Characterizing Topological Properties Of Low-Dimensional Correlated Quantum Matter By Means Of Quantum Entanglement

One of the main focus in modern condensed matter physics is the novel quantum states of matter in low dimensional strongly correlated systems.The topological nontriviality of such quantum matter is largely encoded in its ground state wavefunction,especially in its entanglement properties.By taking the logarithm of the reduced density matrix of one subsystem,the entanglement Hamiltonian(EH)can be defined.EH together with its eigenvalue spectrum,the so-called entanglement spectrum(ES),are the quantitative tools to describe entanglement.Under left/right bipartition,the resulting edge ES has the low-lying levels in one-to-one correspondence with the edge excitation spectrum of original system,thus being able to reveal bulk topological information.In this thesis,we focus on 1D symmetry-protected-topological(SPT)phase,i.e.the shortrange-entangled phase with symmetry,whose exactly solvable fixed point has valancebond-solid(VBS)configuration.We use the entanglement in the VBS type ground state wavefunction to characterize the topological properties of SPT phase.In the first part,we start with the Haldane phase in AFM integer spin chain,and introduce AKLT model as its exactly-solvable fixed point,then we make clear that only the odd-spin Haldane phase belongs to non-trivial SPT phase.By taking the odd-spin AKLT state as a concrete example,we prove that,the edge ES degeneracy of SPT phase contains both topological part and non-universal part,the former can be isolated by a non-local unitary transformation called topological disentangler,which is determined solely by the protecting symmetry.Our results clarify the structure of ES in SPT phase,thus playing a guiding role in employing ES to characterize topological phase.From the second part we focus on extracting the information about the quantum critical point(QCP)separating original SPT phase and trivial phase from the VBS type ground state wavefunction.This QCP cannot be described by Landau's classical theory,with the low-energy elementary excitation being the edge degree of freedom in original VBS wavefunction.We introduce the symmetric extensive bipartition to the VBS wavefunction: by grouping every l sites into a block and tracing out all even blocks,we show the resulting bulk entanglement Hamiltonian(EH)describes the percolation between deconfined edge degrees of freedom when block length is properly chosen,thus being able to reveal full spectrum of QCP.We prove that,the bulk EH of spin-S AKLT wavefunction is the Heisenberg model of spin-S /2,for the non-trivial odd-spin S with l = even,it describes the QCP separating Haldane phase and trivial phase,whose low-energy effective theory is S U(2)1Wess-Zumino-Witten(WZW)conformal field theory(CFT).In the third part we apply extensive bipartition to the S O(5)symmetric spin-2 VBS state with virtual spin-3/2,and show the bulk EH with block length l = odd corresponds to the QCP separating original SPT and trivial phase,the low-energy effective theory is S U(4)1WZW CFT.In the last part we employ S U(2)spins to construct a novel type of VBS with S U(N)(N ? 3)structure: when system length is even,its edges are conjugate to each other,thus breaking inversion symmetry;when system length is odd,the edges are the same,thus being the real SPT phase,its corresponding QCP can emerge from bulk EH when block length l = odd,and the low-energy effective theory is described by S U(N)1WZW CFT.The symmetric extensive bipartition is an effective way to not only get the effective field theory of QCP,but also the corresponding Hamiltonian.We hence generalize the well-known bulk-edge correspondence to bulk-edge-QCP correspondence,which provides valuable insights into the understanding of topological phases.