Abstract The Haldane phase is the prototype of symmetry protected topological (SPT) phases of spin chain systems. It can be protected by several symmetries having in common the degeneracy of the entanglement spectrum (ES). Here we explore in depth this degeneracy for the spin-1 Affleck–Kennedy–Lieb–Tasaki and bilinear-biquadratic Hamiltonians and show the emergence of a bulk-edge correspondence that relates the low energy properties of the entanglement Hamiltonian of a periodic chain and that of the physical Hamiltonian of an open chain. We find that the ES can be described in terms of two spins-1/2 behaving as the effective spins at the end of the open chain. In the case of non-contiguous partitions, we find that the entanglement Hamiltonian is given by the spin-1/2 Heisenberg Hamiltonian, which suggests a relationship between SPT phases and conformal field theory. We finally investigate the string order parameter and the relation with the bulk-edge correspondence.
Abstract The Haldane phase is the prototype of symmetry protected topological (SPT) phases of spin chain systems. It can be protected by several symmetries having in common the degeneracy of the entanglement spectrum (ES). Here we explore in depth this degeneracy for the spin-1 Affleck–Kennedy–Lieb–Tasaki and bilinear-biquadratic Hamiltonians and show the emergence of a bulk-edge correspondence that relates the low energy properties of the entanglement Hamiltonian of a periodic chain and that of the physical Hamiltonian of an open chain. We find that the ES can be described in terms of two spins-1/2 behaving as the effective spins at the end of the open chain. In the case of non-contiguous partitions, we find that the entanglement Hamiltonian is given by the spin-1/2 Heisenberg Hamiltonian, which suggests a relationship between SPT phases and conformal field theory. We finally investigate the string order parameter and the relation with the bulk-edge correspondence.
Under an appropriate symmetric extensive bipartition in a one-dimensional symmetry-protected topological (SPT) phase, a bulk critical-entanglement spectrum can be obtained, resembling the excitation spectrum of the critical point separating the SPT phase from the trivial (vacuum) state. Such a critical point is beyond the standard Landau-Ginzburg-Wilson paradigm for symmetry-breaking phase transitions. For the S=1 SPT (Haldane) phase with the Affleck-Kennedy-Lieb-Tasaki exact wave function, the resulting critical-entanglement spectrum has a residual entropy per lattice site sr=0.67602, showing a delocalized version of the edge excitations in the SPT phase. From the wave function corresponding to the lowest entanglement energy level, the central charge of the critical point can be extracted $c\ensuremath{\approx}1.01\ifmmode\pm\else\textpm\fi{}0.01$. The critical theory can be identified as the same effective field theory as the spin-1/2 antiferromagnetic Heisenberg chain or the spin-1/2 Haldane-Shastry model with inverse-square long-range interaction.
Applying a symmetric bulk bipartition to the one-dimensional Affleck-Kennedy-Lieb-Tasaki valence-bond solid (VBS) states for the integer spin-S Haldane gapped phase, we can create an array of fractionalized spin-S/2 edge states with the super unit cell l in the reduced bulk system, and the topological properties encoded in the VBS wave functions can be revealed. The entanglement Hamiltonian (EH) with even l corresponds to the quantum antiferromagnetic Heisenberg spin-S/2 model. For the even integer spins, the EH still describes the Haldane gapped phase. For the odd integer spins, however, the EH just corresponds to the quantum antiferromagnetic Heisenberg half-odd integer-spin model with spinon excitations, characterizing the critical point separating the topological Haldane phase from the trivial gapped phase. Our results thus demonstrate that the topological bulk property not only determines its fractionalized edge states but also the quantum criticality associated with the topological phase, where the elementary excitations are precisely those fractionalized edge degrees of freedom confined in the bulk of the topological phase.
