[challenge]: Closed-form spectrum of the physical defect-sector Majorana matrix

Released by

Kun Chen, Institute of Theoretical Physics, Chinese Academy of Sciences

Contact email

chenkun0228@gmail.com

Method

Other

Challenge issue

Background

In the quantum asymmetric exclusion process of Robertson and Essler, operator space fragments into dynamically invariant sectors. On a defect-free segment the Lindblad generator becomes quadratic in Majorana superoperators, reducing the many-body dynamics to the spectrum of a finite $2M\times2M$ matrix.

Let $K\in\mathbb{C}^{M\times M}$ have entries $K_{jk}=\delta_{j,k-1}$, and define

C=\begin{pmatrix}
J_-&-2iJ_+\\
0&J_-
\end{pmatrix}.

The physical defect-sector matrix is

A=K\otimes C-K^T\otimes C^T,

with the tensor-product basis ordered by site and the two Majorana components. Hence $A^T=-A$.

The nonnegative rates obey

2J_+=J_1+J_2=J_3+J_4,\qquad 2J_-=J_1-J_2.

On the balanced line, $J_3=J_4$, equivalently

J_1=J_++J_-,\quad J_2=J_+-J_-,\quad
J_3=J_4=J_+,\quad J_+\ge|J_-|.

The source diagonalized the physical $A$ numerically and found an analytic spectrum only after replacing it by a distinct, unphysical boundary-deformed matrix $A'$. The closed-form spectrum of the undeformed physical matrix was left open.

Research objective

Give explicit closed-form analytic expressions for the complete multiset of all $2M$ eigenvalues of $A$, for every positive integer $M$ and every admissible balanced-rate parameter choice.

The answer must include:

• algebraic multiplicities;
• explicit indexing;
• branch conventions;
• degeneracies and exceptional parameter cases; and
• the mandatory $\lambda\leftrightarrow-\lambda$ and complex-conjugation symmetries.

An unevaluated characteristic polynomial, “the roots of” a determinant, an unsolved recurrence or quantization condition, formulas only for $A'$, asymptotics, or isolated numerical instances do not satisfy the challenge. Eigenvectors and Jordan structure are outside scope unless needed to define the eigenvalues correctly.

Success and verification gate

The submitted closed-form spectrum is itself the result. An independent reviewer should be able to:

1. form the undeformed matrix $A$ directly from the definition;
2. check that the formula produces exactly $2M$ roots with the stated algebraic multiplicities;
3. verify for general $M$ and admissible parameters the exact identity

\det(\lambda I-A)=\prod_{j=1}^{2M}(\lambda-\lambda_j),

or an equivalent named terminating recurrence identity;
4. audit branches, degeneracies, and exceptional rate choices; and
5. compare exact small-$M$ characteristic polynomials as secondary checks.

The solver's derivation and search process need not be reconstructed. The reviewer primarily checks the formula against the exact characteristic-polynomial identity.

Why this may lead to research output

The result would close a concrete structural gap in an exact solution of a fragmented open quantum many-body system, exposing every defect-sector mode and its dependence on system size and rates. The scope is intentionally precise: it resolves this Majorana-matrix family, not every fragmented Lindbladian or the model's broader entanglement and initial-state questions.

Current status and references

The problem was audited on 28 July 2026 and assessed as likely open with medium confidence. A 2024 thesis by the source paper's first author retains the same limitation and boundary deformation; the inspected citation chain through 2026 does not supply the physical spectrum.

1. J. A. Robertson and F. H. L. Essler, Exact solution of a quantum asymmetric exclusion process with particle creation and annihilation.
2. J. A. Robertson, Quantum quenches in closed and open spin chains: a thesis in two parts.

Start from the displayed definitions of K, C, and the undeformed matrix A, then consult the Robertson–Essler paper and cited thesis for the existing boundary-deformed result. Done means an explicit formula for all 2M eigenvalues, including multiplicities, branches, degeneracies, exceptional rates, and required symmetries, verified against the characteristic polynomial for general M.