Topological Term Research Articles

Unified theory of the anomalous and topological Hall effects with phase-space Berry curvatures.

Spontaneously broken time-reversal symmetry in magnetic materials leads to a Hall response, with a nonzero voltage transverse to an applied current, even in the absence of external magnetic fields. It is common to analyze the Hall resistivity of chiral magnets as the sum of two terms: an anomalous Hall effect arising from spin-orbit coupling and a topological Hall signal coming from skyrmions, which are topologically nontrivial spin textures. The theoretical justification for such a decomposition has long remained an open problem. Using a controlled semiclassical approach that includes all phase-space Berry curvatures, we show that the solution of the Boltzmann equation leads to a Hall resistivity that is just the sum of an anomalous term arising from momentum-space curvature and a topological term related to the real-space curvature. We also present numerically exact results from a Kubo formalism that complement the semiclassical approach.

Tuning the Topological θ -Angle in Cold-Atom Quantum Simulators of Gauge Theories

The topological θ-angle in gauge theories engenders a series of fundamental phenomena, including violations of charge-parity (CP) symmetry, dynamical topological transitions, and confinement-deconfinement transitions. At the same time, it poses major challenges for theoretical studies, as it implies a sign problem in numerical simulations. Analog quantum simulators open the promising prospect of treating quantum many-body systems with such topological terms, but, contrary to their digital counterparts, they have not yet demonstrated the capacity to control the θ-angle. Here, we demonstrate how a tunable topological θ-term can be added to a prototype theory with U(1) gauge symmetry, a discretized version of quantum electrodynamics in one spatial dimension. As we show, the model can be realized experimentally in a single-species Bose-Hubbard model in an optical superlattice with three different spatial periods, thus requiring only standard experimental resources. Through numerical calculations obtained from the time-dependent density-matrix renormalization group method and exact diagonalization, we benchmark the model system, and illustrate how salient effects due to the θ-term can be observed. These include charge confinement, the weakening of quantum many-body scarring, as well as the disappearance of Coleman’s phase transition due to explicit breaking of CP symmetry. This work opens the door towards studying the rich physics of topological gauge-theory terms in large-scale cold-atom quantum simulators.4 MoreReceived 15 April 2022Revised 19 September 2022Accepted 26 September 2022DOI:https://doi.org/10.1103/PRXQuantum.3.040316Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasColor confinementColor deconfinementConfinementQuantum simulationPhysical Systems1-dimensional systemsBose gasesTechniquesBose-Hubbard modelDensity matrix renormalization groupExact diagonalizationGauge symmetriesLattice gauge theoryPerturbation theoryAtomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Symplectic cohomology and a conjecture of Viterbo

We identify a new class of closed smooth manifolds for which there exists a uniform bound on the Lagrangian spectral norm of Hamiltonian deformations of the zero section in a unit cotangent disk bundle. This settles a well-known conjecture of Viterbo from 2007 as the special case of \(T^n,\) which has been completely open for \(n>1\). Our methods are different and more intrinsic than those of the previous work of the author first settling the case \(n=1\). The new class of manifolds is defined in topological terms involving the Chas–Sullivan algebra and the BV-operator on the homology of the free loop space. It contains spheres and is closed under products. We discuss generalizations and various applications, to \(C^0\) symplectic topology in particular.

Field theory of higher-order topological crystalline response, generalized global symmetries and elasticity tetrads

I discuss aspects of higher-order topological field theory of crystalline insulators with no other symmetries. I show how the topology and geometry of the crystalline lattice is organized in terms of so-called elasticity tetrads which are ground state degrees of freedom labelling translational lattice topological charges, higher-form conservation laws and responses on sub-dimensional manifolds of the bulk insulator. The quasitopological responses obtained in this way depend on the lattice and its embedding in space, as expected for weak topology. In a topological crystalline insulator, they classify higher-order responses and global symmetries in a transparent fashion in generic dimensions. This hierarchy coincides with the dimensional hierarchy of topological terms, the multipole expansion, and anomaly inflow, related to a mixed number of elasticity tetrads and electromagnetic gauge fields. In the continuum limit of the elasticity tetrads, the semi-classical expansion in momentum space can be used to derive the higher-order or subdimensional topological responses to local U(1) symmetries, such as electromagnetic gauge fields, with explicit formulas for the higher-order quasi-topological invariants in terms of the elasticity tetrads and Green’s functions. The topological responses in arbitrary dimensions are readily generalized to parameter space to allow for e.g. multipole pumping. The simple results further bridge the recently appreciated connections between topological field theory, higher-form symmetries and gauge fields and their relation to fractonic excitations and topological defects with restricted mobility in the elasticity of crystalline insulators.

