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Necessity of orthogonal basis vectors for the two-anyon problem in a one-dimensional lattice*

  • Cuicui Zheng ,
  • Jiahui Xie ,
  • Ming Zhang ,
  • Yajiang Chen , ** ,
  • Yunbo Zhang , **
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  • Zhejiang Key Laboratory of Quantum State Control and Optical Field Manipulation, Department of Physics, Zhejiang Sci-Tech University, Hangzhou 310018, China

**Authors to whom any correspondence should be addressed.

Received date: 2024-07-24

  Revised date: 2024-08-22

  Accepted date: 2024-08-26

  Online published: 2024-10-22

Supported by

*National Science Foundation of China(12074340)

National Science Foundation of China(12474492)

Science Foundation of Zhejiang Sci-Tech University(20062098-Y)

Science Foundation of Zhejiang Sci-Tech University(19062463-Y)

Science Foundation of Zhejiang Sci-Tech University(22062344-Y)

Copyright

© 2024 Institute of Theoretical Physics CAS, Chinese Physical Society and IOP Publishing

Abstract

Few-body physics for anyons has been intensively studied within the anyon-Hubbard model, including the quantum walk and Bloch oscillations of two-anyon states. Recently, theoretical and experimental simulations of two-anyon states in a one-dimensional lattice have been carried out by expanding the wavefunction in terms of non-orthogonal basis vectors, resulting in non-physical degrees of freedom. In the present work, we deduce finite difference equations for the two-anyon state in a one-dimensional lattice by solving the Schrödinger equation with orthogonal and complete basis vectors. Such an orthogonal scheme gives all the orthogonal physical eigenstates, while the conventional (non-orthogonal) method produces many non-physical redundant eigensolutions whose components violate the anyonic commutation relations. The dynamical property of the two-anyon states in a sufficiently large lattice is investigated and compared in both the orthogonal and conventional schemes. For initial states with two anyons at the same site or two (next-)neighboring sites, we observe the same dynamical behavior in both schemes, including the revival probability, probability density function and two-body correlation. For other initial states, the conventional scheme produces erroneous states that no longer obey the anyonic relations. The period of Bloch oscillations in the pseudo-fermionic limit has been found to be twice that in the bosonic limit, while these oscillations disappear at other statistical parameters. Our findings are vital for quantum simulations of few-body anyonic physics in lattice models.

Cite this article

Cuicui Zheng , Jiahui Xie , Ming Zhang , Yajiang Chen , Yunbo Zhang . Necessity of orthogonal basis vectors for the two-anyon problem in a one-dimensional lattice*[J]. Communications in Theoretical Physics, 2024 , 76(12) : 125103 . DOI: 10.1088/1572-9494/ad7372

1. Introduction

Over the years, exotic anyonic systems have continuously stimulated the research interests of theoretical and experimental physicists. As a type of identical particle in low dimensions, anyons obey fractional statistics, i.e. the wave function acquires a phase factor when exchanging two anyons [14]. They often arise as quasi-particle excitations in two-dimensional condensed-matter systems, for example the fractional quantum Hall state [511] and topologically nontrivial chiral spin liquids [12, 13], or as virtual particles simulated in one-dimensional systems [1425] and even in arbitrary dimensions [26]. In particular, non-Abelian anyons have been constructed on a trapped-ion processor [27], which is a promising building block for a fault-tolerant topological quantum computer [2830].
Due to the rapid advancement of controllable anyonic platforms [3134], few-body physics for Abelian anyons has been intensively studied within the anyon-Hubbard model, including the quantum walk [3538] and Bloch oscillations (BOs) of two-anyon states [3941]. The quantum walk behavior of identical particles reflects the effects of the exchange statistics [42] and can be a promising candidate for realizing quantum computing [43, 44]. A two-anyon quantum walk exhibits a statistics-dependent bunching/anti-bunching phenomenon in the two-body correlation functions [35] and spatially asymmetric transport [37, 45]. Even a statistical boundary, which is transparent to single anyons and the exchange of multiple particles arriving together, has been introduced to engineer the two-anyon quantum walk [38]. Under a linear external potential, the energy spectrum of ‘bound' two-anyon states in a sufficiently large lattice can have a uniform energy interval between neighboring states for both non-interacting bosons and pseudo-fermions [31, 37], and the energy interval for pseudo-fermions is half that for bosons, which leads to the same relation between the frequencies of their BOs [39]. Such anyonic BOs have been observed in the simulators using a photonic lattice [40] and electric circuits [41].
Theoretical studies on the two-anyon quantum walk [35, 37, 38] have ignored the energy spectrum of the system and only focused on the dynamical properties by solving the time-dependent Schrödinger equation based on the fractional Jordan–Wigner transformation [31, 46], which maps anyons to interacting bosons. Information on the energy spectrum of few-body anyonic systems can be crucial in many circumstances, for example anyonic BOs [39], where another routine to obtain the dynamic behavior of few-body anyonic systems has employed the commutation relations of the anyonic creation and annihilation operators and directly solved the time-dependent two-anyonic Schrödinger equation. However, the wave functions of the two-anyon states in [39] have been expanded by a set of non-orthogonal basis vectors in the Fock space, which introduces extra non-physical degrees of freedom into the systems, for example non-physical eigenstates, and may in some cases result in some erroneous dynamical features. More seriously, this routine has been followed in the important experimental simulations [40, 41]. Therefore, it is necessary and urgent to provide a rectification for the anyonic few-body physics in the anyon-Hubbard model.
In this work, we obtain the finite difference equations for the two-anyon state in a one-dimensional (1D) lattice by solving the Schrödinger equation with orthogonal basis vectors, which is vital for various anyonic quantum simulations. Our orthogonal scheme produces all the orthogonal physical eigenstates, while the conventional (non-orthogonal) method leads to a lot of extra non-physical eigensolutions. Both schemes show that the two-anyon state in a sufficiently large lattice exhibits anyonic BOs in the time-dependent revival probability and probability density function. However, for non-anyonic initial states, the conventional scheme gives an incorrect prediction of the BOs, leading to a probability density distribution that violates the conservation law in the fermionic limit.
This paper is organized as follows: in section 2 we introduce the 1D anyon-Hubbard model with two-anyon states described in both conventional and orthogonal spaces; section 3 shows the results and discussions for stationary two-anyon states, while their dynamics features are given in section 4; section 5 concludes the paper.

