Fidelity mechanics is formalized as a framework for investigating critical phenomena in quantum many-body systems. Fidelity temperature is introduced for quantifying quantum fluctuations, which, together with fidelity entropy and fidelity internal energy, constitute three basic state functions in fidelity mechanics, thus enabling us to formulate analogues of the four thermodynamic laws and Landauer's principle at zero temperature. Fidelity flows, which are irreversible, are defined and may be interpreted as an alternative form of renormalization group flows. Thus, fidelity mechanics offers a means to characterize both stable and unstable fixed points: divergent fidelity temperature for unstable fixed points and zero-fidelity temperature and (locally) maximal fidelity entropy for stable fixed points. In addition, fidelity entropy behaves differently at an unstable fixed point for topological phase transitions and at a stable fixed point for topological quantum states of matter. A detailed analysis of fidelity mechanical-state functions is presented for six fundamental models-the quantum spin-1/2 XY model, the transverse-field quantum Ising model in a longitudinal field, the quantum spin-1/2 XYZ model, the quantum spin-1/2 XXZ model in a magnetic field, the quantum spin-1 XYZ model, and the spin-1/2 Kitaev model on a honeycomb lattice for illustrative purposes. We also present an argument to justify why the thermodynamic, psychological/computational, and cosmological arrows of time should align with each other, with the psychological/computational arrow of time being singled out as a master arrow of time.
Fidelity mechanics is formalized as a framework for investigating critical phenomena in quantum many-body systems. Fidelity temperature is introduced for quantifying quantum fluctuations, which, together with fidelity entropy and fidelity internal energy, constitute three basic state functions in fidelity mechanics, thus enabling us to formulate analogues of the four thermodynamic laws and Landauer's principle at zero temperature. Fidelity flows, which are irreversible, are defined and may be interpreted as an alternative form of renormalization group flows. Thus, fidelity mechanics offers a means to characterize both stable and unstable fixed points: divergent fidelity temperature for unstable fixed points and zero-fidelity temperature and (locally) maximal fidelity entropy for stable fixed points. In addition, fidelity entropy behaves differently at an unstable fixed point for topological phase transitions and at a stable fixed point for topological quantum states of matter. A detailed analysis of fidelity mechanical-state functions is presented for six fundamental models-the quantum spin-1/2 XY model, the transverse-field quantum Ising model in a longitudinal field, the quantum spin-1/2 XYZ model, the quantum spin-1/2 XXZ model in a magnetic field, the quantum spin-1 XYZ model, and the spin-1/2 Kitaev model on a honeycomb lattice for illustrative purposes. We also present an argument to justify why the thermodynamic, psychological/computational, and cosmological arrows of time should align with each other, with the psychological/computational arrow of time being singled out as a master arrow of time.
It is shown that the fidelity, a basic notion of quantum information science, may be used to characterize quantum phase transitions, regardless of what type of internal order is present in quantum many-body states. If the fidelity of two given states vanishes, then there are two cases: (1) they are in the same phase if the distinguishability results from irrelevant local information; or (2) they are in different phases if the distinguishability results from relevant long-distance information. The different effects of irrelevant and relevant information are quantified, which allows us to identify unstable and stable fixed points (in the sense of renormalization group theory). A physical implication of our results is the occurrence of the orthogonality catastrophe near the transition points.
This dissertation puts forward the concept of fidelity as a way to understand and characterize quantum phase transitions involving topological phases. It lies at the interphase between Condensed Matter and Quantum Phase Transitions. ❧ We find that the fidelity metric, which characterizes the degree of distinghuishability between two quantum states, can serve as a useful tool in the understanding of transitions involving topological phases. ❧ The methodology does not require a complete knowledge of the physical mechanism involved in the occurrence of the critical point, such as the identification of a local order parameter and a symmetry breaking mechanism, the fundamental pillars of critical phenomena. Topological phases are unique in that they cannot be classified under these methodologies, due to their lack of local order parameters. It is one of the purposes of this thesis to understand how well we can characterize the phase boundaries between these phases using fidelity concepts. ❧ In the first part, we introduce two quantum models that exhibit a phase transition between an ordered and a topological phase, as quantified by the topological entanglement entropy. In the thermodynamic limit, the different components of the fidelity metric exhibit a divergence, signalling the occurrence of the critical point. ❧ In the second part, we introduce models that have phase boundaries between two distinct topological phases, and we show that the fidelity is successful at identifying the occurrence of a critical region. Moreover, a finite size scaling analysis is carried out, from which we determine the value of critical exponents.
