Yu-Cheng Wang, Xiong-Jun Liu, Shu Chen. Properties and applications of one dimensional quasiperiodic lattices [J]. Acta Physica Sinica, 2019, 68(4): 040301-1
- Acta Physica Sinica
- Vol. 68, Issue 4, 040301-1 (2019)
Fig. 1. IPR of ground states as a function of
for this system with
and
. The left and right insets show the distribution of the ground state with
and
respectively.
基态的倒参与率随
的变化，这里固定
和
. 左右的插图分别展示了
和
时系统的基态波函数的分布
Fig. 2. Sketch of the quasiperiodic lattice realized in the experiment. J describes the hopping between the nearest-neighbor sites of the primary lattice and
is the maximum shift of the on-site energy induced by the secondary lattice.
实验实现准周期晶格的原理示意图. J 描述的是主晶格最近邻格点之间的跃迁,
是由次晶格导致的在位能最大的差别
Fig. 3. Schematics of the experiment. Schematic illustration of the initial CDW state and the states reached after time evolution in the localized, intermediate, and extended phase, respectively: (a) Initial state: CDW state (
,
); (b) localized phase (
,
); (c) the intermediate phase, extended and localized states coexist at different energies (
,
); (d) extended phase (
,
).
实验原理图. 制备的初始CDW态，以及在局域、中间和扩展相中，经过一段时间演化后，分别对应的系统的末态 (a)初态分布，制备为CDW态(根据定义，有
,
); (b)局域态(
,
);(c)中间态，对应于不同的能量存在局域态和扩展态(
,
); (d)扩展态(
,
)
Fig. 4. (a)
as a function of
. Here we use 50 samples for
and
, 30 samples for
, and 20 samples for
; (b) averaged entanglement entropy
and
versus
. Here we use 500 samples for
and
, 100 samples for
and 30 samples for
. The interaction strength is fixed at
. Here a sample is specified by choosing an initial phase
[21].
(a)
随
的变化. 当系统尺寸为
和
时用的样品数是
, 当
时用的样品数是
, 当
时用的样品数是
; (b)平均的纠缠熵
和
随
的变化. 当
和
时用
个样品, 当
时用
个样品, 当
时用
个样品. 相互作用强度始终被固定为
. 这里一个样品指的是任选一个初相位
[21]
Fig. 5. Log-log plot of the width
vs time t for several values of
in the AA model with
,
and
.
AA模型中取不同的
时
随时间t 的变化的对数-对数图, 这里固定
, 跃迁强度
, 以及系统尺寸
Fig. 6. The mean information entropy as a function of
for this system with
and
. The left up inset shows the derivative of the mean information entropy as a function of
with fixed
(blue),
(red), and
(green). The right down inset shows the derivative of the mean information entropy as a function of
with
(blue),
(red), and
(green)[55].
固定
和
, 平均信息熵随周期
的变化. 左上角的插图展示了平均纠缠熵的导数随周期
的变化, 这里固定
(蓝色),
(红色), 和
(绿色). 右下角的插图展示了平均纠缠熵随
的变化, 这里分别固定
(蓝色),
(红色),
(绿色)[55]
Fig. 7. The mean information entropy versus both
and
for the system with
[55].
固定系统尺寸
, 平均信息熵随
和
的变化[55]
Fig. 8. Evolution of Loschmidt echo in a long time with different
s. The initial state is chosen to be the ground state of the Hamiltonian with
((a), (b))and
((c), (d))[66].
取不同值时Loschmidt echo的演化. 初态选准周期势强度为
((a), (b))和
((c), (d))的哈密顿量的基态[66]
Fig. 9. The behavior of
versus
for the system with
,
and
: (a) Different colors correspond to different
s and the initial state is chosen to be the ground state of the initial Hamiltonian; (b) different choice of initial state with
standing for the
eigenstates of the initial Hamiltonian
. A clear boundary can be seen at
. Here we choose
[66].
固定系统参数
,
和
时
随
的变化: (a)不同的颜色对应不同的
值, 这里的初态是初始哈密顿量的基态; (b) 选取不同的初态,
表示初始哈密顿量的第n 个本征态. 在
处, 可以清晰地看到一个相边界. 这里固定
[66]
Fig. 10. (a) MIPR as a function of the incommensurate potential strength
at two p-wave pairing strength
and
. Here use
; (b) phase diagram of this system with a p-wave pairing strength
and incommensurate potential strength
. I: extended phase, II: critical phase and III: localized phase. Here fix
.
(a) 固定两个p波配对强度
和
时，MIPR随准周期势强度
的变化，这里用的系统尺寸是
; (b) 系统随p波配对强度
和准周期势强度
变化的相图，I:扩展相，II:临界相，III:局域相. 这里固定
Fig. 11. (a) Energy spectra of this system with
and
under OBC. The distributions of
(b) and
(c) for the lowest excitation with different
[16].
(a) 在开边界条件下, 固定
和
时系统的能谱; (b), (c)不同的
值时最低激发模的
((b))和
((c))的分布[16]
Fig. 12. IPR((a))and MIPR((b)) as a function of
and
with fixed
; IPR((c)) and MIPR((d)) as a function of
and
with fixed
; IPR((e)) and MIPR((f)) as a function of
and
with fixed
. The lattice size is
and
[91].
第N 个本征态的IPR((a))和MIPR((b))随
和
的变化, 这里固定
; 第N 个本征态的IPR((c))和MIPR((d))随
和
的变化, 这里固定
; 第N 个本征态的IPR((e))和MIPR((f))作为
和
的函数, 这里固定
. 其他参数是
和
[91]
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