Quantum spin models for the SU(n)_1 Wess-Zumino-Witten model

We propose 1D and 2D lattice wave functions constructed from the SU(n)_1 Wess-Zumino-Witten (WZW) model and derive their parent Hamiltonians. When all spins in the lattice transform under SU(n) fundamental representations, we obtain a two-body Hamiltonian in 1D, including the SU(n) Haldane-Shastry model as a special case. In 2D, we show that the wave function converges to a class of Halperin's multilayer fractional quantum Hall states and belongs to chiral spin liquids. Our result reveals a hidden SU(n) symmetry for this class of Halperin states. When the spins sit on bipartite lattices with alternating fundamental and conjugate representations, we provide numerical evidence that the state in 1D exhibits quantum criticality deviating from the expected behaviors of the SU(n)_1 WZW model, while in 2D they are chiral spin liquids being consistent with the prediction of the SU(n)_1 WZW model.

Before constructing the wave functions, let us briefly review the SU(n) 1 WZW model [67]. This rational CFT has n primary fields, denoted by Λ a , with a = 0, 1, . . . , n − 1, corresponding to particular SU(n) irreducible representations. The primary field Λ 0 is an SU(n) singlet, which is also the identity field with conformal weight h(Λ 0 ) = 0. The next primary field Λ 1 is the SU(n) fundamental representation, corresponding to a single box when the SU(n) irreducible representations are represented as the Young tableaux. In general, the primary field Λ a corresponds to a Young tableau with a single column and a rows. Accordingly, Λ a consists of dim Λ a = n a components, and we write these components as Λ a,α , where α ∈ {1, 2, . . . , dim Λ a }.
As we shall discuss further below, the SU(n) 1 WZW model has a free-field representation with n − 1 free bosons. In this representation, the primary fields are conveniently realized using vertex operators.
To build lattice wave functions, we consider N T spins sitting at the fixed positions z j (j = 1, 2, . . . , N T ) in the complex plane. Following Ref. [56], we define lattice wave functions |Ψ = α1,α2,...,αN T 0|Λ a1,α1 (z 1 )Λ a2,α2 (z 2 ) . . . Λ aN T ,αN T (z NT )|0 |α 1 , α 2 , . . . , α NT (2) that are chiral correlators of primary fields. Here, |0 is the vacuum of the CFT and |α j are the basis vectors of the internal state of spin number j. CFT states of the form (2) can be seen as a special type of matrix product states in which the finite-dimensional matrices have been replaced by infinite-dimensional conformal fields. They are therefore sometimes referred to as infinite-dimensional-matrix product states (IDMPS). Regarding the wave function (2), there are several comments in order. First, choosing the primary field Λ aj at site j requires that the spin at this site also transforms under the SU(n) irreducible representation corresponding to a Young tableau with one column and a j rows. Note that the SU(n) 1 WZW model does not have primary fields corresponding to a Young tableaux with more than one column. Secondly, the fusion rules in (1) always have a unique fusion outcome, which ensures that the wave function (2) is a unique function. Lastly, to have a nonvanishing wave function, the N T primary fields in (2) must fuse into the identity Λ 0 (i.e. the SU(n) singlet), Λ a1 ⊗ Λ a2 ⊗ · · · ⊗ Λ aN T = Λ 0 .
In this work, we shall focus on the case, where each of the spins belong either to the SU(n) fundamental representation Λ 1 or to the SU(n) conjugate representation Λ n−1 . We shall denote the sublattice of spins transforming under the fundamental (conjugate) representation by A (B), A : Fundamental representation, B : Conjugate representation, (4) and we shall let N (N ) denote the number of spins in A (B) such that N +N = N T . The condition (3) then gives that (N −N )/n must be an integer, and we shall assume this to be the case throughout. Note that the fundamental and conjugate representations are the same for n = 2, so that there is only one state in this particular case. For n ≥ 3, however, they are different. Before we continue with the above case, let us note that other choices for the primary fields are possible. For instance, for even n, one could use the primary field Λ n/2 (self-conjugate representation) to build the wave function, according to the fusion rule Λ n/2 ⊗ Λ n/2 ⊗ · · · ⊗ Λ n/2 = Λ 0 (N T even). For the SU(4) case, one has SU(4) 1 ≃ SO(6) 1 and the SU(4) self-conjugate primary field Λ 2 becomes the vector representation of SO(6) with conformal weight h(Λ 2 ) = 1/2, which can be interpreted as a Majorana field and has been considered in Ref. [59]. Although we only consider states constructed from the fundamental and conjugate representations below, we note that the formalism we develop is general and that other cases can be treated in a similar way.
In the following, we shall find it convenient to use the notation ϕ αj (z j ) = Λ 1,αj (z j ) for j ∈ A Λ n−1,αj (z j ) for j ∈ B .
Since we shall often refer to the wave function, for which all the primary fields belong to the fundamental representation, we shall give this wave function a particular name: |Ψ F . Explicit representations of |Ψ F and |Ψ will be discussed in Secs. III and V, respectively. In the next two subsections, we shall use their abstract forms to derive relevant null fields and their corresponding decoupling equations, which are our starting point for deriving parent Hamiltonians.

