Communications in Mathematical Physics最新文献

We construct an order-sharp theory for a double-porosity model in the full linear elasticity setup. Crucially, we uncover time and frequency dispersive properties of highly oscillatory elastic composites.

We study the Lévy spin glass model, a fully connected model on N vertices with heavy-tailed interactions governed by a power law distribution of order (0<alpha <2.) Our investigation is divided into three cases (0<alpha <1), (alpha =1), and (1<alpha <2.) When (1<alpha <2,) we identify a high temperature regime, in which the limit and fluctuation of the free energy are explicitly obtained and the site and bond overlaps are shown to exhibit concentration, interestingly, while the former is concentrated around zero, the latter obeys a positivity behavior. At any temperature, we further establish the existence of the limiting free energy and derive a variational formula analogous to Panchenko’s framework in the setting of the Poissonian Viana-Bray model. For (alpha =1), the free energy scales super-linearly and converges to a constant proportional to (beta ) in probability at any temperature. In the case of (0<alpha <1), the scaling for the free energy is again super-linear, however, it converges weakly to the sum of a Poisson Point Process at any temperature. Additionally, we show that the Gibbs measure puts most of its mass on the configurations that align with signs of the polynomially many heaviest edge weights.

In this paper we consider non-intersecting Brownian bridges, under fairly general upper and lower boundaries, and starting and ending data. Under the assumption that these boundary data induce a smooth limit shape (without empty facets), we establish two results. The first is a nearly optimal concentration bound for the Brownian bridges in this model. The second is that the bulk local statistics of these bridges along any fixed time converge to the sine process.

For a rational and (C_2)-cofinite vertex operator algebra V with an automorphism group G of prime order, the fusion rules for twisted V-modules are studied, a twisted Verlinde formula which relates fusion rules for g-twisted modules to the S-matrix in the orbifold theory is established. As an application of the twisted Verlinde formula, a twisted analogue of the Kac-Walton formula is proved, which gives fusion rules between twisted modules of affine vertex operator algebras at positive integer levels in terms of Clebsch–Gordan coefficients associated to the corresponding finite dimensional simple Lie algebras.

Lindblad dynamics and other open-system dynamics provide a promising path towards efficient Gibbs sampling on quantum computers. In these proposals, the Lindbladian is obtained via an algorithmic construction akin to designing an artificial thermostat in classical Monte Carlo or molecular dynamics methods, rather than being treated as an approximation to weakly coupled system-bath unitary dynamics. Recently, Chen, Kastoryano, and Gilyén (arXiv:2311.09207) introduced the first efficiently implementable Lindbladian satisfying the Kubo–Martin–Schwinger (KMS) detailed balance condition, which ensures that the Gibbs state is a fixed point of the dynamics and is applicable to non-commuting Hamiltonians. This Gibbs sampler uses a continuously parameterized set of jump operators, and the energy resolution required for implementing each jump operator depends only logarithmically on the precision and the mixing time. In this work, we build upon the structural characterization of KMS detailed balanced Lindbladians by Fagnola and Umanità, and develop a family of efficient quantum Gibbs samplers using a finite set of jump operators (the number can be as few as one), akin to the classical Markov chain-based sampling algorithm. Compared to the existing works, our quantum Gibbs samplers have a comparable quantum simulation cost but with greater design flexibility and a much simpler implementation and error analysis. Moreover, it encompasses the construction of Chen, Kastoryano, and Gilyén as a special instance.

We introduce group-theoretical fusion 2-categories, a categorification of the notion of a group-theoretical fusion 1-category. Physically speaking, such fusion 2-categories arise by gauging subgroups of a global symmetry. We show that group-theoretical fusion 2-categories are completely characterized by the property that the braided fusion 1-category of endomorphisms of the monoidal unit is Tannakian. Then, we describe the underlying finite semisimple 2-category of group-theoretical fusion 2-categories, and, more generally, of certain 2-categories of bimodules. We also partially describe the fusion rules of group-theoretical fusion 2-categories. Using our previous results, we classify fusion 2-categories admitting a fiber 2-functor. Next, we study fusion 2-categories with a Tambara–Yamagami defect, that is (mathbb {Z}/2)-graded fusion 2-categories whose non-trivially graded factor is (textbf{2Vect}). We classify these fusion 2-categories, and examine more closely the more restrictive notion of Tambara–Yamagami fusion 2-categories. Throughout, we give many examples to illustrate our various results.

