Abstract
Area law violations for entanglement entropy in the form of a square root has recently been studied for one-dimensional frustration-free quantum systems based on the Motzkin walks and their variations. Here, we further modify the Motzkin walks using the elements of a symmetric inverse semigroup as basis states on each step of the walk. This change alters the number of paths allowed in the Motzkin walks and by introducing an appropriate term in the Hamiltonian with a tunable parameter we show that we can jump from a state that violates the area law logarithmically to a state that obeys the area law providing an example of quantum phase transition in a one-dimensional system.
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References
U. Schollwoeck, Rev. Mod. Phys. 77, 259 (2005)
R. Orus, Ann. Phys. 349, 117 (2014)
S.R. White, Phys. Rev. Lett. 69, 2863 (1992)
M.B. Hastings, JSTAT 73, P08024 (2007)
I. Arad, A. Kitaev, Z. Landau, U. Vazirani, https://doi.org/arXiv:1301.1162 [quant-ph] (2013)
T. Nishioka, S. Ryu, T. Takayanagi, J. Phys. A42, 504008 (2009)
J. Eisert, M. Cramer, M.B. Plenio, Rev. Mod. Phys. 82, 277 (2010)
N. Laflorencie, Phys. Rep. 643, 1 (2016)
D.N. Page, Phys. Rev. Lett. 71, 1291 (1993)
S.K. Foong, S. Kanno, Phys. Rev. Lett. 72, 1148 (1994)
S. Sen, Phys. Rev. Lett. 77, 1 (1996)
S. Irani, J. Math. Phys. 51, 022101 (2010)
S. Yan, D.A. Huse, S.R. White, Science 332, 1173 (2011)
D. Gottesman, M.B. Hastings, New J. Phys. 12, 025002 (2010)
G. Vitagliano, A. Riera, J.I. Latorre, https://doi.org/arXiv:1003.1292 [quant-ph] (2010)
G. Ramírez, J.R. Laguna, G. Sierra, J. Stat. Mech. 2014, P10004 (2014)
O. Salberger, V. Korepin, https://doi.org/arXiv:1605.03842 [quant-ph] (2016)
S. Bravyi, L. Caha, R. Movassagh, D. Nagaj, P.W. Shor, Phys. Rev. Lett. 109, 207202 (2012)
L. Huijse, B. Swingle, Phys. Rev. B 87, 035108 (2013)
R. Movassagh, https://doi.org/arXiv:1606.09313 [quant-ph] (2016)
A. Osterloh, L. Amico, G. Falci, R. Fazio, Nature 416, 608 (2002)
P. Calabrese, J. Cardy, J. Phys. A 42, 504005 (2009)
C. Holzhey, F. Larsen, F. Wilczek, Nucl. Phys. B 424, 443 (1994)
C. Callan, F. Wilczek, Phys. Lett. B 333, 55 (1994)
M.M. Wolf, Phys. Rev. Lett. 96, 010404 (2006)
D. Gioev, I. Klich, Phys. Rev. Lett. 96, 100503 (2006)
R. Movassagh, P.W. Shor, Proc. Natl. Acad. Sci. 113, 201605716 (2016)
O. Salberger, T. Udagawa, Z. Zhang, H. Katsura, I. Klich, V. Korepin, J. Stat. Mech. 1706, 063103 (2017)
Z. Zhang, I. Klich, https://doi.org/arXiv:1702.03581 [cond-mat.stat-mech] (2017)
Z. Zhang, A. Ahmadain, I. Klich, https://doi.org/arXiv:1606.07795 [quant-ph] (2016)
J. Kellendonk, M.V. Lawson, J. Algebra 224, 140 (2000)
M. Senechal, Quasicrystals and Geometry (Cambridge University Press, 1995)
R. Exel, D. Goncalves, C. Starling, https://doi.org/arXiv:1106.4535 [math.OA] (2011)
P. Padmanabhan, S.-J. Rey, D. Teixeira, D. Trancanelli, JHEP 1705, 136 (2017)
V.V. Wagner, J. Lond. Math. Soc. 29, 411 (1954)
M.V. Lawson, Inverse Semigroups – The Theory of Partial Symmetries (World Scientific, 1998)
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Sugino, F., Padmanabhan, P. Quantum phase transitions in a frustration-free spin chain based on modified Motzkin walks. Eur. Phys. J. Spec. Top. 227, 269–284 (2018). https://doi.org/10.1140/epjst/e2018-00080-2
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DOI: https://doi.org/10.1140/epjst/e2018-00080-2