# Experimentally Accessible Witnesses of Many-Body Localization

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## Abstract

**:**

## 1. Introduction

## 2. Probing Disordered Optical Lattice Systems

## 3. Measurements Considered Feasible

**In-situ**: An in-situ measurement detects the occupation of individual lattice sites. This technique only allows resolving the parity of the particle number on each site, which for fermions constitutes no limitation, however. Using the fact that single-shot measurements are performed, higher moments such as density–density correlators can also be extracted from this kind of measurements. Both ramifications are used. This measurement has been used to determine onsite parities in Ref. [36] to show particle localization in two-dimensional disordered optical lattices. Here, we try to additionally witness the interactions necessary to distinguish Anderson from MBL systems.

**Time-of-flight**: The time-of-flight (ToF) measurement extracts position-averaged momentum information of the form

## 4. Phenomenology of Many-Body Localization

**II**, see Ref. [14]. Here, $\mathbf{I}$ corresponds to the ballistic regime and $\mathbf{II}$ captures the slower dephasing. In the context of optical lattices, local excitations seem difficult to implement. Hence, in the following, we focus on the observation of indirect effects on the dynamical evolution in MBL systems.

## 5. Feasible Witnesses

#### 5.1. Absence of Particle Transport

**Measure**

**1**(Particle propagation and phase correlations)

**.**

#### 5.2. Slow Spreading of Information

**Measure**

**2**(Logarithmic information propagation)

**.**

**II**. Hence, an unbounded growth of correlations between distant regions is in principle possible, given sufficient time. Furthermore, we have shown that this built-up of correlations also happens on observable time scales, as can be seen from the evolution of density–density correlations captured by

**Measure 2**.

#### 5.3. Dephasing and Equilibration

**Measure**

**3**(Density evolution: Equilibration of fluctuations)

**.**

**Measure 2**. If we now, however, turn to the interacting model, a local excitation will slowly explore larger and larger parts of the Hilbert space, leading to a slow, but persistent decrease of the fluctuations.

#### 5.4. Present and Future Experimental Realizations

**Measure 1**,

**Measure 3**or the imbalance, which is a measure of particle localization as well [21], the quantum average does in principle commute with the disorder average allowing for simultaneous averaging with fewer realizations. This is however not the case for non-linear quantities such as

**Measure 2**. Here, the full procedure described above needs to be carried out. The repetition rates of optical lattices are on the order of seconds and leading experimentalists assured us that taking reliable data for all our measures is indeed feasible [45].

**Measure 2**are used as well. In Ref. [47], the authors defined a quantity called transport distance which basically coincides with our

**Measure 2**. The difference being that their scaling function is only linear instead of quadratic. However, they dis not employ this measure to show the many-body correlations in these systems. Rather, they calculated the number and configurational entanglement [46]. The system sizes used are very restricted, possibly due to the complicated procedure of obtaining these entropies.

**Measure 2**or

**Measure 3**might complement these results nicely by overcoming these problems and hence being applicable also for larger systems and potentially also higher dimensional systems, where the fate of MBL is still debated.

## 6. Conclusions and Outlook

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Numerical Details

**Figure A1.**Finite size scaling for the evolution of particle density in the middle of the chain for a typical disorder realization. For $L=10,20$, an exact diagonalization code was used. The other system sizes are simulated with a TEBD code [43].

**Figure A2.**Evolution of the discarded weight. This plot varies strongly depending on the chosen disorder realization. From the 100 realizations used for the averaged plots, the realization with the largest discarded weights is shown here.

## Appendix B. Bosonic Model with On-Site Interactions

**Measure 2**for a related model that is used in some of the experimental realizations of MBL. This is the disordered Bose–Hubbard model with on-site interactions given by

**Measure 2**with the exception that the number operators were replaced by parity operators.

**Measure**

**4**(Logarithmic information propagation)

**.**

**Measure 4**for the Anderson ($U=0$) and MBL ($U=2$) case. Similar to the main text, we find that, in the non-interacting case, the measure saturates after few tunneling times. In contrast, for the interacting model, we found that the measure grew in comparable fashion to the fermionic counterpart (grey stars). This suggests that the correlation measure can be employed in similar models as well.

**Figure A3.**Plotted are the results of a TEBD simulation of the dynamical evolution of the parity–parity correlations ${P}_{\mathrm{corr}}$. The initial state $\psi $ is again found in Equation (2) under the Hamiltonian in Equation (A1) for the case of an Anderson insulator with $U=0$ and MBL with $U=2$. We compared the results of the fermionic MBL setting and the bosonic MBL and Anderson setting with a local Hilbert space dimension truncation $k=3$. Every data point corresponds to an average of over 100 realizations.

