Abstract Critical phenomena have been extensively investigated both theoretically and experimentally in many fields, such as condensed matter physics, biology, e.g. brain criticality, and cosmology. In particular, the behavior of response functions right at critical points (CPs) is highly topical. It turns out that in the frame of Boltzmann–Gibbs–von Neumann–Shannon approach, the extensive character of entropy breaks down at CPs. The latter implies diverging susceptibilities, which is at odds with experimental observations. Here, we investigate the influence of the spin magnitude S on the quantum Grüneisen parameter Γ q 0 K right at CPs for the 1D Ising model under a transverse magnetic field. Our findings are fourfold: i) for higher S , Γ q 0 K is increased, but remains finite, reflecting the enhancement of the Hilbert space dimensionality; ii) the Schmidt decomposition theorem recovers the extensivity of the nonadditive q -entropy S q only for a special value of the entropic index q ; iii) the universality class in the frame of S q depends only on the symmetry of the system; iv) we propose an experimental setup to explore finite-size effects in connection with the Hilbert space occupation at CPs. Our findings unveil both universal and non-universal aspects of quantum criticality in terms of Γ q 0 K and S q .
Abstract Critical phenomena have been extensively investigated both theoretically and experimentally in many fields, such as condensed matter physics, biology, e.g. brain criticality, and cosmology. In particular, the behavior of response functions right at critical points (CPs) is highly topical. It turns out that in the frame of Boltzmann–Gibbs–von Neumann–Shannon approach, the extensive character of entropy breaks down at CPs. The latter implies diverging susceptibilities, which is at odds with experimental observations. Here, we investigate the influence of the spin magnitude S on the quantum Grüneisen parameter Γ q 0 K right at CPs for the 1D Ising model under a transverse magnetic field. Our findings are fourfold: i) for higher S , Γ q 0 K is increased, but remains finite, reflecting the enhancement of the Hilbert space dimensionality; ii) the Schmidt decomposition theorem recovers the extensivity of the nonadditive q -entropy S q only for a special value of the entropic index q ; iii) the universality class in the frame of S q depends only on the symmetry of the system; iv) we propose an experimental setup to explore finite-size effects in connection with the Hilbert space occupation at CPs. Our findings unveil both universal and non-universal aspects of quantum criticality in terms of Γ q 0 K and S q .
Critical phenomena have been extensively investigated both theoretically and experimentally in many fields, such as condensed matter physics, biology, e.g., brain criticality, and cosmology. In particular, the behaviour of response functions right at critical points (CPs) is highly topical. It turns out that in the frame of Boltzmann-Gibbs-von Neumann-Shannon approach, the extensive character of entropy breaks down at CPs. The latter implies diverging susceptibilities, which is at odds with experimental observations. Here, we investigate the influence of the spin magnitude S on the quantum Grüneisen parameter Γq0K right at CPs for the 1D Ising model under a transverse magnetic field. Our findings are fourfold: i) for higher S, Γq0K is increased, but remains finite, reflecting the enhancement of the Hilbert space dimensionality; ii) the Schmidt decomposition theorem recovers the extensivity of the nonadditive q-entropy Sq only for a special value of the entropic index q; iii) the universality class in the frame of Sq depends only on the symmetry of the system; iv) we propose an experimental setup to explore finite-size effects in connection with the Hilbert space occupation at CPs. Our findings unveil both universal and non-universal aspects of quantum criticality in terms of Γq0K and Sq.
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.
The magnetic Grüneisen ratio, defined as Γ g ≡ (1/ T )(∂ T /∂ g ) S , characterizes the magnetocaloric effect (MCE) driven by an external field g and serves as a highly sensitive probe for field-induced quantum critical points (QCPs). Near a QCP, Γ g displays diverging behaviors described by a universal scaling function depending on the universality class. Here, we systematically investigate the universal scaling functions of Grüneisen ratio Γ g in typical one-dimensional (1D) and two-dimensional (2D) quantum spin systems, including the transverse-field Ising model, XY model, q -state quantum Potts model ( q = 3, 4), and the J 1 - J 2 columnar dimer model. Our approach employs the linearized tensor renormalization group (LTRG) method for infinite-size 1D systems and the stochastic series expansion quantum Monte Carlo (SSE QMC) simulations for 2D systems, enabling precise calculations of the Γ g near QCPs. We extract universal scaling functions via data collapse analysis. This universal magnetocaloric framework offers a quantitative tool for analyzing measurements in quantum materials and for guiding the search for quantum critical cooling applications.
