Critical Scaling Behaviors of Entanglement Spectra
Qi-Cheng Tang1,2 , Wei Zhu1,2**
1 School of Science, Westlake University, Hangzhou 3100242 Institute of Natural Sciences, Westlake Institute of Advanced Study, Hangzhou 310024
Abstract :We investigate the evolution of entanglement spectra under a global quantum quench from a short-range correlated state to the quantum critical point. Motivated by the conformal mapping, we find that the dynamical entanglement spectra demonstrate distinct finite-size scaling behaviors from the static case. As a prototypical example, we compute real-time dynamics of the entanglement spectra of a one-dimensional transverse-field Ising chain. Numerical simulation confirms that the entanglement spectra scale with the subsystem size $l$ as $\sim$$l^{-1}$ for the dynamical equilibrium state, much faster than $\propto$ $\ln^{-1} l$ for the critical ground state. In particular, as a byproduct, the entanglement spectra at the long time limit faithfully gives universal tower structure of underlying Ising criticality, which shows the emergence of operator-state correspondence in the quantum dynamics.
收稿日期: 2019-10-25
出版日期: 2019-11-08
:
03.65.Ud
(Entanglement and quantum nonlocality)
11.25.Hf
(Conformal field theory, algebraic structures)
[1] Di Francesco P , Mathieu P, Sénéchal D 1997 Conformal Field Theory, Graduate Texts in Contemporary Physics (New York: Springer) [2] Belavin A A, Polyakov A M and Zamolodchikov A B 1984 J. Stat. Phys. 34 763 [3] Belavin A, Polyakov A and Zamolodchikov A 1984 Nucl. Phys. B 241 333 [4] Friedan D, Qiu Z and Shenker S 1984 Phys. Rev. Lett. 52 1575 [5] Cardy J L 1986 Nucl. Phys. B 270 186 [6] Cardy J L 1986 Nucl. Phys. B 275 200 [7] Srednicki M 1993 Phys. Rev. Lett. 71 666 [8] Holzhey C, Larsen F and Wilczek F 1994 Nucl. Phys. B 424 443 [9] Calabrese P and Cardy J 2004 J. Stat. Mech. 2004 P06002 [10] Korepin V E 2004 Phys. Rev. Lett. 92 096402 [11] Calabrese P and Cardy J 2005 J. Stat. Mech. 2005 P04010 [12] Calabrese P and Cardy J 2006 Phys. Rev. Lett. 96 136801 [13] Fradkin E and Moore J E 2006 Phys. Rev. Lett. 97 050404 [14] Calabrese P and Cardy J 2007 J. Stat. Mech. 2007 P10004 [15] Calabrese P and Lefevre A 2008 Phys. Rev. A 78 032329 [16] Hsu B, Mulligan M, Fradkin E and Kim E A 2009 Phys. Rev. B 79 115421 [17] Calabrese P and Cardy J 2009 J. Phys. A 42 504005 [18] Nienhuis B, Campostrini M and Calabrese P 2009 J. Stat. Mech. 2009 P02063 [19] Alba V, Tagliacozzo L and Calabrese P 2010 Phys. Rev. B 81 060411 [20] Calabrese P, Campostrini M, Essler F and Nienhuis B 2010 Phys. Rev. Lett. 104 095701 [21] Calabrese P, Cardy J and Tonni E 2012 Phys. Rev. Lett. 109 130502 [22] Calabrese P, Cardy J and Tonni E 2013 J. Stat. Mech. 2013 P02008 [23] Cardy J 2014 Phys. Rev. Lett. 112 220401 [24] Calabrese P, Cardy J and Tonni E 2015 J. Phys. A 48 015006 [25] Coser A, Tonni E and Calabrese P 2014 J. Stat. Mech. 2014 P12017 [26] Cardy J 2016 J. Stat. Mech. 2016 023103 [27] Calabrese P and Cardy J 2016 J. Stat. Mech. 2016 064003 [28] Cardy J and Tonni