We present a general symmetry-based framework for obtaining many-body Hamiltonians with scarred eigenstates that do not obey the eigenstate thermalization hypothesis. Our models are derived from parent Hamiltonians with a non-Abelian (or q-deformed) symmetry, whose eigenspectra are organized as degenerate multiplets that transform as irreducible representations of the symmetry (`tunnels'). We show that large classes of perturbations break the symmetry, but in a manner that preserves a particular low-entanglement multiplet of states -- thereby giving generic, thermal spectra with a `shadow' of the broken symmetry in the form of scars. The generators of the Lie algebra furnish operators with `spectrum generating algebras' that can be used to lift the degeneracy of the scar states and promote them to equally spaced `towers'. Our framework applies to several known models with scars, but we also introduce new models with scars that transform as irreducible representations of symmetries such as SU(3) and $q$-deformed SU(2), significantly generalizing the types of systems known to harbor this phenomenon. Additionally, we present new examples of generalized AKLT models with scar states that do not transform in an irreducible representation of the relevant symmetry. These are derived from parent Hamiltonians with enhanced symmetries, and bring AKLT-like models into our framework.
We revisit the $η$-pairing states in Hubbard models and explore their connections to quantum many-body scars to discover a universal scars mechanism. $η$-pairing occurs due to an algebraic structure known as a Spectrum Generating Algebra (SGA), giving rise to equally spaced towers of eigenstates in the spectrum. We generalize the original $η$-pairing construction and show that several Hubbard-like models on arbitrary graphs exhibit SGAs, including ones with disorder and spin-orbit coupling. We further define a Restricted Spectrum Generating Algebra (RSGA) and give examples of perturbations to …
Quantum many-body scar states are many-body states with finite energy density in non-integrable models that do not obey the eigenstate thermalization hypothesis. Recent works have revealed "towers" of scar states that are exactly known and are equally spaced in energy, specifically in the AKLT model, the spin-1 XY model, and a spin-1/2 model that conserves number of domain walls. We provide a common framework to understand and prove known exact towers of scars in these systems, by evaluating the commutator of the Hamiltonian and a ladder operator. In particular we provide a simple proof of the…
We construct a 1D model Hamiltonian of spinless fermions for which the spinless analogue of $η$-pairing states are quantum many-body scars of the model. These states are excited states and display subvolume entanglement entropy scaling; they form a tower of states that are equally spaced in energy (resulting in periodic oscillations in the Loschmidt echo and in the time evolution of observables for initial states prepared in a superposition of them) and are atypical in the sense that they weakly break the eigenstate thermalization hypothesis, while the other excited states are thermal. We exte…
Motivated by the existence of exact many-body quantum scars in the AKLT chain, we explore the connection between Matrix Product State (MPS) wavefunctions and many-body quantum scarred Hamiltonians. We provide a method to systematically search for and construct parent Hamiltonians with towers of exact eigenstates composed of quasiparticles on top of an MPS wavefunction. These exact eigenstates have low entanglement in spite of being in the middle of the spectrum, thus violating the strong Eigenstate Thermalization Hypothesis (ETH). Using our approach, we recover the AKLT chain starting from the…
Periodic orbits and quantum many-body scars in integrable spin chains
Institute of Science and Technology Austria (ISTA)（奥地利科学技术研究所）；Technical University of Munich（慕尼黑工业大学）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）
We obtain multiple exact results on the entanglement of the exact excited states of non-integrable models we introduced in arXiv:1708.05021. We first discuss a general formalism to analytically compute the entanglement spectra of exact excited states using Matrix Product States and Matrix Product Operators and illustrate the method by reproducing a general result on single-mode excitations. We then apply this technique to analytically obtain the entanglement spectra of the infinite tower of states of the spin-$S$ AKLT models in the zero and finite energy density limits. We show that in the zer…
The eta-pairing states are a set of exactly known eigenstates of the Hubbard model on hypercubic lattices, first discovered by Yang [Phys. Rev. Lett. 63, 2144 (1989)]. These states are not many-body scar states in the Hubbard model because they occupy unique symmetry sectors defined by the so-called "eta-pairing SU(2)" symmetry. We study an extended Hubbard model with bond-charge interactions, popularized by Hirsch [Physica C 158, 326 (1989)], where the eta-pairing states survive without the eta-pairing symmetry and become true scar states. We also discuss similarities between the eta-pairing …
