1.3 Quantum Phase Transitions (QPT)
1.3.2 The Pseudogap Kondo Problems
Here we define the pseudogap kondo problems. These are special Kondo problems where conduction electron density of states (DOS) vanishes in a power law fashion at ω = 0 (ρc(w) ∼ |w|r, 0 ≤ r ≤ 1). Pseudogap Kondo problems have been extensively studied in recent years, by RG, NRG [41, 43, 44, 45, 46, 47] ,and slave-boson large N technique [48, 61]. By tuning the exponent r of the pseudogap DOS, ground state of pseudogap
1.3. QUANTUM PHASE TRANSITIONS (QPT)
Kondo systems may undergo a QPT between Kondo and local moment (LM) state. If exponent r is too large, the conduction DOS is not sufficient to Kondo screening, leading to LM ground state. On the other hand, Kondo screening was observed at T < TK if the exponent r is small enough that conduction electrons are sufficient to screen the impurity.
Pseudogap DOS in single channel and two-channel Kondo systems leads to QPT with particle-hole symmetry and asymmetry. The QPT in pseudogap Kondo can be analyzed by renormalization group (RG) techniques. We briefly introduce the rich ground phase diagrams of pseudogap Kondo system given by Matthias Vojta et. al. via perturbative RG and numerical renormalization group (NRG) approach [41, 43, 44].
Renormalization Group And Numerical Renormalization Group
Renormalization group (RG) approach originally comes from quantum field theory, and has been applied to condensed matter system. Anderson et. al. applied so called ”poor man scaling RG” to Kondo problem [49], where all the leading to logarithmic terms were summed up via perturbation theory. However, perturbation theory breaks down for T < TK as the system reaches the strong coupling Kondo ground state. K.G. Wilson use a non-perturbative technique: numerical renormalization group (NRG) approach to analyze Kondo physics at T < Tk [50, 51, 52]. There is the other non-perturbative method called Bethe ansatz, confirming Wilson’s NRG calculation [53]. The s-d model was given a definitive result for ground state by NRG calculation. In condensed matter system, NRG can accurately describe magnetic doped metals , while methods can not. The QPT of pseudogap Kondo problems can be studied by using perturbative RG and NRG approach.
RG phase diagrams for 1CK single impurity pseudogap Anderson model In results by Matthisa Vojta et. al. [41, 43, 44]. Pseudogap Kondo problems analyzed by perturbative RG leads to non-trivial fixed point and associated phase transitions.
Here, we introduce the single impurity Anderson model coupled to single channel electron reservoir with particle-hole symmetry and asymmetry. We briefly introduce particle-hole symmetry and asymmetry below. The full symmetry of 1CK single impurity pseudogap
1.3. QUANTUM PHASE TRANSITIONS (QPT)
Figure 1.13: The RG flow phase diagram of particle-hole symmetric single impurity pseu-dogap Anderson model. The details are in text. Adapted from [43].
Anderson model with particle-hole symmetry is SU(2)spin× SU(2)charge, where particle-hole symmetry is SU(2) pseudospin symmetry. The 1CK single impurity Anderson model as Eq.(1.1) is coupled to pseudogap conduction electron density of states,
H =X
The second term of in Eq.(1.1) was replaced by the bath Hamiltonian of pseudogap host conduction electron DOS. The other terms of Eq.(1.14) are the same as Eq.(1.1).
The Λ is the untraviolet (UV) cutoff. In the presence of particle-hole symmetry, the Coulomb potential is assumed as U0 = −2ǫd. And the Hamiltonian is invariant by below transformation,
d†σ −→ dσ, c†kσ −→ c−kσ.
