy scale 1/L by increasing the system size until we reach the thermody-namic limit L → ∞ (or zero temperature). For the sake of convenience, at a fixed K we may define the crossover region by two finite sizes Lc1 and Lc2 (similar to the case with L fixed), where L_c1 is the critical size which separates the Kondo and the crossover regime and L_c2 is the critical size distinguishing the crossover regime and singlet state.

First, we describe the qualitative behaviors of T_k at finite sizes. We devide our discus-sions in the two cases mentioned above. For case (I) (represented by L = 4n, see Fig.4.4) T_k increases with increasing L as the crossover region is for K < K_c where the Kondo resonance is recovered at large system size; while for case (II) (L = 4n + 2, see Fig.4.5) since the crossover occurs for K > Kc, Tk decreases with increasing L. Note that in case (I) with large K and case (II) with small K, the ground state remains at local singlet and Kondo state, respectively; therefore, no crossover behaviors are identified.

0.001 0.01

Figure 4.4: T_k/T_k⁰ as functions of 1/L: (a) L=4n, K varies from 0.0025 to 0.015, distance:

0.0025. (b) L=4n+2, K varies from 0.036 to 0.04, distance: 0.001. Other parameters of data: ²d = -0.8, tR = tL = 0.4, φ = 0, in units of t.

Next, we perform the finite-size scaling for Tk near the transition to investigate the nature of the phase transition in the thermodynamic limit. We find T_k in case (I) and (II) follows its own unique universal scaling function of 1/(T^∗L) as K → K_c (see Fig. 4.4

and Fig. 4.5), where

T^∗ = cfT_kexp (−πfT_k/ | K − K_c |) (4.7)

with fT_k = c⁰T_k has the same form in both case (I) and (II). Here, c, c⁰, K_c are non-universal prefactors depending on the initial parameters of the Hamiltonian, which can be fitted by the numerics: c ≈ 5.5, K_c ≈ 0.0271 in both case (I) and (II); c⁰ ≈ 0.65 in case (I), c⁰ ≈ 0.5 for case (II). Note that here K can be either smaller (case (I)) or larger (case (II)) than Kc. The universal scaling at finite sizes and the exponential form for the crossover energy scale T^∗ indicate that in the thermodynamic limit the system exhibits the Kosterlitz- Thouless transition at a finite critical AF coupling strength K_c. We have checked the consistency of our result from our finite-size scaling that K_c indeed reaches to the same value in the thermodynamic limit for both case (I) and (II) even though the corresponding crossover regions are on the opposite side of the transition (K < K_c for case (I) and K > Kc for case (II)).

Note that unlike the similar side-coupled double-dot system studied previously where the KT transition between the Kondo and spin-singlet phase occurs at K_c= 0, we find in our setup a finite K_cfor the same KT transition. This difference can be understood in the following. In the previous side-coupled double-dot setup, the conduction electron bath consists of Fermi-sea with continuous spectrum and a constant density of state. However, in our current setup the conduction electrons are made of a tight-binding ring with a finite size. where a is lattice constant. Therefore, the density of state D(²) in 1D tigh-binding ring is given by

D(²) = dn

d² ∝ 1

ap

(4t)²− ²²(k) (4.8)

where ²(k) = −2tcos(ka) is the energy for 1D tight-binding ring and a is the lattice constant. It is clear that the DOS for the tight-binding ring exhibits singular behaviors for (4t)² − ²²(k) ≈ 0. It is expected that the more singular DOS of the conduction

electron bath in the current setup makes the Kondo resonance more robust against the AF coupling and therefore leads to a finite K_c.

0.001 0.01 distance: 0.0025. (b) L=4n+2, K varies from 0.036 to 0.04, distance: 0.001. T_k follows universal scaling functions of 1/(T^∗L). Other parameters of data: ²_d = -0.8, t_R = t_L = 0.4, φ = 0, in units of t.

Section 4.2 is summarized in this paragraph. Phase transition which describes the competition between spin-singlet state and Kondo state in the system is the most impor-tant thing we get from this section. The phase transition we get in this section help us understand the phase diagrams in thermodynamic limit. Since sign of L dependence in case II is different with that in case I, the majority crossover regimes in phase diagrams of two cases are in the opposite side of K_C. The phase diagram of case I and II are sketched in Fig.(4.1(a)) and Fig.(4.1(b)), respectively. Although the crossover regime is different between the case I and case II, they merge into a whole and follow K–T transition with finite K_C when L → ∞ which we get from scaling methods. In 4.3 and 4.4, we will discuss the density of state and the persistent current and use these two parameters to check the phase diagrams that we sketched in section 4.1.