We establish the existence of quasi-periodic traveling wave solutions for the (beta )-plane equation on ({mathbb {T}}^2) with a large quasi-periodic traveling wave external force. These solutions exhibit large sizes, which depend on the frequency of oscillations of the external force. Due to the presence of small divisors, the proof relies on a nonlinear Nash-Moser scheme tailored to construct nonlinear waves of large size. To our knowledge, this is the first instance of constructing quasi-periodic solutions for a quasilinear PDE in dimensions greater than one, with a 1-smoothing dispersion relation that is highly degenerate - indicating an infinite-dimensional kernel for the linear principal operator. This degeneracy challenge is overcome by preserving the traveling-wave structure, the conservation of momentum and by implementing normal form methods for the linearized system with sublinear dispersion relation in higher space dimension.

This paper delves into a fundamental aspect of quantum statistical mechanics—the absence of thermal phase transitions in one-dimensional (1D) systems. Originating from Ising’s analysis of the 1D spin chain, this concept has been pivotal in understanding 1D quantum phases, especially those with finite-range interactions, as extended by Araki. In this work, we focus on quantum long-range interactions and successfully derive a clustering theorem applicable to a wide range of interaction decays at arbitrary temperatures. This theorem applies to any interaction forms that decay faster than (r^{-2}) and does not rely on translation invariance or infinite system size assumptions. Also, we rigorously established that the temperature dependence of the correlation length is given by (e^{mathrm{const.} beta }), which is the same as the classical cases. Our findings indicate the absence of phase transitions in 1D systems with super-polynomially decaying interactions, thereby expanding upon previous theoretical research. To overcome significant technical challenges originating from the divergence of the imaginary-time Lieb–Robinson bound, we utilize the quantum belief propagation to refine the cluster expansion method. This approach allowed us to address divergence issues effectively and contributed to a deeper understanding of low-temperature behaviors in 1D quantum systems.

We initiate the study of utilizing quantum Langevin dynamics (QLD) to solve optimization problems, particularly those nonconvex objective functions that present substantial obstacles for traditional gradient descent algorithms. Specifically, we examine the dynamics of a system coupled with an infinite heat bath. This interaction induces both random quantum noise and a deterministic damping effect to the system, which nudge the system towards a steady state that hovers near the global minimum of objective functions. We theoretically prove the convergence of QLD in convex landscapes, demonstrating that the average energy of the system can converge to zero in the low temperature limit with an exponential convergence rate. Numerically, we first show the energy dissipation capability of QLD by retracing its origins to spontaneous emission. Furthermore, we conduct detailed discussion of the impact of each parameter. Finally, based on the observations when comparing QLD with the classical Fokker-Plank-Smoluchowski equation, we propose a time-dependent QLD by setting temperature and (hbar ) as time-dependent parameters, which can be theoretically proven to converge better than the time-independent case and also outperforms a series of state-of-the-art quantum and classical optimization algorithms in many nonconvex landscapes.

We constrain the low-energy spectra of Laplace operators on closed hyperbolic manifolds and orbifolds in three dimensions, including the standard Laplace--Beltrami operator on functions and the Laplacian on powers of the cotangent bundle. Our approach employs linear programming techniques to derive rigorous bounds by leveraging two types of spectral identities. The first type, inspired by the conformal bootstrap, arises from the consistency of the spectral decomposition of the product of Laplace eigensections, and involves the Laplacian spectra as well as integrals of triple products of eigensections. We formulate these conditions in the language of representation theory of (textrm{PSL}_2(mathbb {C})) and use them to prove upper bounds on the first and second Laplacian eigenvalues. The second type of spectral identities follows from the Selberg trace formula. We use them to find upper bounds on the spectral gap of the Laplace--Beltrami operator on hyperbolic 3-orbifolds, as well as on the systole length of hyperbolic 3-manifolds, as a function of the volume. Further, we prove that the spectral gap (lambda _1) of the Laplace--Beltrami operator on all closed hyperbolic 3-manifolds satisfies (lambda _1 < 47.32). Along the way, we use the trace formula to estimate the low-energy spectra of a large set of example orbifolds and compare them with our general bounds, finding that the bounds are nearly sharp in several cases.