Eigenstate properties of the disordered Bose−Hubbard chain

doi:10.1007/s11467-023-1384-1

Front. Phys.

2024, Vol. 19

Issue (4) : 43207 https://doi.org/10.1007/s11467-023-1384-1

Eigenstate properties of the disordered Bose−Hubbard chain

Jie Chen¹(

), Chun Chen¹(

), Xiaoqun Wang^2,³(

)

¹. Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
². School of Physics, Zhejiang University, Hangzhou 310058, China
³. Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China

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Abstract

Many-body localization (MBL) of a disordered interacting boson system in one dimension is studied numerically at the filling faction one-half. The von Neumann entanglement entropy $S v N$ is commonly used to detect the MBL phase transition but remains challenging to be directly measured. Based on the $U (1)$ symmetry from the particle number conservation, $S v N$ can be decomposed into the particle number entropy $S N$ and the configuration entropy $S C$ . In light of the tendency that the eigenstate’s $S C$ nears zero in the localized phase, we introduce a quantity describing the deviation of $S N$ from the ideal thermalization distribution; finite-size scaling analysis illustrates that it shares the same phase transition point with $S v N$ but displays the better critical exponents. This observation hints that the phase transition to MBL might largely be determined by $S N$ and its fluctuations. Notably, the recent experiments [A. Lukin, et al., Science 364, 256 (2019); J. Léonard, et al., Nat. Phys. 19, 481 (2023)] demonstrated that this deviation can potentially be measured through the $S N$ measurement. Furthermore, our investigations reveal that the thermalized states primarily occupy the low-energy section of the spectrum, as indicated by measures of localization length, gap ratio, and energy density distribution. This low-energy spectrum of the Bose model closely resembles the entire spectrum of the Fermi (or spin $X X Z$ ) model, accommodating a transition from the thermalized to the localized states. While, owing to the bosonic statistics, the high-energy spectrum of the model allows the formation of distinct clusters of bosons in the random potential background. We analyze the resulting eigenstate properties and briefly summarize the associated dynamics. To distinguish between the phase regions at the low and high energies, a probing quantity based on the structure of $S v N$ is also devised. Our work highlights the importance of symmetry combined with entanglement in the study of MBL. In this regard, for the disordered Heisenberg $X X Z$ chain, the recent pure eigenvalue analyses in [J. Šuntajs, et al., Phys. Rev. E 102, 062144 (2020)] would appear inadequate, while methods used in [A. Morningstar, et al., Phys. Rev. B 105, 174205 (2022)] that spoil the $U (1)$ symmetry could be misleading.

Keywords entanglement entropy decomposition U(1) symmetry thermalization-to-localization transition

Corresponding Author(s): Jie Chen,Chun Chen,Xiaoqun Wang

Issue Date: 29 February 2024

Cite this article:

Jie Chen,Chun Chen,Xiaoqun Wang. Eigenstate properties of the disordered Bose−Hubbard chain[J]. Front. Phys. , 2024, 19(4): 43207.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-023-1384-1
https://academic.hep.com.cn/fop/EN/Y2024/V19/I4/43207

Fig.1 (a) Particle number entropy

S N

versus disorder intensity

μ

for different sizes

L

at an energy density

ε = 0.25

. (b) Results for configuration entropy

S C

at the same energy density. Companion results for

p n

and

S v N n

with respect to differing sizes are shown in the insets of (a) and (b) for

μ = 2

Fig.2 Half-chain particle number fluctuation

F N

versus

μ

at an energy density

ε = 0.25

for various lengths

L

Fig.3 (a) Variation of the total entropy

S v N

with the disorder intensity for different sizes; the inset gives the result of its data collapse, yielding the estimate of the phase transition point at

μ c ≈ 7.8

. (b) is the corresponding result of

| d (S N) |

for different chain lengths when disorder increases; its data collapse is given also in the inset, showing the same phase transition point at around

μ c ≈ 7.8

. The energy density here is always fixed to be

ε = 0.25

Fig.4 (a) Numerical results on the two-body density-density correlation function

G 2

for

L = 12

from the ED calculations.

G 2

versus

d

shows an exponential decay for

μ = 3.5

and

7.0

ε = 0.3

, from which the localization lengths might be extracted. (b) Phase diagram on the plane of energy density

ε

and disorder strength

μ

is constructed based on the color contour of the localization length

ξ

and consists of the ETH and MBL regimes, separated by a line derived from the scaling analyses of the total entropy

S v N

and

| d (S N) |

Fig.5 Landscape of the phase diagram from the adjacent gap ratio

r

, using a dBH chain of

L = 12

. The right color bar labels

r

approximately between the GOE limit

r G O E ≈ 0.536

and the Poisson limit

r P o i . ≈ 0.386

. A blue square in the bottom-left corner is an artifact that may occur from an insufficient number of eigenstates, particularly in such an area where both

ε

and

μ

are small, i.e., essentially originating from the finite-size effect. In addition, schematics of the different MBL regions are indicated: scatter MBL and cluster MBL. The red (from

S v N

) and cyan (from

| d (S N) |

) squared symbols with the horizontal error bars for

μ

mark the position estimates of the phase boundaries at

ε = 0.05, 0.1, 0.15, 0.2,

0.25, 0.3, 0.35, 0.4

Fig.6 Particle number distributions of the 11 eigenstates, 5 above and 5 below the eigenstate closest in energy to the given initial state. (a) corresponds to the target energy

E = ? l | H | l ? μ = 2

realized in the region of low energy density and small disorder strength. (b) corresponds to

E = ? l | H | l ? μ = 20

within the low-energy and strong-disorder region. (c) gives the result of

E = ? p | H | p ? μ = 2

for the high energy density and weak disorder. (d) targets

E = ? p | H | p ? μ = 20

upon the high energy density and large disorder strength. The chain size here is fixed to be

L = 14

Fig.7 3D plot of

| d (S v N n) |

L = 12

, where the phase transition boundary and the three different phases can be perceived.

Fig. A1 The distributions of the local compressibility

K i

ε = 0.3

for several disorder strengths with

L = 14

and

K i ∈ [0, 1.5]

. The histograms are obtained using ED and are accompanied by the fitting solid curves. For

μ = 0.4, 4.0

, the distributions are well fitted by a Gaussian function, and a two-peak structure arises at

μ = 5.0

, where the main peak can still be fitted by a Gaussian function. When

μ = 10.0

, however, it fulfills a power law within the range of small

K i

, meanwhile exhibiting a satellite peak at

K i ≈ 0.25

Fig. A2 Energy density distributions for various disorder strengths

μ ∈ [2, 20]

, obtained from ED using a chain of length

L = 12

Fig. A3 The average energy densities for the two kinds of initial states as a function of

μ

for different chain lengths. In the present study, the

p

-state corresponds to an initial state where all

L / 2

bosons are loaded at the first site and zero occupancy for the remainder of the system. At the same time, one particle occupies each site of the left part of the system, and no particle is in all sites of the right part, comprising the

l

-state.

Fig. A4 (a) The correlation function

G 2 (d)

for

L = 22

with

μ = 0.9, 1.5, 2.0, 3.0, 6.0, 10.0

. The result for

L = 32

and

μ = 6.0

is also included for comparison; the insets of (a) are for

G 2

μ = 6.0

, left with

L = 12

and right with

L = 18

. (b) The localization length

ξ

for

L = 12, 14, 16, 18, 20, 22

at different disorder strengths. The data are extracted from

G 2 (d)

evaluated in the many-body states which have been evolved up to

t = 100 τ

. Here, the initial product state is chosen to be the

p

-state.

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