2. Model

We consider a pair of non-interacting anyons hopping on a one-dimensional (1D) lattice under a uniform external force F. The corresponding Hamiltonian is given as [39, 41]
$\begin{eqnarray}H=-J\displaystyle \sum _{l=1}^{L}({a}_{l}^{\dagger }{a}_{l+1}+{a}_{l+1}^{\dagger }{a}_{l})+F\displaystyle \sum _{l=1}^{L}l\,{n}_{l},\end{eqnarray}$
where the operator al (${a}_{l}^{\dagger }$) annihilates (creates) one anyon at site l, ${n}_{l}={a}_{l}^{\dagger }{a}_{l}$ denotes the particle number operator, L is the number of lattice sites and J is the tunneling amplitude between the nearest-neighbor sites. The anyonic operators satisfy the following commutation relations [39, 4749]:
$\begin{eqnarray}\begin{array}{rcl}{a}_{l}{a}_{k}^{\dagger } & = & {{\rm{e}}}^{-{\rm{i}}\chi \pi \epsilon (l-k)}{a}_{k}^{\dagger }{a}_{l}+{\delta }_{{lk}},\\ {a}_{l}{a}_{k} & = & {{\rm{e}}}^{{\rm{i}}\chi \pi \epsilon (l-k)}{a}_{k}{a}_{l},\\ {a}_{l}^{\dagger }{a}_{k}^{\dagger } & = & {{\rm{e}}}^{{\rm{i}}\chi \pi \epsilon (l-k)}{a}_{k}^{\dagger }{a}_{l}^{\dagger },\end{array}\end{eqnarray}$
where χ ∈ [0, 1] is the anyonic statistical parameter and the sign function ε(x) gives 1, 0 or −1 depending on whether x is positive, zero or negative. Two anyons with χ = 0 behave as conventional bosons, while those with χ = 1 are pseudo-fermions which act like fermions off-site and bosons on-site.
The analysis of the eigenenergy spectrum and the BO dynamical motion of two correlated anyons greatly depends on the choice of the basis vectors of the Hilbert space. Since [N, H] = 0, with N being the total particle number N = ∑lnl, the system evolves in the two-particle Hilbert space. One conventional choice, employed in [39, 41], is the Fock states with one anyon at site l and the other at site k of the lattice, i.e. ${a}_{l}^{\dagger }{a}_{k}^{\dagger }\left|0\right\rangle $ with $\left|0\right\rangle $ the vacuum state and l, k = 1, 2, … , L. The eigenstates of the Hamiltonian equation (1) are expanded as
$\begin{eqnarray}\left|\psi \right\rangle =\displaystyle \sum _{l,k}\displaystyle \frac{1}{\sqrt{2}}{c}_{{lk}}{a}_{l}^{\dagger }{a}_{k}^{\dagger }\left|0\right\rangle ,\end{eqnarray}$
with clk the probability amplitude of two anyons occupying the corresponding Fock state. Note that no restrictions are put on the site labels l and k, which means that they can independently go through from 1 to L. In this case, the dimension of the Hilbert space is L2. Substituting equations (1) and (3) into the stationary Schrödinger equation $H\left|\psi \right\rangle =E\left|\psi \right\rangle $, we obtain the eigenequation with respect to the coefficients clk as
$\begin{eqnarray}\begin{array}{rcl}{{Ec}}_{{lk}} & = & -J[{{\rm{e}}}^{{\rm{i}}\chi \pi ({\delta }_{{lk}}+{\delta }_{l(k-1)})}{c}_{l(k-1)}+{{\rm{e}}}^{-{\rm{i}}\chi \pi ({\delta }_{{lk}}+{\delta }_{l(k+1)})}{c}_{l(k+1)}\\ & & +{c}_{(l-1)k}+{c}_{(l+1)k}]+F(l+k){c}_{{lk}},\end{array}\end{eqnarray}$
which is exactly the relation equation (4) in [41]. In this conventional scheme of basis vectors, the probability amplitudes clk form a matrix, as shown in figure 1(b), and clk and ckl are related to each other as follows:
$\begin{eqnarray}{c}_{{lk}}={c}_{{kl}}{{\rm{e}}}^{{\rm{i}}\chi \pi \epsilon (k-l)},\end{eqnarray}$
due to the commutation relation (2).
Figure 1. (a) Schematic mapping of the one-dimensional two-anyon system into single-particle motion on a two-dimensional lattice in the presence of an external force, according to equations (4) and (8). The entire square lattice constitutes the conventional Hilbert space, with each site being a probability amplitude clk, while the folded triangular area in light green represents the orthogonal Hilbert space with lk. The gray gradient of the sites shows the on-site potential energy given by the external force, and the left/right arrows correspond to jumps with complex phases e±iχπ. Panels (b) and (c) show the expansion coefficients in equations (3) and (6) for conventional and orthogonal Hilbert spaces with dimensions L2 and L(L + 1)/2, respectively. The colored stars indicate three initial states that we choose for the study of dynamical properties.
Since the Fock states ${a}_{l}^{\dagger }{a}_{k}^{\dagger }\left|0\right\rangle $ and ${a}_{k}^{\dagger }{a}_{l}^{\dagger }\left|0\right\rangle $ are linearly dependent, it is not appropriate to simultaneously have both of them serving as basis vectors, which are typically linearly independent state vectors and form a complete set for the state space. Thus an alternative and more intuitive choice for the basis vectors is the Fock states with one anyon at site l and the other one at site k of the lattice, but with the restriction of lk. It can be easily shown that these new basis vectors, ${(1+{\delta }_{{lk}})}^{-\tfrac{1}{2}}{a}_{l}^{\dagger }{a}_{k}^{\dagger }\left|0\right\rangle $ for lk, are orthogonal, normal and complete as required by quantum theory. So, the dimension of the Hilbert space, i.e. the number of expansion coefficients, is reduced to L(L + 1)/2. The eigenstates of the Hamiltonian equation (1) are expanded in this orthogonal scheme of basis vectors as
$\begin{eqnarray}\left|\psi \right\rangle =\displaystyle \sum _{l\geqslant k}\displaystyle \frac{1}{\sqrt{1+{\delta }_{{lk}}}}c{{\prime} }_{{lk}}{a}_{l}^{\dagger }{a}_{k}^{\dagger }\left|0\right\rangle .\end{eqnarray}$
The normalizations of the two-anyon states ∣ψ⟩ in equations (3) and (6) require that the coefficients c and $c^{\prime} $ should satisfy the following relation:
$\begin{eqnarray}\displaystyle \sum _{l,k}| {c}_{{lk}}{| }^{2}=\displaystyle \sum _{l\geqslant k}| c{{\prime} }_{{lk}}{| }^{2}=1.\end{eqnarray}$
The eigenequation for the coefficients $c{{\prime} }_{{lk}}$ is obtained from the stationary Schrödinger equation in just the same way as for equation (4)
$\begin{eqnarray}\begin{array}{rcl}{Ec}{{\prime} }_{{lk}} & = & -J\left[\sqrt{1+{\delta }_{{lk}}}{{\rm{e}}}^{{\rm{i}}\chi \pi {\delta }_{{lk}}}c{{\prime} }_{l(k-1)}\right.\\ & & +(1-{\delta }_{{lk}})\sqrt{1+{\delta }_{l\left(k+1\right)}}{{\rm{e}}}^{-{\rm{i}}\chi \pi {\delta }_{l(k+1)}}c{{\prime} }_{l(k+1)}\\ & & +(1-{\delta }_{{lk}})\sqrt{1+{\delta }_{\left(l-1\right)k}}c{{\prime} }_{(l-1)k}\\ & & \left.+\sqrt{1+{\delta }_{{lk}}}c{{\prime} }_{(l+1)k}\right]+F(l+k)c{{\prime} }_{{lk}}.\end{array}\end{eqnarray}$
We illustrate schematically the difference in and connection between the structure of two two-dimensional (2D) Hilbert spaces, the conventional one with coefficients c and the orthogonal one with coefficients $c^{\prime} $, as shown in figure 1. Here, the open boundary condition is applied, i.e. hopping from 1 (L) to 0 (L + 1) is forbidden. Clearly, the orthogonal space in the triangular area happens to be the conventional one in the square area folded along the diagonal elements. The eigenstate in both schemes is written as a column vector of coefficients c or $c^{\prime} $ in an arranged order (see below for examples). The eigenequations for two anyons on a 1D lattice, equations (4) and (8), which equivalently describe a single particle jumping between the nearest neighbors on a 2D lattice, have exactly the same form in these two schemes, with the only difference being the phases and delta functions in front of the coefficients. For instance, the probability amplitude with one anyon at site l and the other at site k is indicated by a lattice site (l, k) in two dimensions. Note that in the conventional basis vector where the lattice sites l and k are unrestricted, the result of this 1D-to-2D mapping covers the entire 2D square lattice. However, in the orthogonal basis vector where they are restricted, i.e. lk, one is left with only the lower left triangular region of the 2D square lattice. Hopping occurs only between the nearest-neighbor sites, that is, the coefficients clk can only jump to their four nearest-neighbor sites. The normal jumps are indicated by ordinary bars, while those with complex phases are denoted by arrows. It is found that only those jumps in and out of the diagonal elements along the horizontal k direction, i.e. (l, l) → (l, l ± 1), carry complex phases e∓iχπ (shown as solid/hollow arrows in figure 1(a)). Note that only one of the two cases of non-zero complex phases in the conventional jumps is preserved in the orthogonal jumps, as can be seen from the remaining Kronecker δ in the phases in the first two terms of equation (8). The site-dependent potential is manifested by the increasing darkening of the lattice site along the diagonal elements, which is the effect of the external force.

