Appendix to “Phase diagram of the spin \(S\), SU(\(N\)) antiferromagnet on a square lattice”

The appendix for Chapter 3.

The quadratic Casimir eigenvalue in terms of the Young tableau

In this appendix, we discuss the relation between the Young tableau of an irreducible representation and the corresponding eigenvalue of the quadratic Casimir operator. For an irreducible representation, whose Young tableau has \(n_l\) rows of length \(\{l_i\}\) and \(n_c\) columns of length \(\{c_i\}\), the eigenvalue of the quadratic Casimir operator is [99]

(7)\[ C = \frac{1}{2} \left[ r\left(N-\frac{r}{N}\right)+\sum_i^{n_l}l_i^2-\sum_i^{n_c}c_i^2\right],\]

where \(r=\sum_i l_i=\sum_i c_i\) is the total number of boxes.

In this appendix, we also derive Eq. (7), which is stated in the Appendix of Ref. [99] without an explicit proof. First, we notice that in Eq. (7) there is an implicit choice of normalization. As we show below, such a normalization is consistent with Eq. (3.10).

For an irreducible representation, the value of \(C\) can be easily computed with Weyl’s formula [94][1],

(8)\[ C = \langle\Lambda|\Lambda + 2\delta\rangle,\]

where \(\Lambda\) is the maximum weight of the representation and \(\delta\) the Weyl vector. In the Dynkin representation the metric tensor of the scalar product is, up to a normalization \(\cal N\), the inverse of the transpose of the Cartan matrix \(A\) [94],

(9)\[\begin{split} \begin{split} G^{ij} &= {\cal N}\left[\left(A^T\right)^{-1}\right]_{ij},\\ \left[\left(A^T\right)^{-1}\right]_{ij} &= \text{min}(i,j)-\frac{ij}{N}, \end{split}\end{split}\]

and the Weyl vector is \(\delta=(1,1,\ldots, 1)\). To fix the normalization \({\cal N}\), we compute \(C\) for the defining representation, and match it with Eq. (3.10). For the defining representation, the maximum Dynkin weight is \(\Lambda_{\alpha_i}=\delta_{i,1}\), hence

(10)\[\begin{split} \begin{split} C &= {\cal N} \sum_{i,j=1}^{N-1}\delta_{i,1}\left[\text{min}(i,j)-\frac{ij}{N}\right]\left(\delta_{j,1} + 2\right)\\ &= {\cal N}\frac{N^2-1}{N}. \end{split}\end{split}\]

On the other hand, by taking the trace on both hand sides of Eq. (3.10), \(C\) is readily computed as \(C=(N^2-1)/(2N)\). Therefore the normalization constant is \({\cal N}=1/2\).

Employing Eq. (8), we first compute \(\langle \Lambda|\Lambda\rangle\). Using Eq. (3.2)

(11)\[ \langle \Lambda|\Lambda\rangle = \sum_{i,j=1}^{N-1}\left(l_i-l_{i+1}\right)G^{ij}\left(l_j-l_{j+1}\right),\]

where the metric tensor \(G^{ij}\) is given in Eq. (9). By developing the products and employing change of variables \(i\rightarrow i-1\), \(j\rightarrow j-1\), Eq. (11) can be written as

(12)\[\begin{split} \begin{split} \langle \Lambda|\Lambda\rangle =& l_1G^{11}l_1\\ &+l_1\sum_{j=2}^{N-1}\left(G^{1,j}-G^{1,j-1}\right)l_j\\ &+\sum_{i=2}^{N-1}l_i\left(G^{i,1}-G^{i-1,1}\right)l_1\\ &+\sum_{i,j=2}^{N-1}l_i\left(G^{i,j}-G^{i-1,j}-G^{i,j-1}+G^{i-1,j-1}\right)l_j, \end{split}\end{split}\]

where we have used that \(l_N=0\) for Young tableaux of \(\mathfrak{su}(N)\) representations. Using Eq. (9), we have

