We define a Banach space M 1 \mathcal {M}_{1} of models for fermions or quantum spins in the lattice with long range interactions and make explicit the structure of (generalized) equilibrium states for any m ∈ M 1 \mathfrak {m}\in \mathcal {M}_{1} . In particular, we give a first answer to an old open problem in mathematical physics – first addressed by Ginibre in 1968 within a different context – about the validity of the so–called Bogoliubov approximation on the level of states. Depending on the model m ∈ M 1 \mathfrak {m}\in \mathcal {M}_{1} , our method provides a systematic way to study all its correlation functions at equilibrium and can thus be used to analyze the physics of long range interactions. Furthermore, we show that the thermodynamics of long range models m ∈ M 1 \mathfrak {m}\in \mathcal {M}_{1} is governed by the non–cooperative equilibria of a zero–sum game, called here thermodynamic game.
We define a Banach space M 1 \mathcal {M}{1} of models for fermions or quantum spins in the lattice with long range interactions and make explicit the structure of (generalized) equilibrium states for any m ∈ M 1 \mathfrak {m}\in \mathcal {M}{1} . In particular, we give a first answer to an old open problem in mathematical physics – first addressed by Ginibre in 1968 within a different context – about the validity of the so–called Bogoliubov approximation on the level of states. Depending on the model m ∈ M 1 \mathfrak {m}\in \mathcal {M}{1} , our method provides a systematic way to study all its correlation functions at equilibrium and can thus be used to analyze the physics of long range interactions. Furthermore, we show that the thermodynamics of long range models m ∈ M 1 \mathfrak {m}\in \mathcal {M}{1} is governed by the non–cooperative equilibria of a zero–sum game, called here thermodynamic game.
Realistic effective interparticle interactions of quantum many-body systems are widely seen as being short-range. However, the rigorous mathematical analysis of this type of model turns out to be extremely difficult, in general, with many important fundamental questions remaining open still nowadays. By contrast, mean-field models come from different approximations or Ansätze, and are thus less realistic, in a sense, but are technically advantageous, by allowing explicit computations while capturing surprisingly well many real physical phenomena. Here, we establish a precise mathematical relation between mean-field and short-range models, by using the long-range limit that is known in the literature as the Kac limit. If both attractive and repulsive long-range forces are present then it turns out that the limit mean-field model is not necessarily what one traditionally guesses. One important innovation of our study, in contrast with previous works on the subject, is the fact that we are able to show the convergence of equilibrium states, i.e., of all correlation functions. This paves the way for studying phase transitions, or at least important fingerprints of them like strong correlations at long distances, for models having interactions whose ranges are finite, but very large. It also sheds a new light on mean-field models. Even on the level of pressures, our results go considerably further than previous ones, by allowing, for instance, a continuum of long-range interaction components, as well as very general short-range Hamiltonians for the "free" part of the model.
This paper presents a novel mathematical framework for modeling phase transitions in quantum spin systems with long-range interactions using nonlinear variational inequalities (NVIs). Traditional methods such as mean-field theory and Landau-Ginzburg models often fall short in capturing spatial inhomogeneities, critical phenomena, and constraints inherent in realistic spin systems. Our approach addresses these limitations by reformulating the energy-based governing equations as an NVI problem defined over a suitable convex set of physically admissible spin configurations. We incorporate both Ising and Heisenberg-type quantum lattice models with power-law decaying interactions to accurately represent nonlocal effects. This research provides rigorous mathematical preliminaries, including operator-theoretic foundations in Hilbert and Banach spaces, and prove existence and uniqueness results under monotonicity and coercivity assumptions. Analytical insights include bifurcation analysis, critical point theory, and regularity of solutions, offering a robust theoretical foundation for understanding phase transitions. A finite element–Galerkin discretization is developed to simulate the variational system numerically. Simulation results demonstrate the emergence of ordered phases, spin textures, and domain formation, consistent with known physical behavior. Comparisons with classical models reveal the advantages of the NVI framework in capturing phase boundaries and metastable states with higher fidelity. The proposed framework also opens pathways for extending the analysis to quantum entanglement, topological order, and non-equilibrium phenomena. This research thus contributes both to the mathematical theory of variational inequalities and to the physical modeling of quantum spin systems, providing tools for deeper exploration of complex condensed matter phenomena.
We study correlations in fermionic lattice systems with long-range interactions in thermal equilibrium. We prove a bound on the correlation decay between anti-commuting operators and generalize a long-range Lieb-Robinson type bound. Our results show that in these systems of spatial dimension D with, not necessarily translation invariant, two-site interactions decaying algebraically with the distance with an exponent α≥2D, correlations between such operators decay at least algebraically with an exponent arbitrarily close to α at any non-zero temperature. Our bound is asymptotically tight, which we demonstrate by a high temperature expansion and by numerically analyzing density-density correlations in the 1D quadratic (free, exactly solvable) Kitaev chain with long-range pairing.
