TY - JOUR AB - We show that recent results on adiabatic theory for interacting gapped many-body systems on finite lattices remain valid in the thermodynamic limit. More precisely, we prove a generalized super-adiabatic theorem for the automorphism group describing the infinite volume dynamics on the quasi-local algebra of observables. The key assumption is the existence of a sequence of gapped finite volume Hamiltonians, which generates the same infinite volume dynamics in the thermodynamic limit. Our adiabatic theorem also holds for certain perturbations of gapped ground states that close the spectral gap (so it is also an adiabatic theorem for resonances and, in this sense, “generalized”), and it provides an adiabatic approximation to all orders in the adiabatic parameter (a property often called “super-adiabatic”). In addition to the existing results for finite lattices, we also perform a resummation of the adiabatic expansion and allow for observables that are not strictly local. Finally, as an application, we prove the validity of linear and higher order response theory for our class of perturbations for infinite systems. While we consider the result and its proof as new and interesting in itself, we also lay the foundation for the proof of an adiabatic theorem for systems with a gap only in the bulk, which will be presented in a follow-up article. AU - Henheik, Sven Joscha AU - Teufel, Stefan ID - 10600 IS - 1 JF - Journal of Mathematical Physics KW - mathematical physics KW - statistical and nonlinear physics SN - 0022-2488 TI - Adiabatic theorem in the thermodynamic limit: Systems with a uniform gap VL - 63 ER - TY - JOUR AB - Based on a result by Yarotsky (J Stat Phys 118, 2005), we prove that localized but otherwise arbitrary perturbations of weakly interacting quantum spin systems with uniformly gapped on-site terms change the ground state of such a system only locally, even if they close the spectral gap. We call this a strong version of the local perturbations perturb locally (LPPL) principle which is known to hold for much more general gapped systems, but only for perturbations that do not close the spectral gap of the Hamiltonian. We also extend this strong LPPL-principle to Hamiltonians that have the appropriate structure of gapped on-site terms and weak interactions only locally in some region of space. While our results are technically corollaries to a theorem of Yarotsky, we expect that the paradigm of systems with a locally gapped ground state that is completely insensitive to the form of the Hamiltonian elsewhere extends to other situations and has important physical consequences. AU - Henheik, Sven Joscha AU - Teufel, Stefan AU - Wessel, Tom ID - 10642 IS - 1 JF - Letters in Mathematical Physics KW - mathematical physics KW - statistical and nonlinear physics SN - 0377-9017 TI - Local stability of ground states in locally gapped and weakly interacting quantum spin systems VL - 112 ER - TY - JOUR AB - We prove a generalised super-adiabatic theorem for extended fermionic systems assuming a spectral gap only in the bulk. More precisely, we assume that the infinite system has a unique ground state and that the corresponding Gelfand–Naimark–Segal Hamiltonian has a spectral gap above its eigenvalue zero. Moreover, we show that a similar adiabatic theorem also holds in the bulk of finite systems up to errors that vanish faster than any inverse power of the system size, although the corresponding finite-volume Hamiltonians need not have a spectral gap. AU - Henheik, Sven Joscha AU - Teufel, Stefan ID - 10643 JF - Forum of Mathematics, Sigma KW - computational mathematics KW - discrete mathematics and combinatorics KW - geometry and topology KW - mathematical physics KW - statistics and probability KW - algebra and number theory KW - theoretical computer science KW - analysis TI - Adiabatic theorem in the thermodynamic limit: Systems with a gap in the bulk VL - 10 ER - TY - JOUR AB - We investigate the BCS critical temperature Tc in the high-density limit and derive an asymptotic formula, which strongly depends on the behavior of the interaction potential V on the Fermi-surface. Our results include a rigorous confirmation for the behavior of Tc at high densities proposed by Langmann et al. (Phys Rev Lett 122:157001, 2019) and identify precise conditions under which superconducting domes arise in BCS theory. AU - Henheik, Sven Joscha ID - 10623 IS - 1 JF - Mathematical Physics, Analysis and Geometry KW - geometry and topology KW - mathematical physics SN - 1385-0172 TI - The BCS critical temperature at high density VL - 25 ER - TY - JOUR AB - We compute the deterministic approximation of products of Sobolev functions of large Wigner matrices W and provide an optimal error bound on their fluctuation with very high probability. This generalizes Voiculescu's seminal theorem from polynomials to general Sobolev functions, as well as from tracial quantities to individual matrix