[1] | Basko D M, Aleiner I L and Altshuler B L 2006 Ann. Phys. 321 1126 | Metal–insulator transition in a weakly interacting many-electron system with localized single-particle states
[2] | Gornyi I V, Mirlin A D and Polyakov D G 2005 Phys. Rev. Lett. 95 206603 | Interacting Electrons in Disordered Wires: Anderson Localization and Low- Transport
[3] | Imbrie J Z 2016 Phys. Rev. Lett. 117 027201 | Diagonalization and Many-Body Localization for a Disordered Quantum Spin Chain
[4] | Imbrie J Z 2016 J. Stat. Phys. 163 998 | On Many-Body Localization for Quantum Spin Chains
[5] | Oganesyan V and Huse D A 2007 Phys. Rev. B 75 155111 | Localization of interacting fermions at high temperature
[6] | Žnidarič M, Prosen T and Prelovšek P 2008 Phys. Rev. B 77 064426 | Many-body localization in the Heisenberg magnet in a random field
[7] | Pal A and Huse D A 2010 Phys. Rev. B 82 174411 | Many-body localization phase transition
[8] | Monthus C and Garel T 2010 Phys. Rev. B 81 134202 | Many-body localization transition in a lattice model of interacting fermions: Statistics of renormalized hoppings in configuration space
[9] | Canovi E, Rossini D, Fazio R, Santoro G E and Silva A 2011 Phys. Rev. B 83 094431 | Quantum quenches, thermalization, and many-body localization
[10] | Kjäll J A, Bardarson J H and Pollmann F 2014 Phys. Rev. Lett. 113 107204 | Many-Body Localization in a Disordered Quantum Ising Chain
[11] | Luitz D J, Laflorencie N and Alet F 2015 Phys. Rev. B 91 081103 | Many-body localization edge in the random-field Heisenberg chain
[12] | Tang B, Iyer D and Rigol M 2015 Phys. Rev. B 91 161109 | Quantum quenches and many-body localization in the thermodynamic limit
[13] | Bauer B and Nayak C 2013 J. Stat. Mech. 2013 P09005 | Area laws in a many-body localized state and its implications for topological order
[14] | Bardarson J H, Pollmann F and Moore J E 2012 Phys. Rev. Lett. 109 017202 | Unbounded Growth of Entanglement in Models of Many-Body Localization
[15] | Serbyn M, Papić Z and Abanin D A 2013 Phys. Rev. Lett. 110 260601 | Universal Slow Growth of Entanglement in Interacting Strongly Disordered Systems
[16] | Kim I H, Chandran A and Abanin D A 2014 arXiv:1412.3073 [cond-mat.dis-nn] | Local integrals of motion and the logarithmic lightcone in many-body localized systems
[17] | Ros V, Müller M and Scardicchio A 2015 Nucl. Phys. B 891 420 | Integrals of motion in the many-body localized phase
[18] | Deutsch J M 1991 Phys. Rev. A 43 2046 | Quantum statistical mechanics in a closed system
[19] | Srednicki M 1994 Phys. Rev. E 50 888 | Chaos and quantum thermalization
[20] | Tasaki H 1998 Phys. Rev. Lett. 80 1373 | From Quantum Dynamics to the Canonical Distribution: General Picture and a Rigorous Example
[21] | Rigol M, Dunjko V, Yurovsky V and Olshanii M 2007 Phys. Rev. Lett. 98 050405 | Relaxation in a Completely Integrable Many-Body Quantum System: An Ab Initio Study of the Dynamics of the Highly Excited States of 1D Lattice Hard-Core Bosons
[22] | Rigol M, Dunjko V and Olshanii M 2008 Nature 452 854 | Thermalization and its mechanism for generic isolated quantum systems
[23] | Wootton J R and Pachos J K 2011 Phys. Rev. Lett. 107 030503 | Bringing Order through Disorder: Localization of Errors in Topological Quantum Memories
[24] | Stark C, Pollet L, Imamoğlu A and Renner R 2011 Phys. Rev. Lett. 107 030504 | Localization of Toric Code Defects
[25] | Huse D A, Nandkishore R, Oganesyan V, Pal A and Sondhi S L 2013 Phys. Rev. B 88 014206 | Localization-protected quantum order
[26] | Bahri Y, Vosk R, Altman E and Vishwanath A 2013 arXiv:1307.4092 [cond-mat.dis-nn] | Localization and topology protected quantum coherence at the edge of 'hot' matter
[27] | Serbyn M, Papić Z and Abanin D A 2013 Phys. Rev. Lett. 111 127201 | Local Conservation Laws and the Structure of the Many-Body Localized States
[28] | Huse D A, Nandkishore R and Oganesyan V 2014 Phys. Rev. B 90 174202 | Phenomenology of fully many-body-localized systems
[29] | Rademaker L and M Ortu no 2016 Phys. Rev. Lett. 116 010404 | Explicit Local Integrals of Motion for the Many-Body Localized State
[30] | Chandran A, Kim I H, Vidal G and Abanin D A 2015 Phys. Rev. B 91 085425 | Constructing local integrals of motion in the many-body localized phase
[31] | Inglis S and Pollet L 2016 Phys. Rev. Lett. 117 120402 | Accessing Many-Body Localized States through the Generalized Gibbs Ensemble
[32] | Anderson P W 1958 Phys. Rev. 109 1492 | Absence of Diffusion in Certain Random Lattices
[33] | ${\boldsymbol R} = f(g({\boldsymbol E}))$. Every element of ${\boldsymbol E}$ is 0 or 1. It is 1 only when it is permuted via ${\boldsymbol P}^† $ to a diagonal element of $[{\boldsymbol U}^{\rm d} | {\boldsymbol E}] {\boldsymbol P}^† $. Function $g({\boldsymbol E})$ orthonomalizes every column of ${\boldsymbol E}$ to all columns of ${\boldsymbol U}^{\rm d}$. Function $f({\boldsymbol S})$ unitarizes ${\boldsymbol S}$ in the space spanned by the columns of ${\boldsymbol S}$ via repeated applications of ${\boldsymbol S} \leftarrow \frac 32 {\boldsymbol S} - \frac 12 {\boldsymbol S} {\boldsymbol S}^† {\boldsymbol S}$ until convergence, i. e., ${\boldsymbol S} = \frac 32 {\boldsymbol S} - \frac 12 {\boldsymbol S} {\boldsymbol S}^† {\boldsymbol S}$. In other words, function $f({\boldsymbol S})$ produces a matrix close to ${\boldsymbol S}$ and so that $[{\boldsymbol U}^{\rm d} | {\boldsymbol R}]$ is unitary |
[34] | The number of steps in this minimization is 1 in this work, which is found to be the most efficient for the whole LIOM construction. The step size $|\delta {\boldsymbol X}|$ is set to $|\delta {\boldsymbol X}| / |{\boldsymbol I}| \sim 0.08$, to which the minimization efficiency is not very sensitive |
[35] | He R Q and Lu Z Y 2016 arXiv:1606.09509v1 [cond-mat.dis-nn] | Interaction-Induced Characteristic Length in Strongly Many-Body Localized Systems
[36] | The data points of $n = 8, 9$ and $n = 7, 8, 9$ with $L = 12$ are used to calculate $\xi$ and $\xi_+$, respectively |