Papers

We study general maximal hard-core models on periodic lattice-type graphs. While for standard models, phase coexistence may arise for high activity, for maximal models this behavior may occur also for sufficiently low activity values. Relying on the concept of volume allocation, we develop a unified set of assumptions that imply the Peierls condition, a convergent cluster expansion for the partition function, and hence the conclusions of Pirogov-Sinai theory for this class of models. We further check these assumptions for a number of examples including standard lattice-type periodic graphs, proving crystallization for both high and low activity. We also derive estimates for the values of activity that bound the phase coexistence regions in the phase diagram, rewriting for this purpose the proof of a technical result in the derivation of Pirogov-Sinai theory.

Let $(\mathcal N,\nu)$ be a finite tracial von Neumann algebra and set $\mathcal{K}_r(P,Q)=\nu((P^{1/2}QP^{1/2})^r)$. We prove that $$ [\mathcal{K}_r(P_i,P_j)]_{i,j=1}^3\ge0, \qquad 1/2\le r\le1. $$ The range $1/2<r<1$ is new even for matrices, and order three is sharp for $1/2\le r<1$. This single mechanism resolves two problems. First, if $(\mathcal R,\omega)$ is finite tracial and $H=[H_{ij}]_{i,j=1}^3\in M_3(\mathcal R)_+$, then $$ [\omega(|H_{ij}|^p)]_{i,j=1}^3\ge0, \qquad 1\le p\le2, $$ settling the Lin–van den Driessche conjecture on Schatten norm compression. Second, for a semifinite tracial von Neumann algebra $(\mathcal M,\tau)$, define $$ d_t(A,B)^2=\frac{\tau(A)+\tau(B)}2- \|B^{t/4}A^{t/4}\|_{2/t}^{2/t}. $$ We prove that $d_t$ is a complete metric on $L^1(\mathcal M,\tau)_+$ for $1\le t\le2$, including the previously open range $1<t<2$ between the Hellinger and Bures endpoints, and that $$ d_t\text{ is a metric on }L^1(\mathcal M,\tau)_+ \quad\Longleftrightarrow\quad t\in[1,2]\ \text{or}\ \mathcal M\text{ is abelian}. $$ Its topology and Cauchy sequences agree with those of $L^1$. The same classification holds for unital tracial $C^*$-algebras and, with the known $t=0$ obstruction, resolves Problem~2 of Komálovics and Molnár.

Let $(M^n,g)$, $n\ge2$, be a compact connected Riemannian manifold with smooth nonempty boundary, $\operatorname{Ric}_M\ge(n-1)Kg$ for $K>0$, and nonnegative outward boundary mean curvature. We derive an explicit algebraic lower bound $B_\eta$ for the first Dirichlet eigenvalue from Kasue's mixed inradius model. With $R$ the inradius and $s=4KR^2/\pi^2$, the bound involves one square root and approximates the model eigenvalue with relative error $O_n(s^4)$ as $s\downarrow0$ for fixed $n$. The construction uses the first two moments of a positive representation of the averaged model potential and a sharp pointwise corrector estimate. For the chosen spectral transfer, the first two moments determine an optimal bound. We also show that the model eigenvalue is the geometric infimum at fixed $(n,K,R)$. For $0<s<1$, it is approached by smooth collapsing balls with sectional curvature at least $K$ and totally geodesic boundary. At the maximal inradius, the model value is attained by the round hemisphere. The bound strictly improves the variance refinement throughout $0<s\le1$. We also give a direct algebraic comparison with an optimized power-test benchmark.

For the $H^{2|2}$ model on $\mathbb Z^2$, we prove that, at energy zero, fractional moment of Green functions, $\mathbb E[ G(x,y)^s]$ decays exponentially in the distance between $x$ and $y$, for every constant edge weight $W>0$ and every $0<s<1/2$. Here $G$ is the infinite-volume Green function at energy zero of the associated random Schrödinger operator. The main ingredient is a quantitative lower bound on the derivative of $\mathbb E[e^{u_x/2}]$ with respect to the edge weights, where $u$ is the pinned effective field.

