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Reflection Positivity in Heisenberg and Ice-Type Models

J.J.A.M. de Groot

Published Jan 1, 2020• 0 citations
OpenAlex

Abstract

This thesis gives a thorough description of the mathematical tool called reflection positivity, which can be used to prove the occurrence of phase transitions in physical models. A major result, although already known, is a theorem that gives tractable conditions on the Hamiltonian such that the Boltzmann functional is reflection positive. In this thesis, the theorem is used to give conditions on free parameters in four different models, such that the model is reflection positive with respect to certain chosen reflections. The evaluated models are (a) the antiferromagnetic quantum Heisenberg model; (b) the spin ice model; (c) the 6-vertex model and (d) the 16-vertex model. For the Heisenberg model we found that for reflections in a reflection plane there are certain parameter values such that the Boltzmann functional is reflection positive, this is an already known and published result. For the spin ice model we found that for the spin invariant reflection there is no symmetry that yields a reflection positive Boltzmann functional, this is a new result. For both the 6-vertex and 16-vertex model we showed that, for certain energy values, the Boltzmann functional is reflection positive with respect to reflections in the diagonal, which are also new results. In the case of the 16-vertex model, this boils down to checking whether or not a matrix is positive semidefinite. Using this result we showed that energy values that allow for the existence of magnetic monopoles do not yield a reflection positive Boltzmann functional. A topic for further research is investigating the occurrence of phase transitions in the models that are shown to be reflection positive, for which chessboard estimates seem to be a promising approach. Furthermore, in this thesis it was not rigorously proved that the spin ice model with a spin inverting reflection gives a reflection positive Boltzmann functional for rotational symmetry or symmetry in a reflection plane. This is believed to be true, but does require further investigation.

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Theory of spin-reflection positivity developed in recent years is reviewed. This theory makes use of symmetries in an electron system and theory of matrix to investigate the ground state properties. Existences of anti- and ferromagnetic long-range orders in itinerant electron systems, and of off-diagonal long-range order are two successful applications of the theory. In this article, the author attempt to summarize exact results proved by utilizing this theory and related topics. First a general theory and basic theorems are introduced. Second, based on the band structures of conduction electrons, existences of a singlet state with strongly antiferromagnetic correlation, a state with both anti- and ferromagnetic long-range orders, and a fully saturated ferromagnetic state are proved. The theory is applied to several of the main theoretical models for strongly correlated electron systems, such as the Heisenberg model, the Hubbard model, the Anderson model, the single- and multichannel Kondo model, and the generalized Hubbard model, and a series of rigorous results are found in these models. Third, it is proved that off-diagonal long-range order and charge-density wave exist in the ground states of the attractive Hubbard model and the generalized Hubbard model. A relation between pseudospin symmetry and the uniform density theorem is introduced. Fourth, the theory is applied successfully to explain experimental observations of oscillatory interlayer magnetic coupling in ultrathin magnetic films. Finally several unsolved problems are discussed. All results introduced in this article are mathematically exact.

International Journal of Modern Physics B•Mar 30, 1998•46 citations

Reflection Positivity, Magnetic Lattice Laplacians and Emergent Phases in Lattice Fermionic Systems

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Emergent phenomena in interacting fermionic systems often arise as collective many-body effects whose formation can be understood by studying the interplay between geometry and symmetry. Understanding such phenomena from first principles remains one of the central challenges of Mathematical Physics, particularly in regimes where perturbative methods fail and non-trivial phases emerge through strongly correlated dynamics. This thesis investigates a class of lattice fermionic systems exhibiting \textit{Spontaneous Symmetry Breaking}, \textit{Flux ordering} and \textit{Topological Order}. Although the models considered belong to different physical frameworks, including Euclidean lattice field theories formulated in terms of Grassmann variables and Quantum Lattice Systems described by operators acting on Hilbert spaces, they share a common mathematical structure. In each case, geometric features encoded by lattice Dirac operators, magnetic fluxes, and discrete gauge fields interact with Reflection Positivity properties that allow for a rigorous non-perturbative analysis. A central role throughout the thesis is indeed played by \textit{Reflection Positivity} and one of its most striking consequences: the \textit{Chessboard Estimates}. These methods provide powerful tools for converting geometric information into quantitative bounds on energies, correlation functions, and phase stability. The first part of the thesis, based on \cite{fabbri2026chirallongrangeordereuclidean}, is devoted to lattice Gross–Neveu models in two dimensional Euclidean Spacetime. After a Hubbard–Stratonovich transformation, the fermionic theories are mapped to effective bosonic systems for which Reflection Positivity can be established. Combining Reflection Positivity methods with Peierls' Argument, we prove the existence of Chiral Long-Range Order for a class of lattice regularizations and obtain quantitative bounds relating the order parameter to the minimizers of the effective potential. The second part, based on \cite{GP}, focuses on fermions coupled to dynamical $\mathbb{Z}_2$-gauge fields. We prove the stability of the $\pi$-flux phase under gauging by showing that monopole excitations possess a strictly positive energy cost. These results imply the emergence of effective Dirac fermions at low energies and provide a rigorous characterization of the semimetallic phase of the model. Finally, we investigate the topological phase obtained by introducing a fermionic mass gap in the $\pi$-flux phase, following \cite{bachmann2026anyonspifluxphasefermionic}. We prove the existence of an almost four-fold degenerate low-energy sector on the torus, separated from the rest of the spectrum by a uniform gap, and construct quasi-local loop and string operators whose algebra exhibits the characteristic toric-code braiding phases. This provides a rigorous realization of topological order emerging from interacting fermions coupled to a dynamical gauge field. Taken together, these results illustrate how Reflection Positivity methods and spectral properties of the Magnetic Laplacian can be combined to obtain rigorous information on the phase diagram of interacting fermionic systems. They reveal a common mechanism underlying symmetry breaking, flux stabilization, and topological order across a variety of lattice models.

