# Many body localization

Many body localization (MBL) is a dynamical phenomenon occurring in isolated many-body quantum systems. It is characterized by the system failing to reach thermal equilibrium, and retaining a memory of its initial condition in local observables for infinite times.[1]

## Thermalization and localization

Textbook quantum statistical mechanics[2] assumes that systems go to thermal equilibrium (thermalization). The process of thermalization erases local memory of the initial conditions. In textbooks, thermalization is ensured by coupling the system to an external environment or "reservoir," with which the system can exchange energy. What happens if the system is isolated from the environment, and evolves according to its own Schrödinger equation? Does the system still thermalize?

Quantum mechanical time evolution is unitary and formally preserves all information about the initial condition in the quantum state at all times. However, a quantum system generically contains a macroscopic number of degrees of freedom, but can only be probed through few-body measurements which are local in real space. The meaningful question then becomes whether accessible local measurements display thermalization.

This question can be formalized by considering the quantum mechanical density matrix ρ of the system. If the system is divided into a subregion A (the region being probed) and its complement B (everything else), then all information that can be extracted by measurements made on A alone is encoded in the reduced density matrix ρA = TrB ρ(t). If in the long time limit ρA(t) approaches a thermal density matrix at a temperature set by the energy density in the state, then the system has "thermalized," and no local information about the initial condition can be extracted from local measurements. This process of "quantum thermalization" may be understood in terms of B acting as a reservoir for A. In this perspective, the entanglement entropy S=-Tr ρA log ρA of a thermalizing system in a pure state plays the role of thermal entropy.[3][4][5] Thermalizing systems therefore generically have extensive or "volume law" entanglement entropy at any non-zero temperature.[6][7][8] They also generically obey the Eigenstate Thermalization Hypothesis (ETH).[9][10][11]

In contrast, if ρA(t) fails to approach a thermal density matrix even in the long time limit, and remains instead close to its initial condition ρA(0), then the system retains forever a memory of its initial condition in local observables. This latter possibility is referred to as "many body localization," and involves B failing to act as a reservoir for A. A system in a many body localized phase exhibits MBL, and continues to exhibit MBL even when subject to arbitrary local perturbations. Eigenstates of systems exhibiting MBL do not obey the ETH, and generically follow an "area law" for entanglement entropy (i.e. the entanglement entropy scales with the surface area of subregion A). A brief list of properties differentiating thermalizing and MBL systems is provided below.

• In thermalizing systems, a memory of initial conditions is not accessible in local observables at long times. In MBL systems, memory of initial conditions remains accessible in local observables at long times.
• In thermalizing systems, energy eigenstates obey ETH. In MBL systems, energy eigenstates do not obey ETH.
• In thermalizing systems, energy eigenstates have volume law entanglement entropy. In MBL systems, energy eigenstates have area law entanglement entropy.
• Thermalizing systems generically have non-zero thermal conductivity. MBL systems have zero thermal conductivity.
• Thermalizing systems have continuous local spectra. MBL systems have discrete local spectra.[12]
• In thermalizing systems, entanglement entropy grows as a power law in time starting from low entanglement initial conditions.[13] In MBL systems, entanglement entropy grows logarithmically in time starting from low entanglement initial conditions.[14][15][16]
• In thermalizing systems, the dynamics of out-of-time-ordered correlators forms a linear light cone which reflects the ballistic propagation of information. In MBL systems, the light cone is logarithmic.[17][18][19][20][21]

## History

MBL was first proposed by P.W. Anderson in 1958[22] as a possibility that could arise in strongly disordered quantum systems. The basic idea was that if particles all live in a random energy landscape, then any rearrangement of particles would change the energy of the system. Since energy is a conserved quantity in quantum mechanics, such a process can only be virtual and cannot lead to any transport of particle number or energy.

While localization for single particle systems was demonstrated already in Anderson's original paper (coming to be known as Anderson localization), the existence of the phenomenon for many particle systems remained a conjecture for decades. In 1980 Fleishman and Anderson[23] demonstrated the phenomenon survived the addition of interactions to lowest order in perturbation theory. In a 1998 study,[24] the analysis was extended to all orders in perturbation theory, in a zero dimensional system, and the MBL phenomenon was shown to survive. In 2005[25] and 2006,[26] this was extended to high orders in perturbation theory in high dimensional systems. MBL was argued to survive at least at low energy density. A series of numerical works[27][14][28] [29] provided further evidence for the phenomenon in one dimensional systems, at all energy densities (“infinite temperature”). Finally, in 2014[30] Imbrie presented a proof of MBL for certain one dimensional spin chains with strong disorder, with the localization being stable to arbitrary local perturbations – i.e. the systems were shown to be in a many body localized phase.

