Symmetry-protected Topological States Research Articles

Entanglement-enabled symmetry-breaking orders

A spontaneous symmetry-breaking order is conventionally described by a tensor-product wavefunction of some few-body clusters; some standard examples include the simplest ferromagnets and valence bond solids. We discuss a type of symmetry-breaking orders, dubbed entanglement-enabled symmetry-breaking orders, which cannot be realized by any such tensor-product state. Given a symmetry breaking pattern, we propose a criterion to diagnose if the symmetry-breaking order is entanglement-enabled, by examining the compatibility between the symmetries and the tensor-product description. For concreteness, we present an infinite family of exactly solvable gapped models on one-dimensional lattices with nearest-neighbor interactions, whose ground states exhibit entanglement-enabled symmetry-breaking orders from a discrete symmetry breaking. In addition, these ground states have gapless edge modes protected by the unbroken symmetries. We also propose a construction to realize entanglement-enabled symmetry-breaking orders with spontaneously broken continuous symmetries. Under the unbroken symmetries, some of our examples can be viewed as symmetry-protected topological states that are beyond the conventional classifications.

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.

Winding θ and destructive interference of instantons

While the θ dependence of field theories is 2π periodic, the ground-state wavefunctions at θ and θ + 2π often belong to different classes of symmetry-protected topological states. When this is the case, a continuous change of the θ parameter can introduce an interface that supports a nontrivial field theory localized on the wall. We consider the 2d ℂPN−1 sigma model as an example and construct a weak-coupling setup of this interface theory by considering the small S1 compactification with nonzero winding θ parameter and a suitable symmetry-twisted boundary condition. This system has N classical vacua connected by fractional instantons, but the anomaly constraint tells us that the fractional-instanton amplitudes should vanish completely to have N-fold degeneracy at the quantum level. We show how this happens in this purely bosonic system, uncovering that the integration over the zero modes annihilates the fractional instanton amplitudes, in sharp contrast to what happens when the θ angle is constant. Moreover, we provide another explanation of this selection rule by showing that the N perturbative vacua acquire different charges under the global symmetry with the activation of the winding θ angle. We also demonstrate a similar destructive interference between instanton effects in the ℂPN−1 quantum mechanics with the Berry phase.

Study of gapped phases of 4d gauge theories using temporal gauging of the ℤN 1-form symmetry

To study gapped phases of 4d gauge theories, we introduce the temporal gauging of ℤN 1-form symmetry in 4d quantum field theories (QFTs), thereby defining effective 3d QFTs with overset{sim }{mathbb{Z}} N × ℤN 1-form symmetry. In this way, spatial fundamental Wilson and ’t Hooft loops are simultaneously genuine line operators. Assuming a mass gap and Lorentz invariant vacuum of the 4d QFT, the overset{sim }{mathbb{Z}} N × ℤN symmetry must be spontaneously broken to an order-N subgroup H, and we can classify the 4d gapped phases by specifying H. This establishes the 1-to-1 correspondence between the two classification schemes for gapped phases of 4d gauge theories: one is the conventional Wilson-’t Hooft classification, and the other is the modern classification using the spontaneous breaking of 4d 1-form symmetry enriched with symmetry-protected topological states.

Deterministic Constant-Depth Preparation of the AKLT State on a Quantum Processor Using Fusion Measurements

A class of symmetry-protected topological states relevant for measurement-based quantum computation is shown to be constructible using only constant-depth quantum circuits by leveraging symmetries, midcircuit measurements, and feed forward.

Tunable symmetry-protected higher-order topological states with fermionic atoms in bilayer optical lattices

Higher-order topological states that possess gapped bulk energy bands and exotic topologically protected boundary states with at least two dimension lower than the bulk have significantly opened a new perspective for understanding of topological quantum matters. Here, we propose to generate two-dimensional topological boundary states for implementing synthetic magnetic flux of ultracold atoms trapped in bilayer optical lattices, which includes Chern insulator, Dirac semimetals, and second-order topological phase (SOTP) by the interplay of the two-photon detuning and effective Zeeman shift. These observed topological phases can be well characterized by the energy gap of bulk, Wilson loop spectra, and the spin textures at the higher symmetric points of system. We show that the SOTP exhibits a pair of $0$D boundary states. While the phases of Dirac semimetals and Chern insulator support the conventional $1$D boundary states due to the principle of bulk-boundary correspondence. Strikingly, the emerged boundary states for Dirac semimetals and SOTP are topologically protected by $\cal P T$-symmetry and chiral-mirror symmetry ($\mathcal{\widetilde{M}}_{\alpha}$), respectively. In particular, the location of $0$D corner states for SOTP which are associated with existing $\mathcal{\widetilde{M}}_{\alpha}$-symmetry can be highly manipulated by tuning magnetic flux. Our scheme herein provides a platform for emerging exotic topological boundary states, which may facilitate the study of higher-order topological phases in ultracold atomic gases.

