In their 1972 study of approach to equilibrium, Lanford and Robinson showed that gauge-invariant quasi-free states of lattice fermions maximize entropy among all translation-invariant states with a fixed two-point function, and suggested that the maximizer is unique. In subsequent work on this topic, the uniqueness question re-emerged, together with the problem of whether such quasi-free states are weak Gibbs states. We provide a positive answer to both questions within a class of states whose momentum-space two-point function $\widehat C$ satisfies $0<\widehat C(k)<1$ and belongs to the Wiener algebra of the Brillouin zone. The proof reveals that both the entropy maximization principle and weak Gibbsianity follow directly from the thermodynamic formalism for lattice fermions.In their 1972 study of approach to equilibrium, Lanford and Robinson showed that gauge-invariant quasi-free states of lattice fermions maximize entropy among all translation-invariant states with a fixed two-point function, and suggested that the maximizer is unique. In subsequent work on this topic, the uniqueness question re-emerged, together with the problem of whether such quasi-free states are weak Gibbs states. We provide a positive answer to both questions within a class of states whose momentum-space two-point function $\widehat C$ satisfies $0<\widehat C(k)<1$ and belongs to the Wiener algebra of the Brillouin zone. The proof reveals that both the entropy maximization principle and weak Gibbsianity follow directly from the thermodynamic formalism for lattice fermions.

math-phCross-listedarXiv:2603.14020Mar 17, 2026

Reduced density matrices and entanglement entropy in free lattice models

Ingo Peschel, Viktor Eisler

We review the properties of reduced density matrices for free fermionic or bosonic many-particle systems in their ground state. Their basic feature is that they have a thermal form and thus lead to a quasi-thermodynamic problem with a certain free-particle Hamiltonian. We discuss the derivation of this result, the character of the Hamiltonian and its eigenstates, the single-particle spectra and the full spectra, the resulting entanglement and in particular the entanglement entropy. This is done for various one- and two-dimensional situations, including also the evolution after global or local quenches.We review the properties of reduced density matrices for free fermionic or bosonic many-particle systems in their ground state. Their basic feature is that they have a thermal form and thus lead to a quasi-thermodynamic problem with a certain free-particle Hamiltonian. We discuss the derivation of this result, the character of the Hamiltonian and its eigenstates, the single-particle spectra and the full spectra, the resulting entanglement and in particular the entanglement entropy. This is done for various one- and two-dimensional situations, including also the evolution after global or local quenches.

arXiv:0906.1663Journal of Physics A Mathematical and TheoreticalDec 2, 2009803 citations

Entropy of systems of bosons

Don S. Lemons

Cambridge University Press eBooksAug 29, 20130 citations

Quasi-free states

Jan Dereziński, Christian Gérard

Suppose that we have a state ψ on the polynomial algebra generated by the fields ϕ ( y ) satisfying the CCR or CAR relations. For simplicity, assume that ψ is even , that is, vanishes on odd polynomials. Clearly, this state determines a bilinear form on Y given by the “2-point function” We say that a state ψ is quasi-free if all expectation values can be expressed in terms of (17.1) by the sum over all pairings. This chapter is devoted to a study of (even) quasi-free states, both bosonic and fermionic. This is an important class of states, often used in physical applications. Fock vacuum states belong to this class. It also includes Gibbs states of quadratic Hamiltonians. Representations obtained by the GNS construction from quasi-free states will be called quasi-free representations . They are usually reducible. Many interesting concepts from the theory of von Neumann algebras can be nicely illustrated in terms of quasi-free representations. Quasi-free states can be easily realized on Fock spaces, using the so-called Araki–Woods, resp. Araki–Wyss representations in the bosonic, resp. fermionic case. Under some additional assumptions, in particular in the case of a finite number of degrees of freedom, these representations can be obtained as follows. First we consider a Fock space equipped with a quadratic Hamiltonian. Then we perform the GNS construction with respect to the corresponding Gibbs state. Finally, we apply an appropriate Bogoliubov rotation.Suppose that we have a state ψ on the polynomial algebra generated by the fields ϕ ( y ) satisfying the CCR or CAR relations. For simplicity, assume that ψ is even , that is, vanishes on odd polynomials. Clearly, this state determines a bilinear form on Y given by the “2-point function” We say that a state ψ is quasi-free if all expectation values can be expressed in terms of (17.1) by the sum over all pairings. This chapter is devoted to a study of (even) quasi-free states, both bosonic and fermionic. This is an important class of states, often used in physical applications. Fock vacuum states belong to this class. It also includes Gibbs states of quadratic Hamiltonians. Representations obtained by the GNS construction from quasi-free states will be called quasi-free representations . They are usually reducible. Many interesting concepts from the theory of von Neumann algebras can be nicely illustrated in terms of quasi-free representations. Quasi-free states can be easily realized on Fock spaces, using the so-called Araki–Woods, resp. Araki–Wyss representations in the bosonic, resp. fermionic case. Under some additional assumptions, in particular in the case of a finite number of degrees of freedom, these representations can be obtained as follows. First we consider a Fock space equipped with a quadratic Hamiltonian. Then we perform the GNS construction with respect to the corresponding Gibbs state. Finally, we apply an appropriate Bogoliubov rotation.

Cambridge University Press eBooksMar 6, 20130 citations

\widehat C

satisfies

0<\widehat C(k)<1

and belongs to the Wiener algebra of the Brillouin zone. The proof reveals that both the entropy maximization principle and weak Gibbsianity follow directly from the thermodynamic formalism for lattice fermions.