Massive discrete Dirac-type operators on periodic planar graphs are investigated. Using Floquet–Bloch decomposition, we first show that neglecting the faces leads to spurious flat bands that are unrelated to the graph's geometry. Then, for the correct Dirac–Hodge operator $D_m$, we establish a spectral reduction expressing $\sigma(D_m)$ through the graph Laplacian on vertices $-\Delta_V$ and the Laplacian on faces $-\Delta_F$. As a consequence, $D_m$ carries a flat band precisely when one of these scalar Laplacians does. We finally apply this reduction to the study of flat bands of the operator $D_m$ on Archimedean lattices and contrast the outcome with that on the Lieb lattice.

Quantum fault tolerance is a prerequisite to harness the full potential of quantum computers. However, achieving fault tolerance remains a major challenge since it requires optimizing tradeoffs between several intertwined theoretical and experimental parameters, including the encoding rate, code distance, parity-check weights, code planarity, and routing. Surface codes remain strong candidates for fault tolerance due to their simple geometric structure, established decoding methods, locality, and experimental attainability. In this work, we present finite families of hyperbolic surface codes that leverage the optimal efficiency scaling $kd^2/n = C (\log{k})^2$ attainable by surface codes with bounded local geometry. We show that optimizing the periodic identifications can double the code distance and thereby increase the code's efficiency by a factor of four, at fixed qubit count, encoding rate, and parity-check weight. We also address the routing problem of hyperbolic surface codes by introducing a topology-aware algorithm that partitions a hyperbolic code into bounded planar modules while reducing long-range communications demand. Circuit-level Monte Carlo simulations yield finite-size threshold estimates for both modular and monolithic layouts, showing that tripling the long-range gate error probability mildly lowers, but does not eliminate, the estimated error-correction thresholds. Together, these results strengthen the case for hyperbolic surface codes as structured and highly efficient large-scale quantum error correction codes for future fault-tolerant architectures.

In quantum mechanics of genus-1 potentials, perturbation theory around the topologically trivial saddle and non-perturbative instanton saddle are constructively related via the P/NP relation. Whether such a relation persists for higher-genus potentials has remained an open problem. In this work, we resolve it using tools from complex algebraic geometry combined with exact WKB. The classical energy conservation relation $p^2 = 2(E-V(q))$ defines a genus-g Riemann surface $X$, whose periods are naturally organized by its Jacobian variety $J(X) = \mathbb{C}^{\text{g}}/\Lambda$ via the Abel-Jacobi map, and independently by the solution basis of the associated Picard-Fuchs equations. We show that the physically relevant, WKB-active cycles are generally related to the Picard-Fuchs basis by a linear transformation that lies in $\text{SL}(2\text{g},\mathbb{Z})$ but not in the symplectic group $\text{Sp}(2\text{g},\mathbb{Z})$ once $\text{g}\geq2$, in contrast to the genus-1 case, where the two groups coincide. This mismatch is the obstruction that has made an explicit higher-genus P/NP relation elusive. We show that it can be resolved by passing to a modified Riemann bilinear identity, built from a suitably transformed intersection matrix, which the WKB-active periods do satisfy exactly. The result is an all-orders quantum P/NP relation valid for potentials of arbitrary genus, which we derive explicitly for genus-2 (quintic and sextic) and genus-3 (septic and octic) anharmonic oscillators.

We prove that the sine beta point process is tail trivial for every beta >0. The key ingredient in the proof is a certain uniform entropy bound, that makes use of the DLR equations of Dereudre et al. (2021) and the bounded two-point correlation density obtained by Qu and Valko (2025) and Assiotis and Najnudel (2026), that allows us to show that tail events are translation invariant. Tail triviality then follows either from the uniqueness theorem of Assiotis (2026) or from translation mixing, which we derive from the correlation estimates of Dumaz and Malvy (2026). As consequences, we obtain the following applications: (a) extremality of sine beta in the full convex class of fixed kernel Gibbs measures without stationarity and finite-energy conditions; (b) L^2-ergodicity of the associated infinite-particle reversible diffusion for every beta>0; and (c) the almost-sure collision/noncollision dichotomy with threshold beta=1 for the diffusion.

