Ground-state properties of anyons in a one-dimensional lattice

Using the Anyon-Hubbard Hamiltonian, we analyze the ground-state properties of anyons in a one-dimensional lattice. To this end we map the hopping dynamics of correlated anyons to an occupation-dependent hopping Bose-Hubbard model using the fractional Jordan-Wigner transformation. In particular, we calculate the quasi-momentum distribution of anyons, which interpolates between Bose-Einstein and Fermi-Dirac statistics. Analytically, we apply a modified Gutzwiller mean-field approach, which goes beyond a classical one by including the influence of the fractional phase of anyons within the many-body wavefunction. Numerically, we use the density-matrix renormalization group by relying on the ansatz of matrix product states. As a result it turns out that the anyonic quasi-momentum distribution reveals both a peak-shift and an asymmetry which mainly originates from the nonlocal string property. In addition, we determine the corresponding quasi-momentum distribution of the Jordan-Wigner transformed bosons, where, in contrast to the hard-core case, we also observe an asymmetry for the soft-core case, which strongly depends on the particle number density.


Introduction
A fundamental principle of quantum statistical mechanics in three dimensions is the existence of two types of particles: bosons obeying Bose-Einstein statistics and fermions obeying Fermi-Dirac statistics. However, in a two-dimensional electron liquid, quasiparticles made of electrons in the fractional quantum Hall effect are charged anyons obeying fractional statistics [1][2][3][4][5]. While exchanging two anyons, the many-body wavefunction acquires a fractional phase e iθ , where the statistical parameter θ varies with the interval 0 < θ < π and corresponds to a fractional statistics. As compared to bosons and fermions, anyons exhibit a wide range of previously unexpected properties and the concept of anyons plays an important role in numerous studies of condensed matter physics and of topological quantum computation [6][7][8][9][10][11][12][13]. In order to generalize the Bose-Einstein and Fermi-Dirac statistics by allowing a maximal finite integer particle number occupying the same quantum state, Gentile supplied an intermediate statistics that had been proven to be also valid for a q-fermion [14][15][16][17][18][19]. However, Y. Shen et al. pointed out that the anyon statistics was not a complete analogue of the Gentile statistics [20]. Based on a generalized Pauli exclusion principle, Haldane provided a useful concept of fractional statistics in arbitrary dimensions [21]. Polychronakos also suggested another form of the fractional exclusion statistics [22]. However, it was shown via the virial expansion that all these fractional statistics proposed in the literature do not apply for anyons [20,23].
In one dimension (1D), anyons were realized as low-energy elementary excitations of the Hubbard model of fermions with correlated hopping processes [24]. Alternatively, it was suggested to create anyons by bosons with complex-valued occupation-dependent hopping amplitudes by photon-assisted tunneling in 1D optical lattices [25,26]. Recently, S. Greschner and L. Santos proposed a Raman scheme to improve the proposal of T. Keilmann et al. in Ref. [25] and deduced a rich ground-state physics including Mott-insulators with attractive interactions, pair-superfluids, dimer phases, and multicritical points [27]. An even simpler scheme for realizing the physics of 1D anyons with ultracold bosons in an optical lattice has recently been proposed in Ref. [28]. It relies on lattice-shaking-induced resonant tunneling against the energy off-sets created by the combination of both a potential tilt and on-site repulsion. In contrast to the above mentioned former proposals based on internal atomic degrees of freedom, no lasers additional to those already used for the creation of the optical lattice are required [28].
The physical properties of 1D anyons are intriguing and complicated. Theoretically, using the generalized coordinate Bethe ansatz method, Kundu obtained the exact solution of a 1D anyon gas with the delta-function potential, which shows that the effective interaction among the anyons is modified by the anyonic statistical parameter [29]. Furthermore, the 1D anyon gas with a special interaction potential has been investigated in various 1D systems in order to get the ground-state properties of the onedimensional anyon gas [30][31][32][33][34][35]. For anyons in the 1D lattice systems one investigated, for instance, the statistically induced phase transition [25], quantum scaling properties [24], dynamics properties of hard-core anyons [36,37], quantum walks [38,39], and anyonic Bloch oscillations [40]. For the hard-core anyons in the 1D optical lattice, Y. Hao et al. investigated how the fractional statistics affected the ground-state properties by mapping the hard-core anyonic Hamiltonian to a noninteracting fermionic system with a generalized Jordan-Wigner transformation [41]. By calculating the one-particle Greens function of the ground state, the reduced one-body density matrix and, thus, the quasimomentum distributions can be obtained for different statistical parameters of anyons. The results showed that the momentum distributions in the Bose and Fermi limit turned out to be symmetric, but those of anyons were in general asymmetric and shifted due to the fractional statistics that anyons obey.
