Showing papers on "Symmetry (geometry) published in 2020"

On gauging finite subgroups

Abstract: We study in general spacetime dimension the symmetry of the theory obtained by gauging a non-anomalous finite normal Abelian subgroup $A$ of a $\Gamma$-symmetric theory. Depending on how anomalous $\Gamma$ is, we find that the symmetry of the gauged theory can be i) a direct product of $G=\Gamma/A$ and a higher-form symmetry $\hat A$ with a mixed anomaly, where $\hat A$ is the Pontryagin dual of $A$; ii) an extension of the ordinary symmetry group $G$ by the higher-form symmetry $\hat A$; iii) or even more esoteric types of symmetries which are no longer groups. We also discuss the relations to the effect called the $H^3(G,\hat A)$ symmetry localization obstruction in the condensed-matter theory and to some of the constructions in the works of Kapustin-Thorngren and Wang-Wen-Witten.

Higher-form symmetries in 5d

Abstract: We study higher-form symmetries in 5d quantum field theories, whose charged operators include extended operators such as Wilson line and ’t Hooft operators. We outline criteria for the existence of higher-form symmetries both from a field theory point of view as well as from the geometric realization in M-theory on non-compact Calabi-Yau threefolds. A geometric criterion for determining the higher-form symmetry from the intersection data of the Calabi-Yau is provided, and we test it in a multitude of examples, including toric geometries. We further check that the higher-form symmetry is consistent with dualities and is invariant under flop transitions, which relate theories with the same UV-fixed point. We explore extensions to higher-form symmetries in other compactifications of M-theory, such as G2-holonomy manifolds, which give rise to 4d $$ \mathcal{N} $$ = 1 theories.

Algebraic higher symmetry and categorical symmetry: A holographic and entanglement view of symmetry

Abstract: We introduce the notion of algebraic higher symmetry, which generalizes higher symmetry and is beyond higher group. We show that an algebraic higher symmetry in a bosonic system in $n$-dimensional space is characterized and classified by a local fusion $n$-category. We find another way to describe algebraic higher symmetry by restricting to symmetric sub Hilbert space where symmetry transformations all become trivial. In this case, algebraic higher symmetry can be fully characterized by a non-invertible gravitational anomaly (i.e. an topological order in one higher dimension). Thus we also refer to non-invertible gravitational anomaly as categorical symmetry to stress its connection to symmetry. This provides a holographic and entanglement view of symmetries. For a system with a categorical symmetry, its gapped state must spontaneously break part (not all) of the symmetry, and the state with the full symmetry must be gapless. Using such a holographic point of view, we obtain (1) the gauging of the algebraic higher symmetry; (2) the classification of anomalies for an algebraic higher symmetry; (3) the equivalence between classes of systems, with different (potentially anomalous) algebraic higher symmetries or different sets of low energy excitations, as long as they have the same categorical symmetry; (4) the classification of gapped liquid phases for bosonic/fermionic systems with a categorical symmetry, as gapped boundaries of a topological order in one higher dimension (that corresponds to the categorical symmetry). This classification includes symmetry protected trivial (SPT) orders and symmetry enriched topological (SET) orders with an algebraic higher symmetry.

Higher-Form Symmetries in 5d

Abstract: We study higher-form symmetries in 5d quantum field theories, whose charged operators include extended operators such as Wilson line and 't Hooft operators. We outline criteria for the existence of higher-form symmetries both from a field theory point of view as well as from the geometric realization in M-theory on non-compact Calabi-Yau threefolds. A geometric criterion for determining the higher-form symmetry from the intersection data of the Calabi-Yau is provided, and we test it in a multitude of examples, including toric geometries. We further check that the higher-form symmetry is consistent with dualities and is invariant under flop transitions, which relate theories with the same UV-fixed point. We explore extensions to higher-form symmetries in other compactifications of M-theory, such as $G_2$-holonomy manifolds, which give rise to 4d $\mathcal{N}=1$ theories.

Symmetry-based indicators for topological Bogoliubov-de Gennes Hamiltonians

Abstract: Determining the topological phases in crystalline systems requires a complete set of topological invariants, whose definition and evaluation is, in general, a complicated task. Symmetry-based indicators, such as the Fu-Kane formula for inversion-symmetric topological insulators, utilize the crystalline symmetry to define easy-to-compute topological invariants. These indicators can be generalized to extract the maximal information about the topology of the band structure and of the associated anomalous boundary states from data at the high-symmetry momenta only. Using the concept of relative topology, the authors show how to construct symmetry-based indicators for band superconductors.

