Investigation To Topological Phase In Parafermion Chain And Relaxation In Boundary Dissipative XY Model

Strongly correlated quantum many-body system continue to pose some of the most interesting challenges in modern condensed matter physics.Some important physi-cal phenomena in condensed matter,including fractional quantum hall effect,high-temperature superconductivity,and spin liquids,are associated with strongly correlated interactions.Except for a few analytical solutions(Bethe ansatz),there has been a lack of exact solutions for such systems.Several methods are successful for solving strongly correlated systems,including quantum monte carlo method,dynamic mean field,den-sity matrix renormalization group method,etc.Among them,the effective method to solve one-dimensional lattice model is density matrix renormalization group.In the past 20 years,on the other hand,important progresses take place in quantum manipu-lation technology,especially in the cold atomic system,where atomic interaction can be controlled by Feshbach resonance.By using optical lattice,quantum simulation of condensed matter system with cold atomic system is possible,which opens a new way for the research of quantum many-body system.With the development of quantum manipulation technology,some interesting quantum many-body toy models have been proposed,including the parafermion models(or clock models).In strongly correlated quantum systems,topological order is newly discovered in recent years,and the first discovered topological state is fractional quantum hall state.The classification of topological phase surpasses the traditional classification of phase based on symmetry breaking theory proposed by Landau.Systems with topological order are generally strongly correlated many-body systems,which can be described by topological quantum field theory.These systems often have anyonic quasi-particle excitation and nontrivial ground state degeneracy.The quantum information can be stored in the degenerate quantum state encoded by these quasi-particles.The unitary gates which are necessary for topological quantum computation are carried by braiding these quasi-particles.The fault tolerance of a topological quantum computer arises from the nonlocal encoding of the quasiparticle states,which are immune to errors caused by local pertur-bations.However,this picture require an assumption that the system is a close quantum system.As physical systems in real experiments are all open systems,it remians a prob-lem whether topological systems are immune to environmental noise in open quantum systems.For example,the influence of dissipation on topological quantum information storage is an interesting topic.At the same time,the evolution of open quantum many-body system itself is an important problem in quantum physics.The open quantum system described by the exact solvable master equation is still limited to single particle,single spin or a harmonic oscillator.A recent studies have shown that some many-body quantum master equations can be analytically solved with the single-particle method,but this is restricted to a small type of the whole systems.In our work,we explore the emergent phenomena in an extended Z3 parafermion model using the density matrix renormalization group method and exact diagonalization method,which can be mapped to a clock model through Jordan-Wigner transforma-tion.We find the ground state in each site should be a single state or twofold degenerate states.In the later case,we can see this model can be smoothly connected to the conven-tional spin 1/2 system,indicating the the asymptotic behaviour of parafermions toward canonical fermions or the emergent phenomena from Z2 spin models to Z3 models.By virtue of this approach,we can generalize the measurement tools,such as order param-eters and correlation functions,in the Z2 spin model to characterize the phases in the parafermion model and map out the whole phase diagram.Various phases have been unveiled,including the topological ferromagnetic parafermion phase,trivial paramag-netic parafermion phase,spin-fluid phase,dimer phase,chiral phase and commensurate phase.Strikingly,all the phase boundaries between these phases finally merge to a sin-gle supercritical point with infinite degeneracy,in which the surrounding phases can be regarded as different types of symmetry breaking over it.We suggest a general princi-ple to find these supercritical points.In the subsequent reaseach,we investigate another alternating bond parafermion model,in which we find an emergent Haldane phase.To characterize this symetry-protected phase,we generalized the string order in the Z2 spin model to the Z3 system.We using entanglement spectrum and ground state degener-acy to strengthen this conclusion.Following the story of symetry-protected topological phase,we find another Z3 x Z3 clock model,can be transformed to two parafermion chains by a generalized Jordan-Wigner transformation.Taking advantage of this tool,we find anomalous gapless symetry-protected topological phases in this model.This phase is characterized by the three-fold degeneracy in ground states as well as low-lying excited states as long as the symmetry is not destroyed.In addition,the ground state supports different degeneracy denpending on open or close boundary conditions,indicating the emerge of edge states.Forthermore,we also examine the three-fold de-generacy in the spectrum.These works open another way for finding new exotic phases in parefermion and clock models.On the other hand,we investigated the relaxation and multiply time scales in the quantum XY model with boundary dissipation.This model can be transformed to a p-wave superconducting model with boundary dissipation.This topological model support edge Majorana zero modes,which can be used as topological quantum memory.The study of boundary dissipative XY model is equivelent mathematically to the topolog-ical quantum state suffered with boundary decoherence.We rewrite the model in the representatin of Majorana operators,finding that each subspace with different number of Majorana fermion is fully decoupled.In the long time limit,we find the relaxation is fully determined by single paticle dynamics.By mapping the the single paticle dynam-ics equation to a non-Hermition Schrodinger equation,we can understand the single particle relaxation both in the large and weak dissipation limit using perturbation the-ory.We find an anomalous behavior in the large dissipation which induces a longer relaxation time.Finally,we examined the role played by the boundary and bulk modes,in which we find the edge modes are most vulnerable to the boundary dissipation while the bulk modes give the longest relaxation time.These results shade new insight into the dynamics of topological qubits in environment,which requires some further reseach.