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Measuring out quasi-local integrals of motion from entanglement

B. Lu, C. Bertoni, S. J. Thomson, J. Eisert

TL;DR

This work tackles the challenge of experimentally accessing quasi-local integrals of motion in many-body localized systems by introducing a spatially-resolved entanglement probe based on negativity. Using the XXZ spin chain with random fields as a testbed, the authors derive rigorous bounds and show through tensor-network simulations that a well-defined l-bit length scale emerges from the time evolution of entanglement between spatially separated regions. They demonstrate logarithmic-in-time growth of the negativity with a distance-induced exponential suppression, enabling extraction of a localization length that agrees with independent l-bit analyses and distinguishing MB localisation from Anderson localisation. The proposed approach provides a practical, experimentally feasible pathway to characterize emergent length scales in MB localisation and potentially in other quantum many-body phenomena via spatially-resolved entanglement.

Abstract

Quasi-local integrals of motion are a key concept underpinning the modern understanding of many-body localisation, an intriguing phenomenon in which interactions and disorder come together. Despite the existence of several numerical ways to compute them - and astoundingly in the light of the observation that much of the phenomenology of many properties can be derived from them - it is not obvious how to directly measure aspects of them in real quantum simulations; in fact, the smoking gun of their experimental observation is arguably still missing. In this work, we propose a way to extract the real-space properties of such quasi-local integrals of motion based on a spatially-resolved entanglement probe able to distinguish Anderson from many-body localisation from non-equilibrium dynamics. We complement these findings with a new rigorous entanglement bound and compute the relevant quantities using tensor networks. We demonstrate that the entanglement gives rise to a well-defined length scale that can be measured in experiments.

Measuring out quasi-local integrals of motion from entanglement

TL;DR

This work tackles the challenge of experimentally accessing quasi-local integrals of motion in many-body localized systems by introducing a spatially-resolved entanglement probe based on negativity. Using the XXZ spin chain with random fields as a testbed, the authors derive rigorous bounds and show through tensor-network simulations that a well-defined l-bit length scale emerges from the time evolution of entanglement between spatially separated regions. They demonstrate logarithmic-in-time growth of the negativity with a distance-induced exponential suppression, enabling extraction of a localization length that agrees with independent l-bit analyses and distinguishing MB localisation from Anderson localisation. The proposed approach provides a practical, experimentally feasible pathway to characterize emergent length scales in MB localisation and potentially in other quantum many-body phenomena via spatially-resolved entanglement.

Abstract

Quasi-local integrals of motion are a key concept underpinning the modern understanding of many-body localisation, an intriguing phenomenon in which interactions and disorder come together. Despite the existence of several numerical ways to compute them - and astoundingly in the light of the observation that much of the phenomenology of many properties can be derived from them - it is not obvious how to directly measure aspects of them in real quantum simulations; in fact, the smoking gun of their experimental observation is arguably still missing. In this work, we propose a way to extract the real-space properties of such quasi-local integrals of motion based on a spatially-resolved entanglement probe able to distinguish Anderson from many-body localisation from non-equilibrium dynamics. We complement these findings with a new rigorous entanglement bound and compute the relevant quantities using tensor networks. We demonstrate that the entanglement gives rise to a well-defined length scale that can be measured in experiments.
Paper Structure (18 sections, 7 theorems, 61 equations, 14 figures)

This paper contains 18 sections, 7 theorems, 61 equations, 14 figures.

Table of Contents

  1. Abstract
  2. Introduction
  3. Results and Discussion
  4. Conclusion
  5. Methods
  6. Data availability
  7. Code availability
  8. Author contributions
  9. Acknowledgements
  10. Supplementary information
  11. Supplementary note 1: TEBD accuracy benchmarks
  12. Supplementary note 2: Comparison of disorder strengths
  13. Supplementary note 3: Bond dimension
  14. Supplementary note 4:Comparison of different measurement times
  15. Supplementary note 5: Negativity between subsystems of fixed size
  16. ...and 3 more sections

Key Result

Theorem 1

Let $\rho$ be an initial product state. Let $H$ be a many-body localised Hamiltonian as per Definition def:mbl_hamiltonian with localisation length $\xi<{1}/({4\log(2)})$ and $2(1/\kappa-\log(2))> 1/\xi$, consider three blocks $A,C,B$ such that $C$ divides $A$ from $B$, with $|C|=r+1$. The growth of for times $t\geq e^{r/(4\xi)}$, while for $t<e^{r/(4\xi)}$,