On the other hand, if U0 6= −2ǫd, the particle-hole symmetry is broken. For example, strong Coulomb potential leads to particle-hole asymmetry. From NRG calculations, the
1.3. QUANTUM PHASE TRANSITIONS (QPT)
fixed-point structure changes at r∗and r = 1/2 , the relevant case of r=1 case is inaccessi-ble from weak coupling. [43, 45] r∗ = 0.375 is given by NRG. P-h symmetry is restored for 0 < r < r∗. The RG flows of the particle-hole symmetric 1CK single impurity pseudogap Anderson model is shown in Fig.(1.13). The horizontal axis denotes the renormalized dot level ǫ, where U = −2ǫ; the vertical axis is the renormalized hybridization V, hoping of dot and leads. The continuous boundary phase transitions were represented by the thick lines; the full (open) circles are stable (unstable) fixed points. Now, we introduce the fixed points corresponded to the phases,
LM : local moment ground state,
SC : strong coupling as Kondo-screened fixed point, SSC : symmetric strong coupling fixed point,
SCR : symmetry critical region fixed point, FImp : free impurity fixed point.
When r=0, the flow is towards to SC fixed point at any finite U. For 0 < r < 1/2 case, LM fixed points are stable, the SSC fixed point which is stable is located at ǫ = 0, and SCR (SCR’) fixed point control the phase transition between SSC and LM (LM’). The SSC fixed point becomes unstable as ≤ r < 1, and SCR (SCR’) fixed point disappears.
The phase transition between LM and LM’ is controlled by SSC fixed point. When r ≥ 1, there is no QPT. The first order phase transition with level crossing between LM and LM’.
The FImp fixed point is located U = 0, meaning that no hoping between leads and dot. As particle-hole asymmetric 1CK single impurity pseudogap Anderson model: The RG flow phase diagram was shown in Fig.(1.14). Particle-hole asymmetric Anderson model can be realized as Coulomb potential is too large, where U0 −→ ∞. The horizontal axis denotes the on-site dot energy levelǫ; the vertical axis is the fermionic coupling v. The bare on-site repulsion is fixed at u0 = ∞. Here, we introduce the fixed points corresponded to the phases,
LM : local moment ground state,
1.3. QUANTUM PHASE TRANSITIONS (QPT)
LM
ε v2
0 ∞
ASC a) r* < r < 1
ε v2
b) r ≥ 1
−∞ 0 ∞
LM
−∞
ASC VFl
ACR
Figure 1.14: The RG flow phase diagram of particle-hole asymmetric single impurity pseudogap Anderson model. The details are in text. Adapted from [43].
ASC : asymmetry strong coupling fixed point, ACR : asymmetry critical region fixed point, VFI : the valence fluctuation fixed point.
The hybridization V0 which is small leaves the moment unscreened, whereas large V0
directs the flow towards ASC fixed point. When r=0, the constant DOS, where the strong-coupling fixed point is the same as in the p-h symmetric situation. For 0 < r < r∗ case, particle-hole symmetry is restored. The phase transition is controlled by ACR unstable fixed point as r∗ < r < 1. When r ≥ 1, there is no QPT. The first order phase transition with level crossing controlled by VFL. The VFI fixed point is located at ǫ = U = 0.
1.3. QUANTUM PHASE TRANSITIONS (QPT)
Figure 1.15: The RG flow phase diagram of 2CK particle-hole asymmetric single impurity pseudogap Anderson model. The details are in text. Adapted from [44].
RG phase diagrams for 2CK single impurity pseudogap Anderson and Kondo model
In the results of Matthisa Vojta et. al. [41, 43, 44], the non-Fermi liquid (NFL) phase in 2CK Kondo model only survives in 0 < rmax region, where rmax = 0.23. The p-h asymmetry is irrelevant for r > 0.23. We will discuss pseudogap 2CK quantum phase transition by Kondo model and Anderson model as shown Fig.(1.15) and Fig.(1.16). The full (open) circles in diagrams are stable (unstable) fixed points and LM fixed means that local moment state. NFL represents non-Fermi liquid, ACR (SCR) represent critical p-h asymmetric (symmetric) point. For r=0 case, the lines of NFL fixed point represent non-Fermi liquid (over-screened Kondo effect, 2CK), it shows that flow is always towards NFL fixed point at any finite coupling. In the RG flows of 2CK pseudogap single impurity Kondo model as shown in Fig.(1.1.5): the horizontal axis denotes the renormalized Kondo coupling j, and renormalized potential scattering v which is representing particle hole asymmetry is the vertical axis. Dashed lines symbolize a flow out of the plane shown here.
