Comment on "systematic Construction of Counterexamples to the Eigenstate Thermalization Hypothesis"

Rubem Mondaini; Krishnanand Mallayya; Lea F. Santos; Marcos Rigol

doi:10.1103/PhysRevLett.121.038901

Comment on "systematic Construction of Counterexamples to the Eigenstate Thermalization Hypothesis"

Rubem Mondaini
, Krishnanand Mallayya
, Lea F. Santos
, Marcos Rigol

Research output: Contribution to journal › Comment/debate › peer-review

Original language	English (US)
Article number	038901
Journal	Physical review letters
Volume	121
Issue number	3
DOIs	https://doi.org/10.1103/PhysRevLett.121.038901
State	Published - Jul 17 2018

All Science Journal Classification (ASJC) codes

General Physics and Astronomy

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10.1103/PhysRevLett.121.038901

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@article{288b9365343a4709a53fa33f6ddb3616,

title = "Comment on {"}systematic Construction of Counterexamples to the Eigenstate Thermalization Hypothesis{"}",

author = "Rubem Mondaini and Krishnanand Mallayya and Santos, \{Lea F.\} and Marcos Rigol",

note = "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 {\textcopyright} 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 0031-9007 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 PHY-1707482 DMR-1603418 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 block-diagonal Hamiltonians with nonlocal many-body 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 many-body operators associated with lattice translations, point-group symmetries, and particle-hole transformations [7,9–15] also generate block-diagonal 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 few-body operators are usually the same (up to finite-size effects) in different symmetry sectors [see Fig. 1(a) ]. The need to analyze each sector separately becomes apparent when studying the off-diagonal matrix elements [15] , an equally important part of the ETH [6] . At any energy, the average magnitude of off-diagonal matrix elements of few-body 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) off-diagonal (for | E α + E β | / N ≤ 0.1 ) matrix elements of the nearest-neighbor spin correlations σ \textasciicircum{} i z σ \textasciicircum{} nn i z for the ferromagnetic transverse-field 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 off-diagonal 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 \textasciicircum{} 2 , λ S \textasciicircum{} x , λ S \textasciicircum{} y , λ S \textasciicircum{} x y are the eigenvalues of the spin-flip, 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 off-diagonal 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 many-body 