The Heisenberg uncertainty principle is one of the most important ideals in quantum mechanics.^[1] It is often expressed by no-go statements.^[2] In a general expression, two noncommuting observables do not simultaneously have well-defined values. This property has no classical analogue, so it is a basic difference between classical and quantum mechanics. As is well known, the principle plays a fundamental role in quantum theory, such as nonlocality,^[3] quantum measurements,^[4] and quantum phase transitions.^[5] Furthermore, it has important applications in current quantum technology, for instance, entanglement detection,^[6] quantum cryptography,^[7] and quantum metrology.^[8]

In fact, the principle was first proposed by Heisenberg,^[1] and originally formulated precisely by Kennard^[9] as ΔxΔp ≥ ℏ/2, where the uncertainties are quantified by standard deviations of the position and momentum. Subsequently, Robertson^[10] generalized it to arbitrary pairs of noncommuting observables A and B, which is given by ΔAΔB ≥ 〈Ψ|[A,B]|Ψ〉/2 in state |Ψ〉. Just recently, several uncertainty relations with respect to the clock operator U and shift operator V were derived.^[11] The expressions of U and V are given by^[11–14] 1 2 where [Z] is the integer part of Z. Here, and are complete orthonormal bases in d-dimensional Hilbert space, respectively. The two bases are related by the discrete Fourier transform (DFT) Based on the DFT, the operators U and V can be rewritten as and . In fact, the expressions of U and V in Eqs. (B2) and (B4) of Ref. [11] are originally from Eqs. (5) and (7) of Ref. [14]. They should be the same. However, they are not completely the same. We have checked them and found that there exist typo errors in Eqs. (B2) and (B4) of Ref. [11]. Physically, the operators U and V are the non-Hermitian generalization of the traditional 2-dimensional Pauli operators to higher-dimensional ones.^[12,13] The definitions of clock operator U and shift operator V are independent of the Hamiltonian, so it can be applied to general quantum systems.

The uncertainties of U and V are respectively given by^[11,14] 3 4 where |Ψ〉 is the wave function of a quantum state and 0 ≤ ΔU² ≤ 1 (0 ≤ ΔV² ≤ 1). One of the tighter uncertainty relations that they obey is^[11] 5 where Δ⁽³⁾ = 〈Ψ|Ψ_U〉〈Ψ_U|Ψ_V〉〈Ψ_V|Ψ〉, |Ψ_U〉 = U|Ψ〉, |Ψ_V〉 = V|Ψ〉 and Φ = arg Δ⁽³⁾. We define the right side in the inequation is the lower bound 6

On the other hand, Massar and Spindel pointed out that the uncertainties ΔU² and ΔV² can measure the localization properties of quantum states.^[14] However, they do not give the corresponding explicit relation. Besides, Bagchi and Pati pointed out that LB depends on quantum states.^[11] However, they did not study the relation between the values of LB and state localization properties. In this paper, we fill the gaps just mentioned. In the Anderson localization phenomenon,^[15–20] a state |Ψ〉 may be localized, critical or delocalized, which depend on models and model parameters. There are many typical and extensively studied one-dimensional models, such as the slowly varying potential model,^[21–23] the deterministic nearest-neighbor tight-binding models with exact mobility edges,^[24] and the random-dimer potential model.^[25–27] It is interesting how these quantities, including ΔU² and ΔV² and LB, are related to Anderson localization.

The rest of the paper is organized as follows. In Section 2, we introduce the tight-binding Hamiltonian for a single electron in one-dimensional (1D) lattice systems. In Section 3, we analyze the behaviors of ΔU² and ΔV², and LB in three typical models. Finally, we give a summary of our main results.

A single electron in 1D lattice systems is considered. The corresponding tight-binding Hamiltonian is described by 7 where ε_j is the on-site potential on the jth site, t is a nearest-neighbor hopping integral, and d is the lattice size. Further, , where is the creation operator of jth site and |0〉 is the vacuum. Models are specified by the choice rules of {ε_j}. The system can also be described by the tight-binding equation 8 The general wave function, i.e., eigenstate |Ψ_β〉 with eigenenergy E_β for Hamiltonian in Eq. (7) can be written as the superposition 9 where is the amplitude of the βth wave function at jth site.