We prove an exact identity, as convergent logarithmic germs at large Euclidean momentum, between the equal-mass two-dimensional $(n-1)$-loop sunset Feynman integral, with $n\geq4$, and the genus-zero one-point descendant invariants of the local Calabi–Yau $n$-fold $X_n=\operatorname{Tot}(K_{F_n})$, where $F_n\subset(\mathbb P^1)^{n}$ is a smooth $(1,\ldots,1)$ hypersurface. Writing $s$ for the momentum variable and $y=-1/s$, the identity reads \[ I^{(n-1)}_{\circleddash}(s)=(-1)^{n-1}yA\,\Bigl[nR^{n-1}-\theta_Q{\mathcal D}_0^\Delta\,R^{n-2} +\sum_{j=0}^{n-3}c_jR^j\Bigr], \] where $R=\log y+O(y)$ is the local mirror coordinate, $A=\theta_yR$, $Q=e^R$, ${\mathcal D}_0^\Delta(Q)$ is the primary one-point potential, and each $c_j$ has the form \[ c_j=c_j^{(0)}+\sum_{a=0}^{n-2}\bigl(c_{j,a}\,{\mathcal D}_a^\Delta(Q) +c'_{j,a}\,\theta_Q{\mathcal D}_a^\Delta(Q)\bigr) \] with the coefficients in $\mathbb Q[\zeta(3),\zeta(5),\ldots]$ the odd zeta values coming from the Gamma class. The descendant series ${\mathcal D}_a^\Delta(Q)$ are given explicitly by the ambient hypergeometric series and the inverse mirror map. We also prove global univalence of the principal mirror map, with explicit upper and lower bounds on the $Q$-Taylor radii, extract the full primary evaluation divisor and its integral cover transform, and reduce the ambient endpoint constants to a finite universal Gamma-class part and $\lfloor(n-1)/2\rfloor$ real parameters.

We determine the equal-mass two-dimensional three- and four-loop sunset Feynman integrals $I^{(3)}_\circleddash$ and $I^{(4)}_\circleddash$ at large Euclidean momentum in terms of the genus-zero Gromov–Witten theory of the local Calabi–Yau fourfold and fivefold $X_L=\operatorname{Tot}(K_{F_L})$, where $F_L\subset(\mathbb P^1)^{L+1}$, $L=3,4$, is a smooth hypersurface of multidegree $(1,\ldots,1)$. At two loops the sunset integral is governed by the prepotential of a local Calabi–Yau threefold. For $L\geq3$ the moduli spaces of unmarked stable maps have positive virtual dimension and there is no prepotential, therefore the Feynman integral is governed instead by the one-point potential $\mathcal F_{\gamma_L}$ with insertion $\gamma_L=H^{L-1}/(L-1)!$, by its gravitational descendants, and by $\zeta(3)$ boundary terms coming from the Gamma class. At both loop orders the coefficient of $R^{L-1}$ in the Frobenius expansion in the mirror coordinate $R$ is $-\theta_Q\mathcal F_{\gamma_L}$ with the Novikov variable $Q=\exp(R)$. At three loops the whole Feynman integral is an explicit expression in $\mathcal F_{\gamma_3}$ alone, because genus-zero recursion determines all descendant series from it; at four loops two diagonal series suffice, and the $\zeta(3)$ term is itself corrected by $\theta_Q\mathcal F_{\gamma_4}$. At three loops we also derive the analytic normalization independently, from the Picard–Fuchs equation and the polar part of a Mellin transform, prove integrality of the multiple-cover transforms for arbitrary integral insertions together with an explicit quantum-$K$ formula, and compute the exact period connection polynomials. Finally we give endpoint values along the Euclidean axis, bounds on the Taylor radii in the Novikov variable $Q$, a computer-assisted proof of the four-loop endpoint nonanalyticity, and a comparison of the normalizations at two, three and four loops.

