arXiv subjects

Dimitrios Ampelogiannis

Publications and source records attributed to Dimitrios Ampelogiannis.

5 recordsLinked to original sources

arXiv2026

Application of the theory of C*-algebras to the emergence of hydrodynamics in quantum many-body systems

This Ph.D. thesis reports on progress in rigorously establishing hydrodynamic principles from the microscopic Hamiltonian dynamics of quantum many-body systems in a general, non-model-specific manner. Using the C*-algebra framework of statistical mechanics, we treat systems directly in the thermodynamic limit, primarily focusing on quantum lattice models where tools such as Lieb-Robinson bounds yield rigorous statements. We thus provide a proof-of-principle that large-scale behaviours can indeed be seen as emerging from microscopic dynamics, with mathematical proof. We first report on ergodicity results in short-range models with exponentially decaying or finite-range interactions. We show that time-averaged observables converge to their ensemble averages and decorrelate from all other observables almost everywhere within the light-cone defined by Lieb-Robinson bounds. This relaxation property indicates the loss of information at large scales, from which we prove a Boltzmann-Gibbs principle: at the Euler scaling limit of large time and distance, observables project onto hydrodynamic modes (extensive conserved quantities), within correlation functions. These results hold independently of microscopic details, capturing the physical idea that such details are lost at large space-time scales. Regarding finer scales of hydrodynamics, we discuss rigorous lower bounds on the strength of diffusion. We establish a general result on the clustering of n-th order connected correlations within C* dynamical systems. These results are applied to obtain a strictly positive lower bound on the diffusion constant of chaotic open quantum spin chains with nearest-neighbor interactions. This thesis underlines the universality of hydrodynamic principles, provides a framework for establishing them rigorously, and sets the stage for future progress toward the goal of proving the hydrodynamic equations.

math-ph↗

arXiv2025

Rigorous bound on hydrodynamic diffusion for chaotic open spin chains

The emergence of diffusion is one of the deepest physical phenomena observed in many-body interacting, chaotic systems. But establishing rigorously that correlation functions, say of the spin, expand diffusively, remains one of the most important problems of mathematical physics. We establish for the first time, with Lindbladian evolution, a lower bound on spin diffusion in chaotic, translation-invariant, nearest-neighbor open quantum spin-1/2 chain satisfying a local detailed-balance condition and strong conservation of magnetisation. The bound is strictly positive if and only if the local quantum jumps transport spin. Physically, the bound comes from the spreading effects of initial-state macroscopic fluctuations, a mechanism which occurs whenever spin is an interacting ballistic mode. Chaoticity means that the Hilbert space of extensive charges is spanned by magnetisation; we expect this to be generic. Our main tool is the Green-Kubo formula, the mathematical technique of projection over quadratically extensive charges, and appropriate correlation decay bounds recently established. Because Lindbladian dynamics is not reversible, the Green-Kubo spin diffusion strength includes a contribution due to irreversibility, which we interpret as encoding the hydrodynamic entropy production that may occur in the forgotten environment. This, we show, vanishes for certain choices of interaction parameters, for which the Lindbladian dynamics becomes reversible. Our methods can be extended to finite or short ranges, higher spins, and other non-Hamiltonian systems such as quantum circuits. As we argue, according to the theory of nonlinear fluctuating hydrodynamics, we further expect these systems to display superdiffusion, and thus have infinite diffusivity; however this is still beyond the reach of mathematical rigour.

cond-mat.stat-mech↗

arXiv2024

Clustering of higher order connected correlations in C$^*$ dynamical systems

In the context of $C^*$ dynamical systems, we consider a locally compact group $G$ acting by $^*$-automorphisms on a C$^*$ algebra $\mathfrak{U}$ of observables, and assume a state of $\mathfrak{U}$ that satisfies the clustering property with respect to a net of group elements of $G$. That is, the two-point connected correlation function vanishes in the limit on the net, when one observable is translated under the group action. Then we show that all higher order connected correlation functions (Ursell functions, or classical cumulants) and all free correlation functions (free cumulants, from free probability) vanish at the same rate in that limit. Additionally, we show that mean clustering, also called ergodicity, extends to higher order correlations. We then apply those results to equilibrium states of quantum spin lattice models. Under certain assumptions on the range of the interaction, high temperature Gibbs states are known to be exponentially clustering w.r.t. space translations. Combined with the Lieb-Robinson bound, one obtains exponential clustering for space-time translations outside the Lieb-Robinson light-cone. Therefore, by our present results, all the higher order connected and free correlation functions will vanish exponentially under space-time translations outside the Lieb-Robinson light cone, in high temperature Gibbs states. Another consequence is that their long-time averaging over a space-time ray vanishes for almost every ray velocity.

math-ph↗

arXiv2023

Almost everywhere ergodicity in quantum lattice models

We rigorously examine the ergodic properties of quantum lattice models with short range interactions, in the C* algebra formulation of statistical mechanics. Ergodicity results, in the context of group actions on C* algebras, assume that the algebra is asymptotically abelian, which is not the case for time evolution. The Lieb-Robinson bound tells us that the spatial extent of the time evolution of operators is exponentially small outside a light-cone, defined by the Lieb-Robinson velocity. More precisely, this means that the algebra of observables is asymptotically abelian only in a space-like region. This implies a form of ergodicity outside the light-cone, but what happens within it? We show that the long-time average of connected correlation functions of observables, along space-time rays of almost every speed, vanishes. This holds in any state that is invariant under space-time translations and that satisfies weak clustering properties in space. Further, we show that the long-time limit of the n-th moment, with respect to the state, of a ray-averaged observable converges to the n-th power of its ensemble average, which means that there are no fluctuations of ray averages in the long time limit. This is a statement of ergodicity. The ray averages can also be performed in a way that accounts for oscillations, showing that correlations (and moments) of the ray averaged observables cannot sustainably oscillate in the long time limit. Finally, we show that in the GNS representation of the algebra of observables, for any KMS state with the above properties, the long-time limit of the ray averaged commutator converges (in the strong operator topology (SOT)) to 0 and the ray average of any observable converges (in the SOT) to a multiple of the identity. This indicates that observables get "thinner" almost everywhere within the light-cone.

math-ph↗

arXiv2023

Long-time dynamics in quantum spin lattices: ergodicity and hydrodynamic projections at all frequencies and wavelengths

Obtaining rigorous and general results about the non-equilibrium dynamics of extended many-body systems is a difficult task. In quantum lattice models with short-range interactions, the Lieb-Robinson bound tells us that the spatial extent of operators grows at most linearly in time. But what happens within this light-cone? We discuss rigorous results on ergodicity and the emergence of the hydrodynamic scale, which establish fundamental principles at the root of non-equilibrium physics. One key idea of the present work is that general structures of hydrodynamics at the Euler scale follow independently from the details of the microscopic dynamics, and in particular do not necessitate chaos; they are consequences of "extensivity". Another crucial observation is that these apply at arbitrary frequencies and wavelengths. That is, long-time oscillatory behaviours can be reproduced from a natural extension of standard hydrodynamic notions, thus enlarging the hydrodynamic paradigm beyond the zero-frequency / infinite-wavelength point that it traditionally addresses.

cond-mat.stat-mech↗