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arXiv:1210.4384 · Physical Review A

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

M. S. Ramkarthik, Vinod Chandra, Arul Lakshminarayan

Published Jan 3, 2013• 5 citations
arXivHTMLPDF

Abstract

The transition from a gapless liquid to a gapped dimerized ground state that occurs in the frustrated antiferromagnetic Majumdar-Ghosh (or ${J}_{1}\ensuremath{-}{J}_{2}$ 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 Majumdar-Ghosh (MG) point where the ground states are degenerate and known exactly suggest 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 as the MG point is approached. 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 decreases on both 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.

DOI
10.1103/physreva.87.012302

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The transition from a gapless liquid to a gapped dimerized ground state that occurs in the frustrated antiferromagnetic Majumdar-Ghosh (or J1\ensuremath−J2{J}_{1}\ensuremath{-}{J}_{2}J1​\ensuremath−J2​ 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 Majumdar-Ghosh (MG) point where the ground states are degenerate and known exactly suggest 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 as the MG point is approached. 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 decreases on both 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.
The von Neumann entanglement entropy of exact valence-bond ground states is studied in two frustrated one-dimensional spin chains: the spin-1/2 Majumdar-Ghosh (MG) model and the spin-3/2 J1-J2-J3 chain in its fully dimerized (FD) and partially dimerized (PD) phases. Using matrix-product-state representations, the entropy is computed as a function of system size for three complementary bipartitions - half-chain, single-site, and pairwise - under both open and periodic boundary conditions. In all cases, the entropy saturates to a finite constant in the thermodynamic limit, confirming area-law behavior. The saturation values, extracted via finite-size scaling, are directly related to the underlying virtual-spin bond structure. The MG model and FD phase exhibit similar entanglement behavior, differing primarily in saturation magnitude determined by spin value and bond multiplicity, and both display even-odd oscillations and exponential convergence with system size. In contrast, the PD phase shows qualitatively distinct signatures, including multiple half-chain saturation values depending on the bond type at the cut, asymmetric edge contributions in the single-site entropy, and a multi-band structure in the pairwise entropy reflecting the coexistence of single- and double-singlet bonds. These results establish entanglement entropy as a robust signature of frustrated bond architecture, enabling clear distinction among dimerized phases with different spin magnitude, bond multiplicity, and dimerization patterns.
We present an analysis of the entanglement characteristics in the Majumdar-Ghosh (MG) or ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model. For a system consisting of up to 28 spins, there is a deviation from the scaling behavior of the entanglement entropy characterizing the unfrustrated Heisenberg chain above ${J}_{2}\ensuremath{\approx}0.25$. This feature can be used as an indicator of the dimer phase transition occurring in this model. Additionally, we also consider entanglement at the MG point J2=0.5J1{J}_{2}=0.5{J}_{1}J2​=0.5J1​.
Momentum-Space Entanglement Signatures and Spinon Breakdown in the J1J_1J1​-J2J_2J2​ Zig-Zag Heisenberg Chain
We investigate the resilience of spinon quasiparticles in the J1J_1J1​-J2J_2J2​ zig-zag spin chain (J2>0J_2>0J2​>0) from the viewpoint of momentum-space entanglement. For small J2J_2J2​, we show that deconfined spinons survive well past the liquid-dimer transition before eventually collapsing towards the Majumdar-Ghosh point. In the highly frustrated zig-zag regime (J2≫∣J1∣J_2 \gg |J_1|J2​≫∣J1​∣), we model the system as two coupled Heisenberg chains and by Fourier transforming each subchain individually, a framework we dub the double-spinon description. While continuum field theories predict that this decoupled phase is strictly unstable to any finite inter-chain coupling, our analysis reveals that the double-spinon description remains robust over an extensive parameter regime. Notably, we find a stark asymmetry in spinon stability reflecting the underlying renormalization group flow: ferromagnetic