(13)\[ G^{1,j}-G^{1,j-1} = G^{i,1}-G^{i-1,1} = -\frac{1}{2N},\]

where we have employed the normalization \({\cal N}=1/2\) obtained after Eq. (10). Further, using Eq. (9), the difference in the parenthesis in the last term of Eq. (12) is computed as

(14)\[\begin{split} \begin{split} &G^{i,j}-G^{i-1,j}-G^{i,j-1}+G^{i-1,j-1} = \\ &\frac{1}{2}\Big[\text{min}(i,j)-\text{min}(i-1,j)\\ &-\text{min}(i,j-1)-\text{min}(i-1,j-1)-\frac{1}{N}\Big]. \end{split}\end{split}\]

By enumerating the various cases, it is easy to see that

(15)\[ \text{min}(i,j)-\text{min}(i-1,j)-\text{min}(i,j-1)-\text{min}(i-1,j-1)=\delta_{ij}.\]

Using Eqs. (13), (14) and (15) in Eq. (12), we obtain the first term in Weyl’s formula,

(16)\[\begin{split} \begin{split} \langle \Lambda|\Lambda\rangle =& \frac{l_1^2}{2}\left(1-\frac{1}{N}\right) -\frac{1}{N}l_1\left(r-l_1\right)\\ &+\frac{1}{2}\sum_{i=1}^{N-1}l_i^2-\frac{1}{2}l_1^2+\frac{1}{2N}\left(r-l_1\right)^2\\ =&\frac{1}{2}\left(\sum_{i=1}^{n_l}l_i^2-\frac{r^2}{N}\right), \end{split}\end{split}\]

where \(r=\sum_i l_i\) is the total number of boxes and the sum over \(l_i^2\) can be restricted to the \(n_l\) nonzero row lengths.

To compute the second term \(\langle \Lambda|2\delta\rangle\) in Weyl’s formula, we use a different parametrization of \(\Lambda\). Since in the Dynkin representation the components of \(\Lambda_{\alpha_i}\) are positive integers [see Eq. (3.2)], we can parametrize \(\Lambda_{\alpha_i}\) as

(17)\[ \Lambda_{\alpha_i} = \sum_{a=1}^{n_c}\delta_{i,c_a}.\]

The set \(\{c_a\}\) represents the position of the rows in the corresponding Young tableau where the number of boxes decreases on the following row. Such a decrease corresponds to the end of the column, hence \(\{c_a\}\) are the column lengths. Using Eq. (17) and Eq. (9) we have

(18)\[\begin{split} \begin{split} \langle\Lambda | 2\delta\rangle &=\frac{1}{2}\sum_{i,j=1}^{N-1}\sum_{a,b=1}^{n_c}\left(\delta_{i,c_a}\text{min}(i,j)2 - \delta_{i,c_a} \frac{ij}{N}2\right)\\ &=\sum_{a=1}^{n_c}\sum_{j=1}^{N-1}\text{min}(c_a,j)-\sum_{a=1}^{n_c}\sum_{j=1}^{N-1}\frac{c_aj}{N} \end{split}\end{split}\]

The first sum in Eq. (18) can be written as

(19)\[\begin{split} \begin{split} \sum_{a=1}^{n_c}\sum_{j=1}^{N-1}\text{min}(c_a,j) &= \sum_{a=1}^{n_c}\left(\sum_{j=1}^{c_a}j + \sum_{j=c_a+1}^{N-1}c_a\right)\\ &=\left(N-\frac{1}{2}\right)r-\frac{1}{2}\sum_{a=1}^{n_c}c_a^2, \end{split}\end{split}\]

where we have used \(\sum_a c_a=r\). The second sum in Eq. (18) can be computed as

(20)\[ \sum_{a=1}^{n_c}\sum_{j=1}^{N-1}\frac{c_aj}{N} = \frac{1}{2}r(N-1)\]

Inserting Eqs. (19) and (20) in Eq. (18), we obtain the second term of Weyl’s formula

(21)\[ \langle\Lambda | 2\delta\rangle = \frac{1}{2}\left(rN-\sum_{a=1}^{n_c}c_a^2\right)\]

Finally, employing Eqs. (16) and (21) in Eq. (8) one obtains Eq. (7).

