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arXiv:1103.2735 · Physical Review B

Matrix product state based algorithm for determining dispersion relations of quantum spin chains with periodic boundary conditions

B. Pirvu, Jutho Haegeman, Frank Verstraete

Published Jan 31, 2012• 42 citations
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Abstract

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-$1/2$, and the bilinear-biquadratic spin-1 models.

DOI
10.1103/physrevb.85.035130

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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.
A variational ansatz for momentum eigenstates of translation-invariant quantum spin chains is formulated. The matrix product state ansatz works directly in the thermodynamic limit and allows for an efficient implementation (cubic scaling in the bond dimension) of the variational principle. Unlike previous approaches, the ansatz includes topologically nontrivial states (kinks, domain walls) for systems with symmetry breaking. The method is benchmarked using the spin-\textonehalf{} XXZXXZXXZ antiferromagnet and the spin-1 Heisenberg antiferromagnet, and we obtain surprisingly accurate results.
We present a matrix product state (MPS) algorithm to approximate ground states of translationally invariant systems with periodic boundary conditions. For a fixed value of the bond dimension DDD of the MPS, we discuss how to minimize the computational cost to obtain a seemingly optimal MPS approximation to the ground state. In a chain with NNN sites and correlation length $\ensuremath{\xi}$, the computational cost formally scales as $g(D,\ensuremath{\xi}/N){D}^{3}$, where $g(D,\ensuremath{\xi}/N)$ is a nontrivial function. For $\ensuremath{\xi}\ensuremath{\ll}N$, this scaling reduces to D3{D}^{3}D3, independent of the system size NNN, making our method NNN times faster than previous proposals. We apply the algorithm to obtain MPS approximations for the ground states of the critical quantum Ising and Heisenberg spin-1/21/21/2 models as well as for the noncritical Heisenberg spin-111 model. In the critical case, for any chain length NNN, we find a model-dependent bond dimension D(N)D(N)D(N) above which the polynomial decay of correlations is faithfully reproduced throughout the entire system.
Based on the matrix product state (MPS) formalism, we introduce an ansatz for capturing excited states in finite systems with open boundary conditions, providing a very efficient method for computing, e.g., the spectral gap of quantum spin chains. This method can be straightforwardly implemented on top of an existing density-matrix renormalization group or MPS ground-state code. Although this approach is built on open-boundary MPSs, we also apply it to systems with periodic boundary conditions. Despite the explicit breaking of translation symmetry by the MPS representation, we show that momentum emerges as a good quantum number and can be exploited for labeling excitations on top of MPS ground states. We apply our method to the critical Ising chain on a ring and the classical Potts model on a cylinder. Finally, we apply the same idea to compute excitation spectra for 2D quantum systems on infinite cylinders. Again, despite the explicit breaking of translation symmetry in the periodic direction, we recover momentum as a good quantum number for labeling excitations. We apply this method to the 2D transverse-field Ising model and the half-filled Hubbard model; for the latter, we obtain accurate results for, e.g., the hole dispersion for cylinder circumferences up to eight sites.
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.
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.