Communications in Mathematical Physics最新文献

Consider D random systems that are modeled by independent (Ntimes N) complex Hermitian Wigner matrices. Suppose they are lying on a circle and the neighboring systems interact with each other through a deterministic matrix A. We prove that in the asymptotic limit (Nrightarrow infty ), the whole system exhibits a quantum chaos transition when the interaction strength (Vert AVert _{{textrm{HS}}}) varies. Specifically, when (Vert AVert _{{textrm{HS}}}ge N^{{varepsilon }}), we prove that the bulk eigenvalue statistics match those of a (DNtimes DN) GUE asymptotically and each bulk eigenvector is approximately equally distributed among the D subsystems with probability (1-textrm{o}(1)). These phenomena indicate quantum chaos of the whole system. In contrast, when (Vert AVert _{{textrm{HS}}}le N^{-{varepsilon }}), we show that the system is integrable: the bulk eigenvalue statistics behave like D independent copies of GUE statistics asymptotically and each bulk eigenvector is localized on only one subsystem. In particular, if we take (Drightarrow infty ) after the (Nrightarrow infty ) limit, the bulk statistics converge to a Poisson point process under the DN scaling.

Many low temperature disordered systems are expected to exhibit Poisson–Dirichlet (PD) statistics. In this paper, we focus on the case when the underlying disorder is a logarithmically correlated Gaussian process (phi _N) on the box ([0,N]^dsubset mathbb {Z}^d). Canonical examples include branching random walk, (*)-scale invariant fields, with the central example being the two dimensional Gaussian free field (GFF), a universal scaling limit of a wide range of statistical mechanics models. The corresponding Gibbs measure obtained by exponentiating (beta ) (inverse temperature) times (phi _N) is a discrete version of the Gaussian multiplicative chaos (GMC) famously constructed by Kahane (Ann Sci Math Québec 9(2): 105–150, 1985). In the low temperature or supercritical regime, i.e., (beta ) larger than a critical (beta _c,) the GMC is expected to exhibit atomic behavior on suitable renormalization, dictated by the extremal statistics or near maximum values of (phi _N). Moreover, it is predicted going back to a conjecture made in 2001 in Carpentier and Le Doussal (Phys Rev E 63(2): 026110, 2001), that the weights of this atomic GMC has a PD distribution. In a series of works culminating in Biskup and Louidor (Adv Math 330, 589–687, 2018), Biskup and Louidor carried out a comprehensive study of the near maxima of the 2D GFF, and established the conjectured PD behavior throughout the super-critical regime ((beta > 2)). In another direction Ding et al. (Ann Probab 5(6A), 3886–3928, 2017), established universal behavior of the maximum for a general class of log-correlated Gaussian fields. In this paper we continue this program simply under the assumption of log-correlation and nothing further. We prove that the GMC concentrates on an O(1) neighborhood of the local extrema and the PD prediction made in Carpentier and Le Doussal (Phys Rev E 63(2): 026110, 2001) holds, in any dimension d, throughout the supercritical regime (beta > sqrt{2d}), significantly generalizing past results. While many of the arguments for the GFF make use of the powerful Gibbs–Markov property, in absence of any Markovian structure for general Gaussian fields, we develop and use as our key input a sharp estimate of the size of level sets, a result we believe could have other applications.

We provide an explicit description of the recurrent configurations of the sandpile model on a family of graphs ({widehat{G}}_{mu ,nu }), which we call clique-independent graphs, indexed by two compositions (mu ) and (nu ). Moreover, we define a delay statistic on these configurations, and we show that, together with the usual level statistic, it can be used to provide a new combinatorial interpretation of the celebrated shuffle theorem of Carlsson and Mellit. More precisely, we will see how to interpret the polynomials (langle nabla e_n, e_mu h_nu rangle ) in terms of these configurations.

This article presents a new relation between the basic representation of split real simply-laced affine Kac–Moody algebras and finite dimensional representations of its maximal compact subalgebra (mathfrak {k}). We provide infinitely many (mathfrak {k})-subrepresentations of the basic representation and we prove that these are all the finite dimensional (mathfrak {k})-subrepresentations of the basic representation, such that the quotient of the basic representation by the subrepresentation is a finite dimensional representation of a certain parabolic algebra and of the maximal compact subalgebra. By this result we provide an infinite composition series with a cosocle filtration of the basic representation. Finally, we present examples of the results and applications to supergravity.

Vanishing gradients can pose substantial obstacles for high-dimensional optimization problems. Here we consider energy minimization problems for quantum many-body systems with extensive Hamiltonians and finite-range interactions, which can be studied on classical computers or in the form of variational quantum eigensolvers on quantum computers. Barren plateaus correspond to scenarios where the average amplitude of the energy gradient decreases exponentially with increasing system size. This occurs, for example, for quantum neural networks and for brickwall quantum circuits when the depth increases polynomially in the system size. Here we prove that the variational optimization problems for matrix product states, tree tensor networks, and the multiscale entanglement renormalization ansatz are free of barren plateaus. The derived scaling properties for the gradient variance provide an analytical guarantee for the trainability of randomly initialized tensor network states (TNS) and motivate certain initialization schemes. In a suitable representation, unitary tensors that parametrize the TNS are sampled according to the uniform Haar measure. We employ a Riemannian formulation of the gradient based optimizations which simplifies the analytical evaluation.

