Trivial State Research Articles

Combinatorial NLTS From the Overlap Gap Property

In an important recent development, Anshu, Breuckmann, and Nirkhe \cite{anshu2022nlts} resolved positively the so-called No Low-Energy Trivial State (NLTS) conjecture by Freedman and Hastings. The conjecture postulated the existence of linear-size local Hamiltonians on n qubit systems for which no near-ground state can be prepared by a shallow (sublogarithmic depth) circuit. The construction in \cite{anshu2022nlts} is based on recently developed good quantum codes. Earlier results in this direction included the constructions of the so-called Combinatorial NLTS – a weaker version of NLTS – where a state is defined to have low energy if it violates at most a vanishing fraction of the Hamiltonian terms \cite{anshu2022construction}. These constructions were also based on codes. In this paper we provide a "non-code" construction of a class of Hamiltonians satisfying the Combinatorial NLTS. The construction is inspired by one in \cite{anshu2022construction}, but our proof uses the complex solution space geometry of random K-SAT instead of properties of codes. Specifically, it is known that above a certain clause-to-variables density the set of satisfying assignments of random K-SAT exhibits an overlap gap property, which implies that it can be partitioned into exponentially many clusters each constituting at most an exponentially small fraction of the total set of satisfying solutions. We establish a certain robust version of this clustering property for the space of near-satisfying assignments and show that for our constructed Hamiltonians every combinatorial near-ground state induces a near-uniform distribution supported by this set. Standard arguments then are used to show that such distributions cannot be prepared by quantum circuits with depth o(log n). Since the clustering property is exhibited by many random structures, including proper coloring and maximum cut, we anticipate that our approach is extendable to these models as well.

Interplay of correlations and Majorana mode from local solution perspective

We study the quasiparticle spectrum of a hybrid system, comprising a correlated (Anderson-type) quantum dot coupled to a topological superconducting nanowire hosting the Majorana boundary modes. From the exact solution of the low-energy effective Hamiltonian, we uncover a subtle interplay between Coulomb repulsion and the Majorana mode. Our analytical expressions show that the spectral weight of the leaking Majorana mode is sensitive to both the quantum dot energy level and the repulsive potential. We compare our results with estimations by [L.S. Ricco et al. Phys. Rev. B 99, 155159 (2019)] obtained for the same hybrid structure using the Hubbard-type decoupling scheme, and analytically quantify the spectral weight of the zero-energy (topological) mode coexisting with the finite-energy (trivial) states of the quantum dot. We also show that empirical verification of these spectral weights could be feasible through spin-polarized Andreev spectroscopy.

Spectroscopic Evidence of Ultrafast Topological Phase Transition by Light-Driven Strain.

Enabling reversible control over the topological invariants, transitioning them from nontrivial to trivial states, has fundamental implications for quantum information processing and spintronics. It offers a promising avenue for establishing an efficient on/off switch mechanism for robust and dissipationless spin-currents. While mechanical strain has traditionally been advantageous for such manipulation of topological invariants, it often comes with the drawback of in-plane fractures, rendering it unsuitable for high-speed, time-dependent operations. This study employs ultrafast optical and THz spectroscopy to explore topological phase transitions induced by light-driven strain in Bi2Se3. Bi2Se3 requires substantial strain for Z2 switching. Our observations provide experimental evidence of ultrafast switching behavior, demonstrating a transition from a topological insulator with spin-momentum-locked surfaces to hybridized states and normal insulating phases under ambient conditions. Notably, applying light-induced strong out-of-plane strain effectively suppresses surface-bulk coupling, facilitating the differentiation of surface and bulk conductance even at room temperature─significantly surpassing the Debye temperature. We expect various time-dependent sequences of transient hybridization and manipulation of topological invariant through photoexcitation intensity adjustments. The sudden surface and bulk transport alterations near the transition point enable coherent conductance modulation at hypersound frequencies. Our findings on the potential of light-triggered ultrafast switching of topological invariants hold promise for high-speed topological switching and its related applications.

