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Recent researches on disorder-driven many-body localization (MBL) in non-Hermitian quantum systems have aroused great interest. In this work, we investigate the non-Hermitian MBL in a one-dimensional hard-core Bose model induced by random two-body dissipation, which is described by $ \hat{H}=\displaystyle\sum\limits_{j}^{L-1}\left[ -J\left( \hat{b}_{j}^{\dagger}\hat{b}_{j+1}+\hat {b}_{j+1}^{\dagger}\hat{b}_{j}\right) +\frac{1}{2}\left( U-{\mathrm{i}}\gamma_{j}\right) \hat{n}_{j}\hat{n}_{j+1}\right] \notag,$ with the random two-body loss $\gamma_j\in\left[0,W\right]$ . By the level statistics, the system undergoes a transition from the AI$^{\dagger}$ symmetry class to a two-dimensional Poisson ensemble with the increase of disorder strength. This transition is accompanied by the changing of the average magnitude (argument)$\overline{\left\langle {r}\right\rangle}$ ($\overline{-\left\langle \cos {\theta}\right\rangle }$ ) of the complex spacing ratio, shifting from approximately 0.722 (0.193) to about 2/3 (0). The normalized participation ratios of the majority of eigenstates exhibit finite values in the ergodic phase, gradually approaching zero in the non-Hermitian MBL phase, which quantifies the degree of localization for the eigenstates. For weak disorder, one can see that average half-chain entanglement entropy$\overline{\langle S \rangle}$ follows a volume law in the ergodic phase. However, it decreases to a constant independent of L in the deep non-Hermitian MBL phase, adhering to an area law. These results indicate that the ergodic phase and non-Hermitian MBL phase can be distinguished by the half-chain entanglement entropy, even in non-Hermitian system, which is similar to the scenario in Hermitian system. Finally, for a short time, the dynamic evolution of the entanglement entropy exhibits linear growth with the weak disorder. In strong disorder case, the short-time evolution of$\overline{S(t)}$ shows logarithmic growth. However, when$t\geqslant10^2$ ,$\overline{S(t)}$ can stabilize and tend to the steady-state half-chain entanglement entropy$\overline{ S_0 }$ . The results of the dynamical evolution of$\overline{S(t)}$ imply that one can detect the occurrence of the non-Hermitian MBL by the short-time evolution of$\overline{S(t)}$ , and the long-time behavior of$\overline{S(t)}$ signifies the steady-state information.-
Keywords:
- disorder /
- non-Hermitian /
- random-matrix theory /
- many-body localization
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[3] Nandkishore R, Huse D A 2015 Annu. Rev. Condens. Matter Phys. 6 15Google Scholar
[4] Kjäll J A, Bardarson J H, Pollmann F 2014 Phys. Rev. Lett. 113 107204Google Scholar
[5] Bera S, Schomerus H, Heidrich-Meisner F, Bardarson J H 2015 Phys. Rev. Lett. 115 046603Google Scholar
[6] Rademaker L, Ortuño M 2016 Phys. Rev. Lett. 116 010404Google Scholar
[7] Khemani V, Sheng D N, Huse D A 2017 Phys. Rev. Lett. 119 075702Google Scholar
[8] Macé N, Alet F, Laflorencie N 2019 Phys. Rev. Lett. 123 180601Google Scholar
[9] Bar Lev Y, Cohen G, Reichman D R 2015 Phys. Rev. Lett. 114 100601Google Scholar
[10] Bairey E, Refael G, Lindner N H 2017 Phys. Rev. B 96 020201Google Scholar
[11] Decker K S C, Karrasch C, Eisert J, Kennes D M 2020 Phys. Rev. Lett. 124 190601Google Scholar
[12] Giamarchi T, Schulz H J 1988 Phys. Rev. B 37 325Google Scholar
[13] De Luca A, Altshuler B L, Kravtsov V E, Scardicchio A 2014 Phys. Rev. Lett. 113 046806Google Scholar
[14] Deutsch J M 2018 Rep. Prog. Phys. 81 082001Google Scholar
[15] De Luca A, Scardicchio A 2013 EPL 101 37003Google Scholar
[16] Bar Lev Y, Reichman D R 2014 Phys. Rev. B 89 220201
[17] Luitz D J, Laflorencie N, Alet F 2016 Phys. Rev. B 93 060201
[18] Abanin D A, Altman E, Bloch I, Serbyn M 2019 Rev. Mod. Phys. 91 021001Google Scholar