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This thesis gives a thorough description of the mathematical tool called reflection positivity, which can be used to prove the occurrence of phase transitions in physical models. A major result, although already known, is a theorem that gives tractable conditions on the Hamiltonian such that the Boltzmann functional is reflection positive. In this thesis, the theorem is used to give conditions on free parameters in four different models, such that the model is reflection positive with respect to certain chosen reflections. The evaluated models are (a) the antiferromagnetic quantum Heisenberg model; (b) the spin ice model; (c) the 6-vertex model and (d) the 16-vertex model. For the Heisenberg model we found that for reflections in a reflection plane there are certain parameter values such that the Boltzmann functional is reflection positive, this is an already known and published result. For the spin ice model we found that for the spin invariant reflection there is no symmetry that yields a reflection positive Boltzmann functional, this is a new result. For both the 6-vertex and 16-vertex model we showed that, for certain energy values, the Boltzmann functional is reflection positive with respect to reflections in the diagonal, which are also new results. In the case of the 16-vertex model, this boils down to checking whether or not a matrix is positive semidefinite. Using this result we showed that energy values that allow for the existence of magnetic monopoles do not yield a reflection positive Boltzmann functional. A topic for further research is investigating the occurrence of phase transitions in the models that are shown to be reflection positive, for which chessboard estimates seem to be a promising approach. Furthermore, in this thesis it was not rigorously proved that the spin ice model with a spin inverting reflection gives a reflection positive Boltzmann functional for rotational symmetry or symmetry in a reflection plane. This is believed to be true, but does require further investigation.
Theory of spin-reflection positivity developed in recent years is reviewed. This theory makes use of symmetries in an electron system and theory of matrix to investigate the ground state properties. Existences of anti- and ferromagnetic long-range orders in itinerant electron systems, and of off-diagonal long-range order are two successful applications of the theory. In this article, the author attempt to summarize exact results proved by utilizing this theory and related topics. First a general theory and basic theorems are introduced. Second, based on the band structures of conduction electrons, existences of a singlet state with strongly antiferromagnetic correlation, a state with both anti- and ferromagnetic long-range orders, and a fully saturated ferromagnetic state are proved. The theory is applied to several of the main theoretical models for strongly correlated electron systems, such as the Heisenberg model, the Hubbard model, the Anderson model, the single- and multichannel Kondo model, and the generalized Hubbard model, and a series of rigorous results are found in these models. Third, it is proved that off-diagonal long-range order and charge-density wave exist in the ground states of the attractive Hubbard model and the generalized Hubbard model. A relation between pseudospin symmetry and the uniform density theorem is introduced. Fourth, the theory is applied successfully to explain experimental observations of oscillatory interlayer magnetic coupling in ultrathin magnetic films. Finally several unsolved problems are discussed. All results introduced in this article are mathematically exact.
Emergent phenomena in interacting fermionic systems often arise as collective many-body effects whose formation can be understood by studying the interplay between geometry and symmetry. Understanding such phenomena from first principles remains one of the central challenges of Mathematical Physics, particularly in regimes where perturbative methods fail and non-trivial phases emerge through strongly correlated dynamics. This thesis investigates a class of lattice fermionic systems exhibiting \textit{Spontaneous Symmetry Breaking}, \textit{Flux ordering} and \textit{Topological Order}. Although the models considered belong to different physical frameworks, including Euclidean lattice field theories formulated in terms of Grassmann variables and Quantum Lattice Systems described by operators acting on Hilbert spaces, they share a common mathematical structure. In each case, geometric features encoded by lattice Dirac operators, magnetic fluxes, and discrete gauge fields interact with Reflection Positivity properties that allow for a rigorous non-perturbative analysis. A central role throughout the thesis is indeed played by \textit{Reflection Positivity} and one of its most striking consequences: the \textit{Chessboard Estimates}. These methods provide powerful tools for converting geometric information into quantitative bounds on energies, correlation functions, and phase stability. The first part of the thesis, based on \cite{fabbri2026chirallongrangeordereuclidean}, is devoted to lattice Gross–Neveu models in two dimensional Euclidean Spacetime. After a Hubbard–Stratonovich transformation, the fermionic theories are mapped to effective bosonic systems for which Reflection Positivity can be established. Combining Reflection Positivity methods with Peierls' Argument, we prove the existence of Chiral Long-Range Order for a class of lattice regularizations and obtain quantitative bounds relating the order parameter to the minimizers of the effective