Odd Dimensions Research Articles

Thouless-Kohmoto-Nightingale-den Nijs formula for a general Hamiltonian

Topological insulators in odd dimensions are characterized by topological numbers. We prove the well-known relation between the topological number given by the Chern character of the Berry curvature and the Chern-Simons level of the low energy effective action for a general class of Hamiltonians bilinear in the fermion with general U(1) gauge interactions including non-minimal couplings by an explicit calculation. A series of Ward-Takahashi identities are crucial to relate the Chern-Simons level to a winding number, which could then be directly reduced to Chern character of Berry curvature by carrying out the integral over the temporal momenta.

Variable coefficient Wolff-type inequalities and sharp local smoothing estimates for wave equations on manifolds

The sharp Wolff-type decoupling estimates of Bourgain--Demeter are extended to the variable coefficient setting. These results are applied to obtain new sharp local smoothing estimates for wave equations on compact Riemannian manifolds, away from the endpoint regularity exponent. More generally, local smoothing estimates are established for a natural class of Fourier integral operators; at this level of generality the results are sharp in odd dimensions, both in terms of the regularity exponent and the Lebesgue exponent.

A remark on black holes of Chern–Simons gravities in 2n + 1 dimensions: n = 1,2,3

A systematic prescription for constructing Chern–Simons gravities in all odd dimensions is given, and it is shown that Chern–Simons gravities in [Formula: see text] dimensions admit solutions described by the same lapse function which describes the BTZ black hole in the [Formula: see text] case. This has been carried out explicitly for [Formula: see text]. Moreover, it is seen that these solutions are unique.

Detecting exotic spheres in low dimensions using coker J

Building off of the work of Kervaire–Milnor and Hill–Hopkins–Ravenel, Wang and Xu showed that the only odd dimensions n for which S n has a unique differentiable structure are 1, 3, 5, and 61. We show that the only even dimensions below 140 for which S n has a unique differentiable structure are 2, 6, 12, 56, and perhaps 4.

Phase-space-simulation method for quantum computation with magic states on qubits

We propose a method for classical simulation of finite-dimensional quantum systems, based on sampling from a quasiprobability distribution, i.e., a generalized Wigner function. Our construction applies to all finite dimensions, with the most interesting case being that of qubits. For multiple qubits, we find that quantum computation by Clifford gates and Pauli measurements on magic states can be efficiently classically simulated if the quasiprobability distribution of the magic states is non-negative. This provides the so far missing qubit counterpart of the corresponding result [V. Veitch et al., New J. Phys. 14, 113011 (2012)] applying only to odd dimension. Our approach is more general than previous ones based on mixtures of stabilizer states. Namely, all mixtures of stabilizer states can be efficiently simulated, but for any number of qubits there also exist efficiently simulable states outside the stabilizer polytope. Further, our simulation method extends to negative quasiprobability distributions, where it provides amplitude estimation. The simulation cost is then proportional to a robustness measure squared. For all quantum states, this robustness is smaller than or equal to robustness of magic.

Distribution of Distances between Elements in a Compact Set

In this article, we propose a review of studies evaluating the distribution of distances between elements of a random set independently and uniformly distributed over a region of space in a normed R -vector space (for example, point events generated by a homogeneous Poisson process in a compact set). The distribution of distances between individuals is present in many situations when interaction depends on distance and concerns many disciplines, such as statistical physics, biology, ecology, geography, networking, etc. After reviewing the solutions proposed in the literature, we present a modern, general and unified resolution method using convolution of random vectors. We apply this method to typical compact sets: segments, rectangles, disks, spheres and hyperspheres. We show, for example, that in a hypersphere the distribution of distances has a typical shape and is polynomial for odd dimensions. We also present various applications of these results and we show, for example, that variance of distances in a hypersphere tends to zero when space dimension increases.

Mass and angular momentum of charged rotating Gödel black holes in five-dimensional minimal supergravity

In this paper, for the sake of providing a concrete comparison between the usual Abbott–Deser–Tekin (ADT) formalism and its off-shell extension, as well as comparing the latter with the Barnich–Brandt–Compere (BBC) approach, we carry out these methods to compute the mass and angular momentum of the rotating charged Gödel black holes in five-dimensional minimal supergravity. We first present the off-shell ADT potential of the supergravity theories in arbitrary odd dimensions, which is consistent with the superpotential via the BBC approach. Then the off-shell generalized ADT method is applied to evaluate the mass and angular momentum of the Gödel-type black holes by including the contribution from the gauge field. Finally, we strictly obey the rules of the original ADT formalism to incorporate the contribution from the gauge field within the potential. With the help of the modified potential, we try to seek for appropriate reference backgrounds to produce the mass and angular momentum. It is observed that the ADT formalism has to incorporate the contribution from the matter fields to yield physical charges for the Gödel-type black holes.

