Entanglement-spectrum statistics of ergodic eigenstates and the entanglement critical length at the many-body localization transition: analytical theory and critical scaling

For the ergodic (thermalizing) phase of the one-dimensional random-field Heisenberg chain, the distribution $P(\lambda)$ of the rescaled Schmidt eigenvalues of half-chain reduced density matrices of infinite-temperature eigenstates was found numerically by Pietracaprina, Parisi, Mariano, Pascazio and Scardicchio (2017) to approach, with increasing system size $L$, a thermodynamic limit close to but apparently distinct from the Marchenko–Pastur (MP) law of random states: the fitted thermodynamic-limit constant $s_\infty\approx 0.66$ in the half-chain entanglement entropy $S=(L/2)\ln 2-s_L$ exceeds the MP value $1/2$, the finite-size corrections decay as $e^{-mL/2}$ with $m\approx 0.20$ instead of $m=\ln 2$, and the logarithmic derivative of $P(\lambda)$ develops a boundary-related slope $\beta_1\to -0.8$ (apparently $\to -1$ near the transition). The same analysis defines an entanglement critical length $L_s(h)$ — the size at which bulk and boundary contributions to the spectrum separate — which diverges at the many-body localization (MBL) transition as $L_s(h)\sim(h_c-h)^{-\nu}$ with $h_c\approx 3.7$ and a tentative $\nu\approx 0.9$, in tension with the Harris/Chayes bound $\nu\ge 2$ and strong-disorder renormalization-group estimates $\nu\approx 3$. The open problem is to give an analytical account of these observations: the exact thermodynamic-limit form of $P(\lambda)$ (or a proof that it coincides with MP), the origin and values of the subleading finite-size corrections and of $\beta_1$, and the critical scaling of $L_s(h)$ including its true exponent and consistency with rigorous bounds. No resolution is known in the literature to date: later numerics confirm persistent deviations from MP (Gumbel-type rather than Tracy–Widom extreme-value statistics of the largest Schmidt eigenvalue — respectively the classical law for maxima of independent random variables and the random-matrix law for the largest Wishart eigenvalue), existing scaling theories describe entanglement at the critical point but not these ergodic-side observables, and the critical exponent remains actively contested.

Background

Many-body localization (MBL) is the extension of Anderson localization — the absence of diffusion of single-particle wave functions in a sufficiently disordered medium — to interacting quantum systems: in a disordered, isolated spin chain, sufficiently strong disorder prevents thermalization even for highly excited eigenstates, because local integrals of motion obstruct the eigenstate thermalization hypothesis (ETH), the statement that individual eigenstates of a generic interacting system encode thermal values of local observables. The standard model is the spin-1/2 random-field Heisenberg chain, with Hamiltonian $H=-J\sum_i \mathbf{S}_i\cdot\mathbf{S}_{i+1}-\sum_i h_i S_i^z$, where the fields $h_i$ are drawn uniformly from $[-h,h]$; the disorder strength $h$ tunes a transition near $h_c\approx 3.7$ (with $J=1$) between an ergodic and an MBL phase. The computations use periodic boundary conditions, the sector of zero total magnetization $S^z_{\mathrm{tot}}=0$, and eigenstates at the center of the energy spectrum, corresponding to infinite temperature. Entanglement diagnostics distinguish the phases sharply: in the ergodic phase the entanglement entropy of an eigenstate obeys a volume law (proportional to subsystem size), while in the MBL phase it obeys an area law (proportional to the length of the cut separating the subsystem, not to its volume), and the entanglement spectrum — the set of Schmidt eigenvalues $\lambda_i$, the eigenvalues of the reduced density matrix obtained by bisecting the chain — shows power-law statistics in the MBL phase, as found by De Luca and Scardicchio (2013).

For a subsystem in a pure state that is completely random (Haar-typical), the Schmidt eigenvalue density follows the Marchenko–Pastur (MP) law, $P(\lambda)=\frac{1}{2\pi}\sqrt{\frac{4-\lambda}{\lambda}}$ on $[0,4]$, with $\lambda$ the Schmidt eigenvalue rescaled by the subsystem dimension $2^{L/2}$ so that $\langle\lambda\rangle=1$; this is the random-matrix (Wishart) benchmark — the eigenvalue statistics of covariance matrices of Gaussian random vectors — underlying Page's theorem that the entanglement entropy of a Haar-typical pure state is close to maximal. Since ETH suggests that ergodic eigenstates behave as random states, one expects the MP law in the ergodic phase, and early numerics found rough agreement. Luitz, Laflorencie and Alet (2015) mapped the transition region and the eigenstate entanglement entropy across it in the same model.

