Theoretical Study Of The Topological Phases And Phase Transitions In The Two-dimensional Strongly Correlated Quantum Many-body System

Under strong interaction between particles,there can emerge novel correlated phases of matter and critical phenomena in a quantum many-body system under low temperature.Conventional theories classify phases and describe phase transitions in terms of symmetry.Then there came the prosperity of a large class of phases of matter whose descriptions are rooted in topology instead of symmetry.This dissertation aims at unveiling novel topological phases and phase transitions emerging in the low energy theories of several prototypical correlated many-body systems concerning the high Tc superconductivity and quantum magnet.Motivated by some recent experiments on the Cuprate,we study the t-J model under staggered magnetic field,and analyze its impact on the pairing symmetry and the superconducting gap nodes.It is found that d-wave pairing symmetry is mostly favourable under electron doping,but the Dirac nodes could annihilate driven by staggered magnetic field,which results in the topological phase transition from the nodal d-wave superconductor into the nodeless d-wave superconductor.In contrast,in the optimal hole doping scenario,there emerges the s-wave pairing condensate under sufficient staggered field.The complex d+is pairing symmetry in between,however,is equivalent to p+ip and p-ip symmetry when viewed from the two low-energy Fermi pockets respectively.Therefore,our model shows that proximity to antiferromagnet could drive the d-wave superconductor into the non-chiral topological superconductor protected by the low-energy emergent valley conservation symmetry.Among the most recent experiments which open a broad new avenue to the high Tc unconventional superconductor based on graphene system,the Moirésuperlattice emerging upon the ABC-stacked trilayer graphene aligned with hexagonal boron-nitride also harbours the half-filled insulating phase.We calculate the effective band structure by means of low energy effective Dirac model coupled with Moirémodulated potential,and construct the minimal tight binding model on the triangular superlattice.From the perspective of weak coupling scenario,the half-filled Fermi-surface exhibits strong instability towards the inter-valley spiral order and gap opening.From the perspective of a strong coupling scenario,the effective valley exchange interaction stabilizes the 120?valley antiferromagnetic order,which is adiabatically equivalent to the weak coupling order.By doping the insulating phase,we predict that the most energetically favourable pairing symmetry is the inter-valley topologically nontrivial p±ip.Above all,the most novel ingredient of our minimal model is a valley-contrasting staggered flux,which could be experimentally tuned by gating and could lead to the topological phase transition between p±ip pairing and d?id pairing state.The exactly solvable Kitaev Toric Code model characterizes the Z₂quantum spin liquid,upon which there emerges the 2-dimensional Z₂electromagnetic gauge theory with electric-magnetic self-duality.Coupling with external field can drive a phase transition by either condensing electric charge or magnetic charge.However,along the path strictly pre-serving the electric-magnetic duality,simultaneous condensation of electric and magnetic charge is frustrated by the Aharonov-Bohm effect,leaving the phase transition along this path a long-standing mystery.We take a different path to address this question by tuning the exact ground state of toric code model instead of the Hamiltonian to directly probe the essential phase transition and the fate of anyons.Utilizing the tensor-network repre-sentation,we map the overlap of quantum wave-function into the classical Ashkin-Teller model which has been exactly solved on the critical lines.Therefore we could accurately pinpoint the critical point on our phase diagram and extract the exact scaling behaviour of the anyon correlation functions.It turns out that the Z₂deconfined topological phase undergoes the quantum Kosterlitz-Thouless phase transition into the gapless deconfined Coulomb state,which characterizes the charge-2e Anderson-Higgs transition in a reverse direction.To summarize,we unveil a novel non-chiral valley topological superconductor that could be realized in high Tc Cuprate system,and we predict inter-valley p+ip topological superconductor on the Moirésuperlattice of graphene,and last but not least,we discover a novel topological quantum phase transition along the electric-magnetic self-dual path in the Z₂ quantum spin liquid.