Abstract We study the generalization of the Sachdev-Ye-Kitaev (SYK) model to a 1 + 1 dimensional chiral SYK model of N flavors of right-moving chiral Majorana fermions with all-to-all random 4-fermion interactions. The interactions in this model are exactly marginal, leading to an exact scaling symmetry. We show the Schwinger-Dyson equation of this model in the large N limit is exactly solvable. In addition, we show this model is integrable for small N ≤ 6 by bosonization. Surprisingly, the two point function in the large N limit has exactly the same form as that for N = 4, although the four point functions of the two cases are quite different. The ground state entropy in the large N limit is the same as that of N free chiral Majorana fermions, leading to a zero ground state entropy density. The OTOC of the model in the large N limit exhibits a non-trivial spacetime structure reminscent of that found by Gu and Kitaev [1] for generic SYK-like models. Specifically we find a Lyapunov regime inside an asymmetric butterfly cone, which are signatures of quantum chaos, and that the maximal velocity dependent Lyapunov exponent approaches the chaos bound 2π/β as the interaction strength approaches its physical upper bound. Finally, the model is integrable for (at least) N ≤ 6 but chaotic in the large N limit, leading us to conjecture that there is a transition from integrability to chaos as N increases past a critical value.
The emergence of quantum chaos for interacting Fermi systems is investigated by numerical calculation of the level spacing distribution
P(s) as a function of interaction strength U and the excitation energy $\ensuremath{\epsilon}$ above the Fermi level. As U increases,
P(s) undergoes a transition from Poissonian (nonchaotic) to Wigner-Dyson (chaotic) statistics and the transition is described by a single scaling parameter given by $Z=(U{\ensuremath{\epsilon}}^{\ensuremath{\alpha}}\ensuremath{-}{u}_{0}){\ensuremath{\epsilon}}^{1/(2\ensuremath{\nu})},$ where
u0 is a constant. While the exponent $\ensuremath{\alpha},$ which determines the global change of the chaos border, is indecisive within a broad range of $0.9\ensuremath{\sim}2.0,$ the finiteness of $\ensuremath{\nu},$ which comes from the increase of the Fock space size with $\ensuremath{\epsilon},$ suggests that the transition becomes sharp as $\ensuremath{\epsilon}$ increases.
A non-perturbative lattice regularization of chiral fermions and bosons with anomaly-free symmetry
G in 1+1D spacetime is proposed. More precisely, we ask whether there is a local short-range quantum Hamiltonian with a finite Hilbert space for a finite system realizing onsite symmetry
G defined on a 1D spatial lattice with continuous time, such that its low energy physics produces a 1+1D anomaly-free chiral matter theory of symmetry
G? In particular, we show that the 3
L-5
R-4
L-0
R U(1) chiral fermion theory, with two left-moving fermions of charge-3 and 4, and two right-moving fermions of charge-5 and 0 at low energy, can be put on a 1D spatial lattice where the U(1) symmetry is realized as an onsite symmetry, if we include properly designed multi-fermion interactions with intermediate strength. In general, we propose that any 1+1D U(1)-anomaly-free chiral matter theory can be defined as a finite system on a 1D lattice with onsite symmetry by using a quantum Hamiltonian with continuous time, but without suffered from Nielsen-Ninomiya theorem's fermion-doubling, if we include properly-designed interactions between matter fields. We propose how to design such interactions by looking for extra symmetries via bosonization/fermionization. We comment on the new ingredients and the differences of ours compared to Ginsparg-Wilson fermion, Eichten-Preskill, and Chen-Giedt-Poppitz (CGP) models, and suggest modifying CGP model to have successful mirror-decoupling. As an additional remark, we show a topological non-perturbative proof on the equivalence relation between the 't Hooft anomaly matching conditions and the boundary fully gapping rules (e.g. Haldane's stability conditions for Luttinger liquid) of U(1) symmetry. Our proof holds universally independent from Hamiltonian or Lagrangian/path integral formulation of quantum theory.
