Observation of Many-Body Localization of Interacting Fermions in a Quasirandom Optical Lattice
Science, 2015 · paper · Physics
Observes many-body localization of interacting fermions in a one-dimensional quasi-random optical lattice. An initial density wave decays under weak disorder but partly persists above a critical disorder strength, which depends on interaction strength, in agreement with simulations.
Checked: PDF from arxiv.org, fetched 4 Oct 2026. Page numbers are those of that PDF.
4 findings
Substance Supplement, Disorder strength: the text gives "an undisturbed tunnel coupling of J = h×540 Hz", but Eq. (4), introduced with "Combining these numbers", divides by "h · 550 Hz"; this is the same quantity given two values.
- p. 7 Supplement, Disorder strength, sentence before Eq. (4)
yielding an undisturbed tunnel coupling of J = h×540 Hz
- p. 7 Supplement, Eq. (4) denominator
h · 550 Hz
- p. 7
yielding an undisturbed tunnel coupling of J = h×540 Hz.
- p. 7 Disorder strength, Eq. (4)
≈6.67 · sd
Only the authors can say what is right here.
Judge's note: I checked page 7 of the image. The sentence before Eq. (4) gives J = h×540 Hz for s_s = 8. Eq. (4) uses h·550 Hz in the denominator, and no second, modified J is defined anywhere. The only occurrences of 540 and 550 in the text are these two lines. The authors need to say which value is correct. About the factor 6.67: Eq. (3) gives the coefficient only as ≈0.2, and 6.67 corresponds to a coefficient of about 0.208 with J = 550 Hz. The printed 6.67 may therefore come from the unrounded integral, so I do not treat it as a separate error. The reviewers' check against the Fig. 3 times (13.7τ, 17.1τ, 20.5τ) does not settle the question: τ = ħ/J gives 13.57τ at 4 ms with 540 Hz and 13.82τ with 550 Hz, so neither value reproduces 13.7τ exactly.
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Reference or label Fig. 13A y-axis (Oscillation Amplitude δI_rms): the upper tick is labelled "10^-1" and the tick below it "5·10^-1"; the lower label is presumably 5·10^-2.
- p. 13 Fig. 13A, y-axis ticks
5·10-1
Judge's note: Viewed page 13 and the crop. The log axis has 10^-1 at the top and a tick labelled 5·10^-1 below it, with four minor ticks between them (consistent with 9, 8, 7, 6 × 10^-2). The curves decay downward and the caption says "decreasing oscillation amplitudes", so the axis is not inverted; 0.5 cannot sit below 0.1. Mislabelled exponent.
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Wording "However, we do not take into account the periodic trap, but rather consider a system with L = 40 sites and open boundary conditions." There is no periodic trap; the confinement is the harmonic trap.
- p. 12 Supplement, Time evolution of the CDW in an interacting model
we do not take into account the periodic trap
Judge's note: Viewed page 12. The text is printed as quoted. Elsewhere the paper speaks of "the 3D harmonic trap", "a confining harmonic potential" and, for these simulations, "a single homogeneous tube without any trapping potentials". The quasi-periodic potential is explicitly included in the simulation in the preceding sentence, so "periodic" cannot refer to the lattice. The evident fix is "harmonic trap".
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Wording "Fig. 13B compares the fitted exponent s∗ to the slope of the logarithmic growth of the entanglement entropy showing the direct correspondence between the two", but Eq. (30) gives δI_rms(t) ∼ (1/√L)(ħ/Ut)^{s∗/2}, so the fitted decay exponent is s∗/2, not s∗.
- p. 13 Supplement, paragraph after Fig. 13
compares the fitted exponent s∗ to the slope of the logarithmic growth
- p. 13 Supplement, Eq. (30)
1 2 s∗
- p. 12 Eq. (30)
(ħ/Ut)^{s*/2}
Only the authors can say what is right here.
Judge's note: Viewed page 13. Eqs. (29) and (30) both print the exponent as (1/2)s*. The text nonetheless calls the fitted power-law exponent s* and Fig. 13B plots "Decay of Oscillations" on the same axis as the entropy slope, where the two sets of points coincide at small U/J at values matching s* in Fig. 5B. Neither the caption nor the text says whether the plotted decay exponent has been doubled. Only the authors can say whether the factor 2 was applied.
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15 minor (typos and the like; most checked by the judge only)
Wording Eq. (4) "0.2 · sd · ER,532 / (h · 550 Hz) ≈ 6.67 · sd" does not reproduce from the printed inputs (ER,532 = h × 17.64 kHz): they give 6.41 with 550 Hz or 6.53 with 540 Hz.
