It is well-known that symmetry-protected topological (SPT) phases can be obtained from the trivial phase by an entangler, a finite-depth unitary operator U U . Here, we consider obtaining the entangler from a local ‘pivot’ Hamiltonian H_\text{pivot} H pivot such that U = e^{i\pi H_\text{pivot}} U = e i π H pivot . This perspective of Hamiltonians pivoting between the trivial and SPT phase opens up two new directions: (i) Since SPT Hamiltonians and entanglers are now on the same footing, can we iterate this process to create other interesting states? (ii) Since entanglers are known to arise as discrete symmetries at SPT transitions, under what conditions can this be enhanced to U(1) U ( 1 ) pivot symmetry generated by H_\text{pivot} H pivot ? In this work we explore both of these questions. With regard to the first, we give examples of a rich web of dualities obtained by iteratively using an SPT model as a pivot to generate the next one. For the second question, we derive a simple criterion for when the direct interpolation between the trivial and SPT Hamiltonian has a U(1) U ( 1 ) pivot symmetry. We illustrate this in a variety of examples, assuming various forms for H_\text{pivot} H pivot , including the Ising chain, and the toric code Hamiltonian. A remarkable property of such a U(1) U ( 1 ) pivot symmetry is that it shares a mutual anomaly with the symmetry protecting the nearby SPT phase. We discuss how such anomalous and non-onsite U(1) U ( 1 ) symmetries explain the exotic phase diagrams that can appear, including an SPT multicritical point where the gapless ground state is given by the fixed-point toric code state.
It is well-known that symmetry-protected topological (SPT) phases can be obtained from the trivial phase by an entangler, a finite-depth unitary operator U U . Here, we consider obtaining the entangler from a local ‘pivot’ Hamiltonian H_\text{pivot} H pivot such that U = e^{i\pi H_\text{pivot}} U = e i π H pivot . This perspective of Hamiltonians pivoting between the trivial and SPT phase opens up two new directions: (i) Since SPT Hamiltonians and entanglers are now on the same footing, can we iterate this process to create other interesting states? (ii) Since entanglers are known to arise as discrete symmetries at SPT transitions, under what conditions can this be enhanced to U(1) U ( 1 ) pivot symmetry generated by H_\text{pivot} H pivot ? In this work we explore both of these questions. With regard to the first, we give examples of a rich web of dualities obtained by iteratively using an SPT model as a pivot to generate the next one. For the second question, we derive a simple criterion for when the direct interpolation between the trivial and SPT Hamiltonian has a U(1) U ( 1 ) pivot symmetry. We illustrate this in a variety of examples, assuming various forms for H_\text{pivot} H pivot , including the Ising chain, and the toric code Hamiltonian. A remarkable property of such a U(1) U ( 1 ) pivot symmetry is that it shares a mutual anomaly with the symmetry protecting the nearby SPT phase. We discuss how such anomalous and non-onsite U(1) U ( 1 ) symmetries explain the exotic phase diagrams that can appear, including an SPT multicritical point where the gapless ground state is given by the fixed-point toric code state.
Progress in understanding symmetry-protected topological (SPT) phases has been greatly aided by our ability to construct lattice models realizing these states. In contrast, a systematic approach to constructing models that realize quantum critical points between SPT phases is lacking, particularly in dimension d>1 d > 1 . Here, we show how the recently introduced notion of the pivot Hamiltonian—generating rotations between SPT phases—facilitates such a construction. We demonstrate this approach by constructing a spin model on the triangular lattice, which is midway between a trivial and SPT phase. The pivot Hamiltonian generates a U(1) U ( 1 ) pivot symmetry which helps to stabilize a direct SPT transition. The sign-problem free nature of the model—with an additional Ising interaction preserving the pivot symmetry—allows us to obtain the phase diagram using quantum Monte Carlo simulations. We find evidence for a direct transition between trivial and SPT phases that is consistent with a deconfined quantum critical point with emergent SO(5) S O ( 5 ) symmetry. The known anomaly of the latter is made possible by the non-local nature of the U(1) U ( 1 ) pivot symmetry. Interestingly, the pivot Hamiltonian generating this symmetry is nothing other than the staggered Baxter-Wu three-spin interaction. This work illustrates the importance of U(1) U ( 1 ) pivot symmetries and proposes how to generally construct sign-problem-free lattice models of SPT transitions with such anomalous symmetry groups for other lattices and dimensions.