Classification and construction of higher-order symmetry-protected topological phases of interacting bosons

Abstract: Motivated by the recent discovery of higher-order topological insulators, we study their counterparts in strongly interacting bosons: ``higher-order symmetry-protected topological (HOSPT) phases'' While the usual (first-order) SPT phases in $d$ spatial dimensions support anomalous $(d\ensuremath{-}1)$-dimensional surface states, HOSPT phases in $d$ dimensions are characterized by topological boundary states of dimension $(d\ensuremath{-}2)$ or smaller, protected by certain global symmetries and robust against disorders Based on a dimensional reduction analysis, we show that HOSPT phases can be built from lower-dimensional SPT phases in a way that preserves the associated crystalline symmetries When the total symmetry is a direct product of global and crystalline symmetry groups, we are able to classify the HOSPT phases using the K\"unneth formula of group cohomology Based on a decorated domain-wall picture of the K\"unneth formula, we show how to systematically construct the HOSPT phases, and demonstrate our construction with many examples in two and three dimensions

Reflection and Time Reversal Symmetry Enriched Topological Phases of Matter: Path Integrals, Non-orientable Manifolds, and Anomalies

Abstract: We study symmetry-enriched topological (SET) phases in 2+1 space-time dimensions with spatial reflection and/or time-reversal symmetries. We provide a systematic construction of a wide class of reflection and time-reversal SET phases in terms of a topological path integral defined on general space-time manifolds. An important distinguishing feature of different topological phases with reflection and/or time-reversal symmetry is the value of the path integral on non-orientable space-time manifolds. We derive a simple general formula for the path integral on the manifold $$\Sigma ^2 \times S^1$$, where $$\Sigma ^2$$ is a two-dimensional non-orientable surface and $$S^1$$ is a circle. This also gives an expression for the ground state degeneracy of the SET on the surface $$\Sigma ^2$$ that depends on the reflection symmetry fractionalization class, generalizing the Verlinde formula for ground state degeneracy on orientable surfaces. Consistency of the action of the mapping class group on non-orientable manifolds leads us to a constraint that can detect when a time-reversal or reflection SET phase is anomalous in (2+1)D and, thus, can only exist at the surface of a (3+1)D symmetry protected topological (SPT) state. Given a (2+1)D reflection and/or time-reversal SET phase, we further derive a general formula that determines which (3+1)D reflection and/or time-reversal SPT phase hosts the (2+1)D SET phase as its surface termination. A number of explicit examples are studied in detail.

Discovering Symmetry Invariants and Conserved Quantities by Interpreting Siamese Neural Networks

Abstract: The authors train a Siamese neural network to decide whether two different descriptions describe the same physical object. The neural network learns to identify the objects by calculating the underlying invariants and conserved quantities.

Probing symmetries of quantum many-body systems through gap ratio statistics

Abstract: The statistics of gap ratios between consecutive energy levels is a widely used tool, in particular in the context of many-body physics, to distinguish between chaotic and integrable systems, described respectively by Gaussian ensembles of random matrices and Poisson statistics. In this work we extend the study of the gap ratio distribution $P(r)$ to the case where discrete symmetries are present. This is important, since in certain situations it may be very impractical, or impossible, to split the model into symmetry sectors, let alone in cases where the symmetry is not known in the first place. Starting from the known expressions for surmises in the Gaussian ensembles, we derive analytical surmises for random matrices comprised of several independent blocks. We check our formulae against simulations from large random matrices, showing excellent agreement. We then present a large set of applications in many-body physics, ranging from quantum clock models and anyonic chains to periodically-driven spin systems. In all these models the existence of a (sometimes hidden) symmetry can be diagnosed through the study of the spectral gap ratios, and our approach furnishes an efficient way to characterize the number and size of independent symmetry subspaces. We finally discuss the relevance of our analysis for existing results in the literature, as well as point out possible future applications and extensions.

Symmetry and Group in Attribute-Object Compositions

Abstract: Attributes and objects can compose diverse compositions. To model the compositional nature of these general concepts, it is a good choice to learn them through transformations, such as coupling and decoupling. However, complex transformations need to satisfy specific principles to guarantee the rationality. In this paper, we first propose a previously ignored principle of attribute-object transformation: Symmetry. For example, coupling peeled-apple with attribute peeled should result in peeled-apple, and decoupling peeled from apple should still output apple. Incorporating the symmetry principle, a transformation framework inspired by group theory is built, i.e. SymNet. SymNet consists of two modules, Coupling Network and Decoupling Network. With the group axioms and symmetry property as objectives, we adopt Deep Neural Networks to implement SymNet and train it in an end-to-end paradigm. Moreover, we propose a Relative Moving Distance (RMD) based recognition method to utilize the attribute change instead of the attribute pattern itself to classify attributes. Our symmetry learning can be utilized for the Compositional Zero-Shot Learning task and outperforms the state-of-the-art on widely-used benchmarks. Code is available at this https URL.

