Open problems for the Movassagh–Shor supercritical-entanglement spin chains: excited-state spectrum, continuum limit, sharp gap asymptotics, and the entanglement limits of physically reasonable one-dimensional local systems

Movassagh and Shor (PNAS 2016) constructed one-dimensional, translationally invariant, frustration-free spin chains with a unique ground state whose half-chain entanglement entropy grows as $\sqrt{n}$, an exponential improvement over the previously believed logarithmic bound on area-law violations, together with a spectral gap that closes only polynomially in the chain length $n$. In the full-length version of that work the authors listed eight open problems about this model class and about the limits of entanglement in physically reasonable local systems. Ground-state correlation functions and a low-local-dimension ($d=4$) realization with a proven polynomial gap are now settled, and volume-law entanglement has been achieved in translationally invariant chains but only with an exponentially small gap. The surviving open core is a precise, multi-part program: the spectral structure (excited states, degeneracies, integrability) of the interacting chain beyond first-order perturbation theory; a rigorous polynomial gap bound for the Hamiltonian variant whose interaction creates maximally entangled states out of the vacuum; the entanglement-versus-gap phase boundary (whether more than $\sqrt{n}$, up to volume-law, entanglement is compatible with a polynomially closing gap); the continuum (field-theoretic) limit of the Hamiltonian class; and sharp gap asymptotics for the model with boundaries, where the gap is known to be $O(n^{-2})$ and was conjectured to close as $O(n^{-3})$.

Background

Entanglement entropy is the standard measure of quantum correlation in many-body ground states: for a pure state on a chain, one writes the Schmidt decomposition across a cut, $|\psi\rangle=\sum_i p_i |f_i\rangle_A|g_i\rangle_B$, and the entropy of the Schmidt spectrum, $S=-\sum_i p_i\log p_i$, is the entanglement between the two halves. The area law conjecture asserts that $S$ of a subregion scales with the boundary length rather than the volume; in one dimension it means $S$ bounded by a constant, a fact proved for gapped chains and responsible for the efficiency of tensor-network simulation, while gapless critical chains described by conformal field theory (the scale-invariant continuum theory describing critical points) violate it by at most a logarithm $\log n$. An argument of Swingle and Senthil, relating ground-state entanglement to the thermal entropy of a subregion at an effective entanglement temperature, underpinned the two-decade folklore that physically reasonable quantum matter could not beat a logarithmic violation. Movassagh and Shor refuted this folklore with an explicit construction. Take spin-$s$ particles (local Hilbert-space dimension $d=2s+1$ per site, carrying $s$ colors) on a chain of $2n$ sites, and let the Hamiltonian be a sum of local nearest-neighbor positive semidefinite projectors whose joint zero-energy space, with boundary projectors, is the uniform superposition of all $s$-colored Motzkin walks; a Motzkin walk is a lattice path from $(0,0)$ to $(2n,0)$ taking steps $(1,\pm1)$ and $(1,0)$ that never dips below the horizontal axis, and the colors decorate the non-flat steps. Such a Hamiltonian is frustration-free (every local term is minimized simultaneously by the ground state) and translationally invariant in the bulk, with a unique ground state. The spin-1 (colorless) case is the earlier chain of Bravyi, Caha, Movassagh, Nagaj and Shor, whose half-chain entanglement is $\frac{1}{2}\log n+O(1)$. For $s>1$ (local dimension $d\geq 5$) Movassagh and Shor proved the half-chain entanglement is $\Theta(\sqrt{n})$ — a power-law, supercritical violation of the area law — and that the spectral gap (the energy difference between the ground state and the first excited level) closes only polynomially in $n$, with an upper bound of $O(n^{-2})$ for the model with boundaries. The proofs map the Hamiltonian to the Glauber dynamics (the standard single-step local update rule) of a Markov chain on colored Motzkin paths, control its mixing by the canonical-path and fractional-matching comparison techniques of Markov-chain analysis (which bound mixing times by routing the state space along chosen transition paths), and use the Brownian excursion (the scaling limit of nonnegative random walks, whose functionals govern the height statistics of random Motzkin paths) to extract the entanglement asymptotics. In the full-length version of the paper the authors posed eight open problems about this class: (i) the excited states and degeneracies beyond their first-order perturbation analysis; (ii) a poly(1/n) gap lower bound for a variant whose interaction creates maximally entangled states out of the vacuum, $|\varphi\rangle=\frac{1}{\sqrt{2}}\left(|00\rangle-\frac{1}{\sqrt{s}}\sum_{i=1}^s|u^i d^i\rangle\right)$, noting their own technique fails because the associated Markov-chain transition probabilities can become negative; (iii) whether similar power-law violation with a polynomially closing gap is possible at local dimension $d<5$; (iv) whether $\sqrt{n}$ entanglement is the maximum achievable in physically reasonable (local, translationally invariant, unique-ground-state) models; (v) the continuum limit of the Hamiltonian class; (vi) improving the gap upper bound from $O(n^{-2})$ to $O(n^{-3})$ for the model with boundaries; (vii) extensions of the combinatorial gap techniques; and (viii) the scaling of the ground-state spin-spin correlation functions $\langle\sigma_i\sigma_k\rangle$ and $\langle\sigma_i\sigma_{i+1}\sigma_k\sigma_{k+1}\rangle$. Two later families are needed to understand what became of these questions. The Fredkin chain of Salberger and Korepin is a half-integer-spin chain with three-site interactions (built from the Fredkin controlled-swap gate) whose ground state is the uniform superposition of $s$-colored Dyck paths, i.e. Motzkin paths without the flat step; its colored versions have local dimension $d=2s+1$ and $\sqrt{n}$ entanglement for $s>1/2$. The area-weighted (or $q$-deformed, weight $t$) variants weight each walk by $t$ raised to its area, driving transitions between bounded, $\log n$, $\sqrt{n}$ and volume-law entanglement scaling.