Propagators and Gaussian effective actions in various patches of de Sitter space

We consider time-ordered (or Feynman) propagators between two different $\alpha-$states of a linear de Sitter Quantum Field in the global de Sitter manifold and in the Poincar\'e patch. We separately examine $\alpha-\beta$, In-In and In-Out propagators and find the imaginary contribution to the effective actions. The In-In propagators are real in both the Poincar\'e patch and in the global de Sitter manifold. On the other side the In-Out propagators at coincident points contain finite imaginary contributions in both patches in even dimensions, but they are not equivalent. In odd dimensions in both patches the imaginary contributions are zero. For completeness, we also consider the Static patch and identify in our construction the state that is equivalent to the Bunch-Davies one in the Poincar\'e patch.

Some conjectures and their consequences for tensor products of modules over a finite p-group

We make some conjectures about tensor products of modular representations for a finite p-group in characteristic p, for p=2 and p=3. The easiest to state is that if M is a module of odd dimension for a 2-group over an algebraically closed field of characteristic two, then k is the only direct summand of M⊗M⁎ of odd dimension, and the remaining indecomposable summands have dimension divisible by four. As a consequence, the odd dimensional modules form an abelian group, where the composition law is to take the unique indecomposable summand of odd dimension of the tensor product, and where the inverse of an odd dimensional module is its dual. In characteristic three the conjectures are more complicated, and in larger characteristics it is not at all clear what the corresponding conjectures should be. We prove a number of theorems relating to these conjectures, and describe examples illustrating them. The examples were computed using the computer algebra package Magma.

Quantifying the magic of quantum channels

To achieve universal quantum computation via general fault-tolerant schemes, stabilizer operations must be supplemented with other non-stabilizer quantum resources. Motivated by this necessity, we develop a resource theory for magic quantum channels to characterize and quantify the quantum ‘magic’ or non-stabilizerness of noisy quantum circuits. For qudit quantum computing with odd dimension d, it is known that quantum states with non-negative Wigner function can be efficiently simulated classically. First, inspired by this observation, we introduce a resource theory based on completely positive-Wigner-preserving quantum operations as free operations, and we show that they can be efficiently simulated via a classical algorithm. Second, we introduce two efficiently computable magic measures for quantum channels, called the mana and thauma of a quantum channel. As applications, we show that these measures not only provide fundamental limits on the distillable magic of quantum channels, but they also lead to lower bounds for the task of synthesizing non-Clifford gates. Third, we propose a classical algorithm for simulating noisy quantum circuits, whose sample complexity can be quantified by the mana of a quantum channel. We further show that this algorithm can outperform another approach for simulating noisy quantum circuits, based on channel robustness. Finally, we explore the threshold of non-stabilizerness for basic quantum circuits under depolarizing noise.

Holography in de Sitter and anti-de Sitter spaces and Gel'fand Graev Radon transform

Bulk reconstruction formulas similar to HKLL are obtained for de Sitter and anti-de Sitter spaces as the inverse Gel'fand Graev Radon transform. While these generalize our previous result on the Euclidean anti-de Sitter space, their validity in here is restricted only to odd dimensions in both instances. The exact Wightman function for the de Sitter space is then derived. The GGR transform fixes the coefficient of the Wightman function. For the anti-de Sitter space it is shown that a reconstruction formula exists for the case of timelike boundary as well. The restriction on the domain of integration on the boundary is derived. As a special case, we point out that the formula is valid for the BTZ black hole as well.

Aligned SICs and embedded tight frames in even dimensions

Alignment is a geometric relation between pairs of Weyl–Heisenberg SICs, one in dimension d and another in dimension , manifesting a well-founded conjecture about a number-theoretical connection between the SICs. In this paper, we prove that if d is even, the SIC in dimension of an aligned pair can be partitioned into (d − 2)2 tight d2-frames of rank and, alternatively, into d2 tight (d − 2)2-frames of rank . The corresponding result for odd d is already known, but the proof for odd d relies on results which are not available for even d. We develop methods that allow us to overcome this issue. In addition, we provide a relatively detailed study of parity operators in the Clifford group, emphasizing differences in the theory of parity operators in even and odd dimensions and discussing consequences due to such differences. In a final section, we study implications of alignment for the symmetry of the SIC.