B. Null vectors
As a rational CFT, the SU(n) 1 WZW model has null vectors in its Verma modules of the Kac-Moody algebra. According to Ref. [56], identifying proper null vectors and deriving decoupling equations for the chiral correlators are the key for constructing parent Hamiltonians of the wave functions. In this subsection, we derive the null vectors relevant for (7).
The SU(n) 1 Kac-Moody algebra is defined by where J a m = 0 dz 2πi z m J a (z) is the mth mode of the Kac-Moody current J a (z) and f abc are the structure constants of the SU(n) Lie algebra. Here and later on, we shall always assume that repeated indices are summed over. The operator product expansion (OPE) between the Kac-Moody currents and a primary field is [67] where the matrices t a with elements (t a ) αβ are the generators of SU(n) in the representation of the primary field. Let us note here that the generators in the fundamental and conjugate representations are related though a complex conjugation and a multiplication by a minus sign, i.e., SU (2) SU (3) SU(4) where τ a are the generators in the fundamental representation (see Appendix A).
To the primary field ϕ α (z), one associates a primary state |ϕ α satisfying the following properties [67]: and descendant states are obtained by multiplying |ϕ α by any number of current operators J a n with n < 0. A null state is a state that is at the same time a descendant and a primary state. Since the wave function (7) only involves primary fields belonging to the fundamental or the conjugate representation, we shall here only need to deal with the two Verma modules formed by the corresponding primary states, as well as their descendants.
Let us first consider the primary field Λ 1,α (z) belonging to the fundamental representation. In Virasoro level m = 1, we look for null vectors with the following form: where W q,aα can be interpreted as Clebsch-Gordan coefficients satisfying aα W * q,aα W q ′ ,aα = δ qq ′ . They come from the tensor product decomposition of the (n 2 − 1)-dimensional SU(n) adjoint representation (carried by J a −1 ) and the fundamental representation (carried by |Λ 1,α ), where the irreducible representations are denoted by their dimensions (they are not distinguished with their complex conjugate representations). Fig. 1(a) shows the tensor product decomposition (13) for n = 2, 3, 4, using the Young tableaux. We have found that, for SU(n) 1 WZW model with all n, null vectors indeed exist in Virasoro level m = 1, and they belong to the SU(n) representation with dimension 1 2 n(n− 1)(n+ 2) in (13). In practice, the Clebsch-Gordan coefficients W q,aα in (12) can be determined by requiring the null vector condition For our purpose, we redefine the null vectors as [56] |χ a,α = q W * q,aα |χ q where (K F ) aα bβ is given by K F can be viewed as a matrix with its entries being (K F ) aα,bβ = (K F ) aα bβ , and it is a projector (i.e. K 2 F = K F ) onto the SU(n) irreducible representation with dimension 1 2 n(n − 1)(n + 2). This also lead to an additional equation, These two equations are sufficient for determining the explicit form of (K F ) a b . For general n, we obtain where d abc is a totally symmetric tensor (see Appendix A). If we build the null vector at Virasoro level m = 1 using the primary state |Λ n−1,α in the conjugate representation, the representations appearing in (13) would be their complex conjugate representations. See Fig. 1(b) for this tensor product decomposition for n = 3 and 4. As a result, the Clebsch-Gordan coefficients in (12) for obtaining the null vectors would be their complex conjugate. Then, the corresponding null vectors can be written as where (K C ) aα bβ = (K * F ) aα bβ . Utilizing (10), we can combine (14) and (17) into a single expression where K aα bβ are the matrix elements of Here, r = +1 for the fundamental representation, r = −1 for the conjugate representation, and t c are the generators in the considered representation.

C. Decoupling equations
Following Ref. [56], a set of decoupling equations can be derived for the chiral correlator (7) using the null vectors (18). These decoupling equations provide operators annihilating the wave functions, which can be used to build parent Hamiltonians.
The null state (18) corresponds to the following null field: By definition of the null field, substituting it into the wave function (7), one obtains a vanishing expression After deforming the integral contour and using the OPE (9) between the Kac-Moody currents and primary fields, we arrive at where (K (i) ) a b denotes the operator K a b in (19) acting on spin number i and (t b j ) αj α ′ j denote the matrix elements of the operator t b acting on spin number j. (Note that the representation chosen for t b j is the same as the representation of spin number j.) Thus, the resulting decoupling equation yields a set of operators which annihilate the wave function |Ψ , i.e. P a i (z 1 , . . . , z N )|Ψ = 0 ∀i, a. Together with the fact that |Ψ is a global SU(n) singlet, T a |Ψ = 0 with T a = i t a i , we obtain where C a i (z 1 , . . . , z N ) is given by and w ij = (z i + z j )/(z i − z j ). For SU(2), we have d abc = 0 and (25) recovers the result in Ref. [56]. Utilizing the formulas in Appendix A, we get which is a convenient form for constructing parent Hamiltonians.

