Symmetry-protected Topological Research Articles

Multipole and fracton topological order via gauging foliated symmetry protected topological phases

Spurred by the recent development of fracton topological phases, unusual topological phases possessing fractionalized quasiparticles with mobility constraints, the concept of symmetries has been renewed. In particular, in accordance with the progress of multipole symmetries, associated with conservation of multipoles, such as dipole or quadruple moments as well as global charges, there have been proposed topological phases with such symmetries. These topological phases are unconventional because excitations are subject to mobility constraints corresponding to the multipole symmetries. We demonstrate a way to construct such phases by preparing layers of symmetry protected topological (SPT) phases and implementing gauging a global symmetry. After gauging, the statistics of a fractional excitation is altered when crossing the SPT phases, resulting in topological phases with the multipole symmetries. The way we construct the phases allows us to have a comprehensive understanding of field theories of topological phases with the multipole symmetries and other fracton models. Published by the American Physical Society 2024

Topological phases in the dynamics of the simple exclusion process.

We study the dynamical large deviations of the classical stochastic symmetric simple exclusion process (SSEP) by means of numerical matrix product states. We show that for half filling, long-time trajectories with a large enough imbalance between the number hops in even and odd bonds of the lattice belong to distinct symmetry-protected topological (SPT) phases. Using tensor network techniques, we obtain the large deviation (LD) phase diagram in terms of counting fields conjugate to the dynamical activity and the total hop imbalance. We show the existence of high activity trivial and nontrivial SPT phases (classified according to string order parameters) separated by either a critical phase or a critical point. Using the leading eigenstate of the tilted generator, obtained from infinite-system density-matrix renormalization group simulations, we construct a near-optimal dynamics for sampling the LDs, and show that the SPT phases manifest at the level of rare stochastic trajectories. We also show how to extend these results to other filling fractions, and discuss generalizations to asymmetric SEPs.

Continuous Phase Transitions between Fractional Quantum Hall States and Symmetry-Protected Topological States.

We study quantum phase transitions in Bose-Fermi mixtures driven by interspecies interaction in the quantum Hall regime. In the absence of such an interaction, the bosons and fermions form their respective fractional quantum Hall (FQH) states at certain filling factors. A symmetry-protected topological (SPT) state is identified as the ground state for strong interspecies interaction. The phase transitions between them are proposed to be described by Chern-Simons-Higgs field theories. For a simple microscopic Hamiltonian, we present numerical evidence for the existence of the SPT state and a continuous transition to the FQH state. It is also found that the entanglement entropy between the bosons and fermions exhibits scaling behavior in the vicinity of this transition.

Generalized homology and Atiyah–Hirzebruch spectral sequence in crystalline symmetry protected topological phenomena

Abstract We propose that symmetry-protected topological (SPT) phases with crystalline symmetry are formulated by an equivariant generalized homology $h^G_n(X)$ over a real space manifold X with G a crystalline symmetry group. The Atiyah–Hirzebruch spectral sequence unifies various notions in crystalline SPT phases, such as the layer construction, higher-order SPT phases, and Lieb–Schultz–Mattis-type theorems. This formulation is applicable to not only free fermionic systems but also interacting systems with arbitrary onsite and crystal symmetries.

Topological invariants for symmetry-protected topological phase entanglers

Broad classes of symmetry protected topological (SPT) phases can be prepared from a product state by locality preserving unitary operators that respect the symmetry as a whole, but cannot be written as symmetric finite depth quantum circuits. This work provides a framework for obtaining explicit topological invariants of such unitary operators, based on a special easily computable flux insertion operator. For unitary operators that are SPT entanglers, these invariants are related to the cocycle that labels the SPT phase.

