Entanglement signatures for the dimerization transition in the Majumdar-Ghosh model

Abstract

The transition from a gapless liquid to a gapped dimerized ground state that occurs in the frustrated antiferromagnetic Majumdar-Ghosh (or J 1 − J 2 Heisenberg) model is revisited from the point of view of entanglement. We study the evolution of entanglement spectra, a "projected subspace" block entropy, and concurrence in the Schmidt vectors through the transition. The standard tool of Schmidt decomposition along with the existence of the unique MG point where the ground states are degenerate and known exactly, suggests the projection into two orthogonal subspaces that is useful even away from this point. Of these, one is a dominant five dimensional subspace containing the complete state at the MG point and the other contributes marginally, albeit with increasing weight as the number of spins is increased. We find that the marginally contributing subspace has a minimum von Neumann entropy in the vicinity of the dimerization transition. Entanglement content between pairs of spins in the Schmidt vectors, studied via concurrence, shows that those belonging to the dominant five dimensional subspace display a clear progress towards dimerization, with the concurrence vanishing on odd/even sublattices, again in the vicinity of the dimerization, and maximizing in the even/odd sublattices at the MG point.In contrast, study of the Schmidt vectors in the marginally contributing subspace, as well as in the projection of the ground state in this space, display pair concurrence which decrease on both the sublattices as the MG point is approached. The robustness of these observations indicate their possible usefulness in the study of models that have similar transitions, and have hitherto been difficult to study using standard entanglement signatures. a

Momentum-Space Entanglement Spectrum of Bosons and Fermions with Interactions

“…For example, the entanglement entropy (EE) is given by Tr(ρ A lnρ A ) = Tr(ρ B lnρ B ). The entanglement spectrum (ES) [2] (defined as the set of eigenvalues of a fictitious entanglement Hamiltonian, H e , with ρ A written as e −H e ) is a useful tool in understanding topological states of matter and strongly correlated systems, including fractional quantum Hall (FQH) systems [2-10], quantum spin chains [11][12][13][14][15][16][17] and ladders [18][19][20][21][22][23][24][25][26], topological insulators [27][28][29][30][31][32], symmetry broken phases [33,34], and other systems in one [35][36][37][38][39] and two [40][41][42][43][44][45][46][47][48][49][50][51][52] spatial dimensions. These studies predominantly focused on real / orbital space entanglement.…”

Exact dimer phase with anisotropic interaction for one dimensional magnets

“…In the second group, which is most relevant to the research in this work, only the ground states (GSs) of the Hamiltonian can be obtained. For example, in the most representative spin-1/2 Majumdar–Ghosh (MG) model 24 – 29 , which reads as with This model can be obtained from the Fermi-Hubbard by second-order exchange interaction, thus for anti-ferromagnetic interaction. The GSs of the above model can be expressed exactly as the product of singlet dimers.…”

Entanglement and avoided crossing dynamics in the disordered Majumdar-Ghosh model

“…However, author Min-Fong Yang [22] have shown that in general onset of QPT need not necessarily be detected by concurrence, block entropy or by any other entanglement measure. The Majumdar-Ghosh model is an extensively studied spin chain model, and a plethora of research articles has been written on the entanglement study of the Majumdar-Ghosh model [23][24][25].…”