Permutationless Many-Jet Event Reconstruction with Symmetry Preserving Attention Networks.

Abstract: Top quarks, produced in large numbers at the Large Hadron Collider, have a complex detector signature and require special reconstruction techniques. The most common decay mode, the "all-jet" channel, results in a 6-jet final state which is particularly difficult to reconstruct in $pp$ collisions due to the large number of permutations possible. We present a novel approach to this class of problem, based on neural networks using a generalized attention mechanism, that we call Symmetry Preserving Attention Networks (SPA-Net). We train one such network to identify the decay products of each top quark unambiguously and without combinatorial explosion as an example of the power of this technique.This approach significantly outperforms existing state-of-the-art methods, correctly assigning all jets in $93.0%$ of $6$-jet, $87.8%$ of $7$-jet, and $82.6%$ of $\geq 8$-jet events respectively.

$\mathbb{Z}_2$-enriched symmetry indicators for topological superconductors in the 1651 magnetic space groups

Abstract: While the symmetry-based diagnosis of topological insulators and semimetals has enabled large-scale discovery of topological materials candidates, the extension of these approaches to the diagnosis of topological superconductors remains a major open question. One important new ingredient in the analysis of topological superconductivity is the presence of $\mathbb Z_2$-valued Pfaffian invariants associated with certain high-symmetry momenta. Such topological invariants lie beyond the conventional scope of symmetry representation theory for band structures, and as such they are nontrivial to incorporate into the systematic calculations of the symmetry indicators of band topology. Here, we overcome this challenge and report the full computation of the $\mathbb Z_2$-enriched symmetry indicators for superconductors in all symmetry settings. Our results indicate that incorporating the $\mathbb Z_2$ band labels enhance the diagnostic power of the scheme in roughly $60\%$ of the symmetry settings. Our framework can also be readily integrated with first-principles calculations to elucidate on the possible properties of unconventional superconductivity in a given compound. As a demonstration, we analyze explicitly the interplay between pairing symmetry and topological superconductivity in the recently discovered superconductors CaPtAs and CaSb$_2$.

From tunnels to towers: quantum scars from Lie Algebras and q-deformed Lie Algebras

Abstract: We present a general symmetry-based framework for obtaining many-body Hamiltonians with scarred eigenstates that do not obey the eigenstate thermalization hypothesis. Our models are derived from parent Hamiltonians with a non-Abelian (or q-deformed) symmetry, whose eigenspectra are organized as degenerate multiplets that transform as irreducible representations of the symmetry (`tunnels'). We show that large classes of perturbations break the symmetry, but in a manner that preserves a particular low-entanglement multiplet of states -- thereby giving generic, thermal spectra with a `shadow' of the broken symmetry in the form of scars. The generators of the Lie algebra furnish operators with `spectrum generating algebras' that can be used to lift the degeneracy of the scar states and promote them to equally spaced `towers'. Our framework applies to several known models with scars, but we also introduce new models with scars that transform as irreducible representations of symmetries such as SU(3) and $q$-deformed SU(2), significantly generalizing the types of systems known to harbor this phenomenon. Additionally, we present new examples of generalized AKLT models with scar states that do not transform in an irreducible representation of the relevant symmetry. These are derived from parent Hamiltonians with enhanced symmetries, and bring AKLT-like models into our framework.

Single-stage gradient-based stellarator coil design: Optimization for near-axis quasi-symmetry.

Abstract: We present a new coil design paradigm for magnetic confinement in stellarators. Our approach directly optimizes coil shapes and coil currents to produce a vacuum quasi-symmetric magnetic field with a target rotational transform on the magnetic axis. This approach differs from the traditional two-stage approach in which first a magnetic configuration with desirable physics properties is found, and then coils to approximately realize this magnetic configuration are designed. The proposed single-stage approach allows us to find a compromise between confinement and engineering requirements, i.e., find easy-to-build coils with good confinement properties. Using forward and adjoint sensitivities, we derive derivatives of the physical quantities in the objective, which is constrained by a nonlinear periodic differential equation. In two numerical examples, we compare different gradient-based descent algorithms and find that incorporating approximate second-order derivative information through a quasi-Newton method is crucial for convergence. We also explore the optimization landscape in the neighborhood of a minimizer and find many directions in which the objective is mostly flat, indicating ample freedom to find simple and thus easy-to-build coils.

