Abstract
The scaling of entanglement entropy for the nearest neighbor antiferromagnetic Heisenberg spin model is studied computationally for clusters joined by a single bond. Bisecting the balanced three legged Bethe cluster, gives a second Renyi entropy and the valence bond entropy which scales as the number of sites in the cluster. For the analogous situation with square clusters, i.e. two \(L \times L\) clusters joined by a single bond, numerical results suggest that the second Renyi entropy and the valence bond entropy scales as L. For both systems, the environment and the system are connected by the single bond and interaction is short range. The entropy is not constant with system size as suggested by the area law.
This is a preview of subscription content, access via your institution.
Similar content being viewed by others
References
Bombelli, L., Koul, R.K., Lee, J., Sorkin, R.D.: Quantum source of entropy for black holes. Phys. Rev. D 34, 373–383 (1986)
Srednicki, M.: Entropy and area. Phys. Rev. Lett. 71, 666–669 (1993)
Plenio, M.B., Eisert, J., Dreissig, J., Cramer, M.: Entropy, entanglement, and area: analytical results for harmonic lattice systems. Phys. Rev. Lett. 94, 060503 (2005)
Holzhey, C., Larsen, F., Wilczek, F.: Geometric and renormalized entropy in conformal field theory. Nucl. Phys. B 424, 443–467 (1994)
Vidal, G., Latorre, J.I., Rico, E., Kitaev, A.: Entanglement in quantum critical phenomena. Phys. Rev. Lett. 90, 227902 (2003)
Latorre, J.I., Rico, E., Vidal, G.: Ground state entanglement in quantum spin chains. Quantum Inf. Comput. 4, 48–92 (2004)
Latorre, J.I., Lütken, C.A., Rico, E., Vidal, G.: Fine-grained entanglement loss along renormalization-group flows. Phys. Rev. A 71, 034301–4p (2005)
Calabrese, P., Cardy, J.: Entanglement entropy and quantum field theory. J. Stat. Mech., P06002 (2004)
Caravan, B., Friedman, B.A., Levine, G.C.: Scaling of entanglement entropy in point-contact, free-fermion systems. Phys. Rev. A 89, 052305–7p (2014)
Song, H.F., Laflorencie, N., Rachel, S., LeHur, K.: Entanglement entropy of the two-dimensional Heisenberg antiferromagnet. Phys. Rev. B 83, 224410–7p (2011)
Sandvik, A.W.: Ground state projection of quantum spin systems in the valence-bond basis. Phys. Rev. Lett. 95, 207203 (2005)
Hastings, M.B., Gonzalez, I., Kallin, A.B., Melko, R.G.: Measuring Renyi entanglement entropy in quantum Monte Carlo simulations. Phys. Rev. Lett. 104, 157201 (2010)
Alet, F., Capponi, S., Laflorencie, N., Mambrini, M.: Valence bond entanglement entropy. Phys. Rev. Lett. 99, 117204 (2007)
Chhajlany, R.W., Tomczak, P., Wojcik, A.: Topological estimator of block entanglement for Heisenberg antiferromagnets. Phys. Rev. Lett. 99, 167204 (2007)
Chandran, A., Khemani, V., Sondhi, S.L.: How universal is the entanglement spectrum? Phys. Rev. Lett. 113, 060501 (2014)
Lieb, E.H., Mattis, D.: Ordering of energy levels of interacting spin systems. J. Math. Phys. 3, 749–751 (1962)
Liang, S., Doucot, B., Anderson, P.W.: Some new variational resonating-valence-bond-type wave functions for the spin- 1/2 antiferromagnetic Heisenberg model on a square lattice. Phys. Rev. Lett. 61, 365–368 (1988)
Levine, G.C., Miller, D.J.: Zero-dimensional area law in a gapless fermionic system. Phys. Rev. B 77, 205119 (2008)
Kallin, A.B., Hastings, M.B., Melko, R.G., Singh, R.R.P.: Anomalies in the entanglement properties of the square-lattice Heisenberg model. Phys. Rev. B 84, 165134 (2011). arXiv:1107.2840v2
Kumar, M., Ramasesha, S., Soos, Z.G.: Density matrix renormalization group algorithm for Bethe lattices of spin-1/2 or spin-1 sites with Heisenberg antiferromagnetic exchange. Phys. Rev. B 85, 134415 (2012)
Changlani, H.J., Ghosh, S., Henley, C.L., Lauchli, A.M.: Heisenberg antiferromagnet on Cayley trees: low-energy spectrum and even/odd site imbalance. Phys. Rev. B 87, 085107 (2013)
Eisert, J., Cramer, M., Plenio, M.B.: Colloquium: area laws for the entanglement entropy. Rev. Mod. Phys. 82, 277–306 (2010)
Bravyi, S., Caha, L., Movassagh, R., Nagaj, D., Schor, P.W.: Criticality without frustration for quantum spin-1 chains. Phys. Rev. Lett. 109, 207202 (2012)
Movassagh, R., Schor, P.W.: Power law violation of the area law in quantum spin chains (2014). arXiv:1408.1657v2
Aharonov, D., Harrow, A.W., Landau, Z., Nagaj, D., Szegedy, M., Vazirani, U.: Local tests of global entanglement and a counterexample to the generalized area law (2014). arXiv:1410.0951v1
Hastings, M.B.: Random MERA states and the tightness of the Brandao-Horodecki entropy bound (2015). arXiv:1505.06468v1
DeGennes, P.G., Hervet, H.J.: Statistics of<<starburst>> polymers. J. Phys. Lett. 44, 351–360 (1983)
Kallin, A.B., Gonzalez, I., Hastings, M.B., Melko, R.G.: Valence bond and von Neumann entanglement entropy in Heisenberg ladders. Phys. Rev. Lett. 103, 117203 (2009)
Islam, R., Ma, R., Preiss, P.M., Eric Tai, M., Lukin, A., Rispoli, M., Griener, M.: Measuring entanglement entropy through the interference of quantum many-body twins (2015). arXiv:1509.01160v1
Marien, M., Audenaert, K.M.R., Acoleyen, K.V., Verstraete, F.: Entanglement rates and the stability of the area law for the entanglement entropy (2014). arXiv:1411.0680v1
Monroe, C., Raussendorf, R., Ruthven, A., Brown, K.R., Maunz, P., Duan, L.-M., Kim, J.: Large-scale modular quantum-computer architecture with atomic memory and photonic interconnects. Phys. Rev. A 89, 022317 (2014)
Monroe, C., Schoelkopf, R.J., Lukin, M.D.: Quantum connections. Sci. Am. 314, 50–57 (2016)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Friedman, B.A., Levine, G.C. Scaling of Entanglement Entropy for the Heisenberg Model on Clusters Joined by Point Contacts. J Stat Phys 165, 727–739 (2016). https://doi.org/10.1007/s10955-016-1640-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-016-1640-7