Multi-scale competition in the Majorana-Kondo system

TL;DR

This paper investigates a Majorana-Kondo system with a side-coupled Majorana zero mode, using full-density-matrix NRG to study how the Kondo screening cloud evolves with temperature under competing perturbations. It introduces a temperature-dependent observable, the integral of the screening cloud $I_L$, to quantify spin and charge screening and reveals a crossover from conventional spin Kondo screening to spin-charge-entangled (SCE) screening governed by an $A\otimes N$ boundary condition. The analysis identifies multiple energy scales, $T_K$, $T_{\rm SCE}$, $T_M$, and $T_{A\otimes N}$, and shows how MZM-MZM coupling and ${\rm SU}_{\bf L}(2)$ symmetry breaking modify the boundary conditions and spectral properties, including the LDoS and impurity entropy. The results demonstrate that the SCE state can persist even when the $A\otimes N$ BC is weakened or broken, with the non-Fermi liquid fixed point protected by topology and symmetry, offering a clear framework for interpreting Majorana-related signatures in hybrid impurity systems.

Abstract

A side-coupled Majorana zero mode in Kondo systems realizes a simple yet nontrivial hybridization that profoundly alters the low-energy physics, making such setups promising candidates for detecting Majorana zero modes. Recently, we demonstrated that the low-energy behavior of this system can be captured by a spin-charge-entangled screening process with an \(A\otimes N\) boundary condition. Here, we investigate the evolution of both the screening cloud and the boundary condition in the presence of competing terms that could break either the spin-charge-entangled \({\rm SU}_{\bf L}(2)\) rotation symmetry or the topological degeneracy. We introduce a temperature-dependent spatial integral of the screening cloud, which can be obtained from the numerical renormalization group. This quantity serves as a proper observable that unambiguously captures the properties of the screening process across temperatures. A clear crossover from conventional Kondo spin screening to spin-charge-entangled screening is observed. Taking into account the overlap between Majorana zero modes, the \(A\otimes N\) boundary condition reduces to a normal one, yet the spin-charge-entangled screening is protected by the \({\rm SU}_{\bf L}(2)\) symmetry. On the other hand, perturbation that breaks the \({\rm SU}_{\bf L}(2)\) symmetry can destroy the screening singlet, while leaving the low-temperature \(A\otimes N\) boundary condition intact.

Figures (4)

• Figure 1: Schematic plots of the system. (a) The 1d setup consists of a fermion bath, a QD, and a TSC with two MZMs $\gamma_{-1}$ and $\gamma_{-2}$. The curves above are schematic diagrams of the density of states (DoS) of the two bulks. (b-d) Local energy spectrum of the MZM-QD sites for different cases. (b) In $\mathbbm{Z}_2\times {\rm SU}_{\bf L}(2)$ symmetric case, the 4-fold degenerate local spectrum is two pairs of SU(2) doublets, labeled by the parity and pseudospin $z$-components $(P,L_z)$. (c) When the Majorana overlap $\epsilon_m$ is turned on, the $\mathbbm{Z}_2$ topological degeneracy disappears and a gap opens between the odd parity states and the even parity states. (d) When the effective Zeeman field $h_U$ is turned on, the ${\rm SU}_{\bf L}(2)$ is broken and a gap opend between the $L_z$-up and -down states.
• Figure 2: Results for the $\mathbbm{Z}_2\times {\rm SU}_{\bf L}(2)$ symmetric point. Schematic illustration of (a1) the spin Kondo singlet, where the MZMs are decoupled from the QD-NL subsystem, and (a2) the SCE screening cloud with an $A\otimes B$ BC. (b-d) Numerical results for different $J_M$. $J_K$ is chosen as $0.2$ to display the local moment fixed point at $S_{\rm imp}=2{\rm ln2}$. (b) The impurity entropy $S_{\rm imp}$. (c) The spin and charge components of the correlation function. Both of them are normalized to the zero-temperature result $\tilde{I}_i=I_i/I_i(T=0)$.
• Figure 3: Parity asymmetric point results. Physical parameters are set to $J_K=1$ and $J_M={\rm 1e-4}$. Schematic illustration of (a1) spin Kondo singlet where the decoupled MZMs become bound due to the overlap, and (a2) the fully screened SCE singlet without the $A\otimes N$ BC. (b-d) Numerical results with tuning $\epsilon_m$. (b) Nonzero coupling between MZMs breaks the IR fixed point $S_{\rm imp}=\frac{1}{2}{\rm ln}2$. The far MZM $\gamma_{-2}$ is screened under low temperature. Large overlap merge the MZMs pair to one normal spinless fermion and removes the non-Fermi liquid stage. (c) The correlation functions verify that the SCE screening cloud establishes when the $S_{\rm imp}={\rm ln2}$ plateau vanishes, regardless of the non-Fermi liquid fixed point. (d) The local density of states.
• Figure 4: ${\rm SU}_{\bf L}(2)$ asymmetric results. (a) Schematic picture of the polarized pseudospin. (b-d) Numerical results with tuning $h_U$. (b) The effective Zeeman field polarizes the pseudospin ${\bf L}$ and eliminate the free moment fixed point, also makes the $S_{\rm imp}={\rm ln}2$ crossover from a Kondo singlet to a single-body state. (c) The decrease of the spin correlation function $I_S$ indicates the disappearance of the ${\bf L}$-singlet. The upper right pannel clearly shows the variation of the zero-temperature correlation function $I_S(T=0)$. (d) The splitting of LDoS for finite Zeeman field. The main plot is curves of the summation over all components $\rho(\omega)=\rho_{\uparrow}$+$A_{\uparrow}$+$\rho_{\downarrow}$, which are also plotted separately in the upper right pannel by dotted curves.