Generalized charge pumps are topological obstructions to trivializing loops in the space of symmetric gapped Hamiltonians. We show that given mild conditions on such pumps, the associated loop has high-symmetry points that must be in distinct symmetry-protected topological (SPT) phases. To further elucidate the connection between pumps and SPTs, we focus on closed paths, “pivot loops”, defined by two Hamiltonians, where the first is unitarily evolved by the second “pivot” Hamiltonian. While such pivot loops have been studied as entanglers for SPTs, here we explore their connection to pumps. We construct families of pivot loops that pump charge for various symmetry groups, often leading to SPT phases—including dipole SPTs. Intriguingly, we find examples where nontrivial pumps do not lead to genuine SPTs but still entangle representation-SPTs (RSPTs). We use the anomaly associated with the nontrivial pump to explain the “unnecessary” criticality between these RSPTs. We also find that particularly nice pivot families form circles in Hamiltonian space, which we show are equivalent to the Hamiltonians satisfying the Dolan-Grady relation—known from the study of integrable models. This additional structure allows us to derive more powerful constraints on the phase diagram. Natural examples of such circular loops arise from pivoting with the Onsager-integrable chiral clock models, containing the aforementioned RSPT example. In fact, we show that these Onsager pivots underlie general group cohomology-based pumps in one spatial dimension. Finally, we recast the above in the language of equivariant families of Hamiltonians and relate the invariants of the pump to the candidate SPTs. We also highlight how certain SPTs arise in cases where the equivariant family is labeled by spaces that are not manifolds.
1+1D symmetry-protected topological (SPT) phases with invertible symmetry can be characterized by the symmetric entanglers: globally symmetric finite-depth local unitary circuits that generate representative ground states from a product state. For non-invertible symmetries, however, distinct SPT phases, such as the three Rep(D8) SPT phases, may not be connected by any symmetric entangler. We show that the three Rep(D8) SPT phases can nevertheless be connected by \emph{symmetry-permuting} finite-depth local unitary circuits, realizing dualities predicted by topological quantum field theory (TQFT). We construct an explicit matrix product unitary connecting two of the Rep(D8) SPT phases, and show that its action on string order parameters reproduces the anyon permutation predicted by the TQFT duality.
The Onsager algebra, invented to solve the two-dimensional Ising model, can be used to construct conserved charges for a family of integrable N N -state chiral clock models. We show how it naturally gives rise to a “pivot” procedure for this family of chiral Hamiltonians. These Hamiltonians have an anti-unitary CPT symmetry that when combined with the usual \mathbb{Z}_N ℤ N clock symmetry gives a non-Abelian dihedral symmetry group D_{2N} D 2 N . We show that this symmetry gives rise to symmetry-protected topological (SPT) order in this family for all even N N , and representation-SPT (RSPT) physics for all odd N N . The simplest such example is a next-nearest-neighbour chain generalising the spin-1/2 cluster model, an SPT phase of matter. We derive a matrix-product state representation of its fixed-point ground state along with the ensuing entanglement spectrum and symmetry fractionalisation. We analyse a rich phase diagram combining this model with the Onsager-integrable chiral Potts chain, and find trivial, symmetry-breaking and (R)SPT orders, as well as extended gapless regions. For odd N N , the phase transitions are “unnecessarily” critical from the SPT point of view.
Symmetry-protected topological (SPT) phases exhibit nontrivial short-ranged entanglement protected by symmetry and cannot be adiabatically connected to trivial product states while preserving the symmetry. In contrast, intrinsic topological phases do not need ordinary symmetry to stabilize them and their ground states exhibit long-range entanglement. It is known that for a given symmetry group G, the 2D SPT phase protected by G is dual to the 2D topological phase exemplified by the twisted quantum double model ${D}^{\ensuremath{\omega}}(G)$ via gauging the global symmetry G. Recently it was realized that such a general gauging map can be implemented by some local unitaries and local measurements when G is a finite, solvable group. Here, we review the general approach to gauging a $G\text{\ensuremath{-}}\mathrm{SPT}$ starting from a fixed-point ground-state wave function and applying a N-step gauging procedure. We provide an in-depth analysis of the intermediate states emerging during the N-step gauging and provide tools to measure and identify the emerging symmetry-enriched topological order (SET) of these states. We construct the generic lattice parent Hamiltonians for these intermediate states and show that they form an entangled superposition of a twisted quantum double (TQD) with an SPT-ordered state. Notably, we show that they can be connected to the TQD through a finite-depth, local quantum circuit which does not respect the global symmetry of the SET order. We introduce the so-called symmetry branch line operators and show that they can be used to extract the symmetry fractionalization classes (SFC) and symmetry defectification classes (SDC) of the SET phases with the input data G and $[\ensuremath{\omega}]\ensuremath{\in}{H}^{3}(G,U(1))$ of the pregauged SPT-ordered state. We illustrate the procedure of preparing and characterizing the emerging SET-ordered states for some Abelian and non-Abelian examples such as dihedral groups Dn and the quaternion group Q8.
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.