A sharp limit on density fluctuations in free-fermion lattices

The question: how evenly are particles distributed?

Tristan Benoist and colleagues study fermions on a cubic lattice with periodic boundaries: moving past one edge brings a particle back through the opposite edge. Particles can hop between neighboring sites but do not interact. The model retains fermionic exclusion, which prevents two particles from occupying the same one-particle state.

The researchers ask what happens when particles are counted in a chosen set of sites. If that region contains a fraction μ of all sites, uniform density predicts that it contains μ times the total particle number, N. Quantum measurements need not return exactly that count.

Their result applies to every normalized energy eigenstate with fixed particle number, including arbitrary superpositions of states sharing exactly the same energy. This matters because restricting attention to a convenient eigenstate basis could miss unusual states allowed by energy degeneracies.

What the bound says

For any fixed dimension d and lattice side length L, the authors bound the mean squared deviation of the regional count from Nμ by a dimension-dependent constant times Nμ(1−μ)L^(d−1). The region can be any subset of sites, without assumptions about its shape or connectedness.

Strictly, this quantity is a second moment around the uniform-density prediction, rather than necessarily the variance around the state’s own mean. It controls both measurement fluctuations and any shift of the expected count away from that prediction.

Chebyshev’s inequality converts the result into a probability statement. The chance that the regional count differs from Nμ by more than ηN is at most a constant times μ(1−μ)L^(d−1)/(Nη²). Here η measures the tolerated deviation as a fraction of the total particle number.

At fixed positive particle density, N grows proportionally to L^d. For fixed η, the probability bound therefore falls as 1/L in every fixed dimension. The corresponding bound on the root-mean-square deviation, divided by N, falls as 1/√L. These are guaranteed upper bounds, not predictions that every eigenstate fluctuates that strongly.

How the proof works

The work uses mathematical proofs, not experiments or numerical simulations. Its key step separates the centered counting operator into components according to the energy change involved in moving one particle between momentum states.

When these components act on an energy eigenstate, different energy changes lead to mutually orthogonal states. Their squared contributions can therefore be added without interference between distinct changes.

For nonzero energy changes, the challenge is counting how many momentum pairs share precisely the same energy gap. The authors express comparisons between gaps as relations among roots of unity—complex numbers whose integer powers equal one. Mann’s theorem restricts the possible elementary relations. A graph-based counting argument then bounds the largest nonzero-gap multiplicity by a constant times L^(d−1).

Zero-energy changes require a separate treatment. An elementary trigonometric count, combined with Fourier identities and a matrix-norm estimate, controls contributions within degenerate energy spaces.

What improves—and what is optimal

Compared with earlier estimates discussed in the paper, the main bound removes logarithmic factors. In one dimension it also removes a prime-length restriction; in higher dimensions it covers arbitrary regions rather than only cubes.

The authors construct examples showing that the size dependence cannot generally be improved. Their explicit construction uses half filling and a region consisting of alternating lattice slices, then selects exact energy eigenstates with sufficiently large fluctuations.

There is an important qualification: sharpness of the probability estimate is established when the tolerance may shrink with system size. It does not establish that every fixed-tolerance probability actually decays as 1/L, or that every region and eigenstate saturates the bound.

Scope and publication status

This is a specific form of macroscopic eigenstate thermalization: a regional particle count concentrates near its uniform-density value. It does not establish thermal behavior for every observable, interacting particles, or arbitrary lattice models. The paper also proves related extensions, including boundaries where hopping stops at the edges and selected quadratic observables.

Nor does this proof determine how quickly an initially uneven distribution relaxes. The authors explicitly distinguish their eigenstate result from finite-time equilibration.

The supplied source is an arXiv preprint posted in November 2026; the record provided does not establish peer-reviewed publication.