Scanning Probe Microscopy of Quantum Chains

Quantum strongly correlated spin systems exhibit energy spectra related to their underlying classical dynamics. For integrable systems, the level-spacing is Poissonian, while for chaotic dynamics, it follows a Wigner-Dyson distribution, corresponding to localized and metallic phases [1,2] Linear spin chains and braids have been considered recently [3] showing that a certain degree of disorder or the presence of impurities along the chain induces a transition from one statistic to the other. We consider a quantum Heisenberg system of two perpendicular linear chains (T-shaped) interacting through the points of closest approach between them. The motivation is to model the tip of a Magnetic Force Microscope, the separation between tip and sample playing the role of an impurity (a different hopping parameter). We study the level-spacing statistics of the whole system as a function of tip-sample separation, observing a transition in level statistics, inducing a change in the total energy, detectable by monitoring the force between tip and sample. We propose a tool to search for signatures of an integrable-to-chaotic transition by looking at the force-separation curves in Scanning Probe Microscopy. [1] Shklovskii B I, Shapiro B, Sears B R, Lambrianides P and Shore H B 1993 \textit{Phys. Rev. }B \textbf{47 }11487 [2] Braun D, Montambaux G and Pascaud M 1998 \textit{Phys. Rev. Lett. }\textbf{81 }1062 [3] Lea F. Santos, Phys Rev E78, 031125 (2008)