SU(3) quantum critical model emerging from a spin-1 topological phase

Abstract

Different from the spin-1 Haldane gapped phase, we propose a novel SO(3) spin-1 matrix product state (MPS), whose parent Hamiltonian includes three-site spin interactions. From the entanglement spectrum of a single block with l sites, an enlarged SU(3) symmetry is identified in the edge states, which are conjugate to each other for the l = even block but identical for the l = odd block. By blocking this novel state, the blocked MPS explicitly displays the SU(3) symmetry with two distinct structures. Under a symmetric bulk bipartition with a sufficient large block length l = even, the entanglement Hamiltonian (EH) of the reduced system characterizes a spontaneous dimerized phase with two-fold degeneracy. However, for the block length l = odd, the corresponding EH represents an SU(3) quantum critical point with delocalized edge quasiparticles, and the critical field theory is described by the SU(3) level-1 Wess-Zumino-Witten conformal field theory. Topological phases of matter have become one of the most important subjects in physics, because their lowenergy excitations have potential use for fault-tolerant quantum computation. Symmetry protected topological (SPT) phases belong to a new type of topological phases with robust gapless edge excitations [1][2][3]. Without breaking the protecting symmetry or closing the energy gap, these SPT phases can not be continuously connected to the trivial phase. A topological quantum critical point (TQCP) exists to separate an SPT phase from its adjacent trivial phase, and the corresponding critical theory is beyond the Landau-Ginzburg-Wilson paradigm [4,5]. The simplest example of SPT phases is the Haldane gapped phase of the antiferromagnetic Heisenberg spin-1 chain [6]. Recently it has been shown that the symmetric bulk bipartition of the SO(3) symmetric Affleck-Kennedy-Lieb-Tasaki (AKLT) wave function for the Haldane phase [7] is an effective way to create an array of fractionalized spin-1/2 edge spins in the bulk subsystem, and the corresponding bulk entanglement spectrum (ES) represents a TQCP described by the SU(2) level-1 Wess-Zumino-Witten (WZW) theory with spinon excitations [8,10]. The TQCP is argued to characterize the quantum critical state between the Haldane gapped phase and its adjacent trivial phase [8][9][10].It is well-known that the quantum spin-1 chain may exhibit an SU(3) symmetry. The SU(3) symmetric matrix product state (MPS) has been constructed [11][12][13][14][15][16]. The on-site physical space is spanned by the SU(3) adjoint representation 8 consisting of a fundamental representation 3 (quarks) and a conjugate representation 3 (antiquarks), while the adjacent lattice sites are connected by the SU(3) quark-antiquark singlet bonds. Different from the Haldane gapped phase, the zero-energy states localized on the edges are conjugate to each other. The natural question arises whether a distinct spin-1 MPS can be found so that the symmetric bulk bipartition leads to the SU(3) quantum criticality for the reduced bulk system. According to the complete class...

One-dimensional symmetry protected topological phases and their transitions

“…Again our conjecture is confirmed, although now our lower bound log 2 d is much less tight -it only grows logarithmically with N , whereas c grows linearly. We note that the analysis of Roy and Quella agrees with complementary approaches, such as the work by Rao et al 22 reporting an SPT phase with SU (3) symmetry and a phase transition to the trivial phase described by WZW SU (3) 1 .…”

“…This unified set of models provides a framework for our second topic: "What characterizes the critical theory between SPT phases?". There have been various works studying the transitions between particular SPT phases [14][15][16][17][18][19][20][21][22][23][24] , but it has proven difficult to make quantitative statements about the general case [25][26][27] . The latter works have led to the intuitive picture that the gapless fields at the transition are in some sense the delocalized boundary excitations.…”

Fidelity Mechanics: Analogues of the Four Thermodynamic Laws and Landauer’s Principle

“…This will be illustrated in Section 9 , Section 10 and Section 11 , when we discuss fidelity mechanical-state functions for the quantum spin- XXZ model in an external magnetic field, the quantum spin- XYZ model, and the quantum spin-1 XYZ model. In addition, the SPT phases [ 108 , 109 , 110 , 111 , 112 ], the SPt phases [ 113 , 114 ], and the gapless and gapped spin liquids [ 31 ] appear to be topological in the control parameter space. As it turns out, a nonlocal order parameter is essential for characterizing a topologically ordered state (cf.…”

Dynamical spin excitations of topological Haldane gapped phase in the $S=1$ Heisenberg antiferromagnetic chain with single-ion anisotropy

“…This quantum phase transition is a Gaussian-type transition [9] described by conformal field theory with a central charge c = 1 [10,11]. The Haldane gapped phase is now understood as a symmetry-protected topological phase [12], while it will undergo a phase transition to a topologically trivial gapped phase, the large-D phase, with increasing the single-ion anisotropy D. When D is strong enough, the ground state in the large-D phase is known as the product state with S z = 0 at each site. A symmetry-protected topological phase cannot be continuously changed into a trivial gapped phase without closing the energy gap [13], so the transition, between the Haldane phase and the large-D phase, is a topological phase transition that possesses a gapless critical point and does not fit into the Landau-Ginzburg-Wilson paradigm [14].…”