Modular matrices from universal wave function overlaps
in Gutzwiller-projected parton wave functions
We implement the universal wave function overlap (UWFO) method to extract modular and matrices for topological orders in Gutzwiller-projected parton wave functions (GPWFs). The modular and matrices generate a projective representation of on the degenerate-ground-state Hilbert space on a torus and may fully characterize the 2+1D topological orders, i.e. the quasi-particle statistics and chiral central charge (up to bosonic quantum Hall states). We used the variational Monte Carlo method to computed the and matrices of the chiral spin liquid (CSL) constructed by the GPWF on the square lattice, and confirm that the CSL carries the same topological order as the bosonic Laughlin state. We find that the non-universal exponents in UWFO can be small and direct numerical computation is able to be applied on relatively large systems. We also discuss the UWFO method for GPWFs on other Bravais lattices in two and three dimensions by using the Monte Carlo method. UWFO may be a powerful method to calculate the topological order in GPWFs.
Topological orderWen (1990, 1989); Wen and Niu (1990) connotes the pattern of long-range entanglement in gapped many-body wave functionsLevin and Wen (2006); Kitaev and Preskill (2006); Chen et al. (2010). It describes gapped quantum phases of matter that lie beyond the Landau symmetry breaking paradigmWen (2004). Local unitary transformations on many-body wave functions can remove local entanglement, however, preserve the long-range topological entanglement. Therefore, a topological ordered state is not smoothly connected to a trivial (direct product) state by local unitary transformationsChen et al. (2010). Physically, topological order is described through topological quantum numbers, such as non-trivial ground state structures and fractional excitations.Laughlin (1983); Wilczek and Zee (1984); Arovas et al. (1984); Wen (1989); Wen and Niu (1990); Wen (1990) These topological properties are fully characterized by the quasi-particle (anyon in the bulk) statisticsLaughlin (1983); Wilczek and Zee (1984); Arovas et al. (1984) and the chiral central charge which encodes information about chiral gapless edge statesWen (1992, 1995).
Both the fusion rule and the topological spin of quasi-particles as well as the chiral central charge are characterized in the non-Abelian geometric phases encoded in the degenerate ground statesKeski-Vakkuri and Wen (1993); Wen (1990, 1989); Wen and Niu (1990); Wen (2012); Liu et al. (2013); Moradi and Wen (2014); He et al. (2014), and vise versa. The non-Abelian geometric phases form a representation of , that is generated by 90°rotation and Dehn twist on a torus, which are called modular and matrices, respectivelyKeski-Vakkuri and Wen (1993); Wen (2012). The element of the modular matrix determines the mutual statistics of quasi-particles while the element of the matrix determines the topological spin and the chiral central chargeKeski-Vakkuri and Wen (1993); Wen (1990, 2012).
Given the fusion coefficients and the topological spin , we can write down the modular and matrices as the following expressions, and .Wang (2010) Here (called the quantum dimension of quasiparticle ) is the largest eigenvalue of matrix which is defined as and is the total quantum dimension, . We see that .
From Verlinde formulaVerlinde (1988), we can reconstruct the fusion coefficients, . Therefore, and provide a complete desciption and can be taken as the order parameter of topological ordersWen (2012); Liu et al. (2013); Moradi and Wen (2014); He et al. (2014).The modular and matrices satisfy the relations, and , where is a so-called charge conjugation matrix that satisfies . The central charge determines the thermal current of the edge state, , at temperature Affleck (1986) and is fixed up to bosonic quantum Hall states.
To fully characterize topological order, various numerical methods are proposed to access the modular and matricesZhang et al. (2012); Cincio and Vidal (2013); Tu et al. (2013); Zaletel et al. (2013); Zhu et al. (2013); Zhang and Qi (2014). Recently, one of us proposed the universal wave function overlap (UWFO) method to calculate modular matricesHung and Wen (2014); Moradi and Wen (2014). For a given set of degenerate ground-state wave functions, it provides us a practical method to extract the modular and matrices
where and are the operators that generate the 90° rotation and Dehn twist, respectively, on a torus with the lattice size. The exponentially small prefactor makes it difficult to numerically calculate the UWFO in (Modular matrices from universal wave function overlaps in Gutzwiller-projected parton wave functions). To avoid the exponential smallness, a gauge-symmetry preserved tensor renormalization method has been developed for the tensor-network wave functionsMoradi and Wen (2014); He et al. (2014), where the system size is effectively reduced as zero after the tensor renormalization.
