The principle of symmetry serves as a cornerstone of modern physics, yet its role in condensed matter many-body systems has largely remained qualitative, acting primarily as a classificatory tool. In this work, we establish, for the first time, a universal framework of hierarchical fractional quantization (HFQ) for superconducting and superfluid systems with continuous symmetries and quantum fluctuations. We demonstrate that symmetry can impose rigorous, computable quantitative constraints on the numerical values of physical observables.We apply this framework to predict the response of the d-wave order parameter in high-temperature superconductors under uniaxial stress (predicting R = 1 and deriving new constraints on superfluid density anisotropy) and the excitation spectrum splitting of a Bose superfluid in an optical lattice under lattice distortion (predicting R = 1/2). Systematic quantitative validation using multiple published experimental datasets shows excellent agreement with theoretical predictions. This work elevates symmetry from a qualitative organizing principle to a quantitative predictive tool, offering a new paradigm for understanding competing orders and symmetry breaking in strongly correlated quantum materials.
The principle of symmetry serves as a cornerstone of modern physics, yet its role in condensed matter many-body systems has largely remained qualitative, acting primarily as a classificatory tool. In this work, we establish, for the first time, a universal framework of hierarchical fractional quantization (HFQ) for superconducting and superfluid systems with continuous symmetries and quantum fluctuations. We demonstrate that symmetry can impose rigorous, computable quantitative constraints on the numerical values of physical observables.We apply this framework to predict the response of the d-wave order parameter in high-temperature superconductors under uniaxial stress (predicting R = 1 and deriving new constraints on superfluid density anisotropy) and the excitation spectrum splitting of a Bose superfluid in an optical lattice under lattice distortion (predicting R = 1/2). Systematic quantitative validation using multiple published experimental datasets shows excellent agreement with theoretical predictions. This work elevates symmetry from a qualitative organizing principle to a quantitative predictive tool, offering a new paradigm for understanding competing orders and symmetry breaking in strongly correlated quantum materials.
The principle of symmetry is a cornerstone of modern physics, yet its role in condensed matter many-body systems has largely been qualitative, serving as a classificatory tool. In this work, we establish, for the first time, a universal framework of hierarchical fractional quantization (HFQ) for superconducting and superfluid systems with continuous symmetries and quantum fluctuations.We demonstrate that symmetry can impose rigorous, computable quantitative constraints on the numerical values of physical observables.We apply this framework to predict the response of the d-wave order parameter in high-temperature superconductors under uniaxial stress (predicting R = 1 and deriving new constraints on superfluid density anisotropy) and the excitation spectrum splitting of a Bose superfluid in an optical lattice under lattice distortion (predicting R = 1/2). Systematic quantitative validation using multiple published experimental datasets shows excellent agreement with theoretical predictions. This work elevates symmetry from a qualitative organizing principle to a quantitative predictive tool, offering a new paradigm for understanding competing orders and symmetry breaking in strongly correlated quantum materials.
This paper presents a systematic extension of the Global Fractional Quantization (GFQ) framework from particle physics to condensed matter systems. We establish a rigorous theoretical foundation for the quantization of collective observables in many-body quantum systems, demonstrating that sums of specific physical quantities over entire bands, ensembles, or collective states exhibit discrete, rational relationships. The framework is built upon four fundamental postulates—Gauge Constraint, Gauge Invariance, Gauge Periodicity, and Global State Symmetry—applied to appropriately defined U(N) collective multiplets in condensed matter contexts. We provide detailed mathematical derivations with complete proofs and demonstrate the framework’s applicability to several key condensed matter systems: topological insulators, superconductors, magnetic systems,and correlated electron materials. The theory predicts novel quantization conditions for sums of Berry curvatures, total polarization, collective spin moments, and superconducting phase differences,which we validate through both analytical models and numerical calculations with comprehensive error analysis. Our results reveal a hierarchical structure of quantization laws that correlate with the system’s dimensionality, symmetry, and fundamental nature (topological vs. trivial, gapped vs. gapless). The GFQ framework offers a new unifying perspective on emergent quantization phenomena across diverse condensed matter platforms and provides testable predictions for future experiments.
We study superfluidity in strongly correlated systems by focusing three features as follows. First, interplay between superfluidity and other order parameters near quantum critical points is studied. The onset positions of the order parameters with or without superconductivity determines how they interact with each other. For the spin density wave (SDW) transition, we find that the quantum critical point shifts by order |∆|, where ∆ is pairing amplitude, so that the region of SDW order is smaller in the superconductor than in the metal. This shift is larger than the |∆|2 shift predicted by theories of competing orders which ignore Fermi surface effects. Second, we study a doped antiferromagnetic Mott insulator motivated by copper based high temperature superconductors. Based on spin and charge fractionalization, the algebraic charge liquid phase whose bosonic spinons have energy gap but fermionic holons form Fermi surfaces and its superconductivity are studied. We show fluctuations of local SDW order orientation lead to d-wave superconductivity, and its quantum critical pairing mechanisms are investigated. Also, a theory of bound states between spinons and holons is studied. We show that the fractionalization induces a doubling electron-like quasiparticles, which naturally induces the fractionalized Fermi liquid. The bound state theory describes exotic properties of the underdoped Cuprates such as “nodal-antinodal dichotomy” at a mean field level description. At last, we study superfluidity of fermions in optical lattices near the Feshbach resonance. Due to divergence of scattering length at the resonance, universality is observed and characteristics of superfluidity has been studied. We extend such universal properties by considering superfluid-insulator transition with an even integer number of fermions per unit cell. The resulting insulator interpolates between a band insulator of fermions and a Mott insulator of fermion pairs.
