Flat-spectrum, zero-mode, and maximal-value conjectures for the fermionic interaction distance underlying free-fermion descriptions of $\mathbb{Z}_N$ topological states

The interaction distance $D_\mathcal{F}(\rho)$ — the minimum trace distance between a subsystem density matrix $\rho$ and the manifold of free-fermion (Gaussian) density matrices, computable from the entanglement spectrum — was applied by Meichanetzidis, Turner, Farjami, Papić and Pachos [Phys. Rev. B 97, 125104 (2018)] to $\mathbb{Z}_N$ parafermion chains, string-net and Walker-Wang models, showing that all $\mathbb{Z}_{2^n}$ ground states are exactly Gaussian ($D_\mathcal{F}=0$) while flat spectra of rank $N\neq 2^n$ can approach the conjectured maximal distance $3-2\sqrt{2}$ from free fermions. The analytic core of that work rests on unproven conjectures: (A) for flat spectra of rank $N$, $D_\mathcal{F}(\bar\rho(N))=3-\frac{N}{2^n}-\frac{2^{n+1}}{N}$ with $2^n\le N<2^{n+1}$, proven only for $N\le 6$ and verified numerically to $N=2^8$; (B) the zero-mode identity $D_\mathcal{F}\left(\frac{1}{2}(\rho\oplus\rho)\right)=D_\mathcal{F}(\rho)$ for arbitrary spectra; (C) $3-2\sqrt{2}$ is the global maximum of $D_\mathcal{F}$; and (D) whether the free-fermion description of $\mathbb{Z}_{2^n}$ chains extends to the spectrum away from the fixed point. Follow-up work either assumes these conjectures or bypasses them with alternative non-Gaussianity measures; none of them has been proven or refuted, so the problem is open.

Background

Whether a strongly correlated quantum state can be faithfully mimicked by noninteracting fermions is a basic question of many-body physics. Free (Gaussian) fermion states are Slater determinants — states of Hamiltonians bilinear in fermionic creation and annihilation operators — and their correlations obey Wick's theorem, meaning all higher correlation functions factorize into two-point functions. Turner, Meichanetzidis, Papić and Pachos introduced the interaction distance to quantify how far a given state is from this free class: for a system bipartitioned into a region $A$ and its complement, $D_\mathcal{F}(\rho)=\min_{\sigma\in\mathcal{F}}D(\rho,\sigma)$, where $\rho$ is the reduced density matrix of $A$ (obtained by tracing out the complement), $\sigma$ ranges over reduced density matrices of free-fermion states, and $D$ is the trace distance (half the $\ell_1$ distance between the two eigenvalue lists). Because the minimum is over all $\sigma$, the eigenvalues of $\rho$ and $\sigma$ can be sorted in decreasing order and compared entrywise, so $D_\mathcal{F}$ depends only on the entanglement spectrum — the eigenvalue list of $\rho$, introduced by Li and Haldane as a fingerprint of topological order. Free entanglement spectra have the restricted form $\sigma_a=e^{-\epsilon_0-\sum_j\epsilon_j n_j(a)}$ with mode occupations $n_j(a)\in\{0,1\}$, i.e. they are normalized tensor products of two-level factors with variational mode energies $\epsilon_j\ge 0$; $D_\mathcal{F}$ measures how far a spectrum deviates from every such product.

Meichanetzidis, Turner, Farjami, Papić and Pachos applied this measure to prototypical topological models: $\mathbb{Z}_N$ parafermion chains (one-dimensional $\mathbb{Z}_N$ clock-model generalizations of the Majorana chain, built from parafermion operators $\alpha_j$ satisfying $\alpha_j\alpha_k=\omega\alpha_k\alpha_j$ with $\omega=e^{2\pi i/N}$), two-dimensional string-net models (Levin–Wen fixed-point lattice models whose edge labels are charges of a fusion category, supporting anyonic excitations), three-dimensional Walker–Wang models, and Kitaev's honeycomb model. The standard $\mathbb{Z}_N$-chain Hamiltonian is $H=-\sum_j\alpha^\dagger_{2j}\alpha_{2j+1}-f\sum_j\alpha^\dagger_{2j-1}\alpha_{2j}+\mathrm{h.c.}$, exactly solvable at the fixed point $f=0$ and nonintegrable for $f\neq0$. At the fixed point, a bipartition across a $(2j,2j+1)$ link yields a flat entanglement spectrum $\bar\rho(N)$: every nonzero eigenvalue equals $1/N$, of rank $N$. The same flat spectra govern $\mathbb{Z}_N$ Abelian string nets, with degeneracy $N^{|\partial A|-1}$ for boundary length $|\partial A|$. Guessing that the optimal free spectrum exactly reproduces the largest $2^n$ levels (where $2^n\le N$, $n=\lfloor\log_2 N\rfloor$), with one further gapped mode fixed by normalization, gives the closed-form upper bound

