Publications

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• Anti-holomorphic modes in vortex lattices

  Brook J. Hocking, Thomas Machon

  SciPost Phys. 14, 080 (2023) · published 24 April 2023 | Toggle abstract · pdf

  A continuum theory of linearized Helmholtz-Kirchoff point vortex dynamics about a steadily rotating lattice state is developed by two separate methods: firstly by a direct procedure, secondly by taking the long-wavelength limit of Tkachenko's exact solution for a triangular vortex lattice. Solutions to the continuum theory are found, described by arbitrary anti-holomorphic functions, and give power-law localized edge modes. Numerical results for finite lattices show excellent agreement to the theory.

• Topological insulators and geometry of vector bundles

  Alexander S. Sergeev

  SciPost Phys. Lect. Notes 67 (2023) · published 12 April 2023 | Toggle abstract · pdf

  For a long time, band theory of solids has focused on the energy spectrum, or Hamiltonian eigenvalues. Recently, it was realized that the collection of eigenvectors also contains important physical information. The local geometry of eigenspaces determines the electric polarization, while their global twisting gives rise to the metallic surface states in topological insulators. These phenomena are central topics of the present notes. The shape of eigenspaces is also responsible for many intriguing physical analogies, which have their roots in the theory of vector bundles. We give an informal introduction to the geometry and topology of vector bundles and describe various physical models from this mathematical perspective.

• Codimension-2 defects and higher symmetries in (3+1)D topological phases

  Maissam Barkeshli, Yu-An Chen, Sheng-Jie Huang, Ryohei Kobayashi, Nathanan Tantivasadakarn, Guanyu Zhu

  SciPost Phys. 14, 065 (2023) · published 11 April 2023 | Toggle abstract · pdf

  (3+1)D topological phases of matter can host a broad class of non-trivial topological defects of codimension-1, 2, and 3, of which the well-known point charges and flux loops are special cases. The complete algebraic structure of these defects defines a higher category, and can be viewed as an emergent higher symmetry. This plays a crucial role both in the classification of phases of matter and the possible fault-tolerant logical operations in topological quantum error-correcting codes. In this paper, we study several examples of such higher codimension defects from distinct perspectives. We mainly study a class of invertible codimension-2 topological defects, which we refer to as twist strings. We provide a number of general constructions for twist strings, in terms of gauging lower dimensional invertible phases, layer constructions, and condensation defects. We study some special examples in the context of $\mathbb{Z}_2$ gauge theory with fermionic charges, in $\mathbb{Z}_2 \times \mathbb{Z}_2$ gauge theory with bosonic charges, and also in non-Abelian discrete gauge theories based on dihedral ($D_n$) and alternating ($A_6$) groups. The intersection between twist strings and Abelian flux loops sources Abelian point charges, which defines an $H^4$ cohomology class that characterizes part of an underlying 3-group symmetry of the topological order. The equations involving background gauge fields for the 3-group symmetry have been explicitly written down for various cases. We also study examples of twist strings interacting with non-Abelian flux loops (defining part of a non-invertible higher symmetry), examples of non-invertible codimension-2 defects, and examples of the interplay of codimension-2 defects with codimension-1 defects. We also find an example of geometric, not fully topological, twist strings in (3+1)D $A_6$ gauge theory.

  5 citations

• Supersymmetric ground states of 3d $\mathcal{N}=4$ SUSY gauge theories and Heisenberg algebras

  Andrea E. V. Ferrari

  SciPost Phys. 14, 063 (2023) · published 6 April 2023 | Toggle abstract · pdf

  We consider 3d $\mathcal{N}=4$ theories on the geometry $\Sigma \times \mathbb{R}$, where $\Sigma$ is a closed and connected Riemann surface, from the point of view of a quantum mechanics on $\mathbb{R}$. Focussing on the elementary mirror pair in the presence of real deformation parameters, namely SQED with one hypermultiplet (SQED[1]) and the free hypermulitplet, we study the algebras of local operators in the respective quantum mechanics as well as their action on the vector space of supersymmetric ground states. We demonstrate that the algebras can be described in terms of Heisenberg algebras, and that they act in a way reminiscent of Segal-Bargmann (B-twist of the free hypermultiplet) and Nakajima (A-twist of SQED[1]) operators.

  1 citation

• Replica method for eigenvalues of real Wishart product matrices

  Jacob A. Zavatone-Veth, Cengiz Pehlevan

  SciPost Phys. Core 6, 026 (2023) · published 5 April 2023 | Toggle abstract · pdf

  We show how the replica method can be used to compute the asymptotic eigenvalue spectrum of a real Wishart product matrix. For unstructured factors, this provides a compact, elementary derivation of a polynomial condition on the Stieltjes transform first proved by Müller [IEEE Trans. Inf. Theory. 48, 2086-2091 (2002)]. We then show how this computation can be extended to ensembles where the factors are drawn from matrix Gaussian distributions with general correlation structure. For both unstructured and structured ensembles, we derive polynomial conditions on the average values of the minimum and maximum eigenvalues, which in the unstructured case match the results obtained by Akemann, Ipsen, and Kieburg [Phys. Rev. E 88, 052118 (2013)] for the complex Wishart product ensemble.