Problem Statement

Resolve the remaining open problems of the Movassagh-Shor supercritical-entanglement program, whose common objective is a complete mathematical and physical characterization of the exactly solvable Motzkin-chain Hamiltonian class and of the entanglement that physically reasonable one-dimensional local systems can carry. Two of the authors' eight original questions are already settled by the later literature (the ground-state spin-spin correlation scaling, item viii, and the $d<5$ construction with a polynomially closing gap, item iii, both achieved via the Fredkin chain and the correlation-function analyses); the problem is the surviving core, and a complete resolution requires all five of the following independent open sub-problems: (1) Spectrum: characterize the excited states of the interacting (supercolored) Motzkin Hamiltonian beyond first-order degenerate perturbation theory, including the degeneracy structure of the low-energy levels, the energies of the imbalanced sectors (subspaces of walks that end a fixed net height above the axis, i.e. with unequal numbers of up and down steps), and whether the full model is integrable or admits any complete solution. (2) Maximally entangling variant: prove (or disprove) a poly(1/n) lower bound on the spectral gap of the Hamiltonian variant whose interaction term creates maximally entangled states out of the vacuum, $|\varphi\rangle=\frac{1}{\sqrt{2}}\left(|00\rangle-\frac{1}{\sqrt{s}}\sum_{i=1}^s|u^i d^i\rangle\right)$, for which the canonical-path Markov-chain technique of the original proof fails because transition probabilities can become negative. (3) Entanglement-gap phase boundary: determine whether entanglement growing faster than $\sqrt{n}$ — in particular, volume-law entanglement — is achievable in local, translationally invariant one-dimensional Hamiltonians with a unique ground state while the gap closes only polynomially in $n$, or whether the constructions with exponential entanglement necessarily force an exponentially small gap. (4) Continuum limit: construct (or rule out) the continuum limit of the class of Motzkin (and Fredkin) Hamiltonians, identifying the effective field theory of their gapless low-energy physics, which the literature indicates is not a relativistic conformal field theory. (5) Sharp gap asymptotics: determine the sharp scaling of the spectral gap of the boundary model, in particular whether the upper bound improves from $O(n^{-2})$ to $O(n^{-3})$ in the system size, together with a matching (ideally tight) polynomial lower bound. The source additionally lists, as a further desideratum rather than a separate completion condition, that the combinatorial gap techniques of these proofs (Brownian excursions, canonical paths, fractional matching) may have broader applications in proving the gap of other local Hamiltonians.

References

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