Communications in Mathematical Physics最新文献

We study the first chiral homology group of elliptic curves with coefficients in vacuum insertions of a conformal vertex algebra V. We find finiteness conditions on V guaranteeing that these homologies are finite dimensional, generalizing the (C_2)-cofinite, or quasi-lisse condition in the degree 0 case. We determine explicitly the flat connections that these homologies acquire under smooth variation of the elliptic curve, as insertions of the conformal vector and the Weierstrass (zeta ) function. We construct linear functionals associated to self-extensions of V-modules and prove their convergence under said finiteness conditions. These linear functionals turn out to be degree 1 analogs of the n-point functions in the degree 0 case. As a corollary we prove the vanishing of the first chiral homology group of an elliptic curve with values in several rational vertex algebras, including affine (mathfrak {sl}_2) at non-negative integral level, the ((2,2k+1))-minimal models and arbitrary simple affine vertex algebras at level 1. Of independent interest, we prove a Fourier space version of the Borcherds formula.

In this paper, we combine the profound Pesin theory with the sophisticated approach for addressing singular flows devised by Liao and prove a closing lemma for (C^{1+alpha }) non-uniform hyperbolic singular flows. As an application, we prove that every ergodic hyperbolic measure which is not supported on singularities can be approximated by periodic measures.

Using a smooth version of the Connes–Thom isomorphism in Grensing’s bivariant K-theory for locally convex algebras, we prove an equivariant version of the Connes–Thom isomorphism in periodic cyclic homology. As an application, we prove that periodic cyclic homology is invariant with respect to equivariant strict deformation quantizations.

We show that for families of 1d lattice systems in an invertible phase, the cohomology class of the higher Berry curvature can be refined to an integral degree-3 class on the parameter space. Similarly, for families of U(1)-invariant 2d lattice systems in an invertible phase, the higher Thouless pump can be refined to an integral degree-2 class on the parameter space. We show that the 2d Thouless pump can be identified with an excess Berry curvature of a flux insertion.

Clifford quantum circuits are elementary invertible transformations of quantum systems that map Pauli operators to Pauli operators. We study periodic one-parameter families of Clifford circuits, called loops of Clifford circuits, acting on ({textsf{d}})-dimensional lattices of prime p-dimensional qudits. We propose to use the notion of algebraic homotopy to identify topologically equivalent loops. We calculate homotopy classes of such loops for any odd p and ({textsf{d}}=0,1,2,3), and 4. Our main tool is the Hermitian K-theory, particularly a generalization of the Maslov index from symplectic geometry. We observe that the homotopy classes of loops of Clifford circuits in (({textsf{d}}+1))-dimensions coincide with the quotient of the group of Clifford Quantum Cellular Automata modulo shallow circuits and lattice translations in ({textsf{d}})-dimensions.

We consider Schrödinger operators over a class of circle maps including (C^{2+epsilon })-smooth circle maps with finitely many break points, where the derivative has a jump discontinuity. We show that in a region of the Lyapunov exponent—determined by the geometry of the dynamical partitions and (alpha )—the spectrum of Schrödinger operators over every such map, is purely singular continuous, for every (alpha )-Hölder-continuous potential V. As a corollary, we obtain that for every sufficiently smooth such map, with an invariant measure (mu ) and with rotation number in a set (mathcal {S}), and (mu )-almost all (xin {mathbb {T}}^1), the corresponding Schrödinger operator has a purely continuous spectrum, for every Hölder-continuous potential V. Set (mathcal {S}) includes some Diophantine numbers of class (D(delta )), for any (delta >1).

The two-dimensional (2-D) Euler equations of a perfect fluid possess a beautiful geometric description: they are reduced geodesic equations on the infinite-dimensional Lie group of symplectomorphims with respect to a right-invariant Riemannian metric. This structure enables insights to Eulerian and Lagrangian stability via sectional curvature and Jacobi equations. The Zeitlin model is a finite-dimensional analogue of the 2-D Euler equations; the only known discretization that preserves the rich geometric structure. Theoretical and numerical studies indicate that Zeitlin’s model provides consistent long-time behaviour on large scales, but to which extent it truly reflects the Euler equations is mainly open. Towards progress, we give here two results. First, convergence of the sectional curvature in the Euler–Zeitlin equations on the Lie algebra (mathfrak {su}(N)) to that of the Euler equations on the sphere. Second, (L^2)-convergence of the corresponding Jacobi equations for Lagrangian and Eulerian stability. The results allow geometric conclusions about Zeitlin’s model to be transferred to Euler’s equations and vice versa, which could expedite the ultimate aim: to characterize the generic long-time behaviour of perfect 2-D fluids.

We prove that for a large class of quantum Fisher information, a quantum channel is sufficient for a family of quantum states, i.e., the input states can be recovered from the output by some quantum operation, if and only if, the quantum Fisher information is preserved under the quantum channel. This class, for instance, includes Winger–Yanase–Dyson skew information. On the other hand, interestingly, the SLD quantum Fisher information, as the most popular example of quantum analogs of Fisher information, does not satisfy this property. Our recoverability result is obtained by studying monotone metrics on the quantum state space, i.e. Riemannian metrics non-increasing under the action of quantum channels, a property often called data processing inequality. For two quantum states, the monotone metric gives the corresponding quantum (chi ^2) divergence. We obtain an approximate recovery result in the sense that, if the quantum (chi ^2) divergence is approximately preserved by a quantum channel, then two states can be approximately recovered by the Petz recovery map. We also obtain a universal recovery bound for the (chi _{frac{1}{2}}) divergence. Finally, we discuss applications in the context of quantum thermodynamics and the resource theory of asymmetry.

We investigate the stationary distribution of asymmetric and weakly asymmetric simple exclusion processes with open boundaries. We project the stationary distribution onto a subinterval, whose size is allowed to grow with the length of the underlying segment. Depending on the boundary parameters of the exclusion process, we provide conditions such that the stationary distribution projected onto a subinterval is close in total variation distance to a product measure.

Let ({mathcal {A}}) be the adjacency matrix of a random d-regular graph on N vertices, and we denote its eigenvalues by (lambda _1geqslant lambda _2cdots geqslant lambda _{N}). For (N^{2/3+o(1)}leqslant dleqslant N/2), we prove optimal rigidity estimates of the extreme eigenvalues of ({mathcal {A}}), which in particular imply that

with very high probability. In the same regime of d, we also show that

where (textrm{TW}_1) is the Tracy–Widom distribution for GOE; analogue results also hold for other non-trivial extreme eigenvalues.