In this paper we complement the previous studies of the ground-state properties of a 1D quantum gas of anyons confined in optical lattices. To this end we map the Anyon-Hubbard model to an occupation-dependent hopping Bose-Hubbard model with the help of a fractional version of the Jordan-Wigner transformation. This mapping has the consequence that the Hilbert space of anyons can be constructed from that of bosons, so that one has access to the two-point correlation function of either the original anyonic or the Jordan-Wigner transformed bosonic creation and annihilation operators. With this we investigate the quasi-momentum distributions of either bosons or anyons, the latter interpolating between Bose-Einstein statistics and Fermi-Dirac statistics. Firstly, in the hard-core limit, we determine the quasi-momentum distributions of anyons with density-matrix renormalization group calculations, which numerically reproduces the results of Y. Hao et al. [41]. In addition, by suggesting a modified Gutzwiller mean-field approach to include the influence of the fractional phase of anyons within the many-body wavefunction, we obtain an approximative analytic expression for the numerical results of Ref. [41] both for a finite system and in the thermodynamic limit. In particular, we analyze in detail how peak-shift and asymmetry of the quasi-momentum distribution of anyons depend on both the fractional phase θ and the particle number density n 0 . Furthermore, we extend the findings of Ref. [41] by working out also the more general soft-core case. The quasi-momentum distribution of anyons reveals the pseudofermion property at the fractional phase θ = π. Surprisingly, the quasi-momentum distributions of the Jordan-Wigner transformed bosons shows a density-dependent asymmetry which is not found for the hard-core case.
The outline of this paper is as follows: in Sec. 2 the Anyon-Hubbard model and the mapping between anyons and bosons are discussed. Both the classical and the modified Gutzwiller mean-field approach is introduced in Sec. 3. The ground-state properties of the 1D Anyon-Hubbard model are determined by studying the quasi-momentum distribution of anyons and bosons in both the hard-core and the soft-core case in Sec. 4 and Sec. 5 respectively. Conclusions are given in Section 6.

Anyon-Hubbard model
The hopping dynamics of correlated anyons on a 1D lattice is described by the Anyon-Hubbard (AH) Hamiltonian [25] where J > 0 denotes the tunneling amplitude connecting two neighbouring sites, U stands for the on-site interaction energy,n j =â † jâ j represents the number operator at site j, and the operatorsâ † j ,â j create or annihilate an anyon on site j. For 1D anyons, the operatorsâ † j andâ j obey the generalized commutation relations [25,29] a jâ † where θ is the statistical exchange phase, the sign function sgn(j − k) = −1, 0, +1 for j < k, j = k, and j > k, respectively. Note that, since the sign function sgn(j − k) = 0 for j = k, two particles on the same site behave as ordinary bosons irrespective of the statistical parameter θ. Moreover, anyons with the statistical exchange phase θ = π are pseudofermions, i.e. while being bosons on-site, they are fermions off-site.
Here we use an exact Anyon-Boson mapping in 1D in terms of a fractional version of the Jordan-Wigner transformation [25] a j =b j exp iθ where the number operator Inserting the Anyon-Boson mapping (3) into Eq. (1), the HamiltonianĤ a can be rewritten as [25] where the conditional hopping of bosons from right to left, i.e. j + 1 → j, occurs with an occupation-dependent amplitude Je iθn j . If the target site j is unoccupied, the hopping amplitude is simply J. If it is occupied by one boson, the amplitude becomes complex and reads Je iθ , and so on. We emphasize that the hard-core limit of anyons in Eq. (1) coincides with that of bosons in Eq. (4) due ton i =â † iâ i =b † ib i . It is also clear that the mapped bosonic Hamiltonian (4) describes local occupation numbers beyond the hard-core limit n j > 1.