We study gauge-invariant two-matrix polynomials of degree $L$, corresponding to the states in the $su(2)$ sector of $\mathcal{N}=4$ super Yang-Mills theory with gauge group $U(N)$ or $SU(N)$ at finite $N$. For the $U(N)$ theory, the state space has the structure of an algebra $\mathcal{A}_N(L)$, which is a direct sum of permutation centraliser algebras, i.e. sub-algebras of the group algebra $\mathbb{C}[S_L]$ of $S_L$. For $N \ge L$, $\mathcal{A}_N (L) = \mathcal{A} (L)$ is independent of $N$, and the one-loop dilatation operator acts on $\mathcal{A}(L)$ with $N$ as a parameter. For $N < L$, $\mathcal{A}_N (L)$ is a subspace of $\mathcal{A}(L)$ defined using a finite-$N$ cutoff on Young diagrams with $L$ boxes. We establish two results on the eigenspaces of the one-loop dilatation operator. Our first main result is that the characteristic polynomial of the one-loop dilatation operator on $\mathcal{A}_N(L)$ divides the characteristic polynomial from $\mathcal{A}(L)$, continued to $N < L$. Our second result is that each eigenstate of the one-loop dilatation operator in the $SU(N)$ theory generates an infinite tower of eigenstates in the $U(N)$ theory, through the multiplication of single-letter traces. These towers define an eigenspace factorisation of the $U(N)$ theory labelled by the number of such traces. Further, we conjecture that the space of $SU(N)$-invariant polynomials in two traceless matrices at finite $N$ is isomorphic to the intersection of $\mathcal{A}_N(L)$ with the subspace of $\mathbb{C}[S_L]$ spanned by derangements. This intersection conjecture is supported by dimension counting. Finally, we propose a one-loop dilatation operator on the intersection, thus providing a direct permutation framework for the $SU(N)$ spectral problem at finite $N$. The computed spectra based on the proposal agree with the prediction from the eigenspace factorisation in the $U(N)$ theory.

For the two-dimensional quantum Heisenberg ferromagnet with asymptotically radial power-law exchange, we prove the leading low-temperature free energy at every fixed spin, uniformly on the weak-field scale. We obtain fractional and logarithmically corrected laws associated with the dispersions predicted in [16, 17]. Quantum Monte Carlo simulations support the associated anomalous magnon dispersions for spin one-half [20]. For kernels with finite second moment, we also extend the nearest-neighbour theorem [14], generalizing Takahashi's quadratic law [21] and its weak-field dependence [11].

Superconductivity and intrinsic topological order can intertwine in ways that go beyond simple coexistence. Here we develop a classification framework for (2+1)D gapped superconductors with condensate charge $2ne$. By focusing on the topological properties of superconducting vortices, we precisely characterize this interplay. We obtain a classification with increasing levels of intertwinement, from the well known invertible superconductors without topological order, to untwisted and twisted SC$^\ast$, and finally to SC$^{\sharp}$. Our main finding is the family SC$^{\sharp}$, in which the elementary vortex carrying $h/(2ne)$ flux cannot be factorized into an invertible superconductor vortex dressed by bulk anyons, and every elementary vortex is necessarily non-Abelian. For such phases, we also obtain no-go results: the elementary vortex sector has quantum dimension at least $\sqrt{6}$ and cannot satisfy Fibonacci fusion rules. We develop a systematic algebraic framework to obtain all these superconducting phases through anyon condensation, allowing us to construct several infinite families of examples and make connections to anyon superconductivity. Finally, we introduce primitive SC$^\sharp$ phases for which factorization of every nontrivial vortex sector is obstructed, and argue that they cannot arise by disordering the relative phases of lower-charge superconducting condensates. Our classification scheme based on vortex properties allows us to uncover new distinctions between topologically ordered superconductors and constrain how they can be realized.

We study magnets composed of molecules with an arbitrary number $m$ of magnetically active atoms in a chain configuration interacting with nearest-neighbor couplings. We derive the thermodynamics of a chain of spin-$j$ $SU(2)$ magnetic atoms using a mean-field approach and determine the equilibrium states and stability conditions. The system exhibits a rich phase structure with a cascade of ferromagnetic phase transitions, their number ranging from $m$, for maximal ferromagnets, to zero, for paramagnets, depending on the relative strengths of the couplings. In the presence of impurities (modified couplings) at the ends of the chain, boundary states emerge that coexist with the paramagnetic phase or the fully ferromagnetic phase. For long chains, a continuum approximation can be taken that clarifies the structure of the phases. In the continuum description, magnetized states at temperatures lower than the critical temperature at which they appear correspond to kink solitons.

