Numerical Study on Phase Transitions in Long-range Quantum Chain and Flat-band Superconductor

Abstract

This dissertation is composed of two main subjects: one is the criticality of the long-range quantum Ising chains, and the other is the phase transitions of the flat-band superconductors. In the first main subject, using the variational Monte Carlo method embedding the restricted Boltzmann machine as a trial wavefunction, we characterize the criticality of the antiferromagnetic transverse-field Ising chain (LR-TFIC) with the long-range interaction algebraically decaying with respect to distance. For various long- range decaying exponent α_LR, we examine the critical exponents and central charge, and we identify the consistent results with the previous studies: the critical exponents are very close to the short-range Ising criticality, but the central charge deviates from the short-range limit one. In a further test of the critical Binder ratio, we find that the universal Binder ratio in the short-range limit does not hold in α_LR < 2. Moreover, we perform the conformal invariance test with the spin-spin correlation function and find that the deviation of the form of the measured correlation function from the conformal-field-theory (CFT) description is pronounced as α_LR decreases, although a precise threshold of the conformal invariance breakdown is yet to be determined. As a further study to obtain a more precise threshold of the conformal symmetry breakdown, we examine the crosscap overlap, which is another indicator of the conformal symmetry, for the antiferromagnetic and ferromagnetic LR-TFIC by using the exact diagonalization (ED) method for up to 36-site chains. Although we should account for the finite-size effects caused by the limitation of the accessible system size of the ED method, we find more concrete evidence that the threshold is located below α_LR < 3 for the antiferromagnetic LR-TFIC. For the ferromagnetic LR-TFIC, we identify that our estimates of the critical crosscap overlap deviate from the universal value of the 2D Ising criticality in α_LR < 4, whereas the previous analytic functional renormalization group studies and other numerical studies refer to the range breaking the 2D Ising criticality as α_LR ≲ 3. In the long-range Kitaev chain, we obtain the exact closed-form expression of the crosscap overlap. As the long-range decaying exponent α varies, we determine the threshold deviating from the value of the 2D Ising universal crosscap overlap on the critical line: α > 1 at µ = 1 and α > 0 at µ = −1. In the second main subject, we investigate flat-band superconductivity exhibiting (1) the features of the superconductor-insulator transition induced by spatial disorders by using the mean-field approach and (2) the Berezinskii-Kosterlitz-Thouless (BKT) type transition from superconductor to normal phase induced by the thermal fluctuation by using the determinant quantum Monte Carlo (DQMC) method. For the first one, on the Kagome lattice, we find that preserving the flat-band degeneracy against disorder makes the flat-band superconductivity more robust than the random hopping (HOP) disorder counterpart, breaking the flat-band degeneracy, although both disorder types eventually exhibit characteristics of the superconductor-insulator transition within the mean-field theory. Moreover, it turns out that the flat-band-preserving (FBP) disorder keeps the flat-band characteristic linear behavior of the superfluid weight as a function of interaction strength; in contrast, the HOP disorder shows the exponential behavior expected from a dispersive band system. Additionally, using the exact diagonalization, we propose the occupation-spectrum structure of the one-particle density matrix (OPDM) linked to the non-interacting flat-band states. We identify the connection between the OPDM spectral structure and the robustness of the flat-band superconducting state. Finally, we determine a more precise superconducting critical temperature of the Lieb-lattice Hubbard model at flat-band half filling by using the DQMC method. We estimate the critical temperature based on the universal jump relation, called the Nelson-Kosterlitz criterion, of the superfluid weight: T_c = 0.032(1) at |U | = 1 and T_c = 0.050(2) at |U | = 2. These results are much lower than the temperatures in the previous mean-field studies T_c ≈ 0.06 and the dynamical mean-field theory study T_c ≈ 0.047, indicating that the quantum fluctuation plays an important role in determining the critical temperature in the flat-band superconductivity on the Lieb lattice. Moreover, we calculate the pair structure factor and analyze its finite-size scaling behavior corresponding to the BKT transition. From this finite-size scaling analysis, the scaled pair structure factor fits the functional form of the BKT-transition type, indicating that our estimate of the critical temperature is reliable.