Papers

Let $\mathbf{c}=(\lambda,\alpha,\mu,\nu)\in\mathbb{R}^{4}$ denote the four running couplings of four-dimensional nonplanar Euclidean scalar $\varphi^{4}$ theory, and let $B(\mathbf{c})=\sum_{p\ge2}\sum_{|\mathbf{m}|=p}\beta(\mathbf{m})\mathbf{c}^{\mathbf{m}}$ be its formal beta-function series. Assuming the factorial coefficient bounds $|\beta(\mathbf{m})|\le (p-1)!C^{p-1}$ for $|\mathbf{m}|=p$, we construct an explicit summation rule $\mathcal{S}$, given by a cut Borel–Laplace transform whose cut is placed by a scale-covariant real-analytic gauge. We prove that it assigns a well-defined real value to $B$ on an explicit nonempty open domain, in fact on all of $\mathbb{R}^{4}$. The summed function $\mathcal{S}B$ is linear and real, real-analytic on $\mathbb{R}^{4}\setminus\{0\}$, $C^\infty$ on $\mathbb{R}^{4}$ with Taylor series $B$ at the origin, and Gevrey-$1$ asymptotic to $B$ to all orders with an explicit remainder bound. At optimal truncation the remainder is $O(e^{-\eta/(C\|\mathbf{c}\|_\infty)})$ for any $\eta<1$. We show that $\mathcal{S}B$ reproduces the classical Borel sum and the ordinary sum of a convergent series up to errors of the same exponentially small order. Exact reproduction is impossible for any linear rule defined on the whole admissible class: regularity forces all weights to equal $1$, leading to divergence. Finally, we prove optimality under the stated hypothesis. Admissible series may diverge at every $\mathbf{c}\ne0$, and their Borel transforms may have the circle $|\tau|=1/(C\|\mathbf{c}\|_\infty)$ as a natural boundary, so classical Borel-type summation need not be available on the class. Moreover, there exist two functions indistinguishable at the level of the coefficient bounds yet differing by exactly $(2\pi/C)e^{-1/(C\|\mathbf{c}\|_\infty)}$. All constants are explicit and all arguments are self-contained.

We develop a phase space description of large $N$ Hermitian one matrix models directly from the recursions satisfied by the orthogonal polynomials. In the one-cut phase, the recursion lattice naturally gives rise to a semiclassical action-angle pair, which can be mapped canonically to eigenvalue and momentum variables. We show that the resulting momentum profile is determined by the density of zeros of the orthogonal polynomials, and that the same phase space structure is reproduced from the Wigner transform of the Christoffel-Darboux projector and from the planar spectral curve. The Gaussian model provides a simple consistency check of the proposal. We then extend the construction to the symmetric quartic two-cut phase, where a period-two Jacobi recursion produces two Bloch bands and two disconnected phase space components with actions given by the partial 't Hooft couplings. Finally, we outline the finite-gap generalization appropriate to multicut phases.

Pseudorandom structures are structures that behave like random ones, without necessarily being random themselves. It is a prominent, and evidently very difficult, conjecture that $$ V(n) = \lambda \cos \Big( 2 \pi \Big( x_1 + n x_2 + \frac{n(n-1)}{2} \alpha \Big) \Big) $$ is pseudorandom from a Schrödinger operator perspective in that the Schrödinger operator in $\ell^2(\mathbb{Z})$ with potential $V$ displays Anderson localization, that is, pure point spectrum with exponentially decaying eigenfunctions for almost all parameter values — the same spectral features as those produced by random potentials. We show that $V$ is pseudorandom in terms of its diffraction properties, that is, the associated diffraction measure is purely absolutely continuous for all $\lambda \not= 0$, all irrational $\alpha$, and all $x_1,x_2$ — which is the case as well for the random case. Our result gives further evidence for the conjecture in the Schrödinger case and it elucidates the apparent dual behavior of Schrödinger spectral measures and diffraction measures.