State space geometry of the spin-1 antiferromagnetic Heisenberg chain

We study the phase diagram of the spin-1 antiferromagnetic Heisenberg chain with uniaxial anisotropy and applied magnetic field in terms of the genuine multipartite entanglement as witnessed by the mean quantum Fisher information density. By considering the manifold generated by the staggered magnetization operator, we connect the state space curvature in the vicinity of the ground state of the Heisenberg chain to the genuine multipartite entanglement. Our analysis demonstrates that the quantum critical points and symmetry protected topological (SPT) phase exhibit large state space curvature, while the separable phases are completely flat, offering insight into the physical interpretation of state space curvature. We further show that the entanglement in the SPT phase is enhanced by the presence of uniaxial anisotropy and undiminished in the presence of uniform magnetic fields. The magnon condensate phase induced by large fields is shown to emanate from the Gaussian critical point and exhibits massive multipartite entanglement over a robust region of the parameter space.

Realizing symmetry-protected topological phases in a spin-1/2 chain with next-nearest-neighbor hopping on superconducting qubits

Quantum simulation on near-term quantum hardware is a topic of intense interest. The preparation of novel quantum states of matter provides a quantitative assessment of the capabilities of near-term digital quantum computers to implement circuits with structure of relevance to quantum simulation. Here, we conduct a benchmark study by realizing symmetry-protected topological (SPT) phases of a spin-1/2 Hamiltonian with next-nearest-neighbor hopping on up to 11 qubits on a programmable superconducting quantum processor using adiabatic state preparation. Using recompilation techniques to reduce the gate count to around 50 two-qubit gates, we observe clear signatures of the two distinct SPT phases, such as excitations localized to specific edges and finite string-order parameters. We identify a parasitic phase associated with the two-qubit gate as the dominant imperfection that limits the depth of the circuits, indicating a research topic of interest for future hardware development.

The asymmetric valence-bond-solid states in quantum spin chains: The difference between odd and even spins

The qualitative difference in low-energy properties of spin S quantum antiferromagnetic chains with integer S and half-odd-integer S discovered by Haldane [F. D. M. Haldane, arXiv:1612.00076 (1981); Phys. Lett. A 93, 464–468 (1983); Phys. Rev. Lett. 50, 1153–1156 (1983)] and Tasaki [Tasaki, Graduate Texts in Physics (Springer, 2020)] can be intuitively understood in terms of the valence-bond picture proposed by Affleck et al. [I. Affleck, Phys. Rev. Lett. 59, 799 (1987); Commun. Math. Phys. 115, 477–528 (1988)]. Here, we develop a similarly intuitive diagrammatic explanation of the qualitative difference between chains with odd S and even S, which is at the heart of the theory of symmetry-protected topological (SPT) phases. (There is a 24 min video in which the essence of the present work is discussed: https://youtu.be/URsf9e_PLlc.) More precisely, we define one-parameter families of states, which we call the asymmetric valence-bond solid (VBS) states, that continuously interpolate between the Affleck–Kennedy–Lieb–Tasaki (AKLT) state and the trivial zero state in quantum spin chains with S = 1 and 2. The asymmetric VBS state is obtained by systematically modifying the AKLT state. It always has exponentially decaying truncated correlation functions and is a unique gapped ground state of a short-ranged Hamiltonian. We also observe that the asymmetric VBS state possesses the time-reversal, the Z2×Z2, and the bond-centered inversion symmetries for S = 2 but not for S = 1. This is consistent with the known fact that the AKLT model belongs to the trivial SPT phase if S = 2 and to a nontrivial SPT phase if S = 1. Although such interpolating families of disordered states were already known, our construction is unified and is based on a simple physical picture. It also extends to spin chains with general integer S and provides us with an intuitive explanation of the essential difference between models with odd and even spins.

