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arXiv:1005.2427 · Physical Review B

Exact renormalization in quantum spin chains

Hong-Hao Tu, Mikel Sanz

Published Sep 3, 2010• 8 citations
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Abstract

We introduce a real-space exact renormalization-group method to find exactly solvable quantum spin chains and their ground states. This method allows us to provide a complete list for exact solutions within SU(2) symmetric quantum spin chains with $S\ensuremath{\le}4$ and nearest-neighbor interactions, as well as examples with $S=5$. We obtain two classes of solutions. One of them converges to the fixed points of renormalization group and the ground states are matrix-product states. Another one does not have renormalization fixed points and the ground states are partially ferromagnetic states.

DOI
10.1103/physrevb.82.104404

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We introduce a real-space exact renormalization-group method to find exactly solvable quantum spin chains and their ground states. This method allows us to provide a complete list for exact solutions within SU(2) symmetric quantum spin chains with S\ensuremath≤4S\ensuremath{\le}4S\ensuremath≤4 and nearest-neighbor interactions, as well as examples with S=5S=5S=5. We obtain two classes of solutions. One of them converges to the fixed points of renormalization group and the ground states are matrix-product states. Another one does not have renormalization fixed points and the ground states are partially ferromagnetic states.
We present a numerical method based on real-space renormalization that outputs the exact ground space of ``frustration-free'' Hamiltonians. The complexity of our method is polynomial in the degeneracy of the ground spaces of the Hamiltonians involved in the renormalization steps. We apply the method to obtain the full ground spaces of two spin systems. The first system is a spin-1/2 Heisenberg model with four-spin cyclic-exchange interactions defined on a square lattice. In this case, we study finite lattices of up to 160 spins and find a triplet ground state that differs from the singlet ground states obtained by Batista and Trugman, [Phys. Rev. Lett. 93, 217202 (2004)]. We characterize such a triplet state as consisting of a triplon that propagates in a background of fluctuating singlet dimers. The second system is a family of spin-1/2 Heisenberg chains with uniaxial exchange anisotropy and next-nearest-neighbor interactions. In this case, the method finds a ground-space degeneracy that scales quadratically with the system size and outputs the full ground space efficiently. Our method can substantially outperform methods based on exact diagonalization and is more efficient than other renormalization methods when the ground-space degeneracy is large.
We introduce a family of spin-1/2 quantum chains, and show that their exact ground states break the rotational and translational symmetries of the original Hamiltonian. We also show how one can use projection to construct a spin-3/2 quantum chain with nearest neighbor interaction, whose exact ground states break the rotational symmetry of the Hamiltonian. Correlation functions of both models are determined in closed form. Although we confine ourselves to examples, the method can easily be adapted to encompass more general models.
We study the ground state of one- and two-dimensional (square-lattice) spin-1/2 quantum antiferromagnets using a numerical real-space renormalization-group (RG) approach. In our RG approach we consider blocks of various sizes but with an odd number of sites; we retain only the doublet ground state and we integrate out the higher-energy states by means of second-order quasidegenerate perturbation theory. That is, we assume that the role of the excited states of a block, in the RG iteration process, is to renormalize the effective coupling parameters between blocks. We compute the ground-state energy of a spin-1/2 linear chain for various block sizes and find close agreement with the Bethe-ansatz exact solution. In the case of the spin-1/2 square-lattice quantum antiferromagnet, the obtained ground-state energy is in reasonable agreement with the available numerical estimates.
We present a real-space renormalization group approach for the corner Hamiltonian, which is relevant to the reduced density matrix in the density matrix renormalization group. A set of self-consistent equations that the renormalized Hamiltonian should satisfy in the thermodynamic limit is also derived from the fixed point of the recursion relation for the corner Hamiltonian. We demonstrate the renormalization group algorithm for the S=1/2S=1/2S=1/2 XXZ spin chain and show that the results are consistent with the exact solution. We further examine the renormalization group for the S=1 Heisenberg spin chain and then discuss the nature of the eigenvalue spectrum of the corner Hamiltonian for the non-integrable model.
An interaction-round-a-face density-matrix renormalization-group (IRF-DMRG) method is developed for higher integer spin chain models which are rotational invariant. The expressions of the IRF weights associated with the nearest-neighbor spin-SSS interaction ${\mathbf{S}}_{i}\ensuremath{\cdot}{\mathbf{S}}_{i+1}$ are explicitly derived. Using the IRF-DMRG with these IRF weights, the Haldane gaps $\ensuremath{\Delta}$ and the ground state energy densities e0{e}_{0}e0​ for both S=1S=1S=1 and S=2S=2S=2 isotropic antiferromagnetic Heisenberg quantum spin chains are calculated by keeping up to only m=90m=90m=90 states.
A real-space, zero-temperature renormalization-group method is used to study a Heisenberg antiferromagnetic spin chain with long-range interactions varying as ${(\mathrm{distance})}^{\ensuremath{-}p}$. Simple renormalization-group equations are obtained and studied analytically, and the results are checked against a more accurate numerical calculation. The calculations provide evidence for a phase transition at $p\ensuremath{\approx}1.85$ characterized by a change in the behavior of zero-temperature correlation functions. For $p\ensuremath{\gtrsim}1.85$ the large-distance physics is that of the nearest-neighbor antiferromagnet, while there is a line of fixed points for $p\ensuremath{\lesssim}1.85$. I conjecture that $p\ensuremath{\approx}1.85$ also marks an order-disorder transition at finite temperature such as occurs in the Ising model with power-law interactions.