We investigate the presence of quantum chaos in the spectrum of the bidimensional Fermi liquid by means of analytical and numerical methods. This model is integrable in a certain limit by bosonization of the Fermi surface. We study the effect on the level statisticsof the momentum cutoff
Λ present in the bidimensional bosonization procedure. We first analyse the level spacing statistics in the
Λ-restricted Hilbert space in one dimension. With
g2 and
g4 interactions, the level statistics are found to be Poissonian at low energies, and G.O.E. at higher energies, for a given cut-off
Λ. In order to study this cross-over, a finite temperature is introduced as a way of focussing, for a large inverse temperature
β, on the low energy many-body states, and driving the statistics from G.O.E. to Poissonian. As far as two dimensions are concerned, we diagonalize the Fermi liquid Hamiltonian with a small number of orbitals. The level spacing statistics are found to be Poissonian in the
Λ-restricted Hilbert space, provided the diagonal elements are of the same order of magnitude as the off-diagonal matrix elements of the Hamiltonian.
A nonperturbative lattice regularization of chiral fermions and bosons with anomaly-free chiral symmetry
G in
1+1D spacetime is proposed. More precisely, we ask ``whether there is a local short-range quantum Hamiltonian with a finite Hilbert space for a finite system realizing on-site symmetry
G defined on a 1D spatial lattice with continuous time, such that its low-energy physics produces a
1+1D anomaly-free chiral matter theory of symmetry
G?'' In particular, we propose that the chiral fermion theory with chiral U(1) ${3}_{L}\text{\ensuremath{-}}{5}_{R}\text{\ensuremath{-}}{4}_{L}\text{\ensuremath{-}}{0}_{R}$ symmetry, with two left-moving fermions of charge 3 and 4, and two right-moving fermions of charge 5 and 0 at IR low energy, can emerge from a 1D UV spatial lattice with a chiral U(1) symmetry, if we include properly designed multi-fermion interactions with intermediate strength (i.e., the dimensionless coupling constant is naturally order 1). In general, we propose that any
1+1D U(1)-anomaly-free chiral matter theory can be defined as a finite system on a 1D lattice with on-site symmetry by using a quantum Hamiltonian with continuous time, but without suffering from Nielsen-Ninomiya theorem's fermion doubling, if we include properly designed interactions between matter fields. We propose how to design such interactions by looking for extra anomaly-free symmetries via bosonization/fermionization. We comment on the new ingredients and the differences of ours compared to Ginsparg-Wilson fermion, Eichten-Preskill and Chen-Giedt-Poppitz (CGP) models, and suggest modifying CGP model to have successful mirror decoupling. Since a lattice on-site internal symmetry can be gauged, we thus can further define a nonperturbative regularization of any anomaly-free U(1) chiral gauge theory in
1+1D. As an additional remark, we show a topological nonperturbative proof of the equivalence relation between the
G-symmetric 't Hooft anomaly cancellation conditions and the
G-symmetry-preserving gapping rules (e.g. Haldane's stability conditions for Luttinger liquid) for multiple U(1) symmetries. We expect that our result holds universally regardless of spatial Hamiltonian or spacetime Lagrangian/path integral formulation of quantum theory. Numerical tests on our proposal are demanding tasks but highly desirable for future work.
Our current understanding of quantum chaos in many-body quantum systems hinges on the random matrix theory (RMT) behavior of eigenstates and their energy level statistics. Although RMT has been remarkably successful in describing "coarse" features of many-body quantum Hamiltonians in chaotic regimes, such as the Wigner-Dyson level spacing statistics or the volume-law behavior of eigenstate entanglement entropy, it remains a challenge to describe their "finer" features, particularly those arising from spatial locality. Here, we show that we can accurately describe the statistical behavior of eigenstate ensembles in many-body Hamiltonians by using pure random states with physical constraints that capture the essential features of the Hamiltonian, specifically spatial locality and symmetries. We demonstrate our approach on local spin Hamiltonians with a scalar U(1) charge. By constructing ensembles of constrained random states that account for two commuting scalar charges playing the role of energy and magnetization, we describe the patterns of entanglement of midspectrum eigenstates beyond their average volume-law behavior, including O(1) corrections and fluctuations, analytically and numerically. When defining the correspondence between quantum chaotic eigenstates in many-body Hamiltonians and RMT ensembles, our work highlights the important role played by spatial locality in describing universal features beyond the volume-law behavior.