Topological Superconductivity Intertwined with Broken Symmetries

Recently the superconductor and topological semimetal PbTaSe$_2$ was experimentally found to exhibit surface-only lattice rotational symmetry breaking below $T_c$. We exploit the Ginzburg-Landau free energy and propose a microscopic two-channel model to study possible superconducting states on the surface of PbTaSe$_2$. We identify two types of topological superconducting states. One is time-reversal invariant and preserves the lattice hexagonal symmetry while the other breaks both symmetries. We find that such time-reversal symmetry breaking is unavoidable for a superconducting state in a two dimensional irreducible representation of crystal point group in a system where the spatial inversion symmetry is broken and the strong spin-orbit coupling is present. Our findings will guide the search for topological chiral superconductors.Comment: 4+5 pages, 5 figure

Floquet topological insulator phase in a Weyl semimetal thin film with disorder

We investigate the effects of periodic fields and disorder on topological properties of a Weyl-semimetal thin film. The two periodic fields, i.e., a periodic magnetic field and elliptically polarized light, are discussed respectively. By use of the Floquet theory, we find that both the two periodic drives can resonantly induce the topological transitions from normal insulator (NI) phases to Floquet topological insulator (FTI) phases. The Floquet topological transitions are characterized by variation of Chern number. Moreover, we show that the Floquet topological transitions can be explained by a combination of the quantum well approximation and the rotating wave approximation. In the disordered Weyl-semimetal thin film model under periodic fields, we calculate the Bott index to characterize topological phase. It is found that the FTI phase is robust against weak disorder, and collapses for strong disorder strength. Interestingly, we find that disorder can also induce a topological transition from a topological trivial phase to an FTI phase, establishing the Floquet topological Anderson insulator (FTAI) phase. Finally, an effective-medium theory based on the Born approximation further confirms the numerical conclusions

Topological Anderson insulator phase in a Dirac-semimetal thin film

The recently discovered topological Dirac semimetal represents a new exotic quantum state of matter. Topological Dirac semimetals can be viewed as three dimensional analogues of graphene, in which the Dirac nodes are protected by crystalline symmetry. It has been found that quantum confinement effect can gap out Dirac nodes and convert Dirac semimetal to a band insulator. The band insulator is either normal insulator or quantum spin Hall insulator depending on the thin film thickness. We present the study of disorder effects in thin film of Dirac semimetals. It is found that moderate Anderson disorder strength can drive a topological phase transition from normal band insulator to topological Anderson insulator in Dirac semimetal thin film. The numerical calculation based on the model parameters of Dirac semimetal Na$_{3}$Bi shows that in the topological Anderson insulator phase a quantized conductance plateau occurs in the bulk gap of band insulator, and the distributions of local currents further confirm that the quantized conductance plateau arises from the helical edge states induced by disorder. Finally, an effective medium theory based on Born approximation fits the numerical data

Disorder-induced topological phase transitions on Lieb lattices

Motivated by the very recent experimental realization of electronic Lieb lattices and research interest on topological states of matter, we study the topological phase transitions driven by Anderson disorder on spin-orbit coupled Lieb lattices in the presence of spin-independent and dependent potentials. By combining the numerical transport and self-consistent Born approximation methods, we found that both time-reversal invariant and broken Lieb lattices can host disorder-induced gapful topological phases, including the quantum spin Hall insulator (QSHI) and quantum anomalous Hall insulator (QAHI) phases. For the time-reversal invariant case, this disorder can induce a topological phase transition directly from normal insulator (NI) to the QSHI. While for the time-reversal broken case, the disorder can induce either a QAHI-QSHI phase transition or a NI-QAHI-QSHI phase transition. Remarkably, the time-reversal broken QSHI phase can be induced by Anderson disorder on the spin-orbit coupled Lieb lattices without time-reversal symmetry.Comment: accepted for publication in Phys. Rev.

The effect of in-plane magnetic field and applied strain in quantum spin Hall systems: application to InAs/GaSb quantum wells

Motivated by the recent discovery of quantized spin Hall effect in InAs/GaSb quantum wells\cite{du2013}$^,$\cite{xu2014}, we theoretically study the effects of in-plane magnetic field and strain effect to the quantization of charge conductance by using Landauer-Butikker formalism. Our theory predicts a robustness of the conductance quantization against the magnetic field up to a very high field of 20 tesla. We use a disordered hopping term to model the strain and show that the strain may help the quantization of the conductance. Relevance to the experiments will be discussed.Comment: 8 pages, 10 figures. Comments are welcome

Optimal Actuator Location of the Minimum Norm Controls for Heat Equation with General Controlled Domain