D. Vertex operator representation
After working out the decoupling equations for (7) using an abstract form of the primary fields, we now turn to an explicit representation of these primary fields, using chiral vertex operators. This is possible, since SU(n) 1 WZW model is equivalent to a free theory of n − 1 massless bosons.
For our purpose, it is convenient to label the spin states in each site by their weights (eigenvalues of the Cartan generators). The state |α , α ∈ {1, 2, . . . , n}, in the fundamental representation is therefore characterized by n − 1 quantum numbers, which we collect into the vector m α given explicitly by , . . .
In the conjugate representation, the state |α , α ∈ {1, 2, . . . , n}, is characterized by the quantum numbers − m α . The SU(3) and SU(4) weight diagrams are shown in Fig. 2 as examples. Using the weights, the primary field ϕ α (z) can be expressed as where r = +1 for the fundamental representation and r = −1 for the conjugate representation as above. The colons denote normal ordering and φ(z) is a vector of n − 1 independent fields of free, massless bosons. The factor κ α is a Klein factor, commuting with the vertex operators and satisfying Majorana-like anticommutation relations Here, (m1, m2, m3) is shorthand notation for the components of the vectors mα and − mα, respectively.
Note that κ α is the same in the fundamental and in the conjugate representation. At this moment, the meaning of these Klein factors is not clear. In fact, their role is to ensure that the wave function (7) is an SU(n) singlet. We will go back to this point when discussing the wave functions in Sec. III and Sec. V. Let us note that the vertex operators in (28) have the anticipated conformal weights, since Another quantity, which will be used in later sections, is m α · m α ′ with α = α ′ . It is easy to convince ourselves that this value does not depend on the states we choose. For α = α ′ , we find Altogether, we thus conclude

III. QUANTUM STATES FROM THE FUNDAMENTAL REPRESENTATION OF SU(n)
In this section, we analyze the wave function (7) in detail, both theoretically and numerically, for the case where all spins transform under the fundamental representation. First, the chiral correlator can be evaluated and expressed in terms of a product of Jastrow factors [67] Ψ F (α 1 , α 2 , . . . , α N ) = χ(α 1 , α 2 , . . . , α N )δ where χ(α 1 , α 2 , . . . , α N ) = κ α1 κ α2 · · · κ αN is a z j -independent phase factor to be determined below and the Kronecker delta function δ i mα i =0 , which is 1 for i m αi = 0 and zero otherwise, ensures charge neutrality. Referring to Eq. (27), we observe that the charge neutrality forces the number of spins N α in the state |α to fulfill N 1 = N 2 = . . . = N n . This gives N α = N/n for all α, and we shall therefore assume N/n to be an integer whenever we consider states constructed from only the fundamental representation of SU(n). Utilizing (32), we note that (33) simplifies to We shall also find it useful to express the state |Ψ F in another notation. For a given spin configuration j , where j = 1, 2, . . . , N/n, be the position within the ket of the jth spin in the state |α . For example, if we choose n = 3 and N = 9 and consider the state ket |1, 2, 1, 3, 2, 3, 3, 2, 1 , we would have x . We can then express |Ψ F as where S N is the symmetric group over the elements {1, 2, . . . , N } and Let us next determine χ from the condition that |Ψ F should be an SU(n) singlet. We shall find below that the wave function |Ψ F is proportional to the ground state of the SU(n) HS model if we choose z j = e 2πij/N and where the right-hand side of (37) is the sign of the permutation needed to transform x Since the ground state of the SU(n) HS model is an SU(n) singlet, it follows that (37) is the correct choice of χ for all choices of z j . The result (37) can be obtained from χ = κ α1 κ α2 · · · κ αN by choosing the factors κ α to be Klein factors, which satisfy the Majorana-like anticommutation relation (29), and choosing to work in a sector, in which κ 1 κ 2 · · · κ n = 1. This follows from The proof given in Appendix B shows directly that the state (34) with χ given by (37) and z j arbitrary is an SU(n) singlet without referring to the SU(n) HS model.