As is known from the rules of Young tableaux of \(\mathfrak{su}(N)\) representations, columns of length \(N\) can be deleted since they correspond to an invariant under SU(\(N\)). This is consistent with the formula of Eq. (7). Indeed, by adding to a Young tableau a column of length \(N\), we have

(22)\[\begin{split} \begin{split} r&\rightarrow r + N,\\ l_i&\rightarrow l_i+1,\\ \sum_{i=1}^{n_c} c_i^2 &\rightarrow N^2 + \sum_{i=1}^{n_c} c_i^2. \end{split}\end{split}\]

Inserting the substitutions of Eq. (22) in Eq. (7), one can check that the Casimir eigenvalue is left unchanged.

Finally, it is easy to check that in the case of the defining representation, whose Young tableau is a single box, Eq. (7) gives the expected result, with the normalization consistent with Eq. (3.10).

Bound on the eigenvalue of the quadratic Casimir operator

The tensor product of \(2S\) self-adjoint antisymmetric representations given in Eq. (3.16) decomposes into different irreducible representations. In this appendix we prove that among those representations, the maximally symmetric one of Fig. 3.2 has the maximum Casimir eigenvalue, which we compute.

Due to the rules for the composition of Young tableaux, each of the irreducible representations arising from the tensor product has a Young tableau whose total number of boxes is \(r\le(2S)(N/2)=NS\) and whose row lengths cannot exceed \(2S\), \(l_i\le 2S\). Thus an upper bound for \(\sum_i l_i^2\) appearing in Eq. (7) is

(23)\[ \sum_i^{n_l}l_i^2 \le \sum_i^{n_l}2Sl_i = 2Sr.\]

This bound is saturated by

(24)\[ l_i=2S, \qquad n_l=r/(2S).\]

On the other hand, an upper bound for the second sum in Eq. (7) is found using the Cauchy-Schwartz inequality on the \(n_c-\)component vectors \((c_1,\ldots,c_n)\) and \((1,\ldots,1)\):

(25)\[ (c_1^2+\ldots+c_n^2)(1+\ldots+1) \ge (c_1+\ldots+c_n)^2.\]

The number of columns in the Young tableau \(n_c\) is bounded by \(n_c=\text{max}(\{l_i\})\le 2S\), and their sum is \(\sum c_i = r\). Hence, Eq. (25) gives

(26)\[ \sum_i^{n_c}c_i^2 \ge \frac{r^2}{n_c}\ge\frac{r^2}{2S}.\]

This bound is saturated by

(27)\[ c_i = r / (2S), \qquad n_c = 2S.\]

Inserting Eqs. (23) and (26) in Eq. (7) we get

(28)\[ C \le \frac{N+2S}{2} \left(-\frac{r^2}{2SN}+r\right)\le \frac{NS(2S+N)}{4},\]

where the upper bound is obtained for \(r=NS\). Together with Eqs. (24) and (27) this precisely corresponds to the Young tableau of Fig. 3.2 Its Casimir eigenvalue is most easily computed using Weyl’s formula (Eq. (8)) and Eq. (3.3), obtaining Eq. (3.22), which saturates the upper bound of Eq. (28). Alternatively, the Casimir eigenvalue can be obtained from Eq. (7), and \(r=NS\), \(l_1=l_2=\ldots=l_{N/2}=2S\) and \(c_1=c_2=\ldots=c_{2S}=N/2\).

Finally, we observe that, since the variables \(\{l_i\}\) and \(\{c_i\}\) in Eq. (7) are positive integers, as soon as we deviate from the solution maximizing \(C\), we decrease the Casimir eigenvalue by a finite integer amount. In other words, there is a finite gap \(O(1)\) in the eigenvalues of the quadratic Casimir operators between the subspace of the representation of Fig. 3.2 and the other irreducible representations arising from the tensor product of \(2S\) self-adjoint antisymmetric representations.