Pietracaprina et al. (2017) made this comparison quantitative. They tracked $P(\lambda)$ and the subleading constant $s_L$ of the half-chain entanglement entropy, defined by $S=(L/2)\ln 2-s_L$ (the O(1) deficit below the maximal volume-law entropy), with $L$ up to 18 (exact diagonalization), fitting $s_L\simeq s_\infty + c\,e^{-mL/2}$: for MP one has $s_\infty=1/2$, $m=\ln 2$, $c\approx 0.38$ (the values of $m$ and $c$ come from the exact diagonalization of random Wishart matrices), while the data give $s_\infty\approx 0.66$ and $m\approx 0.20$ around $h=1$, so MP is not the limit of the observed distribution. They also studied the logarithmic derivative $\beta=\mathrm{d}\ln P/\mathrm{d}\ln\lambda$: for small $L$, $\beta$ decreases monotonically from $-1/2$ (its MP small-$\lambda$ limit) to $-\infty$ at the spectrum edge, but beyond a size $L=L_s(h)$ a local minimum $\beta_1$ (at $-\ln\lambda=\ell_1$) and maximum $\beta_2$ (at $\ell_2$) are born (the positions $\ell_{1,2}$ are read as length scales: an eigenvalue of order $2^{-\ell}$ reflects entanglement spread over roughly $\ell$ spins). As $L\to\infty$ at fixed $h$, $\beta_2\to-1/2$ with $\ell_2=O(1)$ (a bulk, local-dynamics scale), while $\beta_1$ tends to an $h$-dependent value ($\approx -0.8$ at $h=1$, apparently $\to -1$ as $h\to h_c$) with $\ell_1\to\infty$ (a boundary-related scale). $L_s(h)$, the size at which these two scales separate, diverges at the transition as $L_s(h)\approx a(h_c-h)^{-\nu}$ with $a\approx 23$ and $\nu=0.9\pm 0.2$, and is naturally identified with the correlation length of the ergodic phase.

Three theoretical tensions frame the open problem. First, the numerical exponent $\nu\approx 1$ violates the Harris/Chayes bound, the rigorous requirement $\nu\ge 2/d=2$ for critical points of disordered systems in one dimension (disorder fluctuations over one correlation volume must stay subextensive), while real-space renormalization-group treatments of the transition (Vosk, Huse and Altman 2015; Potter, Vasseur and Parameswaran 2015) give $\nu\approx 3$; whether this reflects enormous subleading (irrelevant-operator) corrections — corrections from perturbations that die out under renormalization but dominate at finite size — at accessible sizes $L\lesssim 24$ or a deeper failure of single-parameter scaling is unknown. Second, even the analytic random-matrix derivation of the MP finite-size corrections (the constants $m$ and $c$ for Wishart matrices, so far obtained only numerically) remains open, and there is no explanation for the observed $m\approx 0.20$, $s_\infty\approx 0.66$, or the value of $\beta_1$. Third, no field theory or renormalization-group description of the MBL critical point at infinite temperature exists that would predict the entanglement-spectrum scaling and its subleading terms; entanglement has been proposed as the natural route towards such a theory. Subsequent work (Šuntajs et al. 2020; avalanche mechanisms, the growth of rare thermal inclusions that destabilize the localized phase) even questions whether the one-dimensional MBL transition survives in the thermodynamic limit, which bears directly on whether $L_s(h)$ has a divergent asymptotic form.

Problem Statement

For the ergodic phase of the spin-1/2 random-field Heisenberg chain, $H=-J\sum_i \mathbf{S}_i\cdot\mathbf{S}_{i+1}-\sum_i h_i S_i^z$ with $J=1$, fields $h_i$ uniform in $[-h,h]$, periodic boundary conditions, the $S^z_{\mathrm{tot}}=0$ sector, and eigenstates at the center of the spectrum (infinite temperature), provide an analytical theory of the entanglement-spectrum statistics of infinite-temperature eigenstates and of the length scales they define, specifically:

1. Determine the exact thermodynamic-limit distribution $P(\lambda)$ of the Schmidt eigenvalues of the half-chain reduced density matrix, rescaled by $2^{L/2}$ so that $\langle\lambda\rangle=1$: either derive its closed form (showing whether it is a new universal law distinct from the Marchenko-Pastur distribution, consistent with the fitted constant $s_\infty\approx 0.66$ in the half-chain entropy $S=(L/2)\ln 2-s_L$, versus $1/2$ for Marchenko-Pastur) or prove that the limit coincides with the Marchenko-Pastur law.

2. Derive the subleading finite-size corrections: the origin and values of the exponential decay rate $m\approx 0.20$ (versus $\ln 2$ for Marchenko-Pastur) in $s_L\simeq s_\infty+c\,e^{-mL/2}$, the constant offsets $c$ in these fits, and the asymptotic value of the boundary-related slope $\beta_1$ of the logarithmic derivative $\beta=\mathrm{d}\ln P/\mathrm{d}\ln\lambda$ (numerically $\approx -0.8$ at $h=1$, apparently approaching $-1$ near $h_c$), including the analytic random-matrix derivation of the Marchenko-Pastur finite-size corrections.

3. Establish the critical scaling of the entanglement critical length $L_s(h)$ — the system size at which the local minimum and maximum of $\beta$ are born, separating the bulk scale $\ell_2$ from the boundary scale $\ell_1$ — across the MBL transition: whether it diverges as $L_s(h)\sim(h_c-h)^{-\nu}$ in the thermodynamic limit, the true value of $\nu$ (versus the tentative $\nu\approx 0.9$ from sizes $L\le 18$), the form of the subleading (irrelevant-operator) corrections that distort finite-size estimates, and the reconciliation of the result with the Harris/Chayes bound $\nu\ge 2$ in one dimension and with renormalization-group estimates $\nu\approx 3$ — or a demonstration that $L_s(h)$ does not diverge in the thermodynamic limit or that no such single-parameter scaling exists.

These three deliverables form one coherent objective: a quantitative theory of the ergodic-phase entanglement spectrum and of the entanglement critical length at the many-body localization transition, in which the limiting law, its finite-size corrections, and the diverging length scale (if it exists) and its exponent are predicted rather than fitted. The relation of $L_s(h)$ to transport or dynamical correlation lengths in the ergodic phase is outside the scope of this problem.

References

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