Symmetries of quantum systems provide a natural way to organize the eigenspaces of many-body Hamiltonians. When degeneracies exist, however, they play a more subtle role as different sets of symmetry operators can resolve the same energy subspace into states with very different correlation properties. We study this interplay using a sequence of FeO and Fe$_2$O$_2$ models. For a Hubbard model of FeO, the ground space is 16-fold degenerate. Resolving it by local magnetizations gives a basis of Slater determinants, while total-spin quantum numbers select entangled and non-Gaussian states from the same ground manifold. Adding density-density Hund interactions breaks the independent spin rotations and reduces the ground space to a spin-reversal pair with maximal magnetization. These states are extremely close to Slater determinants in their natural occupations, yet have finite two-body cumulants and conserved-operator algebras whose orbit dimensions differ substantially from the Slater values. Fe$_2$O$_2$ provides a larger example in which spatial mode entanglement can be substantial while fermionic non-Gaussianity remains weak, with the conserved-operator structure again distinguishing the state from the corresponding Slater transformation class. Together, these examples show that symmetry resolution, mode entanglement, reduced-density-matrix correlations, and conserved-operator structure provide complementary information about interacting and degenerate electronic ground states.
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The paper asks how symmetry resolution within a degenerate electronic ground space affects its assigned correlations, and how mode entanglement, fermionic non-Gaussianity, and transformation classes differ. It studies low-energy FeO and Fe$_2$O$_2$ Hamiltonians using symmetry analysis, exact diagonalization, one- and two-body reduced density matrices, and state-dependent one-body stabilizer algebras computed from covariance-matrix null spaces.
For Hubbard FeO with 12 electrons in 16 spin orbitals, the ground space contains one hole in each of four disconnected orbital groups and is 16-dimensional. Resolving local magnetizations yields Slater determinants; resolving total spin instead includes entangled, non-Gaussian states. Within the $S=2$ quintet, spin-mode entropy ranges from zero at $M=\pm2$ to $\log_2 6$ at $M=0$, while the projective $GL(16,\mathbb C)$ orbit dimension increases from 48 to 111.
Density-density Hund interactions leave a maximally polarized spin-reversal ground-state pair. Its natural occupations are nearly Slater-like: the one-body idempotency defect is $3.83\times10^{-7}$, but its computed orbit dimension is 104 rather than 48. Fe$_2$O$_2$ similarly has an idempotency defect of $1.30\times10^{-7}$ while exhibiting 2.09 bits of entanglement between one Fe center and the remainder. Symmetry reduction takes its 10,518,300-dimensional fixed-particle-number space to blocks of at most 18,400, with the ground-state calculation further reducible to dimension 72.
The interaction approximation omits spin-flip exchange and pair hopping and breaks total-spin conservation. Connections between these diagnostics and concrete state-preparation, sampling, or dynamical-simulation costs remain future work.
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Symmetries of quantum systems provide a natural way to organize the eigenspaces of many-body Hamiltonians. When degeneracies exist, however, they play a more subtle role as different sets of symmetry operators can resolve the same energy subspace into states with very different correlation properties. We study this interplay using a sequence of FeO and Fe2O2 models. For a Hubbard model of FeO, the ground space is 16-fold degenerate. Resolving it by local magnetizations gives a basis of Slater determinants, while total-spin quantum numbers select entangled and non-Gaussian states from the same ground manifold. Adding density-density Hund interactions breaks the independent spin rotations and reduces the ground space to a spin-reversal pair with maximal magnetization. These states are extremely close to Slater determinants in their natural occupations, yet have finite two-body cumulants and conserved-operator algebras whose orbit dimensions differ substantially from the Slater values. Fe2O2 provides a larger example in which spatial mode entanglement can be substantial while fermionic non-Gaussianity remains weak, with the conserved-operator structure again distinguishing the state from the corresponding Slater transformation class. Together, these examples show that symmetry resolution, mode entanglement, reduced-density-matrix correlations, and conserved-operator structure provide complementary information about interacting and degenerate electronic ground states.