elements. Applying the result to eitW for large t, we obtain a precise decay rate for the overlaps of several deterministic matrices with temporally well separated Heisenberg time evolutions; thus we demonstrate the thermalisation effect of the unitary group generated by Wigner matrices. AU - Cipolloni, Giorgio AU - Erdös, László AU - Schröder, Dominik J ID - 10732 IS - 8 JF - Journal of Functional Analysis SN - 0022-1236 TI - Thermalisation for Wigner matrices VL - 282 ER - TY - JOUR AB - We consider a correlated NxN Hermitian random matrix with a polynomially decaying metric correlation structure. By calculating the trace of the moments of the matrix and using the summable decay of the cumulants, we show that its operator norm is stochastically dominated by one. AU - Reker, Jana ID - 11135 IS - 4 JF - Random Matrices: Theory and Applications KW - Discrete Mathematics and Combinatorics KW - Statistics KW - Probability and Uncertainty KW - Statistics and Probability KW - Algebra and Number Theory SN - 2010-3263 TI - On the operator norm of a Hermitian random matrix with correlated entries VL - 11 ER - TY - JOUR AB - We show that the fluctuations of the largest eigenvalue of a real symmetric or complex Hermitian Wigner matrix of size N converge to the Tracy–Widom laws at a rate O(N^{-1/3+\omega }), as N tends to infinity. For Wigner matrices this improves the previous rate O(N^{-2/9+\omega }) obtained by Bourgade (J Eur Math Soc, 2021) for generalized Wigner matrices. Our result follows from a Green function comparison theorem, originally introduced by Erdős et al. (Adv Math 229(3):1435–1515, 2012) to prove edge universality, on a finer spectral parameter scale with improved error estimates. The proof relies on the continuous Green function flow induced by a matrix-valued Ornstein–Uhlenbeck process. Precise estimates on leading contributions from the third and fourth order moments of the matrix entries are obtained using iterative cumulant expansions and recursive comparisons for correlation functions, along with uniform convergence estimates for correlation kernels of the Gaussian invariant ensembles. AU - Schnelli, Kevin AU - Xu, Yuanyuan ID - 11332 JF - Communications in Mathematical Physics SN - 0010-3616 TI - Convergence rate to the Tracy–Widom laws for the largest Eigenvalue of Wigner matrices VL - 393 ER - TY - JOUR AB - We consider the quadratic form of a general high-rank deterministic matrix on the eigenvectors of an N×N Wigner matrix and prove that it has Gaussian fluctuation for each bulk eigenvector in the large N limit. The proof is a combination of the energy method for the Dyson Brownian motion inspired by Marcinek and Yau (2021) and our recent multiresolvent local laws (Comm. Math. Phys. 388 (2021) 1005–1048). AU - Cipolloni, Giorgio AU - Erdös, László AU - Schröder, Dominik J ID - 11418 IS - 3 JF - Annals of Probability SN - 0091-1798 TI - Normal fluctuation in quantum ergodicity for Wigner matrices VL - 50 ER - TY - JOUR AB - A recently proposed approach for avoiding the ultraviolet divergence of Hamiltonians with particle creation is based on interior-boundary conditions (IBCs). The approach works well in the non-relativistic case, i.e., for the Laplacian operator. Here, we study how the approach can be applied to Dirac operators. While this has successfully been done already in one space dimension, and more generally for codimension-1 boundaries, the situation of point sources in three dimensions corresponds to a codimension-3 boundary. One would expect that, for such a boundary, Dirac operators do not allow for boundary conditions because they are known not to allow for point interactions in 3D, which also correspond to a boundary condition. Indeed, we confirm this expectation here by proving that there is no self-adjoint operator on a (truncated) Fock space that would correspond to a Dirac operator with an IBC at configurations with a particle at the origin. However, we also present a positive result showing that there are self-adjoint operators with an IBC (on the boundary consisting of configurations with a particle at the origin) that are away from those configurations, given by a Dirac operator plus a sufficiently strong Coulomb potential. AU - Henheik, Sven Joscha AU - Tumulka, Roderich ID - 12110 IS - 12 JF - Journal of Mathematical Physics SN - 0022-2488 TI - Interior-boundary conditions for the Dirac equation at point sources in three dimensions VL - 63 ER - TY - JOUR AB - We prove a general local law for Wigner matrices that optimally handles observables of arbitrary rank and thus unifies the well-known averaged and isotropic local laws. As an application, we prove a central limit theorem in quantum unique ergodicity (QUE): that is, we show that the quadratic forms of a general deterministic matrix A on the bulk eigenvectors of a Wigner matrix have approximately Gaussian fluctuation. For the bulk spectrum, we thus generalise our previous result [17] as valid