[19] Guhr T, Müller–Groeling A, Weidenmüller H A 1998 Phys. Rep. 299 189Google Scholar
[20] Atas Y Y, Bogomolny E, Giraud O, Roux G 2013 Phys. Rev. Lett. 110 084101Google Scholar
[21] Bardarson J H, Pollmann F, Moore J E 2012 Phys. Rev. Lett. 109 017202Google Scholar
[22] Serbyn M, Papić Z, Abanin D A 2013 Phys. Rev. Lett. 110 260601Google Scholar
[23] Bauer B, Nayak C 2013 J. Stat. Mech. 2013 P09005Google Scholar
[24] Serbyn M, Michailidis A A, Abanin D A, Papić Z 2016 Phys. Rev. Lett. 117 160601Google Scholar
[25] Guo Q, Cheng C, Sun Z H, et al. 2021 Nat. Phys. 17 234Google Scholar
[26] Guo Q, Cheng C, Li H, et al. 2021 Phys. Rev. Lett. 127 240502Google Scholar
[27] Ros V, Müller M, Scardicchio A 2015 Nucl. Phys. B 891 420Google Scholar
[28] Bertoni C, Eisert J, Kshetrimayum A, Nietner A, Thomson S J 2023 arXiv: 2208.14432 v4 [cond-mat.dis-nn]
[29] Schreiber M, Hodgman S S, Bordia P, Lüschen H P, Fischer M H, Vosk R, Altman E, Schneider U, Bloch I 2015 Science 349 842Google Scholar
[30] Bordia P, Lüschen H P, Hodgman S S, Schreiber M, Bloch I, Schneider U 2016 Phys. Rev. Lett. 116 140401Google Scholar
[31] Kohlert T, Scherg S, Li X, Lüschen H P, Das Sarma S, Bloch I, Aidelsburger M 2019 Phys. Rev. Lett. 122 170403Google Scholar
[32] Smith J, Lee A, Richerme P, Neyenhuis B, Hess P W, Hauke P, Heyl M, Huse D A, Monroe C 2016 Nat. Phys. 12 907Google Scholar
[33] Roushan P, Neill C, Tangpanitanon J, et al. 2017 Science 358 1175Google Scholar
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[35] Ashida Y, Gong Z, Ueda M 2020 Adv. Phys. 69 249
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[41] Yokomizo K, Murakami S 2019 Phys. Rev. Lett. 123 066404Google Scholar
[42] Wang Y C, You J S, Jen H H 2022 Nat. Commun. 13 4598Google Scholar
[43] Xu Z, Chen S 2020 Phys. Rev. B 102 035153Google Scholar
[44] Hatano N, Nelson D R 1996 Phys. Rev. Lett. 77 570Google Scholar
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[50] Gong Z, Ashida Y, Kawabata K, Takasan K, Higashikawa S, Ueda M 2018 Phys. Rev. X 8 031079
[51] Tomita T, Nakajima S, Danshita I, Takasu Y, Takahashi Y 2017 Sci. Adv. 3 e1701513Google Scholar
[52] Sponselee K, Freystatzky L, Abeln B, et al. 2018 Quantum Sci. Technol. 4 014002Google Scholar
[53] Wang C, Liu C, Shi Z Y 2022 Phy. Rev. Lett. 129 203401Google Scholar
[54] Berry M V, Tabor M 1977 Proc. R. Soc. London, Ser. A 256 375
[55] Bohigas O, Giannoni M J, Schmit C 1984 Phys. Rev. Lett. 52 1Google Scholar
[56] Casati G, Valz-Gris F, Guarnieri I 1980 Lett. Nuovo Cimento 28 279Google Scholar
[57] Rigol M, Dunjko V, Olshanii M 2008 Nature 452 854Google Scholar
[58] Hamazaki R, Kawabata K, Kura N, Ueda M 2020 Phys. Rev. Res. 2 023286Google Scholar
[59] Sá L, Ribeiro P, Prosen T 2020 Phys. Rev. X 10 021019
[60] García-García A M, Sá L, Verbaarschot J J M 2022 Phys. Rev. X 12 021040
[61] Ginibre J 1965 J. Math. Phys. 6 440Google Scholar
[62] Peron T, De Resende B M F, Rodrigues F A, Costa L D F, Méndez-Bermúdez J A 2020 Phys. Rev. E 102 062305Google Scholar
[63] Liu J, Xu Z 2023 Phys. Rev. B 108 184205Google Scholar
[64] Oganesyan V, Huse D A 2007 Phys. Rev. B 75 155111Google Scholar
[65] Ghosh S, Gupta S, Kulkarni M 2022 Phys. Rev. B 106 134202Google Scholar
[66] Li X, Ganeshan S, Pixley J H, Das Sarma S 2015 Phys. Rev. Lett. 115 186601Google Scholar
[67] Suthar K, Wang Y C, Huang Y P, Jen H H, You J S 2022 Phys. Rev. B 106 064208
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图 1 当L = 14时, 哈密顿量(1)式平均的最近邻能级间距s的统计分布 (a) W = 2; (b) W = 20. 黑色虚线、红色实线和绿色点线分别表示A, AI$^{\dagger}$类和二维泊松分布
Fig. 1. Mean unfolded nearest-level-spacing distributions of the Hamiltonian Eq. (1) with L = 14: (a) W = 2; (b) W = 20. Black dash, red solid, and green dotted lines represent A, AI${\mathrm{}}^{\dagger}$ classes, and two dimensional (2D)-Poisson distributions, respectively.
