Communications in Mathematical Physics最新文献

We prove the existence of macroscopic loops in the loop (textrm{O}(2)) model with (frac{1}{2}le x^2le 1) or, equivalently, delocalisation of the associated integer-valued Lipschitz function on the triangular lattice. This settles one side of the conjecture of Fan, Domany, and Nienhuis (1970 s–1980 s) that (x^2 = frac{1}{2}) is the critical point. We also prove delocalisation in the six-vertex model with (0<a,,ble cle a+b). This yields a new proof of continuity of the phase transition in the random-cluster and Potts models in two dimensions for (1le qle 4) relying neither on integrability tools (parafermionic observables, Bethe Ansatz), nor on the Russo–Seymour–Welsh theory. Our approach goes through a novel FKG property required for the non-coexistence theorem of Zhang and Sheffield, which is used to prove delocalisation all the way up to the critical point. We also use the ({mathbb {T}})-circuit argument in the case of the six-vertex model. Finally, we extend an existing renormalisation inequality in order to quantify the delocalisation as being logarithmic, in the regimes (frac{1}{2}le x^2le 1) and (a=ble cle a+b). This is consistent with the conjecture that the scaling limit is the Gaussian free field.

We develop a general framework for studying non-uniqueness of the Riemann problem for the isentropic compressible Euler system in two spatial dimensions, and in this paper we present the most delicate result of our method: non-uniqueness of the contact discontinuity. Our approach is computational, and uses the pressure law as an additional degree of freedom. The stability of the contact discontinuities for this system is a major open problem (see Chen and Wang, in: Nonlinear partial differential equations, Abel Symposia, vol 7, Springer, Heidelberg, 2012). We find a smooth pressure law p, verifying the physically relevant condition (p'>0), such that for the isentropic compressible Euler system with this pressure law, contact discontinuity initial data is wildly non-unique in the class of bounded, admissible weak solutions. This result resolves the question of uniqueness for contact discontinuity solutions in the compressible regime. Moreover, in the same regularity class in which we have non-uniqueness of the contact discontinuity, i.e. (L^infty ), with no BV regularity or self-similarity, we show that the classical contact discontinuity solution to the two-dimensional isentropic compressible Euler system is in fact unique in the class of bounded, admissible weak solutions if we restrict to 1-D solutions.

In 1993 Keski-Vakkuri and Wen introduced a model for the fractional quantum Hall effect based on multilayer two-dimensional electron systems satisfying quasi-periodic boundary conditions. Such a model is essentially specified by a choice of a complex torus E and a symmetric positively definite matrix K of size g with non-negative integral coefficients, satisfying some further constraints. The space of the corresponding wave functions turns out to be (delta )-dimensional, where (delta ) is the determinant of K. We construct a hermitian holomorphic bundle of rank (delta ) on the abelian variety A (which is the g-fold product of the torus E with itself), whose fibres can be identified with the space of wave function of Keski-Vakkuri and Wen. A rigorous construction of this “magnetic bundle” involves the technique of Fourier–Mukai transforms on abelian varieties. The constructed bundle turns out to be simple and semi-homogeneous and it can be equipped with two different (and natural) hermitian metrics: the one coming from the center-of-mass dynamics and the one coming from the Hilbert space of the underlying many-body system. We prove that the canonical Bott–Chern connection of the first hermitian metric is always projectively flat and give sufficient conditions for this property for the second hermitian metric.

We construct polynomials (mathbb {S}_{mu }(z)) parameterized by Young diagrams (mu ), whose coefficients are central elements of the quantized enveloping algebra (textrm{U}_q(mathfrak {gl}_n)). Their constant terms coincide with the central elements provided by the general construction of Drinfeld and Reshetikhin. For another special value of z, we get q-analogues of Okounkov’s quantum immanants for (mathfrak {gl}_n). We show that the Harish-Chandra image of (mathbb {S}_{mu }(z)) is a factorial Schur polynomial. We derive quantum analogues of the higher Capelli identities by calculating the images of the q-immanants in the braided Weyl algebra. We also give a symmetric function interpretation and new proof of the Newton identities of Gurevich, Pyatov and Saponov.