## References

- Basko, D.M.; Aleiner, I.L.; Altshuler, B.L. Metal-insulator transition in a weakly interacting many-electron system with localized single-particle states. Ann. Phys.
**2006**, 321, 1126. [Google Scholar] [CrossRef] - Anderson, P.W. Absence of Diffusion in Certain Random Lattices. Phys. Rev.
**1958**, 109, 1492. [Google Scholar] [CrossRef] - Nandkishore, R.; Huse, D.A. Many-Body Localization and Thermalization in Quantum Statistical Mechanics. Ann. Rev. Cond. Mat. Phys.
**2015**, 6, 15–38. [Google Scholar] [CrossRef][Green Version] - Pal, A.; Huse, D.A. The many-body localization transition. Phys. Rev. B
**2010**, 82, 174411. [Google Scholar] [CrossRef] - Oganesyan, V.; Huse, D.A. Localization of interacting fermions at high temperature. Phys. Rev. B
**2007**, 75, 155111. [Google Scholar] [CrossRef][Green Version] - Polkovnikov, A.; Sengupta, K.; Silva, A.; Vengalattore, M. Non-equilibrium dynamics of closed interacting quantum systems. Rev. Mod. Phys.
**2011**, 83, 863. [Google Scholar] [CrossRef] - Eisert, J.; Friesdorf, M.; Gogolin, C. Quantum many-body systems out of equilibrium. Nat. Phys.
**2015**, 11, 124–130. [Google Scholar] [CrossRef][Green Version] - Gogolin, C.; Eisert, J. Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems. Rep. Prog. Phys.
**2016**, 79, 056001. [Google Scholar] [CrossRef][Green Version] - Znidaric, M.; Prosen, T.; Prelovsek, P. Many-body localization in the Heisenberg XXZ magnet in a random field. Phys. Rev. B
**2008**, 77, 064426. [Google Scholar] [CrossRef] - Bardarson, J.H.; Pollmann, F.; Moore, J.E. Unbounded Growth of Entanglement in Models of Many-Body Localization. Phys. Rev. Lett.
**2012**, 109, 017202. [Google Scholar] [CrossRef][Green Version] - Goold, J.; Clark, S.R.; Gogolin, C.; Eisert, J.; Scardicchio, A.; Silva, A. Total correlations of the diagonal ensemble herald the many-body localisation transition. Phys. Rev. B
**2015**, 92, 180202(R). [Google Scholar] [CrossRef] - Eisert, J.; Cramer, M.; Plenio, M.B. Area laws for the entanglement entropy. Rev. Mod. Phys.
**2010**, 82, 277. [Google Scholar] [CrossRef] - Bauer, B.; Nayak, C. Area laws in a many-body localised state and its implications for topological order. J. Stat. Mech.
**2013**, 2013, P09005. [Google Scholar] [CrossRef] - Friesdorf, M.; Werner, A.H.; Brown, W.; Scholz, V.B.; Eisert, J. Many-body localisation implies that eigenvectors are matrix-product states. Phys. Rev. Lett.
**2015**, 114, 170505. [Google Scholar] [CrossRef] [PubMed] - Srednicki, M. Chaos and quantum thermalization. Phys. Rev. E
**1994**, 50, 888–901. [Google Scholar] [CrossRef][Green Version] - Kim, I.H.; Chandran, A.; Abanin, D.A. Local integrals of motion and the logarithmic light cone in many-body localised systems. Phys. Rev. B
**2015**, 91, 085425. [Google Scholar] - Chandran, A.; Carrasquilla, J.; Kim, I.H.; Abanin, D.A.; Vidal, G. Spectral tensor networks for many-body localisation. Phys. Rev. B
**2015**, 92, 024201. [Google Scholar] [CrossRef] - Friesdorf, M.; Werner, A.H.; Goihl, M.; Eisert, J.; Brown, W. Local constants of motion imply transport. New J. Phys.
**2015**, 17, 113054. [Google Scholar] [CrossRef] - Serbyn, M.; Papić, Z.; Abanin, D.A. Local Conservation Laws and the Structure of the Many-Body Localized States. Phys. Rev. Lett.
**2013**, 111, 127201. [Google Scholar] [CrossRef] [PubMed][Green Version] - Huse, D.A.; Nandkishore, R.; Oganesyan, V. Phenomenology of fully many-body-localized systems. Phys. Rev. B
**2014**, 90, 174202. [Google Scholar] [CrossRef][Green Version] - Schreiber, M.; Hodgman, S.S.; Bordia, P.; Lüschen, H.P.; Fischer, M.H.; Vosk, R.; Altman, E.; Schneider, U.; Bloch, I. Observation of many-body localization of interacting fermions in a quasi-random optical lattice. Science
**2015**, 349, 842. [Google Scholar] [CrossRef] [PubMed] - Bordia, P.; Lüschen, H.P.; Hodgman, S.S.; Schreiber, M.; Bloch, I.; Schneider, U. Coupling Identical one-dimensional Many-Body Localized Systems. Phys. Rev. Lett.
**2016**, 116, 140401. [Google Scholar] [CrossRef] [PubMed] - Wiersma, D.S.; Bartolini, P.; Lagendijk, A.; Righini, R. Localization of light in a disordered medium. Nature
**1997**, 390, 671–673. [Google Scholar] [CrossRef] - Bloch, I.; Dalibard, J.; Nascimbene, S. Quantum simulations with ultracold quantum gases. Nat. Phys.
**2012**, 8, 267. [Google Scholar] [CrossRef] - Eisert, J.; Brandao, F.G.; Audenaert, K.M. Quantitative entanglement witnesses. New J. Phys.
