In dieser Arbeit wird eine mögliche Erweiterung des Adiabatischen Theorems auf seine Gültigkeit für quantenmechanische Quenches, also nicht-adiabatische Änderungen, überprüft. Speziell werden dabei das Transverse Field Ising Model (TFIM) und eine seiner Erweiterungen, das Axial Next Nearest Neighbour Ising Model (ANNNI) untersucht. Die angesprochene Erweiterung des adiabatischen Theorems wird wie folgt formuliert: Solange Zustände derselben Phase betrachtet werden, sind die Überlapps zwischen instantanen Grundzuständen des anfänglichen und zeitentwickelten Hamilton-Operators die größtmöglichen. Überlapps zwischen dem Grundzustand des ersten Systems und anderen Zuständen des zweiten Systems sind demnach kleiner. Im TFIM wird diese conjecture für beide der vorkommenden magnetischen Phasen, der paramagnetischen (PM) und der ferromagnetischen (FM) Phase, bestätigt. Die Ergebnisse im ANNNI-Modell sind uneindeutig. Einerseits können Verletzungen der conjecture in der FM-Phase numerischen Fehlern zugeschrieben werden, weshalb sie durch Anlegen eines kleinen Störfeldes aufgehoben werden können. Ergebnisse in der Antiphase (AP) stimmen mit den Vorhersagen der conjecture überein. Dasselbe gilt für die Ergebnisse der Übergangsphase (FP) in L=12 mit Ausnahme von Quench-Startpunkten, die nahe am Tripelpunkt liegen. Andererseits stellen finite-size Effekte, die die Ergebnisse signifikant verzerren, die einzig mögliche Erklärung für die Abweichungen in der PM-Phase dar. Um diese Annahme zu bestätigen oder zu widerlegen, bedarf es jedoch einer anderen numerischen Methode, die in der Lage ist größere Systemgrößen zu berechnen.
We have derived a twelfth-order, high-temperature series for the axial next-nearest-neighbor Ising (ANNNI) models on the fcc and the sc lattices, and we present detailed results for one of the fcc models. We have determined the phase boundaries of these models and located the Lifshitz point from the intersection of the phase boundary with the ``disorder'' line, where the maximum in the susceptibility \ensuremath{\chi}(q) begins to occur at nonzero q signaling the onset of modulated correlations. As expected, the Lifshitz point appears to be an inflection point of the phase boundary. The effective susceptibility index shows the predicted universal values for definite regions of the parameter space, e.g., XY-like in the midrange modulated regime, with the expected smooth crossover near the multicritical points. Throughout the modulated regime from the Lifshitz point to the decoupling point, the variation in the critical wave vector shows only small deviations from mean-field predictions. Our best estimates for the Lifshitz-point indices, ${\ensuremath{\gamma}}_{\mathit{l}}$=1.62\ifmmode\pm\else\textpm\fi{}0.12, 4${\ensuremath{\nu}}_{\mathit{l}4}$=1.63\ifmmode\pm\else\textpm\fi{}0.10, and ${\mathrm{\ensuremath{\alpha}}}_{\mathit{l}}$=0.20\ifmmode\pm\else\textpm\fi{}0.15, differ from previous estimates. We are confident that our estimates are more reliable not only because we have longer series for a close-packed lattice but also primarily because we have analyzed our series along the appropriate scaling direction.
Describing and understanding the consequences of competing interactions remains profoundly challenging in both classical and quantum systems, as it is difficult to identify suitable order parameters, thereby hindering the characterization of certain phases such as the floating phase found in the one-dimensional axial next-nearest-neighbor Ising (ANNNI) model in the presence of the frustration interactions between the nearest and next-nearest neighbor sites. In this work, we employ the pattern picture to explore the frustration physics in such a model. This picture has been comprehensively detailed in our previous article [Yang and Luo, Phys. Rev. E \textbf{112}, 044102 (2025)]. Here, we apply it to the ANNNI model with periodic boundary conditions, considering system sizes ranging from
L=16 to
L=128. Our results demonstrate that the ground state of the system comprises four phases: ferromagnetic, paramagnetic, floating, and
⟨2,2⟩ antiphase. The transition from the ferromagnetic to the paramagnetic phase is continuous, analogous to that in the transverse-field Ising model, while the transition from the floating phase to the antiphase is first order. Furthermore, the floating phase exhibits particularly intriguing characteristics: states with distinct domain structures emerge successively as the frustration parameter increases. As the system size grows, this succession becomes progressively denser, leading to the reasonable inference that it eventually approaches a continuous variation in the thermodynamic limit. To validate the effectiveness of our picture, we computed the second derivative of the ground-state energy, which exhibits multiple dips within the floating phase
−consistent with the pattern language.
We investigate the effectiveness of the stabilizer Rényi entropy (SRE), a quantifier associated with non-stabilizer resources (quantum magic), as an indicator of quantum phase transitions. Specifically, we analyze the behavior of the purity-corrected SRE of reduced density matrices in the ground states of two one-dimensional spin models: the transverse axial next-nearest-neighbor Ising (TANNNI) model and the quantum compass model (QCM). The ground state of the TANNNI model is obtained using exact diagonalization techniques, while the QCM is analyzed using the Jordan--Wigner (JW) transformation followed by Bogoliubov diagonalization of the resulting quadratic fermionic Hamiltonian. For the TANNNI model, the purity-corrected SRE successfully detects the antiphase--floating phase transition in the high-frustration regime, while in the low-frustration regime the raw (purity-uncorrected) SRE reproduces the known ferromagnetic--paramagnetic phase boundaries more accurately. For the QCM, the purity-corrected SRE exhibits a clear signature near the isotropic point
Jx/Jz=1, where the system undergoes a first-order quantum phase transition. Our results establish SRE of reduced states as a complementary probe of quantum criticality and provide further insight into the role of non-stabilizer resources in many-body quantum phase transitions.