The gauging procedure and carrollian gravity

We discuss a gauging procedure that allows us to construct lagrangians that dictate the dynamics of an underlying Cartan geometry. In a sense to be made precise in the paper, the starting datum in the gauging procedure is a Klein pair corresponding to a homogeneous space. What the gauging procedure amounts to is the construction of a Cartan geometry modelled on that Klein geometry, with the gauge field defining a Cartan connection. The lagrangian itself consists of all gauge-invariant top-forms constructed from the Cartan connection and its curvature. After demonstrating that this procedure produces four-dimensional General Relativity upon gauging Minkowski spacetime, we proceed to gauge all four-dimensional maximally symmetric carrollian spaces: Carroll, (anti-)de Sitter-Carroll and the lightcone. For the first three of these spaces, our lagrangians generalise earlier first-order lagrangians. The resulting theories of carrollian gravity all take the same form, which seems to be a manifestation of model mutation at the level of the lagrangians. The odd one out, the lightcone, is not reductive and this means that although the equations of motion take the same form as in the other cases, the geometric interpretation is different. For all carrollian theories of gravity we obtain analogues of the Gauss-Bonnet, Pontryagin and Nieh-Yan topological terms, as well as two additional terms that are intrinsically carrollian and seem to have no lorentzian counterpart. Since we gauge the theories from scratch this work also provides a no-go result for the electric carrollian theory in a first-order formulation.

Quantum nucleation of topological solitons

The chiral soliton lattice is an array of topological solitons realized as ground states of QCD at finite density under strong magnetic fields or rapid rotation, and chiral magnets with an easy-plane anisotropy. In such cases, topological solitons have negative energy due to topological terms originating from the chiral magnetic or vortical effect and the Dzyaloshinskii-Moriya interaction, respectively. We study quantum nucleation of topological solitons in the vacuum through quantum tunneling in 2 + 1 and 3 + 1 dimensions, by using a complex ϕ4 (or the axion) model with a topological term proportional to an external field, which is a simplification of low-energy theories of the above systems. In 2 + 1 dimensions, a pair of a vortex and an anti-vortex is connected by a linear soliton, while in 3 + 1 dimensions, a vortex is string-like, a soliton is wall-like, and a disk of a soliton wall is bounded by a string loop. Since the tension of solitons can be effectively negative due to the topological term, such a composite configuration of a finite size is created by quantum tunneling and subsequently grows rapidly. We estimate the nucleation probability analytically in the thin-defect approximation and fully calculate it numerically using the relaxation (gradient flow) method. The nucleation probability is maximized when the direction of the soliton is perpendicular to the external field. We find a good agreement between the thin-defect approximation and direct numerical simulation in 2 + 1 dimensions if we read the vortex tension from the numerics, while we find a difference between them at short distances interpreted as a remnant energy in 3 + 1 dimensions.

Boundary effects on orbital magnetization for a bilayer system with different Chern numbers

The real space formalism of orbital magnetization (OM) is an average of the local OM over some appropriate region of the system. Previous studies prefer a bulk average (i.e., without including boundaries). Based on a bilayer model with an adjustable Chern number at half filling, we numerically investigate the effects of boundaries on the real space expressions of OM. The size convergence processes of its three constituent terms ${M}_{\mathrm{LC}}$, ${M}_{\mathrm{IC}}$, and ${M}_{\mathrm{BC}}$ are analyzed. The topological term ${M}_{\mathrm{BC}}$ makes a nonnegligible contribution from boundaries as a manifestation of edge states, especially in the case of nonzero Chern numbers. However, we show that the influence of the boundary on ${M}_{\mathrm{LC}}$ and ${M}_{\mathrm{IC}}$ exactly compensates that on ${M}_{\mathrm{BC}}$. This compensation effect leads to the conclusion that the whole sample average is also a correct algorithm in the thermodynamic limit, which gives the same value as those from the bulk average and the $k$-space formula. This clarification will be beneficial to further studies on orbitronics, as well as the orbital magnetoelectric effects in higher dimensions.