Using the nonlocal exact mapping (3) between anyons and bosons, the Anyon-Hubbard Hamiltonian (1) leads to a bosonic Hamiltonian Eq. (4) that can be solved either analytically or numerically in order to determine the ground-state properties of anyons in 1D lattice systems. Theoretically, since the reflection parity symmetry in the Hamiltonian is broken, we suggest below a modified Gutzwiller mean-field, which goes beyond the classical one found in literature [42][43][44][45] in order to include the influence of the fractional phase of anyons on the hopping dynamics. To get numerical results, we use the density-matrix renormalization group (DMRG) [46][47][48][49][50], which was already applied to the realm of anyons, for instance, in Ref. [25]. To this end, we rely on the ansatz of matrix product states (MPS) with the system length L and open boundary conditions, which is more efficient than periodic boundaries [51][52][53][54][55]. The code is based on a variational ansatz using MPS with the restricted subspace of integer filling, where our simulations admit a maximum of five particles per site and the maximum bond dimension of MPS equals to 1000 [56]. Moreover, Refs. [53,54,56] show the respective details how to calculate the expectation of the string operator in Eq. (3) and the correlation functions in DMRG with MPS ansatz.

Mean-field approximation
At first we work out a Gutzwiller mean-field approach, which turns out to provide qualitative satisfactory results for the quasi-momentum distributions. In a Gutzwiller (GW) approach for bosons, the many-particle state |G is generically approximated by a product state of single lattice-site states |Φ j , which can be expressed as a superposition of different number states on a lattice site [45] Here n max represents a truncation at some sufficiently large maximal particle number and f (j) n denotes the Gutzwiller probability amplitude of finding n bosons on site j, which is normalized such that The total energy of the system then follows from determining the ground-state expectation values of the respective operators according to the Appendix. As a consequence, the interaction energy (A.1), the chemical potential term (A.2), and the kinetic energy (A.3) depend on the Gutzwiller coefficients f (j) n . Due to the polar decomposition f n e iα (j) n , the total energy (7) has to be minimized with respect to both the absolute values F (j) n and the phases α In the classical GW mean-field approach of a homogeneous system, to which we refer as cGW in the following, one assumes that the ground state is a product of identical states on each of the L lattice sites [42][43][44][45], i.e., This has the consequence i.e. one has the identical wave function on each lattice site. Due to the occupation-dependent amplitude Je iθn j for nearest-neighbour hopping in the bosonic Hamiltonian (4), the reflection parity symmetry is broken. Therefore, we suggest to drop the cGW constraint of having identical states on each site to a modified GW mean-field approach (mGW). To this end, we note that the kinetic energy Eq.
n+1 of the phases of the probability amplitudes f (j) n and f (j) n+1 of finding n and n+1 bosons on site j, respectively. If all phase differences would be identical, we would recover cGW. Therefore, we define mGW according to In order to fix the values of the phase differences β (j) n , we complement (10) by the additional assumption This means that the absolute value F n is identical on each site and the non-vanishing phase difference ∆β (j) n between two nearest-neighbour sites is the same in the whole chain.
The application of cGW and mGW is performed in the Appendix for both the hard-core case in Sec. Appendix A.1, which means n max = 1, and the soft-core case by assuming n max = 2 in Sec. Appendix A.2. Provided that Gutzwiller amplitudes and phases have been determined, one can calculate the two-point correlation function of bosons, which has the rather simple expression The correlation function of anyons can be written as For the case i < j and i > j we get correspondingly, where the respective expectation values depend on the Gutzwiller amplitudes f Furthermore, in order to investigate the effect of the anyon statistical parameter θ, we will focus on density distributions of bosons and anyons in the quasi-momentum space, which are believed to be interesting [25] and are defined via the Fourier transformation of the correlation function In the following, we are going to find the ground state of a homogeneous system for a given particle number density n 0 > 0, ratio J/U, and varying statistical parameter θ. Thus, combining the analytic approaches cGW, mGW with numerical results from DMRG, we determine the correlation functions and, thus, investigate the quasimomentum distributions of the ground state.