Emergent higher-symmetry protected topological orders in the confined phase of U(1) gauge theory

We consider compact ${U}^{\ensuremath{\kappa}}(1)$ gauge theory in $3+1$ dimensions with a general $2\ensuremath{\pi}$-quantized topological term ${\ensuremath{\sum}}_{I,J=1}^{\ensuremath{\kappa}}\frac{{K}_{IJ}}{4\ensuremath{\pi}}{\ensuremath{\int}}_{{M}^{4}}{F}^{I}\ensuremath{\wedge}{F}^{J}$, where $K$ is an integer symmetric matrix with even diagonal elements and ${F}^{I}=d{A}^{I}$. At energies below the gauge charges' gaps but above the monopoles' gaps, this field theory has an emergent ${\mathbb{Z}}_{{k}_{1}}^{(1)}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{k}_{2}}^{(1)}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdots}$ 1-symmetry, where ${k}_{i}$ are the diagonal elements of the Smith normal form of $K$ and ${\mathbb{Z}}_{0}^{(1)}$ is regarded as a $U(1)$ 1-symmetry. In the ${U}^{\ensuremath{\kappa}}(1)$ confined phase, the boundary can have a phase whose infrared (IR) properties are described by Chern-Simons field theory. Such a phase has a ${\mathbb{Z}}_{{k}_{1}}^{(1)}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{k}_{2}}^{(1)}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdots}$ 1-symmetry that can be anomalous. To show these results, we develop a bosonic lattice model whose IR properties are described by this continuum field theory, thus acting as its ultraviolet completion. The lattice model in the aforementioned limit has an exact ${\mathbb{Z}}_{{k}_{1}}^{(1)}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{k}_{2}}^{(1)}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdots}$ 1-symmetry. We find that the short-range entangled gapped phase of the lattice model, corresponding to the confined phase of the ${U}^{\ensuremath{\kappa}}(1)$ gauge theory, is a symmetry protected topological (SPT) phase for the ${\mathbb{Z}}_{{k}_{1}}^{(1)}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{k}_{2}}^{(1)}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdots}$ 1-symmetry, whose SPT invariant is $\phantom{\rule{1.0pt}{0ex}}{e}^{\phantom{\rule{1.0pt}{0ex}}i\phantom{\rule{1.0pt}{0ex}}\ensuremath{\pi}{\ensuremath{\sum}}_{I,J}{K}_{IJ}{\ensuremath{\int}}_{{\mathcal{M}}^{4}}{B}_{I}\ensuremath{\smile}{B}_{J}+{B}_{I}\underset{1}{\ensuremath{\smile}}\phantom{\rule{1.0pt}{0ex}}d{B}_{J}}\phantom{\rule{1.0pt}{0ex}}{e}^{\phantom{\rule{1.0pt}{0ex}}i\phantom{\rule{1.0pt}{0ex}}\ensuremath{\pi}{\ensuremath{\sum}}_{I<J}{K}_{IJ}{\ensuremath{\int}}_{{\mathcal{M}}^{4}}\phantom{\rule{1.0pt}{0ex}}d{B}_{I}\underset{2}{\ensuremath{\smile}}\phantom{\rule{1.0pt}{0ex}}d{B}_{J}}$. Here, the background $\mathbb{R}/\mathbb{Z}$-valued 2-cochains ${B}_{I}$ satisfy $\phantom{\rule{1.0pt}{0ex}}d{B}_{I}={\ensuremath{\sum}}_{I}{B}_{I}{K}_{IJ}=0\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{0.28em}{0ex}}1$ and describe the symmetry twist of the ${\mathbb{Z}}_{{k}_{1}}^{(1)}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{k}_{2}}^{(1)}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdots}$ 1-symmetry. We apply this general result to a few examples with simple $K$ matrices. We find the nontrivial SPT order in the confined phases of these models and discuss its classifications using the fourth cohomology group of the corresponding 2-group.

Pivot Hamiltonians as generators of symmetry and entanglement

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 UU. Here, we consider obtaining the entangler from a local ‘pivot’ Hamiltonian H_\text{pivot}Hpivot such that U = e^{i\pi H_\text{pivot}}U=eiπHpivot. 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}Hpivot? 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}Hpivot, 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.

Building models of topological quantum criticality from pivot Hamiltonians

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>1d>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)SO(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.