Actually, in this letter, we will show that the non-universal exponent can be small such that the UWFO can be directly numerically calculated on relatively large systems. We will take a chiral spin liquid (CSL) wave function on the square latticeWen et al. (1989) as an explicit example to extract the modular and matrices from the UWFO. We construct the set of the ground states for a CSL by using Gutzwiller-projected parton wave functions (GPWF).Kalmeyer and Laughlin (1987); Wen (1991); Wen et al. (1989); Wen (1999); Zhang et al. (2012) We use the variational Monte Carlo to calculate the UWFO for the CSL wave functions. The hopping parameters are set as for the CSL on the -flux square lattice, where and for nearest neighbor and next nearest neighbor links, respectively. Since symmetry, the overlap in Eq. (Modular matrices from universal wave function overlaps in Gutzwiller-projected parton wave functions) has a vanishing exponent . in Eq. (Modular matrices from universal wave function overlaps in Gutzwiller-projected parton wave functions) has the relatively small non-universal complex exponent and the direct numeric computation is carried out on relatively large systems up to lattice size in this letter. The CSL is the lattice analogy of bosonic Laughlin stateKalmeyer and Laughlin (1987); Wen et al. (1989). Our numerical results confirm the analogy by directly extracting the modular and matrices from the UWFO.
In the parton construction, the spin operator is written in terms of fermionic parton operators, . Here is the Pauli matrices and () is the fermionic parton operator. We take the complex variables for the -site coordinate, , on a lattice. We have to impose the one-particle-per-site constraint for the partons, , such that the fermionic partons have the same Hilbert space on -site as the spin operators . The GPWF for the spin system can be read as
where the spin configuration and is the Gutzwiller projection operator to impose the one-particle-per-site constraint for the fermionic partons.
The GPWF can be put on a torus by implying the equivalence conditions: and , as shown in Fig. 1. The principal region of a torus is bounded by the four points . The torus is defined by two primitive vectors and . The shape of the torus is invariant under the transformations with and the generators ( and ) have the expressions
Two different constructions of GPWF for a CSL in the lattice analogy of bosonic Laughlin state can be found in Refs. Kalmeyer and Laughlin, 1987; Wen et al., 1989. In Ref. Kalmeyer and Laughlin, 1987, the parton wave functions are discretized integer quantum Hall states and we call it ideal GPWF for a CSL. On a torus, we can explicitly write down the ideal GPWF in terms the Laughlin-Jastrow wave functionsHaldane and Rezayi (1985)
where is the theta function and is the center-of-mass coordinate. Different ground states are specified by the different zeros, , in the center-of-mass wave functions. The zeros are determined by the general boundary conditions.Haldane and Rezayi (1985); Niu et al. (1985) The modular and matrices for the ideal GPWF in Eq.(4) can be analytically calculated by deformation the mass matrixWen (2012)
with the central charge , the same as those for the bosonic Laughlin state.
In Ref. Wen et al., 1989, the general GPWF for a CSL is written as
where is the determinate wave function for the fermionic partons filling the valence bands of the tight binding model
on the -flux square lattice with both nearest neighbor and next nearest neighbor hopping amplitude.Wen et al. (1989) There are flux in every triangle in the plaqutte, e.g. in in Fig. 1, . Different ground state wave functions can be obtained by different general boundary conditions. For the spin operator, the boundary condition is
with for . When we increase from to , the spin operators is invariant, however, the parton wave functions do not go back to themselves and lead to another ground state for GPWF. Therefore, we have different ground states for a CSL labeled by the spin fluxes in the holes of a torus ,
with . Actually only two of them are linearly independent.
