Physical meanings

Discussing the physical meanings of each step in the minimization routine helps us to know more about why the iteration routine goes. In the following, we discuss the physical meanings step by step.

(a) Guess the basis is in the mountain side of the valley of energy.

(b) If the point 3R is in the lower position respected to point 1, the minimum may in a farther position in the 3R direction. So, the method chooses Reflection & Expansion as next step.

(i) If E_3E&R < E_3R, the lowest point changes to 3E&R and then the new simplex is set by point 1, 2, 3E&R.

(ii) If E_3E&R > E_3R, the lowest point is still 3R and then the new simplex is set by points 1, 2, 3R.

(c) If the point 3R is in the higher position in the other side of the valley, point 1 may be close to minimum. So, the method chooses Contraction as next step.

(i) If E3C < E3, the contraction works and the new simplex is set by points 1, 2, 3C.

(ii) If E3C > E3, the contraction fails which means point 1 might be the minimum.

So, the initial points of simplex shrinks toward point 1 and the new initial simplex is set by points 1, 2S, 3S.

Chapter 4 Results

Before we present our new results, it is useful to summarize the behavior of the model at K = 0 (no direct antiferromagnetic spin-exchange coupling), which has been well studied[12][13]. The Kondo resonance of a quantum dot embedded in a mesoscopic ring strongly depends on the finite size L (mod 4), the magnetic flux threading the ring. In particular, the Kondo temperature shows an universal scaling of ξ_k⁰/L at a given magnetic flux. In the Kondo regime, all physical observables, such as: Kondo temperature T_k, the Kondo screening cloud size ξk, persistent current I are enhenced as the size L increases, but with different crossover bahaviors in all four cases of L = 4n, L = 4n + 1, L = 4n + 2, and L = 4n + 3. The magnetic flux dependence of persistent current exhibits a symmetry between size L and L + 2: I_L(φ) = I_L+2(φ + π), indicating that adding the magnetic flux of π is equivalent to switch the behavior of the PC from a system with size L to L + 2.

With the previous results in mind, we may discuss the general properties for K > 0.

Due to the direct antiferromagnetic spin-exchange coupling (AF coupling), we expect in this case the competition between the Kondo and the local spin-singlet ground states, leading to the quantum phase transition. In fact, quantum phase transitions in double-quantum-dot systems with AF coupling have been intensively studied in recent years [7][8]

where the quantum dot coupled to the conduction electron Fermi sea with continuous spectrum. Two types of quantum phase transitions were indentified: the phase transition

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with a quantum critical point and the one of the Kosterlitz-Thouless type. The former type is realized in the double-dot systems where each of the dot couples to an independent conduction electron reservoir, and the critical point seperating the Kondo from the local spin-singlet phase is the well-known two-impurity Kondo fixed point[14]. The latter type exsits in a side-coupled double-dot system where only one of the dots coupled to the electron reserviour[8]. The Kondo resonance becomes more fragile in the side-coupled system so that an infinitesmall AF coupling is sufficient to suppress the Kondo effect and leads to the spin-singlet ground state. We expect the similar Kosterlitz-Thouless transition to occur in our side-coupled double-dot system. However, the mesoscopic ring in our setup consists of tight-binding electrons with a finite size, the details of the transition might be different (see below).

4.1 Mean-field phase diagram

Figure 4.1: Schematic mean-field phase diagram of the model for (a)L=4n, 4n+1, 4n+3 (case (I) in the text) and (b)L=4n+2 (case (II) in the text). The red line indicates the phase variation with K being fixed and the green line indicates the phase variation with L being fixed.

After solving the mean-field equations, we summarize our main results in the schematic mean-field phase diagram as shown in Fig.4.1. We find indeed the transition between the Kondo and spin-singlet phases is of the Kosterlitz-Thouless type. However, our new finding is that the critical point K_c separating the two phases is not at zero as shown in the similar side-coupled double-dot system studied previuously but at a finite value:

K_c > 0. (The details in deriving that K–T transition exist in our system are in the below section)

There are three regions in the phase diagram, corresponding to different mean-field solutions:

1. For small K, the ground state in the thermodynamic limit is the Kondo phase, which has been studied previously: b₀ 6= 0, λ 6= 0, χ = 0.

2. For large K, the ground state for L → ∞ is the local spin-singlet phase: b₀ = 0, λ 6= 0, χ 6= 0.

3. The crossover region between Kondo and spin-singlet phases indicated in the shaded region in Fig.4.1: b₀ = 0, λ 6= 0, χ 6= 0. This region is defined by K_c1 < K < K_c2 for a fixed size L where Kc1 is the critical value seperating the Kondo and the crossover region and Kc2 the critical value seperating the crossover and the local spin-singlet phase. There are two different crossover paths, depending on the finite size L (mod 4):

1. Case (I) (Fig.4.1 (a)) holds for L = 4n, 4n + 1, 4n + 3 where the crossover behavior occurs for K < K_c.

2. Case (II) (Fig.4.1 (b)) occurs for L = 4n + 2 where the crossover ranges mainly for K > Kc.

In both cases, the crossover energy scale T^∗ follows the behavior of the typical

Kosterlitz-Thouless transition:

T^∗ = cfT_kexp [−πfT_k/(k − kc)] (4.1)

where c is a non-universal constant.

We investigate the Kondo effect in our setup at finite sizes by either changing the size L at a fixed K or changing K at a fixed size L. Our scaling at finite size indicates that in the thermodynamic limit L → ∞, K_c1 and K_c2 converge to a single critical value K_c: K_c1 = K_c2→ K_c.