Quantum Mechanics was established as the theory of the microscopic world, which allowed to understand processes in atoms and molecules. Its emergence led to a new scientific paradigm that quickly spread to different research fields. Two relevant examples are Quantum Thermodynamics and Quantum Many-Body Theory, where the former aims to characterize thermodynamic processes in quantum systems and the latter intends to understand the properties of quantum many-body systems. In this thesis, we tackle some of the questions in the overlap between these disciplines, focusing on the concepts of temperature and correlations. Specifically, it contains results on the following topics: locality of temperature, correlations in long-range interacting systems and thermometry at low temperature. The problem of locality of temperature is considered for a system at thermal equilibrium and consists in studying whether it is possible to assign a temperature to any of the subsystems of the global system such that both local and global temperatures are equal. We tackle this problem in two different settings, for generic one-dimensional spin chains and for a bosonic system with a phase transition at non-zero temperature. In the first case, we consider generic one-dimensional translation-invariant spin systems with short-range interactions and prove that it is always possible to assign a local temperature equal to the global one for any temperature, including at criticality. For the second case, we consider a three-dimensional discretized version of the Bose-Einstein model at the grand canonical ensemble for some temperature and particle density, and characterize its non-zero-temperature phase transition. Then, we show that temperature is locally well-defined at any temperature and at any particle density, including at the phase transition. Additionally, we observe a qualitative relation between correlations and locality of temperature in the system. Moving to correlations, we consider fermionic two-site long-range interacting systems at thermal equilibrium. We show that correlations between anti-commutative operators at non-zero temperature are upper bounded by a function that decays polynomially with the distance and with an exponent that is equal to the interaction exponent, which characterizes the interactions in the Hamiltonian. Moreover, we show that our bound is asymptotically tight and that the results extend to density-density correlations as well as other types of correlations for quadratic and fermionic Hamiltonians with long-range interactions. Regarding the results on thermometry, we consider a bosonic model and prove that strong coupling between the probe and the system can boost the thermal sensitivity for low temperature. Furthermore, we provide a feasible measurement scheme capable of producing optimal estimates at the considered regime.
The study of quantum dynamics is an increasingly important aspect of atomic, molecular and optical physics, as well as condensed-matter physics and other disciplines. It is essential in order to understand non-equilibrium phenomena which do not fit within the framework of equilibrium thermodynamics, such as many-body localization, dynamical quantum phase transitions, and non-equilibrium steady states in driven-dissipative systems. However, the exact calculation of dynamical observables is extremely difficult in general for several reasons, which makes it challenging for theoretical predictions and analyses to keep pace with rapidly advancing experiments. In this thesis I use and develop several computational methods to investigate the dynamics of quantum many-body systems, with the two goals of better understanding the evolution of local observables and correlation functions in spin and Fermion systems as well as creating and testing more effective numerical methods for solving quantum dynamics problems. I explore quantum dynamics by using Hamiltonian parameter quenches in several commonly-used models in AMO physics and elsewhere: the Fermi-Hubbard model, the XXZ model, and the transverse-Ising model. The first represents a system of short-range interacting Fermions, while the latter two represent spin systems with different spin-spin couplings and magnetic field terms. All of these models can be realized in current experiments. First, I derive an exact solution for an extreme interaction quench in the Fermi-Hubbard model, from a strongly-interacting initial state to a noninteracting final Hamiltonian. From this result I show that there are nontrivial transient connected correlations, despite the fact that the system has very high temperature and a noninteracting Hamiltonian. Those features would suppress such correlations in equilibrium, but the dynamical correlations are still significant. Then, I develop and examine two numerical methods for solving dynamics of the spin expectation values and correlation functions in the transverse-Ising and XXZ models. Both methods rely on approximating a thermodynamically large system with a much smaller and more computationally tractable cluster of lattice sites, or a series of clusters. The first method is a dynamical numerical linked cluster expansion (NLCE). NLCEs have already been used successfully on systems in thermodynamic equilibrium, and I extended the technique to directly compute the dynamics of an observable after a quantum quench. I review the general NLCE method, describe my dynamical variant of it, and compute the quench dynamics of spin and correlation observables in the transverse-Ising and XXZ models. The second method, which I term the adaptive-boundary cluster method, relies on recently refined bounds on information propagation in quantum systems to choose a cluster of lattice sites that minimizes the finite size error bound of a dynamical computation. I briefly review these bounds, and describe a new algorithm which uses them to construct a cluster whose boundary is chosen according to a related error estimate. Such an adaptive-boundary cluster has the property that there exists a computable upper bound on the finite size error of observables measured at the cluster center, and this error bound is nearly minimal when compared to that of other clusters of the same size. I then compute the quench dynamics of spin and correlation observables in the transverse-Ising model using the adaptive boundary clusters. I apply this method to both uniform and strongly disordered systems. To evaluate the effectiveness of both dynamical NLCE and adaptive-boundary clusters, I compare the results obtained using them to results from a standard and general purpose numerical method: exact solvers applied a rectangular cluster with periodic boundary conditions. Both of the new methods usually converge faster and are more accurate than the standard method for comparable cluster sizes. This thesis illustrates that choosing numerical methods which take advantage of the geometric and entanglement structure of the initial state and Hamiltonian can substantially enhance our ability to study quantum dynamics, by reducing the computational cost of numerically exact solutions. I conclude with a discussion of various avenues for future research to build on this work.
Abstract We take advantage of the simplifications made possible by assuming the system to be infinite from the beginning, to study some exactly solvable models in statistical mechanics. (1) For fermi lattice systems with quadratic hamiltonians the unifying concept of our treatment is a precise formulation of the idea of a linear system. This concept makes it possible to define a dynamical matrix which incorporates all the information one needs for equilibrium and non‐equilibrium problems. As an example, we consider the alternating X Y chain. Furthermore we pay some attention to ergodicity, pointing out the similarity to the infinite harmonic crystal. (2) For molecular field models, including the BCS model, the method is to consider an arbitrary extremal homogeneous state, to construct the dynamics in the representation determined by that state, and then to use the KMS condition to single out the equilibrium state of the infinite system. A synthesis of these methods – (1) and (2) – enables us to give a new simple treatment of the BCS model and a direct analysis of its mathematical structure. In addition we show the equivalence of Kac and van der Waals (spin) systems in the so‐called van der Waals limit. Finally we indicate briefly how to deal with molecular field models with polynomial and more general interactions.