E 2016 J. Stat. Mech. 2016 123103 [29] Alba V, Calabrese P and Tonni E 2018 J. Phys. A 51 024001 [30] Wen X, Ryu S and Ludwig A W W 2018 J. Stat. Mech. 2018 113103 [31] Giudici G, Mendes-Santos T, Calabrese P and Dalmonte M 2018 Phys. Rev. B 98 134403 [32] Giulio G D, Arias R and Tonni E 2019 arXiv:1905.01144 [33] Vidal G, Latorre J I, Rico E and Kitaev A 2003 Phys. Rev. Lett. 90 227902 [34] Pollmann F, Mukerjee S, Turner A M and Moore J E 2009 Phys. Rev. Lett. 102 255701 [35] Metlitski M A, Fuertes C A and Sachdev S 2009 Phys. Rev. B 80 115122 [36] Whitsitt S, Witczak-Krempa W and Sachdev S 2017 Phys. Rev. B 95 045148 [37] Zhu W, Chen X, He Y C and Witczak-Krempa W 2018 Sci. Adv. 4 eaat5535 [38] Li H and Haldane F D M 2008 Phys. Rev. Lett. 101 010504 [39] Laflorencie N 2016 Phys. Rep. 646 1 [40] Fidkowski L 2010 Phys. Rev. Lett. 104 130502 [41] Prodan E, Hughes T L and Bernevig B A 2010 Phys. Rev. Lett. 105 115501 [42] Turner A M, Zhang Y and Vishwanath A 2010 Phys. Rev. B 82 241102 [43] Qi X L, Katsura H and Ludwig A W W 2012 Phys. Rev. Lett. 108 196402 [44] Thomale R, Arovas D P and Bernevig B A 2010 Phys. Rev. Lett. 105 116805 [45] De Chiara G , Lepori L, Lewenstein M and Sanpera A 2012 Phys. Rev. Lett. 109 237208 [46] Lepori L, De Chiara G and Sanpera A 2013 Phys. Rev. B 87 235107 [47] Giampaolo S M, Montangero S, Dell'Anno F, De Siena S and Illuminati F 2013 Phys. Rev. B 88 125142 [48] Lundgren R, Blair J, Laurell P, Regnault N, Fiete G A, Greiter M and Thomale R 2016 Phys. Rev. B 94 081112 [49] Schuler M, Whitsitt S, Henry L P, Sachdev S and Läuchli A M 2016 Phys. Rev. Lett. 117 210401 [50] Whitsitt S, Schuler M, Henry L P, Läuchli A M and Sachdev S 2017 Phys. Rev. B 96 035142 [51] Stojevic V, Haegeman J, McCulloch I P, Tagliacozzo L and Verstraete F 2015 Phys. Rev. B 91 035120 [52] Läuchli A M 2013 arXiv:1303.0741 [53] Laflorencie N and Rachel S 2014 J. Stat. Mech. 2014 P11013 [54] Lieb E, Schultz T and Mattis D 1961 Ann. Phys. 16 407 [55] Peschel I and Eisler V 2009 J. Phys. A 42 504003 [56] Torlai G, Tagliacozzo L and Chiara G D 2014 J. Stat. Mech. 2014 P06001 [57] Vidal G 2004 Phys. Rev. Lett. 93 040502 [58] Chepiga N and Mila F 2017 Phys. Rev. B 96 054425 [59] Milsted A and Vidal G 2017 Phys. Rev. B 96 245105 [60] Zou Y, Milsted A and Vidal G 2018 Phys. Rev. Lett. 121 230402 [61] Zou Y and Vidal G 2019 arXiv:1907.10704 [62] Surace J, Tagliacozzo L and Tonni E 2019 arXiv:1909.07381
[1]
. [J]. 中国物理快报, 2022, 39(11): 110301-.
[2]
. [J]. 中国物理快报, 2022, 39(7): 70302-.
[3]
. [J]. 中国物理快报, 2022, 39(3): 30302-030302.
[4]
. [J]. 中国物理快报, 2022, 39(3): 30301-.
[5]
. [J]. 中国物理快报, 2021, 38(9): 90301-.
[6]
. [J]. 中国物理快报, 2020, 37(9): 90302-.
[7]
. [J]. 中国物理快报, 0, (): 60301-.
[8]
. [J]. 中国物理快报, 2020, 37(6): 60301-.
[9]
. [J]. 中国物理快报, 2019, 36(10): 100301-.
[10]
. [J]. 中国物理快报, 2019, 36(8): 80303-.
[11]
. [J]. 中国物理快报, 2019, 36(8): 80304-.
[12]
. [J]. 中国物理快报, 2019, 36(6): 60301-.
[13]
. [J]. 中国物理快报, 2018, 35(11): 110302-.
[14]
. [J]. 中国物理快报, 2018, 35(10): 100301-.
[15]
. [J]. 中国物理快报, 2018, 35(9): 90303-.