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arXiv:1512.02007 · Physical review. B./Physical review. B

Symmetries and entanglement in the one-dimensional spin- 12 XXZ model

Mykhailo V. Rakov, Michael Weyrauch, Briiissuurs Braiorr-Orrs

Published Feb 22, 2016• 11 citations
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Abstract

An efficient and stable algorithm for $U(1)$ symmetric matrix product states (MPS) with periodic boundary conditions (PBCs) is proposed. It is applied to a study of correlation and entanglement properties of the eigenstates of the spin-$\frac{1}{2}$ XXZ model with different spin projections. Convergence properties and accuracy of the algorithm are studied in detail.

DOI
10.1103/physrevb.93.054417

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An efficient and stable algorithm for U(1)U(1)U(1) symmetric matrix product states (MPS) with periodic boundary conditions (PBCs) is proposed. It is applied to a study of correlation and entanglement properties of the eigenstates of the spin-12\frac{1}{2}21​ XXZ model with different spin projections. Convergence properties and accuracy of the algorithm are studied in detail.
We discuss in detail a modified variational matrix product state algorithm for periodic boundary conditions, based on a recent work by Pippan et al (2010 Phys. Rev. B 81 081103(R)), which enables one to study large systems on a ring (composed of N ∼ 10 2 sites). In particular, we introduce a couple of improvements allowing us to enhance the algorithm in terms of stability and reliability. We employ such a method to compute the stiffness of one-dimensional strongly correlated quantum lattice systems. The accuracy of our calculations is tested in the exactly solvable spin-1/2 Heisenberg chain.
We study a matrix product state algorithm to approximate excited states of translationally invariant quantum spin systems with periodic boundary conditions. By means of a momentum eigenstate ansatz generalizing the one of \"Ostlund and Rommer [see S. \"Ostlund and S. Rommer, Phys. Rev. Lett. 75, 3537 (1995);S. Rommer and S. \"Ostlund, Phys. Rev. B 55, 2164 (1997)], we separate the Hilbert space of the system into subspaces with different momentum. This gives rise to a direct sum of effective Hamiltonians, each one corresponding to a different momentum, and we determine their spectrum by solving a generalized eigenvalue equation. Surprisingly, many branches of the dispersion relation are approximated to a very good precision. We benchmark the accuracy of the algorithm by comparison with the exact solutions and previous numerical results for the quantum Ising, the antiferromagnetic Heisenberg spin-1/21/21/2, and the bilinear-biquadratic spin-1 models.
We investigate the use of matrix product states (MPS) to approximate ground states of critical quantum spin chains with periodic boundary conditions (PBC). We identify two regimes in the (N,D)(N,D)(N,D) parameter plane, where NNN is the size of the spin chain and DDD is the dimension of the MPS matrices. In the first regime, MPS can be used to perform finite size scaling (FSS). In the complementary regime, the MPS simulations show instead the clear signature of finite entanglement scaling (FES). In the thermodynamic limit (or large NNN limit), only MPS in the FSS regime maintain a finite overlap with the exact ground state. This observation has implications on how to correctly perform FSS with MPS as well as on the performance of recent MPS algorithms for systems with PBC. It also gives clear evidence that critical models can actually be simulated very well with MPS by using the right scaling relations; in the appendixes, we give an alternative derivation of the result of Pollmann et al. [Phys. Rev. Lett. 102, 255701 (2009)] relating the bond dimension of the MPS to an effective correlation length.
We introduce an efficient method to calculate the ground state of one-dimensional lattice models with periodic boundary conditions. The method works in the representation of matrix product states (MPS), related to the density matrix renormalization group (DMRG) method. It improves on a previous approach by Verstraete et al. We introduce a factorization procedure for long products of MPS matrices, which reduces the computational effort from m5{m}^{5}m5 to m3{m}^{3}m3, where mmm is the matrix dimension, and $m\ensuremath{\simeq}100--1000$ in typical cases. We test the method on the S=12S=\frac{1}{2}S=21​ and S=1S=1S=1 Heisenberg chains. It is also applicable to nontranslationally invariant cases. The method makes ground-state calculations with periodic boundary conditions about as efficient as traditional DMRG calculations for systems with open boundaries.
Efficient MPS Algorithm for Periodic Boundary Conditions and Applications Section: Solid matter Original Author's Text: English Abstract: We present the implementation of an efficient algorithm for the calculation of the spectrum of one-dimensional quantum systems with periodic boundary conditions. This algorithm is based on a matrix product representation for quantum states (MPS) and a similar representation for Hamiltonians and other operators (MPO). It is significantly more efficient for systems of about 100 sites and more than for small quantum systems. We apply the formalism to calculate the ground state and the first excited state of a spin-1 Heisenberg ring and deduce the size of a Haldane gap. The results are compared to previous high-precision DMRG calculations. Furthermore, we study the spin-1 systems with a biquadratic nearest-neighbor interaction and show the first results of an application to a mesoscopic Hubbard ring of spinless fermions, which carries a persistent current. Key words: matrix product representation for quantum states (MPS) and Hamiltonians (MPO), spin-1 Heisenberg ring, density matrix renormalization group (DMRG). List Issues Configure
We present the implementation of an efficient algorithm for the calculation of the spectrum of one-dimensional quantum systems with periodic boundary conditions. This algorithm is based on a matrix product representation for quantum states (MPS) and a similar representation for Hamiltonians and other operators (MPO). It is significantly more efficient for systems of about 100 sites and more than for small quantum systems. We apply the formalism to calculate the ground state and the first excited state of a spin-1 Heisenberg ring and deduce the size of a Haldane gap. The results are compared to previous high-precision DMRG calculations. Furthermore, we study the spin-1 systems with a biquadratic nearest-neighbor interaction and show the first results of an application to a mesoscopic Hubbard ring of spinless fermions, which carries a persistent current.