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 hard-core 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. PHY-1707482. L. F. S. is supported by the NSF Grant No. DMR-1603418. [1] 1 N. Shiraishi and T. Mori , Phys. Rev. Lett. 119 , 030601 ( 2017 ). PRLTAO 0031-9007 10.1103/PhysRevLett.119.030601 [2] 2 J. M. Deutsch , Phys. Rev. A 43 , 2046 ( 1991 ). PLRAAN 1050-2947 10.1103/PhysRevA.43.2046 [3] 3 M. Srednicki , Phys. Rev. E 50 , 888 ( 1994 ). PLEEE8 1063-651X 10.1103/PhysRevE.50.888 [4] 4 M. Rigol , V. Dunjko , and M. Olshanii , Nature (London) 452 , 854 ( 2008 ). NATUAS 0028-0836 10.1038/nature06838 [5] 5 M. Rigol and M. Srednicki , Phys. Rev. Lett. 108 , 110601 ( 2012 ). PRLTAO 0031-9007 10.1103/PhysRevLett.108.110601 [6] 6 L. D{\textquoteright}Alessio , Y. Kafri , A. Polkovnikov , and M. Rigol , Adv. Phys. 65 , 239 ( 2016 ). ADPHAH 0001-8732 10.1080/00018732.2016.1198134 [7] 7a L. F. Santos and M. Rigol , Phys. Rev. E 81 , 036206 ( 2010 ); PRESCM 1539-3755 10.1103/PhysRevE.81.036206 7b L. F. Santos and M. Rigol Phys. Rev. E 82 , 031130 ( 2010 ). PRESCM 1539-3755 10.1103/PhysRevE.82.031130 [8] 8 A. Gubin and L. F. Santos , Am. J. Phys. 80 , 246 ( 2012 ). AJPIAS 0002-9505 10.1119/1.3671068 [9] 9a M. Rigol , Phys. Rev. Lett. 103 , 100403 ( 2009 ); PRLTAO 0031-9007 10.1103/PhysRevLett.103.100403 9b M. Rigol Phys. Rev. A 80 , 053607 ( 2009 ). PLRAAN 1050-2947 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 0031-9007 10.1103/PhysRevLett.112.130403 [11] 11 S. Sorg , L. Vidmar , L. Pollet , and F. Heidrich-Meisner , Phys. Rev. A 90 , 033606 ( 2014 ). PLRAAN 1050-2947 10.1103/PhysRevA.90.033606 [12] 12 H. Kim , T. N. Ikeda , and D. A. Huse , Phys. Rev. E 90 , 052105 ( 2014 ). PRESCM 1539-3755 10.1103/PhysRevE.90.052105 [13] 13a W. Beugeling , R. Moessner , and M. Haque , Phys. Rev. E 89 , 042112 ( 2014 ); PRESCM 1539-3755 10.1103/PhysRevE.89.042112 13b W. Beugeling , R. Moessner , and M. Haque Phys. Rev. E 91 , 012144 ( 2015 ). PRESCM 1539-3755 10.1103/PhysRevE.91.012144 [14] 14 R. Mondaini , K. R. Fratus , M. Srednicki , and M. Rigol , Phys. Rev. E 93 , 032104 ( 2016 ). PRESCM 2470-0045 10.1103/PhysRevE.93.032104 [15] 15 R. Mondaini and M. Rigol , Phys. Rev. E 96 , 012157 ( 2017 ). PRESCM 2470-0045 10.1103/PhysRevE.96.012157 [16] 16 T. Mori and N. Shiraishi , Phys. Rev. E 96 , 022153 ( 2017 ). PRESCM 2470-0045 10.1103/PhysRevE.96.022153 [17] 17 Z. Lan and S. Powell , Phys. Rev. B 96 , 115140 ( 2017 ). PRBMDO 2469-9950 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 0305-4470 10.1088/0305-4470/32/7/007 [20] 20 E. Khatami , G. Pupillo , M. Srednicki , and M. Rigol , Phys. Rev. Lett. 111 , 050403 ( 2013 ). PRLTAO 0031-9007 10.1103/PhysRevLett.111.050403 ",