- p. 7 Supplement, Eq. (4)
≈ 6.67 · sd
- p. 7 Supplement, text above Eq. (4)
ER,532 = h × 17.64 kHz
- p. 7 Supplement, Eq. (4)
≈ 6.67 · sd.
Only the authors can say what is right here.
Judge's note: Recomputed from the page 7 image. The prefactor in Eq. (3) is itself printed as '≈0.2', and 6.67 needs a prefactor of about 0.204 with J = 540 Hz or 0.208 with 550 Hz, both of which round to 0.2. So the result was probably computed from an unrounded prefactor and is not shown to be wrong, but the chain as printed is off by 2-4% and quotes three digits from a one-digit input. Downgraded from an error to a wording/consistency point.
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Wording "All experiments described in this work employ absorption imaging after 8 ms time of flight (TOF)" but the doublon measurement uses "standard absorbtion imaging after 6ms TOF".
- p. 7 Supplement, General sequence
All experiments described in this work employ absorption
- p. 9 Supplement, Doublon fractions
standard absorbtion imaging after 6ms TOF
- p. 9
measured using standard absorbtion imaging after 6ms TOF.
- p. 7 General sequence
absorption imaging after 8 ms time of flight (TOF)
Judge's note: Both statements seen on the page 7 and page 9 images. The doublon-fraction measurement is a separate procedure and may legitimately use 6 ms, so the 'All experiments' statement is over-broad, not a wrong number. This cluster is kept for the TOF point only; the misspellings 'absorbtion' and 'supress' raised here are covered by C012 and C013.
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Wording Eq. (6) writes "I_0 = (1/L) Σ_l (−1)^l ⟨n_l⟩_0 = ⟨N_e − N_o⟩_0/(N_e + N_o)" with L defined as "the total number of sites"; the two sides agree only if N_e + N_o = L. Eq. (9) also introduces an undefined "n_0" that is absent from Eq. (10).
- p. 10 Supplement, Eq. (6)
(−1)l⟨ˆnl⟩0 = ⟨Ne −No⟩0/(Ne + No)
- p. 10 Eq. (6)
= ⟨Ne −No⟩0/(Ne + No)
Judge's note: Viewed page 10. The text below Eq. (6) defines L as the total number of sites and N_e (N_o) as fermion numbers on even (odd) sites. (1/L)Σ(−1)^l⟨n_l⟩ is (N_e−N_o)/L, which equals the main-text definition (Eq. 1) only when the atom number equals L. For the even-sites-only initial state the two differ by the filling factor. No statement in the section fixes N = L. n_0 in Eq. (9) is not defined and is dropped in Eq. (10). This is a normalisation looseness in a derivation, not a wrong result, so rated minor.
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Typo Fig. 3 caption: "averaged over similar evolutions times to the experiment" should read "evolution times".
- p. 2 Fig. 3 caption
averaged over similar evolutions times to the experiment
Judge's note: Read the Fig. 3 caption on the page 2 image; the plural slip is in the PDF. The side remark about the ED window (15τ to 30τ on page 10) against the experimental 13.7τ-20.5τ is covered by the word 'similar' and is not treated as a defect.
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Typo Fig. 13B y-axis label reads "Entr. Slope σ* / Decay Expon.", whereas the same quantity is s* in the text, Eqs. (29)–(30) and Fig. 5B ("Ent. Entr. Slope s*").
- p. 13 Fig. 13B y-axis label
Entr. Slope σ* / Decay Expon.
- p. 4 Fig. 5B y-axis label
Ent. Entr. Slope s*
- p. 13 text
compares the fitted exponent s∗to the slope
- p. 13 text after Fig. 13
compares the fitted exponent s∗
Judge's note: Viewed the Fig. 13B crop and the Fig. 5B crop. The Fig. 13B glyph is a sigma with a subscript asterisk; Fig. 5B clearly prints s with a subscript asterisk. The caption and text identify the quantity as the entanglement-entropy slope s*, so the meaning is recoverable; likely a symbol-font substitution in the figure. Cosmetic label inconsistency. The exponent-versus-s*/2 point raised in one note is handled in C027.
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Typo Reference [9] prints the third author as "B. L. Altschuler"; the usual spelling of this author's name is Altshuler.