For most of the last century, condensed matter physics has been dominated by band theory and Landau's symmetry breaking theory. In the last twenty years, however, there has been an emergence of a new paradigm associated with fractionalization, emergent gauge bosons and fermions, topological order, string-net condensation, and long range entanglements. These new physical concepts are so fundamental that they may even influence our understanding of the origin of light and electrons in the universe. This book is a pedagogical and systematic introduction to the new concepts and quantum field theoretical methods in condensed matter physics. It discusses many basic notions in theoretical physics which underlie physical phenomena in nature, including a notion that unifies light and electrons. Topics covered include dissipative quantum systems, boson condensation, symmetry breaking and gapless excitations, phase transitions, Fermi liquids, spin density wave states, Fermi and fractional statistics, quantum Hall effects, topological/quantum order, and spin liquid and string-net condensation. Methods discussed include the path integral, Green's functions, mean-field theory, effective theory, renormalization group, bosonization in one- and higher dimensions, non-linear sigma-model, quantum gauge theory, dualities, projective construction, and exactly soluble models beyond one-dimension.
This thesis is dedicated to the derivation and benchmarking of self-consistent numerical methods that can be applied to interacting bosonic lattice models. The central goal is to derive methods with low numerical complexity but high accuracy, to be applied to complex systems which are out-of reach for established methods such as path integral quantum Monte Carlo (QMC) or the density matrix renormalization group. In the first part we derive the self-energy functional theory (SFT) for bosons. Building upon previous works on lattice systems without U(1)-symmetry-breaking, we systematically extend SFT to the possibility of a broken U(1)-symmetry and the presence of disorder. SFT incorporates bosonic dynamical mean-field theory as a certain limit, and represents a general non-perturbative framework, enabling the construction of diagrammatically sound approximations in the thermodynamical limit that are controlled in the number of optimization parameters. Using just three variational parameters, we are able to study the Bose-Hubbard model both in its clean version and in the presence of local disorder, showing excellent agreement with numerically exact QMC results. We systematically analyze the corresponding spectral functions, which cannot be fully captured by QMC. In particular, we find that in the presence of disorder the phase transition from the Bose glass to the superfluid phase at strong interactions is driven by the percolation of superfluid lakes which form around doubly occupied sites, leading to a small condensate fraction over a strongly-localized background. The second part is dedicated to the derivation of reciprocal cluster mean-field theory (RCMF) and its application to the strongly-interacting Harper-Hofstadter-Mott model (HHMm). In RCMF the full lattice in the thermodynamical limit is projected onto finite-size clusters, which are decoupled in reciprocal space through a mean-field decoupling approximation, crucially preserving the symmetries of the non-interacting dispersion. The resulting groundstate phase diagram of the HHMm exhibits band insulating, striped superfluid, and supersolid phases. Furthermore, we observe gapless uncondensed liquid phases at integer fillings, and a metastable competing fractional quantum Hall (fQH) phase. The fQH phase, predicted as the groundstate by other methods, is most likely underestimated by RCMF. We then show how a quasi-one-dimensional geometry stabilizes gapped topologically non-trivial groundstates in the HHMm. We observe quasi-one-dimensional analogues of fQH phases at fillings 1/2 and 3/2, and unconventional gapped non-degenerate groundstates at integer filling with quantized Hall responses. By systematically comparing results computed with RCMF and exact diagonalization (ED), we are able to give conclusive quantitative answers on the phase boundaries of the system, as the two methods approach the thermodynamical limit from opposite sides, since RCMF favours gapless and ED gapped phases.
The Landau Fermi liquid theory and the Ginzburg-Landau phase transition theory stand as two pivotal cornerstones in traditional condensed matter physics, achieving significant success in addressing crucial physical phenomena such as BCS superconductors and liquid helium superfluids. However, marked by the discoveries of the quantum Hall effect and high-temperature superconductivity in the 1980s, it gradually became evident that for a broad class of novel quantum states, such as fractional quantum Hall states and quantum spin liquids, their properties transcend the Landau Fermi liquid theory and Ginzburg-Landau phase transition theory. Topological order and its related concepts of long-range many-body quantum entanglement and fractionalized excitation have become the key concepts to understand these exotic quantum states. Designing and identifying topologically ordered states of matter in quantum materials and quantum simulation systems, and probing and manipulating their fractionalized excitations, are important research directions in modern condensed matter physics. In recent years, great progress has been made in the quantum simulation and manipulation of topological order on highly controllable quantum simulation platforms, such as Rydberg atomic systems, superconducting quantum processors, and two-dimensional moiré superlattices. This article provides a brief overview of recent research advances and challenges in the study of topological order in traditional condensed matter systems and quantum simulation experimental platforms. It also provides prospects for the future developments of this field.