It is now believed that MBL can arise also in periodically driven "Floquet" systems where energy is conserved only modulo the drive frequency.[31][32][33]

## Emergent Integrability

Many body localized systems exhibit a phenomenon known as emergent integrability. Recall that in a non-interacting Anderson insulator, the occupation number of each localized single particle orbital is separately a local integral of motion. It was conjectured[34][35] (and proven by Imbrie) that a similar extensive set of local integrals of motion should also exist in the MBL phase. Consider for specificity a one dimensional spin-1/2 chain with Hamiltonian

${\displaystyle H=\sum _{i}\left[J\left(X_{i}X_{i+1}+Y_{i}Y_{i+1}\right)+J^{\prime }Z_{i}Z_{i+1}+h_{i}Z_{i}\right]}$

Where the X, Y and Z are Pauli operators, and the hI are random variables drawn from a distribution of some width W. When the disorder is strong enough (W>Wc) that all eigenstates are localized, then there exists a local unitary transformation to new variables τ such that

${\displaystyle H=\sum _{i}h_{i}^{\prime }\tau _{i}^{z}+\sum _{ij}J_{ij}\tau _{i}^{z}\tau _{j}^{z}+\sum _{ijk}K_{ijk}\tau _{i}^{z}\tau _{j}^{z}\tau _{k}^{z}+\cdots }$

Where the τ are Pauli operators that are related to the physical Pauli operators by a local unitary transformation, the … indicates additional terms which only involve τz operators, and where the coefficients fall off exponentially with distance. This Hamiltonian manifestly contains an extensive number of localized integrals of motion or "l-bits" (the operators τzi, which all commute with the Hamiltonian). If the original Hamiltonian is perturbed, the l-bits get redefined, but the integrable structure survives.

## MBL as a platform for exotic orders

MBL enables the formation of exotic forms of quantum order that could not arise in thermal equilibrium, through the phenomenon of localization protected quantum order.[36] A form of localization protected quantum order, arising only in periodically driven systems, is the Floquet time crystal.[37][38][39][40][41]

## Experimental realizations

A number of experiments have been reported observing the MBL phenomenon.[42][43][44][45] Most of these experiments involve synthetic quantum systems, such as assemblies of ultracold atoms or trapped ions.[46] Experimental explorations of the phenomenon in solid state systems are still in their infancy.