year = "2018",

month = jul,

day = "17",

doi = "10.1103/PhysRevLett.121.038901",

language = "English (US)",

volume = "121",

journal = "Physical review letters",

issn = "0031-9007",

publisher = "American Physical Society",

number = "3",

}

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 0031-9007 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 PHY-1707482 DMR-1603418 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 block-diagonal Hamiltonians with nonlocal many-body 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 many-body operators associated with lattice translations, point-group symmetries, and particle-hole transformations [7,9–15] also generate block-diagonal 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 few-body operators are usually the same (up to finite-size effects) in different symmetry sectors [see Fig. 1(a) ]. The need to analyze each sector separately becomes apparent when studying the off-diagonal matrix elements [15] , an equally important part of the ETH [6] . At any energy, the average magnitude of off-diagonal matrix elements of few-body 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) off-diagonal (for | E α + E β | / N ≤ 0.1 ) matrix elements of the nearest-neighbor spin correlations σ ^ i z σ ^ nn i z for the ferromagnetic transverse-field 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 off-diagonal 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 spin-flip, 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 off-diagonal 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 many-body 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 hard-core 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. PHY-1707482. L. F. S. is supported by the NSF Grant No. DMR-1603418. [1] 1 N. Shiraishi and T. Mori , Phys. Rev. Lett. 119 , 030601 ( 2017 ). PRLTAO 0031-9007 10.1103/PhysRevLett.119.030601 [2] 2 J. M. Deutsch , Phys. Rev. A 43 , 2046 ( 1991 ). PLRAAN 1050-2947 10.1103/PhysRevA.43.2046 [3] 3 M. Srednicki , Phys. Rev. E 50 , 888 ( 1994 ). PLEEE8 1063-651X 10.1103/PhysRevE.50.888 [4] 4 M. Rigol , V. Dunjko , and M. Olshanii , Nature (London) 452 , 854 ( 2008 ). NATUAS 0028-0836 10.1038/nature06838 [5] 5 M. Rigol and M. Srednicki , Phys. Rev. Lett. 108 , 110601 ( 2012 ). PRLTAO 0031-9007 10.1103/PhysRevLett.108.110601 [6] 6 L. D’Alessio , Y. Kafri , A. Polkovnikov , and M. Rigol , Adv. Phys. 65 , 239 ( 2016 ). ADPHAH 0001-8732 10.1080/00018732.2016.1198134 [7] 7a L. F. Santos and M. Rigol , Phys. Rev. E 81 , 036206 ( 2010 ); PRESCM 1539-3755 10.1103/PhysRevE.81.036206 7b L. F. Santos and M. Rigol Phys. Rev. E 82 , 031130 ( 2010 ). PRESCM 1539-3755 10.1103/PhysRevE.82.031130 [8] 8 A. Gubin and L. F. Santos , Am. J. Phys. 80 , 246 ( 2012 ). AJPIAS 0002-9505 10.1119/1.3671068 [9] 9a M. Rigol , Phys. Rev. Lett. 103 , 100403 ( 2009 ); PRLTAO 0031-9007 10.1103/PhysRevLett.103.100403 9b M. Rigol Phys. Rev. A 80 , 053607 ( 2009 ). PLRAAN 1050-2947 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 0031-9007 10.1103/PhysRevLett.112.130403 [11] 11 S. Sorg , L. Vidmar , L. Pollet , and F. Heidrich-Meisner , Phys. Rev. A 90 , 033606 ( 2014 ). PLRAAN 1050-2947 10.1103/PhysRevA.90.033606 [12] 12 H. Kim , T. N. Ikeda , and D. A. Huse , Phys. Rev. E 90 , 052105 ( 2014 ). PRESCM 1539-3755 10.1103/PhysRevE.90.052105 [13] 13a W. Beugeling , R. Moessner , and M. Haque , Phys. Rev. E 89 , 042112 ( 2014 ); PRESCM 1539-3755 10.1103/PhysRevE.89.042112 13b W. Beugeling , R. Moessner , and M. Haque Phys. Rev. E 91 , 012144 ( 2015 ). PRESCM 1539-3755 10.1103/PhysRevE.91.012144 [14] 14 R. Mondaini , K. R. Fratus , M. Srednicki , and M. Rigol , Phys. Rev. E 93 , 032104 ( 2016 ). PRESCM 2470-0045 10.1103/PhysRevE.93.032104 [15] 15 R. Mondaini and M. Rigol , Phys. Rev. E 96 , 012157 ( 2017 ). PRESCM 2470-0045 10.1103/PhysRevE.96.012157 [16] 16 T. Mori and N. Shiraishi , Phys. Rev. E 96 , 022153 ( 2017 ). PRESCM 2470-0045 10.1103/PhysRevE.96.022153 [17] 17 Z. Lan and S. Powell , Phys. Rev. B 96 , 115140 ( 2017 ). PRBMDO 2469-9950 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 0305-4470 10.1088/0305-4470/32/7/007 [20] 20 E. Khatami , G. Pupillo , M. Srednicki , and M. Rigol , Phys. Rev. Lett. 111 , 050403 ( 2013 ). PRLTAO 0031-9007 10.1103/PhysRevLett.111.050403

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Comment on "systematic Construction of Counterexamples to the Eigenstate Thermalization Hypothesis"

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