Annales Henri Poincaré最新文献

Building upon previous 2D studies, this research focuses on describing 3D tensor renormalisation group (RG) flows for lattice spin systems, such as the Ising model. We present a novel RG map, which operates on tensors with infinite-dimensional legs and does not involve truncations, in contrast to numerical tensor RG maps. To construct this map, we developed new techniques for analysing tensor networks. Our analysis shows that the constructed RG map contracts the region around the tensor (A_*), corresponding to the high-temperature phase of the 3D Ising model. This leads to the iterated RG map convergence in the Hilbert–Schmidt norm to (A_*) when initialised in the vicinity of (A_*). This work provides the first steps towards the rigorous understanding of tensor RG maps in 3D.

We generalize Lévy’s lemma, a concentration-of-measure result for the uniform probability distribution on high-dimensional spheres, to a much more general class of measures, so-called GAP measures. For any given density matrix (rho ) on a separable Hilbert space ({mathcal {H}}), ({textrm{GAP}}(rho )) is the most spread-out probability measure on the unit sphere of ({mathcal {H}}) that has density matrix (rho ) and thus forms the natural generalization of the uniform distribution. We prove concentration-of-measure whenever the largest eigenvalue (Vert rho Vert ) of (rho ) is small. We use this fact to generalize and improve well-known and important typicality results of quantum statistical mechanics to GAP measures, namely canonical typicality and dynamical typicality. Canonical typicality is the statement that for “most” pure states (psi ) of a given ensemble, the reduced density matrix of a sufficiently small subsystem is very close to a (psi )-independent matrix. Dynamical typicality is the statement that for any observable and any unitary time evolution, for “most” pure states (psi ) from a given ensemble the (coarse-grained) Born distribution of that observable in the time-evolved state (psi _t) is very close to a (psi )-independent distribution. So far, canonical typicality and dynamical typicality were known for the uniform distribution on finite-dimensional spheres, corresponding to the micro-canonical ensemble, and for rather special mean-value ensembles. Our result shows that these typicality results hold also for ({textrm{GAP}}(rho )), provided the density matrix (rho ) has small eigenvalues. Since certain GAP measures are quantum analogs of the canonical ensemble of classical mechanics, our results can also be regarded as a version of equivalence of ensembles.

We describe in a simple setting how to extract a braided tensor category from a collection of superselection sectors of a two-dimensional quantum spin system, corresponding to abelian anyons. We extract from this category its fusion ring as well as its F and R-symbols. We then construct the double semion state in infinite volume and extract the braided tensor category describing its semion, anti-semion, and bound state excitations. We verify that this category is equivalent to the representation category of the twisted quantum double (mathcal {D}^{phi }(mathbb {Z}_2)).

We consider the ground state of a Bose gas of N particles on the three-dimensional unit torus in the mean-field regime that is known to exhibit Bose–Einstein condensation. Bounded one-particle operators with law given through the interacting Bose gas’ ground state correspond to dependent random variables due to the bosons’ correlation. We prove that in the limit (N rightarrow infty ) bounded one-particle operators with law given by the ground state satisfy large deviation estimates. We derive a lower and an upper bound on the rate function that match up to second order and that are characterized by quantum fluctuations around the condensate.

We prove an upper bound on the energy density of the dilute spin-(frac{1}{2}) Fermi gas capturing the leading correction to the kinetic energy (8pi a rho _uparrow rho _downarrow ) with an error of size smaller than (arho ^{2}(a^3rho )^{1/3-varepsilon }) for any (varepsilon > 0), where a denotes the scattering length of the interaction. The result is valid for a large class of interactions including interactions with a hard core. A central ingredient in the proof is a rigorous version of a fermionic cluster expansion adapted from the formal expansion of Gaudin et al. (Nucl Phys A 176(2):237–260, 1971. https://doi.org/10.1016/0375-9474(71)90267-3).

We consider dilute Bose gases on the three-dimensional unit torus that interact through a pair potential with scattering length of order ( N^{kappa -1}), for some (kappa >0). For the range ( kappa in [0, frac{1}{43})), Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) proves complete BEC of low energy states into the zero momentum mode based on a unitary renormalization through operator exponentials that are quartic in creation and annihilation operators. In this paper, we give a new and self-contained proof of BEC of the ground state for ( kappa in [0, frac{1}{20})) by combining some of the key ideas of Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) with the novel diagonalization approach introduced recently in Brooks (Diagonalizing Bose Gases in the Gross–Pitaevskii Regime and Beyond, arXiv:2310.11347), which is based on the Schur complement formula. In particular, our proof avoids the use of operator exponentials and is significantly simpler than Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021).

In this article, we study the Dirac spectrum of typical hyperbolic surfaces of finite area, equipped with a nontrivial spin structure (so that the Dirac spectrum is discrete). For random Weil–Petersson surfaces of large genus g with (o(sqrt{g})) cusps, we prove convergence of the spectral density to the spectral density of the hyperbolic plane, with quantitative error estimates. This result implies upper bounds on spectral counting functions and multiplicities, as well as a uniform Weyl law, true for typical hyperbolic surfaces equipped with any nontrivial spin structure.