## References

1. ^ Nandkishore, Rahul; Huse, David A. (2015). "Many-Body Localization and Thermalization in Quantum Statistical Mechanics". Annual Review of Condensed Matter Physics. 6 (1): 15–38. arXiv:1404.0686. Bibcode:2015ARCMP...6...15N. doi:10.1146/annurev-conmatphys-031214-014726. ISSN 1947-5454. S2CID 118465889.
2. ^ Sakurai JJ. 1985. Modern Quantum Mechanics. Menlo Park, CA: Benjamin/Cummings
3. ^ Deutsch, J M (July 26, 2010). "Thermodynamic entropy of a many-body energy eigenstate". New Journal of Physics. 12 (7): 075021. arXiv:0911.0056. doi:10.1088/1367-2630/12/7/075021. S2CID 119180376.
4. ^ Santos, Lea F.; Polkovnikov, Anatoli; Rigol, Marcos (July 5, 2012). "Weak and strong typicality in quantum systems". Physical Review E. 86 (1): 010102. doi:10.1103/PhysRevE.86.010102. PMID 23005351.
5. ^ Deutsch, J. M.; Li, Haibin; Sharma, Auditya (April 30, 2013). "Microscopic origin of thermodynamic entropy in isolated systems". Physical Review E. 87 (4): 042135. arXiv:1202.2403. doi:10.1103/PhysRevE.87.042135. PMID 23679399. S2CID 699412.
6. ^ Garrison, James R.; Grover, Tarun (April 30, 2018). "Does a Single Eigenstate Encode the Full Hamiltonian?". Physical Review X. 8 (2): 021026. doi:10.1103/PhysRevX.8.021026.
7. ^ Dymarsky, Anatoly; Lashkari, Nima; Liu, Hong (January 25, 2018). "Subsystem eigenstate thermalization hypothesis". Physical Review E. 97 (1): 012140. doi:10.1103/PhysRevE.97.012140. hdl:1721.1/114450. PMID 29448325.
8. ^ Huang, Yichen (January 2019). "Universal eigenstate entanglement of chaotic local Hamiltonians". Nuclear Physics B. 938: 594–604. doi:10.1016/j.nuclphysb.2018.09.013.
9. ^ Deutsch, J. M. (February 1, 1991). "Quantum statistical mechanics in a closed system". Physical Review A. 43 (4): 2046–2049. Bibcode:1991PhRvA..43.2046D. doi:10.1103/PhysRevA.43.2046. PMID 9905246.
10. ^ Srednicki, Mark (August 1, 1994). "Chaos and quantum thermalization". Physical Review E. 50 (2): 888–901. arXiv:cond-mat/9403051. Bibcode:1994PhRvE..50..888S. doi:10.1103/PhysRevE.50.888. PMID 9962049. S2CID 16065583.
11. ^ Rigol, Marcos; Dunjko, Vanja; Olshanii, Maxim (April 2008). "Thermalization and its mechanism for generic isolated quantum systems". Nature. 452 (7189): 854–858. arXiv:0708.1324. Bibcode:2008Natur.452..854R. doi:10.1038/nature06838. PMID 18421349. S2CID 4384040.
12. ^ Nandkishore, Rahul; Gopalakrishnan, Sarang; Huse, David A. (2014). "Spectral features of a many-body-localized system weakly coupled to a bath". Physical Review B. 90 (6): 064203. arXiv:1402.5971. doi:10.1103/PhysRevB.90.064203. ISSN 1098-0121. S2CID 118568500.
13. ^ Kim, Hyungwon; Huse, David A. (2013). "Ballistic Spreading of Entanglement in a Diffusive Nonintegrable System". Physical Review Letters. 111 (12): 127205. arXiv:1306.4306. doi:10.1103/PhysRevLett.111.127205. ISSN 0031-9007. PMID 24093298. S2CID 41548576.
14. ^ a b Žnidarič, Marko; Prosen, Tomaž; Prelovšek, Peter (February 25, 2008). "Many-body localization in the Heisenberg XXZ magnet in a random field". Physical Review B. 77 (6): 064426. arXiv:0706.2539. doi:10.1103/PhysRevB.77.064426. S2CID 119132600.
15. ^ Bardarson, Jens H.; Pollmann, Frank; Moore, Joel E. (2012). "Unbounded Growth of Entanglement in Models of Many-Body Localization". Physical Review Letters. 109 (1): 017202. doi:10.1103/PhysRevLett.109.017202. ISSN 0031-9007. PMID 23031128.
16. ^ Huang, Yichen (May 2017). "Entanglement dynamics in critical random quantum Ising chain with perturbations" (PDF). Annals of Physics. 380: 224–227. doi:10.1016/j.aop.2017.02.018. S2CID 44548875.
17. ^ Huang, Yichen; Zhang, Yong-Liang; Chen, Xie (July 2017). "Out-of-time-ordered correlators in many-body localized systems" (PDF). Annalen der Physik. 529 (7): 1600318. doi:10.1002/andp.201600318. S2CID 42690831.
18. ^ Fan, Ruihua; Zhang, Pengfei; Shen, Huitao; Zhai, Hui (May 2017). "Out-of-time-order correlation for many-body localization". Science Bulletin. 62 (10): 707–711. doi:10.1016/j.scib.2017.04.011.