Quantum Locally Testable Code with Constant Soundness

In this paper, we present two constructions of quantum locally testable codes (QLTC) with constant soundness. In the first approach, we introduce an operation called check product, and show how this operation gives rise to QLTCs of constant soundness, constant rate, and distance scaling with locality. In the second approach, we consider hypergraph product of a quantum code and a classical repetition code, and observe a special case in which the soundness of component codes is preserved. This insight leads us to construct QLTCs of constant soundness, scalable rate and distance, and constant average locality. Our work marks a step towards constructing QLTCs of high soundness and distance, which would give a different construction to the No Low-Energy Trivial States (NLTS) theorem.

Emergence of topological phase and non-trivial surface states in rare-earth semimetal GdSb with pressure

Abstract We study the evolution of band topology under external pressure in rare-earth gadolinium mono-antimonide (GdSb) using first-principles calculations. This material crystallizes in a rocksalt-type structure and shows a structural phase transition (SPT) to a CsCl-type structure at 26.1 GPa. The phonon dispersions are analyzed to ascertain the dynamical stability of this material. We use hybrid density functional theory with the inclusion of spin–orbit coupling to investigate the structural, electronic, and topological phase transitions (TPTs). At ambient pressure, GdSb shows a topologically trivial state which is in agreement with existing experimental reports. The first TPT is observed at 6 GPa of hydrostatic pressure (at the high symmetry X-point) which is verified with the help of single-band inversion and surface state analysis along the (001) plane. The non-zero value of the first Z2 topological invariant and the presence of the Dirac cone also confirm the topological phase of this material. A further increase in pressure to 12 GPa results in two band inversions at Γ- as well as X-points, which corresponds to the trivial nature of GdSb. The same is also verified with (0; 000) values of Z2 topological invariants and a pair of Dirac cones in surface states. It is noted that the crystal symmetries are preserved throughout the study and the TPT values are much lower than the SPT pressure, i.e. 26.1 GPa.

Visualization of Unconventional Rashba Bands and Vortex Zero Mode in the Topological Superconductor Candidate AuSn4.

The noble metal alloy AuSn4 has recently been identified as an intrinsic surface topological superconductor, promisingly hosting the Majorana zero mode (MZM) for topological quantum computing. However, the atomic visualization of its nontrivial surface states and MZM remains elusive. Here, we report the direct observation of unconventional surface states and vortex zero mode in AuSn4 by scanning tunneling microscopy/spectroscopy. Unlike the trivial metallic bulk states of Sn-terminated surfaces, the Au-terminated surfaces exhibit pronounced surface states near the Fermi level, arising from unconventional Rashba bands characterized by shared helical spin textures. In the superconducting state, the Sn-terminated surfaces exhibit conventional Caroli-de Gennes-Matricon bound states, while the Au-terminated surfaces display sharp zero-energy core states resembling MZMs in a nonquantum-limit condition. This distinction may result from the dominant contribution of unconventional Rashba bands on the Au-terminated surface. Our results provide a new platform for studying termination-dependent topological surface states and MZM in noble-metal-based superconductors.

Identifying trivial and Majorana zero-energy modes using the Majorana polarization

In this work we consider superconductor-semiconductor hybrids containing both trivial and Majorana zero modes and explore their signatures in the Majorana polarization. In particular, we consider trivial zero-energy states due to confinement and disorder, which seem to be very likely experimental scenarios. We show that the Majorana polarization is able to characterize the topological phase transition as well as the emergence of Majorana zero modes even when trivial zero-energy states proliferate. Notably, the Majorana polarization inherits direct information about spatial correlations which are then the key for distinguishing Majorana and trivial zero modes. We demonstrate the utility of the Majorana polarization in normal-superconductor junctions and superconductor-normal-superconductor Josephson junctions. Our results support the interpretation of the Majorana polarization as a real-space topological indicator. Published by the American Physical Society 2024

Multiple Topological Phases with Electronic Correlation in Intrinsic Ferromagnetic Semimetal VI3 Monolayer.