In this paper, we study the regularity of the Lyapunov exponent for quasiperiodic Schrödinger cocycles with (C^2) cos-type potentials, large coupling constants, and a fixed Diophantine frequency. We obtain the absolute continuity of the Lyapunov exponent. Moreover, we prove the Lyapunov exponent is (frac{1}{2})-Hölder continuous. Furthermore, for any given (rin (frac{1}{2}, 1)), we can find some energy in the spectrum where the local regularity of the Lyapunov exponent is between ((r-epsilon ))-Hölder continuity and ((r+epsilon ))-Hölder continuity.

We consider the Cauchy problem for the logarithmically singular surface quasi-geostrophic (SQG) equation, introduced by Ohkitani,

and establish local existence and uniqueness of smooth solutions in the scale of Sobolev spaces with exponent decreasing with time. Such a decrease of the Sobolev exponent is necessary, as we have shown in the companion paper (Chae et al. in Illposedness via degenerate dispersion for generalized surface quasi-geostrophic equations with singular velocities, arXiv:2308.02120) that the problem is strongly ill-posed in any fixed Sobolev spaces. The time dependence of the Sobolev exponent can be removed when there is a dissipation term strictly stronger than log. These results improve wellposedness statements by Chae et al. (Comm Pure Appl Math 65(8):1037–1066, 2012). This well-posedness result can be applied to describe the long-time dynamics of the (delta )-SQG equations, defined by

for all sufficiently small (delta >0) depending on the size of the initial data. For the same range of (delta ), we establish global well-posedness of smooth solutions to the dissipative SQG equations.

A recent development in our understanding of the theory of quantum fields is the fact that familiar gauge theories in spacetime dimensions greater than two can have non-invertible symmetries generated by topological defects. The hallmark of these non-invertible symmetries is that the fusion rule deviates from the usual group-like structure, and in particular the fusion coefficients take values in topological field theories (TFTs) rather than in mere numbers. In this paper we begin an exploration of the associativity structure of non-invertible symmetries in higher dimensions. The first layer of associativity is captured by F-symbols, which we find to assume values in TFTs that have one dimension lower than that of the defect. We undertake an explicit analysis of the F-symbols for the non-invertible chiral symmetry that is preserved by the massless QED and explore their physical implications. In particular, we show the F-symbol TFTs can be detected by probing the correlators of topological defects with ’t Hooft lines. Furthermore, we derive the Ward–Takahashi identity that arises from the chiral symmetry on a large class of four-dimensional manifolds with non-trivial topologies directly from the topological data of the symmetry defects, without referring to a Lagrangian formulation of the theory.

For any generalized Frobenius manifold with non-flat unity, we construct a bihamiltonian integrable hierarchy of hydrodynamic type which is an analogue of the Principal Hierarchy of a Frobenius manifold. We show that such an integrable hierarchy, which we also call the Principal Hierarchy, possesses Virasoro symmetries and a tau structure, and the Virasoro symmetries can be lifted to symmetries of the tau-cover of the integrable hierarchy. We derive the loop equation from the condition of linearization of actions of the Virasoro symmetries on the tau function, and construct the topological deformation of the Principal Hierarchy of a semisimple generalized Frobenius manifold with non-flat unity. We also give two examples of generalized Frobenius manifolds with non-flat unity and show that they are closely related to the well-known integrable hierarchies: the Volterra hierarchy, the q-deformed KdV hierarchy and the Ablowitz–Ladik hierarchy.

A quantum cellular automaton (QCA) or a causal unitary is by definition an automorphism of a local operator algebra, by which local operators are mapped to nearby local operators. Quantum circuits of small depth, local Hamiltonian evolutions for short time, and translations (shifts) are examples. A Clifford QCA is one that maps any Pauli operator to a finite tensor product of Pauli operators. Here, we obtain a complete table of groups ({mathfrak {C}}({textsf{d}},p)) of translation invariant Clifford QCA in any spatial dimension ({textsf{d}}ge 0) modulo Clifford quantum circuits and shifts over prime p-dimensional qudits, where the circuits and shifts are allowed to obey only coarser translation invariance. The group ({mathfrak {C}}({textsf{d}},p)) is nonzero only for ({textsf{d}}= 2k+3) if (p=2) and ({textsf{d}}= 4k+3) if p is odd where (k ge 0) is any integer, in which case ({mathfrak {C}}({textsf{d}},p) cong {widetilde{mathfrak {W}}}({mathbb {F}}_p)), the classical Witt group of nonsingular quadratic forms over the finite field ({mathbb {F}}_p). It is well known that ({widetilde{mathfrak {W}}}({mathbb {F}}_2) cong {mathbb {Z}}/2{mathbb {Z}}), ({widetilde{mathfrak {W}}}({mathbb {F}}_p) cong {mathbb {Z}}/4{mathbb {Z}}) if (p = 3 bmod 4), and ({widetilde{mathfrak {W}}}({mathbb {F}}_p)cong {mathbb {Z}}/2{mathbb {Z}}oplus {mathbb {Z}}/2{mathbb {Z}}) if (p = 1 bmod 4). The classification is achieved by a dimensional descent, which is a reduction of Laurent extension theorems for algebraic L-groups of surgery theory in topology.