The Non-Abelian Density Matrix Renormalization Group Algorithm

We describe here the extension of the density matrix renormalization group algorithm to the case where Hamiltonian has a non-Abelian global symmetry group. The block states transform as irreducible representations of the non-Abelian group. Since the representations are multi-dimensional, a single block state in the new representation corresponds to multiple states of the original density matrix renormalization group basis. We demonstrate the usefulness of the construction via the spin 2 Heisenberg antiferromagnet with SU(2) symmetry. PACS No. 75.10.Jm, 75.40.Mg Typeset using REVTEX 1 In past years, the density matrix renormalization group (DMRG) method [1] has been extensively used to study one and two dimensional strongly correlated electron systems [2]. This method became very popular when it was realised that it enabled a level of numerical accuracy for one dimensional systems that was not possible using other methods [3]. One major drawback of DMRG is that calculations are performed in a subspace of purely abelian symmetries, such as the U(1) symmetries of total particle number and the z component of the total spin. Bearing this in mind, most of the low-lying states corresponding to higher total spin values of one dimensional fermionic Hamiltonians cannot be directly calculated [3]. One can only obtain a few states in different total particle number and z component of total spin sectors [4]. For models where ferromagnetism emerges the situation worsens, that is, to determine magnetization, a combination of methods must be employed which will artificially raise the energy of the higher spin state [5] within the chosen z component total spin sector. In recognising the imperative need, to introduce a DMRG method which has a total spin quantum number naturally implemented, a number of unsuccessful attempts were previously made (e.g., for the spin 1 Heisenberg model [6] and t-t’-U model [7]). The focus of our study was to seriously investigate this issue. As a result, we successfully present a DMRG algorithm which explicitly uses non-Abelian global symmetries. We will not attempt here to give a complete description of the density matrix renormalization group (DMRG) algorithm, instead we refer the reader to the original description by White [1] and more recent reviews [3]. Instead we concentrate on the essential elements of the algorithm that require modification when using non-Abelian symmetries. These are the construction of tensor product basis and operators (whether it is through adding a single site to a block, or joining blocks to construct a superblock), and the truncation of block states via the reduced density matrix. We introduce the method by way of the Lie group SU(2). This symmetry is readily applicable to all quantum spin systems that can be written in a form that does not break rotational symmetry. In principle, it is not difficult to calculate eigenstates of SU(2) for a 2 finite system by using the Clebsch-Gordan transformation [8]. Especially in DMRG, when the system is built one or two lattice sites at a time, constructing SU(2) eigenstates in this fashion presents no major difficulty. In this form, the tensor product of two basis vectors, labelled here by subscripts 1 and 2, is