For the general class of quasifree fermionic right mover/left mover systems over the infinitely extended two-sided discrete line introduced in Aschbacher [Rev. Math. Phys. 35, 2330001 (2023)] within the algebraic framework of quantum statistical mechanics, we study the von Neumann entropy of a contiguous subsystem of finite length in interaction with its environment. In particular, under the assumption of spatial translation invariance, we analyze the asymptotic behavior of the von Neumann entropy for large subsystem lengths and prove that its leading order density is, in general, nonvanishing and displays the signature of a mixture of the independent thermal species underlying the right mover/left mover system. As special cases, the formalism covers so-called nonequilibrium steady states, thermal equilibrium states, and ground states. Moreover, for general Fermi functions, we derive a necessary and sufficient criterion for the von Neumann entropy density to vanish.
For the general class of quasifree fermionic right mover/left mover systems over the infinitely extended two-sided discrete line introduced in Aschbacher [Rev. Math. Phys. 35, 2330001 (2023)] within the algebraic framework of quantum statistical mechanics, we study the von Neumann entropy of a contiguous subsystem of finite length in interaction with its environment. In particular, under the assumption of spatial translation invariance, we analyze the asymptotic behavior of the von Neumann entropy for large subsystem lengths and prove that its leading order density is, in general, nonvanishing and displays the signature of a mixture of the independent thermal species underlying the right mover/left mover system. As special cases, the formalism covers so-called nonequilibrium steady states, thermal equilibrium states, and ground states. Moreover, for general Fermi functions, we derive a necessary and sufficient criterion for the von Neumann entropy density to vanish.
Abstract We consider the ideal Fermi gas of indistinguishable particles without spin but with electric charge, confined to a Euclidean plane R2 R 2 perpendicular to an external constant magnetic field of strength B>0 B > 0 . We assume this (infinite) quantum gas to be in thermal equilibrium at zero temperature, that is, in its ground state with chemical potential μ≥B μ ≥ B (in suitable physical units). For this (pure) state we define its local entropy S(Λ) S ( Λ ) associated with a bounded (sub)region Λ⊂R2 Λ ⊂ R 2 as the von Neumann entropy of the (mixed) local substate obtained by reducing the infinite-area ground state to this region Λ Λ of finite area ∣Λ∣ | Λ | . In this setting we prove that the leading asymptotic growth of S(LΛ) S ( L Λ ) , as the dimensionless scaling parameter L>0 L > 0 tends to infinity, has the form LB∣∂Λ∣ L B | ∂ Λ | up to a precisely given (positive multiplicative) coefficient which is independent of Λ Λ and dependent on B and μ μ only through the integer part of (μ/B−1)/2 ( μ / B - 1 ) / 2 . Here we have assumed the boundary curve ∂Λ ∂ Λ of Λ Λ to be sufficiently smooth which, in particular, ensures that its arc length ∣∂Λ∣ | ∂ Λ | is well-defined. This result is in agreement with a so-called area-law scaling (for two spatial dimensions). It contrasts the zero-field case B=0 B = 0 , where an additional logarithmic factor ln(L) ln ( L ) is known to be present. We also have a similar result, with a slightly more explicit coefficient, for the simpler situation where the underlying single-particle Hamiltonian, known as the Landau Hamiltonian, is restricted from its natural Hilbert space L2(R2) L 2 ( R 2 ) to the eigenspace of a single but arbitrary Landau level. Both results extend to the whole one-parameter family of quantum Rényi entropies. As opposed to the case B=0 <m
In a remarkable paper [Phys. Rev. Lett. 96, 100503 (2006)], Gioev and Klich conjectured an explicit formula for the leading asymptotic growth of the spatially bipartite von Neumann entanglement entropy of noninteracting fermions in multidimensional Euclidean space at zero temperature. Based on recent progress by one of us (A. V. S.) in semiclassical functional calculus for pseudodifferential operators with discontinuous symbols, we provide here a complete proof of that formula and of its generalization to Rényi entropies of all orders α>0. The special case α=1/2 is also known under the name logarithmic negativity and often considered to be a particularly useful quantification of entanglement. These formulas exhibiting a "logarithmically enhanced area law" have been used already in many publications.
The universal dynamical phenomenon of von Neumann entropy plays an important role in understanding the nature of open many-body systems, as it uncovers general collective properties driven by correlation and quantum dynamics. Here, we reveal a universal time evolution behavior of bipartite von Neumann entropy in open quantum systems with dissipation into the surrounding environments. The coupling with the environment leads to a rapid growth of the entropy, resulting in an exotic short-time evolution law, S ( t ) ∼ − t ln t . Such a fast growth behavior is proved to be a general phenomenon for generic many-body systems in dissipative environments, as long as the systems exhibit well-defined fermionic quasiparticles. Interestingly, even for strongly interacting Luttinger liquids, which display vanishing fermionic but emergent bosonic quasiparticles, the same dynamical phenomenon is still found to persist. These results demonstrate a universal Liouvillian dynamical feature that has no counterpart in isolated Hamiltonian systems.
Abstract Within the rigorous axiomatic framework for the description of quantum mechanical systems with a large number of degrees of freedom, we construct the so-called nonequilibrium steady state for the quasifree fermionic system corresponding to the isotropic XY chain in which a finite sample, subject to a local gauge breaking anisotropy perturbation, is coupled to two thermal reservoirs at different temperatures. Using time dependent and stationary scattering theory, we rigorously prove, from first principles, that the nonequilibrium system under consideration is thermodynamically nontrivial, i. e., that its entropy production rate is strictly positive.
The leading asymptotic large-scale behaviour of the spatially bipartite entanglement entropy (EE) of the free Fermi gas infinitely extended in multidimensional Euclidean space at zero absolute temperature, T = 0, is by now well understood. Here, we present and discuss the first rigorous results for the corresponding EE of thermal equilibrium states at Tgt0. The leading large-scale term of this thermal EE turns out to be twice the first-order finite-size correction to the infinite-volume thermal entropy (density). Not surprisingly, this correction is just the thermal entropy on the interface of the bipartition. However, it is given by a rather complicated integral derived from a semiclassical trace formula for a certain operator on the underlying one-particle Hilbert space. But in the zero-temperature limit Tdownarrow0, the leading large-scale term of the thermal EE considerably simplifies and displays a mathrmln(1/T)-singularity which one may identify with the known logarithmic enhancement at T = 0 of the so-called area-law scaling.
We propose a generalization of the quantum entropy introduced by Wigner and von Neumann [Z. Phys. 57, 30 (1929)10.1007/BF01339852]. Our generalization is applicable to both quantum pure states and mixed states. When the dimension N of the Hilbert space is large, this generalized Wigner-von Neumann (GWvN) entropy becomes independent of the choices of basis and is asymptotically equal to lnN in the sense of typicality. The dynamic evolution of our entropy is also typical, and is reminiscent of quantum H theorem proved by von Neumann. For a composite system, the GWvN entropy is typically additive; for the microcanonical ensemble, it is equivalent to the Boltzmann entropy; and for a system entangled with environment, it is consistent with the familiar von Neumann entropy, which is zero for pure states. In addition, the GWvN entropy can be used to derive the Gibbs ensemble.