3. The structure of the energy spectrum

The eigenequations (4) and (8) are well understood in the language of matrix mechanics by expressing the eigenstates in column vectors. The probability amplitudes clk are arranged in the column vector in the order of left to right and top to bottom, in both schemes. We first take a small-sized system of L = 3 in the absence of external force F = 0 as an example, for which the eigenvector and the Hamiltonian in the conventional Hilbert space of dimension 32 read as
$\begin{eqnarray}| \psi \rangle ={\left({c}_{11},{c}_{12},{c}_{13},{c}_{21},{c}_{22},{c}_{23},{c}_{31},{c}_{32},{c}_{33}\right)}^{{\rm{T}}},\end{eqnarray}$
and
$\begin{eqnarray}{H}_{(L=3)}=-J\left(\begin{array}{ccccccccc}0 & {{\rm{e}}}^{-{\rm{i}}\chi \pi } & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ {{\rm{e}}}^{{\rm{i}}\chi \pi } & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & {{\rm{e}}}^{-{\rm{i}}\chi \pi } & 0 & 1 & 0 & 0\\ 0 & 1 & 0 & {{\rm{e}}}^{{\rm{i}}\chi \pi } & 0 & {{\rm{e}}}^{-{\rm{i}}\chi \pi } & 0 & 1 & 0\\ 0 & 0 & 1 & 0 & {{\rm{e}}}^{{\rm{i}}\chi \pi } & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & {{\rm{e}}}^{-{\rm{i}}\chi \pi }\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & {{\rm{e}}}^{{\rm{i}}\chi \pi } & 0\end{array}\right),\end{eqnarray}$
and in the orthogonal space of dimension 3 × (3 + 1)/2 as
$\begin{eqnarray}| \psi \rangle ={\left(c{{\prime} }_{11},c{{\prime} }_{21},c{{\prime} }_{22},c{{\prime} }_{31},c{{\prime} }_{32},c{{\prime} }_{33}\right)}^{{\rm{T}}},\end{eqnarray}$
and
$\begin{eqnarray}{H}_{(L=3)}^{{\prime} }=-J\left(\begin{array}{cccccc}0 & \sqrt{2} & 0 & 0 & 0 & 0\\ \sqrt{2} & 0 & \sqrt{2}{{\rm{e}}}^{-{\rm{i}}\chi \pi } & 1 & 0 & 0\\ 0 & \sqrt{2}{{\rm{e}}}^{{\rm{i}}\chi \pi } & 0 & 0 & \sqrt{2} & 0\\ 0 & 1 & 0 & 0 & 1 & 0\\ 0 & 0 & \sqrt{2} & 1 & 0 & \sqrt{2}{{\rm{e}}}^{-{\rm{i}}\chi \pi }\\ 0 & 0 & 0 & 0 & \sqrt{2}{{\rm{e}}}^{{\rm{i}}\chi \pi } & 0\end{array}\right),\end{eqnarray}$
respectively. Note that the Hamiltonian matrices in both spaces are Hermitian, which ensures the reality of the eigenvalues and the orthogonality of the eigenvectors. While the high-order algebraic equation hinders us from providing an analytical expression for the eigenvalues of a 9 × 9 matrix, luckily the Hamiltonian in the orthogonal space is analytically solvable in the absence of external force and the six eigenvalues for arbitrary χ are given by
$\begin{eqnarray}{E}_{\mathrm{1,2}}=-\sqrt{5\pm \sqrt{5+4\cos \chi \pi }},\end{eqnarray}$
$\begin{eqnarray}{E}_{\mathrm{3,4}}=0,\end{eqnarray}$
$\begin{eqnarray}{E}_{\mathrm{5,6}}=\sqrt{5\mp \sqrt{5+4\cos \chi \pi }},\end{eqnarray}$
each pair of which is negative, zero and positive, respectively. We compare these solutions with the numerically obtained eigenvalues in nine-dimensional space in figure 2. Three additional eigenvalues emerge in the conventional scheme of the eigenequation, which are all χ-independent, as illustrated by the blue horizontal lines. The zero eigenvalue is degenerate with the other two zero eigenvalues (E3,4) in the whole range of χ, and the other two merge into E2 and E5 in the bosonic limit χ = 0. To understand more clearly the nature of these statistically independent eigenstates, we pay attention to the eigenvalues and their corresponding eigenvectors in the pseudo-Fermion limit χ = 1, which are listed in table 1. For physical states with eigenvalues $\pm \sqrt{6},\pm 2,{0}_{1},{0}_{2}$, the coefficients in the 9-vector $\left|\psi \right\rangle $ satisfy the relation required by the anyonic commutation relation, that is clk = − ckl from equation (5).
Figure 2. The energy spectrum for the two-particle anyon-Hubbard Hamiltonian versus statistical parameter χ calculated numerically in the conventional Hilbert space (blue dots) and analytical results (13) in the orthogonal space (red lines) with L = 3 and F/J = 0. Out of the nine eigenvalues for each χ three are non-physical with the coefficients violating the anyonic relation (5). The energies are in units of J.
Table 1. Eigenvalues and corresponding eigenvectors in conventional and orthogonal Hilbert spaces with L = 3 and F = 0 for χ = 1.
Eigenvalues 9-vector ∣ψ 6-vector ∣ψ
$-\sqrt{6}$ ${\left(\tfrac{1}{2\sqrt{3}},-\tfrac{1}{2\sqrt{2}},0,\tfrac{1}{2\sqrt{2}},-\tfrac{1}{\sqrt{3}},\tfrac{1}{2\sqrt{2}},0,-\tfrac{1}{2\sqrt{2}},\tfrac{1}{2\sqrt{3}}\right)}^{{\rm{T}}}$ ${\left(\tfrac{1}{2\sqrt{3}},\tfrac{1}{2},-\tfrac{1}{\sqrt{3}},0,-\tfrac{1}{2},\tfrac{1}{2\sqrt{3}}\right)}^{{\rm{T}}}$
$\sqrt{6}$ ${\left(\tfrac{1}{2\sqrt{3}},\tfrac{1}{2\sqrt{2}},0,-\tfrac{1}{2\sqrt{2}},-\tfrac{1}{\sqrt{3}},-\tfrac{1}{2\sqrt{2}},0,\tfrac{1}{2\sqrt{2}},\tfrac{1}{2\sqrt{3}}\right)}^{{\rm{T}}}$ ${\left(\tfrac{1}{2\sqrt{3}},-\tfrac{1}{2},-\tfrac{1}{\sqrt{3}},0,\tfrac{1}{2},\tfrac{1}{2\sqrt{3}}\right)}^{{\rm{T}}}$
−2 $\tfrac{1}{2\sqrt{2}}{(-1,1,1,-1,0,1,-1,-1,1)}^{{\rm{T}}}$ $\tfrac{1}{2}{\left(-\tfrac{1}{\sqrt{2}},-\mathrm{1,0},-1,-1,\tfrac{1}{\sqrt{2}}\right)}^{{\rm{T}}}$
2 $\tfrac{1}{2\sqrt{2}}{(-1,-1,1,1,0,-1,-1,1,1)}^{{\rm{T}}}$ $\tfrac{1}{2}{\left(-\tfrac{1}{\sqrt{2}},1,0,-1,1,\tfrac{1}{\sqrt{2}}\right)}^{{\rm{T}}}$
01 $\tfrac{1}{\sqrt{3}}{(1,0,0,0,1,0,0,0,1)}^{{\rm{T}}}$ $\tfrac{1}{\sqrt{3}}{(1,0,1,0,0,1)}^{{\rm{T}}}$
02 $\tfrac{1}{2}{(-1,0,-1,0,0,0,1,0,1)}^{{\rm{T}}}$ $\tfrac{1}{2}{(-1,0,0,\sqrt{2},0,1)}^{{\rm{T}}}$