Symmetry-resolved entanglement of two-dimensional symmetry-protected topological states

Entanglement spectra and their symmetries provide unique fingerprints of symmetry-protected topological states (SPTs). Unfortunately, using such calculations to identify SPTs numerically is notoriously difficult. Here, the authors develop a numerical method mapping this computation to a one-dimensional problem by utilizing the local nature of the quantum circuits that generate any SPT. Its application to models with Ising symmetry fully confirms field-theoretic predictions for their entanglement spectrum. Additionally, nontrivial deformations of the SPT manifest as many-body Aharonov-Bohm fluxes in the entanglement spectrum.

Probing sign structure using measurement-induced entanglement

The sign structure of quantum states is closely connected to quantum phases of matter, yet detecting such fine-grained properties of amplitudes is subtle. Here we employ as a diagnostic measurement-induced entanglement (MIE): the average entanglement generated between two parties after measuring the rest of the system. We propose that for a sign-free state, the MIE upon measuring in the sign-free basis decays no slower than correlations in the state before measurement. Concretely, we prove that MIE is upper bounded by mutual information for sign-free stabilizer states (essentially CSS codes), which establishes a bound between scaling dimensions of conformal field theories describing measurement-induced critical points in stabilizer systems. We also show that for sign-free qubit wavefunctions, MIE between two qubits is upper bounded by a simple two-point correlation function, and we verify our proposal in several critical ground states of one-dimensional systems, including the transverse field and tri-critical Ising models. In contrast, for states with sign structure, such bounds can be violated, as we illustrate in critical hybrid circuits involving both Haar or Clifford random unitaries and measurements, and gapless symmetry-protected topological states.

Emergent fermionic gauge theory and foliated fracton order in the Chamon model

The Chamon model is an exactly solvable spin Hamiltonian exhibiting nontrivial fracton order. In this paper, we dissect two distinct aspects of the model. First, we show that it exhibits an emergent fractonic gauge theory coupled to a fermionic subsystem symmetry-protected topological state under four stacks of ${\mathbb{Z}}_{2}$ planar symmetries. Second, we show that the Chamon model hosts 4-foliated fracton order by describing an entanglement renormalization group transformation that exfoliates four separate stacks of 2D toric codes from the bulk system.

Non-Abelian bosonization of topological insulators and superconductors

Applying the method of [Nucl. Phys. B 972, 115565 (2021)], which bosonizes massless relativistic free fermions, we derive the (non-Abelian) bosonized theory for free fermion topological insulators and superconductors that have, in addition to the U(1) charge, time reversal and charge conjugation symmetries and flavor symmetries. For the case we consider, the flavor symmetries render the topological classification $\mathbb{Z}$. The results are nonlinear $\ensuremath{\sigma}$ models with the topological $\ensuremath{\theta}$ term. In addition, we present the theory of a class of bosonic symmetry-protected topological states, whose boundaries are critical spin liquids.

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.

Persistent corner spin mode at the quantum critical point of a plaquette Heisenberg model

Gapless edge states are the hallmark of a large class of topological states of matter. Recently, intensive research has been devoted to understanding the physical properties of the edge states at the quantum phase transitions of the bulk topological states. A higher-order symmetry-protected topological state is realized in a plaquette Heisenberg model on the square lattice. In its disordered phase, the lattice with an open boundary hosts either dangling corner states with spin-$1/2$ degeneracy characterizing the topological phase, or nondangling corner states without degeneracy, which depends on the bond configuration near the corners. In this work, we study the critical behavior of these corner states at the quantum critical point (QCP), and find that the spin-$1/2$ corner state induces a new universality class of the corner critical behavior, which is distinct from the ordinary transition of the nondangling corners. In particular, we find that the dangling spin-$1/2$ corner state persists at the QCP despite its coupling to the critical spin fluctuations in the bulk. This shows the robustness of the corner state of the higher-order topological state.