Hard-core anyons
In order to illustrate the applicability of the Gutzwiller mean-field theory, we investigate at first the simplest case of hard-core anyons. This means that we assume U/J → ∞, where bosons are impenetrable and each site contains at most one particle, i.e., n max = 1. In the following, we present only the calculations of the correlation function and the quasi-momentum distribution, while the mathematical details for minimizing the total energy are relegated to Sec. Appendix A.1 of the Appendix.
According to Eq. (12), the correlation function of bosons reads , the ground-state quasi-momentum distribution of bosons can be written as which is θ-independent and the number of particles in the condensate is N 0 = N(1−n 0 ). Fig. 1 shows the resulting ground-state quasi-momentum distribution in a hard-core system at density n 0 = 0.5. Since n (18) is θ-independent in the GW approach, if one changes θ from 0 to π, all quasi-momentum distributions with different θ coincide as shown in Fig. 1 (a). The quasi-momentum distribution of bosons n (b) k in DMRG as shown in Fig. 1(b) turns out to be also θ-independent because the phase factor disappears in the bosonic Hamiltonian (4) in the hard-core limit and the systems reduces to a hard-core Bose-Hubbard model. As a consequence, bosons condense at zero quasi-momentum in both the GW approach and DMRG. However, the bosonic population N(1 − n 0 ) of k = 0 in the GW approach is much larger than that in DMRG because the mean-field approach does not contain quantum fluctuations which generally broaden the quasi-momentum distributions.
In the Fermi limit θ = π, we obtain from (21) n where n +1) if 0 < n 0 < 1. In the thermodynamic limit, this reduces to For the system at n 0 = 0.5, the GW approach gives which is shown as the black solid line in Fig. 1(a). However, in Fig. 1(b) from DMRG, the quasi-momentum distribution of anyons in the Fermi limit represents a step-like distribution, which is the characteristic feature of free fermions. This shows that DMRG can grasp the fermion-like feature of anyons with statistics θ = π, whereas the GW approach achieves this only approximately. If K In the fractional phase interpolating between the Bose and Fermi limit 0 < θ < π, we find from (21) n (a) k In the thermodynamic limit, this reduces to where r = |1 − n 0 + n 0 e iθ | and ϕ = arg(1 − n 0 + n 0 e iθ ). Fig. 1 clearly show that: i) the respective peak of the quasi-momentum distribution is shifted to positive momentum, ii) the quasi-momentum distribution is asymmetric about its own peak, and iii) the quasi-momentum distributions broaden and flatten with increasing θ. Comparing Fig.  1 (a) with (b) or b † ib j in Eq. (17) with â † iâ j in Eq. (20), we conclude that the nonlocal string property of anyons, which is denoted by i<l<j (1 − n 0 ) + n 0 e iθ or j<l<i (1 − n 0 ) + n 0 e −iθ in Eq. (20), is the reason why anyons prefer to appear on positive momentum. This is explained numerically in Ref. [41].
In order to find the maximum K where we have introduced the abbreviations K = arcsin sin θ(2n 0 − n 2 0 ) + n 2 0 sin θ cos θ (1 − cos θ)(1 − n 0 ) 2 + (1 + cos θ) . (bosons) and come back to K (a) max (θ = π) = 0 (fermions) while increasing θ. The GW approach shows the same linearly increasing property at small θ, i.e. near bosons. This can be quantitatively understood by taking the limit θ → 0 in Eqs. (27) and (29): At the Fermi side θ → π, we have correspondingly where the slope of K (a) max from GW is smaller than that from DMRG. In general, the GW approach always yields a smaller shift of positive momentum in comparison with the DMRG results for any statistical parameter of anyons between the Fermi and the Bose case. Moreover, we find that both DMRG and GW give a larger peak shift for a bigger particle number density n 0 .