03 $\tfrac{1}{2}{(0,-1,0,-1,0,1,0,1,0)}^{{\rm{T}}}$
$-\sqrt{2}$ $\tfrac{1}{2}{\left(0,\tfrac{1}{\sqrt{2}},1,\tfrac{1}{\sqrt{2}},0,\tfrac{1}{\sqrt{2}},1,\tfrac{1}{\sqrt{2}},0\right)}^{{\rm{T}}}$
$\sqrt{2}$ $\tfrac{1}{2}{\left(0,\tfrac{1}{\sqrt{2}},-1,\tfrac{1}{\sqrt{2}},0,\tfrac{1}{\sqrt{2}},-1,\tfrac{1}{\sqrt{2}},0\right)}^{{\rm{T}}}$
For instance, the second and fourth amplitudes are opposite to each other, i.e. c12 = − c21. The relation still holds for the two vectors for zero eigenvalues 01 and 02, in which case c12 = –c21 = 0. This correctly related coefficient pair assures that we can fold the 9-vectors into 6-vectors in the Hilbert space, as listed in the last column in table 1. However, the other three eigenstates in the last three rows in table 1 violate this basic coefficient relation. We always find that clk = + ckl, i.e. the second and fourth amplitudes take equal values of $-1,1/\sqrt{2},1/\sqrt{2}$ in the 9-vectors for eigenvalues ${0}_{3},-\sqrt{2},\sqrt{2}$, respectively. This makes the folding of the 9-vectors into 6-vectors impossible. We conclude that these statistically independent eigenstates which violate the basic anyon coefficients relation are non-physical and redundant and should be excluded from the two-anyon Hilbert space in the related calculations.
To see how these non-physical eigenstates would change the structure of the energy spectrum of the two-anyon system, we consider a finite lattice size of L = 41 and numerically compute the eigenvalues and corresponding eigenvectors in the orthogonal space and compare them with those in conventional space, as shown in figures 3 and 4. This comparison is crucial in explaining the BO period of pseudo-fermions and clarifying the roles of χ and F in the onset of BOs in the probability densities and correlation functions. The external force leads to the on-site potential energy and we take two force parameters F/J = 0.3 and F/J = 1 as examples. To avoid finite-size effects, let us first pay attention to the middle energy range (−4F, 4F). For a better visual effect, we make some eigenvalue selections, and the eigenvalues are plotted in two different styles. Those whose eigenvectors show the largest overlap with the center of the lattice are plotted in blue solid dots in the conventional space and red open circles in the orthogonal spaces, respectively, i.e. we choose the largest coefficient in each eigenvector in such a way that both the distance between any two particles at sites l and k and that between each particle and the central lattice site are within a certain range Δr. Those out of this range are plotted in light gray dots and dark gray open circles. The equally spaced energy spectral structures as signs of BO can be manifested by choosing the appropriate range Δr for different force parameters F, for example Δr = 14 for F=0.3 and Δr = 8 for F = 1, respectively, in figures 3(a) and (b). Interestingly we adopt the same selection rules for the spectrum in the conventional and orthogonal schemes and find no difference in the middle energy range, i.e. in both schemes there exists this evenly spaced structure in the spectrum for χ = 1 and F/J = 0.3 with spacing F/2 rather than F. This suggests that the orthogonality of the basis vectors does not affect the BO phenomenon for a large enough lattice size. Note that there exist a large number of redundant eigenstates in the middle energy range such as that with eigenvalue 03 in the L = 3 case, which are all degenerate with some specific normal states, for example those with eigenvalues 01, 02. These redundant non-physical states hidden in the high-fold degeneracy, however, will be excluded in the calculation of dynamical evolution in the next section. On the other hand, when we carried out the calculation in the full energy spectrum, a significant difference occurred at the edge of the spectrum between the two schemes, where a large number of non-physical eigenvalues from the conventional diagonalization scheme persist with the anyonic coefficients relation broken in the eigenvectors. We show in figure 4 the numerical results of the edge range of the energy spectrum for L = 41 with F/J = 0.3 and F/J = 1. Note that the selection rule has not been applied to the energy levels as the BO phenomenon does not appear in the upper or lower edges of the spectrum.
Figure 3. Comparison of the energy spectrum of the two-anyon lattice system for L = 41 versus the statistical parameter χ for (a) F/J = 0.3, and (b) F/J = 1, numerically computed from the two choices of basis vectors. For clarity the energies in the range (4F, 4F) are shown in units of F. To avoid finite-size effects, we kept only those eigenvalues whose eigenvectors showed the largest overlap with the center of the lattice, where the gray dots are the energy values of this criterion that were screened out.
Figure 4. The upper edge of the energy spectrum for L = 41 versus the statistical parameter χ with external forces (a) F/J = 0.3, and (b) F/J = 1.
One may wonder how many of these redundant non-physical states there are and how the number of these states increases with the lattice size. It is possible to classify these redundant states into two classes, those that never appear in our orthogonal scheme (Class I, denoted by pure blue dots in the spectrum) and those that are degenerate with the eigenvalues in the orthogonal scheme (Class II, denoted by blue dots and red circles overlapping lines). For instance, in the case of L = 3, the eigenvalue 03 of E = 0 belongs to Class II, while eigenvalues $\pm \sqrt{2}$ are Class I. We consider again the pseudo-fermion statistical limit of χ = 1 in which case it is easy to identify whether the redundant eigenvalues are degenerate with those of the physical states. By checking the opposite relation of any pair of coefficients on transposed positions, say c12 and c21, the redundant eigenvalues whose coefficients violate this relation are shown with the increasing lattice size in figure 5 for external forces F/J = 0, 0.3, 1, respectively. For clarity, we only show the Class I redundant eigenvalues on the right column and count the numbers of Class I and Class II eigenvalues on the left. Clearly, the larger lattice hosts more redundant eigenvalues according to L(L − 1)/2 and the ratio of redundant to orthogonal eigenvalues increases with the number of lattice sites L as (L − 1)/(L + 1) approaching unity. We observed that the distribution of these redundant eigenvalues is significantly influenced by the external force F. For F = 0 almost all redundant eigenvalues belong to Class I and are distributed evenly in the whole spectrum. As F increases, the Class I redundant states are expelled out of the middle energy range for larger lattice sizes. Their number tends to saturate to a lattice-size independent value and Class II eigenvalues shall dominate. These redundant eigenvalues will take part in the dynamical evolution of the system when one starts from an initial state whose coefficients do not meet the requirement for anyons, as discussed in the next section.
Figure 5. The number of redundant eigenvalues and the distribution of Class I redundant eigenvalues with increasing lattice size L for χ = 1 with external forces F/J = 0, 0.3 and 1 from top to bottom.

