Original language  English (US) 

Article number  038901 
Journal  Physical review letters 
Volume  121 
Issue number  3 
DOIs 

State  Published  Jul 17 2018 
All Science Journal Classification (ASJC) codes
 General Physics and Astronomy
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In: Physical review letters, Vol. 121, No. 3, 038901, 17.07.2018.
Research output: Contribution to journal › Comment/debate › peerreview
TY  JOUR
T1  Comment on "systematic Construction of Counterexamples to the Eigenstate Thermalization Hypothesis"
AU  Mondaini, Rubem
AU  Mallayya, Krishnanand
AU  Santos, Lea F.
AU  Rigol, Marcos
N1  Funding Information: Mondaini Rubem 1 Mallayya Krishnanand 2 Santos Lea F. 3 Rigol Marcos 2 1 Beijing Computational Science Research Center , Beijing 100193, China Department of Physics, 2 Pennsylvania State University , University Park, Pennsylvania 16802, USA Department of Physics, 3 Yeshiva University , New York, New York 10016, USA 17 July 2018 20 July 2018 121 3 038901 15 May 2018 15 November 2017 © 2018 American Physical Society 2018 American Physical Society A Comment on the Letter by N. Shiraishi and T. Mori , Phys. Rev. Lett. 119 , 030601 ( 2017 ). PRLTAO 00319007 10.1103/PhysRevLett.119.030601 The authors of the Letter offer a Reply. National Natural Science Foundation of China 10.13039/501100001809 U1530401 11674021 11650110441 National Science Foundation 10.13039/100000001 PHY1707482 DMR1603418 The Letter [1] claims to provide a general method for constructing local Hamiltonians that do not fulfill the Eigenstate Thermalization Hypothesis (ETH) [2–6] . We argue that the claim is misguided. The Letter [1] reports the construction of blockdiagonal Hamiltonians with nonlocal manybody conserved quantities. In the second example, one such quantity is used to construct a Hamiltonian with two exponentially large symmetry sectors (the on site magnetic fields and local spin interactions were chosen to be different in the two sectors). It is not surprising that the ETH is violated when mixing them. Random matrix theory, the base of our understanding of the ETH [6] , only applies within each symmetry sector and not to the entire Hamiltonian [6–8] . Consequently, the ETH should be studied within each sector separately. Conserved nonlocal manybody operators associated with lattice translations, pointgroup symmetries, and particlehole transformations [7,9–15] also generate blockdiagonal Hamiltonians. In the chaotic regime of such models, in contrast to models with local conserved quantities (e.g., total particle number [9–13] ) and the models in Refs. [1,16–18] , the eigenstate expectation values of fewbody operators are usually the same (up to finitesize effects) in different symmetry sectors [see Fig. 1(a) ]. The need to analyze each sector separately becomes apparent when studying the offdiagonal matrix elements [15] , an equally important part of the ETH [6] . At any energy, the average magnitude of offdiagonal matrix elements of fewbody operators that do not break symmetries of the Hamiltonian is generally different within different symmetry sectors. Also, they vanish between eigenstates that belong to different sectors. Hence, mixing different symmetry sectors may lead one to conclude that the ETH is violated while it is not. 1 10.1103/PhysRevLett.121.038901.f1 FIG. 1. (a) Diagonal and (b) offdiagonal (for  E α + E β  / N ≤ 0.1 ) matrix elements of the nearestneighbor spin correlations σ ^ i z σ ^ nn i z for the ferromagnetic transversefield Ising model ( g = J ) in two dimensions [14,15] . Continuous lines in (b) depict running averages. The inset in (a) shows that the fluctuations of the diagonal matrix elements are different in the two sectors shown. Inset in (b): ratio R between the running averages of the offdiagonal matrix elements. The dashed line shows that, as expected from the ETH [6,19] , the ratio R is very close to the square root of the inverse ratio between the Hilbert space dimension D of the sectors. λ Z ^ 2 , λ S ^ x , λ S ^ y , λ S ^ x y are the eigenvalues of the spinflip, mirror x , mirror y , and mirror along the x = y line symmetries, respectively. The results shown are for the zero momentum sector of a lattice with N = 5 × 5 sites (see Ref. [15] for further details). In Fig. 1(b) , we plot offdiagonal matrix elements, and their running average, within the same symmetry