1.3. QUANTUM PHASE TRANSITIONS (QPT)
For the metallic case (r=0), all lines flow into NFL fixed point at any finite coupling, 2CK governs the behaviors everywhere. LM is a unstable fixed point in the metallic case. In Fig.(1.15.b), the other non-Fermi liquid fixed point, NFL’ represents a phase at large couplings and asymmetries. For 0 < r < rmax, p-h asymmetry is irrelevant in non-Fermi liquid phase, a single p-h asymmetric NFL fixed point which separates NFL and NFL’
fixed points is a asymmetric critical region (ACR). The LM fixed point is stable here, and a critical p-h asymmetric fixed point (SCR) controls the phase transition between LM and NFL fixed points. For r > rmax = 0.23 case, there is no NFL phase as shown in Fig.(1.15.b). The other LM phase, LM’ fixed point represents a free local moment. The ACR fixed point controls phase transition between LM and LM’ fixed points. The RG flows of the 2CK pseudogap single impurity Anderson model is shown in Fig.(1.16). The horizontal axis denotes the energy difference between spin and flavor impurity levels, the renormalized hybridization g is the vertical axis. The diagrams represent cuts, taken at v = 0, through the full RG flow. LM and LM’ fixed points represent unscreened spin and free local moment phases, respectively. For r=0 case, the lines accessed to NFL fixed point at finite coupling ǫ, will access to LM fixed point (unstable) as ǫ → ±∞. For 0 < r < rmax, LM (LM’) fixed point becomes a stable fixed point. The NFL fixed point in Fig.(1.16.a) is replaced by ACR fixed point here, and two isolated p-h symmetric fixed point are located outside the u = 0 plane. The two SCR fixed points control phase transition between ACR and LM (LM’) fixed point. For rmax < r < 1, the phase transition between LM and LM’ can be controlled by ACR fixed point. The flow is towards to ACR fixed point as at ǫ = 0. For r ≥ 1, the transition is a level crossing, in other words, there is no QPT.
Free impurity fixed point(FIMP) is located at g = ǫ = 0, and flow is towards to FIMP at ǫ = 0.
1.3. QUANTUM PHASE TRANSITIONS (QPT)
Figure 1.16: The RG flow phase diagram of 2CK particle-hole asymmetric single impurity pseudogap Kondo model. The details are in text. Adapted from [44].
Chapter 2
Large N Approaches To 2CK Anderson Model
The ground state and thermodynamic behaviors of s-d model represented by the the non-degenerate Anderson model can be understood by the Fermi liquid theory [5], conformal field theory [32, 54], the Bethe ansatz solutions [55], renormalization group, and numerical renormalization group [51, 56]. The QPT of pseudogap Kondo problems with the N-fold degenerate Anderson model analyzed by RG and NRG have been introduced in Chapter 1 [41, 43, 44]. Here, we introduce one of large N approaches, so called non-crossing approximation (NCA) to solve the N-fold degenerate Anderson model, where N → ∞ is number of different spin flavor of fermions [57, 58, 60, 61]. The N-fold degenerate Anderson model with infinite U Coulomb potential can be solved by salve-boson representation [62].
The NCA approach has been successfully applied to Anderson impurity modes to address quantum field theory and critical phenomena. In this chapter, we will solve two-channel pseudogap Anderson model with infinite U Coulomb potential via NCA and salve-boson representation.
2.1. METHODS TO N-FOLD DEGENERATE ANDERSON MODEL