4. The dynamical behavior

The nonlocal feature of anyons may exhibit many-body effects in the time evolution of the system even in the absence of on-site interaction. Now we turn to the dynamics of the two correlated anyons in a 1D lattice and explore the time evolution of several observables such as the revival probability, the probability density and the correlation functions. With this aim, let us first introduce the time-dependent probability amplitude clk(t) of finding one anyon at site l and the other one at site k of the lattice at time t. The expansion of any state $\left|\psi (t)\right\rangle $ upon the basis vector takes the same form as equations (3) and (6) except that the coefficients clk(t) are now time-dependent. The evolution equations for the amplitudes clk(t), as obtained from the Schrödinger equation ${\rm{i}}{\partial }_{t}\left|\psi (t)\right\rangle =H\left|\psi (t)\right\rangle $ with = 1, have the same form as equations (4) and (8) by simply replacing the stationary terms on the left-hand side Eclk by the time derivative terms i∂tclk or ${\rm{i}}{\partial }_{t}c{{\prime} }_{{lk}}$, respectively.
We first introduce the revival probability, or the Loschmidt echo, which quantifies the probability of the time-evolved state in the initial configuration
$\begin{eqnarray}{P}_{\mathrm{rev}}(t)=| \langle \psi (0)| \psi (t)\rangle {| }^{2}.\end{eqnarray}$
The probability density function measures the density distribution on each site of the 1D lattice
$\begin{eqnarray}{P}_{q}(t)=\displaystyle \frac{1}{2}\langle \psi (t)| {n}_{q}| \psi (t)\rangle ,\end{eqnarray}$
for q = 1, 2,…,L. Here the factor 1/2 assures the total probability normalized to unity in the whole lattice. The correlation between the two anyons is described by the two-body correlation function in position space,
$\begin{eqnarray}{{\rm{\Gamma }}}_{{qr}}(t)=\langle \psi (t)| {a}_{q}^{\dagger }{a}_{r}^{\dagger }{a}_{r}{a}_{q}| \psi (t)\rangle ,\end{eqnarray}$
with q, r = 1, 2, …, L. The matrix element of which, Γqr(t), represents the probability of detecting one particle at site q and its twin particle at site r. In the above definitions, the initial state ∣ψ(0)⟩ is represented in a column vector of clk(0) or $c{{\prime} }_{{lk}}(0)$, the state vector ∣ψ(t)⟩ = e−iHtψ(0)⟩ at time t is expressed as a column vector of coefficients clk(t) or $c{{\prime} }_{{lk}}(t)$ (in order from left to right and from top to bottom, similar to the L = 3 case), and H is written as a matrix H(L) or $H{{\prime} }_{(L)}$ in the conventional space with dimension L2 or in the orthogonal space with dimension L(L + 1)/2, respectively.
These dynamical quantities can be calculated directly from the coefficients clk in the conventional scheme with results
$\begin{eqnarray}{P}_{\mathrm{rev}}(t)={\left|\displaystyle \sum _{l,k}{c}_{{lk}}(0){c}_{{lk}}(t)\right|}^{2},\end{eqnarray}$
for the revival probability,
$\begin{eqnarray}\begin{array}{rcl}{P}_{q}(t) & = & \displaystyle \frac{1}{4}\displaystyle \sum _{l}\left\{{| {c}_{{lq}}(t)| }^{2}+{| {c}_{{ql}}(t)| }^{2}\right.\\ & & \left.+\left[{c}_{{lq}}^{* }(t){c}_{{ql}}(t){{\rm{e}}}^{-{\rm{i}}\chi \pi \epsilon (l-q)}+{\rm{h}}.{\rm{c}}.\right]\right\},\end{array}\end{eqnarray}$
for the probability density function, and
$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Gamma }}}_{{qr}}(t) & = & \displaystyle \frac{1}{2}\left\{{| {c}_{{rq}}(t)| }^{2}+{| {c}_{{qr}}(t)| }^{2}\right.\\ & & \left.+\left[{c}_{{qr}}^{* }(t){c}_{{rq}}(t){{\rm{e}}}^{-{\rm{i}}\chi \pi \epsilon (q-r)}+{\rm{h}}.{\rm{c}}.\right]\right\},\end{array}\end{eqnarray}$
for the correlation function, respectively. In the orthogonal scheme, one has instead
$\begin{eqnarray}{P}_{\mathrm{rev}}(t)={\left|\displaystyle \sum _{l\geqslant k}c{{\prime} }_{{lk}}(0)c{{\prime} }_{{lk}}(t)\right|}^{2},\end{eqnarray}$
for the revival probability,
$\begin{eqnarray}{P}_{q}(t)=\displaystyle \frac{1}{2}\left[\displaystyle \sum _{l\geqslant q}{| c{{\prime} }_{{lq}}(t)| }^{2}+\displaystyle \sum _{l\leqslant q}{| c{{\prime} }_{{ql}}(t)| }^{2}\right],\end{eqnarray}$
for the probability density function, and
$\begin{eqnarray}{{\rm{\Gamma }}}_{{qr}}(t)=\left\{\begin{array}{lc}| c{{\prime} }_{{rq}}(t){| }^{2} & r\gt q,\\ 2| c{{\prime} }_{{qq}}(t){| }^{2} & r=q,\\ | c{{\prime} }_{{qr}}(t){| }^{2} & r\lt q.\end{array}\right.\end{eqnarray}$
for the correlation function.
For the study of quantum walking, the two anyons are initially prepared in the same site, often located in the center of the lattice, and the dynamics are observed thereafter. To check the inadequacy of the conventional choice of the Hilbert space, here we discuss three distinct initial states for the simulation of system dynamics, i.e. two anyons on the same site, on the neighboring sites and on the next-neighboring sites with an empty site in between, denoted by red, yellow and green stars in figure 1. We see that these initial states in the 2D lattice are on, one step away from and two steps away from the diagonal line, respectively. (i) The most-studied initial state with coefficients clk(0) = δl,cδk,c, i.e. $\left|{\phi }_{1}\right\rangle \equiv (1/\sqrt{2}){a}_{c}^{\dagger 2}\left|0\right\rangle $, is expressed in our notation as a column vector
$\begin{eqnarray}\left|{\phi }_{1}\right\rangle :{\left(0,\ldots {1}_{c,c},\ldots 0\right)}^{{\rm{T}}}\end{eqnarray}$
of dimension L2 in conventional Hilbert space and a vector in the same form yet of dimension L × (L + 1)/2 in orthogonal scheme, with the subscript indicating that the two anyons are equally prepared in the site c. This arrangement sheds light on how the system behaves when the particles are closely packed together. (ii) The second initial state with clk(0) = δl,(c+1)δk,c, i.e. $\left|{\phi }_{2}\right\rangle \equiv {a}_{c+1}^{\dagger }{a}_{c}^{\dagger }\left|0\right\rangle $, is represented as a vector
$\begin{eqnarray}\left|{\phi }_{2}\right\rangle :\displaystyle \frac{1}{\sqrt{2}}{\left(0,\ldots {1}_{c,c+1}{{\rm{e}}}^{{\rm{i}}\chi \pi },\ldots {1}_{c+1,c}...0\right)}^{{\rm{T}}},\end{eqnarray}$
in the conventional space and ${(0,\ldots {1}_{c+1,c}...0)}^{{\rm{T}}}$ in the orthogonal space with the same dimensions as in the initial state $\left|{\phi }_{1}\right\rangle $, signifying that two particles are positioned at the nearest neighbors in the lattice center. (iii) The third state with clk(0) = δl,(c+1)δk,(c−1), i.e. $\left|{\phi }_{3}\right\rangle \equiv {a}_{c+1}^{\dagger }{a}_{c-1}^{\dagger }\left|0\right\rangle $, is represented in a similar way as a vector
$\begin{eqnarray}\left|{\phi }_{3}\right\rangle :\displaystyle \frac{1}{\sqrt{2}}{\left(0,\ldots {1}_{c-1,c+1}{{\rm{e}}}^{{\rm{i}}\chi \pi },\ldots {1}_{c+1,c-1}...0\right)}^{{\rm{T}}},\end{eqnarray}$
in the conventional space and ${(0,\ldots {1}_{c+1,c-1}...0)}^{{\rm{T}}}$ in the orthogonal space, indicating that two particles are in the next-nearest-neighboring sites. Note that in the representation of the initial states $\left|{\phi }_{2}\right\rangle $ and $\left|{\phi }_{3}\right\rangle $, an appropriate phase factor is included in the transposed position of the non-zero coefficients at position (c + 1, c) or (c + 1, c − 1), which ensures that they describe a possible state function of two anyons. It is easy to check that the projection of this properly represented state onto any one of the non-physical eigenstates is zero. In this sense, the initial state $\left|{\phi }_{1}\right\rangle $ is trivial as the overlapping of its representation in the conventional scheme with the non-physical states is already zero (see the L = 3 case for an example: the only non-zero element in the state $\left|{\phi }_{1}\right\rangle $ is the fifth, while the fifth elements of all three non-physical states are zero). On the other hand, in the simulation of the two-anyon Hamiltonian by means of electric circuits, the wave function satisfying the Schrödinger equation may not carry this phase factor, which leads to a state
$\begin{eqnarray}\left|{\phi }_{2}^{\prime} \right\rangle :\displaystyle \frac{1}{\sqrt{2}}{\left(0,\ldots {1}_{c,c+1},\ldots {1}_{c+1,c}...0\right)}^{{\rm{T}}};\end{eqnarray}$
it is even possible to prepare a state like
$\begin{eqnarray}\left|{\phi }_{2}^{\prime\prime} \right\rangle :{\left(0,\ldots {0}_{c,c+1},\ldots {1}_{c+1,c}...0\right)}^{{\rm{T}}}.\end{eqnarray}$
These initial states, however, do not describe two-anyon states, and their expansions in conventional Hilbert space inevitably involve non-physical states, which leads to distinct dynamical results. For the lattice size L = 41 adopted in the following numerical simulation, the lattice center is located at c = 21.
The dynamics of the system depends crucially on the initial states. With properly represented states φi (i = 1, 2, 3) with the phase factor eiχπ included in the correct positions we observe exactly the same dynamical behavior in either the conventional or orthogonal schemes. The revival probability for the initial state $\left|{\phi }_{1}\right\rangle $ is easily calculated as ${P}_{\mathrm{rev}}(t)=| {c}_{{cc}}(t){| }^{2}=| c{{\prime} }_{{cc}}(t){| }^{2}$ in the two schemes, which exhibits degraded BO behavior [39] for statistical parameter χ away from the bosonic and pseudo-fermionic limits, as shown in figure 6(a). When the two anyons in the initial state are not located on the same lattice site, we have ${P}_{\mathrm{rev}}=2| {c}_{c+1,c}(t){| }^{2}=| c{{\prime} }_{c\,+\,1,c}(t){| }^{2}$ and ${P}_{\mathrm{rev}}\,=2| {c}_{c+1,c-1}(t){| }^{2}=| c{{\prime} }_{c\,+\,1,c-1}(t){| }^{2}$ for states $\left|{\phi }_{2}\right\rangle $ or $\left|{\phi }_{3}\right\rangle $ respectively, the results of which show the similar double-period phenomena of BO, that is, the period of BOs in the pseudo-fermionic limit is twice of that in the bosonic limit TB = 2π/F, while the oscillations disappear for statistical parameters in between, for instance at χ = 1/2, as shown in figures 6(b) and(c). The coefficients at time t still satisfy the anyonic relation such that the off-diagonal probability in the orthogonal space is twice the value in conventional space, while the diagonal ones are equal. For state $\left|{\phi }_{3}\right\rangle $, the only difference that occurs at the revival probability is that the secondary peaks tend to be more prominent. We find that this happens equally for other initial states with further increasing distance between the two anyons in the initial state, i.e. the revival probability may reach as high as 0.7 at the original period times of the BOs.
Figure 6. Evolution of the revival probability of two correlated anyons undergoing BOs on a 1D lattice for F/J = 0.3, L = 41 and for increasing values of the statistical phase exchange χ = 0, 0.5 and 1 from top to bottom (χ = 0 corresponds to non-interacting bosons, whereas χ = 1 corresponds to pseudo-fermions). The results calculated in the conventional and orthogonal spaces are exactly the same and we show the revival probability evolution for three initial states, i.e. $\left|{\phi }_{1}\right\rangle $ with two anyons on the same site (a), $\left|{\phi }_{2}\right\rangle $ on the neighboring sites (b) and $\left|{\phi }_{3}\right\rangle $ on the next-neighboring sites with an empty site in between (c).