图 2 当$L=14$时, 平均的径向强度分布$\overline{\mathcal{P}(r)}$和相应的幅角分布$\overline{\mathcal{P} (\theta )}$ (a), (b) $W=2$; (c), (d) $W=20$. 红色实线是通过统计对应的随机矩阵($1000\times 1000$)的结果, 其无序次数选取为1000. (e), (f)径向强度的平均值$\overline{\left\langle {r}\right\rangle }$和相应的幅角的平均值$\overline{-\left\langle \cos{\theta}\right\rangle} $随无序强度变化曲线. 上(下)虚线对应于AI$^{\dagger}$对称类(2D-Poisson)统计极限值, $\overline{\langle {r}\rangle}_{{\text{AI}}^{\dagger}}\approx0.722$, $\overline{-\langle \cos{{\theta}} \rangle}_{{\text{AI}}^{\dagger}}\approx $$ 0.193$ $(\overline{\langle {r}\rangle}_{\text{Pois}}=2/3$, $\overline{-\langle \cos{{\theta}} \rangle}_{\text{Pois}}=0)$
Fig. 2. (a), (b) Mean marginal distributions $\overline{\mathcal{P}(r)}$ and $\overline{\mathcal{P} (\theta )}$ with $W=2$ for the complex energy spectrum for $L=14$; (c), (d) the marginal distributions $\overline{\mathcal{P}(r)}$ and $\overline{\mathcal{P} (\theta )}$ with $W=20$ for the complex energy spectrum. The red solid lines are obtained by calculating $\overline{\mathcal{P}(r)}$ and $\overline{\mathcal{P} (\theta )}$ of the $1000\times 1000$ random matrices with the corresponding random matrix ensembles averaged 1000 realizations. (e), (f) The averages $\overline{\left\langle {r}\right\rangle} $ and $\overline{-\left\langle \cos {\theta}\right\rangle }$ as a function of the disorder strength W. The upper (lower) dash line corresponds to the ${\mathrm{AI}}^{\dagger}$ symmetry class (2D-Poisson) expectation, $\overline{\langle {r}\rangle}_{{\text{AI}}^{\dagger}}\approx0.722$, $\overline{-\langle \cos{{\theta}} \rangle}_{{\text{AI}}^{\dagger}}\approx 0.193$ $(\overline{\langle {r}\rangle}_{\text{Pois}}=2/3$, $\overline{-\langle \cos{{\theta}} \rangle}_{\text{Pois}}=0)$
图 3 当$L=14$时, 在复平面上, 系统所有本征态的$\eta$随重整后能谱$\varepsilon_{i}$的分布情况(红点表示能谱的中心) (a) $W=2$; (c) $W=20$. 归一化的参与率$\eta$统计直方图 (b) $W=2$; (d) $W=20$
Fig. 3. Distribution of $\eta$ for all eigenstates versus the rescaled spectrum $\varepsilon_{i}$ with $L=14$ (Red dots represent the center of the energy spectrum): (a) $W=2$; (c) $W=20$. Histogram of the normalized participation ratio $\eta$: (b) $W=2$; (d) $W=20$.