In order to determine the thermodynamic limit of the quasi-momentum distribution, Fig. 2 (b) and (c) show the system size dependence of K (a) max and n L max /n L→∞ max at density n 0 = 0.75 in the GW approach, where n L max is the maximum value of the quasimomentum distribution with the finite system size L and n L→∞ max corresponds to the thermodynamic limit. Considering the uncertainty of the quasi-momentum in the finite system ∆k = 2π/L, K (a) max turns out to be independent of the system size. According to Fig. 1, the bosons condense at zero momentum with a sharp distribution and the fermions have a flat distribution. In Fig. 2(c), with θ increasing from the Bose to the Fermi case, we find that the condensate population near the Bose side slightly depends on the system size, but the maximum particle number near the Fermi side is almost independent on the system size, since the behavior of anyons with θ ∼ π looks much more like fermions without condensation. Note that we do not show here the case of bosons θ = 0 because bosons condense at zero momentum.

Soft-core anyons
Before going to the soft-core case, we remind that within the Gutzwiller theory b † ib j and n (b) k in Eqs. (17) and (18) turn out to be independent of the fractional phase θ of hard-core anyons because ∆β 0 = 2mπ and the local state |2 is not occupied, i.e. F 2 = 0. Considering the occupation-dependent hopping in the soft-core case, we are going to discuss the influence of ∆β 1 and non-vanishing F 2 in the following. The respective details of minimizing the total energy are given in Sec. Appendix A.2 of the Appendix.

Boson Quasi-momentum Distribution
According to Eq. (12) and supposing n max = 2, the correlation function of bosons is where 2 2 ) and A + B = C. Using Eq. (15a), the quasi-momentum distribution of bosons can be written as and yields in the thermodynamic limit Here, compared with the θ-independent n (b) k of the hard-core case in Eq. (18), the quasi-momentum distribution of bosons from mGW has a background distribution n 0 −C and two peaks at k = 0 and k = −θ with the intensities A and B, respectively.
Noticeably, if comparing the intensity A = F 2 1 (F 2 0 + √ 2F 0 F 2 ) of the k = 0 peak with the intensity B = F 2 1 ( √ 2F 0 F 2 + 2F 2 2 ) of the k = −θ peak, we find that the discrepancy between the intensities A and B is determined by the occupation F 2 0 of |0 and F 2 2 of |2 . As these occupations F 2 0 and F 2 2 change with the density n 0 , the resulting quasi-momentum distribution of bosons reveals a characteristic density dependency. Therefore, we discuss now different density regimes. In the dilute limit n 0 ≪ 1, the positive Hubbard U depresses the state |2 and are similar to that of the hard-core case. Fig. 3(a) shows the quasi-momentum distribution of bosons at density n 0 = 0.25 and J/U = 0.1 in the mGW approach. For θ = 0, we have n (b) k = n 0 − C + Cδ(k), which demonstrates that there exists only one peak of the intensity C = A + B at k = 0 and the quasimomentum distribution is symmetric. For the case of 0 < θ ≤ π, we have (34), which demonstrates that there exists one much lower peak of the intensity B at the negative momentum k = −θ than that of the intensity A at k = 0 due to F 2 2 ≪ F 2 0 . Here, we should note that n (b) k (θ = π) , which is plotted for slightly asymmetric interval −π ≤ k < π in Fig. 3(a), is also a symmetric distribution since n (b) k has period 2π. The results of DMRG with the same parameters are shown in Fig. 3(d), where the symmetric quasi-momentum distribution of θ = 0 is a little sharper than that of θ = π. But for the case of 0 < θ < π, there exists an asymmetric quasi-momentum distribution and more bosons have negative momentum according to the inset of Fig. 3(d). In general, quantum fluctuations broaden n (b) k from mGW. Taking this into account, if we would add the effect of quantum fluctuations on the mGW quasi-momentum distribution, the peak at k = 0 and k = −θ would broaden. Since the peak at k = −θ from mGW is pretty low, a broadening would be merged into the background of the quasi-momentum distribution. Therefore, at small particle densities, the mGW approach gives qualitatively the same quasi-momentum distribution as DMRG, i.e. the peaks locate at k = 0 and more bosons have negative momentum.