The absence of BOs in the case of statistical parameter 0 < χ < 1 can be more clearly seen in the evolution of the probability density function Pq(t) starting from different initial states introduced above. The two anyons will return to their original positions periodically when the BOs occur in the bosonic limit, while it takes twice the time in the pseudo-fermionic limit, no matter when they are put in the same site (φ1) or different sites (φ2) and (φ3) (see the top and bottom panels of figure 7). Away from these two limiting cases, for instance, χ = 1/2, the anyons would never return to their initial positions for all three initial states. For the initial state $\left|{\phi }_{3}\right\rangle $ with an empty site in between, in the pseudo-fermion limit we still find a very large probability that the two anyons to return to their original positions in the ordinary period, which is consistent with the period in the revival probability in figure 6. We find this happens for other initial states when the anyons are separated by more empty sites.
Figure 7. Evolution of the probability density function of two correlated anyons undergoing BOs on a 1D lattice for F/J = 0.3, L = 41 and for increasing values of the statistical phase exchange χ = 0, 0.5 and 1 from top to bottom. The results calculated in the conventional and orthogonal spaces are exactly the same and we show the probability density evolution for three initial states, i.e. $\left|{\phi }_{1}\right\rangle $ with two anyons on the same site (a), $\left|{\phi }_{2}\right\rangle $ on the neighboring sites (b) and $\left|{\phi }_{3}\right\rangle $ on the next-neighboring sites with an empty site in between (c), respectively.
It is necessary to examine carefully the details of the dynamics of those 'non-anyonic' initial states. For the statistical parameter 0 < χ < 1, the newly calculated results in the conventional Hilbert space show drastically different behavior for the three states. Specifically, we have Prev = (1/2)∣cc,c+1(t) + cc+1,c(t)∣2 and Prev = ∣cc+1,c(t)∣2 for states $\left|\phi {{\prime} }_{2}\right\rangle $ and $\left|\phi {{\prime\prime} }_{2}\right\rangle $, respectively. Note that the coefficients here do not satisfy the relation (5), and it is not possible to further simplify this probability. To see more clearly the difference from their anyonic partner state $\left|{\phi }_{2}\right\rangle $, we illustrate the revival probability of these states in figure 8(a) for F/J = 0.3 and in the pseudo-fermionic limit where the most significant discrepancy occurs. It can be seen that all states experience a full revival at 2TB and the anyonic state shows a revival probability of only 0.1 at the time of BO period TB. This means the wave function has little chance of returning to the initial state at time TB when the two anyons are initially prepared in different sites. The non-anyon states $\phi {{\prime} }_{2}$, on the other hand, have a 100% revival probability at this specific moment TB, and even the state $\left|\phi {{\prime\prime} }_{2}\right\rangle $ will have a 50% revival probability. This conclusion holds equally for the initial state $\left|{\phi }_{3}\right\rangle $ where the two anyons are initially separated by an empty site.
Figure 8. (a) The dynamics of revival probability and (b) probability density at the central site (here c = 21) from the initial states $\left|{\phi }_{2}\right\rangle $, $\left|\phi {{\prime} }_{2}\right\rangle $ and $\left|\phi {{\prime\prime} }_{2}\right\rangle $ with F/J = 0.3 and L = 41 in the fermionic limit χ = 1.
The probability density at the central site demonstrates even more discernible features for the experiment simulation. It is shown that while the robust BOs at 2TB still occur in the central site density for anyonic initial state $\left|{\phi }_{2}\right\rangle $, those for non-anyonic initial state $\left|\phi {{\prime} }_{2}\right\rangle $ vanish completely in the entire dynamical evolution, as shown by the constant zero blue-dot-dashed line in figure 8(b). This is easy to understand as the probability density (17) is defined to measure the number of anyons in the lattice, which is the expectation value of the number operator nq on the state φ(t) at time t. Note that the number operator does not even take a diagonal form in the conventional space, as the Fock states basis ${a}_{l}^{\dagger }{a}_{k}^{\dagger }\left|0\right\rangle $ and ${a}_{k}^{\dagger }{a}_{l}^{\dagger }\left|0\right\rangle $ contribute equally to the expectation value of number operator nq at q = k. When we design the initial state in such a way that the coefficients satisfy the anyonic relation, such as $\left|{\phi }_{2}\right\rangle $, the measurement of the probability at the central site c = 21 gives 1/2 as the two anyons are initially located at c = 21 and c + 1 = 22, which may oscillate as the anyons walk to other lattice sites. Starting from a non-anyonic state such as $\left|\phi {{\prime} }_{2}\right\rangle $, we have no chance of finding any anyon in the system at t = 0 or at the later time t > 0. The state $\left|\phi {{\prime\prime} }_{2}\right\rangle $ is a mixture of the pure anyonic and non-anyonic initial state, which proves to be a superposition of $\left|{\phi }_{2}\right\rangle $ and $\left|\phi {{\prime} }_{2}\right\rangle $, leaving only half of the probability for site q or for the whole lattice. In this sense, the simulation of these wave functions no longer describes the behavior of anyons.
Below we continue to consider the correlation function between two anyons with different statistical parameters from three typical initial states in the context of a one-dimensional lattice as shown in figure 9. Anyons with fixed parameter χ exhibit different correlation behaviors for various initial states, while particles starting from the same initial state may behave distinctly for different χ [37]. In the study of BOs, two anyons are initially stacked onto the same lattice site, and only the bosonic and pseudo-fermionic limits are considered in experiments with electric circuits [24]. Here we show that the BOs in the presence of external force depend crucially on both the statistical parameter and the initial states. The correlation functions at a specific time of half the BO period TB are rescaled by the maximum value such that ${{\rm{\Gamma }}}_{{qr}}({T}_{B}/2)/{{\rm{\Gamma }}}_{{qr}}^{\max }({T}_{B}/2)$ are presented in figure 9 for three initial states introduced above with F/J = 0.3. Again we find a perfect coincidence between the conventional and orthogonal schemes for the intentionally prepared initial state $\left|{\phi }_{\mathrm{1,2,3}}\right\rangle $ with any statistical parameter χ. Specifically, the case of bosonic limit has been studied in a waveguide lattice system [21, 50, 51] where the quantum correlations of photon pairs lead to nontrivial quantum interference and strongly depend on the input states. We choose to illustrate the correlation at a time of half the oscillation period TB/2, when the density peaks are far enough away. When two bosons are initially stacked onto the same lattice site, i.e. the $\left|{\phi }_{1}\right\rangle $ state, the numerical results show that no interference occurs between the particles, i.e. the correlation function is just a product of two single-particle probability densities. The correlation matrix is characterized by four peaks at the corners of the matrix in the top panel in figure 9(a), which is a result of the ballistic propagation of particles. When the two bosons are prepared in two neighboring sites, i.e. the $\left|{\phi }_{2}\right\rangle $ state, the correlation map changes considerably, as shown in the top panel of figure 9(b). The most obvious feature is the vanishing of the two off-diagonal lobes: the bosons tend to bunch to the same lobe. This can be explained as a generalized Hong–Ou–Mandel interference: two paths lead to a coincidence measurement between site q and site r. This interference phenomenon is also known as Hanbury Brown–Twiss interference [52, 53]. The four lobes are recovered when the bosons are initiated in the $\left|{\phi }_{3}\right\rangle $ state, i.e. with one empty site between the occupied sites. This state contains strong non-classical features with very clear differences from that for the $\left|{\phi }_{1}\right\rangle $ state, as shown in the top panel of figure 9(c). The boson pair exhibits bunching behavior but with a different symmetry: if one particle is detected in between the lobes, the probability of detecting the second one in a lobe vanishes, even though a single particle is most likely to reach the lobes. Similarly, if one boson is detected in a lobe, it is certain that the other one is also in a lobe. We note that quantum interference emerges since the two bosons are indistinguishable.
Figure 9. Two-body correlation functions of two anyons in position space on a 1D lattice of L = 41 sites for external force F/J = 0.3 and statistical parameters χ = 0, 1/2 and 1 from top to bottom. The results calculated in the conventional and orthogonal spaces are exactly the same and we show the correlation at t = TB/2 for three initial states $\left|{\phi }_{1}\right\rangle $ (a), $\left|{\phi }_{2}\right\rangle $ (b) and $\left|{\phi }_{3}\right\rangle $ (c).
In the pseudo-fermionic limit, we observe that starting from the doubly occupied state $\left|{\phi }_{1}\right\rangle $ the probability of finding both particles at the initial position takes the maximum value as the particles happen to return to and localize at the central lattice site at time TB/2 with a relatively large probability. This is witnessed by the corresponding probability density shown in the lower left panel of figure 7. On the other hand, the dynamics of the $\left|{\phi }_{3}\right\rangle $ state shows the typical anti-bunching behavior of fermions, i.e. if one anyon is detected at one side, most likely the second one will appear at the other side of the lattice. From the density distribution of the particles, we can see that at time TB/2, the anyons move to those locations that are the farthest away from the center of the lattice. This is also the case for anyons in the bosonic limit; however, the correlation is completely different. For $\left|{\phi }_{2}\right\rangle $ the density of the particles is essentially distributed in the central region of the lattice and the correlation reveals fractional statistics, showing both strong diagonal weights and the onset of fermionization. Note that the moment TB/2 is at the first quarter of a full BO period, 2TB in this case.
For anyons with statistical parameter χ = 1/2, interestingly we find the behavior of anyons in state $\left|{\phi }_{3}\right\rangle $ resembles that of pseudo-fermions very closely, showing a very clear anti-bunching phenomenon at this specific time in first BO period. This occurs only once as the probability density enters into a chaotic dynamical process without the BOs hereafter. For the other two initial states $\left|{\phi }_{\mathrm{1,2}}\right\rangle $ with χ = 1/2, more complicated correlation functions follow the chaotic evolution of the probability density due to the absence of BOs. When the anyons complete one full BO period (TB for bosons and 2TB for pseudo-fermions), either the probability density or the correlation function returns to that of the initial states, respectively, and then the system starts to evolve for another period, which does not happen for anyons for arbitrary intermediate statistical parameters 0 < χ < 1.