Planar $k$-uniform states have maximally mixed reductions on every $k$-site interval of a ring. We introduce planar $k$-purity—the mean purity of those intervals—as a faithful cost function for this simultaneous constraint. For a Haar-random pure state of $n$ parties with local dimension $p$, the joint purity of two intervals depends only on their overlap. An exhaustive four-replica calculation therefore yields the complete cyclic covariance kernel and a closed variance for every $1\le k\le\lfloor n/2\rfloor$. In the balanced qubit case, $k=\lfloor n/2\rfloor$ and $N=2^n$, the variance is asymptotic to $20/(3nN^2)$ for even $n$ and $6/(nN^2)$ for odd $n$. A site-factorized permutation representation gives all raw moments and permits exact finite-sum evaluation of the skewness. Fixed-seed simulations validate nonbalanced qubit and qutrit cases, while balanced-qubit simulations through $n=10$ validate the parity formulas and finite-size skewness. Balanced planar and absolute balanced purity have the same Haar mean but different fluctuations; the planar variance is approximately $2.83$ times larger at $n=10$. Thus subsystem incidence, although invisible to every one-cut marginal distribution, controls the collective fluctuations of geometrically related cuts. Together with universal attainability and a linear number of interval constraints at balance, this provides a geometry-aware benchmark for multipartite entanglement beyond AME existence regimes.

We prove a similarity result for non-self-adjoint non-elliptic quadratic $\mathcal{PT}$-symmetric operators with real spectrum satisfying a dynamical averaging condition. In particular, we show that such an operator is weakly similar to a self-adjoint operator precisely when the fundamental matrix of the associated quadratic form is diagonalizable.

We define a relative $\mathbb{Z}_2$-index $\mathcal{N}(\omega_1,\omega_2)$ for locally-comparable, parity-invariant pure states on a unital $C^*$-algebra. We prove that the index is well-defined, multiplicative, invariant under parity-preserving automorphisms, and locally constant in the norm topology. For the one-dimensional self-dual CAR algebra, we apply this construction to the half-chain automorphism $\sigma$ and define the many-body Majorana number $\mathcal{N}\sigma(\omega):=\mathcal{N}(\omega,\omega\circ\sigma)$ for parity-invariant $\sigma$-local pure states. In the quasi-free Hilbert–Schmidt regime, this index agrees with the usual single-body Majorana number. We then introduce symmetric local automorphism paths and prove that $\mathcal{N}_\sigma$ completely classifies the resulting automorphic-path-components. This automorphic-path equivalence retains the bulk Majorana number but may forget a relative zero-dimensional parity obstruction.

In this paper we consider some general aspects of Euclidean conformal field theories on curved spaces of dimension $D\ge 3$. We first look at the OPE of scalar primary fields on conformally flat spaces. Extending the results of \cite{Konechny:2026bqg}, we give a general construction of the descendants' contributions to the OPE in the scalar channel. We then discuss CFTs on non-conformally flat spaces in the ambient space formalism. We investigate the short-distance behaviour of three-point functions on such spaces using the general ansatz proposed by Parisini, Skenderis, and Withers \cite{Parisini:2022wkb, Parisini:2023nbd}. We find that some additional corrections need to be added to the ansatz to ensure the existence of a local covariant OPE.

Leading eigenvalues of the truncated propagator, known as Ruelle-Pollicott (RP) resonances, are an elegant way of addressing the dynamics of unitary many-body systems. We study unitary propagators in their canonical form, known in the mathematical literature as the CMV matrices, and obtain a number of exact results for RP resonances and the associated norm-diverging eigenvectors. For the simplest CMV class describing a unilateral shift with an impurity, motivated by operator dynamics in dual-unitary circuits, we obtain closed-form results and in particular show that the three independent ways of obtaining RP resonances – the truncated propagator, analytic continuation of the resolvent, and the rigged Hilbert space approach – all give the same results. In more realistic CMV matrices, in which shift-like operator dynamics characteristic of chaotic systems is only asymptotic, we rely on the rich theory of orthogonal polynomials on the unit circle and identify two phases. In the first phase, relaxation occurs due to local operators effectively evolving into increasingly nonlocal ones with negligible backflow. Especially interesting is the second phase, which, surprisingly, exhibits faster relaxation because of contributions from the backflow of large operators. Additionally, in the second phase, RP resonances are not equal to the eigenvalues of the truncated propagator, instead, they are “hidden” within a ring of ill-conditioned eigenvalues.