$D_\mathcal{F}(\bar\rho(N))\le 3-\frac{N}{2^n}-\frac{2^{n+1}}{N},$

proven to be exact for $N\le 6$ and confirmed by numerical minimization up to $N=2^8=256$. Since $N/2^n$ can approach $\sqrt{2}$ by rational approximations, the bound approaches $3-2\sqrt{2}\approx 0.1716$, which exhaustive numerical maximization never exceeds; this is conjectured to be the maximum of $D_\mathcal{F}$ over all states. For $N=2^n$ the bound gives $D_\mathcal{F}=0$: all $\mathbb{Z}_{2^n}$ chain, string-net and Walker-Wang ground states are Gaussian — for example, the $\mathbb{Z}_4$ ground state is a local unitary rotation of the $\mathbb{Z}_2\times\mathbb{Z}_2$ Gaussian ground state — even though these states carry anyonic excitations and the chains are nonintegrable away from the fixed point. This coexistence of a free-fermion entanglement spectrum with anyonic statistics, and the prospect of fermionizing the low-energy physics of interacting chains by new types of free-particle descriptions, is the physical motivation for the analytic conjectures that remain open.

Problem Statement

Settle, with mathematical rigor, the analytic structure of the fermionic interaction distance that underlies the free-fermion description of $\mathbb{Z}_N$ topological states, as left open by Meichanetzidis et al. (2018). The problem comprises four linked parts.

(A) Flat-spectrum conjecture. Prove or refute that for every integer $N\ge 2$, with $n=\lfloor\log_2 N\rfloor$,

$D_\mathcal{F}(\bar\rho(N))=3-\frac{N}{2^n}-\frac{2^{n+1}}{N},$

where $\bar\rho(N)$ is the flat spectrum of rank $N$ (all eigenvalues $1/N$, padded with zeros as needed), i.e. that the two-level-factor ansatz matching the largest $2^n$ levels is optimal among all free spectra. Exactness is currently proven only for $N\le 6$ and supported numerically up to $N=2^8$.

(B) Zero-mode conjecture. Prove or refute that

$D_\mathcal{F}\!\left(\tfrac{1}{2}(\rho\oplus\rho)\right)=D_\mathcal{F}(\rho)$

for every spectrum $\rho$, where $\oplus$ denotes the direct sum, so that $\frac{1}{2}(\rho\oplus\rho)$ is the normalized spectrum in which every eigenvalue of $\rho$ appears twice — the spectrum of $\rho$ coupled to one additional decoupled zero-energy fermion mode with occupation probability $\frac{1}{2}$. The inequality $\le$ is already proven (doubling the optimal free state gives a free state at the same distance); the open content is the reverse inequality $\ge$.

(C) Global maximum. Prove or refute that $\sup_{\rho}D_\mathcal{F}(\rho)=3-2\sqrt{2}$ over all states, approached by flat spectra with $N/2^n\to\sqrt{2}$; the maximum of $D_\mathcal{F}$ over rank-4 spectra is proven to be $1/6$, attained by the rank-3 flat spectrum.

(D) Spectral extension away from the fixed point. Determine whether the free-fermion description of the $\mathbb{Z}_{2^n}$ parafermion chains and string nets — exact for ground and excited states at the fixed point — extends beyond the numerically observed Gaussian ground state and low-lying states when the chain is away from its fixed point (coupling $f>0$). In particular: for the general $\mathbb{Z}_{2^n}$ chains and string nets (the $\mathbb{Z}_4$ chain being the first nontrivial instance), decide whether the ground-state $D_\mathcal{F}$ is exactly zero for $f\neq 0$ or only exponentially small, and whether the low-energy sector, the full spectrum, or both admit an exact free-fermion (Gaussian) representation, or establish that none exists.

Relationship among the parts: A–C are the analytic core — A fixes the exact value of $D_\mathcal{F}$ for every Abelian topological fixed-point spectrum, B is the composition rule under which $D_\mathcal{F}$ can be computed for general spectra mode by mode, and C calibrates the global scale of the measure — while D is the physical fermionization target this apparatus serves. A complete answer resolves each of A, B, C (proof or counterexample) and settles D (explicit construction or rigorous no-go).

References

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