We present details on a physical realization, in a many‐body Hamiltonian system, of the abstract probabilistic structure recently exhibited by Gell‐Mann, Sato and one of us (C.T.), that the nonadditive entropy Sq = k[1−Trρ̂q]/[q−1] (ρ̂≡density matrix; S1 = −kTrρ̂ ln ρ̂) can conform, for an anomalous value of q (i.e., q≠1), to the classical thermodynamical requirement for the entropy to be extensive. Moreover, we find that the entropic index q provides a tool to characterize both universal and nonuniversal aspects in quantum phase transitions (e.g., for a L‐sized block of the Ising ferromagnetic chain at its T = 0 critical transverse field, we obtain limL→∞S37−6(L)/L = 3.56±0.03). The present results suggest a new and powerful approach to measure entanglement in quantum many‐body systems. At the light of these results, and similar ones for a d = 2 Bosonic system discussed by us elsewhere, we conjecture that, for blocks of linear size L of a large class of Fermionic and Bosonic d‐dimensional many‐body Hamiltonians with short‐range interaction at T = 0, we have that the additive entropy S1(L)∝[Ld−1−1]/(d−1) (i.e., ln L for d = 1, and Ld−1 for d>1), hence it is not extensive, whereas, for anomalous values of the index q, we have that the nonadditive entropy Sq(L)∝Ld(∀d), i.e., it is extensive. The present discussion neatly illustrates that entropic additivity and entropic extensivity are quite different properties, even if they essentially coincide in the presence of short‐range correlations.
We present details on a physical realization, in a many‐body Hamiltonian system, of the abstract probabilistic structure recently exhibited by Gell‐Mann, Sato and one of us (C.T.), that the nonadditive entropy Sq = k[1−Trρ̂q]/[q−1] (ρ̂≡density matrix; S1 = −kTrρ̂ ln ρ̂) can conform, for an anomalous value of q (i.e., q≠1), to the classical thermodynamical requirement for the entropy to be extensive. Moreover, we find that the entropic index q provides a tool to characterize both universal and nonuniversal aspects in quantum phase transitions (e.g., for a L‐sized block of the Ising ferromagnetic chain at its T = 0 critical transverse field, we obtain limL→∞S37−6(L)/L = 3.56±0.03). The present results suggest a new and powerful approach to measure entanglement in quantum many‐body systems. At the light of these results, and similar ones for a d = 2 Bosonic system discussed by us elsewhere, we conjecture that, for blocks of linear size L of a large class of Fermionic and Bosonic d‐dimensional many‐body Hamiltonians with short‐range interaction at T = 0, we have that the additive entropy S1(L)∝[Ld−1−1]/(d−1) (i.e., ln L for d = 1, and Ld−1 for d>1), hence it is not extensive, whereas, for anomalous values of the index q, we have that the nonadditive entropy Sq(L)∝Ld(∀d), i.e., it is extensive. The present discussion neatly illustrates that entropic additivity and entropic extensivity are quite different properties, even if they essentially coincide in the presence of short‐range correlations.
The Boltzmann-Gibbs-von Neumann entropy of a large part (of linear size L ) of some (much larger) d -dimensional quantum systems follows the so-called area law (as for black holes), i.e., it is proportional to Ld-1. Here we show, for d=1,2 , that the (nonadditive) entropy Sq satisfies, for a special value of q not equal to 1, the classical thermodynamical prescription for the entropy to be extensive, i.e., Sq proportional variant Ld. Therefore, we reconcile with classical thermodynamics the area law widespread in quantum systems. Recently, a similar behavior was exhibited in mathematical models with scale-invariant correlations [C. Tsallis, M. Gell-Mann, and Y. Sato, Proc. Natl. Acad. Sci. U.S.A.102 15377 (2005)]. Finally, we find that the system critical features are marked by a maximum of the special entropic index q.
Entanglement in a pure state of a many-body system can be characterized by the Rényi entropies S^{(\alpha)}=\ln\textrm{tr}(\rho^\alpha)/(1-\alpha) S ( α ) = ln tr ( ρ α ) / ( 1 − α ) of the reduced density matrix \rho ρ of a subsystem. These entropies are, however, difficult to access experimentally and can typically be determined for small systems only. Here we show that for free fermionic systems in a Gaussian state and with particle number conservation, S^{(2)} S ( 2 ) can be tightly bound—from above and below—by the much easier accessible Rényi number entropy S^{(2)}_N=-\ln \sum_n p^2(n) S N ( 2 ) = − ln ∑ n p 2 ( n ) which is a function of the probability distribution p(n) p ( n ) of the total particle number in the considered subsystem only. A dynamical growth in entanglement, in particular, is therefore always accompanied by a growth—albeit logarithmically slower—of the number entropy. We illustrate this relation by presenting numerical results for quenches in non-interacting one-dimensional lattice models including disorder-free, Anderson-localized, and critical systems with off-diagonal (bond) disorder.