2D topological materials with magnetic ordering have become hot topics due to their nontrivial band topology and quantum states. In this work, the second-order topological states and evolution of linear band crossing are successfully predicted utilizing the effective k· p and tight binding models in the intrinsic ferromagnetic VI3 monolayer under various effective Hubble interaction Ueff. Upon inclusion of spin orbit coupling, a small bandgap (Eg-1) of 12.7 meV is opened with a Chern invariant C = -1 at Ueff = 0 eV. The Eg-1 undergoes a transition from the non-trivial state to trivial state at Ueff = 0.80 eV, accompanied by the appearance of Dirac cone. Remarkably, the increase of Ueff causes the band inversion and adjustment of crystal symmetry, resulting in two unreported coexistingtopological bandgaps (Eg-2 and Eg-3). Furthermore, a gapless node-loop appears at Ueff = 1.06 eV and disappears at Ueff = 1.09 eV around Γ point. Moreover, for the first time, the existence of second-order topological states with quantized corner fractional charges (e/3) is also observed in the VI3 monolayer at Ueff ≥0.96 eV. These results make the VI3 monolayer a compelling candidate for exploring topological devices.

Generalized cluster states from Hopf algebras: non-invertible symmetry and Hopf tensor network representation

Cluster states are crucial resources for measurement-based quantum computation (MBQC). It exhibits symmetry-protected topological (SPT) order, thus also playing a crucial role in studying topological phases. We present the construction of cluster states based on Hopf algebras. By generalizing the finite group valued qudit to a Hopf algebra valued qudit and introducing the generalized Pauli-X operator based on the regular action of the Hopf algebra, as well as the generalized Pauli-Z operator based on the irreducible representation action on the Hopf algebra, we develop a comprehensive theory of Hopf qudits. We demonstrate that non-invertible symmetry naturally emerges for Hopf qudits. Subsequently, for a bipartite graph termed the cluster graph, we assign the identity state and trivial representation state to even and odd vertices, respectively. Introducing the edge entangler as controlled regular action, we provide a general construction of Hopf cluster states. To ensure the commutativity of the edge entangler, we propose a method to construct a cluster lattice for any triangulable manifold. We use the 1d cluster state as an example to illustrate our construction. As this serves as a promising candidate for SPT phases, we construct the gapped Hamiltonian for this scenario and provide a detailed discussion of its non-invertible symmetries. We demonstrate that the 1d cluster state model is equivalent to the quasi-1d Hopf quantum double model with one rough boundary and one smooth boundary. We also discuss the generalization of the Hopf cluster state model to the Hopf ladder model through symmetry topological field theory. Furthermore, we introduce the Hopf tensor network representation of Hopf cluster states by integrating the tensor representation of structure constants with the string diagrams of the Hopf algebra, which can be used to solve the Hopf cluster state model.

Approximate Quantum Codes From Long Wormholes

We discuss families of approximate quantum error correcting codes which arise as the nearly-degenerate ground states of certain quantum many-body Hamiltonians composed of non-commuting terms. For exact codes, the conditions for error correction can be formulated in terms of the vanishing of a two-sided mutual information in a low-temperature thermofield double state. We consider a notion of distance for approximate codes obtained by demanding that this mutual information instead be small, and we evaluate this mutual information for the SYK model and for a family of low-rank SYK models. After an extrapolation to nearly zero temperature, we find that both kinds of models produce fermionic codes with constant rate as the number, N, of fermions goes to infinity. For SYK, the distance scales as N1/2, and for low-rank SYK, the distance can be arbitrarily close to linear scaling, e.g. N.99, while maintaining a constant rate. We also consider an analog of the no low-energy trivial states property which we dub the no low-energy adiabatically accessible states property and show that these models do have low-energy states that can be prepared adiabatically in a time that does not scale with system size N. We discuss a holographic model of these codes in which the large code distance is a consequence of the emergence of a long wormhole geometry in a simple model of quantum gravity.