A class of moving boundary problems of Stefan-type for an extension of the 2+1- dimensional Gardner equation is shown to be amenable to exact solution by application of a Ermakov- Painlevé II symmetry reduction.

Markovian open-system dynamics have widespread applications throughout quantum information science, including algorithmic state preparation. Their convergence is commonly quantified using the worst case global trace distance between the evolving and stationary states. However, this criterion can be unnecessarily stringent when only physically relevant observables are of interest. Here we introduce and study observable-specific mixing times. We prove that, for quasi-local, rapidly mixing Lindbladians, sums of geometrically local observables equilibrate in a time independent of system size, in contrast to the logarithmic dependence of global state mixing. This separation reduces the runtime of dissipative quantum algorithms, including quantum Gibbs samplers, for estimating quantities such as the Gibbs state energy and local order parameters, yielding an overall scaling that is linear in system size. Complementing this quantum result, we develop a quantum-inspired classical algorithm for estimating the same quantities. Its runtime is likewise linear in system size, but scaling exponentially in $\mathcal{O}\big(\log(1/\epsilon)^D\big)$, where $D$ denotes the spatial dimension of the lattice. We further analyse non-interacting Lindbladians over qudits, fermions, and bosons, demonstrating that locality of observables is not always necessary for a qualitatively faster mixing. Small-scale simulations of quantum Gibbs samplers reveal no large hidden constants in our asymptotic analysis and show that the theoretical predictions closely capture the finite-size dynamics.

We study the quantum transport of Dirac electrons in graphene subjected to a polyadic Cantor-structured electrostatic potential. Using the superperiodic potential formalism, we obtain a closed-form expression for the transmission probability. As the Cantor stage increases, the transmission spectrum evolves from a sparse set of superlattice resonances to a near-transparent regime, with the polyadic order setting the rate of this evolution. The angular response depends on the doping configuration, showing distinct behavior in the $n$–$n$–$n$, $n$–$p$–$n$, and Dirac-point cases. In the near-transparent regime, the transmission follows double-logarithmic scaling laws with respect to four independent control parameters: the Cantor stage, the potential height, the initiator length, and the angle of incidence. By combining these individual scaling relations, we establish a unified scaling law governing quantum transport. These results show that the hierarchical self-similarity of the potential governs the transport properties of such systems.

We develop a continuum theory for a fluid of elongated particles that advance along their own axes. The same kinematics is ideally shared by flocks of sheep, self-propelled rods, vehicular traffic, and turning flocks of birds. Within the framework of continua with vectorial microstructure, in which each point also carries an orientation, we impose the no-side-slip (skate) condition $\vv=u\,\nn$ as an ideal, non-integrable internal constraint. We derive the pure equations of motion and the equation governing the constraint reaction from the principle of virtual power. Using a constitutive closure provided by the Ericksen-Leslie theory of nematic liquid crystals, we show how the constraint also shapes collective effects. It turns parabolic orientational diffusion into hyperbolic orientation-density waves and forbids any steady simple shear. It fixes the empirical Toner-Tu convective coefficient to the flow-alignment ratio measured in colloidal rollers, thereby providing a mechanical foundation for angular sound in micropolar active hydrodynamics. It also recovers the inertial spin model of bird flocks, with the addition of a banking force and a turn-density coupling. Finally, it recovers the classical macroscopic models of one-dimensional traffic flow, while describing steering, lateral tyre forces, and road geometry in two dimensions.

