Siddhant Midha, Yifan F. Zhang, Daniel Malz, Dmitry A. Abanin, Sarang Gopalakrishnan
Apr 23, 2026·quant-ph·PDF Belief propagation has recently emerged as a powerful framework for evaluating tensor networks in higher dimensions, combining computational efficiency with provable analytical guarantees. In this work, we develop the first end-to-end theory of tensor network belief propagation for a class of projected entangled pair states satisfying \emph{strong injectivity}. We show that when the injectivity parameter exceeds a constant threshold, BP fixed points can be found efficiently, and a cluster-corrected BP algorithm computes physical quantities to $1/\mathrm{poly}(N)$ error in $\mathrm{poly}(N)$ time for an $N$ qubit system. We identify a striking phenomenon we term \emph{algorithmic locality}: local perturbations of the tensor network affect the BP fixed point with an influence decaying rapidly with distance. As a result, updates to the fixed point after a local perturbation can be carried out using only local recomputation. Moreover, through the cluster expansion, this locality extends to observables, implying that local expectation values can be approximated from local data with controlled accuracy. Our results provide the first rigorous guarantee for the effectiveness of tensor-network belief propagation on a wide class of many-body states, bridging a gap between widely used numerical practice and provable algorithmic performance.
Arunaday Gupta, Baisong Sun, Xi He, Bei Zeng
Apr 23, 2026·quant-ph·PDF Exact quantum codes detecting a prescribed set of Pauli errors are approached through algebraic constructions--stabilizer, codeword-stabilized, permutation-invariant, topological, and related families. Geometrically, exact Pauli detection is governed by joint higher-rank numerical ranges of these Pauli operators, whose structure for rank $\geq 2$ is largely uncharted. From this viewpoint, we show that such codes often form connected continuous families rather than collections of disjoint solution regions. These families are characterized by a single scalar derived from the Knill-Laflamme conditions: denoted $λ^*$, it is the Euclidean norm of the signature vector of Pauli expectation values on the maximally mixed code state, and provides a one-parameter summary of the code's joint Pauli variance profile. Within these continuous landscapes, stabilizer codes occupy only discrete, measure-zero subsets of the attainable $λ^*$-spectrum, exposing a largely unexplored continuum of genuinely nonadditive exact codes. We establish this picture by analyzing the geometry of higher-rank operator compressions, and extend it to symmetry-restricted settings where cyclic and permutation symmetries are imposed on both the error model and the code projector. Small-system cases reveal interval, singleton, and empty regimes through eigenvalue interlacing and symmetry-sector decompositions; larger systems are treated numerically via Stiefel-manifold optimization and symmetry-adapted parameterizations. In every unrestricted and symmetry-compatible case analyzed, the attainable $λ^*$-spectrum forms a single closed interval whenever nonempty--although a general proof remains open. These results place stabilizer, symmetric, and nonadditive code families within a unified higher-rank variance framework, suggesting a continuous geometric perspective on the landscape of exact quantum codes.
Sudip Halder, Jaume de Haro, Supriya Pan, Emmanuel N. Saridakis, Tapan Saha, Subenoy Chakraborty
This article opens new window to obtain accelerating scaling attractors without any need of dark energy. We study cosmological dynamics in a two-fluid system where pressureless dark matter (DM) undergoes adiabatic particle creation and exchanges energy with a barotropic fluid. Considering six widely used interaction prescriptions, we formulate the corresponding autonomous systems in a compact phase space and perform a unified dynamical analysis. We find that accelerating scaling attractors, namely late-time states where both fluids coexist with fixed energy fractions, arise only when the interaction is controlled by the DM density and energy flows from DM to the second fluid. Such attractors appear in the global and local DM-based interactions, and in the global mixed case, but are entirely absent when the interaction depends on the second fluid or on local mixed terms, which instead drive the universe to a DM-dominated accelerating phase. These results clarify the unique conditions under which matter creation can mimic dark-energy-like behaviour without introducing a dark-energy component.