Non-Hermitian zero-energy pinning of Andreev and Majorana bound states in superconductor-semiconductor systems

The emergence of Majorana bound states in finite length superconductor-semiconductor hybrid systems has been predicted to occur in the form of oscillatory energy levels with parity crossings around zero energy. Each zero-energy crossing is expected to produce a quantized zero-bias conductance peak but several studies have reported conductance peaks pinned at zero energy over a range of Zeeman fields, whose origin, however, is not clear. In this work, we consider superconducting systems with spin-orbit coupling under a Zeeman field and demonstrate that non-Hermitian effects, due to coupling to ferromagnet leads, induce zero-energy pinning of Majorana and trivial Andreev bound states. We find that this zero-energy pinning effect occurs due to the formation of non-Hermitian spectral degeneracies known as exceptional points, whose emergence can be controlled by the interplay of non-Hermiticity, the applied Zeeman field, and chemical potentials. Moreover, depending on the non-Hermitian spatial profile, we find that non-Hermiticity changes the single point Hermitian topological phase transition into a flattened zero energy line bounded by exceptional points from multiple low energy levels. This seemingly innocent change notably enables a gap closing well below the Hermitian topological phase transition, which can be in principle simpler to achieve. Furthermore, we reveal that the energy gaps separating Majorana and trivial Andreev bound states from the quasicontinuum remain robust for the values that give rise to the zero-energy pinning effect. While reasonable values of non-Hermiticity can be indeed beneficial, very strong non-Hermitian effects can be detrimental as it might destroy superconductivity. Our findings can be therefore useful for understanding the zero-energy pinning of trivial and topological states in Majorana devices. Published by the American Physical Society 2024

Unlocking Quantum Catalysis in Topological Trivial Materials: A Case Study of Janus Monolayer MoSMg

The emerging field of topological catalysis has received significant attention due to its potential for high‐performance catalytic activity in the hydrogen‐evolution reaction (HER). While topological materials often possess fragile surface states, trivial topological materials not only offer a larger pool of candidates but also demonstrate robust surface states. As a result, the search for topological catalysts has expanded to include trivial schemes. In this study, a novel 2D Janus monolayer, MoSMg, which demonstrates exceptional obstructed atomic insulating behavior, is presented. Crucially, this trivial metallic topological state exhibits clean obstructed surface states, leading to a significant enhancement in catalytic performance for the HER in electrochemical processes, particularly under high hydrogen coverage. Moreover, the edge sites of this MoSMg monolayer exhibit even more superior catalytic activity, characterized by near‐zero Gibbs free energies. In these findings, the way is paved for exploring new avenues in the design of quantum electrocatalysts, especially within the realm of trivial topological materials.

Pressure and speed of sound in two-flavor color-superconducting quark matter at next-to-leading order

Deconfined quark matter at asymptotically high densities is weakly coupled, due to the asymptotic freedom of quantum chromodynamics. In this weak-coupling regime, bulk thermodynamic properties of quark matter, assuming a trivial ground state, are currently known to partial next-to-next-to-next-to-leading order. However, the ground state at high densities is expected to be a color superconductor, in which the excitation spectrum of (at least some) quarks exhibit a gap with a nonperturbative dependence on the strong coupling. In this work, we calculate the thermodynamic properties of color-superconducting quark matter at high densities and zero temperature at next-to-leading order (NLO) in the coupling in the presence of a finite gap. We work in the limit of two massless quark flavors, which corresponds to deconfined symmetric nuclear matter, and further assume that the gap is small compared to the quark chemical potential. In these limits, we find that the NLO corrections to the pressure and speed of sound are comparable in size to the leading-order effects of the gap, and further increase both quantities above their values for nonsuperconducting quark matter. We also provide a parametrization of the NLO speed of sound to guide phenomenology in the high-density region, and we furthermore comment on whether our findings should be expected to extend to the case of three-flavor quark matter of relevance to neutron stars. Published by the American Physical Society 2024