for test matrices A of large rank as well as the result of Benigni and Lopatto [7] as valid for specific small-rank observables. AU - Cipolloni, Giorgio AU - Erdös, László AU - Schröder, Dominik J ID - 12148 JF - Forum of Mathematics, Sigma KW - Computational Mathematics KW - Discrete Mathematics and Combinatorics KW - Geometry and Topology KW - Mathematical Physics KW - Statistics and Probability KW - Algebra and Number Theory KW - Theoretical Computer Science KW - Analysis SN - 2050-5094 TI - Rank-uniform local law for Wigner matrices VL - 10 ER - TY - JOUR AB - We review recent results on adiabatic theory for ground states of extended gapped fermionic lattice systems under several different assumptions. More precisely, we present generalized super-adiabatic theorems for extended but finite and infinite systems, assuming either a uniform gap or a gap in the bulk above the unperturbed ground state. The goal of this Review is to provide an overview of these adiabatic theorems and briefly outline the main ideas and techniques required in their proofs. AU - Henheik, Sven Joscha AU - Wessel, Tom ID - 12184 IS - 12 JF - Journal of Mathematical Physics SN - 0022-2488 TI - On adiabatic theory for extended fermionic lattice systems VL - 63 ER - TY - JOUR AB - Motivated by Kloeckner’s result on the isometry group of the quadratic Wasserstein space W2(Rn), we describe the isometry group Isom(Wp(E)) for all parameters 0 < p < ∞ and for all separable real Hilbert spaces E. In particular, we show that Wp(X) is isometrically rigid for all Polish space X whenever 0 < p < 1. This is a consequence of our more general result: we prove that W1(X) is isometrically rigid if X is a complete separable metric space that satisfies the strict triangle inequality. Furthermore, we show that this latter rigidity result does not generalise to parameters p > 1, by solving Kloeckner’s problem affirmatively on the existence of mass-splitting isometries. AU - Gehér, György Pál AU - Titkos, Tamás AU - Virosztek, Daniel ID - 12214 IS - 4 JF - Journal of the London Mathematical Society KW - General Mathematics SN - 0024-6107 TI - The isometry group of Wasserstein spaces: The Hilbertian case VL - 106 ER - TY - JOUR AB - We derive a precise asymptotic formula for the density of the small singular values of the real Ginibre matrix ensemble shifted by a complex parameter z as the dimension tends to infinity. For z away from the real axis the formula coincides with that for the complex Ginibre ensemble we derived earlier in Cipolloni et al. (Prob Math Phys 1:101–146, 2020). On the level of the one-point function of the low lying singular values we thus confirm the transition from real to complex Ginibre ensembles as the shift parameter z becomes genuinely complex; the analogous phenomenon has been well known for eigenvalues. We use the superbosonization formula (Littelmann et al. in Comm Math Phys 283:343–395, 2008) in a regime where the main contribution comes from a three dimensional saddle manifold. AU - Cipolloni, Giorgio AU - Erdös, László AU - Schröder, Dominik J ID - 12232 IS - 11 JF - Annales Henri Poincaré KW - Mathematical Physics KW - Nuclear and High Energy Physics KW - Statistical and Nonlinear Physics SN - 1424-0637 TI - Density of small singular values of the shifted real Ginibre ensemble VL - 23 ER - TY - JOUR AB - We consider the eigenvalues of a large dimensional real or complex Ginibre matrix in the region of the complex plane where their real parts reach their maximum value. This maximum follows the Gumbel distribution and that these extreme eigenvalues form a Poisson point process as the dimension asymptotically tends to infinity. In the complex case, these facts have already been established by Bender [Probab. Theory Relat. Fields 147, 241 (2010)] and in the real case by Akemann and Phillips [J. Stat. Phys. 155, 421 (2014)] even for the more general elliptic ensemble with a sophisticated saddle point analysis. The purpose of this article is to give a very short direct proof in the Ginibre case with an effective error term. Moreover, our estimates on the correlation kernel in this regime serve as a key input for accurately locating [Formula: see text] for any large matrix X with i.i.d. entries in the companion paper [G. Cipolloni et al., arXiv:2206.04448 (2022)]. AU - Cipolloni, Giorgio AU - Erdös, László AU - Schröder, Dominik J AU - Xu, Yuanyuan ID - 12243 IS - 10 JF - Journal of Mathematical Physics KW - Mathematical Physics KW - Statistical and Nonlinear Physics SN - 0022-2488 TI - Directional extremal statistics for Ginibre eigenvalues VL - 63 ER - TY - JOUR AB - We prove local laws, i.e. optimal concentration estimates for arbitrary products of resolvents of a Wigner random matrix with deterministic matrices in between. We find that the size of such products heavily depends on whether some of the deterministic matrices are traceless. Our estimates correctly account for this dependence and they hold