19. ^ He, Rong-Qiang; Lu, Zhong-Yi (February 10, 2017). "Characterizing many-body localization by out-of-time-ordered correlation". Physical Review B. 95 (5): 054201. arXiv:1608.03586. doi:10.1103/PhysRevB.95.054201. S2CID 119268185.
20. ^ Swingle, Brian; Chowdhury, Debanjan (February 21, 2017). "Slow scrambling in disordered quantum systems". Physical Review B. 95 (6): 060201. doi:10.1103/PhysRevB.95.060201. hdl:1721.1/107244. S2CID 53485500.
21. ^ Chen, Xiao; Zhou, Tianci; Huse, David A.; Fradkin, Eduardo (July 2017). "Out-of-time-order correlations in many-body localized and thermal phases". Annalen der Physik. 529 (7): 1600332. arXiv:1610.00220. doi:10.1002/andp.201600332. S2CID 119201477.
22. ^ Anderson, P. W. (1958). "Absence of Diffusion in Certain Random Lattices". Physical Review. 109 (5): 1492–1505. Bibcode:1958PhRv..109.1492A. doi:10.1103/PhysRev.109.1492. ISSN 0031-899X.
23. ^ Fleishman, L.; Anderson, P. W. (1980). "Interactions and the Anderson transition". Physical Review B. 21 (6): 2366–2377. doi:10.1103/PhysRevB.21.2366. ISSN 0163-1829.
24. ^ Altshuler, Boris L.; Gefen, Yuval; Kamenev, Alex; Levitov, Leonid S. (1997). "Quasiparticle Lifetime in a Finite System: A Nonperturbative Approach". Physical Review Letters. 78 (14): 2803–2806. arXiv:cond-mat/9609132. Bibcode:1997PhRvL..78.2803A. doi:10.1103/PhysRevLett.78.2803. ISSN 0031-9007. S2CID 18852288.
25. ^ Gornyi, I. V.; Mirlin, A. D.; Polyakov, D. G. (2005). "Interacting Electrons in Disordered Wires: Anderson Localization and Low-TTransport". Physical Review Letters. 95 (20): 206603. arXiv:cond-mat/0506411. doi:10.1103/PhysRevLett.95.206603. ISSN 0031-9007. PMID 16384079. S2CID 39376817.
26. ^ Basko, D.M.; Aleiner, I.L.; Altshuler, B.L. (2006). "Metal–insulator transition in a weakly interacting many-electron system with localized single-particle states". Annals of Physics. 321 (5): 1126–1205. arXiv:cond-mat/0506617. Bibcode:2006AnPhy.321.1126B. doi:10.1016/j.aop.2005.11.014. ISSN 0003-4916. S2CID 18345541.
27. ^ Oganesyan, Vadim; Huse, David A. (2007). "Localization of interacting fermions at high temperature". Physical Review B. 75 (15): 155111. arXiv:cond-mat/0610854. doi:10.1103/PhysRevB.75.155111. ISSN 1098-0121. S2CID 119488834.
28. ^ Pal, Arijeet; Huse, David A. (2010). "Many-body localization phase transition". Physical Review B. 82 (17): 174411. arXiv:1010.1992. doi:10.1103/PhysRevB.82.174411. ISSN 1098-0121. S2CID 41528861.
29. ^ Serbyn, Maksym; Papić, Z.; Abanin, D. A. (2014). "Quantum quenches in the many-body localized phase". Physical Review B. 90 (17). doi:10.1103/PhysRevB.90.174302. hdl:1721.1/91499. ISSN 1098-0121. S2CID 18658716.
30. ^ Imbrie, John Z. (2016). "On Many-Body Localization for Quantum Spin Chains". Journal of Statistical Physics. 163 (5): 998–1048. arXiv:1403.7837. doi:10.1007/s10955-016-1508-x. ISSN 0022-4715. S2CID 11250762.
31. ^ D’Alessio, Luca; Polkovnikov, Anatoli (2013). "Many-body energy localization transition in periodically driven systems". Annals of Physics. 333: 19–33. arXiv:1210.2791. doi:10.1016/j.aop.2013.02.011. ISSN 0003-4916. S2CID 118476386.
32. ^ Lazarides, Achilleas; Das, Arnab; Moessner, Roderich (2015). "Fate of Many-Body Localization Under Periodic Driving". Physical Review Letters. 115 (3): 030402. arXiv:1410.3455. doi:10.1103/PhysRevLett.115.030402. ISSN 0031-9007. PMID 26230771. S2CID 28538293.
33. ^ Ponte, Pedro; Papić, Z.; Huveneers, François; Abanin, Dmitry A. (2015). "Many-Body Localization in Periodically Driven Systems" (PDF). Physical Review Letters. 114 (14): 140401. doi:10.1103/PhysRevLett.114.140401. ISSN 0031-9007. PMID 25910094. S2CID 38608177.