Symmetry-Enforced Many-Body Separability Transitions

We study quantum many-body mixed states with a symmetry from the perspective of , i.e., whether a mixed state can be expressed as an ensemble of short-range-entangled symmetric pure states. We provide evidence for “symmetry-enforced separability transitions” in a variety of states, where in one regime the mixed state is expressible as a convex sum of symmetric short-range-entangled pure states, while in the other regime, such a representation is not feasible. We first discuss the Gibbs state of Hamiltonians that exhibit spontaneous breaking of a discrete symmetry, and argue that the associated thermal phase transition can be thought of as a symmetry-enforced separability transition. Next we study cluster states in various dimensions subjected to local decoherence, and identify several distinct mixed-state phases and associated separability phase transitions, which also provides an alternative perspective on recently discussed “average symmetry-protected topological order.” We also study decohered p+ip superconductors, and find that if the decoherence breaks the fermion parity explicitly, then the resulting mixed state can be expressed as a convex sum of nonchiral states, while a fermion parity–preserving decoherence results in a phase transition at a nonzero threshold that corresponds to spontaneous breaking of fermion parity. Finally, we briefly discuss systems that satisfy the no low-energy trivial state property, such as the recently discovered good low-density parity-check codes, and argue that the Gibbs state of such systems exhibits a temperature-tuned separability transition. Published by the American Physical Society 2024

Sublattice Structure and Topology in Spontaneously Crystallized Electronic States.

The prediction and realization of the quantum anomalous Hall effect are often intimately connected to honeycomb lattices in which the sublattice degree of freedom plays a central role in the nontrivial topology. Two-dimensional Wigner crystals, on the other hand, form triangular lattices without sublattice degrees of freedom, resulting in a topologically trivial state. Here, we discuss the possibility of spontaneously formed honeycomb-lattice crystals that exhibit the quantum anomalous Hall effect. Starting from a single-band system with nontrivial quantum geometry, we derive the mean-field energy functional of a class of crystal states and express it as a model of sublattice pseudospins in momentum space. We find that nontrivial quantum geometry leads to extra terms in the pseudospin model that break an effective "time-reversal symmetry" and favor a topologically nontrivial pseudospin texture. When the effects of these extra terms dominate over the ferromagnetic exchange coupling between pseudospins, the anomalous Hall crystal state becomes energetically favorable over the trivial Wigner crystal state.

Glassy Word Problems: Ultraslow Relaxation, Hilbert Space Jamming, and Computational Complexity

We introduce a family of local models of dynamics based on “word problems” from computer science and group theory, for which we can place rigorous lower bounds on relaxation timescales. These models can be regarded either as random circuit or local Hamiltonian dynamics and include many familiar examples of constrained dynamics as special cases. The configuration space of these models splits into dynamically disconnected sectors, and for initial states to relax, they must “work out” the other states in the sector to which they belong. When this problem has a high time complexity, relaxation is slow. In some of the cases we study, this problem also has high space complexity. When the space complexity is larger than the system size, an unconventional type of jamming transition can occur, whereby a system of a fixed size is not ergodic but can be made ergodic by appending a large reservoir of sites in a trivial product state. This finding manifests itself in a new type of Hilbert space fragmentation that we call fragile fragmentation. We present explicit examples where slow relaxation and jamming strongly modify the hydrodynamics of conserved densities. In one example, density modulations of wave vector q exhibit almost no relaxation until times O(exp(1/q)), at which point they abruptly collapse. We also comment on extensions of our results to higher dimensions. Published by the American Physical Society 2024

Trivial State Fuzzy Processing for Error Reduction in Healthcare Big Data Analysis towards Precision Diagnosis.