**2007**, 9, 46. [Google Scholar] [CrossRef] - Audenaert, K.M.R.; Plenio, M.B. When are correlations quantum? New J. Phys.
**2006**, 8, 266. [Google Scholar] [CrossRef] - Guehne, O.; Reimpell, M.; Werner, R.F. Estimating entanglement measures in experiments. Phys. Rev. Lett.
**2007**, 98, 110502. [Google Scholar] [CrossRef] - Gring, M.; Kuhnert, M.; Langen, T.; Kitagawa, T.; Rauer, B.; Schreitl, M.; Mazets, I.; Smith, D.A.; Demler, E.; Schmiedmayer, J. Relaxation and Prethermalization in an Isolated Quantum System. Science
**2012**, 337, 1318. [Google Scholar] [CrossRef] - Steffens, A.; Friesdorf, M.; Langen, T.; Rauer, B.; Schweigler, T.; Hübener, R.; Schmiedmayer, J.; Riofrio, C.A.; Eisert, J. Towards experimental quantum field tomography with ultracold atoms. Nat. Commun.
**2015**, 6, 7663. [Google Scholar] [CrossRef] - Luitz, D.J.; Laflorencie, N.; Alet, F. Many-body localisation edge in the random-field Heisenberg chain. Phys. Rev. B
**2015**, 91, 081103. [Google Scholar] [CrossRef] - Singh, R.; Bardarson, J.H.; Pollmann, F. Signatures of the many-body localization transition in the dynamics of entanglement and bipartite fluctuations. New J. Phys.
**2016**, 18, 023046. [Google Scholar] [CrossRef][Green Version] - Serbyn, M.; Papić, Z.; Abanin, D.A. Criterion for many-body localization-delocalization phase transition. Phys. Rev. X
**2015**, 5, 041047. [Google Scholar] [CrossRef] - Serbyn, M.; Knap, M.; Gopalakrishnan, S.; Papic, Z.; Yao, N.Y.; Laumann, C.R.; Abanin, D.A.; Lukin, M.D.; Demler, E.A. Interferometric Probes of Many-Body Localization. Phys. Rev. Lett.
**2014**, 113, 147204. [Google Scholar] [CrossRef] [PubMed] - Roy, D.; Singh, R.; Moessner, R. Probing many-body localisation by spin noise spectroscopy. Phys. Rev. B
**2015**, 92, 180205. [Google Scholar] [CrossRef] - Vasseur, R.; Parameswaran, S.A.; Moore, J.E. Quantum revivals and many-body localization. Phys. Rev. B
**2015**, 91, 140202. [Google Scholar] [CrossRef][Green Version] - Choi, J.Y.; Hild, S.; Zeiher, J.; Schauß, P.; Rubio-Abadal, A.; Yefsah, T.; Khemani, V.; Huse, D.A.; Bloch, I.; Gross, C. Exploring the many-body localization transition in two dimensions. Science
**2016**, 352, 1547–1552. [Google Scholar] [CrossRef][Green Version] - Trotzky, S.; Chen, Y.A.; Flesch, A.; McCulloch, I.P.; Schollwoeck, U.; Eisert, J.; Bloch, I. Probing the relaxation towards equilibrium in an isolated strongly correlated one-dimensional Bose gas. Nat. Phys.
**2012**, 8, 325. [Google Scholar] [CrossRef] - Ros, V.; Müller, M.; Scardicchio, A. Integrals of motion in the many-body localized phase. Nucl. Phys. B
**2015**, 891, 420–465. [Google Scholar] [CrossRef][Green Version] - Lieb, E.H.; Robinson, D.W. The finite group velocity of quantum spin systems. Commun. Math. Phys.
**1972**, 28, 251–257. [Google Scholar] [CrossRef] - Daley, A.J.; Kollath, C.; Schollwoeck, U.; Vidal, G. Time-dependent density-matrix renormalization- group using adaptive effective Hilbert spaces. J. Stat. Mech.
**2004**, 2004, P04005. [Google Scholar] [CrossRef] - Kirsch, W. An invitation to random Schroedinger operators. arXiv
**2007**, arXiv:0709.3707. [Google Scholar] - Germinet, F.; Klein, A. Bootstrap multi-scale analysis and localization in random media. Commun. Math. Phys.
**2001**, 222, 415. [Google Scholar] [CrossRef] - Wall, M.L.; Carr, L.D. Open Source TEBD. 2013. Available online: http://physics.mines.edu/downloads/software/tebd(2009) (accessed on 12 June 2019).
- Burrell, C.K.; Eisert, J.; Osborne, T.J. Information propagation through quantum chains with fluctuating disorder. Phys. Rev. A
**2009**, 80, 052319. [Google Scholar] [CrossRef][Green Version] - Gross, C.; Bloch, I.; (Max-Planck-Institut für Quantenoptik, Garching, Germany). Personal communication, 2018.
- Lukin, A.; Rispoli, M.; Schittko, R.; Tai, M.E.; Kaufman, A.M.; Choi, S.; Khemani, V.; Leonard, J.; Greiner, M. Probing entanglement in a many-body-localized system. arXiv
**2018**, arXiv:1805.09819. [Google Scholar] - Rispoli, M.; Lukin, A.; Schittko, R.; Kim, S.; Tai, M.E.; Léonard, J.; Greiner, M. Quantum critical behavior at the many-body-localization transition. arXiv
**2018**, arXiv:1812.06959. [Google Scholar] - Cramer, M.; Bernard, A.; Fabbri, N.; Fallani, L.; Fort, C.; Rosi, S.; Caruso, F.; Inguscio, M.; Plenio, M. Spatial entanglement of bosons in optical lattices. Nat. Commun.
**2013**, 4, 2161. [Google Scholar] [CrossRef] - Jones, E.; Oliphant, T.; Peterson, P. SciPy: Open Source Scientific Tools for Python; ResearchGate: Berlin, Germany, 2001. [Google Scholar]