5. Conclusions

We have revisited the conventional non-orthogonal theoretical method which has been recently applied to explore the stationary and dynamical properties of a pair of non-interacting anyons hopping on a 1D lattice subject to an external force. To overcome the non-physical degrees of freedom induced by this conventional non-orthogonal method, we develop an orthogonal method that expands the two-anyon state in the orthogonal Hilbert space. The eigenequations and time-evolution equations of the two-anyonic states have been deduced from Schrödinger equation in both schemes.
We find that the orthogonality of the basis vectors is crucial when comparing the stationary energy spectrum and the time-dependent dynamical behavior of two-anyon states. The orthogonal method maps the two-anyon states of a 1D lattice into single-particle motions in a folded triangular region of a 2D square lattice, while it is an entire square lattice for the conventional scheme. And the conventional scheme produces L(L − 1)/2 non-physical redundant eigenstates, whose coefficients violate the anyonic commutation relation. These redundant non-physical eigenvalues affect the overall structure of the energy spectrum.
Moreover, the time-evolution revival probability, probability density function and the two-body correlation in these two schemes strongly depend on the initial states. Three typical initial states with two anyons on the same site, on the neighboring sites and on the next-neighboring sites (i.e. with an empty site in between), are taken as examples to illustrate the necessity of the orthogonal basis vectors. We observed exactly the same dynamical behavior in both schemes for initial states that satisfy the anyonic commutation relation. However, the calculation for the non-anyonic initial states in the conventional space shows erroneous results, for example BOs do not occur at TB for the initial state with two anyons on the (next-)neighboring sites in the fermionic limit, as the conventional scheme predicted. This can be readily checked by the electric circuits simulation by preparing the initial state accordingly. In addition, our calculation shows that the two-anyon correlation function recovers the full interference pattern and the bunching behavior in the bosonic limit, which has been realized in the waveguide lattice system. The dynamics of the $\left|{\phi }_{3}\right\rangle $ state shows interesting anti-bunching behavior of fermions in the pseudo-fermionic limit as well as in the middle value χ = 1/2.

We thank Professor Xiang-Dong Zhang for helpful discussions on the selection rule of the energy levels in order to illustrate the BOs in the spectrum.