Let ({{mathfrak {g}}}) be a symmetrizable Kac–Moody algebra, (U_q({{mathfrak {g}}})) its quantum group, and (U_q({mathfrak {k}})subset U_q({{mathfrak {g}}})) a quantum symmetric pair subalgebra determined by a Lie algebra automorphism (theta ). We introduce a category (mathcal {W}_{theta }) of weight (U_q({mathfrak {k}}))-modules, which is acted on by the category of weight (U_q({{mathfrak {g}}}))-modules via tensor products. We construct a universal tensor K-matrix ({{mathbb {K}}} ) (that is, a solution of a reflection equation) in a completion of (U_q({mathfrak {k}})otimes U_q({{mathfrak {g}}})). This yields a natural operator on any tensor product (Motimes V), where (Min mathcal {W}_{theta }) and (Vin {{mathcal {O}}}_theta ), i.e., V is a (U_q({{mathfrak {g}}}))-module in category ({{mathcal {O}}}) satisfying an integrability property determined by (theta ). Canonically, (mathcal {W}_{theta }) is equipped with a structure of a bimodule category over ({{mathcal {O}}}_theta ) and the action of ({{mathbb {K}}} ) is encoded by a new categorical structure, which we call a boundary structure on (mathcal {W}_{theta }). This generalizes a result of Kolb which describes a braided module structure on finite-dimensional (U_q({mathfrak {k}}))-modules when ({{mathfrak {g}}}) is finite-dimensional. We also consider our construction in the case of the category ({{mathcal {C}}}) of finite-dimensional modules of a quantum affine algebra, providing the most comprehensive universal framework to date for large families of solutions of parameter-dependent reflection equations. In this case the tensor K-matrix gives rise to a formal Laurent series with a well-defined action on tensor products of any module in (mathcal {W}_{theta }) and any module in ({{mathcal {C}}}). This series can be normalized to an operator-valued rational function, which we call trigonometric tensor K-matrix, if both factors in the tensor product are in ({{mathcal {C}}}).

We consider the boundary driven harmonic model, i.e. the Markov process associated to the open integrable XXX chain with non-compact spins. We characterize its stationary measure as a mixture of product measures. For all spin values, we identify the law of the mixture in terms of the Dirichlet process. Next, by using the explicit knowledge of the non-equilibrium steady state we establish formulas predicted by Macroscopic Fluctuation Theory for several quantities of interest: the pressure (by Varadhan’s lemma), the density large deviation function (by contraction principle), the additivity principle (by using the Markov property of the mixing law). To our knowledge, the results presented in this paper constitute the first rigorous derivation of these macroscopic properties for models of energy transport with unbounded state space, starting from the microscopic structure of the non-equilibrium steady state.

Let B be a spacetime region of width (2R >0), and (varphi ) a vector state localized in B. We show that the vacuum relative entropy of (varphi ), on the local von Neumann algebra of B, is bounded by (2pi R)-times the energy of the state (varphi ) in B. This bound is model-independent and rigorous; it follows solely from first principles in the framework of translation covariant, local Quantum Field Theory on the Minkowski spacetime.

In this paper we first prove that the maximal ideal of the universal affine vertex operator algebra (V^k(sl_n)) for (k=-n+frac{n-1}{q}) is generated by two singular vectors of conformal weight 3q if (n=3), and by one singular vector of conformal weight 2q if (ngeqslant 4). We next determine the associated varieties of the simple vertex operator algebras (L_k(sl_3)) for all the non-admissible levels (k=-3+frac{2}{2m+1}), (mgeqslant 0). The varieties of the associated simple affine W-algebras (W_k(sl_3,f)), for nilpotent elements f of (sl_3), are also determined.

The goal of this paper is to analyze how the celebrated phase transitions of the XY model are affected by the presence of a non-elliptic quenched disorder. In dimension (d=2), we prove that if one considers an XY model on the infinite cluster of a supercritical percolation configuration, the Berezinskii–Kosterlitz–Thouless (BKT) phase transition still occurs despite the presence of quenched disorder. The proof works for all (p>p_c) (site or edge). We also show that the XY model defined on a planar Poisson-Voronoi graph also undergoes a BKT phase transition. When (dge 3), we show in a similar fashion that the continuous symmetry breaking of the XY model at low enough temperature is not affected by the presence of quenched disorder such as supercritical percolation (in (mathbb {Z}^d)) or Poisson-Voronoi (in (mathbb {R}^d)). Adapting either Fröhlich–Spencer’s proof of existence of a BKT phase transition (Fröhlich and Spencer in Commun Math Phys 81(4):527–602, 1981) or the more recent proofs (Lammers in Probab Theory Relat Fields 182(1–2):531–550, 2022; van Engelenburg and Lis in Commun Math Phys 399(1):85–104, 2023; Aizenman et al. in Depinning in integer-restricted Gaussian Fields and BKT phases of two-component spin models, 2021. arXiv preprint arXiv:2110.09498; van Engelenburg and Lis in On the duality between height functions and continuous spin models, 2023. arXiv preprint arXiv:2303.08596) to such non-uniformly elliptic disorders appears to be non-trivial. Instead, our proofs rely on Wells’ correlation inequality (Wells in Some Moment Inequalities and a Result on Multivariable Unimodality. PhD thesis, Indiana University, 1977).

We establish the Anderson localization and exponential dynamical localization for a class of quasi-periodic Schrödinger operators on (mathbb Z^d) with bounded or unbounded Lipschitz monotone potentials via multi-scale analysis based on Rellich function analysis in the perturbative regime. We show that at each scale, the resonant Rellich function uniformly inherits the Lipschitz monotonicity property of the potential via a novel Schur complement argument.