**Figure 1.**An overview over the dynamical behavior of MBL systems versus their ergodic and thermalizing and Anderson localized counterparts.

**Measure 1**detects particle propagation and phase correlations and can be implemented using time-of-flight imaging.

**Measure 2**and

**Measure 3**utilize in-situ imaging to observe density–density correlations and equilibration behavior.

**Figure 2.**Plotted are the results of a TEBD simulation [43] of the dynamical evolution of the initial state $\psi $ from Equation (2) under the Hamiltonian in Equation (1) for the case of an Anderson insulators with $U=0$ and MBL with $U=2$. The disorder strength is $I=8$. The three plots are averaged over 100 disorder realizations. (

**Left**) Shown is the time evolution of ${y}_{\mathrm{Phase}}$ defined in

**Measure 1**demonstrating that the phase correlation behavior saturates both for MBL and Anderson localization. (

**Middle**) The plot shows the dynamical evolution of ${y}_{\mathrm{Corr}}$ defined in

**Measure 2**. Information propagation is fully suppressed in an Anderson insulator, resulting in a saturation of this quantity. In contrast, correlations continue to spread in the MBL system beyond all bounds, giving rise to a remarkably strong signal feasible to be detected in experiments. (

**Right**) Shown are the averaged fluctuations ${g}_{\mathrm{Eq}}$ defined in

**Measure 3**as a function of the time T over which the average is performed. The insets show the time evolution of the particle density at the position $L/2$, which enters the calculation of ${g}_{\mathrm{Eq}}$ for one disorder realization, which is identical for the MBL and Anderson localized model. As the insets also show, the local fluctuations continue indefinitely for the Anderson insulator, corresponding to a saturation of ${g}_{\mathrm{Eq}}$, while the MBL system equilibrates and ${g}_{\mathrm{Eq}}$ continues to decrease accordingly.

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**MDPI and ACS Style**

Goihl, M.; Friesdorf, M.; Werner, A.H.; Brown, W.; Eisert, J.
Experimentally Accessible Witnesses of Many-Body Localization. *Quantum Rep.* **2019**, *1*, 50-62.
https://doi.org/10.3390/quantum1010006

**AMA Style**

Goihl M, Friesdorf M, Werner AH, Brown W, Eisert J.
Experimentally Accessible Witnesses of Many-Body Localization. *Quantum Reports*. 2019; 1(1):50-62.
https://doi.org/10.3390/quantum1010006

**Chicago/Turabian Style**

Goihl, Marcel, Mathis Friesdorf, Albert H. Werner, Winton Brown, and Jens Eisert.
2019. "Experimentally Accessible Witnesses of Many-Body Localization" *Quantum Reports* 1, no. 1: 50-62.
https://doi.org/10.3390/quantum1010006