arXiv:cond-mat/0101401 · Physical Review Letters

Ground-State Phase Diagram of 2D Electrons in a High Landau Level: A Density-Matrix Renormalization Group Study

Naokazu Shibata, D. Yoshioka

Published Jun 18, 2001• 148 citations

Abstract

The ground-state phase diagram of 2D electrons in a high Landau level (index $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$) is studied by the density-matrix renormalization group method. Pair correlation functions are systematically calculated for various filling factors from $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/8$ to $1/2$. It is shown that the ground-state phase diagram consists of three different charge density wave states called stripe phase, bubble phase, and Wigner crystal. The boundary between the stripe and the bubble phases is determined to be ${\ensuremath{\nu}}_{c}^{\mathrm{s}\mathrm{\ensuremath{-}}\mathrm{b}}\ensuremath{\sim}0.38$, and that for the bubble phase and Wigner crystal is ${\ensuremath{\nu}}_{c}^{\mathrm{b}\mathrm{\ensuremath{-}}\mathrm{W}}\ensuremath{\sim}0.24$. Each transition is of first order.

DOI: 10.1103/physrevlett.86.5755

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arXiv:cond-mat/0101401 · Physical Review Letters

Ground-State Phase Diagram of 2D Electrons in a High Landau Level: A Density-Matrix Renormalization Group Study

Naokazu Shibata, D. Yoshioka

Published Jun 18, 2001• 148 citations

arXiv HTML PDF

Abstract

The ground-state phase diagram of 2D electrons in a high Landau level (index

N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2

) is studied by the density-matrix renormalization group method. Pair correlation functions are systematically calculated for various filling factors from

\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/8

1/2

. It is shown that the ground-state phase diagram consists of three different charge density wave states called stripe phase, bubble phase, and Wigner crystal. The boundary between the stripe and the bubble phases is determined to be

{\ensuremath{\nu}}_{c}^{\mathrm{s}\mathrm{\ensuremath{-}}\mathrm{b}}\ensuremath{\sim}0.38

, and that for the bubble phase and Wigner crystal is

{\ensuremath{\nu}}_{c}^{\mathrm{b}\mathrm{\ensuremath{-}}\mathrm{W}}\ensuremath{\sim}0.24

. Each transition is of first order.

DOI: 10.1103/physrevlett.86.5755

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The ground-state phase diagram of 2D electrons in a high Landau level (index

N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2

) is studied by the density-matrix renormalization group method. Pair correlation functions are systematically calculated for various filling factors from

\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/8

1/2

{\ensuremath{\nu}}_{c}^{\mathrm{s}\mathrm{\ensuremath{-}}\mathrm{b}}\ensuremath{\sim}0.38

, and that for the bubble phase and Wigner crystal is

{\ensuremath{\nu}}_{c}^{\mathrm{b}\mathrm{\ensuremath{-}}\mathrm{W}}\ensuremath{\sim}0.24

. Each transition is of first order.

We investigate the quantum Hall charge density wave states in the

N=2

Landau level under mass anisotropy. In the absence of anisotropy, there are two first-order phase transitions between the Wigner crystal, the

2

-electron bubble phase, and the stripe phase. By adding mass anisotropy, our analytical and numerical studies show that the

2

-electron bubble phase disappears and the stripe phase significantly enlarges its domain in the phase diagram. Meanwhile, a regime of stripe crystals that may be observed experimentally is unveiled after the bubble phase gets out. Upon increase of the anisotropy, the energy of the phases at the transitions becomes progressively smooth as a function of the filling. We conclude that all first-order phase transitions are replaced by continuous phase transitions, providing a possible realisation of continuous quantum crystalline phase transitions.

U

at ${U}_{\mathrm{c}1}\ensuremath{\sim}7.8t$ and ${U}_{\mathrm{c}2}\ensuremath{\sim}9.9t$ (

t

being the hopping integral), indicating that there are three phases separated by first-order transitions. The absence of any singularity in physical quantities for $0\ensuremath{\le}U<{U}_{\mathrm{c}1}$ implies a metallic phase in this regime. For

U>{U}_{\mathrm{c}2}

, the local spin density induced by an applied pinning magnetic field exhibits a three sublattice feature, which is compatible with the ${120}^{\ensuremath{\circ}}$ N\'eel-ordered state realized in the limit of $U\ensuremath{\rightarrow}\ensuremath{\infty}$. For

{U}_{\mathrm{c}1}<U<{U}_{\mathrm{c}2}

, a response to the applied pinning magnetic field is comparable to that in the metallic phase with a relatively large spin correlation length, but showing neither valence bond nor chiral magnetic order, which therefore resembles gapless spin liquid. However, the spin structure factor for the intermediate phase exhibits the maximum at the

\mathrm{K}

and ${\mathrm{K}}^{\ensuremath{'}}$ points in the momentum space, which is not compatible to spin liquid with a large spinon Fermi surface. The calculation also finds that the pairing correlation function monotonically decreases with increasing

U

and thus the superconductivity is unlikely in the intermediate phase.