  1 citation

• Particle dynamics on torsional galilean spacetimes

  José Figueroa-O'Farrill, Can Görmez, Dieter Van den Bleeken

  SciPost Phys. 14, 059 (2023) · published 4 April 2023 | Toggle abstract · pdf

  We study free particle motion on homogeneous kinematical spacetimes of galilean type. The three well-known cases of Galilei and (A)dS-Galilei spacetimes are included in our analysis, but our focus will be on the previously unexplored torsional galilean spacetimes. We show how in well-chosen coordinates free particle motion becomes equivalent to the dynamics of a damped harmonic oscillator, with the damping set by the torsion. The realization of the kinematical symmetry algebra in terms of conserved charges is subtle and comes with some interesting surprises, such as a homothetic version of hamiltonian vector fields and a corresponding generalization of the Poisson bracket. We show that the Bargmann extension is universal to all galilean kinematical symmetries, but also that it is no longer central for nonzero torsion. We also present a geometric interpretation of this fact through the Eisenhart lift of the dynamics.

  1 citation

• Diagonal fields in critical loop models

  Sylvain Ribault

  SciPost Phys. Core 6, 020 (2023) · published 27 March 2023 | Toggle abstract · pdf

  In critical loop models, there exist diagonal fields with arbitrary conformal dimensions, whose $3$-point functions coincide with those of Liouville theory at $c\leq 1$. We study their $N$-point functions, which depend on the $2^{N-1}$ weights of topologically inequivalent loops on a sphere with $N$ punctures. Using a numerical conformal bootstrap approach, we find that $4$-point functions decompose into infinite but discrete linear combinations of conformal blocks. We conclude that diagonal fields belong to an extension of the $O(n)$ model.

  2 citations

• Lifshitz symmetry: Lie algebras, spacetimes and particles

  José Figueroa-O'Farrill, Ross Grassie, Stefan Prohazka

  SciPost Phys. 14, 035 (2023) · published 15 March 2023 | Toggle abstract · pdf

  We study and classify Lie algebras, homogeneous spacetimes and coadjoint orbits (``particles'') of Lie groups generated by spatial rotations, temporal and spatial translations and an additional scalar generator. As a first step we classify Lie algebras of this type in arbitrary dimension. Among them is the prototypical Lifshitz algebra, which motivates this work and the name "Lifshitz Lie algebras". We classify homogeneous spacetimes of Lifshitz Lie groups. Depending on the interpretation of the additional scalar generator, these spacetimes fall into three classes: 1. ($d+2$)-dimensional Lifshitz spacetimes which have one additional holographic direction; 2. ($d+1$)-dimensional Lifshitz—Weyl spacetimes which can be seen as the boundary geometry of the spacetimes in (1) and where the scalar generator is interpreted as an anisotropic dilation; 3. and $(d+1)$-dimensional aristotelian spacetimes with one scalar charge, including exotic fracton-like symmetries that generalise multipole algebras. We also classify the possible central extensions of Lifshitz Lie algebras and we discuss the homogeneous symplectic manifolds of Lifshitz Lie groups in terms of coadjoint orbits.

  5 citations

• Growth of the Wang-Casati-Prosen counter in an integrable billiard

  Zaijong Hwang, Christoph A. Marx, Joseph J. Seaward, Svetlana Jitomirskaya, Maxim Olshanii

  SciPost Phys. 14, 017 (2023) · published 10 February 2023 | Toggle abstract · pdf

  This work is motivated by an article by Wang, Casati, and Prosen [Phys. Rev. E vol. 89, 042918 (2014)] devoted to a study of ergodicity in two-dimensional irrational right-triangular billiards. Numerical results presented there suggest that these billiards are generally not ergodic. However, they become ergodic when the billiard angle is equal to $\pi/2$ times a Liouvillian irrational, morally a class of irrational numbers which are well approximated by rationals. In particular, Wang et al. study a special integer counter that reflects the irrational contribution to the velocity orientation; they conjecture that this counter is localized in the generic case, but grows in the Liouvillian case. We propose a generalization of the Wang-Casati-Prosen counter: this generalization allows to include rational billiards into consideration. We show that in the case of a $45°\!\!:\!45°\!\!:\!90°$ billiard, the counter grows indefinitely, consistent with the Liouvillian scenario suggested by Wang et al.

• Charge and antipodal matching across spatial infinity

  Federico Capone, Kevin Nguyen, Enrico Parisini

  SciPost Phys. 14, 014 (2023) · published 6 February 2023 | Toggle abstract · pdf

  We derive the antipodal matching relations used to demonstrate the equivalence between soft graviton theorems and BMS charge conservation across spatial infinity. To this end we provide a precise map between Bondi data at null infinity $\mathscr{I}$ and Beig—Schmidt data at spatial infinity $i^0$ in a context appropriate to the gravitational scattering problem and celestial holography. In addition, we explicitly match the various proposals of BMS charges at $\mathscr{I}$ found in the literature with the conserved charges at $i^0$.

  8 citations