图 4 (a)不同尺寸下, 平均半链纠缠熵$\overline{\left\langle {S}\right\rangle }$随无序强度的变化; (b)当$L=14$时, 不同无序强度W对应的$\overline{S\left( t\right) }$随时间的演化. 初态为$\left\vert \psi_{0} \right\rangle =\left\vert 1010\cdots\right\rangle $. 插图展示了平均稳态熵$\overline{S_{0}}$随无序强度的变化
Fig. 4. (a) Mean half-chain entanglement entropy $\overline{\left\langle {S}\right\rangle }$ as a function of the disorder strength W for different L; (b) the dynamics of the mean half-chain entanglement entropy $\overline{S\left( t\right) }$ for different W with $L=14$. The initial state is taken as $\left\vert \psi_{0} \right\rangle =\left\vert 1010\cdots\right\rangle $. The inset displays the mean steady-state entanglement entropy $\overline{S_{0}}$ as a function of W.
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[1] Basko D M, Aleiner I L, Altshuler B L 2006 Ann. Phys. 321 1126Google Scholar
[2] Laumann C R, Pal A, Scardicchio A 2014 Phys. Rev. Lett. 113 200405Google Scholar
[3] Nandkishore R, Huse D A 2015 Annu. Rev. Condens. Matter Phys. 6 15Google Scholar
[4] Kjäll J A, Bardarson J H, Pollmann F 2014 Phys. Rev. Lett. 113 107204Google Scholar
[5] Bera S, Schomerus H, Heidrich-Meisner F, Bardarson J H 2015 Phys. Rev. Lett. 115 046603Google Scholar
[6] Rademaker L, Ortuño M 2016 Phys. Rev. Lett. 116 010404Google Scholar
[7] Khemani V, Sheng D N, Huse D A 2017 Phys. Rev. Lett. 119 075702Google Scholar
[8] Macé N, Alet F, Laflorencie N 2019 Phys. Rev. Lett. 123 180601Google Scholar
[9] Bar Lev Y, Cohen G, Reichman D R 2015 Phys. Rev. Lett. 114 100601Google Scholar
[10] Bairey E, Refael G, Lindner N H 2017 Phys. Rev. B 96 020201Google Scholar
[11] Decker K S C, Karrasch C, Eisert J, Kennes D M 2020 Phys. Rev. Lett. 124 190601Google Scholar
[12] Giamarchi T, Schulz H J 1988 Phys. Rev. B 37 325Google Scholar
[13] De Luca A, Altshuler B L, Kravtsov V E, Scardicchio A 2014 Phys. Rev. Lett. 113 046806Google Scholar
[14] Deutsch J M 2018 Rep. Prog. Phys. 81 082001Google Scholar
[15] De Luca A, Scardicchio A 2013 EPL 101 37003Google Scholar
[16] Bar Lev Y, Reichman D R 2014 Phys. Rev. B 89 220201
[17] Luitz D J, Laflorencie N, Alet F 2016 Phys. Rev. B 93 060201
[18] Abanin D A, Altman E, Bloch I, Serbyn M 2019 Rev. Mod. Phys. 91 021001Google Scholar
[19] Guhr T, Müller–Groeling A, Weidenmüller H A 1998 Phys. Rep. 299 189Google Scholar
[20] Atas Y Y, Bogomolny E, Giraud O, Roux G 2013 Phys. Rev. Lett. 110 084101Google Scholar
[21] Bardarson J H, Pollmann F, Moore J E 2012 Phys. Rev. Lett. 109 017202Google Scholar
[22] Serbyn M, Papić Z, Abanin D A 2013 Phys. Rev. Lett. 110 260601Google Scholar
[23] Bauer B, Nayak C 2013 J. Stat. Mech. 2013 P09005Google Scholar
[24] Serbyn M, Michailidis A A, Abanin D A, Papić Z 2016 Phys. Rev. Lett. 117 160601Google Scholar
[25] Guo Q, Cheng C, Sun Z H, et al. 2021 Nat. Phys. 17 234Google Scholar
[26] Guo Q, Cheng C, Li H, et al. 2021 Phys. Rev. Lett. 127 240502Google Scholar
[27] Ros V, Müller M, Scardicchio A 2015 Nucl. Phys. B 891 420Google Scholar
[28] Bertoni C, Eisert J, Kshetrimayum A, Nietner A, Thomson S J 2023 arXiv: 2208.14432 v4 [cond-mat.dis-nn]
[29] Schreiber M, Hodgman S S, Bordia P, Lüschen H P, Fischer M H, Vosk R, Altman E, Schneider U, Bloch I 2015 Science 349 842Google Scholar