A. Wave functions in the hardcore boson basis
In order to compare the state (33) to known models in particular limits, we shall now express the state in a hardcore boson basis. In this picture, the coordinates z j are lattice sites that can be empty or occupied by at most one hardcore boson. A spin in the state |n is interpreted as an empty site, and a spin in the state |α , with α ∈ {1, 2, . . . , n − 1}, is interpreted as a site occupied by a hardcore boson with color α.
Referring to (27), we observe that the (n − 1)th component m αj ,n−1 of the vector m αj can be written as where p j is one if α j ∈ {1, 2, . . . , n − 1} and zero if α j = n. In other words, we can use this component to distinguish between occupied sites and holes, and we shall use this observation to eliminate the coordinates of the unoccupied sites from the Jastrow factor in (33). The part of this factor that includes the contribution from m αj ,n−1 can be written as The second factor in the above expression can be simplified as [57,60] i<j where Let us next consider the part of the Jastrow factor that includes the contributions from m αj ,l with l = 1, 2, . . . , n − 2.
Utilizing (27) and (32), we find If α or α ′ is equal to n, we instead get n−2 l=1 m α,l m α ′ ,l = 0 as follows immediately from (27). The part of the Jastrow factor that includes the contributions for m αj ,l with l = 1, 2, . . . , n − 2 can therefore be written as Combining (40) and (44), we get the expression for the Jastrow factor. We would like to also remove the hole coordinates from the sign factor χ. Doing so gives rise to a sign factor that compensates the factor j (−1) (j−1)pj in the wave function. The remaining sign factor is then sgn(x N/n ). We note, however, that this factor can be absorbed by rearranging the ordering in the Jastrow factor. Putting everything together, we thus conclude that the state (35) can also be written as where the sum is over all possible distributions of the (n − 1)N/n colored bosons on the N lattice sites with at most one boson per site, and with α, β ∈ {1, 2, . . . , n − 1} and i, j ∈ {1, 2, . . . , N/n}. We shall now comment further on (47) for particular choices of the lattice.

Jastrow wave functions for the uniform 1D lattice
We first consider a uniform lattice in 1D with periodic boundary conditions, which is achieved by choosing z j = e 2πij/N . For this particular case, we have the simple expression f N (z [56]. Inserting this in (47), we see that the wave function for the particular case of a uniform 1D lattice reduces to the ground state of the SU(n) HS Hamiltonian [42][43][44] Let us next consider a regular lattice in 2D. We shall assume that the area of each lattice site (defined as the area of the region consisting of all points that are closer to the given lattice site than to any other lattice site) is the same for all lattice sites. In this case, it has been shown in [60] that (49) for N large. The state (47) can therefore be written as in the thermodynamic limit, where Up to a local phase factor that can be removed with a simple transformation (of both the wave function and the parent Hamiltonian), we thus observe that the wave function (34) reduces to the lattice version of the Halperin state [69], which appeared in the context of the multilayer FQH effect. For example, the SU(3) state corresponds to Halperin's 221 double-layer spin-singlet state. One consequence of this interesting connection is that the wave function (34) describes an SU(n) chiral spin liquid state, supporting Abelian anyonic excitations (the same as those in Halperin states). Another consequence is that the particular series of Halperin FQH states in (50) have a hidden enhanced SU(n) symmetry. For instance, one may expect that the chiral gapless edge excitations of these states are described by the SU(n) 1 WZW model.

B. Numerical results
Since the properties of the uniform 1D SU(n) HS state are already well-known, we shall here only investigate the states in 2D. We compute the TEE −γ by considering the state on an R × L square lattice on the cylinder and using the formula [62,63,70] for the entanglement entropy of half of the cylinder. In (52), we assume the cut to be perpendicular to the cylinder axis, L is the number of spins along the cut, and the formula is valid asymptotically for large L and R. The mapping of the IDMPS (36) to a cylinder is done through a conformal transformation, which amounts to choosing and considering r j and l j as the coordinates rather than Re(z j ) and Im(z j ). This will also change the chiral correlator by a constant factor, but we can ignore this, since the factor does not depend on the state of the spins. The square lattice is then obtained by choosing r j ∈ {−R/2+1/2, −R/2+3/2, . . ., R/2−1/2} and l j ∈ {1, 2, . . . , L} and N = RL.
Since it is easier to compute numerically, we choose to consider the Renyi entropy with index 2, which is defined as S and interpreting as a classical probability distribution. The results of the computations are shown as a function of the number of spins along the cut in Fig. 3. The figure provides evidence for n = 3 and n = 4 that the TEE is −γ = − ln(n)/2. This is consistent with the prediction that the states in 2D are chiral spin liquid states, with the SU(n) 1 WZW model being their corresponding chiral edge CFT: According to the fusion rule (1) of the SU(n) 1 WZW model, the states support n types of Abelian anyons with quantum dimension 1, giving rise to a total quantum dimension √ n.