Suppose $M$ is a planar map with a macroscopic boundary decorated by a spanning tree $T$, such that $(M,T)$ has been sampled from the critical Boltzman measure. We show that $M$ converges to a Liouville quantum gravity disk with parameter $\gamma=\sqrt{2}$ under the Tutte embedding as the length of the boundary of $M$ goes to infinity. Furthermore, conditioned on $(M,T)$, the simple random walk on $M$ converges to a planar Brownian motion modulo reparametrization of time.

We study Bernoulli percolation and its universality classes on hierarchical lattices generated by edge iterated graph systems (EIGS). By examining eight critical exponents, we prove that no finite family of substitution-multiplicative dimensions completely determines universality, providing a rigorous proof of the failure of dimension-based universality discussed by Hauser and Saxena [Phys. Rev. B 34 (1986), 8193-8195]. Although the six exponentials theorem provides number-theoretic criteria for scale commensurability, we further disprove the conjecture that critical-exponent universality requires commensurate discrete renormalisation scales.

We give a polynomial-time algorithm for approximating the partition function of mean-field mixed $p$-spin models to arbitrarily high accuracy throughout the second-moment regime. This improves the quasipolynomial-time algorithm of Bencs, Huang, Lee, Liu, and Regts (arXiv:2507.15616) and answers their open question. In particular, our result covers the entire replica-symmetric regime of the Sherrington–Kirkpatrick model. The algorithm is deterministic, runs in time polynomial in $n$ and $1/\varepsilon$, and succeeds for every typical realization of the disorder. Our main technical contribution is a new algorithmic application of color coding to the classical high-temperature expansion introduced by Aizenman, Lebowitz, and Ruelle in their study of fluctuations of the Sherrington–Kirkpatrick partition function. We combine this approach with the zero-freeness established by Bencs et al. (arXiv:2507.15616) to convert additive approximations into multiplicative ones.

This is a continuation of our previous paper on toric gravitational instantons. We develop a systematic theory of compact simply-connected Einstein four-manifolds with toric symmetry. Applications include (1) The construction of simply connected positive Einstein four-manifolds with arbitrarily large $b_2$. Our method is based on local gluing combined with a global virtual counting argument. The key idea involves a particular design of rod structures to enable a gluing construction and restrict possible degenerations. (2) A topological characterization of algebraically special toric Einstein metrics. This uses special curvature identities for $W^+$ in the toric setting. (3) Uniqueness, up to scaling and isometry, on toric four-manifolds admitting algebraically special Einstein metrics. The proofs uses a variational study of the Einstein–Hilbert functional. (4) Diffeomorphism classification of compact toric Einstein four-manifolds in terms of an improved Hitchin-Thorpe inequality $\chi\geq3|\tau|$. In particular, our construction recovers all the classical toric Einstein metrics, including the Page and Chen–LeBrun–Weber metrics, starting from the round sphere through moduli spaces of conical Einstein metrics.

We construct chiral extensions of the cotangent Yangian using vertex algebroids. At zero level, the construction yields a vertex algebra with a compatible coproduct, whose Zhu reduction recovers the cotangent Yangian of Abedin and Niu. We also construct vertex algebra extensions at non-zero levels and study level-additive coproducts. The zero-level vertex algebra admits two natural quasi-conformal structures. We compute the corresponding sphere one-point conformal blocks and equip each block space with an associative product induced by the coproduct. For the first quasi-conformal structure, the blocks space can be identified with the dual of a certain Lie algebroid homology; for the second, we describe the block algebra and its classical limit in terms of mini-twistor geometry.

In a recent paper, a mathematical analogy was found between two physically distinct classes of two-dimensional free boundary problems in inviscid fluid mechanics. One class involves free surface flows with constant vorticity in the bulk but without boundary surface tension, the other comprises free surface flows without vorticity in the bulk but with non-zero boundary surface tension (capillarity). Several examples were given of pairs of physically-distinct free boundary problems that amount to solving the same mathematical problem, albeit with the possibility of a single constraint arising for the capillary flows. The present work shows that this analogy extends to another pair of physically-distinct problems: a hollow vortex equilibrium in a shear flow with a third order irrotational strain and a steady bubble with surface tension in a simple straining flow. For both problems, we construct an isolated exact solution when a parameter $\gamma=0$ and provide numerical solutions for non-zero $\gamma$. An interesting feature is that non-trivial conformal mapping identities emerge from the class of problems with capillarity that apply equally for the hollow vortex problems when the parameters are appropriately constrained.

