Symmetry-protected Topological States Research Articles

Building 1D lattice models with $G$-graded fusion category

We construct a family of one-dimensional (1D) quantum lattice models based on GG-graded unitary fusion category \mathcal{C}_G𝒞G. This family realize an interpolation between the anyon-chain models and edge models of 2D symmetry-protected topological states, and can be thought of as edge models of 2D symmetry-enriched topological states. The models display a set of unconventional global symmetries that are characterized by the input category \mathcal{C}_G𝒞G. While spontaneous symmetry breaking is also possible, our numerical evidence shows that the category symmetry constrains the models to the extent that the low-energy physics has a large likelihood to be gapless.

Intrinsic symmetry-protected topological mixed state from modulated symmetries and hierarchical structure of boundary anomaly

Intrinsic symmetry-protected topological mixed state from modulated symmetries and hierarchical structure of boundary anomaly

Anomaly inflow, dualities, and quantum simulation of Abelian lattice gauge theories induced by measurements

Previous work [] has demonstrated that quantum simulation of Abelian lattice gauge theories (Wegner models including the toric code in a limit) in general dimensions can be achieved by local adaptive measurements on symmetry-protected topological (SPT) states with higher-form generalized global symmetries. The entanglement structure of the resource SPT state reflects the geometric structure of the gauge theory. In this work we explicitly demonstrate the anomaly inflow mechanism between the deconfining phase of the simulated gauge theory on the boundary and the SPT state in the bulk by showing that the anomalous gauge variation of the boundary state obtained by bulk measurement matches that of the bulk theory. Moreover, we construct the resource state and the measurement pattern for the measurement-based quantum simulation of a lattice gauge theory with a matter field (Fradkin-Shenker model), where a simple scheme to protect gauge invariance of the simulated state against errors is proposed. We further consider taking an overlap between the wave function of the resource state for lattice gauge theories and that of a parameterized product state, and we derive precise dualities between partition functions with insertion of defects corresponding to gauging higher-form global symmetries, as well as measurement-induced phases where states induced by a partial overlap possess different (symmetry-protected) topological orders. Measurement-assisted operators to dualize quantum Hamiltonians of lattice gauge theories and their noninvertibility are also presented. Published by the American Physical Society 2024

Cluster State as a Noninvertible Symmetry-Protected Topological Phase.

We show that the standard 1+1D Z_{2}×Z_{2} cluster model has a noninvertible global symmetry, described by the fusion category Rep(D_{8}). Therefore, the cluster state is not only a Z_{2}×Z_{2} symmetry protected topological (SPT) phase, but also a noninvertible SPT phase. We further find two new commuting Pauli Hamiltonians for the other two Rep(D_{8}) SPT phases on a tensor product Hilbert space of qubits, matching the classification in field theory and mathematics. We identify the edge modes and the local projective algebras at the interfaces between these noninvertible SPT phases. Finally, we show that there does not exist a symmetric entangler that maps between these distinct SPT states.

Engineering Topological States in Nanographenes

Our research focuses on the rational design, deterministic assembly, and detailed investigation of the physical phenomena emerging from quantum confinement effects in graphene nanomaterials. We pursue a highly integrated multidisciplinary program, founded on synthetic bottom-up approaches toward functional materials with precisely defined structure. We control their assembly into hierarchically ordered architectures and evaluate inherent physical properties using modern scanning probe techniques cross multiple length, time, and energy scales. We recently demonstrated the rational design and experimental realization of a graphene nanoribbon (GNR) superlattices that hosts a 1D array of symmetry-protected topological states, thus generating otherwise inaccessible electronic structure.1 This new class of materials can be thought of as an extended form of poly-acetylene wherein highly localized half-filled topological states replace the familiar single occupied p-orbitals along a conjugated π-system. Experimental results and first-principles calculations reveal that the frontier band structure of these GNR superlattices is defined purely by the coupling between adjacent topological interface states and can be tuned all the way from a semiconductor and a metal.2,3 This novel manifestation of 1D topological phases presents an entirely new route to band engineering in 1D materials based on precise control of their electronic topology and is a promising new platform for future studies of 1D quantum spin physics and metallicity in low dimensional carbon nanomaterials. 1. Topological Band Engineering of Graphene Nanoribbons. Rizzo, D. J.; Veber, G.; Cao, T.; Bronner, C.; Chen, T.; Zhao, F.; Rodriguez, H.; Louie, S. G.; Crommie, M. F.; Fischer, F. R. Nature 2018, 560, 204-206. 2. Inducing Metallicity in Graphene Nanoribbons via Zero-Mode Superlattices. Rizzo, D. J.; Veber, G.; Jiang, J.; McCurdy, R.; Cao, T.; Bronner, C.; Chen, T.; Louie, S. G.; Fischer, F. R.; Crommie, M. F. Science 2020, 369, 1597-1603. 3. Robust Metallic Zero-Mode States in Olympicene Graphene Nanoribbons. McCurdy, R. D.; Delgado, A.; Jiang, J.; Zhu, J.; Wen, E. C. H.; Blackwell, R. E.; Veber, G. C.; Wang, S.; Louie, S. G.; Fischer, F. R. J. Am. Chem. Soc. 2023, 145, 15162-15170.

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.

Nonlocal Finite-Depth Circuits for Constructing Symmetry-Protected Topological States and Quantum Cellular Automata

Whether a given target state can be prepared by starting with a simple product state and acting with a finite-depth quantum circuit is a key question in condensed matter physics and quantum information science. It underpins classifications of topological phases, as well as the understanding of topological quantum codes, and has obvious relevance for device implementations. Traditionally, this question assumes that the quantum circuit is made up of unitary gates that are . Inspired by the advent of noisy intermediate-scale quantum devices, we reconsider this question with k- gates, i.e., gates that act on no more than k degrees of freedom but are not restricted to be geometrically local. First, we construct explicit finite-depth circuits of symmetric k-local gates that create symmetry-protected topological (SPT) states from an initial product state. Our construction applies to SPT states protected by global symmetries or subsystem symmetries but not to those with higher-form symmetries, which we conjecture remain nontrivial. Next, we show how to implement arbitrary translationally invariant quantum cellular automata in any dimension using finite-depth circuits of k-local gates. These results imply that the topological classifications of SPT phases and quantum cellular automata both collapse to a single trivial phase in the presence of k-local interactions. We furthermore argue that SPT phases are fragile to k-local symmetric perturbations. We conclude by discussing the implications for other phases, such as fracton phases, and surveying future directions. Our analysis opens a new experimentally motivated conceptual direction examining the stability of phases and the feasibility of state preparation without the assumption of geometric locality. Published by the American Physical Society 2024

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.