IV. PARENT HAMILTONIANS FOR THE STATES FROM THE FUNDAMENTAL REPRESENTATION
In this section, we derive parent Hamiltonians of the states Ψ F in Eq. (33). In 1D, we obtain two-body parent Hamiltonians, including the SU(n) HS model as a special case, and for 2D lattices they are parent Hamiltonians of the SU(n) chiral spin liquid states.
Our starting point is the fact that the operator C a i in (26) annihilates the state (33) as derived above, C a i |Ψ F = 0. It follows that the positive semi-definite Hermitian operator is therefore a parent Hamiltonian of (33), H|Ψ F = 0. Inserting (26) in (56) and utilizing the formulas listed in Appendix A, we obtain the explicit expression which is valid for general z j .

A. Exchange form of the parent Hamiltonian
As we shall now show, H can also be expressed in terms of the exchange operator P ij , which swaps the spin states at sites i and j, i.e., P ij = n α,β=1 |α i , β j β i , α j |. To do so, we define the following fermionic representation of the SU(n) generators with the local constraint n α=1 c † iα c iα = 1 for all i. Using Fierz identity (A9), we can then express the SU(n) Heisenberg interaction in terms of P ij . Inserting this into (57), we get

B. SU(n) Hamiltonian in 1D
In this subsection, we restrict ourselves to 1D systems. This is done by restricting all z j to lie on the unit circle in the complex plane, i.e. |z j | = 1 ∀j. When this is the case, we have w * ij = −w ij , and using (A5), the 1D Hamiltonian therefore takes the form For the particular case n = 2, the three-body term vanishes because d abc = 0, and we recover the Hamiltonian in Eq. (70) of [56].
In the following, we simplify (61). First, by using the cyclic identity w ij w ik + w ji w jk + w ki w kj = 1, we find and i =j =k Inserting these relations into (61), utilizing that T a = i t a i and w 2 ij = 1 + 4zizj (zi−zj ) 2 , the parent Hamiltonian (61) for the state (33) with |z j | = 1, ∀j, can be written as where E 1D is given by Here let us remind that (64) directly comes from (56) and H 1D |Ψ F = 0. Since |Ψ F is an SU(n) singlet, we have d abc T a T b T c |Ψ F = T a T a |Ψ F = 0. Thus, we could get rid of the three-body and two-body Casimirs in (64) and define a pure two-body parent Hamiltonian which has (33) as its ground state with ground-state energy The Hamiltonian (66) is an inhomogenous generalization of the SU(n) HS model. For n = 2, it reduces to the SU(2) inhomogenous HS model derived in [55].

C. 1D uniform Hamiltonian and the SU(n) HS model
We now further restrict z j to be uniformly distributed on the unit circle by choosing z j = e 2πij/N . This gives a uniform 1D lattice with periodic boundary conditions. In this case, i =j The 1D uniform parent Hamiltonian is therefore whose ground-state energy is given by We note that the first term in (70) is given by where H HS is the 1D SU(n) HS model In the thermodynamic limit N → ∞, we have ( N π ) 2 sin 2 π N (i − j) → 1/(i − j) 2 and the strength of SU(n) exchange interaction in (73) is inversely proportional to the square of the distance between the spins.
Then, we can write the uniform 1D parent Hamiltonian as Since T a |Ψ F = 0, the ground-state energy of H HS is given by The 1D uniform parent Hamiltonian thus practically reduces to the 1D SU(n) HS model.

Energy spectra of the SU(n) HS model
For the SU(n) HS model, it has been shown [72] that it has a hidden Yangian symmetry, generated by the total spin operator T a and the operator We note that Λ a = n+1 n+2 i C a i , which thus annihilates |Ψ F as well. It is known [72] that T a and Λ b both commute with H HS , but they do not mutually commute, which is responsible for the huge degeneracies in the spectra of H HS .
The eigenvalues of the SU(n) HS model have been obtained in [72]. Combining (59) and (72), we rewrite the SU(n) HS Hamiltonian as where It has been shown [72] that the complete set of eigenvalues of H Haldane can be obtained by the simple formula where Here m i are distinct integer rapidities satisfying m i ∈ [0, N ] ∀i. Physically, the sum of these rapidities is proportional to the lattice momenta of the energy eigenstate |{m i } [72] According to [72], there is a simple rule for finding physically allowed sets of rapidities Using (79) and (81), the energy and lattice momenta of the ground state are therefore given by and Note that the ground-state energy E Haldane determined in this way is consistent with (75) by taking into account the constant term in (77).