Anomalies in (2+1)D Fermionic Topological Phases and (3+1)D Path Integral State Sums for Fermionic SPTs

Given a (2+1)D fermionic topological order and a symmetry fractionalization class for a global symmetry group $G$, we show how to construct a (3+1)D topologically invariant path integral for a fermionic $G$ symmetry-protected topological state ($G$-FSPT) in terms of an exact combinatorial state sum. This provides a general way to compute anomalies in (2+1)D fermionic symmetry-enriched topological states of matter. Equivalently, our construction provides an exact (3+1)D combinatorial state sum for a path integral of any FSPT that admits a symmetry-preserving gapped boundary, including the (3+1)D topological insulators and superconductors in class AII, AIII, DIII, and CII that arise in the free fermion classification. Our construction uses the fermionic topological order (characterized by a super-modular tensor category) and symmetry fractionalization data to define a (3+1)D path integral for a bosonic theory that hosts a non-trivial emergent fermionic particle, and then condenses the fermion by summing over closed 3-form $\mathbb{Z}_2$ background gauge fields. This procedure involves a number of non-trivial higher-form anomalies associated with Fermi statistics and fractional quantum numbers that need to be appropriately canceled off with a Grassmann integral that depends on a generalized spin structure. We show how our construction reproduces the $\mathbb{Z}_{16}$ anomaly indicator for time-reversal symmetric topological superconductors with ${\bf T}^2 = (-1)^F$. Mathematically, with standard technical assumptions, this implies that our construction gives a combinatorial state sum on a triangulated 4-manifold that can distinguish all $\mathbb{Z}_{16}$ $\mathrm{Pin}^+$ smooth bordism classes. As such, it contains the topological information encoded in the eta invariant of the pin$^+$ Dirac operator, thus giving an example of a state sum TQFT that can distinguish exotic smooth structure.

De Haas-van Alphen effect and the first-principles study of the possible topological stannide Cu3Sn

The search for quantum materials with various symmetry-protected topological states has been the focus of recent research interest, primarily due to their fascinating physical properties and the potential technological utility. In this work, we report on the magnetotransport, de Haas-van Alphen (dHvA) oscillations, and the first-principles calculations of the stannide Cu3Sn that is isostructural with the recently reported topological semimetal Ag3Sn. The magnetoresistance was found to vary quasi-linearly in field. Clear dHvA oscillations were observed under a field as low as 1 Tesla at 2 K, with three major oscillation frequencies Fα = 8.74 T, Fβ = 150.19 T and Fγ = 229.66 T and extremely small effective masses. The analysis of dHvA quantum oscillations revealed a possible nonzero Berry phase, suggestive of the nontrivial band topology. The corroborating evidence for the nontrivial electronic topology also comes from the first-principles calculations which yield a nonzero Z2 topological index. These results collectively suggest that Cu3Sn, in analogy to its homolog Ag3Sn, may be another intermetallic stannide hosting topological Dirac fermions.

Symmetric Mass Generation

The most well-known mechanism for fermions to acquire a mass is the Nambu–Goldstone–Anderson–Higgs mechanism, i.e., after a spontaneous symmetry breaking, a bosonic field that couples to the fermion mass term condenses, which grants a mass gap for the fermionic excitation. In the last few years, it was gradually understood that there is a new mechanism of mass generation for fermions without involving any symmetry breaking within an anomaly-free symmetry group, also applicable to chiral fermions with anomaly-free chiral symmetries. This new mechanism is generally referred to as the symmetric mass generation (SMG). It is realized that the SMG has deep connections with interacting topological insulator/superconductors, symmetry-protected topological states, perturbative local and non-perturbative global anomaly cancellations, and deconfined quantum criticality. It has strong implications for the lattice regularization of chiral gauge theories. This article defines the SMG, summarizes the current numerical results, introduces an unifying theoretical framework (including the parton-Higgs and the s-confinement mechanisms, as well as the symmetry-extension construction), and presents an overview of various features and applications of SMG.

Defectiveness and anomaly from non-Hermitian perturbations in topological quantum states

Topological quantum states are important research fields in condensed matter physics, which can be divided into two categories: (1) symmetry-protected topological states (SPTs) with short-range entanglement and (2) symmetry-enriched topological states (SETs) with long-range entanglement. Recently, research on non-Hermitian topological quantum states (SPTs and SETs) has become a hot topic. In this review paper, we discuss the theory for non-Hermitian topological insulators—a particular type of SPTs and that for non-Hermitian Z$_2$ topological orders—a particular type of SETs based on the progress of our research group.These theories would help understand the physics of non-Hermitian topological quantum states.

