Graph-theoretical search for integrable multistate Landau-Zener models

TL;DR

The paper develops a graph-theoretical program to identify integrable multitime Landau-Zener (MTLZ) models by encoding integrability as cycle- and multipath-based constraints on edge data. An efficient enumeration algorithm tests candidate graphs against four necessary rules, with detailed analyses up to N≤11 showing agreement with the conjecture that only hypercubes, fans, and their Cartesian products host MTLZ models. It further probes larger graphs via (0,2)-graphs and their descendants, finding general non-integrability for many examples but identifying potential avenues for counterexamples at higher N. The work lays groundwork for discovering new solvable MLZ models and sets the stage for future extensions to related time-dependent families and potential experimental realizations in quantum-dot systems.

Abstract

The search for exactly solvable models is an evergreen topic in theoretical physics. In the context of multistate Landau-Zener models -- $N$-state quantum systems with linearly time-dependent Hamiltonians -- the theory of integrability provides a framework for identifying new solvable cases. In particular, it was proved that the integrability of a specific class known as the multitime Landau-Zener (MTLZ) models guarantees their exact solvability. A key finding was that an $N$-state MTLZ model can be represented by data defined on an $N$-vertex graph. While known host graphs for MTLZ models include hypercubes, fans, and their Cartesian products, no other families have been discovered, leading to the conjecture that these are the only possibilities. In this work, we conduct a systematic graph-theoretical search for integrable models within the MTLZ class. By first identifying minimal structures that a graph must contain to host an MTLZ model, we formulate an efficient algorithm to systematically search for candidate graphs for MTLZ models. Implementing this algorithm using computational software, we enumerate all candidate graphs with up to $N = 13$ vertices and perform an in-depth analysis of those with $N \le 11$. Our results corroborate the aforementioned conjecture for graphs up to $11$ vertices. For even larger graphs, we propose a specific family, termed descendants of ``$(0,2)$-graphs'', as promising candidates that may violate the conjecture above. Our work can serve as a guideline to identify new exactly solvable multistate Landau-Zener models in the future.

Figures (10)

• Figure 1: Examples of known of graphs that host MTLZ models: (a) the hypercube graphs $Q_D\equiv(K_2)^D$ with $D\ge 1$; (b) the fan graphs $K_{2,n}$ with $n\ge 3$; (c) a Cartesian product of a $K_{2}$ and $K_{2,3}$.
• Figure 2: Graphs of a $4$-cycle with two types of orientations: (a) the non-bipartite orientation; (b) the bipartite orientation.
• Figure 3: Graphs appeared in the description of the rules for candidate graphs: (a) the complete bipartite graph $K_{3,3}$; (b) the $1221$ graph.
• Figure 4: Arguments showing that for a graph with $d\ge 3$, any inner layer must contain at least $3$ vertices. The dashed lines present different layers.
• Figure 5: The minimal subgraphs with $d\ge 3$: (a) with no 1221 subgraphs; (b) with at least one $1221$ subgraph.