Nikita Kalinin, Ernesto Lupercio, Mikhail Shkolnikov
We define an $\operatorname{SL}_n(\mathbb{Z})$-invariant tropical zeta function of a convex domain. In dimension 2 it admits boundary Dirichlet-series representation with summands indexed by Farey pairs. For $C^3$ strictly convex domains, it extends meromorphically to $\Re(s)>3/5$, holomorphic there except for a simple pole at $s=2/3$, with residue proportional to equiaffine perimeter. A Tauberian argument yields the $t^{1/3}$ wave-front lattice-perimeter asymptotic for $t\rightarrow 0$.
Moritz Dober, Alexander Glazman, Sébastien Ott
The $q$-state Potts model is an archetypical model for various types of phase transitions. We consider it on the square grid and focus on the regime where it undergoes a discontinuous transition, that is $q>4$. At the transition point $T_c(q)$, there are exactly $q+1$ extremal Gibbs measures (pure phases): $q$ ordered (monochromatic) and one disordered (free). This work establishes for the first time the wetting phenomenon in a precise geometric form and in the entire regime of discontinuity $q>4$: at $T_c(q)$, between two ordered phases a disordered layer emerges and, in the diffusive scaling, its boundaries converge to a pair of Brownian motions conditioned not to intersect. This is starkly different from the subcritical ($T<T_c(q)$) behaviour. At $T_c(q)$, previous results (Bricmont--Lebowitz '87, Messager--Miracle-Sole--Ruiz--Shlosman '91) were limited to the construction and properties of the surface tension for large enough $q$. In a companion work, arXiv:2502.04129, we provide a detailed study of the Potts model under order-disorder Dobrushin conditions. That work also develops a ``renewal picture'' à la Ornstein-Zernike for a suitable percolation model, which plays a central part in our study of the Potts interfaces. The latter is the random-cluster representation of an Ashkin--Teller model (ATRC), and is related to the Potts model via a chain of couplings going through the six-vertex model. In the current work, we extend the analysis to a pair of interacting order-disorder interfaces forming the separation between the two ordered phases, and couple them to a pair of well-behaved random walks conditioned not to intersect. The construction of the coupling is based on rigorously deriving entropic repulsion between the two interfaces. We also prove convergence of interfaces in the FK-percolation model at $p_c(q)$ when $q>4$.
Melchior Wirth
We establish a relation between the exponential decay rates of quantum Markov semigroups with respect to different inner products. More precisely, it was conjectured by Fagnola, Poletti, Sasso and Umanità that for a Gaussian quantum Markov semigroup, the exponential decay rate with respect to the KMS inner product is bounded below by the exponential decay rate for the GNS inner product. We show that this is indeed the case and not limited to Gaussian quantum Markov semigroups, but holds for quantum Markov semigroups with a faithful normal invariant state on arbitrary von Neumann algebras. Additionally, the KMS inner product can be replaced by a whole class of inner products induced by operator monotone functions.
Denis S. Grebenkov
Autocatalytic processes underlie diverse systems in which replication is triggered at interfaces, including heterogeneous catalysis on solid substrates, enzyme activity at membranes, viral infections, biofilm growth, and spatially structured ecosystems. In a typical scenario, particles move in a bulk medium and interact with surface regions, where they may either disappear or reproduce through branching, splitting or fission. Here, we develop a general theoretical framework to understand such surface-mediated autocatalytic processes. We show that the interplay between loss and replication at surfaces gives rise to rich population dynamics. For this purpose, we derive a renewal-type nonlinear integral equation for the generating function of the population size, providing access to its full probability distribution and statistical moments. We further establish an equivalent description in terms of a Fokker-Planck equation with nonlinear Robin-type boundary conditions that encode surface reactions. Our results identify distinct dynamical regimes and universal scaling laws, and provide a unified framework to predict when surface activity promotes extinction or explosive growth. These findings offer quantitative insight into catalytic efficiency, metabolic regulation, and population persistence in spatially heterogeneous environments.