Playing nonlocal games with phases of quantum matter

The parity game is an example of a nonlocal game: By sharing a Greenberger-Horne-Zeilinger (GHZ) state before playing this game, the players can win with a higher probability than is allowed by classical physics. The GHZ state of $N$ qubits is also the ground state of the ferromagnetic quantum Ising model on $N$ qubits in the limit of vanishingly weak quantum fluctuations. Motivated by this observation, we examine the probability that $N$ players who share the ground state of a generic quantum Ising model, which exhibits nonvanishing quantum fluctuations, still win the parity game using the protocol optimized for the GHZ state. Our main result is a modified parity game for which this protocol asymptotically exhibits quantum advantage in precisely the ferromagnetic phase of the quantum Ising model. We further prove that the ground state of the exactly soluble $d=1+1$ transverse-field Ising model can provide a quantum advantage for the parity game over an even wider region, which includes the entire ferromagnetic phase, the critical point and part of the paramagnetic phase. By contrast, we find examples of topological phases and symmetry-protected topological (SPT) phases of matter, namely the deconfined phase of the toric code Hamiltonian and the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ SPT phase in one dimension, that do not exhibit an analogous quantum advantage away from their fixed points.

Exactly Solvable Model for a Deconfined Quantum Critical Point in 1D.

We construct an exactly solvable lattice model for a deconfined quantum critical point (DQCP) in (1+1) dimensions. This DQCP occurs in an unusual setting, namely, at the edge of a (2+1) dimensional bosonic symmetry protected topological (SPT) phase with Z_{2}×Z_{2} symmetry. The DQCP describes a transition between two gapped edges that break different Z_{2} subgroups of the full Z_{2}×Z_{2} symmetry. Our construction is based on an exact mapping between the SPT edge theory and a Z_{4} spin chain. This mapping reveals that DQCPs in this system are directly related to ordinary Z_{4} symmetry breaking critical points.

Duality viewpoint of criticality

In this work, we study quantum many-body systems which are self-dual under duality transformation connecting different symmetry protected topological (SPT) phases. We provide a geometric explanation of the criticality of these self-dual models. More precisely, we show a ground state (quasi-)degeneracy under the periodic boundary conditions,i.e., the ingappability of the bulk spectrum. Equivalently, the symmetry group at criticality, including the duality symmetry, has a mixed 't Hooft anomaly. This approach can not only predict the spectrum of the self-dual model with ordinary 0-form symmetry, but also be applied to that with generalized symmetry, such as higher form and subsystem symmetry. As an application, we illustrate our results with several examples in one and two dimensions, which separate two different SPTs.

Towards string order melting of spin-1 particle chains in superconducting transmons using optimal control

Utilizing optimal control to simulate a model Hamiltonian is an emerging strategy that leverages the intrinsic physics of a device with digital quantum simulation methods. Here we evaluate optimal control for probing the nonequilibrium properties of symmetry-protected topological (SPT) states simulated with superconducting hardware. Assuming a tunable transmon architecture, we cast the evolution of these SPT states as a series of one- and two-site pulse optimization problems that are solved in the presence of leakage constraints. From the generated pulses, we classically simulate the time-dependent melting of the perturbed SPT string order across a six-site model with an average state infidelity of ${10}^{\ensuremath{-}3}$. The feasibility of these pulses as well as their efficient application indicate that high-fidelity simulations of string order melting are within reach of current quantum computing systems.

Symmetry Protected Topological Order in Open Quantum Systems

We systematically investigate the robustness of symmetry protected topological (SPT) order in open quantum systems by studying the evolution of string order parameters and other probes under noisy channels. We find that one-dimensional SPT order is robust against noisy couplings to the environment that satisfy a strong symmetry condition, while it is destabilized by noise that satisfies only a weak symmetry condition, which generalizes the notion of symmetry for closed systems. We also discuss "transmutation" of SPT phases into other SPT phases of equal or lesser complexity, under noisy channels that satisfy twisted versions of the strong symmetry condition.

