Full disorder-averaged negativity spectrum of the partial transpose in the random singlet phase

The random singlet phase (RSP) is the infinite-randomness fixed point governing the ground states of strongly disordered antiferromagnetic spin-1/2 chains, where the strong-disorder renormalization group builds the state as a product of singlets joining arbitrarily distant sites. For a tripartition $A_1\cup A_2\cup B$ , the mixed-state entanglement between $A_1$ and $A_2$ is encoded in the eigenvalue distribution of the partially transposed density matrix $\rho_A^{T_2}$ (the negativity spectrum). Turkeshi, Ruggiero and Calabrese [Phys. Rev. B 101, 064207 (2020)] derived, via a renewal equation for the statistics of singlets crossing the cuts, the leading logarithmic asymptotics of the disorder-averaged negativity moments $\overline{\mathrm{tr}(\rho_A^{T_2})^\alpha}$ — but the full disorder-averaged spectral density $\overline{\mathbb{P}(\lambda)}$ , including its $\lambda$ -dependence, finite-size features, and the corrections generated by the multiple-decimation processes neglected in the derivation, has not been derived. This problem asks for that complete derivation, or an equivalent full characterization of the joint singlet-count statistics from which the spectrum follows. The extensions raised in the same source — negativity moments at other infinite-randomness fixed points and quench dynamics of the negativity spectrum — are recorded as separate follow-on open problems.

Background

For a pure state of a spin chain partitioned into three parts $A_1\cup A_2\cup B$ , tracing out $B$ leaves the blocks $A_1$ and $A_2$ in a mixed state whose quantum correlations cannot be separated from classical ones by Rényi entropies. The standard diagnostic is the partial transpose: the operator $\rho_A^{T_2}$ obtained from the reduced density matrix $\rho_A=\mathrm{tr}_B|\Psi\rangle\langle\Psi|$ ( $A=A_1\cup A_2$ ) by transposing only the indices belonging to $A_2$ . In one dimension, $\rho_A^{T_2}$ has a negative eigenvalue if and only if $A_1$ and $A_2$ are entangled. The entanglement negativity $\mathcal{N}=(\|\rho_A^{T_2}\|_1-1)/2$ and the logarithmic negativity $\mathcal{E}=\log\|\rho_A^{T_2}\|_1$ , with $\|\cdot\|_1$ the trace norm, quantify this entanglement. The full spectral density $\mathbb{P}(\lambda)=\sum_i\delta(\lambda-\lambda_i)$ over the eigenvalues $\lambda_i$ of $\rho_A^{T_2}$ — the negativity spectrum — contains strictly more information than either scalar; it is fully characterized by the negativity moments $M^{T_2}_\alpha=\mathrm{tr}(\rho_A^{T_2})^\alpha$ , since all moments determine $\lambda\mathbb{P}(\lambda)$ through an inverse Stieltjes transform.

Strong quenched disorder in antiferromagnetic spin-1/2 chains (random-bond Heisenberg and XX models) drives the ground state into the random singlet phase (RSP): the strong-disorder renormalization group (SDRG) introduced by Ma, Dasgupta and Hu and developed by Fisher iteratively decimates the strongest bond into a singlet and generates weaker effective couplings, producing singlets between arbitrarily distant sites and flowing to an infinite-randomness fixed point (IRFP) — a critical point at which the ratio of disorder to energy scale diverges. Refael and Moore showed that the disorder-averaged entanglement entropy of a block grows logarithmically, $\overline{S}=(\ln 2/3)\ln\ell+\mathrm{const}$ . Ruggiero, Alba and Calabrese subsequently proved, within the SDRG, that the disorder-averaged logarithmic negativity between adjacent intervals $A_1,A_2$ also grows logarithmically, $\overline{\mathcal{E}}=\overline{n}_{A_1:A_2}\ln 2$ with $\overline{n}_{A_1:A_2}=(1/6)\ln(\ell_1\ell_2/(\ell_1+\ell_2))$ : the negativity is proportional to the mean number of singlets shared by the two blocks. Before 2020 the negativity spectrum itself was known only for clean systems — one-dimensional conformal field theories, gapped phases, and free fermions (via a partial time-reversal operation instead of partial transpose).

