Abstract
An exponential deformation of a 1D critical Hamiltonian, with couplings falling on a length scale \(h^{-1}\), gives rise to ground states whose entanglement entropy follows a volume law, i.e. the area law is violated maximally. The ground state is now in the so-called rainbow phase, where valence bonds connect sites on the left half with their symmetric counterparts on the right. Here we discuss some of the most relevant features of this rainbow phase, focusing on the XX and Heisenberg models. Moreover, we show that the rainbow state can be understood either as a thermo-field double of a conformal field theory with a temperature proportional to h or as a massless Dirac fermion in a curved spacetime with constant negative curvature proportional to h. Finally, we introduce a study of the time-evolution of the rainbow state after a quench to a homogeneous Hamiltonian.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
R.J. Baxter, Phys. A. 106(1–2), 18 (1981)
C. Gómez, M. Ruiz-Altaba, G. Sierra, Quantum Groups in Two-Dimensional Physics. Cambridge Lecture Notes in Physics (Cambridge University Press, Cambridge, 1996)
M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition, 10th edn. (Cambridge University Press, Cambridge, 2010)
M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen(De), U. Sen, Adv. Phys. 56(2), 243 (2007)
N.W. Ashcroft, N.D. Mermin, Solid State Physics (Harcourt College Publishers, San Diego, 1976)
P. Hohenberg, W. Kohn, Phys. Rev. 136, B864 (1964)
W. Kohn, L.J. Sham, Phys. Rev. 140, A1133 (1965)
J. Bardeen, L.N. Cooper, J.R. Schrieffer, Phys. Rev. 106, 162 (1957)
J. Bardeen, L.N. Cooper, J.R. Schrieffer, Phys. Rev. 108, 1175 (1957)
R.B. Laughlin, Phys. Rev. B 23, 5632 (1981)
L. Fu, C.L. Kane, Phys. Rev. B 76, 045302 (2007)
I. Bloch, J. Dalibard, W. Zwerger, Rev. Mod. Phys. 80, 885 (2008)
M. Lewenstein, A. Sanpera, V. Ahufinger, Ultracold Atoms in Optical Lattices: Simulating Quantum Many-Body Systems (Oxford University Press, Oxford, 2012)
E. Schmidt, Math. Ann. 63(4), 433 (1907)
C. Shannon, Bell Syst. Tech. J. 27(3), 379 (1948)
G. Vitagliano, A. Riera, J.I. Latorre, New J. Phys. 12(11), 113049 (2010)
K. Okunishi, T. Nishino, Phys. Rev. B 82, 144409 (2010)
H. Ueda, H. Nakano, K. Kusakabe, T. Nishino, Progr. Theor. Phys. 124(3), 389 (2010)
K.G. Wilson, Rev. Mod. Phys. 47, 773 (1975)
G. Ramírez, J. Rodríguez-Laguna, G. Sierra, J. Stat. Mech. 2014(10), P10004 (2014)
B. Jin, V. Korepin, J. Stat. Phys. 116, 79 (2004)
G. Ramírez, J. Rodríguez-Laguna, G. Sierra, J. Stat. Mech. 2015(6), P06002 (2015)
P. Calabrese, J. Cardy, J. Stat. Mech. 2004(06), P06002 (2004)
E. Tonni, J. Rodríguez-Laguna, G. Sierra, J. Stat. Mech. 2018(4), 043105 (2018)
J. Rodríguez-Laguna, J. Dubail, G. Ramírez, P. Calabrese, G. Sierra, J. Stat. Mech. 50(16), 164001 (2017)
J. Dubail, J.M. Stéphan, J. Viti, P. Calabrese, SciPost Phys. 2, 002 (2017)
G. Ramírez, J. Rodríguez-Laguna, G. Sierra, Quenched dynamics of valence bond states. Work in progress
E.H. Lieb, D.W. Robinson, Commun. Math. Phys. 28(3), 251 (1972)
B. Alkurtass, L. Banchi, S. Bose, Phys. Rev. A 90, 042304 (2014)
I. Pitsios, L. Banchi, A.S. Rab, M. Bentivegna, D. Caprara, A. Crespi, N. Spagnolo, S. Bose, P. Mataloni, R. Osellame, F. Sciarrino, Nat. Commun. 8, 1569 (2017)
P. Calabrese, J. Cardy, J. Stat. Mech. 2005(04), P04010 (2005)
Acknowledgements
We would like to thank Pasquale Sodano for organising the conferences hosted at the International Institute of Physics (Natal) and for his efforts to collect the contributions for the Proceedings. We would also like to thank Silvia N. Santalla for useful conversations. We acknowledge financial support from the grants FIS2015-69167-C2-1-P, QUITEMAD+ S2013/ICE-2801 and SEV-2016-0597.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Ramírez, G., Rodríguez-Laguna, J., Sierra, G. (2020). Breaking the Area Law: The Rainbow State. In: Ferraz, A., Gupta, K., Semenoff, G., Sodano, P. (eds) Strongly Coupled Field Theories for Condensed Matter and Quantum Information Theory. Springer Proceedings in Physics, vol 239. Springer, Cham. https://doi.org/10.1007/978-3-030-35473-2_19
Download citation
DOI: https://doi.org/10.1007/978-3-030-35473-2_19
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-35472-5
Online ISBN: 978-3-030-35473-2
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)