For the high density n 0 = 1.25 in Fig. 3(b) and (e), there exists more than one particle per site on average. Thus the vacuum state |0 is almost unoccupied in the ground state, i.e. F 2 0 ≪ F 2 2 . For 0 < θ ≤ π, Fig. 3(b) clearly shows that the high peaks of the quasi-momentum distribution exactly appear at k = −θ in comparison with that at k = 0. The results of mGW qualitatively coincide with that of DMRG in Fig. 3(d) although the momentum value k of the DMRG peaks is a little larger than −θ because the peaks at k = −θ compete with the peaks at k = 0 under the influence of quantum fluctuation. Of course, θ = 0 and θ = π still give the symmetric quasi-momentum distribution of bosonic operators. Fig. 3(c) and (f) show the quasi-momentum distribution at the density n 0 = 0.75 in the middle of n 0 = 0.25 and n 0 = 1.25. The competition between the k = 0 peak of the intensity A and the k = −θ of the intensity B of the quasi-momentum distribution leads to final peak shifts from k = 0 to the negative momentum but comes back to k = 0 while increasing θ from 0 to π as shown in Fig. 3(f). Note that this peculiar behaviour is not reproduced by mGW in Fig. 3(c).
If we think about the occupation-dependent factor e iθn j in (4) again, we recognize in Fig. 3 a clear physical picture of the ground state: i) for a small density n 0 < 1, the states |0 and |1 are preferred and ∆β 0 = 2mπ is the leading term, which gives a quasi-momentum peak of bosons at k ∼ 0; ii) for a density 1 < n 0 < 2, the states |1 and |2 are preferred and ∆β 1 = 2lπ − θ is the leading term, which gives a quasimomentum peak of bosons at k ∼ −θ. In order to deal with the density 2 < n 0 , we should truncate the mGW approach at n max = 3, 4, · · · . For example assuming n max = 3, the quasi-momentum distribution of bosons shows an additional peak of k = −2θ. For the density 2 < n 0 < 3, the leading peak of the quasi-momentum distribution of bosons appears at the momentum k ∼ −2θ.
In the Bose limit θ = 0, the correlation function of anyons is Using Eq. (15b), the ground-state quasi-momentum distribution of anyons can be written as n If θ = 0 and then |w| < 1, the quasi-momentum distribution of anyons is written as where u = we −ik and v = we −i(k+θ) . In the thermodynamic limit, this reduces to where W = |w| and χ = arg(w).
In the hard-core limit F 2 2 = 0, so we have w = z, χ = ϕ, A = n 0 (1 − n 0 ), and B = 0, thus Eq. (39) reduces to Eq. (26). For small densities n 0 ≪ 1, which implies F 2 2 ∼ 0, we expect the quasi-momentum distribution of anyons in the soft-core case to be similar to that in the hard-core case. This can be shown by comparing Fig. 4 (a) and (b) from DMRG or (a ′ ) and (b ′ ) from mGW at the density n 0 = 0.25. However, for the fractional phase θ = π in the soft-case, we also note that there exists a quite small area with the property n (a) k > 1 that is shown as the shaded (yellow in color) zone in the insets of Fig. 4 (b) and (b ′ ). The shaded area increases while increasing the density, for example n 0 = 0.75 is shown in Fig. 4 (d) and (d ′ ). This result reveals that anyons with the statistical phase θ = π are pseudofermions, i.e. possibly more than one identical anyon exist in one state but they behave like a fermion while exchanging two anyons on different sites. Comparing the results from DMRG and mGW in Fig. 4, we demonstrate again that the mGW approach grasps the main features of the quasimomentum distribution of anyons. Finally, we should note that the hard-core case is more suitable to describe the step-like behaviour of anyons (θ = π) than the soft-core case. Fig. 5 shows the fractional phase θ-dependence of anyonic K (a) max at the densities n 0 = 0.25, 0.5, 1.25, and 1.5. In general, for the density 0 < n 0 < 1 and 0 < θ < π, K (a) max at density n 0 = 0.5 is larger than that at n 0 = 0.25, which coincides with the finding of the hard-core case. Moreover, for 1 < n 0 < 2, the DMRG results show that the peak position K (a) max of the quasi-momentum distribution at density n 0 = 1.25 coincides with that at density n 0 = 0.25 within uncertainty. This is also shown by the DMRG results at density n 0 = 1.5 and n 0 = 0.5. However, the mGW approach in the thermodynamic limit shows that K (a) max at density n 0 +1 is a little larger than that at density n 0 , especially for the zone θ ∼ π.