Identifying CFT from finite-size spectra
CFT gives a powerful prediction for the spectra of 1D critical spin chains. In particular, it is known that the eigenenergies of a critical quantum chain with N sites and with periodic boundary conditions are given by [73,74] where ε ∞ is the ground-state energy per site in the thermodynamic limit, v is the spin-wave velocity, c is the central charge, h andh are conformal weights of the primary fields, and n l and n r are non-negative integers. For the SU(n) HS model, the spin-wave velocity and the conformal weights of the primary fields can be determined directly by the finite-size spectra obtained from (79). To show this, we consider the SU(n) HS Hamiltonian in (77). Let us start with an excitation defined through the rapidities which is obtained by removing the particle "1" in the ground-state configuration (82). Using (79) and (81), we obtain the excitation energy E ′ and the lattice momentum P ′ of this excitation and Comparing to the CFT prediction of the finite-size spectra (85), this excited state corresponds to h =h = n r = 0 and n l = 1. Thus, we obtain the spin-wave velocity However, let us note that the central charge c cannot be obtained using (85). The reason is that the SU(n) HS Hamiltonian has long-range interactions, which allow an N -dependent constant term and the ground-state energy as a function of N could violate the CFT prediction (85).
where [qn + a], with q = 0, . . . , N n − 1, denotes the missing rapidities in the rapidity set. By using (79) and (81), we obtain the excitation energies and lattice momenta of the corresponding excited states and Note that these excitations are gapless in the thermodynamic limit N → ∞. Compared to the CFT prediction (85), these excited states correspond to h a =h a and n l = n r = 0. Comparing with (85), we obtain the conformal weights h a = a(n−a) 2n , which correspond to the primary fields Λ a of the SU(n) 1 WZW model. This also agrees with the known results [72,75,76] that the SU(n) 1 WZW model describes the low-energy physics of the SU(n) HS model.
Regarding the excited states of the SU(n) HS model, one remaining interesting question is to obtain their explicit form and to relate them with the rapidity description in (79). Some of these excited states have already been obtained in Refs. [77,78]. As a further remark, we note that the gapless excitations at lattice momenta P = a 2π n with a ∈ {1, . . . , n} are also known to exist in the SU(n) ULS model [39][40][41], which belongs to the same SU(n) 1 WZW universality class [79][80][81][82].

D. SU(n) Hamiltonian in 2D
In this subsection, we discuss the parent Hamiltonian in 2D. After multiplying by an overall constant 2(n+1) (n−1)(n+2) , the 2D Hamiltonian in (57) can be written as Note that this Hamiltonian can be defined on any 2D lattice (both regular or irregular) and does not rely on a particular lattice geometry. For SU(2), we have d abc = 0 and f abc t a i t b j t c k = t i · ( t j × t k ). Then, (93) reduces to the parent Hamiltonian in [57] for the ν = 1/2 lattice Laughlin state. This state is also known as the Kalmeyer-Laughlin state [64,65], whose parent Hamiltonian has been extensively studied [57,[83][84][85][86][87][88][89]. From the parent Hamiltonian, it becomes transparent that the chiral three-spin interaction term t i · ( t j × t k ), which explicitly breaks time-reversal and parity symmetries, stabilizes the spin-1/2 Kalmeyer-Laughlin state. Recently, it has been found [61,90,91] that Hamiltonians with short-range chiral three-spin interactions can already stabilize the Kalmeyer-Laughlin state. This is very encouraging, as such short-range Hamiltonian might be realized in cold atomic systems in optical lattices [61,92].
The SU(n) parent Hamiltonian (93) also has three-body interactions. Compared to the SU(2) case, one remarkable feature is that, the three-body coupling is suppressed by a factor of 1/(n − 1). This gives us a hint that, for large n, one may have a chance to drop the three-body terms and the (long-range) Hamiltonian with two-body Heisenberg interactions may stabilize the lattice Halperin state (50) as its ground state. However, as the number of terms in the three-body interactions d abc t a i t b j t c k and f abc t a i t b j t c k also increases with n, it is unclear whether the parent Hamiltonian can be adiabatically connected to the long-range Heisenberg model without closing the gap. Clarifying whether the gap closes in this interpolation is an interesting problem and certainly deserves further investigation.
Finally, for the 2D SU(n) Heisenberg model on a square lattice with only nearest-neighbor interactions, it has been argued [93,94] that chiral spin liquid supporting Abelian anyons becomes stable in the large n limit. Thus, it would be interesting to further explore its possible connection with our wave function (33).