Systematic errors

Fig. 7 Scaling of systematic \(\Delta_\tau\) error for \(L=4\), \(\Theta = 2\), and \(S=1/2\). For each point, we simulated with a range of different values for \(\Delta\tau\) and fitted the energy to \(E(\Delta_\tau) = E_0 + \alpha {\Delta_\tau}^2\).

In this appendix we show that there is no explicit dependence on the magnitude of the Trotter error as a function of \(N\). To keep the notation simple, we will show this on the basis of the \(S=1/2\) Hamiltonian where \(\hat{H}_{\text{Casimir}}\) [see Eq. (3.19))] as well as the orbital index can be omitted:

(29)\[\begin{split} \begin{split} \hat{H}_{\text{QMC}} =&\hat{H}_J + \hat{H}_U \\ = &- \frac{J}{2N} \sum_{\langle i,j \rangle } \left\{ \hat{D}_{i,j}, \hat{D}^{\dagger}_{i,,j} \right\} + \frac{U}{N} \sum_{i}\left(\nop{i}-\frac{N}{2}\right)^2 \end{split}\end{split}\]

In this appendix, we have normalized the Hamiltonian by the factor \(\frac{1}{N}\), such that total energy differences defining e.g. the spin gap [\(E_0(S=1) - E_0\)] remain constant in the large-\(N\) limit. In particular, with the mean-field ansatz \(\chi_{i,j} = \frac{1}{N} \langle \hat{D}_{i,j} \rangle\) corresponding to the Affleck and Marston saddle point [119], the Hamiltonian reads:

(30)\[\begin{equation} \hat{H}_{\text{MF}} = - \frac{J}{2} \sum_{\langle i,j \rangle } \chi_{i,j} \hat{D}^{\dagger}_{i,,j} + \overline{\chi_{i,j}} \hat{D}^{}_{i,,j}. \end{equation}\]

In this large-\(N\) limit, one will check explicitly that the spin gap on a finite lattice is \(N\) independent, and that the energy is extensive in the volume, \(V\), and in \(N\). Since gaps are \(N\) independent, at least in the large-\(N\) limit, it makes sense comparing results at different \(N\) but at constant temperature or projection parameter.

In the formulation of the AF QMC method, one introduces a checkerboard decomposition, where the interaction terms are grouped into disjoint families of commuting operators. This factorization introduces a Trotter discretization error, whose \(N-\)dependence we estimate as follows. To render the calculation as simple as possible, we will consider as an illustration a one-dimensional chain. In this case, the checkerboard decomposition in even and odd bonds, \(b\), allows us to write the Hamiltonian as:

(31)\[\begin{equation} \hat{H} = \underbrace{\sum_{b \in A } \hat{h}_b}_{\equiv \hat{H}_A} + \underbrace{\sum_{b \in B } \hat{h}_b}_{\equiv \hat{H}_B}. \end{equation}\]

Both \(\hat{H}_A\) and \(\hat{H}_B\) are sums of commuting terms. \(\hat{h}_b\) corresponds to a local Hamiltonian, such that it is extensive in \(N\) but intensive in volume.

An explicit form of \(\hat{h}_b\) on a bond with legs \(i\),\(j\) in the fermion representation would read: \(\frac{1}{N} \left\{ \hat{D}_{(i,j)}, \hat{D}^{\dagger}_{(i,j)} \right\}\). Note that to keep calculations as simple as possible, we implicitly consider a one-dimensional lattice in which \(\hat{H}_A\) is a sum of commuting terms. For the two-dimensional case, the checkerboard bond decomposition necessitates four terms. We will use the symmetric Trotter decomposition,