Turkeshi, Ruggiero and Calabrese gave the first treatment for a disordered critical system. At the SDRG fixed point, a fixed disorder realization gives a factorized $\rho_A^{T_2}$ whose spectrum is exactly discrete: $\lambda_\pm=\pm 2^{-(n_{A:B}+n_{A_1:A_2})}$ with degeneracies $d_-=2^{n_{A:B}+n_{A_1:A_2}-1}(2^{n_{A_1:A_2}}-1)$ and $d_+=2^{n_{A:B}+2n_{A_1:A_2}}-d_-$ , where $n_{A:B}$ and $n_{A_1:A_2}$ count singlets crossing the corresponding cuts; the spectrum depends on the realization only through these two integers. The disorder-averaged spectrum is therefore a functional of the joint distribution $P(n_{A:B},n_{A_1:A_2})$ , equivalently of its generating function $g(t,s)=\log\langle e^{t n_{A:B}+s n_{A_1:A_2}}\rangle$ . The authors write a renewal equation for $g(t,s)$ , keeping only the dominant process of isolated single decimations (formation of an in–out singlet at one bond at a time, with waiting-time distribution inherited from the SDRG flow) and treating the singlet-emission probabilities $p,q$ ( $p+q=1$ ) as constants equal to their asymptotic values — $p=2/3$ , $q=1/3$ for adjacent intervals with $\ell_1\propto\ell_2\gg 1$ . The equation is explicitly an "educated conjecture" whose assumptions, in the authors' words, "can be tested only a posteriori". Solving it at leading order in the RG time $\mu\propto\ln\ell$ yields the universal leading scaling of the disorder-averaged moments, $\log\overline{M^{T_2}_\alpha}=\mathcal{E}_\alpha$ with

\mathcal{E}^{e}_\alpha=\frac{\sqrt{5+(1+q)\,2^{3-\alpha}}-3}{2}\,\mu+\dots \quad(\alpha\ \mathrm{even}), \qquad \mathcal{E}^{o}_\alpha=\frac{\sqrt{5+2^{3-\alpha}}-3}{2}\,\mu+\dots \quad(\alpha\ \mathrm{odd}),

and exposes a structural feature: the disorder average does not commute with the logarithm, so the averaged negativity and averaged logarithmic negativity are not trivially related. The parallel program for the simpler entanglement spectrum (moments of $\rho_A$ rather than $\rho_A^{T_2}$ ), carried out earlier by Fagotti, Calabrese and Moore, stopped at the same stage — leading moment asymptotics — and its full spectrum is likewise still missing. What remains unknown is everything beyond this leading asymptotics: the complete $\lambda$ -dependent distribution $\overline{\mathbb{P}(\lambda)}$ , its nonleading finite-size features, and the corrections induced by the processes (multiple simultaneous decimations, RG-time dependence of $p,q$ , memory effects) discarded by the renewal equation.

Problem Statement

Derive analytically the full disorder-averaged negativity spectrum $\overline{\mathbb{P}(\lambda)}$ of the partial transpose $\rho_A^{T_2}$ for a tripartition of an infinite random spin-1/2 chain in the random singlet phase — canonically, two adjacent intervals $A_1,A_2$ of lengths $\ell_1,\ell_2$ embedded in an infinite chain. Equivalently, determine the complete generating function $g(t,s)$ of the joint distribution $P(n_{A:B},n_{A_1:A_2})$ of singlets crossing the cuts at all RG scales, from which $\overline{\mathbb{P}(\lambda)}$ follows through the fixed-realization discrete spectrum $\lambda_\pm=\pm 2^{-(n_{A:B}+n_{A_1:A_2})}$ with degeneracies $d_\pm$ . A complete answer must: (i) go beyond the leading logarithmic asymptotics of the disorder-averaged moments and reconstruct the entire $\lambda$ -dependent distribution, including nonleading finite-size features and subleading terms in the RG time $\mu\propto\ln\ell$ ; and (ii) either quantify the corrections generated by the processes neglected in the single-decimation renewal equation — multiple simultaneous decimations, the RG-time dependence and activation times of the emission probabilities $p$ and $q$ , and memory effects — or replace that conjectural equation by a controlled derivation. The result must reproduce the known leading even/odd moment coefficients (with $q=1/3$ for adjacent intervals), the normalization $\mathrm{tr}\,\rho_A^{T_2}=1$ , and be consistent with numerical strong-disorder renormalization group and exact free-fermion (random XX chain) computations of the partial transpose. The further directions raised in the same source — negativity moments at other infinite-randomness fixed points, and the quench dynamics of the negativity spectrum — are distinct follow-on problems, not part of this one.

References

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