We compare the roles of fidelity and entanglement in characterizing renormalization group flows and quantum phase transitions. It turns out that the scaling parameter extracted from fidelity for different ground states succeeds to capture nontrivial information including stable and unstable fixed points, whereas the von Neumann entropy as a bipartite entanglement measure (or equivalently, majorization relations satisfied by the spectra of the reduced density matrix along renormalization group flows) often fails, as far as the intrinsic irreversibility-information loss along renormalization group flows-is concerned. We also clarify an intimate connection between the von Neumman entropy, majorization relations, and fidelity. The relevance to Zamolodchikov's c theorem is indicated.
Quantum phase transition arises in general as second order phase transition at zero temperature, tuned by a non-thermal parameter such as pressure, doping or a magnetic field. The point in the phase diagram of the material in which different phases meet is called a quantum critical point (QCP). Physics around QCPs are of extensive current interest because the critical quantum fluctuations influence the physical properties in a wide temperature range (quantum criticality), and are believed to be responsible for many emergent physical properties such as non-Fermi liquids and unconventional superconductivity. In this research we explore dynamics and thermodynamics near QCPs via investigating three classes of models, which all have real material correspondence. Specifically first, we study local dynamics in a perturbed quantum critical Ising chain with E8 symmetry, where we show the local dynamical spin susceptibility has a singular dependence on frequency, but differs from the diffusion form. The nuclear magnetic resonance (NMR) relaxation rate at low temperatures depends exponentially on the inverse temperature, whose prefactor we also determine. We propose NMR experiments as a means to further test the applicability of the E8 description for CoNb2O6. Second, we investigate the thermodynamic properties of itinerant ferromagnets near quantum critical points, described by the quantum Landau-Ginzburg theory. We provide a regularized perturbative renormalization group procedure to calculate the free energy. We further carry out numerical calculations on thermodynamic quantities, capturing not only the leading critical behaviors, but also the subleading and nonsingular contributions. We demonstrate various thermodynamic signatures of quantum criticality, including the entropy accumulation effect and the divergence of the specific heat coefficient. A detailed comparison to the recent experimental results on an itinerant ferromagnet Sr3Ru2O7 is also presented. Third, we explore Ising-nematic and magnetic phases and their transitions in iso-electronically doped iron pnictides by carrying out a large-N study of an effective low-energy Ginzburg-Landau model for these systems. We demonstrate that the magnetic and Ising orders transitions are concurrent at zero temperature, and both transitions are weakly first-order, which is consistent with RG-based prediction and experimental observations.
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 work presented in this thesis meanly addresses two topics in theoretical physics which are quantum thermodynamics and topological order. In the first case, physicists are trying to build up a theory able to describe quite in general phenomena involving heat and energy exchanges in quantum systems. The second topic, instead, is related to exotic phenomena and states of matter like the quantum Hall effect (QHE) or topological insulators and topological superconductors. In the first part od the thesis we define the quantum dynamics for closed and open systems. This is a key ingredient to address the field of quantum thermodynamics. Then, after an introductory part about the quantum thermodynamic transformations, we move toward the field of nonequilibrium fluctuation relations. We address the problem of irreversibility in classical as well as quantum mechanics. Here we present one of our main result. We characterize the "thermodynamic" irreversible adiabatic evolution of a quantum system starting such branch in a thermal equilibrium state at inverse temperature ßi. We give the amount of thermodynamic entropy growth for the process. As direct application of the preceding result we then address a quantum Otto cycle (QOC) working at finite power. We saw that the increasing of irreversible character of the evolution affects the main figures of merit of the cycle. The second part of the thesis addresses the field of topological order. At first we introduce the concept of topological orders, classes and invariants. Then we introduce the well known Kitaev model for 1 D superconductors. This model predicts Majorana zero mode at the ends of the wire (the 1 D system). MZM are topological states showing great resistance against disorder, local perturbations and any dissipative element. Then we consider a generalized Kitaev model where long range interactions are accounted. We get rich topological phase diagrams showing the presence of several MZM per edge. We study the appearing/disappearing dynamics of the modes according to the time reversal symmetry, that is fundamental in the study of topological phase. The phase diagrams we obtained also show the presence of massive edge modes. In this last case the topological invariants do not well describe any transition. At last we focused on a very limit cases where MZM are obtained at finite length of the wire. Such cases are really interesting since the great advance we can get from the finiteness of the wire in an experimental setup. The last part is about single electron tunneling devices. Here we got a different ability to work as "heat-to-current harvester" for a device using quantum dots respect to an analogue one using metallic dots. These different arguments find their unity by considering recent scientific works in which heat transport is addressed in single electron transistor devices where some element of the circuit shows a topological behaviour. This is a perfect system from hich we can get new transport phenomena.