potential. The second part, based on \cite{GP}, focuses on fermions coupled to dynamical Z2\mathbb{Z}_2Z2​-gauge fields. We prove the stability of the π\piπ-flux phase under gauging by showing that monopole excitations possess a strictly positive energy cost. These results imply the emergence of effective Dirac fermions at low energies and provide a rigorous characterization of the semimetallic phase of the model. Finally, we investigate the topological phase obtained by introducing a fermionic mass gap in the π\piπ-flux phase, following \cite{bachmann2026anyonspifluxphasefermionic}. We prove the existence of an almost four-fold degenerate low-energy sector on the torus, separated from the rest of the spectrum by a uniform gap, and construct quasi-local loop and string operators whose algebra exhibits the characteristic toric-code braiding phases. This provides a rigorous realization of topological order emerging from interacting fermions coupled to a dynamical gauge field. Taken together, these results illustrate how Reflection Positivity methods and spectral properties of the Magnetic Laplacian can be combined to obtain rigorous information on the phase diagram of interacting fermionic systems. They reveal a common mechanism underlying symmetry breaking, flux stabilization, and topological order across a variety of lattice models.
Abstract: We develop a novel approach to phase transitions in quantum spin models based on a relation to their classical counterparts. Explicitly, we show that whenever chessboard estimates can be used to prove a phase transition in the classical model, the corresponding quantum model will have a similar phase transition, provided the inverse temperature β and the magnitude of the quantum spins S satisfy β ≪ √ S. From the quantum system we require that it is reflection positive and that it has a meaningful classical limit; the core technical estimate may be described as an extension of the Berezin-Lieb inequalities down to the level of matrix elements. The general theory is applied to prove phase transitions in various quantum spin systems with S ≫ 1. The most notable examples are the quantum orbital-compass model on Z 2 and the quantum 120-degree model on Z 3 which are shown to exhibit symmetry breaking at low-temperatures despite
We analyze the validity of reflection positivity in the classification of invertible phases of quantum spin systems. We provide a mathematical model in which every 2d invertible state admits a reflection-positive representative. We prove that reflection positivity provides a canonical lift from the set of invertible phases to the set of invertible phases protected by a Z/N\mathbb{Z}/NZ/N-rotational symmetry. Using this, we define a refined version of the index recently introduced by the author. This refined version conjecturally provides a microscopic characterization of an invariant that coincides with the chiral central charge c−c_-c−​ when conformal field theory effectively describes the boundary modes.
This thesis investigates properties of classical and quantum spin systems on lattices. These models have been widely studied due to their relevance to condensed matter physics. We identify the ground states of an antiferromagnetic RP2 model, these ground states are very di�erent from the ferromagnetic model and there was some disagreement over their structure, we settle this disagreement. Correlation inequalities are proved for the spin- 1/2 XY model and the ground state of the spin-1 XY model. This provides fresh results in a topic that had been stagnant and allows the proof of some new results, for example existence of some correlation functions in the thermodynamic limit. The occurrence of nematic order at low temperature in a quantum nematic model is proved using the method of reflection positivity and infrared bounds. Previous results on this nematic order were achieved indirectly via a probabilistic representation. This result is maintained in the presence of a small antiferromagnetic interaction, this case was not previously covered. Probabilistic representations for quantum spin systems are introduced and some consequences are presented. In particular, N´eel order is proved in a bilinear-biquadratic spin-1 system at low temperature. This result extends the famous result of Dyson, Lieb and Simon [35]. Dilute spin systems are introduced and the occurrence of a phase transition at low temperature characterised by preferential occupation of the even or odd sublattice of a cubic box is proved. This result is the first of its type for such a mixed classical and quantum system. A probabilistic representation of the spin-1 Bose-Hubbard model is also presented and some consequences are proved.
ABSTRACT. We develop a novel approach to phase transitions in quantum spin models based on a relation to the corresponding classical spin systems. Explicitly, we show that whenever chessboard estimates can be used to prove a phase transition in the classical model, the corresponding quantum model will have a similar phase transition, provided the inverse temperature β and the magnitude of the quantum spins S satisfy β ≪ √ S. From the quantum system we require that it is reflection positive and that it has a meaningful classical limit; the core technical estimate may be described as an extension of the Berezin-Lieb inequalities down to the level of matrix elements. The general theory is further applied to prove phase transitions in various quantum spin systems with S ≫ 1. The most notable examples are the quantum orbital-compass model on Z 2 and the quantum 120degree model on Z 3 which are shown to exhibit symmetry breaking at low-temperatures despite