Elasticity tetrads, mixed axial-gravitational anomalies, and ( 3+1 )-d quantum Hall effect

For two-dimensional topological insulators, the integer and intrinsic (without external magnetic field) quantum Hall effect is described by the gauge anomalous (2+1)-dimensional [2+1d] Chern-Simons (CS) response for the background gauge potential of the electromagnetic U(1) field. The Hall conductance is given by the quantized prefactor of the CS term, which is a momentum-space topological invariant. Here, we show that three-dimensional crystalline topological insulators with no other symmetries are described by a topological (3+1)-dimensional [3+1d] mixed CS term. In addition to the electromagnetic U(1) gauge field, this term contains elasticity tetrad fields $E^{\ a}_{\mu}({\bf r},t) = \partial_{\mu}X^a(\mathbf{r},t)$ which are gradients of crystalline U(1) phase fields $X^a(\mathbf{r},t)$ and describe the deformations of the crystal. For a crystal in three spatial dimensions $a=1,2,3$ and the mixed axial-gravitational response contains three parameters protected by crystalline symmetries: the weak momentum-space topological invariants. The response of the Hall conductance to the deformations of the crystal is quantized in terms of these invariants. In the presence of dislocations, the anomalous 3+1d CS term describes the Callan-Harvey anomaly inflow mechanism. The response can be extended to all odd spatial dimensions. The elasticity tetrads, being the gradients of the lattice U(1) fields, have canonical dimension of inverse length. Similarly, if such tetrad fields enter general relativity, the metric becomes dimensionful, but the physical parameters, such as Newton's constant, the cosmological constant, and masses of particles, become dimensionless.

Examining ODD/ADHD Symptom Dimensions as Predictors of Social, Emotional, and Academic Trajectories in Middle Childhood

The goal of this article is to investigate the symptom dimensions of oppositional defiant disorder (ODD; irritability, defiance) and attention deficit/hyperactivity disorder (ADHD; inattention, hyperactivity-impulsivity) as predictors of academic performance, depressive symptoms, and peer functioning in middle childhood. Children (N = 346; 51% female) were assessed via teacher-report on measures of ODD/ADHD symptoms at baseline (Grades K–2) and academic performance, depressive symptoms, peer rejection, and victimization on 7 occasions over 4 school years (K–2 through 3–5). Self-report and grade point average data collected in Grades 3–5 served as converging outcome measures. Latent growth curve and multiple regression models were estimated using a hierarchical/sensitivity approach to assess robustness and specificity of effects. Irritability predicted higher baseline depressive symptoms, peer rejection, and victimization, whereas defiance predicted higher baseline peer rejection; however, none of these ODD-related effects persisted 3 years later to Grades 3–5. In contrast, inattention predicted persistently poorer academic performance, persistently higher depressive symptoms, and higher baseline victimization; hyperactivity-impulsivity predicted subsequent peer rejection and victimization in Grades 3–5. In converging models, only inattention emerged as a robust predictor of 3-year outcomes (viz., grade point average, depressive symptoms, peer rejection, and relational victimization). Broadly, ODD dimensions—particularly irritability—may be linked to acute disturbances in social-emotional functioning in school-age children, whereas ADHD dimensions may predict more persistent patterns of peer, affective, and academic problems. By examining all 4 ODD/ADHD symptom dimensions simultaneously, the present analyses offer clarity and specificity regarding which dimensions affect what outcomes, and when. Findings underscore the importance of multidimensional approaches to research, assessment, and intervention.

Robust cusps near topological phase transitions: Signatures of Majorana fermions and interactions with fluctuations

In this paper, we study nonanalyticities or cusps near a topological phase transition driven by changes of global topologies rather than by the formation of conventional (local) order. The particular phase transitions between topologically ordered superfluids and nontopological superfluids studied in this paper are always accompanied by the emergence or disappearance of Majorana boundary modes, while at the critical point these boundary modes are liberated and evolve into the bulk. Proliferation of massless Majorana fermions in the bulk at the transition leaves unique thermodynamic signatures. For a very broad class of isotropic superfluids or superconductors (with or without time-reversal symmetry), the resultant transitions in $d$ spatial dimensions will be of $(d+1)\mathrm{th}$ order. The $(d+1)\mathrm{th}$-order derivatives of the grand potential are either discontinuous in even dimensions or logarithmically divergent in odd dimensions. Bulk quantities such as compressibility can develop a cusp along with the change of topology of the superfluids. We compute quantitatively the cusp structures and further investigate the effect of interactions induced by massive fluctuation fields. These cusps are robust as direct manifestations of Majorana fermions and these interactions, and can be potentially considered as an alternative approach for detecting these puzzling particles.