(32)\[\begin{equation} e^{-\Delta \tau \hat{H} + \Delta \tau^3 \hat{R}_3} = e^{-\frac{\Delta \tau}{2} \hat{H}_A} e^{-\Delta \tau \hat{H}_B} e^{-\frac{\Delta \tau}{2}\hat{H}_A} + {\cal O } \left( \Delta \tau^5\right) \end{equation}\]

with \(\hat{R}_3 = \left( \left[ \hat{H}_A, \left[ \hat{H}_A, \hat{H}_B \right] \right] + 2 \left[ \hat{H}_B, \left[ \hat{H}_B, \hat{H}_A \right] \right] \right)/24\). Since \(\hat{H}_A\) and \(\hat{H}_B\) are sums of local operators, \(\hat{R}_3\) is also a sum of local operators. Hence, \(\hat{R}_3\) is extensive in the volume. By explicitly computing the commutators, one will also show, that \(\hat{R}_3 \) is extensive in \(N\). Hence \(\hat{R}_3\) scales as \(\hat{H}\). Note that for non-local Hamiltonians considered in Ref. [120], this does not apply. We can now compute the corrections to the free energy:

(33)\[\begin{equation} F_{\text{QMC}} = F - \Delta\tau^{2} \frac{\text{Tr} e^{-\beta \hat{H}} \hat{R}_3 } { \text{Tr} e^{-\beta \hat{H}} } + {\cal O} ( \Delta \tau^4). \end{equation}\]

Hence the quantity plotted in Fig. 7 corresponds to

(34)\[\begin{equation} \langle \hat{R}_3 \rangle / \langle \hat{H} \rangle. \end{equation}\]

It is intensive in \(N\) and \(V\), such that it has a well defined value in the large-\(N\) limit.

Bounds on the bond observable

In this appendix, we discuss a lower and upper bound for a bond observable \(\sum_a \hat{S}^{(a)}_i \hat{S}^{(a)}_j\), where \(i\) and \(j\) are two distinct lattice sites, not necessarily nearest neighbor. The bond observable can be expressed as

(35)\[\begin{split} \sum_a\hat{S}^{(a)}_i \hat{S}^{(a)}_j = \frac{1}{2}\sum_a \left(\hat{S}^{(a)}_i + \hat{S}^{(a)}_j\right)\left(\hat{S}^{(a)}_i + \hat{S}^{(a)}_j\right)\\ -\frac{1}{2}\sum_a\hat{S}^{(a)}_i\hat{S}^{(a)}_i -\frac{1}{2}\sum_a\hat{S}^{(a)}_j\hat{S}^{(a)}_j.\end{split}\]

With the choice of Eq. (3.9), the first term on the right-hand side of Eq. (35) is the quadratic Casimir element \(\hat{C}_{2,\Gamma_i\otimes \Gamma_j}\) of the tensor product of the two \(\mathfrak{su}(N)\) representations \(\Gamma_i\) and \(\Gamma_j\), at lattice sites \(i\) and \(j\) [compare with Eq. (3.10)]. The spectrum of \(\hat{C}_{2,\Gamma_i\otimes \Gamma_j}\) consists in the eigenvalues of the quadratic Casimir operator of all irreducible representations to which \(\Gamma_i\otimes\Gamma_j\) reduces. An upper bound is readily found by the maximally symmetric composition of \(\Gamma_i\) and \(\Gamma_j\), which corresponds to a Young tableau with \(N/2\) rows and \(4S\) columns; the proof is identical to that of App. B.2. Being a square of a hermitian operator, \(\hat{C}_{2,\Gamma_i\otimes \Gamma_j} \ge 0\). Such lower bound is saturated by the totally antisymmetric composition of \(\Gamma_i\) and \(\Gamma_j\), which corresponds to the trivial \(S=0\) representation. The second and third term on the right-hand side of Eq. (35) are the quadratic Casimir operator of the \(\mathfrak{su}(N)\) representation considered here, and take the value given in Eq. (3.22). Inserting the bounds on \(\hat{C}_{2,\Gamma_i\otimes \Gamma_j}\) discussed above in Eq. (35), we obtain

(36)\[ -C(N,S) \le \Braket{\sum_a\hat{S}^{(a)}_i \hat{S}^{(a)}_j} \le C(N, 2S)/2-C(N,S) = \frac{NS^2}{2}.\]

---

[1]

The Weyl’s formula for the Casimir element, Eq. (8), should not be confused with the Weyl’s formula for the dimension of an irreducible representation, Eq. (3.4).