The paper asks how symmetry resolution within a degenerate electronic ground space affects its assigned correlations, and how mode entanglement, fermionic non-Gaussianity, and transformation classes differ. It studies low-energy FeO and Fe2O2 Hamiltonians using symmetry analysis, exact diagonalization, one- and two-body reduced density matrices, and state-dependent one-body stabilizer algebras computed from covariance-matrix null spaces.
For Hubbard FeO with 12 electrons in 16 spin orbitals, the ground space contains one hole in each of four disconnected orbital groups and is 16-dimensional. Resolving local magnetizations yields Slater determinants; resolving total spin instead includes entangled, non-Gaussian states. Within the S=2 quintet, spin-mode entropy ranges from zero at M=±2 to log26 at M=0, while the projective GL(16,C) orbit dimension increases from 48 to 111.
Density-density Hund interactions leave a maximally polarized spin-reversal ground-state pair. Its natural occupations are nearly Slater-like: the one-body idempotency defect is 3.83×10−7, but its computed orbit dimension is 104 rather than 48. Fe2O2 similarly has an idempotency defect of 1.30×10−7 while exhibiting 2.09 bits of entanglement between one Fe center and the remainder. Symmetry reduction takes its 10,518,300-dimensional fixed-particle-number space to blocks of at most 18,400, with the ground-state calculation further reducible to dimension 72.
The interaction approximation omits spin-flip exchange and pair hopping and breaks total-spin conservation. Connections between these diagnostics and concrete state-preparation, sampling, or dynamical-simulation costs remain future work.
We analyze many-body entanglement in interacting fermionic systems by using the M-body reduced density matrix. We demonstrate that if a particle number conserving fermionic Hamiltonian contains only up to M-body interaction terms, then its N-particle ground state cannot be maximally M-body entangled. As a key step in the proof, we show that the energy expectation value of a maximally M-body mixed state is equal to the spectral mean of the Hamiltonian on the corresponding N-particle subspace. We further demonstrate that the many-body entanglement structure of a ground state can place quantitative constraint on the interaction strength of its parent Hamiltonian. We illustrate the theorem and its implications in Hubbard and extended SYK models. Going beyond ground states, we analyze entanglement generation under unitary dynamics from Slater-determinant initial states in these models. We determine early-time growth and estimate entanglement saturation times. Finally, we derive explicit symmetry-refined saturation upper bounds for M-body entanglement in the presence of an Abelian symmetry.
Understanding the structure of operators that commute with k identical replicas of unitary ensembles, also known as their k-commutants, is an important problem in quantum many-body physics with deep implications for the late-time behavior of physical quantities such as correlation functions and entanglement entropies under unitary evolution. In this work, we study the k-commutants of free-fermion unitary systems, which are heuristically known to contain SO(k) and SU(k) groups without and with particle number conservation respectively, with formal derivations of projectors onto these commutants appearing only very recently. We establish a complementary perspective by highlighting a larger O(2k) replica symmetry (or SU(2k) respectively) that the k-commutant transforms irreducibly under, which leads to a simple geometric understanding of the commutant in terms of coherent states parametrized by a Grassmannian manifold. We derive this structure by mapping the k-commutant to the ground state of effective ferromagnetic Heisenberg models, analogous to the ones that appear in the noisy circuit literature, which we solve exactly using standard representation theory methods. Further, we show that the Grassmannian manifold of the k-commutant is exactly the manifold of fermionic Gaussian states on 2k sites, which reveals a duality between real space and replica space in free-fermion systems. This geometric understanding also provides a compact projection formula onto the k-commutant, based on the resolution of identity for coherent states, which can prove advantageous in analytical calculations of averaged non-linear functionals of Gaussian states, as we demonstrate using some examples for the entanglement entropies. In all, this work provides a geometric perspective on the k-commutant of free-fermions that naturally connects to problems in quantum many-body physics.