1
Laidlaw M G G, DeWitt C M 1971 Feynman functional integrals for systems of indistinguishable particles Phys. Rev. D 3 1375

DOI

2
Leinaas J M, Myrheim J 1977 On the theory of identical particles Nuovo. Cim. B 37 1

DOI

3
Wilczek F 1982 Magnetic flux, angular momentum, and statistics Phys. Rev. Lett. 48 1144

DOI

4
Lerda A 1992 Anyons: Quantum Mechanics of Particles with Fractional Statistics, Lecture Notes in Physics No. m14 Springer

5
Canright G S, Girvin S M 1990 Fractional statistics: quantum possibilities in two dimensions Science 247 1197

DOI

6
Laughlin R B 1983 Anomalous quantum Hall effect: an incompressible quantum fluid with fractionally charged excitations Phys. Rev. Lett. 50 1395

DOI

7
Halperin B I 1984 Statistics of quasiparticles and the hierarchy of fractional quantized Hall states Phys. Rev. Lett. 52 1583

DOI

8
Camino F E, Zhou W, Goldman V J 2005 Realization of a Laughlin quasiparticle interferometer: observation of fractional statistics Phys. Rev. B 72 075342

DOI

9
Kim E-A, Lawler M, Vishveshwara S, Fradkin E 2005 Signatures of fractional statistics in noise experiments in quantum Hall fluids Phys. Rev. Lett. 95 176402

DOI

10
Stern A 2008 Anyons and the quantum Hall effect pedagogical review Ann. Phys. 323 204

DOI

11
Bartolomei H 2020 Fractional statistics in anyon collisions Science 368 173

DOI

12
Yao H, Kivelson S A 2007 Exact chiral spin liquid with non-Abelian anyons Phys. Rev. Lett. 99 247203

DOI

13
Bauer B, Cincio L, Keller B, Dolfi M, Vidal G, Trebst S, Ludwig A 2014 Chiral spin liquid and emergent anyons in a kagome lattice Mott insulator Nat. Commun. 5 5137

DOI

14
Ha Z N C 1994 Exact dynamical correlation functions of Calogero–Sutherland model and one-dimensional fractional statistics Phys. Rev. Lett. 73 1574

DOI

15
Murthy M V N, Shankar R 1994 Thermodynamics of a one-dimensional ideal gas with fractional exclusion statistics Phys. Rev. Lett. 73 3331

DOI

16
Ha Z N C 1995 Fractional statistics in one dimension: view from an exactly solvable model Nucl. Phys. B 435 604

DOI

17
Batchelor M T, Guan X-W, Oelkers N 2006 One-dimensional interacting anyon gas: low-energy properties and Haldane exclusion statistics Phys. Rev. Lett. 96 210402

DOI

18
Batchelor M T, Guan X-W, He J-S 2007 The Bethe ansatz for 1D interacting anyons J. Stat. Mech. 03 P03007

DOI

19
Hao Y, Zhang Y, Chen S 2008 Ground-state properties of one-dimensional anyon gases Phys. Rev. A 78 023631

DOI

20
Hao Y, Zhang Y, Chen S 2009 Ground-state properties of hard-core anyons in one-dimensional optical lattices Phys. Rev. A 79 043633

DOI

21
Bromberg Y, Lahini Y, Morandotti R, Silberberg Y 2009 Quantum and classical correlations in waveguide lattices Phys. Rev. Lett. 102 253904

DOI

22
Greschner S, Santos L 2015 Anyon Hubbard model in one-dimensional optical lattices Phys. Rev. Lett. 115 053002

DOI

23
Alicea J, Fendley P 2016 Topological phases with parafermions: theory and blueprints Annu. Rev. Condens. Matter. Phys. 7 119

DOI

24
Zhang W, Greschner S, Fan E, Scott T C, Zhang Y 2017 Ground-state properties of the one-dimensional unconstrained pseudo-anyon Hubbard model Phys. Rev. A 95 053614

DOI

25
Rossini D, Carrega M, Calvanese Strinati M, Mazza L 2019 Anyonic tight-binding models of parafermions and of fractionalized fermions Phys. Rev. B 99 085113

DOI

26
Haldane F D M 1991 'Fractional statistics' in arbitrary dimensions: a generalization of the Pauli principle Phys. Rev. Lett. 67 937

DOI

27
Iqbal M 2024 Non-Abelian topological order and anyons on a trapped-ion processor Nature 626 505

DOI

28
Kitaev A Y 2003 Fault-tolerant quantum computation by anyons Ann. Phys. 303 2

DOI

29
Nayak C, Simon S H, Stern A, Freedman M, Sarma S D 2008 Non-Abelian anyons and topological quantum computation Rev. Mod. Phys. 80 1083

DOI

30
Lahtinen V, Pachos J 2017 A short introduction to topological quantum computation SciPost Phys. 3 021

DOI

31
Keilmann T, Lanzmich S, McCulloch I, Roncaglia M 2011 Statistically induced phase transitions and anyons in 1D optical lattices Nat. Commun. 2 361

DOI

32
Georgescu I M, Ashhab S, Nori F 2014 Quantum simulation Rev. Mod. Phys. 86 153

DOI

33
Nakamura J, Liang S, Gardner G C, Manfra M J 2020 Direct observation of anyonic braiding statistics Nat. Phys. 16 931

DOI

34
Altman E 2021 Quantum simulators: architectures and opportunities PRX Quantum. 2 017003

DOI

35
Sansoni L, Sciarrino F, Vallone G, Mataloni P, Crespi A, Ramponi R, Osellame R 2012 Two-particle bosonic–fermionic quantum walk via integrated photonics Phys. Rev. Lett. 108 010502

DOI

36
Campagnano G, Zilberberg O, Gornyi I V, Feldman D E, Potter A C, Gefen Y 2012 Hanbury Brown–Twiss interference of anyons Phys. Rev. Lett. 109 106802

DOI

37
Wang L, Wang L, Zhang Y 2014 Quantum walks of two interacting anyons in one-dimensional optical lattices Phys. Rev. A 90 063618

DOI

38
Lau L L, Dutta S 2022 Quantum walk of two anyons across a statistical boundary Phys. Rev. Res. 4 L012007

DOI

39
Longhi S, Valle G D 2012 Anyonic Bloch oscillations Phys. Rev. B 85 165144

DOI

40
Corrielli G, Crespi A, Della Valle G, Longhi S, Osellame R 2013 Fractional Bloch oscillations in photonic lattices Nat. Commun. 4 1555

DOI

41
Zhang W, Yuan H, Wang F, Di H, Sun N, Zheng X, Sun H, Zhang X 2022 Observation of Bloch oscillations dominated by effective anyonic particle statistics Nat. Commun. 13 2392

DOI

42
Hong C K, Ou Z Y, Mandel L 1987 Measurement of subpicosecond time intervals between two photons by interference Phys. Rev. Lett. 59 2044

DOI

43
Childs A M 2009 Universal computation by quantum walk Phys. Rev. Lett. 102 180501

DOI

44
Childs A M, Gosset D, Webb Z 2013 Universal computation by multiparticle quantum walk Science 339 791

DOI

45
Kwan J, Segura P, Li Y, Kim S, Gorshkov A V, Eckardt A, Bakkali-Hassani B, Greiner M 2023 Realization of 1D anyons with arbitrary statistical phase arXiv:2306.01737

46
Fradkin E 1989 Jordan–Wigner transformation for quantum-spin systems in two dimensions and fractional statistics Phys. Rev. Lett. 63 322

DOI

47
Amico L, Osterloh A, Eckern U 1998 One-dimensional XYZ model for particles obeying fractional statistics Phys. Rev. B 58 R1703

DOI

48
Osterloh A, Amico L, Eckern U 2000 Fermionic long-range correlations realized by particles obeying deformed statistics J. Phys. A: Math. Gen. 33 L487

DOI

49
Batchelor M T, Foerster A, Guan X W, Links J, Zhou H Q 2008 The quantum inverse scattering method with anyonic grading J. Phys. A 41 465201

DOI

50
Alberto Peruzzo M L 2010 Quantum walks of correlated photons Science 329 1500

DOI

51
Meinecke J D A, Poulios K, Politi A, Matthews J C F, Peruzzo A, Ismail N, Wörhoff K, O'Brien J L, Thompson M G 2013 Coherent time evolution and boundary conditions of two-photon quantum walks in waveguide arrays Phys. Rev. A 88 012308

DOI

52
Schellekens M, Hoppeler R, Perrin A, Viana Gomes J, Boiron D, Aspect A, Westbrook C I 2005 Hanbury Brown Twiss effect for ultracold quantum gases Science 310 648

DOI

53
Jeltes T 2007 Comparison of the Hanbury Brown-Twiss effect for bosons and fermions Nature 445 402

DOI

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