We study weak ergodicity breaking in a one-dimensional, nonintegrable spin-1 XY model. We construct for it an exact, highly excited eigenstate, which despite its large energy density, can be represented analytically by a finite bond-dimension matrix product state (MPS) with area-law entanglement. Upon a quench to a finite Zeeman field, the state undergoes periodic dynamics with perfect many-body revivals, in stark contrast to other generic initial states which instead rapidly thermalize. This dynamics can be completely understood in terms of the evolution of entangled virtual spin-1/2 degrees …
We introduce a class of spinless fermion models that exhibit quantum many-body scars (QMBS) originating from kinetic constraints in the form of density-assisted hopping. The models can be defined on any lattice in any dimension and allow for spatially varying interactions. We construct a tower of exact eigenstates with finite energy density, and we demonstrate that these QMBS are responsible for the nonthermal nature of the system by studying the entanglement entropy and correlation functions. The quench dynamics from certain initial states is also investigated, and it is confirmed that the QM…
We find exponentially many exact quantum many-body scar states in a two-dimensional PXP model -- an effective model for a two-dimensional Rydberg atom array in the nearest-neighbor blockade regime. Such scar states are remarkably simple valence bond solids despite being at effectively infinite temperature, and thus strongly violate the eigenstate thermalization hypothesis. Given a particular boundary condition, such eigenstates have integer-valued energies. Moreover, certain charge-density-wave initial states give rise to strong oscillations in the Rydberg excitation density after a quantum qu…
The discovery of Quantum Many-Body Scars (QMBS) both in Rydberg atom simulators and in the Affleck-Kennedy-Lieb-Tasaki (AKLT) spin-1 chain model, have shown that a weak violation of ergodicity can still lead to rich experimental and theoretical physics. In this review, we provide a pedagogical introduction to and an overview of the exact results on weak ergodicity breaking via QMBS in isolated quantum systems with the help of simple examples such as the fermionic Hubbard model. We also discuss various mechanisms and unifying formalisms that have been proposed to encompass the plethora of syste…
We study the nature of the ergodicity-breaking "quantum many-body scar" states that appear in the PXP model describing constrained Rabi oscillations. For a {wide class of bipartite lattices} of Rydberg atoms, we reveal that the nearly energy-equidistant tower of these states arises from the Hamiltonian's close proximity to a generalized projector-embedding form, a structure common to many models hosting quantum many-body scars. We construct a non-Hermitian, but strictly local extension of the PXP model hosting exact quantum scars, and show how various Hermitian scar-stabilizing extensions from…
Exact Quantum Many-Body Scars in 2D Quantum Gauge Models
Kavli Institute for the Physics and Mathematics of the Universe (WPI), UTIAS, The University of Tokyo（东京大学宇宙物质前沿研究联合中心）；Department of Physics, Graduate School of Science, The University of Tokyo（东京大学理学研究科物理学专攻）；Institute for Physics of Intelligence, The University of Tokyo（东京大学智能物理研究所）；Trans-scale Quantum Science Institute, The University of Tokyo（东京大学跨尺度量子科学研究所）
Formation of quantum scars in many-body systems provides a novel mechanism for enhancing coherence of weakly entangled states. At the same time, coherence of edge modes in certain symmetry protected topological (SPT) phases can persist away from the ground state. In this work we show the existence of many-body scars and their implications on bulk coherence in such an SPT phase. To this end, we study the eigenstate properties and the dynamics of an interacting spin-$1/2$ chain with three-site "cluster" terms hosting a $\mathbb{Z}_2 \times \mathbb{Z}_2$ SPT phase. Focusing on the weakly interact…
The concept of quantum many-body scars has recently been put forward as a route to describe weak ergodicity breaking and violation of the Eigenstate Thermalization Hypothesis. We propose a simple setup to generate quantum many-body scars in a doubly modulated Bose-Hubbard system which can be readily implemented in cold atomic gases. The dynamics are shown to be governed by kinetic constraints which appear via density assisted tunneling in a high-frequency expansion. We find the optimal driving parameters for the kinetically constrained hopping which leads to small isolated subspaces of scared …