Phan Thành Nam, Rongchan Zhu, Xiangchan Zhu
We derive the classical Gibbs measure on $\mathbb{T}^2$ associated with the fractional Bessel interaction potential $\widehat{v}_β(k)=\langle k\rangle^{-β}$ from a renormalized grand-canonical quantum Bose gas with the same interaction. Our result covers the whole range $\frac32<β\leq2$, where $\widehat{v}_β(k)$ is not summable and the quantum model cannot be written in the usual density-square form, as the associated self-energy diverges. We therefore need to renormalize the zero mode by a centered number-fluctuation term and then develop a detailed analysis for the high-frequency remainders. All this allows us to implement a low-frequency localization and obtain the convergence of the quantum relative free energy to the classical fractional-Bessel free energy, as well as the convergence of the reduced density matrices to the limiting Gibbs measure.
Kai Hippi, Félix Lequen, Søren Mikkelsen, Tuomas Sahlsten, Henrik Ueberschär
Let $-Δ_{\mathbb{H}}+V$ be the Schrödinger operator on $\mathbb{H}$ where $V \in L^p(\mathbb{H}) \cap L^\infty(\mathbb{H})$ for some $p > 0$. If $(X_n)$ is a uniformly discrete sequence of compact hyperbolic surfaces with a uniform spectral gap that Benjamini-Schramm converges to $\mathbb{H}$, we prove quantum mixing for the eigenfunctions of $-Δ_{X_n}+V_n$ in any sufficiently large spectral window $I$, where $V_n$ is the potential on $X_n$ induced by $V$. These apply to large degree lifts of a potential on a base surface such as congruence covers of arithmetic surfaces, with high probability to random hyperbolic surfaces in the Weil-Petersson model of large genus, and to Hartree one-particle operators arising in thermodynamic limit of many-body Bose gas on hyperbolic surfaces. The proof uses the Duhamel formula for the hyperbolic wave equation together with exponential mixing of the geodesic flow on $T^1 X_n$.
Vinayak M. Kulkarni
We develop a mathematically controlled framework for Yang--Baxter integrability in pseudo-Hermitian quantum impurity systems arising from periodic driving of a Dirac-like bath. The effective impurity Hamiltonian possesses a dynamically generated $\PT$ symmetry and exhibits exceptional points (EPs) where it becomes non-diagonalizable. We construct a Lax operator based on a rank-one biorthogonal projector associated with the impurity contact sector and prove that it satisfies an RLL relation within a projector algebra, leading to an $\etamet$-modified RTT structure and a commuting family of transfer matrices. The associated rational projector-type $R$-matrix satisfies the Yang--Baxter equation in the $\PT$-unbroken phase and extends continuously to the EP through a regularization of the biorthogonal projector. Within this framework we derive biorthogonal Bethe equations and show that the Gaudin matrix becomes defective at the EP, motivating the diagnostic $\mathcal{R}=κ(G)\,|\det G|$ that sharply separates EP singularities from Kondo criticality. We further prove that Bethe rapidities exhibit square-root coalescence and $\mathbb{Z}_2$ monodromy at the EP, reflecting the underlying Jordan structure, and that the effective pseudo-Hermitian Hamiltonian emerges from the periodically driven microscopic system with controlled operator-norm error $\mathcal{O}(1/Ω)$ via the Floquet--Magnus expansion.
Igor F. Herbut, Samson C. H. Ling
The internal low-energy symmetry of the massless Lorentz-invariant Dirac Hamiltonian in $2+1$ dimensions is known to be $O(2N)$, where $N$ is the number of two-component Dirac fermions. Here we point out that there exists an analogous internal symmetry of the single-particle quadratic-band-touching Hamiltonian in two spatial dimensions, and it is the unitary symplectic group, $USp(2N)$. All fermionic bilinears belong to one of the three small irreducible representations of this group. The interacting theory that respects the $USp(2N)$ symmetry and the spatial rotations is constructed and found to allow two independent interaction terms. When these interactions are infrared-relevant the symplectic symmetry either remains preserved or becomes spontaneously broken to $USp(N) \times USp(N)$. The symmetry in the lattices such as honeycomb to infinite order in the dispersion's expansion in powers of local momentum is given by the overlap of the symplectic and the orthogonal groups. We show that this overlap is $O(2N) \bigcap USp(2N) = U(N)$.
