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arXiv:2410.22773 · Physical Review Letters

Exact Volume-Law Entangled Zero-Energy Eigenstates in a Large Class of Spin Models

Sashikanta Mohapatra, Sanjay Moudgalya, Ajit Coimbatore Balram

Published May 30, 2025• 7 citations
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Abstract

Exact solutions for excited states in nonintegrable quantum Hamiltonians have revealed novel dynamical phenomena that can occur in quantum many-body systems. This work proposes a method to analytically construct a specific set of volume-law entangled zero-energy exact excited eigenstates in a large class of spin Hamiltonians. In particular, we show that all spin chains that satisfy a simple set of conditions host exact volume law entangled zero-energy eigenstates in the middle of their spectra. Examples of physically relevant spin chains of this type include the transverse-field Ising model, PXP model, spin-S XY model, and spin-S Kitaev chain. Although these eigenstates are highly atypical in their structure, they are thermal with respect to local observables. Our framework also unifies many recent constructions of volume-law entangled eigenstates in the literature. Finally, we show that a similar construction also generalizes to spin models on graphs in arbitrary dimensions.

DOI
10.1103/physrevlett.134.210403

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Exact solutions for excited states in nonintegrable quantum Hamiltonians have revealed novel dynamical phenomena that can occur in quantum many-body systems. This work proposes a method to analytically construct a specific set of volume-law entangled zero-energy exact excited eigenstates in a large class of spin Hamiltonians. In particular, we show that all spin chains that satisfy a simple set of conditions host exact volume law entangled zero-energy eigenstates in the middle of their spectra. Examples of physically relevant spin chains of this type include the transverse-field Ising model, PXP model, spin-S XY model, and spin-S Kitaev chain. Although these eigenstates are highly atypical in their structure, they are thermal with respect to local observables. Our framework also unifies many recent constructions of volume-law entangled eigenstates in the literature. Finally, we show that a similar construction also generalizes to spin models on graphs in arbitrary dimensions.
Exact zero-energy eigenstates built from entangled dimers have recently been found in a large class of spin Hamiltonians. While the individual states have a simple structure, their superpositions can have very different entanglement properties. In this work, we study these superpositions for geometrically local spin-1/2 chains with periodic boundary conditions. Using stabilizer theory and polynomial methods, we characterize the subspace they span and find the minimum entanglement entropy between two halves of the chain. For Hamiltonians generated by a single local Pauli interaction with noncommuting translates, this minimum obeys a volume law at fixed interaction range. The structure of the least-entangled superpositions also suggests further exact eigenstate constructions, including zero-energy states separable between the two halves of the chain and states at nonzero energies.
The eigenstate thermalization hypothesis (ETH) plays a major role in explaining thermalization of isolated quantum many-body systems. However, there has been no proof of the ETH in realistic systems due to the difficulty in the theoretical treatment of thermal energy eigenstates of nonintegrable systems. Here, we write down analytically thermal eigenstates of nonintegrable spin chains. We consider a class of theoretically tractable volume-law states, which we call entangled antipodal pair (EAP) states. These states are thermal, in the most fundamental sense that they are indistinguishable from the Gibbs state with respect to all local observables, with infinite temperature. We then identify Hamiltonians having the EAP state as an eigenstate and rigorously show that some of these Hamiltonians are nonintegrable. Furthermore, a thermal pure state at an arbitrary temperature is obtained by the imaginary time evolution of an EAP state. Our results offer a potential avenue for providing a provable example of the ETH.
Abstract It has been observed recently that various spin chain Hamiltonians admit special zero energy ‘crosscap’ eigenstates. These states are made up of maximally entangled Bell pairs prepared on antipodal sites of a periodic chain. We generalize the states by allowing the antipodal pairs to have non-maximal, tunable entanglement. We give sufficient conditions for such states to be exact zero energy eigenstates of a local Hamiltonian. The conditions are naturally satisfied in many models which have a global U (1) symmetry. These models include well known integrable models such as the XX model, the Bariev model, the folded XXZ model, and also a variety of non-integrable models. Using the zero-energy crosscap states we also derive a family of exact zero modes with sub-volume law entanglement.
In this Letter, we report first exact volume-entangled Einstein-Podolsky-Rosen-type scar states hosted by PXP and related Hamiltonians corresponding to various geometric configurations of Rydberg-blockaded atom systems, including the most extensively studied ones such as the chain with periodic boundary conditions (PBCs) and square lattice. We start by introducing a new zero-energy eigenstate of the PBC chain and proceed by generalizing it to a wide variety of geometries and Hamiltonians. We point out the potential experimental relevance of our states by providing a protocol for their preparation on near-term Rydberg quantum devices, which relies only on strictly local measurements and evolution under native Hamiltonians. We also demonstrate the utility of these states for the study of quantum dynamics by describing a protocol for measuring infinite-temperature out-of-time-order correlator functions.
Teleportation of quantum information over long distances requires robust entanglement on the macroscopic scale. The construction of highly energetic eigenstates with tunable long-range entanglement can provide a new medium for information transmission. Using a symmetric superposition of the antipodal triplet states, we construct polynomially many exact zero-energy eigenstates for a class of nonintegrable spin-1/2 Hamiltonians with two-body interactions. These states exhibit nonthermal correlations, and hence, are genuine quantum many-body scars. By tuning the distribution of triplets, we induce extensive, logarithmic, or area-law entanglement and can observe a second-order entanglement phase transition. Quasiparticle excitations in this manifold converge to be exact quantum many-body scars in the thermodynamic limit. This framework has a natural extension to higher dimensions, where entangled states controlled by lattice geometry and internal symmetries can result in new classes of correlated out-of-equilibrium quantum matter. Our results provide a new avenue for entanglement control and quantum state constructions.
In this work, we present new, highly non-trivial area-law exact zero-energy eigenstates of the one-dimensional (1D) PXP and related models. We formulate sufficient conditions for a matrix product state to represent an exact zero-energy eigenstate of a given 1D kinetically constrained model and use them to prove our new states. We also demonstrate that all previously known exact eigenstates of PXP-type models satisfy these conditions, and, in fact, can be directly deduced from them. We discuss and demonstrate a remarkably effective general numerical technique for discovering finite-bond-dimension eigenstates residing in degenerate subspaces of a broad class of Hamiltonians. Our results highlight a previously unrecognized structure characteristic of the exponentially large nullspaces in kinetically constrained models, suggesting the possibly of extensively many increasingly complex area-law zero-energy eigenstates in the thermodynamic limit. The important implications of these emergent exact eigenstates for the general thermalization phenomenology are exemplified by one of the states introduced in this work, which we propose is a member of the primary Z2\mathbb{Z}_2Z2​ quantum many-body scar tower responsible for long-lived revivals in the Rydberg atom chain experiment.