For the general GPWF in Eq. (6), we use the UWFO in Eq. (Modular matrices from universal wave function overlaps in Gutzwiller-projected parton wave functions) to exact the modular matrices and . To carry out the UWFO, we need calculate the following overlaps
where is the sate in Eq. (8) and , , where and are the 90°rotation and Dehn twist transformations in Eq. (3) on a torus. The matrix has rank 2 with the numerical tolerance less than implying two-fold ground state degeneracy.
Given GPWFs, we implement the “sign trick”Zhang et al. (2011) to calculate the overlap
where is the amplitude wave function of the spin configuration in and the sign term
is calculated by Monte Carlo method according to the weight . The amplitude term is the normalization factor for weight
Actually, we are only interested in the ratios of amplitudes. For example, for matrix in Eq. (9), we evaluate the matrix-element amplitude ratios
according to the Monte Carlo sampling weight .
We set the mean field hopping parameters as , where and are for nearest neighbor and next nearest neighbor links, respectively. The overlap calculations are carried out on the systems with lattice sizes, . From the overlaps in Eq. (9), we follow the steps below to extract the modular and matrices. We first digonalize the matrix
Only two eigenvectors (e.g. and ) have non-zero eigenvalues around . These two states ( and ) are the linearly independent ground sates. In terms of the normalized , the overlaps for and in Eq. (9) turn out to be square matrices
Generally, is not diagonal since and are not the minimum entangled states or eigenstates of the Wilson loop operatorsZhang et al. (2012). We then diagonalize to obtain the minimum entangled states and
where is diagonal and the phases of are fixed according to the conditions and .
Since the CSL wave function has the 90° rotation symmetry, the exponent in in Eq.(16) vanishes, , that is confirmed in the numerical calculations. The UWFO of the matrix has a complex exponent in the prefactor. The real part of the exponent is easily obtained from the amplitude of the in Eq.(16) by fitting with respect to , , as shown in Fig. 2 (a). The phase is defined up , with . For , the corresponding integers are . From the fitting in Fig. 2 (b), we obtain . The central charge is sensitive to the exact value of as shown in Fig. 2 (c). The final result for the modular and matrices is
with the central charge , very close to the exact result for the ideal GPWF in Eq. (5).
Above we apply the UWFO method on the square lattice. For a general Bravais lattice, we can firstly map it onto an equivalent square lattice. We take the kagome lattice as example. We map the unit cell of the kagome lattice onto the one with square unit cell. Different sites within the unit cell are mapped onto different orbitals on the square lattice, as shown in Fig.3. Then we can make the modular transformations and on the square lattice torus. On the square lattice, we can also use Kadanoff block renormalization procedure to reduce the system size . Then the exponents in the prefactors of UWFO can be significantly reduced. Many local unitary transformations on the lattice can potentially reduce the exponents in the UWFO. If different ground state sectors have the same topological spins, we can follow Ref. Liu et al., 2013 to identify the minimum entangled states to diagonalize the modular matrix. The UWFO method is easily generalized to the 3+1D topological orders in the GPWFs. The GPWF for quantum dimer models in 3D has already been constructed in Ref. Ivanov et al., 2014. In 3D, the modular group of the 3-torus is generated by
We can use the UWFO to directly study the topological information in 3+1DMoradi and Wen (2014).
In conclusion, we use the universal wave function overlap method to exact the modular and matrices for the topological order in the Gutzwiller-projected parton wave function for the chiral spin liquid state on the square lattice. The chiral spin liquid is the lattice analogy of bosonic Laughlin state and the analogy is directly confirmed by the modular and matrices from the universal wave function overlap. The exponents in the prefactors of the wave function overlaps are found to be small and the variational Monte Carlo calculations are carried out on relatively large systems. The Monte Carlo calculations of the universal wave function overlap can be easily generalize to other Bravais lattices and 3+1D topological orders.
X-G. W is supported by NSF Grant No. DMR-1005541 and NSFC 11274192. He is also supported by the BMO Financial Group and the John Templeton Foundation. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research.
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