V. QUANTUM STATES FROM THE FUNDAMENTAL AND CONJUGATE REPRESENTATIONS OF SU(n)
In this section, we turn to the more general situation, where we use both the fundamental and the conjugate representation to construct IDMPSs. In this case, the chiral correlator (7) evaluates to where χ(α 1 , α 2 , . . . , α N +N ) = κ α1 κ α2 · · · κ α N +N is a z j -independent phase factor that we shall determine below and By using (32), we can also express the chiral correlator in the simpler form Considering (27), we observe that the charge neutrality condition δ N +N i=1 ri mα i =0 yields . . . where N α (Nᾱ) is the number of spins in the fundamental (conjugate) representation in the state |α . Together with the conditions we thus conclude that must hold for all nonzero terms in the wave function. This is consistent with our previous observation that (N −N )/n must be an integer. χ is determined from the requirement that |Ψ must be a singlet state, i.e. T a |Ψ = 0, where T a = N +N i=1 t a i . We show explicitly in Appendix B that this condition is fulfilled for where x (α,ᾱ) i is the position within the ket of the ith spin that is in the state |α without distinguishing between the fundamental and the conjugate representation. As for the case, where only the fundamental representation is used, we can obtain (103) by demanding κ α in (28) to be Klein factors.

A. Numerical results
We next investigate the states numerically for lattices with alternating fundamental and conjugate representations. We start with the uniform 1D case, where we use the fundamental representation on all the odd sites and the conjugate representation on all the even sites (see Fig. 4). Let us consider the entanglement entropy of a block of L consecutive spins, where L is even. We compute this quantity by Monte Carlo simulations as explained in Sec. III B, and the result is shown in Fig. 5. We observe that the entanglement entropy grows logarithmically. The CFT prediction for the entanglement entropy of a critical 1D system is [95][96][97] and by using this formula as a fit, we obtain the central charges c = 1.5 for n = 3 and c = 1.7 for n = 4, respectively. Next we compute the correlation function by using the Metropolis Monte Carlo algorithm. Here, t 3 is the third SU(n) generator, which we choose such that  with c = n − 1. The numerically estimated critical exponents of the two-point correlation function also differ from 2(n − 1)/n, the expected value for critical spin chains described by the SU(n) 1 WZW model. One possibility for these deviations is that the system is still described by the SU(n) 1 WZW model, but in the presence of marginally irrelevant perturbations. Another possibility is that the system belongs to another universality class which is sharply different from the SU(n) 1 WZW model. In the present framework, it is rather difficult to distinguish these possibilities. In Sec. VI C, we propose a short-range Hamiltonian where critical ground states belonging to the same universality class are likely to appear and which is easier to analyze in practice and may shed light on the correct critical theory. Another integrable U q [sl(2|1)] superspin chain with alternating representations 3 and3 has been studied in [98], which exhibits several critical theories depending on the parameters of the Hamiltonian. There could be a connection between these results and our results.
We now turn to the 2D state on a square lattice on the cylinder, with fundamental and conjugate representations in a checkerboard pattern (see Fig. 4). In Fig. 7, we compute the TEE following the same approach as in Sec. III B. The results are in agreement with −γ = − ln(n)/2 within the precision of the computation. Similar to the SU(n) which is valid for general z j . Note that this reduces to our previous result (57) for r j = +1 ∀j. We also observe that (106) does not depend on r j for n = 2. This happens because the fundamental representation and the conjugate representation are the same representation for n = 2.

A. 1D parent Hamiltonian
We now specialize to 1D by forcing all z j to fulfill |z j | = 1. This gives w * ij = −w ij . We therefore obtain the 1D parent Hamiltonian By using (A5) and (A7), we find and by using (62) and the definition of T a , we get Inserting these expressions in the expression for the Hamiltonian leads to where

