# Something Cool

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# Entanglement Entropy Bounds in the Higher Spin XXZ Chain

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# Entanglement Entropy Bounds in the Higher Spin XXZ Chain

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We consider the Heisenberg XXZ spin-$J$ chain ($J\in\mathbb{N}/2$) with anisotropy parameter $\Delta$. Assuming that $\Delta>2J$, and introducing threshold energies $E_{K}:=K\left(1-\frac{2J}{\Delta}\right)$, we show that the bipartite entanglement entropy (EE) of states belonging to any spectral subspace with energy less than $E_{K+1}$ satisfy a logarithmically corrected area law with prefactor $(2\lfloor K/J\rfloor-2)$.

This generalizes previous results by Beaud and Warzel as well as Abdul-Rahman, Fischbacher and Stolz, who covered the spin-$1/2$ case.

# Smooth quasiperiodic SL(2,\R)-cocycles (III)-Sketch of the proof of the continuity of the Lyapunov exponent

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In this talk, we will sketch how to generalize the large deviation theorem to quasiperiodic Gevrey $SL(2,\R)$-cocycles and use it to prove the joint continuity of the Lyapunov exponent.

# Bose-Einstein condensation in one-dimensional noninteracting Bose gases in the presence of soft Poissonian obstacles

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We study Bose--Einstein condensation (BEC) in one-dimensional noninteracting Bose gases in Poisson random potentials on $\mathbb R$ with single-site potentials that are nonnegative, compactly supported, and bounded measurable functions in the grand-canonical ensemble at positive temperatures and in the thermodynamic limit. For particle densities larger than a critical one, we prove the following: With arbitrarily high probability when choosing the fixed strength of the random potential sufficiently large, BEC where only the ground state is macroscopically occupied occurs. If the strength of the Poisson random potential converges to infinity in a certain sense but arbitrarily slowly, then this kind of BEC occurs in probability and in the $r$th mean, $r \ge 1$. Furthermore, in Poisson random potentials of any fixed strength an arbitrarily high probability for type-I g-BEC is also obtained by allowing sufficiently many one-particle states to be macroscopically occupied.

# Smooth quasiperiodic SL(2,\R)-cocycles (II)-Sharp transition space for the continuity of the Lyapunov exponent.

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We construct discontinuous points of the Lyapunov exponent of quasiperiodic Shr\"odinger cocycles in Gevrey space $G^{s}$ with $s>2$. In contrast, the Lyapunov exponent has been proved to be continuous in $G^{s}$ with $s<2$ by Klein and Cheng-Ge-You-Zhou. This shows that $G^2$ is the transition space for the continuity of the Lyapunov exponent.

# Smooth quasiperiodic SL(2,\R)-cocycles (I)-Global rigidity results for rotations reducibility and Last's intersection spectrum conjecture.

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For quasiperiodic Schr\"odinger operators with one-frequency analytic potentials, from dynamical systems side, it has been proved that the corresponding quasiperiodic Schr\"odinger cocycle is either rotations reducible or has positive Lyapunov exponent for all irrational frequency and almost every energy by Avila-Fayad-Krikorian. From spectral theory side, the ``Schr\"odinger conjecture" has been verified by Avila-Fayad-Krikorian and the ``Last's intersection spectrum conjecture" has been proved by Jitomirskaya-Marx. The proofs of above results crucially depend on the analyticity of the potentials. People are curious about if the analyticity is essential for those problems, some open problems in this aspect were raised by Fayad-Krikorian and Jitomirskaya-Marx. In this paper, we prove the above mentioned results for ultra-differentiable potentials.

# On the abominable properties of the Almost Mathieu operator with Liouville frequencies

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We show that, for sufficiently well approximable frequencies, several spectral characteristics of the Almost Mathieu operator can be as poor as at all possible in the class of all discrete Schroedinger operators. For example, the modulus of continuity of the integrated density of states may be no better than logarithmic. Other characteristics to be discussed are homogeneity, the Parreau-Widom property, and (for the critical AMO) the Hausdorff content of the spectrum. Based on joint work with A. Avila, Y. Last, and Q. Zhou

# THE UMPTEEN OPERATOR

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It was found in the 1990s that special linear maps playing a role in the representation theory of the symmetric group share common features with random matrices. We construct a representation-theoretic operator which shares some properties with the Anderson model (or, perhaps, with magnetic random Schroedinger operators), and show that indeed it boasts Lifshitz tails. The proof relies on a close connection between the operator and the infinite board version of the fifteen puzzle.

No background in the representation theory of the symmetric group will be assumed. Based on joint work with Ohad Feldheim.