There is a significant public health concern regarding medical diagnosis errors, which are a major cause of mortality. Identifying the root cause of these errors is challenging, and even if one is identified, implementing an effective treatment to prevent their recurrence is difficult. Optimization-based analysis in healthcare data management is a reliable method for improving diagnostic precision. Analyzing healthcare data requires pre-classification and the identification of precise information for precision-oriented outcomes. This article introduces a Cooperative-Trivial State Fuzzy Processing method for significant data analysis with possible derivatives. Trivial State Fuzzy Processing operates on the principle of fuzzy logic-based processing applied to structured healthcare data, focusing on mitigating errors and uncertainties inherent in the data. The derivatives are aided by identifying and grouping diagnosis-related and irrelevant data. The proposed method mitigates invertible derivative analysis issues in similar data grouping and irrelevance estimation. In the grouping and detection process, recent knowledge of the diagnosis progression is exploited to identify the functional data for analysis. Such analysis improves the impact of trivial diagnosis data compared to a voluminous diagnosis history. The cooperative derivative states under different data irrelevance factors reduce trivial state errors in healthcare big data analysis.

Subsystem complexity and measurements in holography

We investigate the impact of measuring one subsystem on the holographic complexity of another. While a naive expectation might suggest a reduction in complexity due to the collapse of the state to a trivial product state during quantum measurements, our findings reveal a counterintuitive result: in numerous scenarios, measurements on one subsystem can amplify the complexity of another. We first present a counting argument elucidating this complexity transition in random states. Then, employing the subregion “complexity=volume” (CV) proposal, we identify a complexity phase transition induced by projective measurements in various holographic CFT setups, including CFT vacuum states, thermofield double states, and the joint system of a black hole coupled to a bath. According to the AdS/BCFT correspondence, the post-measurement dual geometry involves an end-of-the-world brane created by the projective measurement. The complexity phase transition corresponds to the transition of the entanglement wedge to the one connected to the brane. In the context of the thermofield double setup, complete projection on one side can transform the other side into a boundary state black hole with higher complexity or a pure AdS with lower complexity. In the joint system of a black hole coupled to a nongraviting bath, where (a part of) the radiation is measured, the BCFT features two boundaries: one for the black hole and the other for the measurement. We construct the bulk dual involving intersecting or non-intersecting branes, and investigate the complexity transition induced by the projective measurement. Notably, for a subsystem that contains the black hole brane, its RT surface may undergo a transition, giving rise to a complexity jump.

Fermionic Hamiltonians without trivial low-energy states

Fermionic Hamiltonians without trivial low-energy states

Introduction of Local Resonators to a Nonlinear Metamaterial With Topological Features

Abstract Recent work in nonlinear topological metamaterials has revealed many useful properties such as amplitude dependent localized vibration modes and nonreciprocal wave propagation. However, thus far, there have not been any studies to include the use of local resonators in these systems. This work seeks to fill that gap through investigating a nonlinear quasi-periodic metamaterial with periodic local resonator attachments. We model a one-dimensional metamaterial lattice as a spring-mass chain with coupled local resonators. Quasi-periodic modulation in the nonlinear connecting springs is utilized to achieve topological features. For comparison, a similar system without local resonators is also modeled. Both analytical and numerical methods are used to study this system. The dispersion relation of the infinite chain of the proposed system is determined analytically through the perturbation method of multiple scales. This analytical solution is compared to the finite chain response, estimated using the method of harmonic balance and solved numerically. The resulting band structures and mode shapes are used to study the effects of quasi-periodic parameters and excitation amplitude on the system behavior both with and without the presence of local resonators. Specifically, the impact of local resonators on topological features such as edge modes is established, demonstrating the appearance of a trivial bandgap and multiple localized edge states for both main cells and local resonators. These results highlight the interplay between local resonance and nonlinearity in a topological metamaterial demonstrating for the first time the presence of an amplitude invariant bandgap alongside amplitude dependent topological bandgaps.