In this paper, we study optimal actuator location of the minimum norm controls for a multi-dimensional heat equation with control defined in the space $L^p(0,T;L^2(\Omega))$. The actuator domain $\omega$ is quite general in the sense that it is required only to have a prescribed Lebesgue measure. A relaxation problem is formulated and is transformed into a two-person zero-sum game problem. By the game theory, we develop a necessary and sufficient condition and the existence of relaxed optimal actuator location for $p\in[2,+\infty]$, which is characterized by the Nash equilibrium of the associated game problem. An interesting case is for the case of $p=2$, for which it is shown that the classical optimal actuator location can be obtained from the relaxed optimal actuator location without additional condition. Finally for $p=2$, a sufficient and necessary condition for classical optimal actuator location is presented.Comment: 41 page

Waiting times and stopping probabilities for patterns in Markov chains

Suppose that $\mathcal C$ is a finite collection of patterns. Observe a Markov chain until one of the patterns in $\mathcal C$ occurs as a run. This time is denoted by $\tau$. In this paper, we aim to give an easy way to calculate the mean waiting time $E(\tau)$ and the stopping probabilities $P(\tau=\tau_A)$ with $A\in\mathcal C$, where $\tau_A$ is the waiting time until the pattern $A$ appears as a run.Comment: 13 page

Theory for Spin Selective Andreev Reflection in Vortex Core of Topological Superconductor: Majorana Zero Modes on Spherical Surface and Application to Spin Polarized Scanning Tunneling Microscope Probe

Majorana zero modes (MZMs) have been predicted to exist in the topological insulator (TI)/superconductor (SC) heterostructure. Recent spin polarized scanning tunneling microscope (STM) experiment$^{1}$ has observed spin-polarization dependence of the zero bias differential tunneling conductance at the center of vortex core, which may be attributed to the spin selective Andreev reflection, a novel property of the MZMs theoretically predicted in 1-dimensional nanowire$^{2}$. Here we consider a helical electron system described by a Rashba spin orbit coupling Hamiltonian on a spherical surface with a s-wave superconducting pairing due to proximity effect. We examine in-gap excitations of a pair of vortices with one at the north pole and the other at the south pole. While the MZM is not a spin eigenstate, the spin wavefunction of the MZM at the center of the vortex core, r = 0, is parallel to the magnetic field, and the local Andreev reflection of the MZM is spin selective, namely occurs only when the STM tip has the spin polarization parallel to the magnetic field, similar to the case in 1-dimensional nanowire2. The total local differential tunneling conductance consists of the normal term proportional to the local density of states and an additional term arising from the Andreev reflection. We also discuss the finite size effect, for which the MZM at the north pole is hybridized with the MZM at the south pole. We apply our theory to examine the recently reported spin-polarized STM experiments and show good agreement with the experiments.Comment: 14 pages, 14 figures, 1 table. Comments are welcome

Optimal operating protocol to achieve efficiency at maximum power of heat engines

The efficiency at maximum power has been investigated extensively, yet the practical control scheme to achieve it remains elusive. We fill such gap with a stepwise Carnot-like cycle, which consists the discrete isothermal process (DIP) and adiabatic process. With DIP, we validate the widely adopted assumption of \mathscr{C}/t relation of the irreversible entropy generation S^{(\mathrm{ir})}, and show the explicit dependence of the coefficient \mathscr{C} on the fluctuation of the speed of tuning energy levels as well as the microscopic coupling constants to the heat baths. Such dependence allows to control the irreversible entropy generation by choosing specific control schemes. We further demonstrate the achievable efficiency at maximum power and the corresponding control scheme with the simple two-level system. Our current work opens new avenues for the experimental test, which was not feasible due to the lack the of the practical control scheme in the previous low-dissipation model or its equivalents.Comment: 7 pages, 4 figures, comments are welcome ([email protected]

Finite-size effects in non-Hermitian topological systems

We systematically investigate the finite-size effects in non-Hermitian one-dimensional (1D) Su-Schrieffer-Heeger (SSH) and two-dimensional (2D) Chern insulator models. Using a combination of analytical and numerical calculations, we show that the non-Hermitian intra-cell hoppings in the SSH model can modify the localization lengths of bulk and end states, giving rise to a complex finite-size energy gap that exhibits an oscillating exponential decay as the chain length grows. However, the imaginary staggered on-site potentials in the SSH model only change the end-state energy, leaving the localization lengths of the system unchanged. In this case, the finite-size energy gap can undergo a transition from real values to imaginary values. We observed similar phenomena for the finite-size effect in 2D Chern insulator systems.Comment: 12 pages, 12 figures. Accepted by Physical Review