Figures (14)

  • Figure 1: Division into subsystems and computation of negativity: a) A sketch showing how a one-dimensional spin chain is partitioned into three subsystems. We are interested in computing the entanglement between subsystems $A$ and $B$ after subsystem $C$ has been traced out, giving rise to a spatially-resolved entanglement measure. b) Sketch of the initial quantum state in matrix product operator (MPO) form, made by taking the outer product of two matrix product state vectors. c) Sketch of how the negativity is computed: the partial transpose of subsystem $A$ corresponds to 'twisting' the MPO legs while tracing out subsystem $C$ corresponds to contracting the relevant MPO indices.
  • Figure 2: Behaviour of entanglement negativity in time and space: Results showing the growth of the negativity $E_N(r,t)$ with time for different distances $r$. Data is shown for a system size $L=24$ and a disorder strength $d=8.0$, averaged over $N_s=100$ disorder realisations. a) The dynamics of $E_N(r,t)$ following a quench from a Nèel state, showing the logarithmic growth at late times. The circular markers are the raw data points, while the solid lines are a smoothed guide to the eye. The error bars indicate the standard error in the mean. We note that these error bars show agreement on average between the various disorder realizations, but they are not fully statistically independent errors, as would be expected in an experiment where each data point would come from a different run. b) The full dynamics of $E_N(r,t)$, reflecting the logarithmic 'light cone'. Each circle maps the point where the negativity grows beyond the corresponding threshold $\varepsilon$ and the lines are linear fits. c) By extracting the behaviour of $E_N(r,t^*) \propto \exp(-r/\xi)$ at fixed times $t^*$ [dashed vertical lines in panel (a), horizontal lines in panel (b)], we can extract a well-defined length scale $\xi(t)$, which depends only weakly on time. The solid lines indicate the fits to the data points which are used to extract the $l$-bit length scale, demonstrating convergence at late times.
  • Figure 3: Extracted l-bit length scale: The characteristic $l$-bit length scale $\xi$ extracted from the entanglement negativity at time $t^* = 500$, shown in blue for $L=24$ with $N_s \in [50,100]$ disorder realisations and various values of the disorder strength $d$. Error bars indicate the fit error and are roughly the same size as the plot markers. Orange lines mark the localisation length obtained through exact diagonalisation following Ref. goihlConstructionExactConstants2018. For further details on calculating the localisation length using exact diagonalisation, see Supplementary Note 7 SM. The black line indicates the localisation length of the corresponding Anderson-localised system, obtained by directly diagonalising the Hamiltonian in the non-interacting limit ($\Delta=0$), for a system of size $L=128$ with $N_s=10000$.
  • Figure S1: Relative error of the simulation: A comparison of the relative error in the energy of the time-evolved state for different values of the disorder strength $d$, shown for system sizes $L=14$ ($\chi=128$, averaged over $N_s=240$ disorder realisations) and - at strong disorder only - $L=24$ ($\chi=192$, averaged over $N_s=100$ disorder realisations). The relative error remains below $1\%$ for all disorder strengths. Error bars indicate the variance over disorder distributions and in most cases, are of comparable size to the plot markers.
  • Figure S2: Entanglement negativity at different disorder strengths: A comparison of the dynamics of the entanglement negativity $E_N(r,t)$ for different values of the disorder strength $d$, shown for $L=14$ with bond dimension $\chi = 128$ and averaged over $N_s=240$ disorder realisations. In the delocalised phase, the negativity saturates to a value determined by the size of the subsystems $A$ and $B$, while in the localised phase the negativity displays a slow $\propto \log(t)$ growth even at late times. In this dephasing regime, we are able to use the data shown here to extract a length scale that characterises the localised phase, as detailed in the main text.
  • ...and 9 more figures

Theorems & Definitions (17)

  • Definition 1: Quasi-local operators
  • Definition 2: Many-body localisation
  • Theorem 1: Rigorous entanglement bounds
  • Definition 3: Quasi-locality
  • Definition 4: Strong quasi-locality
  • Definition 5: Many-body localisation
  • Theorem 2: Entanglement growth bound for sums of quasi-local operators
  • Corollary 1
  • proof
  • Lemma 1: Quasi-local sums
  • ...and 7 more