Let $\alpha>0$ and restrict a finite inverse-power potential with arbitrary real coefficients to a line. Combine terms having the same projected centre and squared height $\left(a, b^2\right)$, delete classes whose coefficient sum is zero, and let $m$ be the number of remaining classes. We prove the following dichotomy. If $m=0$, exact cancellation occurs if and only if every class sum vanishes, and every ordinary point of the original domain is critical. If $m \geq 1$, there are at most $2 m-1$ critical points: when all effective heights are positive the zeros are counted with analytic multiplicity, while in the presence of effective sources on the line the assertion is one global distinct-point bound. This proves the signed line conjecture of Gabrielov-Novikov-Shapiro, valid throughout their range and in fact for every $\alpha>0$. The structural input is a projective paired Haar theorem: for finite $\beta>1$, the full $2 m$-dimensional space $\sum L_j / Q_j^\beta$, with pairwise nonproportional positive-definite binary quadratics and arbitrary real linear numerators, has at most $2 m-1$ projective zeros counted with multiplicity. For positive charges, the substitution $p=2 \alpha$ proves Conjecture 3 of Edelsbrunner-Fillmore-Oliveira throughout its stated range $p \geq 1$ and extends the same conclusion to every $p>0$. For every $n$, an explicit positive $n$-charge configuration attains $2 n-1$ simple critical points.

We prove explicitly the n-point K-identities we proposed previously in the calculation of one tensor hard string scattering amplitudes (HSSA). These K-identities were the key to obtain the stringy scaling behavior in the saddle point calculation of the n-point HSSA and, on the other hand, to consistently match with the calculation of decoupling of zero norm states. Moreover, we introduce a G function to generate an infinite set of generalized K-identities (GKI). The lowest order set of these GKI is the scattering equations (SE) used in the calculation of field theory amplitudes in the CHY formalism. The next to leading order set of these GKI is the K-identities used previously in the calculation of one tensor HSSA. We expect that the higher order sets of these GKI can find applications in calculating other HSSA.

The reduction of mechanical presymplectic structures and its application to the reduction of time-dependent Hamiltonian systems were developed in a recent paper (Gutiérrez Sagredo et al, 2025). This approach overcomes some limitations of the original Albert reduction (Albert, 1989). In this paper we will describe the reduction of time-dependent Hamiltonian systems using the extended formalism. This process combines cotangent bundle reduction with reduction of presymplectic structures of corank 1 and 2. These results generalize previous work (Lacirasella et al, 2012), in the same direction. Several examples will be presented to illustrate the theory from this new perspective.

In this paper, we introduce a generalized framework for star products that preserve associativity while not necessarily obeying the canonical commutation relations. Within this framework, we formulate the augmented star product and investigate the associated augmented star exponentials. We further demonstrate several applications of these constructions to special functions and to problems arising in quantum physics.

Spin-boson models are paradigmatic examples of open quantum systems and serve as the theoretical paradigm for current quantum computers based on single-ion trap technology. Despite their ubiquity, a complete spectral diagonalization of these models remains an open problem, except in highly singular regimes. This paper establishes a rigorous framework for the approximate diagonalization of generalized spin-boson systems. Our approach is inspired by the Brockett-Wegner double-commutator flow, which is a non-linear differential equation governing the evolution of (here unbounded) operators. Unlike relatively recent applications of this flow to quadratic Hamiltonians in quantum field theory, it does not close in the spin-boson context. We overcome this fundamental obstruction by performing a detailed analysis of the resulting non-closed algebraic structure, allowing us to explicitly bound the higher-order error term with respect to the spin-boson coupling strength. Consequently, this work provides the first mathematically rigorous justification for several heuristic diagonalization techniques widely employed in the theoretical physics literature for small-coupling regimes, in the simplest non-trivial cases. More broadly, our framework renders a flow-based algorithm feasible for systematic higher-order diagonalization and self-energy renormalization. This strategy is conceptually akin to multi-scale analysis, or, much more recently, to the iterative, local Lie-Schwinger block-diagonalization method by Fröhlich and Pizzo.

