The conventional description of fermionic matter relies on the dichotomy between Fermi liquid and non‑Fermi liquid behaviors. In both cases the microscopic degrees of freedom are governed by the Pauli exclusion principle, and the many‑body state is constructed by filling single‑particle orbitals up to a Fermi level. Recent theoretical and experimental developments have revealed that under extreme conditions large assemblies of fermions can develop correlations that defy the standard frameworks. Here we propose a new collective fermionic statistic, termed NCFS, which emerges when the number of fermionic excitations in a large system follows a non‑Gaussian probability distribution P(N) that incorporates higher‑order correlations beyond the two‑point function. We formulate the NCFS in terms of a set of collective variables s_1, s_2, …, s_K that encode multi‑particle correlations, derive the constraints imposed by the Pauli principle and conservation laws, and analyze the resulting entropy and correlation functions. The NCFS predicts novel transport anomalies, unusual spectral line shapes, and a distinct scaling of the specific heat. Potential physical realizations include ultracold fermionic gases tuned near a Feshbach resonance, heavy‑fermion compounds, and engineered quantum dot lattices. The emergence of NCFS opens a new frontier for discovering exotic quantum states, with implications that may justify high‑profile recognition, including a Nobel Prize.
The conventional description of fermionic matter relies on the dichotomy between Fermi liquid and non‑Fermi liquid behaviors. In both cases the microscopic degrees of freedom are governed by the Pauli exclusion principle, and the many‑body state is constructed by filling single‑particle orbitals up to a Fermi level. Recent theoretical and experimental developments have revealed that under extreme conditions large assemblies of fermions can develop correlations that defy the standard frameworks. Here we propose a new collective fermionic statistic, termed NCFS, which emerges when the number of fermionic excitations in a large system follows a non‑Gaussian probability distribution P(N) that incorporates higher‑order correlations beyond the two‑point function. We formulate the NCFS in terms of a set of collective variables s_1, s_2, …, s_K that encode multi‑particle correlations, derive the constraints imposed by the Pauli principle and conservation laws, and analyze the resulting entropy and correlation functions. The NCFS predicts novel transport anomalies, unusual spectral line shapes, and a distinct scaling of the specific heat. Potential physical realizations include ultracold fermionic gases tuned near a Feshbach resonance, heavy‑fermion compounds, and engineered quantum dot lattices. The emergence of NCFS opens a new frontier for discovering exotic quantum states, with implications that may justify high‑profile recognition, including a Nobel Prize.
The concept of entanglement in many‑body physics has traditionally focused on spin or particle‑number degrees of freedom. Here we introduce a novel class of fermionic quantum phases in which orbital degrees of freedom of identical particles become entangled across spatially distinct sites. We construct the general form of the many‑body wavefunction for a system of N fermions occupying L orbitals per site, and show that the ground state of certain interacting Hamiltonians can be written as |\Psi> = \sum_{i,j} c_{ij} | \phi_i \phi_j >, where | \phi_i > and | \phi_j > denote single‑particle orbital states localized at different lattice sites. This entangled orbital structure leads to emergent properties distinct from conventional band insulators, Mott insulators, or spin liquids. We discuss the symmetry constraints, the role of fermionic statistics, and the topological aspects that arise when the orbital entanglement is global. Theoretical analysis predicts that FOEPs can host protected edge modes, exotic quasiparticles, and non‑Abelian statistics in two‑dimensional lattices. We also outline experimental signatures that could be observed in engineered cold‑atom systems, photonic lattices, or layered transition‑metal dichalcogenides, and argue that the discovery of FOEPs would open a new frontier in the search for quantum materials with high‑temperature superconductivity or quantum information processing capabilities.
The paradigm of Landau Fermi liquid theory has successfully described the low-energy behavior of numerous metallic systems. However, a growing number of experimental observations in strongly correlated electron materials, such as high-temperature superconductors and heavy fermion compounds, defy this conventional description, exhibiting what is known as non-Fermi liquid (NFL) behavior. These exotic states lack well-defined quasiparticle excitations, displaying anomalous scaling laws and transport properties. Ultracold atomic gases have emerged as highly controllable quantum simulators for strongly correlated matter, offering unprecedented access to tunable interaction strengths, dimensionality, and lattice geometries. This paper explores the potential of ultracold fermionic gases to unveil and characterize NFL behavior. We review the theoretical underpinnings of Landau Fermi liquid theory and its breakdown, survey prominent theoretical models for NFL states, and highlight experimental techniques in ultracold atomic systems pertinent to their detection. Specific attention is given to phenomena such as the BCS-BEC crossover, quantum criticality, and pseudogap phases in ultracold fermions, where strong interactions are expected to lead to deviations from quasiparticle descriptions. We discuss experimental signatures like anomalous spectral functions, linear-in-temperature resistivity analogs, and deviations in thermodynamic properties, proposing how these could be observed and analyzed in current and future ultracold atom platforms. This research aims to leverage the purity and tunability of atomic gases to gain fundamental insights into the nature of NFL states, contributing to a broader understanding of quantum many-body physics beyond the conventional quasiparticle picture.