Z 2 -projective translational symmetry protected topological phases

Abstract: Symmetry is fundamental to topological phases. In the presence of a gauge field, spatial symmetries will be projectively represented, which may alter their algebraic structure and generate novel physics. We show that the ${\mathbb{Z}}_{2}$ projectively represented translational symmetry operators adopt a distinct anticommutation relation. As a result, each energy band is twofold degenerate, and carries a varying spinor structure for translation operators in momentum space, which cannot be flattened globally. Moreover, combined with other internal or external symmetries, they give rise to exotic band topologies. Particularly, with the inherent time-reversal symmetry, a single fourfold Dirac point must be enforced at the Brillouin zone corner. By breaking one primitive translation, the Dirac semimetal is shifted into a special topological insulator phase, where the edge bands have a M\"obius twist. Our work opens an arena of research for exploring topological phases protected by projectively represented space groups.

Linear Transformations on Affine-Connections

Abstract: We state and prove a simple Theorem that allows one to generate invariant quantities in Metric-Affine Geometry, under a given transformation of the affine connection. We start by a general functional of the metric and the connection and consider transformations of the affine connection possessing a certain symmetry. We show that the initial functional is invariant under the aforementioned group of transformations iff its $\Gamma$-variation produces tensor of a given symmetry. Conversely if the tensor produced by the $\Gamma$-variation of the functional respects a certain symmetry then the functional is invariant under the associated transformation of the affine connection. We then apply our results in Metric-Affine Gravity and produce invariant actions under certain transformations of the affine connection. Finally, we derive the constraints put on the hypermomentum for such invariant Theories.

General Lieb-Schultz-Mattis type theorems for quantum spin chains

Abstract: We develop a general operator algebraic method which focuses on projective representations of symmetry group for proving Lieb-Schultz-Mattis type theorems, i.e., no-go theorems that rule out the existence of a unique gapped ground state (or, more generally, a pure split state), for quantum spin chains with on-site symmetry. We first prove a theorem for translation invariant spin chains that unifies and extends two theorems proved by two of the authors in [OT1]. We then prove a Lieb-Schultz-Mattis type theorem for spin chains that are invariant under the reflection about the origin and not necessarily translation invariant.

Detecting Symmetries with Neural Networks

Abstract: Identifying symmetries in data sets is generally difficult, but knowledge about them is crucial for efficient data handling. Here we present a method how neural networks can be used to identify symmetries. We make extensive use of the structure in the embedding layer of the neural network which allows us to identify whether a symmetry is present and to identify orbits of the symmetry in the input. To determine which continuous or discrete symmetry group is present we analyse the invariant orbits in the input. We present examples based on rotation groups $SO(n)$ and the unitary group $SU(2).$ Further we find that this method is useful for the classification of complete intersection Calabi-Yau manifolds where it is crucial to identify discrete symmetries on the input space. For this example we present a novel data representation in terms of graphs.

Recipes for edge-transitive tetravalent graphs

Abstract: This paper presents all constructions known to the authors which result in tetravalent graphs whose symmetry groups are large enough to be transitive on the edges of the graph.

Is preservation of symmetry necessary for coarse-graining?

Abstract: There is a need for theory on how to group atoms in a molecule to define a coarse-grained (CG) mapping This article investigates the importance of preserving symmetry of the underlying molecular graph of a given molecule when choosing a CG mapping 26 CG models of seven alkanes with three different CG techniques were examined We unexpectedly find preserving symmetry has no consistent effect on CG model accuracy regardless of CG method or comparison metric

Euclidean Symmetry and Equivariance in Machine Learning

Abstract: Understanding symmetry’s role in the physical sciences is critical for choosing an appropriate machine learning method. While invariant models are the most prevalent symmetry-aware models, equivariant models can more faithfully represent physical interactions. Until recently, equivariant models had been absent in the literature due to their technical complexity. Now, after two years of active development, fully-equivariant Euclidean neural net- works are ready to take on challenges across the physical sciences.

