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1. On Spectral Bands of Discrete Periodic Operators

   https://doi.org/10.1007/s00220-023-04891-7  

   Filonov, Nikolay; Kachkovskiy, Ilya (February 2024, Communications in Mathematical Physics)

   We consider discrete periodic operator on Z^d with respect to lattices of full rank. We describe the class of lattices for which the operator may have a spectral gap for arbitrarily small potentials. We also show that, for a large class of lattices, the dimensions of the level sets of spectral band functions at the band edges do not exceed d-2. 

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2. Subharmonic spin correlations and spectral pairing in Floquet time crystals

   https://doi.org/10.1103/PhysRevB.111.184308  

   Penner, Alexander-Georg; Schmid, Harald; Glazman, Leonid I; von_Oppen, Felix (May 2025, Physical Review B)

   Floquet time crystals are characterized by the subharmonic behavior of temporal correlation functions. Studying the paradigmatic time crystal based on the disordered Floquet quantum Ising model, we show that its temporal spin correlations are directly related to spectral characteristics and that this relation provides analytical expressions for the correlation function of finite chains, which compare favorably with numerical simulations. Specifically, we show that the disorder-averaged temporal spin correlations are proportional to the Fourier transform of the splitting distribution of the pairs of eigenvalues of the Floquet operator, which differ by $\pi$to exponential accuracy in the chain length. We find that the splittings are well described by a log-normal distribution, implying that the temporal spin correlations are characterized by two parameters. We discuss possible implications for the phase diagram of Floquet time crystals. 

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3. Giant Wilson loops and AdS2/dCFT1

   https://doi.org/10.1007/JHEP11(2020)064  

   Giombi, Simone; Jiang, Jiaqi; Komatsu, Shota (November 2020, Journal of High Energy Physics)

   null (Ed.)

   A bstract The 1/2-BPS Wilson loop in $$ \mathcal{N} $$ N = 4 supersymmetric Yang-Mills theory is an important and well-studied example of conformal defect. In particular, much work has been done for the correlation functions of operator insertions on the Wilson loop in the fundamental representation. In this paper, we extend such analyses to Wilson loops in the large-rank symmetric and antisymmetric representations, which correspond to probe D3 and D5 branes with AdS 2 × S 2 and AdS 2 × S 4 worldvolume geometries, ending at the AdS 5 boundary along a one-dimensional contour. We first compute the correlation functions of protected scalar insertions from supersymmetric localization, and obtain a representation in terms of multiple integrals that are similar to the eigenvalue integrals of the random matrix, but with important differences. Using ideas from the Fermi Gas formalism and the Clustering method, we evaluate their large N limit exactly as a function of the ’t Hooft coupling. The results are given by simple integrals of polynomials that resemble the Q -functions of the Quantum Spectral Curve, with integration measures depending on the number of insertions. Next, we study the correlation functions of fluctuations on the probe D3 and D5 branes in AdS. We compute a selection of three- and four-point functions from perturbation theory on the D-branes, and show that they agree with the results of localization when restricted to supersymmetric kinematics. We also explain how the difference of the internal geometries of the D3 and D5 branes manifests itself in the localization computation. 

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4. A Family of Banach Spaces Over ℝ∞

   https://doi.org/10.1142/S2591722621400020  

   Gill, Tepper L.; Kalita, Hemanta; Hazarika, Bipan (March 2021, Proceedings of the Singapore National Academy of Science)

   null (Ed.)

   In T. L. Gill and W. W. Zachary, Functional Analysis and the Feynman Operator Calculus (Springer, New York, 2016), the topology of [Formula: see text] was replaced with a new topology and denoted by [Formula: see text]. This space was then used to construct Lebesgue measure on [Formula: see text] in a manner that is no more difficult than the same construction on [Formula: see text]. More important for us, a new class of separable Banach spaces [Formula: see text], [Formula: see text], for the HK-integrable functions, was introduced. These spaces also contain the [Formula: see text] spaces and the Schwartz space as continuous dense embeddings. This paper extends the work in T. L. Gill and W. W. Zachary, Functional Analysis and the Feynman Operator Calculus (Springer, New York, 2016) from [Formula: see text] to [Formula: see text]. 

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5. Unveiling Operator Growth Using Spin Correlation Functions

   https://doi.org/10.3390/e23050587  

   Carrega, Matteo; Kim, Joonho; Rosa, Dario (May 2021, Entropy)

   null (Ed.)

   In this paper, we study non-equilibrium dynamics induced by a sudden quench of strongly correlated Hamiltonians with all-to-all interactions. By relying on a Sachdev-Ye-Kitaev (SYK)-based quench protocol, we show that the time evolution of simple spin-spin correlation functions is highly sensitive to the degree of k-locality of the corresponding operators, once an appropriate set of fundamental fields is identified. By tracking the time-evolution of specific spin-spin correlation functions and their decay, we argue that it is possible to distinguish between operator-hopping and operator growth dynamics; the latter being a hallmark of quantum chaos in many-body quantum systems. Such an observation, in turn, could constitute a promising tool to probe the emergence of chaotic behavior, rather accessible in state-of-the-art quench setups. 

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