A Train Protection Logic Based on Topological Manifolds for Virtual Coupling

Virtual coupling is a promising innovation aimed at increasing railway capacity. Compared to current railway signaling systems, it allows two or more trains to run with reduced headway between them. However, such reduced headways are a challenge to safety. In this work we consider this challenge by formally describing and verifying an approach to virtual coupling. We propose a general modeling method based on topological manifolds to describe the protection logic for virtual coupling train control systems. We also describe the basic train control elements in topological terms and analyze the line condition of our virtual coupling logic. We establish that the line condition safety requirements and its representation as a manifold are equivalent and further provide a formal definition of the concept of a movement authority with manifold notations. This allows us to consider the dynamic behavior of trains and a series of theorems that establish the correctness of our protection logic for virtual coupling. Finally, we apply the presented methods to a case study. The results show that the proposed method provides a suitable way to realize a virtual coupling logic safely.

Non-invertible self-duality defects of Cardy-Rabinovici model and mixed gravitational anomaly

We study properties of self-duality symmetry in the Cardy-Rabinovici model. The Cardy-Rabinovici model is the 4d U(1) gauge theory with electric and magnetic matters, and it enjoys the SL(2, ℤ) self-duality at low-energies. SL(2, ℤ) self-duality does not realize in a naive way, but we notice that the STp duality transformation becomes the legitimate duality operation by performing the gauging of ℤN 1-form symmetry with including the level-p discrete topological term. Due to such complications in its realization, the fusion rule of duality defects becomes a non-group-like structure, and thus the self-duality symmetry is realized as a non-invertible symmetry. Moreover, for some fixed points of the self-duality, the duality symmetry turns out to have a mixed gravitational anomaly detected on a K3 surface, and we can rule out the trivially gapped phase as a consequence of anomaly matching. We also uncover how the conjectured phase diagram of the Cardy-Rabinovici model satisfies this new anomaly matching condition.

Topological Co3Sn2S2 magnetic Weyl semimetal: From fundamental understanding to diverse fields of study

A wider scope in topological semimetals accelerates various attractive quantum phenomena for the last two decades. We here report the detailed investigation of magnetic Co 3 Sn 2 S 2 Weyl semimetal based on the cobalt accompanied kagome lattice for the first time. We emphasize the highly motivated theoretical and experimental analyses in each topological terms of Co 3 Sn 2 S 2 that led to intensifying emergent topological materials with inherent properties, particularly in the pursuance of magnetic rich quantum anomalous Hall effect. We additionally highlight the Co 3 Sn 2 S 2 surface band structure, recognized in the diverse fields of study from the perception of topological properties to modern electrochemistry approaches. From materials perspective, this typical kagome Co 3 Sn 2 S 2 Weyl semimetal alone has extensively covered more enriching nontrivial surface states than topological insulators and nonmagnetic semimetals even in the absence of magnetic field and external strain. Some specific examples include Weyl nodes linked flat band magnetism, a large anomalous Hall effect, anomalous Hall angle, giant magnetooptical effect, and thermomagnetic Nernst effect. In addition, we selectively extend the review on the undisputed surface band characteristics of Co 3 Sn 2 S 2 , by which a major concept of magnetic Weyl semimetal could hierarchically meet the contemporary quantum phenomena. This strongly led to more focus on nontrivial magnetic topological studies for gaining insightful perspectives.

Near horizon gravitational charges

In this paper, we study the near horizon symmetry and gravitational charges in the Newman-Penrose formalism. In particular we investigate the effect from topological terms. We find that the Pontryagin term and Gauss-Bonnet term have significant influence on the near horizon charges and bring interesting novel features. We show that the gravitational charge derived from a general class of topological terms including the Pontryagin term and Gauss-Bonnet term can be obtained from the ambiguities of the symplectic potential.

Implications of the topological Chern-Simons mass in the gap equation

In this paper, we solve the gap equation of the Yang-Mills-Gribov-Zwanziger-Chern-Simon theory by considering the first order in the Chern-Simon topological mass term, $M$. As a result, we find three possible solutions to the gap equation, i.e. three different Gribov parameters. In addition, we analyze the regime of the theory for each of these Gribov parameters and we obtain a different result from the literature.