34. ^ Serbyn, Maksym; Papić, Z.; Abanin, Dmitry A. (2013). "Local Conservation Laws and the Structure of the Many-Body Localized States". Physical Review Letters. 111 (12): 127201. arXiv:1305.5554. doi:10.1103/PhysRevLett.111.127201. ISSN 0031-9007. PMID 24093294. S2CID 13006260.
35. ^ Huse, David A.; Nandkishore, Rahul; Oganesyan, Vadim (2014). "Phenomenology of fully many-body-localized systems". Physical Review B. 90 (17): 174202. arXiv:1305.4915. doi:10.1103/PhysRevB.90.174202. ISSN 1098-0121. S2CID 5553355.
36. ^ Huse, David A.; Nandkishore, Rahul; Oganesyan, Vadim; Pal, Arijeet; Sondhi, S. L. (2013). "Localization-protected quantum order". Physical Review B. 88 (1): 014206. arXiv:1304.1158. Bibcode:2013PhRvB..88a4206H. doi:10.1103/PhysRevB.88.014206. ISSN 1098-0121.
37. ^ Khemani, Vedika; Lazarides, Achilleas; Moessner, Roderich; Sondhi, S. L. (2016). "Phase Structure of Driven Quantum Systems". Physical Review Letters. 116 (25): 250401. arXiv:1508.03344. Bibcode:2016PhRvL.116y0401K. doi:10.1103/PhysRevLett.116.250401. ISSN 0031-9007. PMID 27391704.
38. ^ Else, Dominic V.; Bauer, Bela; Nayak, Chetan (2016). "Floquet Time Crystals". Physical Review Letters. 117 (9): 090402. arXiv:1603.08001. Bibcode:2016PhRvL.117i0402E. doi:10.1103/PhysRevLett.117.090402. ISSN 0031-9007. PMID 27610834. S2CID 1652633.
39. ^ von Keyserlingk, C. W.; Khemani, Vedika; Sondhi, S. L. (2016). "Absolute stability and spatiotemporal long-range order in Floquet systems". Physical Review B. 94 (8): 085112. arXiv:1605.00639. Bibcode:2016PhRvB..94h5112V. doi:10.1103/PhysRevB.94.085112. ISSN 2469-9950.
40. ^ Zhang, J.; Hess, P. W.; Kyprianidis, A.; Becker, P.; Lee, A.; Smith, J.; Pagano, G.; Potirniche, I.-D.; Potter, A. C.; Vishwanath, A.; Yao, N. Y.; Monroe, C. (2017). "Observation of a discrete time crystal". Nature. 543 (7644): 217–220. arXiv:1609.08684. Bibcode:2017Natur.543..217Z. doi:10.1038/nature21413. ISSN 0028-0836. PMID 28277505. S2CID 4450646.
41. ^ Choi, Soonwon; Choi, Joonhee; Landig, Renate; Kucsko, Georg; Zhou, Hengyun; Isoya, Junichi; Jelezko, Fedor; Onoda, Shinobu; Sumiya, Hitoshi; Khemani, Vedika; von Keyserlingk, Curt; Yao, Norman Y.; Demler, Eugene; Lukin, Mikhail D. (2017). "Observation of discrete time-crystalline order in a disordered dipolar many-body system". Nature. 543 (7644): 221–225. arXiv:1610.08057. Bibcode:2017Natur.543..221C. doi:10.1038/nature21426. ISSN 0028-0836. PMC 5349499. PMID 28277511.
42. ^ Kondov, S. S.; McGehee, W. R.; Xu, W.; DeMarco, B. (2015). "Disorder-Induced Localization in a Strongly Correlated Atomic Hubbard Gas". Physical Review Letters. 114 (8): 083002. doi:10.1103/PhysRevLett.114.083002. ISSN 0031-9007. PMID 25768762.
43. ^ Schreiber, M.; Hodgman, S. S.; Bordia, P.; Luschen, H. P.; Fischer, M. H.; Vosk, R.; Altman, E.; Schneider, U.; Bloch, I. (2015). "Observation of many-body localization of interacting fermions in a quasirandom optical lattice". Science. 349 (6250): 842–845. arXiv:1501.05661. doi:10.1126/science.aaa7432. ISSN 0036-8075. PMID 26229112. S2CID 5112350.
44. ^ Choi, J.-y.; Hild, S.; Zeiher, J.; Schauss, P.; Rubio-Abadal, A.; Yefsah, T.; Khemani, V.; Huse, D. A.; Bloch, I.; Gross, C. (2016). "Exploring the many-body localization transition in two dimensions". Science. 352 (6293): 1547–1552. arXiv:1604.04178. Bibcode:2016Sci...352.1547C. doi:10.1126/science.aaf8834. ISSN 0036-8075. PMID 27339981. S2CID 35012132.
45. ^ Wei, Ken Xuan; Ramanathan, Chandrasekhar; Cappellaro, Paola (2018). "Exploring Localization in Nuclear Spin Chains". Physical Review Letters. 120 (7): 070501. Bibcode:2018PhRvL.120g0501W. doi:10.1103/PhysRevLett.120.070501. ISSN 0031-9007. PMID 29542978.
46. ^ Smith, J.; Lee, A.; Richerme, P.; Neyenhuis, B.; Hess, P. W.; Hauke, P.; Heyl, M.; Huse, D. A.; Monroe, C. (2016). "Many-body localization in a quantum simulator with programmable random disorder". Nature Physics. 12 (10): 907–911. arXiv:1508.07026. Bibcode:2016NatPh..12..907S. doi:10.1038/nphys3783. ISSN 1745-2473. S2CID 53408060.