The interactions and correlations among the constituents of manybody systems are manifested in characteristic physical properties such as ferromagnetism, superconductivity, etc. A list of some that have been studied in the last century is given in the Box (see below). A parallel development in quantum information was initially slow but over the past two decades progress has been very rapid. Fundamentally this is another aspect of quantum correlations in composite systems arising from the twin features of the superposition principle and the tensor product structure of state space. These features are not utilized in the same manner in quantum many-body physics. In the Box, a corresponding parallel list with properties of many-body systems is given because there has been an interplay between the two research efforts and their understanding. The basic quantum mechanical principles apply to both cases except different aspects are utilized because the goals are different in each. Both areas of investigation are based on a probabilistic foundation with a variational underpinning founded on an “entropy” maximization, which may be called Quantum Statistical Mechanics [1]. The specific form of the entropy as a functional of the density matrix will be made explicit presently. See Table I for definitions of density matrix and associated quantities. The equilibrium properties of the many-body systems are then given by a maximization of von Neumann entropy subject to certain constraints such as the average value of the Hamiltonian of the system “energy”. This leads to the familiar exponential probabilities of the Boltzmann-Gibbs (BG) form. Any nonequilibrium properties are studied by a quantum time evolution equation. Non-equilibrium properties such as the anomalous relaxation in time are often analyzed with the quantum version of the Tsallis entropy with an entropic index, q, which leads to powerlaw probabilities in contrast to the BG type [2]. (See the Box in the Introduction of Tsallis and Boon). In quantum information, the maximum entropy scheme is not of use because its origins are elsewhere as will be made clear presently. The evolution is replaced by processing of information
This paper studies quantum systems with a finite number of degrees of freedom in the context of non-extensive thermodynamics. A trial density matrix, obtained by heuristic methods, is proved to be the equilibrium density matrix. If the entropic index q is larger than 1 then existence of the trial equilibrium density matrix requires that q is less than some critical value qc which depends on the rate by which the eigenvalues of the Hamiltonian diverge. Existence of a unique equilibrium density matrix is proved if in addition q<2 holds. For q between 0 and 1, such that 2 <q+ qc , the free energy has at least one minimum in the set of trial density matrices. If a unique equilibrium density matrix exists then it is necessarily one of the trial density matrices. Note that this is a finite rank operator, which means that in equilibrium high energy levels have zero probability of occupancy. 1. Introduction The formalism of non-extensive thermodynamics started more than 10 years ago with the introduction by C. Tsallis [1] of a family of entropies, parameterized with a parameter q called the entropic index. It has developed gradually to a collection of mostly phenomenological results with a large number of applications, some more convincing than others. Nevertheless, part of the physics community is still sceptic about the need and physical relevance of the theory. By providing mathematical proofs for some of the fundaments of the formalism this paper tries to improve its credibility and to provide the necessary base for further extension. Some of the results presented in the present paper, in particular concerning q>1-statistics, have already been published in [2]. Here, missing details are filled in and the q<1-case is added. The structure of the paper is as follows. In the...
The standard Theory of Critical Phenomena within Boltzmann-Gibbs statistical mechanics is known to successfully predict, at any temperature differing from a continuous-phase-transition critical one, the (finite) values for thermodynamically relevant quantities such as isothermal magnetic susceptibility, specific heat, correlation length, Grüneisen parameter, and similar ones. However, at the precise critical point, empirically unreachable infinities emerge within that approach at the N→∞ thermodynamical limit. Such divergences, although mathematically correct, may be seen as poorly informative ones. Indeed, they make no distinction at all between say the Ising, XY, Heisenberg, and other models. Moreover, for say the XY model, they make no distinction whether the spins are, for instance, 1/2, 1 or 3/2, or whether the interactions are say first-neighbor or first-and-second-neighbor ones. The introduction of nonadditive entropic functionals within a consistently generalized statistical mechanics substantially improves the situation, i.e., it regularizes the theory providing informative finite values instead of mere infinities. This undoubtedly enriching fact is illustrated [Soares et al., Phys. Rev. B 111, L060409 (2025)] for the Grüneisen parameter at the zero-temperature quantum critical point of a spin-1/2 ferromagnetic Ising chain in the presence of a transverse magnetic field, by introducing the nonadditive entropic functional Sq with the unique value q=q⋆≡37-6=0.0828…, which precisely guarantees the thermodynamical extensivity of the entropy as mandated by the preservation of the Legendre-transform structure of classical thermodynamics.
Entanglement in J 1 – J 2 , S = 1/2 quantum spin chains with an impurity is studied using analytic methods as well as large scale numerical density matrix renormalization group methods. The entanglement is investigated in terms of the von Neumann entropy, S = −Trρ A logρ A , for a subsystem A of size r of the chain. The impurity contribution to the uniform part of the entanglement entropy, S imp , is defined and analysed in detail in both the gapless, J 2 ≤ J 2 c , as well as the dimerized phase, J 2 > J 2 c , of the model. This quantum impurity model is in the universality class of the single channel Kondo model and it is shown that in a quite universal way the presence of the impurity in the gapless phase, J 2 ≤ J 2 c , gives rise to a large length scale, ξ K , associated with the screening of the impurity, the size of the Kondo screening cloud. The universality of Kondo physics then implies scaling of the form S imp ( r /ξ K , r / R ) for a system of size R . Numerical results are presented clearly demonstrating this scaling. At the critical point, J 2 c , an analytic approach based on a Fermi liquid picture, valid at distances and energy scales , is developed and analytic results at T = 0 are obtained showing S imp = πξ K [1+π(1− r / R )cot(π r / R )]/(12 R ) for finite R . For T >0, in the thermodynamic limit, we find S imp = [π 2 ξ K T /(6 v )]coth(2π rT / v ), with v the spin-wave velocity. In the dimerized phase an appealing picture of the entanglement is developed in terms of a thin soliton (TS) ansatz and the notions of impurity valence bonds (IVB) and single particle entanglement (SPE) are introduced. The TS-ansatz permits a variational calculation of the complete entanglement in the dimerized phase that appears to be exact in the thermodynamic limit at the Majumdar–Ghosh point, J 2 = J 1 /2, and surprisingly precise even close to the critical point J 2 c . In the appendices the TS-ansatz is further used to calculate and with high precision at the Majumdar–Ghosh point and the relation between the finite temperature entanglement entropy, S ( T ), and the thermal entropy, S th ( T ), is discussed. Finally, the alternating part of S imp is discussed, together with its relation to the boundary induced dimerization.