We study the phase diagram of the SOq(3) quantum group invariant spin-1 bilinear-biquadratic spin chain for real values of q>1. Numerical computations suggest that the chain has at least three clearly distinguished phases: A chiral analog of the Haldane phase, a dimerized phase, and a ferromagnetic phase. In contrast, the counterpart of the extended critical region that is known to exist for q=1 remains elusive. Our results show that the Haldane phase fails to exhibit a twofold degeneracy in the entanglement spectrum but that the degeneracy is restored upon a suitable q deformation of the entanglement Hamiltonian, which can be interpreted as a Zeeman field. The structure of the phase diagram is confirmed through analytical calculations in the extreme anisotropic limit $q\ensuremath{\rightarrow}\ensuremath{\infty}$. Our results suggest that symmetries of the form Uq[su(2)] for distinct choices of q should be interpreted as one single family instead of separate symmetries when defining SPT phases, leading naturally to the notion of a q-SPT phase.
Here we study the emergence of different symmetry-protected topological (SPT) phases in a spin-2 quantum chain. We consider a Heisenberg-like model with bilinear, biquadratic, bicubic, and biquartic nearest-neighbor interactions, as well as uniaxial anisotropy. We show that this model contains four different effective spin-1 SPT phases, corresponding to different representations of the $({\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2})+T$ symmetry group, where Z2 is some $\ensuremath{\pi}$ rotation in the spin internal space and T is time reversal. One of these phases is equivalent to the usual spin-1 Haldane phase, while the other three are different but also typical of spin-1 systems. The model also exhibits an SO(5)-Haldane phase. Moreover, we also find that the transitions between the different effective spin-1 SPT phases are continuous and can be described by a c=2 conformal field theory. At such transitions, indirect evidence suggests a possible effective field theory of four massless Majorana fermions. The results are obtained by approximating the ground state of the system in the thermodynamic limit using matrix product states via the infinite time-evolving block decimation method, as well as by effective field theory considerations. Our findings show that different large effective spin-1 SPT phases separated by continuous quantum phase transitions can be stabilized in a simple quantum spin chain.
Abstract Quantum phase transitions out of a symmetry-protected topological (SPT) phase in (1 + 1) dimensions into an adjacent, topologically distinct SPT phase protected by the same symmetry or a trivial gapped phase, are typically described by a conformal field theory (CFT). At the same time, the low-lying entanglement spectrum of a gapped phase close to such a quantum critical point is known (Cho et al arXiv:1603.04016), very generally, to be universal and described by (gapless) boundary conformal field theory. Using this connection we show that symmetry properties of the boundary conditions in boundary CFT can be used to characterize the symmetry-protected degeneracies of the entanglement spectrum, a hallmark of non-trivial symmetry-protected topological phases. Specifically, we show that the relevant boundary CFT is the orbifold of the quantum critical point with respect to the symmetry group defining the SPT, and that the boundary states of this orbifold carry a quantum anomaly that determines the topological class of the SPT. We illustrate this connection using various characteristic examples such as the time-reversal breaking ‘Kitaev chain’ superconductor (symmetry class D), the Haldane phase, and the Z 8 classification of interacting topological superconductors in symmetry class BDI in (1 + 1) dimensions.
Symmetry-protected topological phases have fundamentally changed our understanding of quantum matter. An archetypal example of such a quantum phase of matter is the Haldane phase, containing the spin-1 Heisenberg chain. The intrinsic quantum nature of such phases, however, often makes it challenging to study them using classical means. Here, we use trapped-ion qutrits to natively engineer spin-1 chains within the Haldane phase. Using a scalable deterministic procedure to prepare the Affleck-Kennedy-Lieb-Tasaki (AKLT) state within the Haldane phase, we study the topological features of this system on a qudit quantum processor. Notably, we verify the long-range string order of the state, despite its short-range correlations, and observe spin fractionalization of the physical spin-1 particles into effective qubits at the chain edges, a defining feature of this system. The native realization of Haldane physics on a qudit quantum processor and the scalable preparation procedures open the door to the efficient exploration of a wide range of systems beyond spin-1/2.