In numerical calculations, for different models, we directly diagonalize the tight-binding equation (8) with the periodic boundary condition at finite system sizes and obtain all eigenenergies E_β and corresponding eigenstates |Ψ_β〉. From Eqs. (3), (4), and (6), we obtain ΔU² and ΔV², as well as LB. For simplicity and convenience, based on Eqs. (3) and (4), we define a reduced relative uncertainty 10

3.1. Slowly varying potential model

For the slowly varying potential (SVP) model,^[21–23] the on-site potential in Eq. (7) can be written as 11 Here, λ is the potential strength and α is a real number; φ is the original phase, which is redundant in localization. It is found that for 0 < υ < 1, there are two mobility edges (MEs) at E_c = ± (2.0 − λ)t provided that λ < 2.0t. Extended states are in the middle of the band (|E_β| < |E_c|). Localized states are at the band edges (|E_β| > |E_c|); states are more localized as eigenenergies E_β are nearer band edges. At MEs, states are critical.

We take λ = 1.0t as an example. At the same time, πα = 0.2 and υ = 0.7 are set, which are the same as that in Refs. [21]–[23]. In this case, two MEs are E_c = ±t. We denote and , respectively. States are extended for |E_β| < t and localized for |E_β| > t. Figure 1(a) shows that at extended states, uncertainties ΔU²s are near ones. At localized states, ΔU²s decrease as states vary from MEs to band edges; at the same eigenenergies E_β, the larger lattice sizes, the smaller ΔU²s are. There are sharp transitions in ΔU²s at MEs, where states are critical. Thus ΔU²s can well characterize the localization properties of the three different kinds of states in position space. At the same time, figure 1(b) shows that on the whole, uncertainties ΔV²s are decreasing as states vary from band center to band edges. There is a deep pit in ΔV²s when eigenenergies E_β are near each ME. The values of ΔV² are almost independent of the lattice sizes Ns. Fortunately, figure 1(c) shows that the reduced relative uncertainty U_r > 0 for extended states and U_r < 0 for localized ones. For larger lattice size N, U_r almost equals to zero for critical states (seen in inset). It means that uncertainties ΔU²s are greater than, equal to, and less than uncertainties ΔV²s for extended, critical, and localized states, respectively. Therefore, different kinds of states can be distinguished from each other with the comparisons of ΔU² and ΔV².

Fig. 1.

Figure Option ViewDownloadNew Window

Fig. 1. (color online) For the slowly varying potential model, (a) ΔU2, (b) ΔV2, (c) Ur, (d) LB, and (e) IPR as functions of eigenenergies Eβ, respectively. Here, λ = 1.0t, πα = 0.2, and υ = 0.7; the vertical dashed lines are for the functions and , respectively; the horizontal dashed line in panel (c) is for Ur = 0; lattice sizes d = 1000, 2000, and 10000, respectively. Inset in panel (c): partial enlarger for Eβ near .

Figure 1(d) shows that for extended states, LBs are near ones. For localized states, on the whole, the more localized the states are, the smaller the LBs are. There are rapid transitions in LBs as states move from the extended into the localized regime. A deep pit is in LBs when eigenenergies E_β are near each ME, so LBs depend on states, and different kinds of states can also be distinguished from each other based on LB.

Furthermore, compared to our results intuitively, we also calculate the inverse participation ratio , which is a traditional and often used quantity to characterize localization properties of states. Roughly, it is proportional to the localization length of states.^[28] Figure 1(e) shows that for extended states, IPRs are relatively large. For localized states, IPRs are relatively small, and at a fixed E_β, the larger the d is, the smaller IPR is. On the whole, all our studied quantities are in good correspondence with IPR.

3.2. Deterministic nearest-neighbor tight-binding models with exact mobility edges

One family of the models are specified by on-site potentials^[24] 12 The on-site potential is a smooth function of in the open interval (−1,1). Each value of corresponds to a different tight-binding model. is the original phase, which is redundant in localization. For a quasiperiodic modulation, b is set to be irrational. There exist self-duality points under a generalized duality transformation.^[24] Based on it, the analytical critical condition about MEs when is 13 where sgn(x) is the sign function.