Sebastian Heller, Franz Pedit, Charles Ouyang
We construct the first smooth embedded compact special Legendrian surfaces in \(\mathbb S^5\) of genus greater than one. More precisely, for every sufficiently large integer \(k\), we construct an embedded special Legendrian surface whose conformal structure is the Fermat curve of degree \(k\) and genus \(\tfrac12(k-1)(k-2)\). Our approach combines an elementary implicit function theorem with the description of special Legendrian surfaces via loop algebra-valued meromorphic connections and a characterization of the unitarizability locus in the ${SL}_{3}(\mathbb C)$-character variety of the thrice-punctured sphere.
Alexander D. Popov
We consider two-dimensional harmonic oscillator in the complex Bargmann-Fock-Segal representation with $T^*{\mathbb R}^{2}={\mathbb C}^2$ as classical phase space. We show that the eigenfunctions $ψ_n$ of the quantum Hamiltonian correspond to complex radial coordinates in the reduced phase space ${\mathbb C}^2/{\mathbb Z}_n\subset{\mathbb C}^2$. They describe ${\mathbb Z}_n$-invariant motion of particle along a circle $S^1$ in lens space $S^3/{\mathbb Z}_n\subset{\mathbb C}^2/{\mathbb Z}_n$, where ${\mathbb Z}_n$ is the cyclic group of rotation by an angle $2π/n$ on the circle $S^1$, $n=1,2,...\,$. Thus the general solution of the Schrödinger equation carries information about an infinite number of admissible classical states $ψ_n$ that can be mapped to other states after lifting into the quantum bundle. We show that in the Kepler/hydrogen atom problem there is a similar correspondence between classical and quantum states.
Renjun Duan, Jinkai Ni
The time-periodic problem on the Boltzmann equation with a given time-periodic external force in the three-dimensional whole space has remained open since it was first studied in [13] for only spatial dimensions not less than five. The goal of this paper is to give an affirmative answer to this problem provided that the external force is sufficiently small in the function space $\mathcal{C}(\mathbb{R};\dot{B}^{-3/2}_{2,\infty}\cap\dot{H}^N)$ with $N\geq 4$. The proof is based on Serrin's method through studying the global-in-time stability of the Cauchy problem with time-periodic external forces. As a direct consequence, the result also yields the existence and stability of stationary solutions to the physically realistic three-dimensional Boltzmann equation when the external force is time-independent.
Mohamed Aarif A, Soumya Chakrabarti
We study the gravitational collapse of a non-interacting mix of perfect fluid and a spatially homogeneous scalar field within a Chiellini-integrable framework. We choose an extended Higgs-type self-interaction potential and reduce the Klein-Gordon equation into a generalized damped Milne-Pinney class of differential equation. We derive a closed-form analytical solution for the scalar field, the scale factor and explore the collapsing branch of the same. We find that it exhibits an asymptotic collapse in which the proper volume decreases monotonically but never reaches zero at finite time. We analyze the energy conditions for the constituent elements of the collapsing sphere. While the scalar field remains canonical in nature, we find that the perfect fluid can violated the Null Energy Condition. We also study the formation of apparent horizon condition and find multiple possibilities depending on the parameter space : either no trapped surface or the formation of multiple apparent horizons. We match the interior homogeneous solution to a generalized Vaidya exterior via the Israel-Darmois junction conditions, yielding the corresponding boundary mass function, ensuring a smooth collapse scenario.