Gapless symmetry-protected topological phase of quantum antiferromagnets on anisotropic triangular strip

We study a three-leg spin-1/2 ladder with geometrically frustrated interleg interactions. We call this model an anisotropic triangular-strip (ATS) model. We numerically and field-theoretically show that its ground state belongs to a gapless symmetry-protected topological (SPT) phase. The numerical approach is based on density-matrix renormalization group analyses of the entanglement entropy and the entanglement spectrum. Whereas the entanglement entropy exhibits a critical behavior, the entanglement spectrum is nontrivially degenerate. These entanglement properties imply that the ground state is a gapless topological phase. We investigate the ATS model using a quantum field theory to support the numerical findings. When the frustrated interchain interaction is deemed a perturbation acting on the three spin chains, the frustrated interchain interaction almost isolates the second chain from the other two chains. However, at the same time, the second chain mediates a ferromagnetic interaction between the first and third chains. Therefore, the ground state of the ATS model is a gapless Tomonaga-Luttinger liquid weakly coupled to a spin-1 Haldane chain with irrelevant interactions. Last but not least, we show that the gapless SPT phase of the ATS model is a symmetry-protected critical phase. We point out that the symmetry protection of criticality is essential in characterization of the gapless SPT phase.

Topological order and entanglement dynamics in the measurement-only XZZX quantum code

We examine the dynamics of a $(1+1)$-dimensional measurement-only circuit defined by the stabilizers of the [[5,1,3]] quantum error correcting code interrupted by single-qubit Pauli measurements. The code corrects arbitrary single-qubit errors and it stabilizes an area law entangled state with a ${D}_{2}={\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry protected topological (SPT) order, as well as a symmetry breaking (SB) order from a twofold bulk degeneracy. The Pauli measurements break the topological order and induce a phase transition into a trivial area law phase. Allowing more than one type of Pauli measurement increases the measurement-induced frustration and the SPT and SB order can be broken either simultaneously or separately at nonzero measurement rate. This yields a rich phase diagram and unanticipated critical behavior at the phase transitions. Although the correlation length exponent $\ensuremath{\nu}=\frac{4}{3}$ and the dynamical critical exponent $z=1$ are consistent with bond percolation, the prefactor of the logarithmic entanglement growth may take noninteger multiples of the percolation value. Remarkably, we identify a robust transient scaling regime for the purification dynamics of $L$ qubits. It reveals a modified dynamical critical exponent ${z}^{*}\ensuremath{\ne}z$, which is observable up to times $t\ensuremath{\sim}{L}^{{z}^{*}}$ and is reminiscent of the relaxation of critical systems into a prethermal state.

Real-space construction of crystalline topological superconductors and insulators in 2D interacting fermionic systems

The construction and classification of crystalline symmetry protected topological (SPT) phases in interacting bosonic and fermionic systems have been intensively studied in the past few years. Crystalline SPT phases are not only of conceptual importance, but also provide great opportunities towards experimental realization since space group symmetries naturally exist for any realistic material. In this paper, we systematically classify the crystalline topological superconductors (TSC) and topological insulators (TI) in 2D interacting fermionic systems by using an explicit real-space construction. In particular, we discover an intriguing fermionic crystalline topological superconductor that can only be realized in interacting fermionic systems (i.e., not in free-fermion or bosonic SPT systems). Moreover, we also verify the recently conjectured crystalline equivalence principle for generic 2D interacting fermionic systems.

Strongly correlated crystalline higher-order topological phases in two-dimensional systems: A coupled-wire study

Coupled-wire constructions have been widely applied to quantum Hall systems and symmetry-protected topological (SPT) phases. In this Letter, we use the coupled one-dimensional nonchiral Luttinger liquids with domain-wall structured mass terms as quantum wires to construct the crystalline higher-order topological superconductors (HOTSC) in two-dimensional interacting fermionic systems by two representative examples: $D_4$-symmetric class-D HOTSC and $C_4$-symmetric class-BD\1 HOTSC, with Majorana corner modes on the edge. Furthermore, based on the coupled-wire constructions, the quantum phase transition between different phases of 2D HOTSC by tuning the inter-wire couplings is investigated in a straightforward way.