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arXiv:0812.4513

Gap Estimation by means of Hyperbolic Deformation

Hiroshi Ueda, Hiroki Nakano, Koichi Kusakabe, Tomotoshi Nishino

Published Dec 24, 2008• 1 citation

Abstract

We present a way of numerical gap estimation applicable for one-dimensional infinite uniform quantum systems. Using the density matrix renormalization group method for a non-uniform Hamiltonian, which has deformed interaction strength of $j$-th bond proportional to $\cosh λj$, the uniform Hamiltonian is analyzed as a limit of $λ\to 0$. As a consequence of the deformation, an excited quasi-particle is weakly bounded around the center of the system, and kept away from the system boundary. Therefore, insensitivity of an estimated excitation gap of the deformed system to the boundary allows us to have the bulk excitation gap $Δ(λ)$, and shift in $Δ(λ)$ from $Δ(0)$ is nearly linear in $λ$ when $λ\ll 1$. Efficiency of this estimation is demonstrated through application to the S=1 antiferromagnetic Heisenberg chain. Combining the above estimation and another one obtained from the technique of convergence acceleration for finite-size gaps estimated by numerical diagonalizations, we conclude that the Haldane gap is in $[0.41047905,~0.41047931]$.

DOI: 10.48550/arxiv.0812.4513

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We present a way of numerical gap estimation applicable for one-dimensional infinite uniform quantum systems. Using the density matrix renormalization group method for a non-uniform Hamiltonian, which has deformed interaction strength of

j

-th bond proportional to

\cosh λj

, the uniform Hamiltonian is analyzed as a limit of

λ\to 0

. As a consequence of the deformation, an excited quasi-particle is weakly bounded around the center of the system, and kept away from the system boundary. Therefore, insensitivity of an estimated excitation gap of the deformed system to the boundary allows us to have the bulk excitation gap

Δ(λ)

, and shift in

Δ(λ)

from

Δ(0)

is nearly linear in

λ

when

λ\ll 1

. Efficiency of this estimation is demonstrated through application to the S=1 antiferromagnetic Heisenberg chain. Combining the above estimation and another one obtained from the technique of convergence acceleration for finite-size gaps estimated by numerical diagonalizations, we conclude that the Haldane gap is in

[0.41047905,~0.41047931]

.

The one-dimensional Heisenberg antiferromagnet of S = 1 spins is studied by numericaldiagonalization method based on the Lanczos algorithm. The Haldane gap of this system isnumerically estimated from finite-size clusters under the twisted boundary condition up tothe 30-site system. The finite-size results are analyzed by a convergence-acceleration technique; the multistep procedure of the convergence acceleration is successfully carried out.The present estimate for the gap magnitude is more precisely than the previously reportedvalue from a Lanczos-diagonalization study. It also agrees well with the result reported ina study based on the density matrix renormalization group calculation. The present studyclarifies that Lanczos-diagonalization calculations can sufficiently contribute for preciselyestimating an energy gap although this method can treat only small clusters.

The one-dimensional Heisenberg antiferromagnets of large-integer-

S

spins are studied; their Haldane gaps are estimated by the numerical diagonalization method for

S=5

and

6

. We successfully obtain a monotonically increasing sequence of finite-size energy difference data corresponding to the Haldane gaps from the huge-scale parallel calculations of diagonalization under the twisted boundary condition and create a monotonically decreasing sequence within the range of system sizes treated in this study from the monotonically increasing sequence. Consequently, the gaps for

S=5

and

6

are estimated to be

0.000050 \pm 0.000005

and

0.0000030 \pm 0.0000005

, respectively. The asymptotic formula of the Haldane gap for

S\rightarrow\infty

is examined from the new estimates to determine the coefficient in the formula more precisely.

We introduce a one-parameter deformation for one-dimensional quantum lattice models, the hyperbolic deformation, where the scale of the local energy is proportional to cosh λj at the jth site. From the quantum-classical correspondence with a two-dimensional classical system, it is expected that the deformation does not modify the ground state conspicuously, if there is a finite excitation gap. Under this situation, the excited quasi particle is weakly confined around the center of the system. We derive scaling relations among the mean-square width 〈w2〉 of confinement, energy correction to the excitation gap Δ, and deformation parameter λ. This scaling is useful for the extraction of the bulk excitation gap in the limit λ = 0.

We carry out finite-size extrapolations of numerical-diagonalization data of the S =1 Heisenberg chain having a nonzero energy gap between the unique singlet ground state and the first excited state, namely the Haldane gap. Very precise estimates of the ground-state energy per site E g /N = −1.4014840447(39) and the staggered component of the magnetic structure factor S π = 3.864356(31) at T =0 are successfully obtained from the finite-size data of system sizes up to N = 24 under the twisted boundary condition by the sequence interval squeeze method, which was applied to a precise estimation of the Haldane gap by Nakano and Terai [J. Phys. Soc. Jpn. 78 (2009) 014003]. The present estimates are compared with other estimates in previous studies from various methods including the quantum Monte Carlo simulation and the density matrix renormalization group calculation.

We propose two methods of estimating a systematic error in extrapolation to the infinite-size limit in the study of measuring the Haldane gaps of the one-dimensional Heisenberg antiferromagnet with the integer spin up to S=5. The finite-size gaps obtained by numerical diagonalizations based on Lanczos algorithm are presented for sizes that have not previously been reported. The changes of boundary conditions are also examined. We successfully demonstrate that our methods of extrapolation work well. The Haldane gap for S=1 is stimated to be

0.4104789 \pm 0.0000013

. We successfully obtain the gaps up to S=5, which make us confirm the asymptotic formula of the Haldane gap in

S\to\infty

.