The partition function, a fundamental tenet in statistical thermodynamics, contains in principle all thermodynamic information about a system. It encapsulates both microscopic information through the quantum energy levels and statistical information from the partitioning of the particles among the available energy levels. For identical particles, this statistical accounting is complicated by the symmetry requirements of the allowed quantum states. In particular, for Fermi systems, the enforcement of the Pauli principle is typically a numerically demanding task, responsible for much of the cost of the calculations. The interplay of these three elements---the structure of the many-body spectrum, the statistical partitioning of the N particles among the available levels, and the enforcement of the Pauli principle---drives the behavior of mesoscopic and macroscopic Fermi systems. In this paper, we develop an approach for the determination of the partition function, a numerically difficult task, for systems of strongly interacting identical fermions and apply it to a model system of harmonically confined, harmonically interacting fermions. This approach uses a recently introduced many-body method that is an extension of the symmetry-invariant perturbation method (SPT) originally developed for bosons. It uses group theory and graphical techniques to avoid the heavy computational demands of conventional many-body methods which typically scale exponentially with the number of particles. The SPT application of the Pauli principle is trivial to implement since it is done ``on paper'' by imposing restrictions on the normal-mode quantum numbers at first order in the perturbation. The method is applied through first order and represents an extension of the SPT method to excited states. Our method of determining the partition function and various thermodynamic quantities is accurate and efficient and has the potential to yield interesting insight into the role played by the Pauli principle and the influence of large degeneracies on the emergence of the thermodynamic behavior of large-N systems.
The collective behavior exhibited by a large number of microscopic quantum particles is at the heart of some of the most striking phenomena in condensed matter physics such as Bose-Einstein condensation and superconductivity. Physicists and mathematicians have made great progress in understanding when and how these collective phenomena emerge through the interplay of particle statistics, particle interaction and the value of thermodynamic parameters like the temperature or the chemical potential. Due to the extreme complexity of realistic many-body systems, it is natural to introduce appropriate simplifications to render their analysis feasible. Three examples of such simplifications which have proven themselves as viable starting points for a fruitful and mathematically rigorous analysis of many-body systems are the following: (a) the study of integrable models; (b) the derivation of effective theories, valid on a macroscopic scale, from more fundamental microscopic theories under appropriate coarse-graining; and (c) the use of quantum information theory to understand general connections between correlation, entanglement and particle statistics. In this thesis, we present mathematically rigorous results that were obtained in these three directions. (1) We prove anomalous quantum many-body transport in XY quantum spin chains for certain choices of the external magnetic field. The anomalous transport is described via new kinds of anomalous Lieb-Robinson bounds, including one of power-law type. We note that the XY spin chain is integrable as it can be mapped to free fermions via the non-local Jordan-Wigner transformation. (2) We derive effective macroscopic theories of Ginzburg-Landau type from the microscopic BCS theory of superconductivity in certain circumstances. We study the case of a multi-component order parameter for translation-invariant systems and the condensation of fermion pairs at zero temperature in a domain with a hard boundary. (3) We use techniques from quantum information-theory to derive bounds on the entropy of fermionic reduced density matrices, a measure of the entanglement inherent to a fermionic quantum state.
The concept of entanglement has long been central to quantum information science, yet most studies have focused on spin, charge, or spatial degrees of freedom. Here we introduce a new class of many‑body states, the Fermionic Orbital Entangled Phases (FOEP), in which the orbital degrees of freedom of itinerant fermions become entangled across the lattice. We construct a general formalism for describing orbital entanglement, derive the structure of the many‑body wavefunction, and discuss the symmetry constraints that allow such phases to emerge. The key feature of FOEP is the formation of a coherent superposition of orbital configurations that cannot be factorized into products of single‑particle orbitals. We propose that two‑dimensional materials with strong spin‑orbit coupling and multiple active orbitals, such as transition‑metal dichalcogenides and perovskite monolayers, provide fertile ground for realizing FOEP. We also outline experimental signatures, including anomalous orbital‑dependent transport, orbital‑selective quasiparticle interference patterns, and the emergence of new collective modes detectable by Raman or resonant inelastic X‑ray scattering. Finally, we discuss the potential impact of FOEP on quantum technologies and the prospects for discovering fundamentally new quantum matter that could attract Nobel‑level recognition.
Strong correlation among electrons under high magnetic field gives rise to an entirely new arena of emergent physics, namely fractional quantum Hall effect. Such systems have entirely different elementary degrees of freedom and generally, demand non-perturbative approaches to develop a better understanding. In the literature, there are several analytical methodologies and numerical toolkits available to study such a system. Clustering of zeros, parent Hamiltonian, off-diagonal order parameter, parton construction, matrix product states are to be named among a few of those popular methodologies in the existing literature. Most of these methods work well in the lowest Landau level or holomorphic wavefunction framework. It is, however, imperative to develop such methodology to study systems with Landau levels mixing to study more exotic as well as experimentally relevant states. In this work, we have developed particular methodologies, which denounce the traditional importance of the analytic properties of first quantized model wavefunction thereby extend the existing parent Hamiltonian, topological order-parameter, matrix product states descriptions to mixed Landau level systems. Such extension produces a deeper, compact and holistic understanding of universal physics of exotic phases in strongly correlated systems from the microscopic viewpoint, as well as produces interesting new results. Our second quantized/ non-analytic approach allows us to construct the ``entangled Pauli principle", a guidebook to extract universal/topological properties such as braiding statistics, fractional charge quantization, topological degeneracy of the ground states starting from a relatively simple many-body wavefunction, ``root pattern" of fractional quantum Hall state. Such an entangled Pauli principle can be derived from a microscopic parent Hamiltonian setting, thereby provide us a potential tool to probe the non-universal physics in quantum Hall fluids as well. Essentially, entangled Pauli principle is the ``DNA" of fractional quantum Hall states. Using this guiding principle, we have shown ground states with non-abelian excitations, such as Majorana fermion or Fibonacci fermion can be stabilized for two-particle interaction. Fibonacci fermion supports universal quantum gates, thereby a potential candidate for the topologically protected universal quantum computer. Entangled Pauli principle, along with a recently developed topological order parameter for composite fermions, gives rise to Parent Hamiltonian description for composite fermions as well.