We establish sharp two-point function asymptotics for the two dimensional Villain model at sufficiently low temperatures, confirming a central prediction in the Berezinskii-Kosterlitz-Thouless theory. We characterise the power law exponent through the renormalisation group flow of a marginal line defect observable in the dual high-temperature Discrete Gaussian model.

We study the new eigenvalues of a fixed point interaction on the cubic three-torus. Their mean-one consecutive spacings converge to a probability law independent of the interaction parameter. We characterize this law by the consecutive zeros of a random meromorphic function built from three-squares congruence densities. Its small-gap distribution has the form $s^5\Psi(\log_2(s^{-2}))+o(s^5)$, where $\Psi$ is continuous, positive, and one-periodic. The proof combines periodic mean-square approximation, a weighted Hilbert-transform estimate, and Fourier estimates that retain the interactions between primes.

The ionization conjecture for the Thomas-Fermi-Dirac-von Weizsäcker model asks how many electrons a given nuclear configuration can bind. So far, only the single-center case has been settled by Frank, Nam, and Van Den Bosch, while the original molecular case has still remained open. We advance this conjecture in the two-center case. We prove that for two distinct nuclei of total charge $Z$, the number $N$ of bound electrons satisfies $N\le Z+C Z^{153/154}(1+\log Z)^{3/8}$, with constants independent of the nuclear positions and the nuclear charges. The proof relies on a comparison with the neutral Thomas-Fermi density and on localization and exterior $L^1$-estimates coupled through the screened potential. Our results could be extended to molecules with any fixed number $K$ of nuclei.

We study normalized ground states for the nonlinear Schrödinger energy on two-dimensional square grids with unbounded defects. We first show that unbounded defects do not necessarily destroy the dimensional crossover, proving its persistence on the half-grid and the quarter-grid. We then consider a prototypical defected grid where a macroscopically two-dimensional region coexists with an unbounded one-dimensional channel. On this graph, normalized ground states do not exist for sufficiently small or sufficiently large masses, showing that the one-dimensional component can dominate the variational problem and induce loss of compactness. Finally, we show that this mechanism is sensitive to the local geometry of the graph: suitable additional defects preserve small-mass nonexistence but restore the existence of ground states for large masses. Our analysis combines small-mass energy asymptotics with a large-mass blow-up argument.

We investigate the spectral statistics of random perturbations of semiclassical Schrödinger operators on compact manifolds, near energy levels corresponding to chaotic classical dynamics. The prototypical example is that of an operator $h^2\Delta+V(x)$ that is perturbed by a smooth potential $h^\alpha V_\omega$ with $\alpha\in (0,1)$, and $V_\omega$ is a random potential that decorrelates on distances $h^\beta$, with $0<\beta<2\alpha$. We show that for a generic perturbation, the spectral fluctuations of the smoothed counting function of eigenvalues obey a universal behavior at a certain mesoscopic scale, which is coherent with the predictions of random matrix theory.

Part I: Semiclassical Gravity Efficiently Solves $\mathsf{NP}$-Complete Problems Assuming the gravitational field is classical and that it couples to quantum fields via the semiclassical Einstein field equations (EFEs), we show that the weak-field dynamics of a massive and non-relativistic qubit can in principle be used to solve an $\mathsf{NP}$-complete problem in polynomial time. We attribute this vast computational power to the non-linear qubit dynamics afforded by the semiclassical EFEs. Consequently, the above two assumptions entail a violation of the Physical Extended Church–Turing Thesis, which we regard as evidence for the quantization of gravity. Part II: Spacetime Quasicrystals We generalize self-similar quasicrystals, such as the Penrose tilings, from Euclidean space to Minkowski spacetime. We construct the first examples of Lorentzian quasicrystals and identify their key novel features. We then explore possible connections to fundamental physics. First, we propose a speculative scenario in which our $(3+1)$D universe is embedded in a particularly symmetric $(9+1)$D torus $\mathbb{T}^{9,1}$, previously identified as the most symmetric toroidal compactification of the superstring. We suggest that this construction may shed light on the seesaw relation $M_{\rm Pl}M_{\rm vac} \approx M_{\rm EW}^{2}$ between the Planck, vacuum-energy, and electroweak scales. Second, we replace the random causal sets of causal set theory with ordered, highly symmetric quasi-crystalline causal sets. We find suggestive evidence that our quasicrystals outperform Poisson random sets in the number-volume correspondence in both $(1+1)$D and $(3+1)$D, where such behaviour has been conjectured to be impossible. Finally, extending hyperuniformity from space to spacetime, we show that our $(1+1)$D quasicrystal is the first demonstrated example of a spacetime-hyperuniform structure.

