For decades, Landau’s theory of phase transition has provided a successful classification for quantum and classical states of matter based on their symmetry broken patterns, except for certain exotic quantum states such as the fractional quantum Hall (FQH) effect. However, such exotic phenomena are crucial for a complete understanding of the nature. This thesis explores new physical principles emerging from topology and topological states. First, we use an example to demonstrate Landau’s theory by studying a pair-density wave system, where the symmetry-breaking paradigm is applicable. It turns out that such a system exhibits the Kosterlitz-Thouless (KT) transition, a phase transition driven by the proliferation of topological defects, i.e. vortex-antivortex pairs. For topological systems, where symmetry broken pattern cannot be used as a classification tool, we prove a theorem that two gapped systems with non-vanishing ground-state overlap must be adiabatically connected, and thus are necessarily in the same topological phase. This theorem provides a simple and generic approach to classify topological band insulators/superconductors, without the needs to calculate any known or yet-to-known topological indices. Once the overlap is found nonzero, the two systems must be topologically identical. After presenting a generic proof, the theorem is also verified through calculating the overlap for several milestone topological band insulators and certain interacting systems. Such an overlap technique is then generalized to (2+1)-D strongly-interacting topological systems at fixed points, including both symmetry-protected topological (SPT) states and intrinsic topological states like FQH. For interacting topological states, the main challenge of utilizing wave-function overlaps to classify them lies in the famous Anderson orthogonality catastrophe (AOC), which states that two different many-body wave functions must have zero overlap in the thermodynamic limit. In this thesis, we found that wave-function overlaps indeed carry critical information about the topological nature of quantum states and this information can be extracted from the finite-size scaling of the overlaps. In the finite-size scaling analysis, we found a universal topological response term as a sub-leading contribution. This term depends on both the central charge of the corresponding conformal field theory (CFT) and the Euler characteristics of the underlying manifolds on which the system is defined. This term reveals a fundamental connection between ground-state overlaps and CFT’s. In addition, surprisingly, the overlap between an intrinsic topological state and a topologically trivial product state shows a decay faster than the exponential behavior expected via a typical AOC analysis. Such finite-size scaling behaviors could be utilized to theoretically detect the gapless edge modes, and to distinguish the topology of quantum states or serve as a signature of topological phase transitions. Possible generalization to higher dimensions and generic non-fixed-point topological systems is also discussed.
We consider the orthogonality catastrophe at the Anderson metal-insulator transition (AMIT). The typical overlap
F between the ground state of a Fermi liquid and the one of the same system with an added potential impurity is found to decay at the AMIT exponentially with system size
L as $F\ensuremath{\sim}\mathrm{exp}(\ensuremath{-}c{L}^{\ensuremath{\eta}})$, where $\ensuremath{\eta}$ is the power of multifractal intensity correlations. Thus, strong disorder typically increases the sensitivity of a system to an added impurity exponentially. We recover, on the metallic side of the transition, Anderson's result that the fidelity
F decays with a power law $F\ensuremath{\sim}{L}^{\ensuremath{-}q({E}_{F})}$ with system size
L. Its power increases as the Fermi energy
EF approaches the mobility edge
EM as $q({E}_{F})\ensuremath{\sim}[({E}_{F}\ensuremath{-}{E}_{M})/{E}_{M}{]}^{\ensuremath{-}\ensuremath{\nu}\ensuremath{\eta}}$, where $\ensuremath{\nu}$ is the critical exponent of the correlation length ${\ensuremath{\xi}}_{c}$. On the insulating side of the transition,
F is constant for system sizes exceeding the localization length $\ensuremath{\xi}$. While these results are obtained for the typical fidelity
F, we find that
logF is widely, log normally, distributed with a width diverging at the AMIT. As a consequence, the mean value of the fidelity
F converges to one at the AMIT, in strong contrast to its typical value which converges to zero exponentially fast with system size
L. This counterintuitive behavior is explained as a manifestation of multifractality at the AMIT.
We present a characterization of quantum phase transitions in terms of the the overlap function between two ground states obtained for two different values of external parameters. On the examples of the Dicke and XY models, we show that the regions of criticality of a system are marked by the extremal points of the overlap and functions closely related to it. Further, we discuss the connections of this approach and with the Anderson orthogonality catastrophe as well as with the dynamical study of the Loschmidt echo for critical systems.
It is well known that the ground states of a Fermi liquid with and without a single Kondo impurity have an overlap that decays as a power law of the system size, expressing the Anderson orthogonality catastrophe. Ground states with two different values of the Kondo couplings have, however, a finite overlap in the thermodynamic limit. This overlap, which plays an important role in quantum quenches for impurity systems, is a universal function of the ratio of the corresponding Kondo temperatures, which is not accessible using perturbation theory or the Bethe ansatz. Using a strategy based on the integrable structure of the corresponding quantum field theory, we propose an exact formula for this overlap, which we check against extensive density matrix renormalization group calculations.
Advances in experimental tools allow to study quantum fidelity in unprecedented controlled settings. While the fidelity in metals, when adding a local impurity, is well known to show the Anderson orthogonality catastrophe (AOC), there remain outstanding questions in other quantum phases and settings. Here, we aim to tackle these by exploring systematically the ground-state fidelity of the Fermi liquid in the (extended) Aubry-Andr\'e (AA) model, which allows to explore the AOC in both localized extended and critical phases. We discover that the AOC is typically exponential in the critical regime of the AA model and at the mobility edge of the extended AA model for an extended impurity, while it decays in the AA model with a power law for a weak single-site impurity. We explain this in terms of critical correlations and multipoint correlations. The OC is found to be exponential in the insulating regime, due to a fundamentally different, statistical mechanism, which is explained in detail. Furthermore, we consider a parametric perturbation to the AA model, and find an exponential OC numerically, in agreement with an analytical derivation which we provide here.
According to Anderson's orthogonality catastrophe, the overlap of the N -particle ground states of a free Fermi gas with and without an (electric) potential decays in the thermodynamic limit. For the finite one-dimensional system various boundary conditions are employed. Unlike the usual setup the perturbation is introduced by a magnetic (vector) potential. Although such a magnetic field can be gauged away in one spatial dimension there is a significant and interesting effect on the overlap caused by the phases. We study the leading asymptotics of the overlap of the two ground states and the two-term asymptotics of the difference of the ground-state energies. In the case of periodic boundary conditions our main result on the overlap is based upon a well-known asymptotic expansion by Fisher and Hartwig on Toeplitz determinants with a discontinuous symbol. In the case of Dirichlet boundary conditions no such result is known to us and we only provide an upper bound on the overlap, presumably of the right asymptotic order.