Realizing quasi-monochromatic switchable thermal emission from electro-optically induced topological phase transitions.

Explorations into the photonic analogs of topological materials have garnered significant research interest due to their application potential. Particularly in planar systems, the prospects of engendering extinguishable topological states can have wide-ranging implications. With an objective of employing these concepts for thermal emission engineering, here, we design and numerically investigate a quasi-monochromatic highly directional mid-infrared source elicited from inversion symmetry-protected topological interface states. Notably, by relying on the architecture of electro-optic effect-induced topological phase transitions, we introduce the possibility of ultrafast switching of thermal radiation. These reversible phase transitions, being free from carrier transport are inherently fast and evoke thermal emission modulation with a modulation depth upto 0.99. Specifically, our platform exhibits a near-perfect extinguishable spectral emission peak at 4~mum with a quality factor of over 18500, displaying negligible parasitic emissions. Furthermore, the optimized interface state manifests itself for only one of the polarization modes, resulting in polarized emission under resonance conditions. To establish a methodical approach to parameter optimization, we also model our platform as a leaky mode resonator using the framework of temporal coupled-mode theory. We believe, our findings can provide a way forward in establishing complete control over the optical characteristics of the infrared thermal emitters.

Anomaly cascade in ( 2+1 )-dimensional fermionic topological phases

We develop a theory of anomalies of fermionic topological phases of matter in (2+1)D with a general fermionic symmetry group $G_f$. In general, $G_f$ can be a non-trivial central extension of the bosonic symmetry group $G_b$ by fermion parity $(-1)^F$. We encounter four layers of obstructions to gauging the $G_f$ symmetry, which we dub the anomaly cascade: (i) An $\mathcal{H}^1(G_b,\mathbb{Z}_{\bf T})$ obstruction to extending the symmetry permutations on the anyons to the fermion parity gauged theory, (ii) An $\mathcal{H}^2(G_b, \ker r)$ obstruction to extending the $G_b$ group structure of the symmetry permutations to the fermion parity gauged theory, where $r$ is a map that restricts symmetries of the fermion parity gauged theory to the anyon theory, (iii) An $\mathcal{H}^3(G_b, \mathbb{Z}_2)$ obstruction to extending the symmetry fractionalization class to the fermion parity gauged theory, and (iv) the well-known $\mathcal{H}^4(G_b, U(1))$ obstruction to developing a consistent theory of $G_b$ symmetry defects for the fermion parity gauged theory. We describe how the $\mathcal{H}^2$ obstruction can be canceled by anomaly inflow from a bulk (3+1)D symmetry-protected topological state (SPT) and also its relation to the Arf invariant of spin structures on a torus. If any anomaly in the above sequence is non-trivial, the subsequent ones become relative anomalies. A number of conjectures regarding symmetry actions on super-modular categories, guided by general expectations of anomalies in physics, are also presented.

Custodial Chiral Symmetry in a Su-Schrieffer-Heeger Electrical Circuit with Memory.

Custodial symmetries are common in the standard model of particle physics. They arise when quantum corrections to a parameter are proportional to the parameter itself. Here, we show that a custodial symmetry of the chiral type is also present in a classical Su-Schrieffer-Heeger (SSH) electrical circuit with memory. In the absence of memory, the SSH circuit supports a symmetry-protected topological edge state. Memory induces nonlinearities that break chiral symmetry explicitly and spread the state across the circuit. However, the resulting state is still protected against perturbations by the ensuing custodial chiral symmetry. These predictions can be verified experimentally and demonstrate the interplay between symmetry and memory.

Corner states, hinge states, and Majorana modes in SnTe nanowires

SnTe materials are one of the most flexible material platforms for exploring the interplay of topology and different types of symmetry breaking. We study symmetry-protected topological states in SnTe nanowires in the presence of various combinations of Zeeman field, s-wave superconductivity and inversion-symmetry-breaking field. We uncover the origin of robust corner states and hinge states in the normal state. In the presence of superconductivity, we find inversion-symmetry-protected gapless bulk Majorana modes, which give rise to quantized thermal conductance in ballistic wires. By introducing an inversion-symmetry-breaking field, the bulk Majorana modes become gapped and topologically protected localized Majorana zero modes appear at the ends of the wire.