- p. 5 Reference [9]
D. M. Basko, I. L. Aleiner, B. L. Altschuler
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: that the cited Ann. Phys. 321, 1126 (2006) paper prints the author's name as 'Altshuler'. The page 5 image does show 'B. L. Altschuler' in reference [9]; the correct spelling cannot be checked from the files.
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Typo Acknowledgments: "the European Commision (UQUAM, AQuS)" should read "European Commission".
- p. 6 Acknowledgments
the European Commision (UQUAM, AQuS)
- p. 6 Acknowledgments
European Commision
Judge's note: Seen on the page 6 image in the Acknowledgments paragraph. Plain misspelling.
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Typo Eq. (3) prints the integrand as "cos(2βksx |w (ksx)|2)", with the Wannier density inside the cosine; it should be cos(2βk_s x) |w(k_s x)|².
- p. 7 Supplement, Eq. (3)
cos(2βksx |w (ksx)|2)
Judge's note: Page 7 image shows the cosine's closing parenthesis after |w(k_s x)|², so this is in the PDF and not an extraction artefact. Misplaced parenthesis; the intended overlap integral is evident.
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Typo Doublon fractions: "standard absorbtion imaging after 6ms TOF" should read "absorption".
- p. 9 Supplement, Doublon fractions
standard absorbtion imaging
- p. 7 General sequence
All experiments described in this work employ absorption imaging after 8 ms time of flight
- p. 9 Doublon fractions
standard absorbtion imaging after 6ms TOF
Judge's note: Seen on the page 9 image; the paper spells 'absorption' correctly on pages 7 and 9 elsewhere. This cluster is kept for the misspelling; the 8 ms against 6 ms TOF point also raised here is covered by C008.
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Typo Doublon fractions: "in 200 µs to supress hopping in all three dimensions" should read "suppress".
- p. 9 Supplement, Doublon fractions
to supress hopping in all three di-
- p. 9
in 200 µs to supress hopping in all three dimensions
- p. 9 Doublon fractions
to supress hopping in all three
Judge's note: Seen on the page 9 image; 'suppresses' is spelled correctly on page 8. Plain misspelling.
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Typo "Therefore the CDW or (imbalance) at time t is": the parenthesis is misplaced (should be "the CDW (or imbalance)"); the sentence after Eq. (10), "which can be calculated numerically", also has no full stop.
- p. 10 Supplement, sentence before Eq. (10)
Therefore the CDW or (imbalance) at time t is
Judge's note: Viewed page 10. The text is printed as quoted, and the line after Eq. (10) ends without a full stop. Cosmetic.
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Typo Eq. (14): "Ĥ[{σ}] = −J Σ_{ij} [f†_i f_{i+1} + h.c.] + Σ_i V_i f†_i f_i": the first sum runs over "ij" but the summand contains no j.
- p. 12 Supplement, Eq. (14)
ˆH[{σ}] = −J X ij
Judge's note: Viewed page 12. The subscript under the first sum is "ij" and the summand depends only on i. Eqs. (7) and (13)/(18) sum the hopping term over i (or i,σ). The j is spurious.
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Typo Eq. (16): "φ_{σ}[n_j] = 0, if σ_n = ↑; π, if σ_n = ↓": the cases use an undefined index n where the argument and following text use n_j.
- p. 12 Supplement, Eq. (16)
if σn =↑
Judge's note: Viewed the crop and page 12. The left-hand side is φ_{σ}[n_j] and the text below says "the n_j-th spin is ↓", so the conditions should read σ_{n_j}. Index slip.
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Typo Eq. (24): the phase factor is printed "e^{−i(E[{σ}]−E[{σ′}]t)}", with t inside the bracket; it should be e^{−i(E[{σ}]−E[{σ′}])t}.
- p. 13 Supplement, Eq. (24) exponent
e−i(E[{σ}]−E[{σ′}]t)
Judge's note: Viewed the crop. The closing parenthesis comes after t, so t multiplies only the second energy as printed. Misplaced parenthesis.
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Typo Eq. (23): "|Ψ_0⟩ = Σ_{σ_1,…,σ_l} Π_{j=1}^{l(t)} A^j_{σ_i} |{σ}⟩": the amplitude subscript is σ_i although the product runs over j; Eq. (24) has A^j_{σ_j}.
- p. 13 Supplement, Eq. (23)
Aj σi|{σ}⟩
Judge's note: Viewed the crop. Eq. (23) prints the subscript σ_i; Eq. (24) just below prints A^j_{σ_j}. Index slip.
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