Conclusions
In summary, we have studied the ground-state property of the 1D Anyon-Hubbard model. With the help of a fractional version of the Jordan-Wigner transformation, the Anyon-Hubbard model is mapped to the occupation-dependent hopping Bose-Hubbard model and, thus, the Hilbert space of anyons can be constructed from that of bosons. By calculating the two-point correlation function of creation and annihilation operators of bosons and anyons, we investigate the quasi-momentum distributions interpolating between Bose-Einstein statistics and Fermi-Dirac statistics. Theoretically, in order to include the influence of the fractional phase of anyons on the many-body wavefunction, we modify the classical Gutzwiller mean-field approach and get an analytic expression for the quasi-momentum distributions of anyons and bosons. In order to test the accuracy of the mGW mean-field approach, we use DMRG for numerical calculations.
In the hard-core case, the results show that the bosons condensate at zero momentum and have a symmetric quasi-momentum distribution around zero momentum. Due to the nonlocal string property, more anyons are shifted to a positive momentum and have an asymmetric quasi-momentum distribution, where the peak position depends on both the fractional phase and the particle number density.
In the soft-core case, the results show that the quasi-momentum peaks of bosons strongly depend on the particle number density. At density n 0 < 1, ∆β 0 = 2mπ leads to a quasi-momentum peak of bosons at k ∼ 0. At density 1 < n 0 < 2, ∆β 1 = 2lπ −θ leads a quasi-momentum peak of bosons at k ∼ −θ. Again, the quasi-momentum distribution of anyons show nonlocal string behaviour and yield similar features as in the hard-core case. Furthermore, anyons with θ = π are pseudofermions, i.e., there exists more than one identical anyon in one state in the soft-core case. However, in the hard-core case, the quasi-momentum distribution of anyons is a typical step-like function.
where the expectation values Ô j ≡ Φ j |Ô j |Φ j yield Using Eq. (A.4), the Gutzwiller kinetic energy reduces to Using the polar decomposition f n+1 denotes the difference between the phase of the probability amplitude f The ground state of the system with given parameters is determined from varying the coefficients f (j) n and minimizing the total energy. This can either be done for a given chemical potential µ, or for a given mean particle number n 0 = n = nmax n=0 nF 2 n according to Eq. (A.4c). In this paper, we assume that n 0 is given, so the chemical potential E µ = −µn 0 L resulting from (A.2) does not depend on f (j) n . Thus, after having determined the Gutzwiller coefficients f (j) n , the chemical potential µ is fixed by Appendix A.1. Hard-Core Case In the hard-core limit, bosons are impenetrable and each site contains at most one particle, i.e., n max = 1. In the GW approach, the normalization condition (6) reduces to F 2 0 + F 2 1 = 1. For a given particle number N, the mean density of particle number is given by n 0 = N/L = F 2 1 according to the Eq. (A.4c). Thus the absolute values of both Gutzwiller amplitudes are determined according to The expectation value of the energy per lattice site E tot = E tot /L is then given by where m = 0, ±1, ±2, · · · . This expression shows that the mGW approach reduces to the cGW approach for hard-core anyons.
Furthermore, if we set ∆β 0 = 2mπ and ∆β 1 = 2lπ in Eqs. (A.12), the mGW approach reduces to the cGW approach, for which the energy is E cGW tot = −µn 0 +UF 2 2 −2JF 2 1 F 2 0 + 2F 2 2 cos(θ) (A.15) Comparing Eq. (A.14) with (A.15), it is straightforward to conclude that E mGW tot ≤ E cGW tot , which shows that the mGW approach is superior to the cGW approach. Therefore, we restrict ourselves to evaluate the mGW approach for the soft-core case.