Sodbaatar Adiya, Khongorzul Dorjgotov, Bayarmagnai Gombodorj, Hiroyuki Ochiai, Uuganbayar Zunderiya
We study invariant solutions of a certain class of time-fractional diffusion-wave equations with variable coefficients via Lie symmetry analysis. In physics, the fractional diffusion equation describes transport dynamics that are governed by anomalous diffusion while the fractional wave equation describes oscillations and wave propagation in various physical systems. In order to obtain exact invariant solutions of these equations, we firstly determine infinitesimal symmetries with respect to the variable coefficients of the equations. With the help of these symmetries, we then find solutions in terms of Mittag-Leffler functions, generalized Wright functions and Fox H-functions.
Fawzi Aly, Alex Kitt, Luke Mohr
Additive manufacturing and welding processes are highly sensitive to heat dissipation, where improper thermal management leads to residual stresses, distortions, and cracking. Existing heat transfer models, such as Rosenthal's solutions, fail to handle finite 3D geometries, cooling effects, or transient behavior, limiting their accuracy. We overcome these limitations by developing an analytical framework that incorporates cooling boundary conditions mimicking Newton's Law of Cooling. Using two different and proven-equivalent approaches, Laplace transform and Fourier series, we derive closed-form solutions for transient and steady-state temperature profiles under various heat sources, including Gaussian, ellipsoidal, double-ellipsoidal, and time-dependent on/off switch sources. We compare our analytical solutions to numerical implementations, demonstrating strong agreement while providing deeper physical insight. This approach significantly reduces computational cost and experimental requirements, making it a scalable tool for optimizing thermal predictions and mitigating residual stresses in metal-based manufacturing. Additionally, our framework enables the generation of synthetic datasets for machine learning models to predict heat distribution efficiently.
Alexander Zlokapa
Apr 22, 2026·quant-ph·PDF Estimating local observables in Gibbs states is a central problem in quantum simulation. While this task is BQP-complete at asymptotically low temperatures, the possibility of quantum advantage at constant temperature remains open. The Sachdev-Ye-Kitaev (SYK) model is a natural candidate: at any constant temperature, its Gibbs states have polynomial quantum circuit complexity and are not described by Gaussian states. Rigorous analyses of the SYK model are difficult due to the failure of known techniques using random matrix theory, cluster expansions, and rigorous formulations of the quantum path integral and replica trick. Despite this, we give a rigorous proof of a quasipolynomial-time classical algorithm that estimates SYK local thermal expectations at sufficiently high constant temperature. Our result introduces a new Wick-pair cluster expansion that we expect to be broadly useful for disordered quantum many-body systems.
A. Mazel, I. Stuhl, Y. Suhov
We prove that the maximal graph-density of an independent set in a Penrose P3 tiling considered as a planar non-directed graph is equal to $(57 - 25 \sqrt{5})/2 \approx 0.54915$ despite the fact that the graph is bipartite. Accordingly, the extreme Gibbs measure of the nearest-neighbor hard core particle model on this graph is unique for sufficiently large values of the particle activity. This invalidates a natural expectation to observe the coexistence of even and odd phases.
Roberto Feola, Riccardo Montalto, Federico Murgante
We study the long-time dynamics of small-amplitude solutions to the three-dimensional gravity-capillary water waves equations for an inviscid and irrotational fluid with periodic boundary conditions. We prove that, for almost all values of the surface tension parameter, solutions with initial size $\varepsilon$ exist and remain small over time intervals of order $\varepsilon^{-2}$. A major difficulty arises from the loss of derivatives caused by the quasilinear nature of the equations combined with severe quadratic and cubic small-divisor interactions in high space dimensions. Classical normal form methods applied to 3D water waves system typically fail to prevent derivative loss due to the accumulation of near-resonances. To overcome this obstruction, we develop a new analytical strategy that combines a sharp frequency partition with a quasi-resonant normal form transformation acting only on selected interaction scales. Our microlocal analysis reveals that the potentially dangerous interactions terms exhibit a block-diagonal structure, which stems from both the geometric properties of the quasi-resonant frequency sets and the Hamiltonian structure of the water waves system. As a consequence, these operators preserve Sobolev norms and do not produce energy growth. This structural insight, together with the quasi-resonant normal-form transformation, allows us to prevent derivative-loss mechanisms while avoiding the accumulation of harmful small denominators.