[30] Bordia P, Lüschen H P, Hodgman S S, Schreiber M, Bloch I, Schneider U 2016 Phys. Rev. Lett. 116 140401Google Scholar
[31] Kohlert T, Scherg S, Li X, Lüschen H P, Das Sarma S, Bloch I, Aidelsburger M 2019 Phys. Rev. Lett. 122 170403Google Scholar
[32] Smith J, Lee A, Richerme P, Neyenhuis B, Hess P W, Hauke P, Heyl M, Huse D A, Monroe C 2016 Nat. Phys. 12 907Google Scholar
[33] Roushan P, Neill C, Tangpanitanon J, et al. 2017 Science 358 1175Google Scholar
[34] Bender C M 2007 Rep. Prog. Phys. 70 947Google Scholar
[35] Ashida Y, Gong Z, Ueda M 2020 Adv. Phys. 69 249
[36] Zhang K, Yang Z, Fang C 2020 Phys. Rev. Lett. 125 126402Google Scholar
[37] Zhang K, Yang Z, Fang C 2022 Nat. Commun. 13 2496Google Scholar
[38] Ou Z, Wang Y, Li L 2023 Phys. Rev. B 107 L161404Google Scholar
[39] Yao S, Wang Z 2018 Phys. Rev. Lett. 121 086803Google Scholar
[40] Borgnia D S, Kruchkov A J, Slager R J 2020 Phys. Rev. Lett. 124 056802Google Scholar
[41] Yokomizo K, Murakami S 2019 Phys. Rev. Lett. 123 066404Google Scholar
[42] Wang Y C, You J S, Jen H H 2022 Nat. Commun. 13 4598Google Scholar
[43] Xu Z, Chen S 2020 Phys. Rev. B 102 035153Google Scholar
[44] Hatano N, Nelson D R 1996 Phys. Rev. Lett. 77 570Google Scholar
[45] Hatano N, Nelson D R 1997 Phys. Rev. B 56 8651Google Scholar
[46] Hatano N, Nelson D R 1998 Phys. Rev. B 58 8384Google Scholar
[47] Hamazaki R, Kawabata K, Ueda M 2019 Phys. Rev. Lett. 123 090603Google Scholar
[48] Tang L Z, Zhang G Q, Zhang L F, Zhang D W 2021 Phys. Rev. A 103 033325Google Scholar
[49] Zhai L J, Yin S, Huang G Y 2020 Phys. Rev. B 102 064206Google Scholar
[50] Gong Z, Ashida Y, Kawabata K, Takasan K, Higashikawa S, Ueda M 2018 Phys. Rev. X 8 031079
[51] Tomita T, Nakajima S, Danshita I, Takasu Y, Takahashi Y 2017 Sci. Adv. 3 e1701513Google Scholar
[52] Sponselee K, Freystatzky L, Abeln B, et al. 2018 Quantum Sci. Technol. 4 014002Google Scholar
[53] Wang C, Liu C, Shi Z Y 2022 Phy. Rev. Lett. 129 203401Google Scholar
[54] Berry M V, Tabor M 1977 Proc. R. Soc. London, Ser. A 256 375
[55] Bohigas O, Giannoni M J, Schmit C 1984 Phys. Rev. Lett. 52 1Google Scholar
[56] Casati G, Valz-Gris F, Guarnieri I 1980 Lett. Nuovo Cimento 28 279Google Scholar
[57] Rigol M, Dunjko V, Olshanii M 2008 Nature 452 854Google Scholar
[58] Hamazaki R, Kawabata K, Kura N, Ueda M 2020 Phys. Rev. Res. 2 023286Google Scholar
[59] Sá L, Ribeiro P, Prosen T 2020 Phys. Rev. X 10 021019
[60] García-García A M, Sá L, Verbaarschot J J M 2022 Phys. Rev. X 12 021040
[61] Ginibre J 1965 J. Math. Phys. 6 440Google Scholar
[62] Peron T, De Resende B M F, Rodrigues F A, Costa L D F, Méndez-Bermúdez J A 2020 Phys. Rev. E 102 062305Google Scholar
[63] Liu J, Xu Z 2023 Phys. Rev. B 108 184205Google Scholar
[64] Oganesyan V, Huse D A 2007 Phys. Rev. B 75 155111Google Scholar
[65] Ghosh S, Gupta S, Kulkarni M 2022 Phys. Rev. B 106 134202Google Scholar
[66] Li X, Ganeshan S, Pixley J H, Das Sarma S 2015 Phys. Rev. Lett. 115 186601Google Scholar
[67] Suthar K, Wang Y C, Huang Y P, Jen H H, You J S 2022 Phys. Rev. B 106 064208
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