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Fermionic Orbital Entangled Phases (FOEP)

Jincheng Zhang

Published Aug 17, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.21973145

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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.
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.
In this dissertation we study the properties of quantum many-body systems whose behavior is governed by strong correlation effects between large numbers of degrees of freedom. We begin with a review of various types of condensed matter settings and a general discussion of how interactions can lead to qualitatively new properties. In Chapter 2 we discuss a class of spin models which generate nontrivial realizations of a global symmetry, both in the presence and absence of topological order. In Chapter 3, we examine a model of a non-Fermi liquid. This model is built out of electron-like degrees of freedom and has a Fermi surface, yet has no stable quasiparticle excitations. In Chapter 4 we discuss some aspects of entanglement entropy in gapless multi-component systems, and suggest a way of measuring the topological properties of Kondo lattice wavefunctions.
We investigate the phase diagram and the spin-orbital entanglement of a one-dimensional $\mathrm{SU}(2)\phantom{\rule{0.16em}{0ex}}\ensuremath{\bigotimes}\phantom{\rule{0.16em}{0ex}}\text{XXZ}$ model with SU(2) spin exchange and anisotropic XXZ\text{XXZ}XXZ orbital exchange interactions and negative exchange coupling constant. As a unique feature, the spin-orbital entanglement entropy in the entangled ground states increases here linearly with system size. In the case of Ising orbital interactions, we identify an emergent phase with long-range spin-singlet dimer correlations triggered by a quadrupling of correlations in the orbital sector. The peculiar translational-invariant spin-singlet dimer phase has finite von Neumann entanglement entropy and survives when orbital quantum fluctuations are included. It even persists in the isotropic $\mathrm{SU}(2)\phantom{\rule{0.16em}{0ex}}\ensuremath{\bigotimes}\phantom{\rule{0.16em}{0ex}}\text{SU}$(2) limit. Surprisingly, for finite transverse orbital coupling, the long-range spin-singlet correlations also coexist in the antiferromagnetic spin and alternating orbital phase making this phase also unconventional. Moreover, we also find a complementary orbital singlet phase that exists in the isotropic case but does not extend to the Ising limit. The nature of entanglement appears essentially different from that found in the frequently discussed model with positive coupling. Furthermore, we investigate the collective spin and orbital wave excitations of the disentangled ferromagnetic-spin/ferro-orbital ground state and explore the continuum of spin-orbital excitations. Interestingly, one finds among the latter excitations two modes of exciton bound states. Their spin-orbital correlations differ from the remaining continuum states and exhibit logarithmic scaling of the von Neumann entropy with increasing system size. We demonstrate that spin-orbital excitons can be experimentally explored using resonant inelastic x-ray scattering, where the strongly entangled exciton states can be easily distinguished from the spin-orbital continuum.
An emergent SU(4) symmetry discovered in the microscopic model for d1 honeycomb materials [M. G. Yamada, M. Oshikawa, and G. Jackeli, ] has enabled us to tailor exotic SU(4) models in real materials. In the honeycomb structure, the emergent SU(4) Heisenberg model would potentially have a quantum spin-orbital liquid ground state due to the , and we can also expect similar spin-orbital liquids in three-dimensional versions of the honeycomb lattice. In such quantum spin-orbital liquids, both the spin and orbital degrees of freedom become fractionalized and entangled together due to the strong frustrated interactions between them. Similarly to spinons in pure quantum spin liquids, quantum spin-orbital liquids can host not only spinon excitations, but also fermionic excitations at low temperature.
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 recent experimental realization of spin-orbit (SO) coupling for ultracold bosons and fermions opens an exciting avenue for engineering quantum matter that may be challenging to realize in solid-state materials such as SO-coupled pseudospin-1 fermions. While one-dimensional SO coupling for spin-1 bosons has been experimentally realized, the generation of two-dimensional (2D) SO coupling and its topological properties are largely unexplored. Here we propose an experimental scheme for realizing a 2D Rashba-type SO coupling in a square lattice for pseudospin-1 Fermi gases. Because of the extended spin degree of freedom, many interesting topological phases could exist without relying on lattice point group symmetries that are crucial in solid-state materials. These exotic phases include triply degenerate points, quadratic band touching, a large Chern number (C=5C=5C=5) superfluid with five Majorana modes, triple-Weyl fermions, and so on. Our scheme can be generalized to larger spins and provides a distinctive route for engineering topological quantum matter by utilizing large spin degrees of freedom, instead of specific lattice symmetries.