In Fig. 2, we take the parameters of λ = −0.9t and as examples. From Eq. (15), an ME E_c = t/3. In this case, all states are localized when eigenenergies E_β > E_c, and almost all states, except some states at subband edges, are extended when E_β < E_c. In calculations, the irrational number b in Eq. (14) is approximated by the ratio of successive Fibonacci numbers: F_j = F_{j − 1} + F_{j − 2}, with F₀ = F₁ = 1. In this way, choosing b = F_{j − 1}/F_j and lattice size d = F_j, we can obtain the periodic approximant for the quasiperiodic potential.

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. (color online) For the deterministic nearest neighbor tight binding models with exact mobility edges, (a) ΔU2, (b) ΔV2, (c) Ur, (d) LB, and (e) IPR as functions of eigenenergies Eβ, respectively. Here, λ = −0.9t and ; the vertical dashed line is for the function Eβ = Ec = t/3; the horizontal dashed line in panel (c) is for Ur = 0; lattice sizes d = 987, 2584, and 10946, respectively.

Similar to that for the SVP model, figure 2 shows that for extended states, uncertainties ΔU²s are near ones, the reduced relative uncertainty U_r > 0, and LBs are near ones. For localized states, ΔU²s are near zeros, U_r < 0, LBs are relatively small. There are good correspondences for them with IPRs. So all these quantities can characterize the localization properties of different kinds of states in the model.

3.3. Random-dimer potential model

In the random-dimer potential (RDP) model,^[25–27] the on-site potential in Eq. (7) is given by 14 where V_a and V_b are dimer energies. The potential is random but with short correlations, which is different from potentials in the above two models where potentials are determined (without disorder). It was found^[25] that delocalized (extended) states exist when |V_a − V_b| < 2.0t and there are only localized states when |V_a − V_b| > 2.0t. At finite lattice sizes d, the number of delocalized states is roughly proportional to ; for infinite lattice sizes, there are only states with eigenenergy E_β = V_a(V_b) are delocalized.^[26]

In Fig. 3, we take V_a = −V_b = −0.75t as examples. At the same time, we choose q = 0.5, which corresponds to the most random situation. In this case, states with eigenenergies E_β near V_a(V_b) are delocalized. The results of Fig. 3 are obtained for average of 500, 250, and 50 samples for lattice sizes d = 1000, 2000, and 10000, respectively. The averages with more samples give the same results.

Fig. 3.

	Figure Option ViewDownloadNew Window
	Fig. 3. (color online) For the random-dimer potential model, (a) ΔU2, (b) ΔV2, (c) Ur, (d) LB, and (e) IPR as functions of eigenenergies Eβ, respectively. Here, Va = −Vb = −0.75t; the vertical dashed lines are for the functions Eβ = Va and Eβ = Vb, respectively; the horizontal dashed line in panel (c) is for Ur = 0; lattice sizes d = 1000, 2000, and 10000, respectively.

Similar to that for the SVP model and the deterministic nearest-neighbor tight-binding models with exact mobility edges, figure 3 shows that for states with eigenenergies E_β very near V_a(V_b) (delocalized states), ΔU²s are near ones, U_r > 0, LBs are also near ones. For other states (localized ones), at the same E_β, ΔU²s decease with lattice sizes Ns; U_r < 0 and LBs are relatively small. There are good correspondences between these quantities and IPRs. Thus, they can reflect the localization properties of different kinds of states in the model.

It is known that the operators U and V are the non-Hermitian generalization of the Pauli operators from two dimensions to higher dimensions. The simplest higher-dimensional quantum states may be states for an electron in lattice systems. In the three studied models, we found that ΔU² can well characterize state localization properties. The reduced relative uncertainty U_r can also do so. At the same time, Resta proposed and used a U-related quantity to study many-body localization.^[29] Here, we found that U itself can well reflect the localization properties of single electron lattice systems. Besides, Bagchi and Pati pointed out that LB [Eq. (6)] depends on states.^[11] Here, we give further results that LBs are relatively large for extended states and relatively small for localized states. Therefore, all these studies present more pictures than existing works.

For the three studied models, we found that uncertainties ΔU² > ΔV², ΔU² = ΔV², and ΔU² < Δ V² for extended, critical, and localized states, respectively. A similar result has been found by contrast of Shannon information entropies of states for an electron in position and momentum space,^[30] or by contrast of the distinguishability and visibility for an electron in lattice systems.^[31] Thus, these quantities can be used as novel indexes to characterize Anderson localization.