The information-geometric origin of fidelity susceptibility and its utility as a universal probe of quantum criticality in many-body settings have been widely discussed. Here we explore the metric response of quantum relative entropy (QRE), by tracing out all but n adjacent sites from the ground state of spin chains of finite length N, as a parameter of the corresponding Hamiltonian is varied. The diagonal component of this metric defines a susceptibility of the QRE that diverges at quantum critical points (QCPs) in the thermodynamic limit. We study two spin-1/2 models as examples, namely the integrable transverse field Ising model (TFIM) and a non-integrable Ising chain with three-spin interactions. We demonstrate distinct scaling behaviors for the peak of the QRE susceptibility as a function of N: namely a square logarithmic divergence in TFIM and a power-law divergence in the non-integrable chain. This susceptibility encodes uncertainty of entanglement Hamiltonian gradients and is also directly connected to other information measures such as Petz-Rényi entropies. We further show that this susceptibility diverges even at finite N if the subsystem size, n, exceeds a certain value when the Hamiltonian is tuned to its classical limits due to the rank of the RDMs being finite; unlike the divergence associated with the QCPs which require N→∞.
The Grüneisen ratio, defined as Γg≡(1/T)(∂T/∂g)S, serves as a highly sensitive probe for detecting quantum critical points (QCPs) driven by an external feild g and for characterizing the magnetocaloric effect (MCE). Near a QCP, the Grüneisen ratio displays a universal divergence which is governed by a universality-class-dependent scaling function stemming from the scale invariance. In this work, we systematically investigate the universal scaling functions of Grüneisen ratio in both one-dimensional (1D) and two-dimensional (2D) quantum spin systems, including the transverse-field Ising model, the spin-1/2 Heisenberg model, the quantum q-state Potts model (q=3,4) and the J1-J2 columnar dimer model. Our approach employs the thermal tensor-network method for infinite-size 1D systems and the stochastic series expansion quantum Monte Carlo (SSE QMC) simulations for 2D systems, enabling precise calculations of the Grüneisen ratio near QCPs. Through data collapse analysis, we extract the corresponding scaling functions, which establish quantitative frameworks to interpret magnetocaloric experiments and guide the development of ultralow-temperature refrigeration.
We consider the scaling behavior of thermodynamic quantities in the one-dimensional transverse field Ising model near its quantum critical point (QCP). Our study has been motivated by the question about the thermodynamical signatures of this paradigmatic quantum critical system and, more generally, by the issue of how quantum criticality accumulates entropy. We find that the crossovers in the phase diagram of temperature and (the nonthermal control parameter) transverse field obey a general scaling ansatz, and so does the critical scaling behavior of the specific heat and magnetic expansion coefficient. Furthermore, the Gr\"uneisen ratio diverges in a power-law way when the QCP is accessed as a function of the transverse field at zero temperature, which follows the prediction of quantum critical scaling. However, at the critical field, upon decreasing the temperature, the Gr\"uneisen ratio approaches a constant instead of showing the expected divergence. We are able to understand this unusual result in terms of a peculiar form of the quantum critical scaling function for the free energy; the contribution to the Gr\"uneisen ratio vanishes at the linear order in a suitable Taylor expansion of the scaling function. In spite of this special form of the scaling function, we show that the entropy is still maximized near the QCP, as expected from the general scaling argument. Our results establish the telltale thermodynamic signature of a transverse-field Ising chain, and will thus facilitate the experimental identification of this model quantum-critical system in real materials.