optimally down to the smallest possible spectral scale. AU - Cipolloni, Giorgio AU - Erdös, László AU - Schröder, Dominik J ID - 12290 JF - Electronic Journal of Probability KW - Statistics KW - Probability and Uncertainty KW - Statistics and Probability TI - Optimal multi-resolvent local laws for Wigner matrices VL - 27 ER - TY - JOUR AB - We study the BCS energy gap Ξ in the high–density limit and derive an asymptotic formula, which strongly depends on the strength of the interaction potential V on the Fermi surface. In combination with the recent result by one of us (Math. Phys. Anal. Geom. 25, 3, 2022) on the critical temperature Tc at high densities, we prove the universality of the ratio of the energy gap and the critical temperature. AU - Henheik, Sven Joscha AU - Lauritsen, Asbjørn Bækgaard ID - 11732 JF - Journal of Statistical Physics KW - Mathematical Physics KW - Statistical and Nonlinear Physics SN - 0022-4715 TI - The BCS energy gap at high density VL - 189 ER - TY - JOUR AB - We study the overlaps between right and left eigenvectors for random matrices of the spherical ensemble, as well as truncated unitary ensembles in the regime where half of the matrix at least is truncated. These two integrable models exhibit a form of duality, and the essential steps of our investigation can therefore be performed in parallel. In every case, conditionally on all eigenvalues, diagonal overlaps are shown to be distributed as a product of independent random variables with explicit distributions. This enables us to prove that the scaled diagonal overlaps, conditionally on one eigenvalue, converge in distribution to a heavy-tail limit, namely, the inverse of a γ2 distribution. We also provide formulae for the conditional expectation of diagonal and off-diagonal overlaps, either with respect to one eigenvalue, or with respect to the whole spectrum. These results, analogous to what is known for the complex Ginibre ensemble, can be obtained in these cases thanks to integration techniques inspired from a previous work by Forrester & Krishnapur. AU - Dubach, Guillaume ID - 10285 JF - Electronic Journal of Probability TI - On eigenvector statistics in the spherical and truncated unitary ensembles VL - 26 ER - TY - GEN AB - We consider a model of the Riemann zeta function on the critical axis and study its maximum over intervals of length (log T)θ, where θ is either fixed or tends to zero at a suitable rate. It is shown that the deterministic level of the maximum interpolates smoothly between the ones of log-correlated variables and of i.i.d. random variables, exhibiting a smooth transition ‘from 3/4 to 1/4’ in the second order. This provides a natural context where extreme value statistics of log-correlated variables with time-dependent variance and rate occur. A key ingredient of the proof is a precise upper tail tightness estimate for the maximum of the model on intervals of size one, that includes a Gaussian correction. This correction is expected to be present for the Riemann zeta function and pertains to the question of the correct order of the maximum of the zeta function in large intervals. AU - Arguin, Louis-Pierre AU - Dubach, Guillaume AU - Hartung, Lisa ID - 9230 T2 - arXiv TI - Maxima of a random model of the Riemann zeta function over intervals of varying length ER - TY - GEN AB - We comment on two formal proofs of Fermat's sum of two squares theorem, written using the Mathematical Components libraries of the Coq proof assistant. The first one follows Zagier's celebrated one-sentence proof; the second follows David Christopher's recent new proof relying on partition-theoretic arguments. Both formal proofs rely on a general property of involutions of finite sets, of independent interest. The proof technique consists for the most part of automating recurrent tasks (such as case distinctions and computations on natural numbers) via ad hoc tactics. AU - Dubach, Guillaume AU - Mühlböck, Fabian ID - 9281 T2 - arXiv TI - Formal verification of Zagier's one-sentence proof ER - TY - JOUR AB - It is well known that special Kubo-Ando operator means admit divergence center interpretations, moreover, they are also mean squared error estimators for certain metrics on positive definite operators. In this paper we give a divergence center interpretation for every symmetric Kubo-Ando mean. This characterization of the symmetric means naturally leads to a definition of weighted and multivariate versions of a large class of symmetric Kubo-Ando means. We study elementary properties of these weighted multivariate means, and note in particular that in the special case of the geometric mean we recover the weighted A#H-mean introduced by Kim, Lawson, and Lim. AU - Pitrik, József AU - Virosztek, Daniel ID - 8373 JF - Linear Algebra and its Applications KW - Kubo-Ando mean KW - weighted multivariate mean KW - barycenter SN - 0024-3795 TI - A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means VL - 609 ER -