Entanglement plays a central role in numerous fields of quantum science. However, as one departs from the typical "Alice versus Bob" setting into the world of indistinguishable fermions, it is not immediately clear how the concept of entanglement is defined among these identical particles. Our endeavor to recover the notion of subsystems, or mathematically speaking, the tensor product structure of the Hilbert space, lead to two natural pictures of defining fermionic entanglement: the particle picture and the mode picture. In the particle picture, entanglement characterizes the deviation of a fermionic quantum state from the non-interacting ones, e.g., single Slater determinants. In the mode picture, we recover the notion of subsystems, by referring to the partitioning of the orbital/mode that the fermions occupy, which allows us to naturally adopt the formalism of entanglement between distinguishable constituents. Both pictures reveal essential and interconnected aspects of fermionic entanglement, and thus offer precise tools for studying electron entanglement in highly relevant systems such as atoms and molecules. We showcase here two applications: i) resolving the correlation paradox in the molecular dissociation limit, ii) quantitative electronic structure analysis with orbital entanglement.
We present an analytical study of the spatial structure of correlations in periodic fermionic systems at the single Slater determinant level, focusing on the cyclic case. Exploiting cyclic symmetry, the transformation from localized orbitals to molecular orbitals is simultaneously a discrete Fourier transform, a Vandermonde matrix on the roots of unity, a complex Hadamard matrix, and the character table of the cyclic group Cm. Within this framework, the spatial structure of correlations is fully determined by the occupied irreducible representations (Fourier modes), independently of the specific Hamiltonian generating them. In particular, the fermionic two-point correlator is expressed as a truncated Fourier sum over the occupied orbitals, leading to exact analytical expressions for symmetric occupation patterns of Fourier modes (ℓ and −ℓ pairs), corresponding in quantum chemistry to aromatic fillings, i.e. fillings satisfying Hückel's 4n+2 rule. For the half filling case, we derive exact spatial selection rules. In particular, the two-point correlator vanishes identically for all even distances along the ring. This property propagates to reduced density matrices and implies a complete absence of mutual information between the corresponding orbitals. These results reveal that, despite the global delocalization of molecular orbitals, the correlation structure in cyclic systems is highly non-uniform and governed by symmetry-induced interference. Although formulated in terms of cyclic lattice systems, the present results extend straightforwardly to one-dimensional fermionic chains with periodic conditions, reflecting the underlying translational symmetry of the problem. The framework therefore provides a Hamiltonian-independent and universal reference for understanding correlation patterns in one-dimensional periodic fermionic systems.
Orbital rotations are used in quantum many-body physics and chemistry to reduce the complexity of quantum states by compressing their correlation into a small set of relevant orbitals. In striking contrast, quantum information processing strives for the opposite: orbitals that maximize entanglement as a resource. Motivated by this, we prove that Slater determinants of N fermions on a lattice of dA+dB=d sites, although free of particle correlation, can carry up to min{dA,dB,N,d−N} ebits of orbital entanglement. Given enough particles and holes, this equals the maximum over all fermionic states. For arbitrary free states, we derive the maximal orbital total correlation in closed form for all bipartitions, the finest partition, and all partitions of up to five sites, using new majorization relations for Hermitian matrices. Taken together, these results turn mean-field states, the starting point of most many-body methods in physics and chemistry, into a viable resource for quantum information processing.
The complicated ways in which electrons interact in many-body systems such as molecules and materials have long been viewed through the lens of local electron correlation and associated correlation functions. However, quantum information science has demonstrated that more global diagnostics of quantum states like the entanglement entropy can provide a complementary and clarifying lens on electronic behavior. One particularly useful measure that can be used to distinguish between quantum and classical sources of entanglement is the accessible entanglement, the entanglement available as a quantum resource for systems subject to conservation laws, such as fixed particle number, due to superselection rules. In this work, we introduce an algorithm and demonstrate how to compute accessible and symmetry-resolved entanglements for interacting fermion systems. This is accomplished by combining an incremental version of the swap algorithm with a recursive auxiliary field quantum Monte Carlo algorithm recently developed by the authors. We apply these tools to study the pairing and charge density waves exhibited in the paradigmatic attractive Hubbard model via entanglement. We find that the particle and spin symmetry-resolved entanglements and their related full probability distribution functions show very clear---and unique---signatures of the underlying electronic behavior even when those features are less pronounced in conventional correlation functions. Overall, this work provides a systematic means of characterizing the entanglement within quantum systems that can grant a deeper understanding of the complicated electronic behavior that underlies quantum phase transitions and crossovers in many-body systems.