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.

arXiv:1902.08047Memoirs of the American Mathematical SocietyNov 14, 20129 citations

Equilibrium states of a semi-quantum lattice system

Akitaka Kishimoto

Reports on Mathematical PhysicsDec 1, 197710 citations

On the equivalence of the KMS condition and the variational principle for quantum lattice systems with mean-field interactions

J.-B. Bru, W. de Siqueira Pedra, Rafael Sussumu Yamaguti Miada

We extend Araki's well-known results on the equivalence of the KMS condition and the variational principle for equilibrium states of quantum lattice systems with short-range interactions, to a large class of models possibly containing mean-field interactions (representing an extreme form of long-range interactions). This result is reminiscent of van Hemmen's work on equilibrium states for mean-field models. The extension was made possible by our recent outcomes on states minimizing the free energy density of mean-field models on the lattice, as well as on the infinite volume dynamics for such models.We extend Araki's well-known results on the equivalence of the KMS condition and the variational principle for equilibrium states of quantum lattice systems with short-range interactions, to a large class of models possibly containing mean-field interactions (representing an extreme form of long-range interactions). This result is reminiscent of van Hemmen's work on equilibrium states for mean-field models. The extension was made possible by our recent outcomes on states minimizing the free energy density of mean-field models on the lattice, as well as on the infinite volume dynamics for such models.

arXiv:2103.00901Advances in Theoretical and Mathematical PhysicsJan 1, 20221 citation

Variational approach to open quantum systems with long-range competing interactions

Dawid A. Hryniuk, Marzena Hanna Szymańska

Competition between short- and long-range interactions underpins many emergent phenomena in nature. Despite rapid progress in their experimental control, computational methods capable of accurately simulating open quantum many-body systems with complex long-ranged interactions at scale remain scarce. Here, we address this limitation by introducing an efficient and scalable approach to dissipative quantum lattices in one and two dimensions, combining matrix product operators and time-dependent variational Monte Carlo. We showcase the versatility, effectiveness, and unique methodological advantages of our algorithm by simulating the non-equilibrium dynamics and steady states of spin- $\frac{1}{2}$ lattices with competing algebraically-decaying interactions for as many as N = 200 sites, revealing the emergence of spatially-modulated magnetic order far from equilibrium. This approach offers promising prospects for advancing our understanding of the complex non-equilibrium properties of a diverse variety of experimentally-realizable quantum systems with long-ranged interactions, including Rydberg atoms, ultracold dipolar molecules, and trapped ions. Far-from-equilibrium many-body quantum systems with complex non-local interactions are of significant fundamental and practical importance. Here the authors demonstrate how variational Monte Carlo in combination with compact tensor network parameterization of the density matrix enables accurate and scalable simulation of such open quantum systems.Competition between short- and long-range interactions underpins many emergent phenomena in nature. Despite rapid progress in their experimental control, computational methods capable of accurately simulating open quantum many-body systems with complex long-ranged interactions at scale remain scarce. Here, we address this limitation by introducing an efficient and scalable approach to dissipative quantum lattices in one and two dimensions, combining matrix product operators and time-dependent variational Monte Carlo. We showcase the versatility, effectiveness, and unique methodological advantages of our algorithm by simulating the non-equilibrium dynamics and steady states of spin-$$\frac{1}{2}$$ lattices with competing algebraically-decaying interactions for as many as N = 200 sites, revealing the emergence of spatially-modulated magnetic order far from equilibrium. This approach offers promising prospects for advancing our understanding of the complex non-equilibrium properties of a diverse variety of experimentally-realizable quantum systems with long-ranged interactions, including Rydberg atoms, ultracold dipolar molecules, and trapped ions. Far-from-equilibrium many-body quantum systems with complex non-local interactions are of significant fundamental and practical importance. Here the authors demonstrate how variational Monte Carlo in combination with compact tensor network parameterization of the density matrix enables accurate and scalable simulation of such open quantum systems.

arXiv:2510.01543Communications PhysicsJan 10, 20261 citation

\frac{1}{2}

lattices with competing algebraically-decaying interactions for as many as N = 200 sites, revealing the emergence of spatially-modulated magnetic order far from equilibrium. This approach offers promising prospects for advancing our understanding of the complex non-equilibrium properties of a diverse variety of experimentally-realizable quantum systems with long-ranged interactions, including Rydberg atoms, ultracold dipolar molecules, and trapped ions. Far-from-equilibrium many-body quantum systems with complex non-local interactions are of significant fundamental and practical importance. Here the authors demonstrate how variational Monte Carlo in combination with compact tensor network parameterization of the density matrix enables accurate and scalable simulation of such open quantum systems.