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arXiv:1601.03990 · Physical review. B./Physical review. B

Read-Green resonances in a topological superconductor coupled to a bath

Pieter W. Claeys, Stijn De Baerdemacker, Dimitri Van Neck

Published Jun 9, 2016• 33 citations
arXivHTMLPDF

Abstract

We study a topological superconductor capable of exchanging particles with an environment. This additional interaction breaks particle-number symmetry and can be modeled by means of an integrable Hamiltonian, building on the class of Richardson-Gaudin pairing models. The isolated system supports zero-energy modes at a topological phase transition, which disappear when allowing for particle exchange with an environment. However, it is shown from the exact solution that these still play an important role in system-environment particle exchanges, which can be observed through resonances in low-energy and low-momentum level occupations. These fluctuations signal topologically protected Read-Green points and cannot be observed within traditional mean-field theory.

DOI
10.1103/physrevb.93.220503

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We study a topological superconductor capable of exchanging particles with an environment. This additional interaction breaks particle-number symmetry and can be modeled by means of an integrable Hamiltonian, building on the class of Richardson-Gaudin pairing models. The isolated system supports zero-energy modes at a topological phase transition, which disappear when allowing for particle exchange with an environment. However, it is shown from the exact solution that these still play an important role in system-environment particle exchanges, which can be observed through resonances in low-energy and low-momentum level occupations. These fluctuations signal topologically protected Read-Green points and cannot be observed within traditional mean-field theory.
Read-Green resonances in a px+ipyp_x+ip_ypx​+ipy​ superfluid interacting with a bath
We study the weak-pairing phase in a finite-size two-dimensional px+ipyp_x+ip_ypx​+ipy​ superfluid interacting with an environment. This interaction breaks particle-number symmetry and can be modelled by means of an integrable Hamiltonian. We present the exact wave function and solve the resulting Bethe ansatz equations, from which it is shown how resonances are introduced at fixed interaction strengths. These cannot be observed within mean-field theory and are argued to be caused by zero-energy excitations at topologically protected Read-Green points.
In this work, we study the full set of eigenstates of a px+ipy{p}_{x}+i{p}_{y}px​+ipy​ topological superconductor coupled to a particle bath, which can be described in terms of an integrable Hamiltonian of the Richardson-Gaudin class. The results derived in this work also characterize the behavior of an anisotropic XXZ central spin model in an external magnetic field since both types of Hamiltonian are known to share the exact same conserved quantities, making them formally equivalent. We show how by ramping the coupling strength (or equivalently the magnetic field acting in the zzz-direction on the central spin), each individual eigenstate undergoes a sequence of gain/loss of excitations when crossing the specific values known as Read-Green points. These features are shown to be completely predictable for every one of the 2N{2}^{N}2N eigenstates, using only two integers easily obtainable from the zero-coupling configuration, which defines the eigenstate in question. These results provide a complete map of the particle-number sectors (superconductor) or magnetization sectors (central spin) involved in the large number of level crossings that occur in these systems at the Read-Green points. It further allows us to define quenching protocols that could create states with remarkably large excitation-number fluctuations.
What distinguishes trivial superfluids from topological superfluids in interacting many-body systems where the number of particles is conserved? Building on a class of integrable pairing Hamiltonians, we present a number-conserving, interacting variation of the Kitaev model, the Richardson-Gaudin-Kitaev chain, that remains exactly solvable for periodic and antiperiodic boundary conditions. Our model allows us to identify fermion parity switches that distinctively characterize topological superconductivity (fermion superfluidity) in generic interacting many-body systems. Although the Majorana zero modes in this model have only a power-law confinement, we may still define many-body Majorana operators by tuning the flux to a fermion parity switch. We derive a closed-form expression for an interacting topological invariant and show that the transition away from the topological phase is of third order.
It is known that, under short-range interactions many topological superconductors (TSC) and topological insulators (TI) are trivialized, which means the boundary state of the system can be trivially gapped out by interaction without leading to symmetry breaking or topological ground state degeneracy. This phenomenon is also referred to as "symmetric mass generation" (SMG), and has attracted broad attentions from both the condensed matter and high energy physics communities. However, after the trivialization caused by interaction, some trace of the nontrivial topology of the system still persists. Previous studies have indicated that interacting topological TSC and TI could be related to the "zero" of Green's function, namely the fermion Green's function G(iω→0)=0G(\mathrm{i} ω\rightarrow 0) = 0G(iω→0)=0. In this work, through the general "decorated defect" construction of symmetry protected topological (SPT) states, we demonstrate the existence of Green's function zero after SMG, by mapping the evaluation of the Green's function to the problem of a single particle path integral. This method can be extended to the cases without spatial translation symmetry, where the momentum space which hosts many quantized topological numbers is no longer meaningful. Using the same method one can demonstrate the existence of the Green's function zero at the "avoided topological transition" in the bulk of the system.
We study symmetry breaking quantum phase transitions in topological insulators and superconductors where the single electron gap remains open in the bulk. Specifically, we consider spontaneous breaking of the symmetry that protects the gapless boundary modes, so that in the ordered phase these modes are gapped. Here we determine the fate of the topological boundary modes right at the transition where they are coupled to the strongly fluctuating order parameter field. Using a combination of exact solutions and renormalization group techniques, we find that the surface fermionic modes either decouple from the bulk fluctuations, or flow to a strongly coupled fixed point which remains gapless. In addition, we study transitions where the critical fluctuations are confined only to the surface and find that in several cases the critical point is naturally supersymmetric. This allows a determination of critical exponents and points to an underlying connection between band topology and supersymmetry. Finally, we study the fate of gapless Majorana modes localized on point and line defects in topological superconductors at bulk criticality, which is analogous to a quantum impurity problem. Again, an interplay of topology and strong correlations causes these modes to remain gapless but in a strongly coupled state. Experimental candidates for realizing these phenomena are discussed.
Many-body interactions play a crucial role in quantum topological systems, being able to impact or alter the topological classifications of noninteracting fermion systems. In open quantum systems, where interactions with the environment cause dissipation and decoherence of the fermionic dynamics, the absence of Hermiticity in the subsystem Hamiltonian drastically reduces the stability of the topological phases of the corresponding closed systems. Here we investigate the nonperturbative effects induced by the environment on the prototype Su-Schrieffer-Heeger chain coupled to local harmonic oscillator baths through either intracell or intercell transfer integrals. Despite the common view, this type of coupling, if suitably engineered, can even induce a transition to topological phases. By using a world-line quantum Monte Carlo technique we determine the phase diagram of the model proving that the bimodality of the probability distribution of the polarization signals the emergence of the topological phase. We show that a qualitative description can be obtained by using an approach based on the cluster perturbation theory providing, in particular, a non-Hermitian Hamiltonian for the fermionic subsystem and insights on the dissipative dynamics. We prove that non-Hermitian effects disappear in the presence of classical heat baths.