When describing entanglement in typical midspectrum eigenstates of many-body lattice Hamiltonians, two paradigms have emerged that capture the behavior observed in integrable and nonintegrable systems, Haar-random fermionic Gaussian states and Haar-random pure states, respectively. Remarkably, the former capture the behavior of interacting integrable systems, whose eigenstates are non-Gaussian. We argue that the paradigm that captures both the entanglement properties and the lack of Gaussianity in integrable systems is that of random superpositions of polynomially many Gaussian states. In contrast, eigenstates of nonintegrable systems are consistent with being described by random superpositions of exponentially many Gaussian states. We gain this understanding by comparing analytical and numerical results for the one-body purity, the non-Gaussianity, and the entanglement entropy of the random superpositions and the Hamiltonian eigenstates.
The entanglement entropy of subsystems of typical eigenstates of quantum many-body Hamiltonians has recently been conjectured to be a diagnostic of quantum chaos and integrability. In quantum chaotic systems it has been found to behave as in typical pure states, while in integrable systems it has been found to behave as in typical pure Gaussian states. In this tutorial, we provide a pedagogical introduction to known results about the entanglement entropy of subsystems of typical pure states and of typical pure Gaussian states. They both exhibit a leading term that scales with the volume of the subsystem, when smaller than one half of the volume of the system, but the prefactor of the volume law is fundamentally different. It is constant (and maximal) for typical pure states, and it depends on the ratio between the volume of the subsystem and of the entire system for typical pure Gaussian states. Since particle-number conservation plays an important role in many physical Hamiltonians, we discuss its effect on typical pure states and on typical pure Gaussian states. We prove that, while the behavior of the leading volume-law terms does not change qualitatively, the nature of the subleading terms can change. In particular, subleading corrections can appear that depend on the square root of the volume of the subsystem. We unveil the origin of those corrections. Finally, we discuss the connection between the entanglement entropy of typical pure states and analytical results obtained in the context of random matrix theory, as well as numerical results obtained for physical Hamiltonians.
We study the fermionic non-Gaussianity in typical quantum states, focusing on Haar random states of qubits with or without a global
U(1) symmetry. Using the Weingarten calculus, we derive analytical predictions for the non-Gaussianity, defined as the relative entropy between the reduced density matrix and its Gaussianized counterpart. We identify two regimes controlled by the ratio between the subsystem and the system size,
ℓ/L. For
ℓ/L<1/2, the non-Gaussianity vanishes in the absence of symmetries, because typical reduced density matrices are exponentially close to the maximally mixed state. In the presence of a global
U(1) symmetry, instead, it remains small but finite. By contrast, in the regime
ℓ/L>1/2, the non-Gaussianity becomes extensive. These results establish the typical scaling of fermionic non-Gaussianity in random states and analyze how this is modified by the presence of global symmetries.
To which degree the average entanglement entropy of midspectrum eigenstates of quantum-chaotic interacting Hamiltonians agrees with that of random pure states is a question that has attracted considerable attention in the recent years. While there is substantial evidence that the leading (volume-law) terms are identical, which and how subleading terms differ between them is less clear. Here we carry out state-of-the-art full exact diagonalization calculations of clean spin-1/2 XYZ and XXZ chains with integrability breaking terms to address this question in the absence and presence of U(1) symmetry, respectively. We first introduce the notion of maximally chaotic regime, for the chain sizes amenable to full exact diagonalization calculations, as the regime in Hamiltonian parameters in which the level spacing ratio, the distribution of eigenstate coefficients, and the entanglement entropy are closest to the random matrix theory predictions. In this regime, we carry out a finite-size scaling analysis of the subleading terms of the average entanglement entropy of midspectrum eigenstates when different fractions ν of the spectrum are included in the average. We find indications that, for ν→0, the magnitude of the negative O(1) correction is only slightly greater than the one predicted for random pure states. For finite ν, following a phenomenological approach, we derive a simple expression that describes the numerically observed ν dependence of the O(1) deviation from the prediction for random pure states.
We provide a pedagogical introduction to eigenstate thermalization. This phenomenon, which occurs in generic quantum many-body systems, allows one to understand why thermalization takes place in isolated systems under unitary dynamics. We motivate eigenstate thermalization using random matrix theory and discuss recent complementary results for the volume-law entanglement entropy of Haar-random states. We discuss numerical results that highlight the corresponding behaviors in quantum many-body systems.
We characterize the magic of random fermionic Gaussian states through their Pauli spectrum and their stabilizer entropies. For the Haar ensemble of Majorana Gaussian states we derive closed finite-
N expressions for the average stabilizer purities and reconstruct the Pauli spectrum exactly, as a mixture of products of beta-distributed variables resolved by Majorana weight. The stabilizer entropies, and their filtered versions, freeze at Rényi index
qc=2: for
q>2 the magic density is
1/(q−1), controlled by rare weight-two Majorana strings. Resolving these contributions requires exponentially many characteristic samples, making direct sampling estimates of the stabilizer entropies inefficient in this frozen regime. For Haar-random Gaussian states at fixed filling, we express the averaged stabilizer purities at positive integer
q as coefficient integrals over
SU(2q), with dimension independent of
N, and evaluate the
q=2 case exactly. We establish that the frozen density describes typical states in the Majorana and half-filled number-conserving ensembles and relate these ensembles to the corresponding
SYK2 ground states. Finally, we show that typical fermionic Gaussian states retain certified magic after discarding any fixed fraction
κ<κ⋆G≃0.7613 of their modes in the thermodynamic limit, remaining magical even beyond the
2/3 threshold for Haar-random states in the full Hilbert space.
Eigenstates of local many-body interacting systems that are far from spectral edges are thought to be ergodic and close to being random states. This is consistent with the eigenstate thermalization hypothesis and volume-law scaling of entanglement. We point out that systematic departures from complete randomness are generically present in midspectrum eigenstates, and focus on the departure of the entanglement entropy from the random-state prediction. We show that the departure is (partly) due to spatial correlations and due to orthogonality to the eigenstates at the spectral edge, which imposes structure on the midspectrum eigenstates.