coupling (J1<0J_{1} < 0J1​<0) is marginally irrelevant and sustains fractionalization deep into the spiral phase, whereas antiferromagnetic coupling (J1>0J_{1} > 0J1​>0) is marginally relevant and drives confinement much earlier. The ultimate breakdown of this fractionalized description is driven by a continuum of inter-chain excitations which manifests itself as a sharp ground-state momentum shift distinct from macroscopic thermodynamic phase boundaries. Our results establish momentum cut entanglement analysis as a tool to trace the quasiparticle resilience of spinons, as we show that treating the zig-zag Heisenberg chain as two coupled SU(2)1_11​ Wess-Zumino-Witten models provides a theoretical framework for strongly frustrated quantum magnets applicable beyond the decoupled limit.
We perform a systematic DMRG investigation of the two natural spin-1 generalizations of the spin-1/2 Majumdar-Ghosh chain, the spin-1 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg chain, where J2{J}_{2}J2​ is a next-nearest-neighbor Heisenberg coupling, and the spin-1 ${J}_{1}\text{\ensuremath{-}}{J}_{3}$ model, where J3{J}_{3}J3​ refers to a three-site interaction defined by ${J}_{3}\left[({\mathbf{S}}_{i\ensuremath{-}1}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i})({\mathbf{S}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i+1})+\mathrm{H}.\mathrm{c}.\right]$. Although both models are rigorously equivalent to the Majumdar-Ghosh chain for spin 1/2, their physics appears to be quite different for spin 1. Indeed, when all couplings are antiferromagnetic, the spin-1 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model undergoes an effective decoupling into two next-nearest-neighbor (NNN) Haldane chains upon increasing J2{J}_{2}J2​, while the ${J}_{1}\text{\ensuremath{-}}{J}_{3}$ chain undergoes a spontaneous dimerization similar to the spin-1/2 Majumdar-Ghosh chain upon increasing J3{J}_{3}J3​. By extending the phase diagram to all signs of the couplings, we show that both the dimerized and the NNN-Haldane phase are actually present in the ${J}_{1}\text{\ensuremath{-}}{J}_{3}$ model, the former one adjacent to the Haldane one, the latter to the ferromagneric one, with an Ising transition between them. By contrast, the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ chain only has a NNN-Haldane phase between the Haldane phase and the ferromagnetic phase for positive J2{J}_{2}J2​. In both cases, our DMRG data are consistent with a continuous Kosterlitz-Thouless transition between the NNN-Haldane and the ferromagnetic phases.
We investigate the nature of low-energy excitations in a spin chain with antiferrmomagnetic nearest-neighbor J1J_1J1​, next-nearest-neighbor J2J_2J2​, and three-site J3J_3J3​ interactions using the time-dependent density matrix renormalization group and the single mode approximation techniques. In the absence of the J2J_2J2​ interaction, we identify clear distinctions in the spectral functions in the fully dimerized phase across the exactly dimerized line for different magnitudes of the spins. In contrast to the spin-1/21/21/2 chain, where the spinon continuum dominates the spectral functions, the magnon modes are prominent in the spectral functions of the spin-111 and spin-3/23/23/2 chains. Through single mode approximation and valence bond solid approaches, we disentangle magnon and spinon contributions to the spectral functions. After including the J2J_2J2​ interactions, for the spin-111 chain we trace the evolution of the dynamical structure factor along the phase transition line between the Haldane phase and the fully dimerized phase. We find that the excitation spectrum is a continuum along this line and the spectral gap closes as the order of the transition changes from first order to second order. Along the line of first-order transitions, the spinon-like domain walls are deconfined, and the model exhibits their confinement into discrete bound states away from the transition line. A similar phenomenon occurs in the spin-3/23/23/2 chain across the phase transition between partially dimerized to fully dimerized phases, revealing a universal spinon confinement phenomenon across first-order phase transitions. This study presents the dynamical structure factor corresponding to the ground state phase diagram and establishes a unified quasiparticle framework for understanding the fundamental nature of excitations across distinct quantum phases in frustrated J1J_1J1​-J2J_2J2​-J3J_3J3​ Heisenberg spin chains.