, what one would expect for the thermodynamics of a one-dimensional system. In this limit, we suggest that the entropy carried by each additional QPC is quantized using the one-dimensional finite size/temperature conformal scaling:

ΔS_A = α\log{(1/E)\sinh{(L_AE)}}

. The sub-extensive entropy in the case of a small number of QPCs should be contrasted with the expectation for both classical, ergodic systems and quantum chaotic systems wherein a restricted geometry might affect the equilibrium relaxation times, but not the equilibrium properties themselves, such as extensive entropy and heat capacity.

DOI: 10.48550/arxiv.2501.03076

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L_A

). For sufficiently low energies (

E

) and small subsystems, it is demonstrated numerically that the entanglement entropy

S_A \sim L_A E

ΔS_A = α\log{(1/E)\sinh{(L_AE)}}

We investigate the eigenstate thermalization hypothesis (ETH) in integrable models, focusing on the $\text{spin}\ensuremath{-}\frac{1}{2}$ isotropic Heisenberg

(XXX)

chain. We provide numerical evidence that the ETH holds for typical eigenstates (weak ETH scenario). Specifically, using a numerical implementation of state-of-the-art Bethe ansatz results, we study the finite-size scaling of the eigenstate-to-eigenstate fluctuations of the reduced density matrix. We find that fluctuations are normally distributed, and their standard deviation decays in the thermodynamic limit as ${L}^{\ensuremath{-}1/2}$, with

L

the size of the chain. This is in contrast with the exponential decay that is found in generic nonintegrable systems. Based on our results, it is natural to expect that this scenario holds in other integrable spin models and for typical local observables. Finally, we investigate the entanglement properties of the excited states of the

XXX

chain. We numerically verify that typical midspectrum eigenstates exhibit extensive entanglement entropy (i.e., volume-law scaling).