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Valence Bond Entanglement Entropy

par Nicolas Laflorencie - 19 octobre 2007

The notion of quantum entanglement in condensed matter systems provides a new route to study and understand the effect of strong correlations and the nature of exotic phases at low temperature. Following these lines, members of the lab have recently shown that the entanglement present in some quantum magnets can be studied in the framework of Resonating Valence Bond physics, known to play a crucial role for instance in the physics of spin liquids or high Tc superconductivity.

When a quantum system is divided in two parts, A and B, the amount of entanglement shared between A and B is usually characterized by the Von Neumann Entropy

S^{\rm VN}=-{\rm{Tr}} \rho_{\rm A}\ln\rho_{\rm A}.


\rho_{\rm A}={\rm Tr}_{\rm B} |GS\rangle\langle GS|

is the reduced density matrix.

As a trivial example one can take two antiferromagnetically coupled spins S=1/2. The ground-state of such a system is a spin singlet :

|S\rangle = |\uparrow\downarrow - \downarrow\uparrow\rangle/\sqrt{2}

If one cuts the system into two parts A and B as depicted in Fig.1, the reduced density matrix is simply

\rho_{\rm A}=\frac{1}{2}{\text{1\!\!\hspace{1.05pt}l}}

, thus giving for the entanglement entropy

S^{\rm VN}=\ln 2


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Fig.1 : The two spins S=1/2 system is divided into two sub-systems A and B.

The idea developped in Phys. Rev. Lett. 99, 117204 (2007) by F. Alet (LPT), S. Capponi (LPT), N. Laflorencie (EPF Lausanne and LPS, Orsay) and M. Mambrini (LPT) is based on the fact that the ground-state of any SU(2) symmetric antiferromagnet can be expressed as a (complicated) linear combination of pair-wise singlets, or more explicitely as a sum of coverings similar to the one depicted in Fig. 2.

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Fig. 2 : Example of singlet covering on a 2D square lattice. The system is divided into two parts.

Then a natural measure of the valence bond entanglement entropy has been constructed by counting the average number of singlet bonds actually crossing the boundary between the two sub-systems (like \Omega and {\overline{\Omega}} in Fig. 2). Such a computation has been achieved using a newly developped Quantum Monte Carlo algorithm (see A. W. Sandvik in Phys. Rev. Lett. 95, 207203 (2005)) which works directly in the Valence Bond basis. Comparisons of with the Von Neumann entropy in 1D shows a suprisingly good agreement. Results are also presented for 2D systems thus providing the first measurement of the block entropy for interacting systems in D>1.

Post-scriptum :

For more detailed, see the original paper Valence Bond Entanglement Entropy, by F. Alet, S. Capponi, N. Laflorencie and M. Mambrini in Phys. Rev. Lett. 99, 117204 (2007) (see also cond-mat/0703027)

Selected for the Virtual Journal of Quantum Information, September 2007 Issue, and for the Virtual Journal of Nanoscale Science & Technology, September 2007 Issue