sectors as in Fig. 1(a) . The mismatch of their magnitudes is apparent. Their ratio is determined by the ratio of the Hilbert space dimensions [6,19] , see inset in Fig. 1(b) . This shows that different symmetry sectors should not be mixed when discussing the ETH. We are also troubled by the statement in Ref. [1] that numerical simulations have shown that the ETH is valid for Hamiltonians with (i) translational invariance, (ii) no local conserved quantity, and (iii) local [ O ( 1 ) support] interactions. None of these conditions is necessary for the onset of quantum chaos and the validity of the ETH. An early discussion on the connection between the ETH and thermalization in manybody lattice Hamiltonians involved a nontranslationally invariant system [4] . Many of the models in which the ETH has been verified have a local conserved quantity: the total particle number or magnetization [9–13] . Finally, in Ref. [20] , the ETH was verified in a model of hardcore bosons with dipolar ( 1 / r 3 ) interactions in the presence of a harmonic trap, which does not satisfy any of the three conditions. R. M. is supported by the NSFC, Grants No. U1530401, No. 11674021, and No. 11650110441. K. M. and M. R. are supported by the NSF, Grant No. PHY1707482. L. F. S. is supported by the NSF Grant No. DMR1603418. [1] 1 N. Shiraishi and T. Mori , Phys. Rev. Lett. 119 , 030601 ( 2017 ). PRLTAO 00319007 10.1103/PhysRevLett.119.030601 [2] 2 J. M. Deutsch , Phys. Rev. A 43 , 2046 ( 1991 ). PLRAAN 10502947 10.1103/PhysRevA.43.2046 [3] 3 M. Srednicki , Phys. Rev. E 50 , 888 ( 1994 ). PLEEE8 1063651X 10.1103/PhysRevE.50.888 [4] 4 M. Rigol , V. Dunjko , and M. Olshanii , Nature (London) 452 , 854 ( 2008 ). NATUAS 00280836 10.1038/nature06838 [5] 5 M. Rigol and M. Srednicki , Phys. Rev. Lett. 108 , 110601 ( 2012 ). PRLTAO 00319007 10.1103/PhysRevLett.108.110601 [6] 6 L. D’Alessio , Y. Kafri , A. Polkovnikov , and M. Rigol , Adv. Phys. 65 , 239 ( 2016 ). ADPHAH 00018732 10.1080/00018732.2016.1198134 [7] 7a L. F. Santos and M. Rigol , Phys. Rev. E 81 , 036206 ( 2010 ); PRESCM 15393755 10.1103/PhysRevE.81.036206 7b L. F. Santos and M. Rigol Phys. Rev. E 82 , 031130 ( 2010 ). PRESCM 15393755 10.1103/PhysRevE.82.031130 [8] 8 A. Gubin and L. F. Santos , Am. J. Phys. 80 , 246 ( 2012 ). AJPIAS 00029505 10.1119/1.3671068 [9] 9a M. Rigol , Phys. Rev. Lett. 103 , 100403 ( 2009 ); PRLTAO 00319007 10.1103/PhysRevLett.103.100403 9b M. Rigol Phys. Rev. A 80 , 053607 ( 2009 ). PLRAAN 10502947 10.1103/PhysRevA.80.053607 [10] 10 R. Steinigeweg , A. Khodja , H. Niemeyer , C. Gogolin , and J. Gemmer , Phys. Rev. Lett. 112 , 130403 ( 2014 ). PRLTAO 00319007 10.1103/PhysRevLett.112.130403 [11] 11 S. Sorg , L. Vidmar , L. Pollet , and F. HeidrichMeisner , Phys. Rev. A 90 , 033606 ( 2014 ). PLRAAN 10502947 10.1103/PhysRevA.90.033606 [12] 12 H. Kim , T. N. Ikeda , and D. A. Huse , Phys. Rev. E 90 , 052105 ( 2014 ). PRESCM 15393755 10.1103/PhysRevE.90.052105 [13] 13a W. Beugeling , R. Moessner , and M. Haque , Phys. Rev. E 89 , 042112 ( 2014 ); PRESCM 15393755 10.1103/PhysRevE.89.042112 13b W. Beugeling , R. Moessner , and M. Haque Phys. Rev. E 91 , 012144 ( 2015 ). PRESCM 15393755 10.1103/PhysRevE.91.012144 [14] 14 R. Mondaini , K. R. Fratus , M. Srednicki , and M. Rigol , Phys. Rev. E 93 , 032104 ( 2016 ). PRESCM 24700045 10.1103/PhysRevE.93.032104 [15] 15 R. Mondaini and M. Rigol , Phys. Rev. E 96 , 012157 ( 2017 ). PRESCM 24700045 10.1103/PhysRevE.96.012157 [16] 16 T. Mori and N. Shiraishi , Phys. Rev. E 96 , 022153 ( 2017 ). PRESCM 24700045 10.1103/PhysRevE.96.022153 [17] 17 Z. Lan and S. Powell , Phys. Rev. B 96 , 115140 ( 2017 ). PRBMDO 24699950 10.1103/PhysRevB.96.115140 [18] 18 Z. Lan , M. van Horssen , S. Powell , and J. P. Garrahan , arXiv:1706.02603 . [19] 19 M. Srednicki , J. Phys. A 32 , 1163 ( 1999 ). JPHAC5 03054470 10.1088/03054470/32/7/007 [20] 20 E. Khatami , G. Pupillo , M. Srednicki , and M. Rigol , Phys. Rev. Lett. 111 , 050403 ( 2013 ). PRLTAO 00319007 10.1103/PhysRevLett.111.050403
PY  2018/7/17
Y1  2018/7/17
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U2  10.1103/PhysRevLett.121.038901
DO  10.1103/PhysRevLett.121.038901
M3  Comment/debate
C2  30085812
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SN  00319007
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JO  Physical review letters
JF  Physical review letters
IS  3
M1  038901
ER 