This paper presents a complete classification of the formality of configuration spaces of $\mathbb{R}^{n'} \times \mathbb{C}^{n}$. We define a constructible de Rham-Dolbeault cohomology theory which provides a constructible CDGA (commutative differential graded algebra) model of $\Conf_m(\mathbb{R}^{n'} \times \mathbb{C}^{n})$. For $(n'=0,n\ge2)$ or $(n'=1,n\ge1)$, the CDGAs are non-formal. For $n'\ge2,n\ge1$, we establish an explicit quasi-isomorphism between the constructible CDGA and its cohomology by using a diagrammatic CDGA of admissible diagrams and a regularized configuration space integral, which leads to the formality. As an application, we show that the local operator algebra of a topological-holomorphic field theory on $\mathbb{R}^{n'} \times \mathbb{C}^{n}$ ($n'\ge2,n\ge1$) is homotopically equivalent to a higher dimensional analog of vertex algebras.

The lattice Green's functions of the cubic lattices are classical: each is a symmetric square of a second-order operator, with a closed form in complete elliptic integrals. The hyperkagome lattice, realized by the iridium sublattice of the spin-liquid candidate Na4Ir3O8, has resisted such a treatment: Varma and Monien reduced its density of states to a threefold integral they found no way to solve exactly. From exact lattice moments we obtain an irreducible third-order linear differential (Picard-Fuchs) operator annihilating the Green's function. It is not a literal symmetric square, yet its symmetric square carries a rational solution: the differential Galois group is orthogonal, and an order-two differential intertwiner makes the operator projectively a symmetric square. Our main result is that the underlying second-order operator is the uniformizing equation of the genus-zero modular curve of Gamma_0(30)+, the level-thirty group generated by Gamma_0(30) and all its Atkin-Lehner involutions, an explicit eta quotient parametrizing the natural variable. That uniformizing equation is itself not new - it is a row of the Lian-Yau table of genus-zero groups - so what is new here is the identification of a lattice Green's function with it, together with a proof: the Schwarzian identity is established by an a priori pole-degree bound rather than checked numerically, which also proves the tabulated entry. The hyperkagome Green's function is therefore modular at level thirty, supplying the closed form Varma and Monien sought. Its weight-two period is obtained explicitly: a depth-one quasimodular form which we prove is not a modular form times an algebraic function at any weight.

We consider a class of discrete multi-time determinantal point processes whose correlation kernels admit a double-contour integral form, containing Schur processes as simple examples. We show that the form of the correlation kernel is preserved under conditioning of the point process to a restricted domain, however, with an integrand which becomes more complicated and which is characterized by a Riemann-Hilbert problem. We apply our general result to domino tilings of reduced Aztec diamonds and to lozenge tilings of hexagons with holes. Our results show a striking analogy with the Its-Izergin-Korepin-Slavnov method for integrable kernels.

We study a two-particle lattice Schrödinger operator describing two identical fermions on the one-dimensional lattice $\mathbb Z$ with nearest-neighbor interaction of strength $\lambda\in\mathbb R$. Using the direct-integral decomposition with respect to the total quasi-momentum $k\in\mathbb T:=(-\pi,\pi]$, we reduce the problem to fiber operators $H_\lambda(k)$ acting in the odd relative-coordinate space. For $k\in(-\pi,\pi)$, the essential spectrum is \[ \sigma_{\mathrm{ess}}(H_\lambda(k)) = [4-4\cos(k/2),\,4+4\cos(k/2)]. \] We give a complete description of the spectral transition at both edges. The operator has a unique simple eigenvalue outside the essential spectrum if and only if \[ |\lambda|>2\cos(k/2), \] in which case \[ E(k,\lambda) = 4+\lambda+\frac{4\cos^2(k/2)}{\lambda}. \] For $|\lambda|<2\cos(k/2)$ there is no discrete eigenvalue. At the critical coupling $|\lambda|=2\cos(k/2)$, the eigenvalue merges with the corresponding spectral edge and becomes a threshold resonance, with a bounded non-square-integrable odd solution. We also derive the threshold and strong-coupling asymptotics of the eigenvalue and eigenfunction. In the strong-coupling regime the eigenfunction localizes at the interaction sites, whereas at critical coupling it converges pointwise to the corresponding resonant solution. The exceptional fiber $k=\pi$, where the hopping vanishes and the essential spectrum collapses to $\{4\}$, is treated separately.