Classically emulated digital quantum simulation of the Schwinger model with a topological term via adiabatic state preparation

We designed a protocol for digital quantum computation of a gauge theory with a topological term in Minkowski spacetime, which is practically inaccessible by standard lattice Monte Carlo simulations. We focus on $1+1$ dimensional quantum electrodynamics with the $\ensuremath{\theta}$ term known as the Schwinger model and test our protocol for this on an IBM simulator. We construct the true vacuum state of a lattice Schwinger model using adiabatic state preparation which, in turn, allows us to compute an expectation value of the fermion mass operator with respect to the vacuum. Upon taking a continuum limit we find that our result in the massless case agrees with the known exact result. In the massive case, we find an agreement with mass perturbation theory in the small-mass regime and deviations in the large-mass regime. We estimate computational costs required to take a reasonable continuum limit. Our results imply that digital quantum simulation appears a promising tool to explore nonperturbative aspects of gauge theories with real time and topological terms.

Representation homology of simply connected spaces

Let G $G$ be an affine algebraic group defined over a field k $k$ of characteristic 0. We study the derived moduli space of G $G$ -local systems on a pointed connected CW complex X $X$ trivialized at the basepoint of X $X$ . This derived moduli space is represented by an affine DG scheme R Loc G ( X , * ) $ \bm{\mathrm{R}}\mathrm{Loc}_G(X,\ast )$ : we call the (co)homology of the structure sheaf of R Loc G ( X , * ) $ \bm{\mathrm{R}}\mathrm{Loc}_G(X,\ast )$ the representation homology of X $X$ in G $G$ and denote it by HR * ( X , G ) $ \mathrm{HR}_\ast (X,G)$ . The 0-dimensional homology, HR 0 ( X , G ) $ \mathrm{HR}_0(X,G)$ , is isomorphic to the coordinate ring of the G $G$ -representation variety Rep G [ π 1 ( X ) ] $ {\rm {Rep}}_G[\pi _1(X)]$ of the fundamental group of X $X$ — a well-known algebro-geometric invariant that plays a role in many areas of topology. The higher representation homology is much less studied. In particular, when X $X$ is simply connected, HR 0 ( X , G ) $ \mathrm{HR}_0(X,G)$ is trivial but HR ∗ ( X , G ) $ \mathrm{HR}_*(X,G)$ is still an interesting rational invariant of X $X$ that depends on the Lie algebra of G $G$ . In this paper, we use Quillen's rational homotopy theory to compute the representation homology of an arbitrary simply connected space (of finite rational type) in terms of its Lie and Sullivan algebraic models. When G $G$ is reductive, we also compute HR * ( X , G ) G $ \mathrm{HR}_\ast (X,G)^G$ , the G $G$ -invariant part of representation homology, and study the question when HR * ( X , G ) G $ \mathrm{HR}_\ast (X,G)^G$ is free of locally finite type as a graded commutative algebra. This question turns out to be related to the so-called Strong Macdonald Conjecture, a celebrated result in representation theory proposed (as a conjecture) by Feigin and Hanlon in the 1980s and proved by Fishel, Grojnowski and Teleman in 2008. Reformulating the Strong Macdonald Conjecture in topological terms, we give a simple characterization of spaces X $X$ for which HR * ( X , G ) G $ \mathrm{HR}_\ast (X,G)^G$ is a graded symmetric algebra for any complex reductive group G $G$ .

Gel′fand-Fuchs cohomology in algebraic geometry and factorization algebras

Let X X be a smooth affine variety over a field k \mathbf k of characteristic 0 0 and T ( X ) T(X) be the Lie algebra of regular vector fields on X X . We compute the Lie algebra cohomology of T ( X ) T(X) with coefficients in k \mathbf k . The answer is given in topological terms relative to any embedding k ⊂ C \mathbf k\subset \mathbb {C} and is analogous to the classical Gel′fand-Fuchs computation for smooth vector fields on a C ∞ C^\infty -manifold. Unlike the C ∞ C^\infty -case, our setup is purely algebraic: no topology on T ( X ) T(X) is present. The proof is based on the techniques of factorization algebras, both in algebro-geometric and topological contexts.