B. 1D uniform parent Hamiltonian
For the 1D uniform case, z j = exp(i 2π NT j). By using (68) and (69), the parent Hamiltonian therefore simplifies to where We plot examples of spectra of H 1D uniform in Fig. 8. The spectra show that the ground state is unique.
One important motivation of studying long-range parent Hamiltonians is that they may shed light on the physics of some short-range realistic Hamiltonians. As we already mentioned, the SU(n) HS Hamiltonian with inversesquare interactions and the SU(n) ULS model with only nearest-neighbor interactions belong to the same SU(n) 1 WZW universality class. For other long-range parent Hamiltonians constructed for the SU(2) k and SO(n) 1 WZW models [56,59,99,100], the corresponding short-range Hamiltonians are the SU(2) spin-k 2 Takhtajan-Babujian models [101,102] and the SO(n) Reshetikhin models [103,104], respectively. Regarding the SU(n) parent Hamiltonian (112) with both fundamental and conjugate representations, the natural question one may ask is whether there exist shortrange Hamiltonians belonging to the same universality class. In fact, finding such short-range Hamiltonians can also be very useful for clarifying the unsolved issue in Sec. V A on identifying the critical theory of these models.
To address this problem, we restrict ourselves to the 1D uniform case with alternating fundamental and conjugate representations (see Fig. 4). Following the strategy in [61], we truncate the long-range interactions in (112) by keeping only two-body interactions between nearest-neighbor and next-nearest-neighbor sites, as well as three-body interaction terms among three consecutive sites. In the thermodynamic limit, N T → ∞, this procedure yields the following Hamiltonian: By using (A4) and (A6), the three-body interaction term can be rewritten as and then the truncated Hamiltonian is expressed as There is no guarantee that the truncated Hamiltonian with precisely the coupling constants in (116) has the same physics as the long-range parent Hamiltonian (112). However, the form of (116) suggests that a candidate short-range Hamiltonian which shares the same physics might be found in the J 2 − J 3 SU(n) spin chain with J 2 , J 3 being close to the couplings in (116). We have performed an exact diagonalization of the Hamiltonian in (117) for n = 3 and N T = 10 sites. Fig. 9 shows the overlap | Ψ J2−J3 |Ψ | between the ground state |Ψ J2−J3 of (117) and the state |Ψ defined in (96). The maximum overlap (marked with a circle in Fig. 9) is 0.9998 and occurs for J 2 = 0.557 and J 3 = −0.536. These values are quite close to J 2 = 0.467 and J 3 = −0.400 predicted by the truncated Hamiltonian (116).
Let us also mention several solvable cases in (117), which are useful for understanding the phase diagram and are also interesting on their own right. One known solvable point in (117) is the pure SU(n) Heisenberg chain with J 2 = J 3 = 0, which has gapped dimerized ground states for n ≥ 3 [105,106]. In Fig. 9, this Heisenberg point is marked with a plus sign. Motivated by a recent work [107], we have also identified another class of solvable cases in (117), which have perfectly dimerized ground states and can be viewed as SU(n) generalizations of the spin-1/2 Majumdar-Ghosh model [108]. These SU(n) Majumdar-Ghosh Hamiltonians are written as where K 1 , K 2 > 0 and which, on a periodic chain with even N T sites, have ground-state energy E MG = [(n + 1)(n − 2)K 1 + (n − 1)(n + 2)K 2 ]N T /(2n 2 ). In Fig. 9, the Majumdar-Ghosh Hamiltonian (118) is shown as a straight line terminated at J 2 = −3, J 3 = 3 and J 2 = 6/5, J 3 = −3/5. This line seems to be at a phase boundary between two different phases. Fully clarifying the phase diagram of (117) requires extensive numerics. This is beyond the scope of the present work and we leave it for a future study.

VII. CONCLUSION
In summary, we have constructed a family of spin wave functions with SU(n) symmetry from CFT, and we have used the CFT properties of the states to derive parent Hamiltonians in both 1D and 2D. The states are defined on arbitrary lattices, and each of the spins transforms under either the fundamental or the conjugate representation of SU(n). For the case, where all spins in the model transform under the fundamental representation, our results provide a natural generalization of the SU(n) HS model from a uniform lattice in 1D to nonuniform lattices in 1D and to 2D. For the nonuniform 1D case, the Hamiltonian can be chosen to consist of only two-body terms. In 2D, the states reduce to Halperin type wave functions in the thermodynamic limit. This suggests that these states are chiral spin liquids with Abelian anyons, and we find numerically that the total quantum dimension is close to √ n. It also shows that a class of Halperin states have an SU(n) symmetry and provides parent Hamiltonians that can stabilize these topological states.
We have also investigated the case with alternating fundamental and conjugate representations numerically. In 1D, our results suggest that the state is critical, but the central charges and the exponents of the correlation functions deviate from the results expected for the SU(n) 1 WZW model. In 2D, we find a nonzero TEE, and the extracted total quantum dimension is √ n, which is consistent with the SU(n) 1 WZW model predictions. For the case with alternating fundamental and conjugate representations, we have proposed a short-range Hamiltonian for the 1D uniform case and solved it exactly for particular choices of the parameters. Given that it is possible in many related models with long-range Hamiltonians to find short-range Hamiltonians that describe practically the same low-energy physics, it is likely that the proposed short-range Hamiltonian has a ground state in the same universality class as the constructed SU(n) wave functions for certain choices of the parameters.
Note added.-During the preparation of this manuscript, we learned that related results have been obtained by R. Bondesan and T. Quella [109]. where all elements that are not shown are zero. For the A sites (fundamental representation), we therefore have and for the B sites (conjugate representation), we have Altogether, Let us define The term in |Ψ ′ having N 1 + 1 spins in the state |1 in the fundamental representation, N1 spins in the state |1 in the conjugate representation, N 2 − 1 spins in the state |2 in the fundamental representation, and N2 spins in the state |2 in the conjugate representation at given positions has coefficient 1→N2−1 }, . . . , {x 1→N1 , x 1→j−1 , x j ) is the index of the jth spin in the state |α in the fundamental (conjugate) representation. We define the order operator O as (B5) Note that 1→N2−1 }, . . . , {x 1→N2 }, . . .) 1→N2−1 }, . . . , {x 1→N2 }, . . .).