We construct regular, spherically symmetric, compact energy-cutoff equilibria for gravitational Vlasov–Poisson systems sourced by positive and negative mass distributions in the context of stellar dynamics. Motivated by Bondi's notion of negative mass and its relation to the weak field limit of Einstein gravity with a signed mass–energy source, we show that the internal structure of the steady states depends qualitatively on the type of negative mass present. In the Bondi convention, the signs of inertial, passive gravitational, and active gravitational mass are reversed together, preserving the equivalence principle. The resulting equilibria consist of a central overlap core containing both mass species, surrounded by a finite positive mass shell and an exterior vacuum. Thus, although a pure Bondi negative mass gas cannot form a compact steady state, Bondi negative mass can be spatially confined by a suitable positive mass distribution. In the gravitational-charge convention, only the passive and active gravitational masses change sign, so that the two species obey opposite free-fall laws and unlike masses repel. The resulting equilibria are spatially segregated into a negative mass core, a vacuum gap, a positive mass shell, and an exterior vacuum. For cutoff exponent $n=-1/2$, the radial matching is explicit, while for the regular cutoff $n=1/2$ the matter regions satisfy Lane–Emden-type equations. These results provide a kinetic framework for investigating the distribution of negative mass in astrophysical and cosmological settings.

We study the holomorphy of the ground state and its energy with respect to the coupling constant for bosonic quadratic Hamiltonians. We prove general theorems establishing this holomorphy, thereby providing a rigorous justification for formal perturbative expansions. This theory is applied to several concrete models, including the single pair interaction model and the Pauli-Fierz model in the dipole approximation. Importantly, for the Pauli-Fierz model with a realistic ultraviolet cutoff, the perturbation expansions in the elementary charge converge at the physical charge value, even when mass renormalization is taken into account. Furthermore, we explicitly determine the radii of convergence of the ground states and their energies for both the single pair interaction model and the fibers of the translation-invariant Pauli-Fierz model.

Exact cumulants of entanglement entropies of random state ensembles have traditionally been studied within the random matrix framework. In this work, we propose an alternative approach based on the intrinsic connection to integrable systems. The central idea is to embed entropic quantities into tau functions satisfying Toda-type lattice equations, which in turn yield linear difference equations for their averages. Directly solving the difference equations recovers exact entropy formulas in the literature. The integrable systems approach bypasses the case-by-case, ensemble-dependent derivations required by random matrix methods. The approach also suggests a possible route towards unified and more efficient higher-order cumulant calculations by exploring integrable hierarchies.

We study the continuum scaling limit of the Langevin dynamics for three-dimensional U(1) lattice Yang–Mills theory. The model is defined on the discrete 3D torus with a general class of plaquette actions that are suitably normalized, including Wilson, Manton, and Villain actions. Under the weak-coupling scaling and in the DeTurck gauge, we prove that the rescaled logarithmic field converges to the solution of the one-form stochastic heat equation in probability. In particular, the limiting dynamics are universal and do not depend on the higher-order details of the plaquette action.

Perturbed plane waves are fundamental objects in scattering theory on Euclidean space and asymptotically Euclidean spaces. In this paper, we investigate the structure of perturbed plane waves in the $\textit{forward}$ direction, in which the outgoing spherical wave is typically singular and conjoined to the incoming plane wave. Melrose & Zworski provided a microlocal description (in the more general setting of asymptotically conic manifolds) using their notion of Lagrangian distributions associated to pairs of intersecting Legendrian submanifolds, on the way to proving that the S-matrix is an FIO. Here, we revisit the problem in the asymptotically Euclidean case, for which the oscillatory integrals used by Melrose–Zworski attain their most complicated form (relative to the more general asymptotically conic case). We seek a more elementary description in terms of physical-space asymptotics. These are specified using a two-faced compactification $X\hookleftarrow \mathbb{R}^d$, with one face for each asymptotic regime. We prove full polyhomogeneity. A transport equation arises as a model problem at the main face (`bf'). The quantum inverted harmonic oscillator arises as a model problem at the front face (`ff').