The main result of the paper states that if $0\neq v\in\mathbb C^m\otimes\mathbb C^n$, $d=\min\{m,n\}$, and $p=SR(v)$, then for every $1\le q\le d$, \begin{equation*} \min\left\{OSR(\mathbf U): \begin{array}{l} \mathbf U\in\mathbb M_m\otimes\mathbb M_n\text{ is unitary}, SR(\mathbf Uv)=q \end{array}\right\} = \left\lceil\max\left\{\frac pq,\frac qp\right\}\right\rceil . \end{equation*} Here $OSR(\mathbf U)$ denotes the operator Schmidt rank of $\mathbf U$. Furthermore, if $\mathbf U\in\mathbb M_m\otimes\mathbb M_n$ is unitary and $OSR(\mathbf U)\le k$, then, for every unit vector $x\in\mathbb C^m\otimes\mathbb C^n$ and every $1\le r\le d$, \begin{equation*} F_{\min\{kr,d\}}(\mathbf Ux)\ge F_r(x) \quad\text{and}\quad F_{\min\{kr,d\}}(x)\ge F_r(\mathbf Ux), \end{equation*} where $F_r(x)$ denotes the sum of the $r$ largest squared Schmidt coefficients of $x$. This result is stronger than rank reachability as it controls the entire cumulative Schmidt profile. We also establish a majorization relation and use it to obtain sharp bounds on Schatten-norm distortion and on the change of Rényi entropies under unitaries with bounded operator Schmidt rank.

A KMS-symmetric quantum Markov semigroup is completely determined by the quadratic form in induces on the GNS Hilbert space. The quadratic forms that arise in this way are called quantum Dirichlet forms. In this article we give a complete description of bounded quantum Dirichlet forms in terms of derivations. In particular, we exhibit an explicit characterization of the bounded quantum Dirichlet forms on type I factors. We also show that any unbounded quantum Dirichlet form can be represented in terms of derivations. The key technical insight is that the modular group always restricts to a strongly continuous group on the domain of a quantum Dirichlet form such that the square root of the analytic generator is accretive with respect to the form norm.

We study graph transformations and their effects on the eigenvectors of three fundamental graph matrices: the adjacency matrix $A$, the signless Laplacian $Q$, and the Laplacian $L$. We extend the Laplacian eigenvector edge and matching principles introduced by Merris to $Q$ and $A$. We also extend some of the known $L$ graph transformations to $Q$ and $A$ and prove that graphs admitting a common eigenvector $\mathbf{x}$ for $A, L$ and $Q$ are such that vertices corresponding to nonzero components of $\mathbf{x}$ have same degree. Following our previous work for $L$, we investigate bivalent graphs for $A$ and $Q$, that is graphs admitting an eigenvector with all components in $\{-1, 1\}$. In particular, for trees, we characterize $Q$-bivalence and establish that $A$-bivalence forces all vertex degrees to be odd.

In this paper, we establish the absence of absolutely continuous spectrum for scalar one-dimensional Laurent operators with multiplicative disorder and critical long-range hoppings. In particular, the requirement of at least quartic off-diagonal decay rate in the existing theory is bypassed by exploiting the finite-difference bound of the convolution kernel, which is available even when the interaction decays only as $|n|^{-1}\log^{-\beta}|n|$ with $\beta>1$. We then apply the abstract main result to a concrete physical model, the capacitance operator in subwavelength physics. For a chain of three-dimensional resonators with one resonator per period, we verify the required decay and finite-difference estimates and conclude the absence of absolutely continuous spectrum. These results leave open the distinction between pure point and singular continuous spectrum, but constitute a key step in proving Anderson localization in wave systems with critical long-range interactions.