Topological orders, braiding statistics, and mixture of two types of twisted BF theories in five dimensions

Topological orders are a prominent paradigm for describing quantum many-body systems without symmetry-breaking orders. We present a topological quantum field theoretical (TQFT) study on topological orders in five-dimensional spacetime (5D) in which topological excitations include not only point-like particles, but also two types of spatially extended objects: closed string-like loops and two-dimensional closed membranes. Especially, membranes have been rarely explored in the literature of topological orders. By introducing higher-form gauge fields, we construct exotic TQFT actions that include mixture of two distinct types of BF topological terms and many twisted topological terms. The gauge transformations are properly defined and utilized to compute level quantization and classification of TQFTs. Among all TQFTs, some are not in Dijkgraaf-Witten cohomological classification. To characterize topological orders, we concretely construct all braiding processes among topological excitations, which leads to very exotic links formed by closed spacetime trajectories of particles, loops, and membranes. For each braiding process, we construct gauge-invariant Wilson operators and calculate the associated braiding statistical phases. As a result, we obtain expressions of link invariants all of which have manifest geometric interpretation. Following Wen’s definition, the boundary theory of a topological order exhibits gravitational anomaly. We expect that the characterization and classification of 5D topological orders in this paper encode information of 4D gravitational anomaly. Further consideration, e.g., putting TQFTs on 5D manifolds with boundaries, is left to future work.

Einstein Gauss-Bonnet theories as ordinary, Wess-Zumino conformal anomaly actions

Recently, the possibility of evading Lovelock's theorem at d=4, via a singular redefinition of the dimensionless coupling of the Gauss-Bonnet term, has been very extensively discussed in the cosmological context. The term is added as a quadratic contribution of the curvature tensor to the Einstein-Hilbert action, originating theories of “Einstein Gauss-Bonnet” (EGB) type. We point out that the action obtained by the dimensional regularization procedure, implemented with the extraction of a single conformal factor, correspond just to an ordinary Wess-Zumino anomaly action, even though it is deprived of the contribution from the Weyl tensor. We also show that a purely gravitational version of the EGB theory can be generated by allowing a finite renormalization of the Gauss-Bonnet topological contribution at d=4+ϵ, as pointed out by Mazur and Mottola. The result is an effective action which is quadratic, rather then quartic, in the dilaton field, and scale free, compared to the previous derivations. The dilaton, in this case can be removed from the spectrum, leaving a pure gravitational theory, which is nonlocal. We comment on the physical meaning of the two types of actions, which may be used to describe such topological terms both below and above the conformal breaking scale.

Tobological feedback for superconducting states

We discuss feedback effects that stabilize the superconducting states by the induced topological term in the effective Lagrangian. The chiral feedback effect due to the Chern-Simons-like term for quasi-two-dimensional system with time-reversal symmetry breaking (TRSB) was studied in [1, 2]. We consider the extension of the chiral feedback to three-dimensional TRSB system and investigate similar feedback effects for quasi-two-dimensional or three-dimensional time-reversal symmetric systems.

Induced gravitational topological term and the Einstein-Cartan modified theory

It is well known that only the axial piece of the torsion couples minimally to fermions in a Riemann-Cartan geometry, while the other ones decouple. In this paper, we consider the Dirac field minimally coupled to a dynamical background with torsion and compute its contribution to the fermionic one-loop effective action. Such a contribution owns topological nature since it can be linked with topological invariants from Riemann-Cartan spaces, like Nieh-Yan and Chern-Simons terms. Furthermore, we propose a novel modified theory of gravity constructed by adding the aforementioned one-loop contribution to the Einstein-Cartan action. The modified field equations reduce to those ones of GR under certain circumstances, providing therefore trivial solutions. However, in particular, we find a non-trivial solution where the modified field equations do not reduce to the GR ones.

Volume-preserving diffeomorphism as nonabelian higher-rank gauge symmetry

We propose nonabelian higher-rank gauge theories in 2+1D and 3+1D. The gauge group is constructed from the volume-preserving diffeomorphisms of space. We show that the intriguing physics of the lowest Landau level (LLL) limit can be interpreted as the consequences of the symmetry. We derive the renowned Girvin-MacDonald-Platzman (GMP) algebra as well as the topological Wen-Zee term within our formalism. Using the gauge symmetry in 2+1D, we derive the LLL effective action of vortex crystal in rotating Bose gas as well as Wigner crystal of electron in an applied magnetic field. We show that the nonlinear sigma models of ferromagnets in 2+1D and 3+1D exhibit the higher-rank gauge symmetries that we introduce in this paper. We interpret the fractonic behavior of the excitations on the lowest Landau level and of skyrmions in ferromagnets as the consequence of the higher-rank gauge symmetry.