Motivated by the geometric interpretation of spatially homogeneous cosmological models, we formulate a spacetime closedness theorem directly at the Lorentzian level. A classical space-form classification theorem organizes homogeneous and isotropic spatial geometries, but by itself it does not yield a Lorentzian statement about the ambient spacetime. The main technical step is to derive finite slice-volume from genuinely Lorentzian control hypotheses on the foliation. This is achieved in a strong version, for maximally regular globally hyperbolic $(n+1)$–spacetimes with a finite-time Big Bang and homogeneous complete spacelike slices, and in a weaker version in which maximal regularity is replaced by time-integrability of the accumulated expansion rate. In both cases, the argument separates an analytic step, deriving finite slice-volume from the Big Bang and temporal control, from a geometric step, upgrading finite volume to compactness by homogeneity and completeness. The theorem is stated in arbitrary spacetime dimension and is accompanied by a Lean~4 formalization of the strong and weak abstract statements.

Anticoncentration and entanglement are two properties used to study classical hardness or easiness of Gaussian boson sampling in varying regimes. In this paper, we consider three directions concerning anticoncentration and entanglement in Gaussian boson sampling: (1) We derive closed-form expressions for the second moment of hafnians of symmetric Gaussian products, and use this to precisely locate the anticoncentration transition as a function of the number of squeezed input modes. (2) We derive closed-form expressions for the Rényi-$\alpha$ Page curves for Gaussian boson sampling. (3) We study unequal input squeezing parameters $(s_i)_i$, and prove estimates as well as monotonicity of the average-case Rényi-2 entropy Page curve in terms of the magnitudes $|s_i|$, which allow for extending equal squeezing results to unequal squeezing.

We provide a rigorous Laughlin-pump argument that establishes the fractional quantization of the Hall conductance directly from the Laughlin state as a topological index. Our proof uses a recently defined index of a pair of pure states together with an infinite matrix product state analysis that is rigorous on a sufficiently thin cylinder. By virtue of the connection to an index, our result implies, in particular, that Laughlin states are representatives of topologically stable fractional quantum Hall phases.

In this thesis, we study the spectral properties of dynamically defined aperiodic metric and discrete graphs. Our goal is to determine to what extent spectral properties of discrete one-dimensional ergodic Schrödinger operators persist when the aperiodicity is manifested through the geometry rather than through a potential. The graphs considered here are inspired by one-dimensional aperiodic tilings, and are called tiling graphs and decorated $\mathbb{Z}$-graphs. For a large family of metric tiling graphs equipped with the standard Laplacian, we show that the spectrum is of zero Lebesgue measure, and is a generalized Cantor set up to a possible discrete set of energies. For decorated $\mathbb{Z}$-graphs, we further show that for a Baire-generic and Lebesgue almost-sure choice of the decoration edge lengths, the spectrum is a generalized Cantor set. We then study the integrated density of states (IDS) for metric and discrete decorated $\mathbb{Z}$-graphs. We prove a gap labelling theorem, which characterizes the set of possible values taken by the IDS inside spectral gaps. We show that the gap labels are contained in the Schwartzman group associated with the dynamical system generating the graph, up to a geometric scaling factor. Lastly, we consider the Dry Ten Martini Problem for discrete Sturmian decorated $\mathbb{Z}$-graphs, asking whether all possible values predicted by the gap labelling theorem are indeed attained by the IDS inside spectral gaps. We answer this question negatively, by identifying a large set of gap labels which are not attained due to jump discontinuities of the IDS. We then show that away from these jump discontinuities, the periodic approximants for Sturmian graphs display the same combinatorial structure as the standard Sturmian Hamiltonians, and use this to obtain an explicit characterization of the realized gap labels for Sturmian comb graphs.