Derivation of matrix product states for the Heisenberg spin chain with open boundary conditions

Using the algebraic Bethe Ansatz, we derive a matrix product representation of the exact Bethe-Ansatz states of the six-vertex Heisenberg chain (either $XXX$ or $XXZ$ and spin-$\frac{1}{2}$) with open boundary conditions. In this representation, the components of the Bethe eigenstates are expressed as traces of products of matrices that act on a tensor product of auxiliary spaces. As compared to the matrix product states of the same Heisenberg chain but with periodic boundary conditions, the dimension of the exact auxiliary matrices is enlarged as if the conserved number of spin-flips considered would have been doubled. This result is generic for any non-nested integrable model, as is clear from our derivation, and we further show this by providing an additional example of the same matrix product state construction for a well-known model of a gas of interacting bosons. Counterintuitively, the matrices do not depend on the spatial coordinate despite the open boundaries, and thus they suggest generic ways of exploiting (emergent) translational invariance both for finite size and in the thermodynamic limit.

Figure 1

Example of a typical MPS coefficient generated by an iterative singular-value decomposition. In the generic case of all nonzero singular values, the $A_i^{{\sigma }_i}$ matrix sizes telescope as $(1\times d),(d\times d^2),\cdots ,(d^{[L/2]}\times d^{[L/2]}),\cdots ,(d^2\times d),(d\times 1)$, where at the center of the chain there will be one single square matrix if the number of sites, $L$, is odd or a pair of matrices of transposed dimensions if $L$ is even [17] (the square brackets denote the integer part of $L/2$). Here $d$ is the dimension of the local Hilbert space for each lattice site—we have $d=2$ for the case of the spin-$\frac{1}{2}$ chains that we study and as depicted in the figure. The dashed matrix in the center represents the product of all the central matrices that, in principle, telescope to very large sizes. In practical implementations, these need to be truncated and are replaced by uniformly square matrices of size $D\times D$, where $D$ is called the bond dimension. This makes the practical OBC-MPS Ansatz in the bulk of the chain look similar to the one used for PBCs discussed in the text.

Figure 2

Schematic depiction of the traced looped monodromy matrix used to define the transfer matrix in the formulation of ABA with an OBC. The vertical lines indicate the local physical Hilbert spaces for the different spins along the chain, and they are labeled with the site index. The thicker horizontal lines correspond to the auxiliary space used in the definition of the monodromy matrix, and the ending dots are the points contracted when tracing over the auxiliary space to obtain the transfer matrix. Each crossing of a physical and an auxiliary space corresponds to an $L$-operator insertion, and the adjoining lines correspond to the respective index contractions. The first drawing (top) depicts the standard formulation using a single auxiliary space. The second drawing (bottom) shows how the vertical lines can be made into straight lines by spitting the auxiliary space into two also straight lines and reversing the order of the products in the lower one via matrix transpositions. The two-auxiliary-space formulation connects to the one-auxiliary-space one via the identification of a tensor $\tilde{Q}$ (green box) that merges the two spaces [see Eq. (21)] in a way that naturally connects with the tracing procedure. The matrix $Q$ was not included in the graphical representation; it can be added when closing the trace and combined with $\tilde{Q}$ into $\mathcal{Q}$; cf. Figs. 3 and 4.

Figure 3

Schematic depiction of the tensor-network structure underlying the ABA construction of exact eigenstates for the six-vertex Heisenberg spin chain with an OBC. The different nodes are labeled using the same notation as in the text. The vertical tensor contraction along each column of the network bracketed with $\left.〈{\sigma }_i\right|$ gives the matrices $A_i^{{\sigma }_i}$ that enter into the MPS rewriting of the eigenstates (i.e., $C_n$ and $D_n$). The horizontal dimension corresponds to the auxiliary space(s) of the ABA and also of the MPS construction—the so-called bond dimension. Those horizontal bonds are eventually all traced out, but in an unusual way due to the introduction of the matrix $\tilde{Q}$. The remaining piece of the $Q$-matrix sits on the leftmost column, is built as a pure tensor product, and has the combined functions of a projector into the given number of excitations ($n$) and boundary-condition-changing operator imposing DWBCs in the auxiliary space.

Figure 4

Another schematic depiction of a tensor-network structure for the six-vertex Heisenberg spin chain, but this time for PBCs. The different nodes are labeled, mutatis mutandis, using a similar notation as in the text. The ABA construction is simpler in this case, and the auxiliary or bond dimension is traced out in the conventional way. Notice one still finds a $Q$-matrix, the leftmost column in the network, that fixes the excitation number and modifies the horizontal boundary conditions of the ABA six-vertex construct.