Differential equations, recurrence relations, and quadratic constraints for L-loop two-point massive tadpoles and propagators.

We consider L-loop two-point tadpole (watermelon) integral with arbitrary masses, regularized both dimensionally and analytically. We derive differential equation system and recurrence relations (shifts of dimension and denominator powers). Since the L-loop sunrise integral corresponds to the (L + 1)-loop watermelon integral with one cut line, our results are equally applicable to the former. The obtained differential system has a Pfaffian form and is linear in dimension and analytic regularization parameters. In general case, the solutions of this system can be expressed in terms of the Lauricella functions FC(L) with generic parameters. Therefore, as a by-product, we obtain, to our knowledge for the first time, the Pfaffian system for FC(L) for arbitrary L. The obtained system has no apparent singularities. Near odd dimension and integer denominator powers the system can be easily transformed into canonical form. Using the symmetry properties of the matrix in the right-hand side of the differential system, we obtain quadratic constraints for the expansion of solutions near integer dimension and denominator powers. In particular, we obtain quadratic constraints for Bessel moments similar to those discovered by Broadhurst and Roberts.

Entanglement entropy in generalised quantum Lifshitz models

We compute universal finite corrections to entanglement entropy for generalised quantum Lifshitz models in arbitrary odd spacetime dimensions. These are generalised free field theories with Lifshitz scaling symmetry, where the dynamical critical exponent z equals the number of spatial dimensions d, and which generalise the 2+1-dimensional quantum Lifshitz model to higher dimensions. We analyse two cases: one where the spatial manifold is a d-dimensional sphere and the entanglement entropy is evaluated for a hemisphere, and another where a d-dimensional flat torus is divided into two cylinders. In both examples the finite universal terms in the entanglement entropy are scale invariant and depend on the compactification radius of the scalar field.

Multiple characters transforms and generalized Boolean functions

In this paper we investigate generalized Boolean functions whose spectrum is flat with respect to a set of Walsh-Hadamard transforms defined using various complex primitive roots of 1. We also study some differential properties of the generalized Boolean functions in even dimension defined in terms of these different characters. We show that those functions have similar properties to the vectorial bent functions. We next clarify the case of gbent functions in odd dimension. As a by-product of our proofs, more generally, we also provide several results about plateaued functions. Furthermore, we find characterizations of plateaued functions with respect to different characters in terms of second derivatives and fourth moments.

Principal tensor strikes again: Separability of vector equations with torsion

Many black hole spacetimes with a 3-form field exhibit a hidden symmetry encoded in a torsion generalization of the principal Killing–Yano tensor. This tensor determines basic properties of such black holes while also underlying the separability of the Hamilton–Jacobi, Klein–Gordon, and (torsion-modified) Dirac field equations in their background. As a specific example, we consider the Chong–Cvetič–Lü–Pope black hole of D=5 minimal gauged supergravity and show that the torsion-modified vector field equations can also be separated, with the principal tensor playing a key role in the separability ansatz. For comparison, separability of the Proca field in higher-dimensional Kerr–NUT–AdS spacetimes (including new explicit formulae in odd dimensions) is also presented.

Hamilton-Jacobi approach to holographic renormalization of massive gravity

Recently, a practical approach to holographic renormalization has been developed based on the Hamilton-Jacobi formulation. Using a simple Einstein-scalar theory, we clarify that this approach does not conflict with the Hamiltonian constraint as it seems. Then we apply it to the holographic renormalization of massive gravity. We assume that the shift vector is falling off fast enough asymptotically. We derive the counterterms up to the boundary dimension d = 4. Interestingly, we find that the conformal anomaly can even occur in odd dimensions, which is different from the Einstein gravity. We check that the counterterms cancel the divergent part of the on-shell action at the background level. At the perturbation level, they are also applicable in several time-dependent cases.