Subsystem symmetry enriched topological order in three dimensions

Abstract: The authors introduce a type of three-dimensional topological orderenriched by planar subsystem symmetries, and show that it ischaracterized by the fractionalization of the symmetries on loop-likeexcitations, an increased value of the topological entanglemententropy, and the emergence of non-abelian fracton excitations upongauging the symmetry

Flavor Symmetry of 5d SCFTs, Part 1: General Setup

Abstract: A large class of 5d superconformal field theories (SCFTs) can be constructed by integrating out BPS particles from 6d SCFTs compactified on a circle. We describe a general method for extracting the flavor symmetry of any 5d SCFT lying in this class. For this purpose, we utilize the geometric engineering of 5d N=1 theories in M-theory, where the flavor symmetry is encoded in a collection of non-compact surfaces.

Symmetry crossover in layered M PS 3 complexes ( M = Mn , Fe , Ni ) via near-field infrared spectroscopy

Abstract: Author(s): Neal, SN; Kim, HS; O'Neal, KR; Haglund, AV; Smith, KA; Mandrus, DG; Bechtel, HA; Carr, GL; Haule, K; Vanderbilt, D; Musfeldt, JL | Abstract: We employ synchrotron-based near-field infrared spectroscopy to reveal the vibrational properties of bulk, few-sheet, and single-sheet members of the MPS3 (M=Mn, Fe, Ni) family of materials and compare our findings with complementary lattice dynamics calculations. MnPS3 and the Fe analog are similar in terms of their symmetry crossovers, from C2/m to P3¯1m, as the monolayer is approached. These states differ as to the presence of a C3 rotation around the metal center. On the other hand, NiPS3 does not show a symmetry crossover, and the lack of a Bu symmetry mode near 450 cm-1 suggests that C3 rotational symmetry is already present, even in the bulk material. We discuss these findings in terms of local symmetry and temperature effects as well as the curious relationship between these symmetry transformations and those that take place under pressure.

Periodic Structures With Higher Symmetries: Their Applications in Electromagnetic Devices

Abstract: Higher symmetries frequently amaze human beings because of the illusions and incredible landscapes such symmetries can produce. For example, imagine the unearthly pictures of the Dutch graphic artist M.C. Escher. He made use of glide symmetry and reflection to produce unbelievable transitions and transformations of objects and beings, as illustrated in Figure 1(a). However, the history of higher symmetries started much earlier. Escher was partially inspired by the Moorish tessellations in the Alhambra in Granada, Spain, such as the ones pictured in Figure 1(b).

The bulk-edge correspondence in three simple cases

Abstract: We present examples in three symmetry classes of topological insulators in one or two dimensions where the proof of the bulk-edge correspondence is particularly simple. This serves to illustrate th...

Topological edge modes without symmetry in quasiperiodically driven spin chains

Abstract: We construct an example of a 1$d$ quasiperiodically driven spin chain whose edge states can coherently store quantum information, protected by a combination of localization, dynamics, and topology. Unlike analogous behavior in static and periodically driven (Floquet) spin chains, this model does not rely upon microscopic symmetry protection: Instead, the edge states are protected purely by emergent dynamical symmetries. We explore the dynamical signatures of this Emergent Dynamical Symmetry-Protected Topological (EDSPT) order through exact numerics, time evolving block decimation, and analytic high-frequency expansion, finding evidence that the EDSPT is a stable dynamical phase protected by bulk many-body localization up to (at least) stretched-exponentially long time scales, and possibly beyond. We argue that EDSPTs are special to the quasiperiodically driven setting, and cannot arise in Floquet systems. Moreover, we find evidence of a new type of boundary criticality, in which the edge spin dynamics transition from quasiperiodic to chaotic, leading to bulk thermalization.

Finding Symmetry Breaking Order Parameters with Euclidean Neural Networks

Abstract: Curie's principle states that "when effects show certain asymmetry, this asymmetry must be found in the causes that gave rise to them" We demonstrate that symmetry equivariant neural networks uphold Curie's principle and can be used to articulate many symmetry-relevant scientific questions into simple optimization problems We prove these properties mathematically and demonstrate them numerically by training a Euclidean symmetry equivariant neural network to learn symmetry-breaking input to deform a square into a rectangle and to generate octahedra tilting patterns in perovskites

Orbifolds from modular orbits

Abstract: Given a two-dimensional conformal field theory with a global symmetry, we propose a method to implement an orbifold construction by taking orbits of the modular group. For the case of cyclic symmetries we find that this approach always seems to be consistent, even in asymmetric orbifold cases where the usual construction does not yield a modular invariant theory; our approach keeps modular invariance manifest but may give a result that is equivalent to the original theory. For the case that the symmetry is a subgroup of a continuous flavor symmetry, we can give explicit constructions of the spectrum, with twisted sectors corresponding to a nonstandard group projection on an enlarged twisted sector Hilbert space.