Physical review. B, Condensed matter

Density-matrix renormalization-group study of the spin-½ XXZ antiferromagnet on the Bethe lattice

Hiromi Otsuka

Published Jun 1, 1996• 39 citations

Abstract

Ground-state properties of a spin-\textonehalf{} $\mathrm{XXZ}$ antiferromagnet on the Bethe lattice with three nearest neighbors are investigated; especially, the nature of the phase transition driven by the anisotropy is examined. In actual calculations, we employ the density-matrix renormalization-group method which is so far used in the studies on the one-dimensional quantum systems; its applicability to the higher-dimensional system is thus exhibited in our calculations. Numerical data on local spin correlations imply that the model undergoes the first-order phase transition at the Heisenberg point ${J}_{\mathrm{xy}}={J}_{z}$ and the ground state is conjectured to be in the Ising-like ordered phase (the ordered phase in $\mathrm{XY}$ plain) for ${J}_{\mathrm{xy}}<{J}_{z}$ (${J}_{\mathrm{xy}}>{J}_{z}$). We also compare our results with those for other systems, say, the $\mathrm{XYZ}$ model on the square lattice to discuss a possible type of symmetry breaking.

DOI: 10.1103/physrevb.53.14004

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Physical review. B, Condensed matter

Density-matrix renormalization-group study of the spin-½ XXZ antiferromagnet on the Bethe lattice

Hiromi Otsuka

Published Jun 1, 1996• 39 citations

Publisher

Abstract

Ground-state properties of a spin-\textonehalf{}

\mathrm{XXZ}

antiferromagnet on the Bethe lattice with three nearest neighbors are investigated; especially, the nature of the phase transition driven by the anisotropy is examined. In actual calculations, we employ the density-matrix renormalization-group method which is so far used in the studies on the one-dimensional quantum systems; its applicability to the higher-dimensional system is thus exhibited in our calculations. Numerical data on local spin correlations imply that the model undergoes the first-order phase transition at the Heisenberg point

{J}_{\mathrm{xy}}={J}_{z}

and the ground state is conjectured to be in the Ising-like ordered phase (the ordered phase in

\mathrm{XY}

plain) for

{J}_{\mathrm{xy}}<{J}_{z}

(

{J}_{\mathrm{xy}}>{J}_{z}

). We also compare our results with those for other systems, say, the

\mathrm{XYZ}

model on the square lattice to discuss a possible type of symmetry breaking.

DOI: 10.1103/physrevb.53.14004

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Ground-state properties of a spin-\textonehalf{}

\mathrm{XXZ}

{J}_{\mathrm{xy}}={J}_{z}

and the ground state is conjectured to be in the Ising-like ordered phase (the ordered phase in

\mathrm{XY}

plain) for

{J}_{\mathrm{xy}}<{J}_{z}

(

{J}_{\mathrm{xy}}>{J}_{z}

). We also compare our results with those for other systems, say, the

\mathrm{XYZ}

model on the square lattice to discuss a possible type of symmetry breaking.

The half-filled Hubbard model on the Bethe lattice with coordination number

z=3

is studied using the density-matrix renormalization group (DMRG) method. Ground-state properties such as the energy

E

, average local magnetization $ $, its fluctuations $ - ^2$ and various spin correlation functions $ - $ are determined as a function of the Coulomb interaction strength

U/t

. The calculated local magnetic moments $ $ increase monotonically with increasing Coulomb repulsion

U/t

forming an antiferromagnetic spin-density-wave state which matches the two sublattices of the bipartite Bethe lattice. At large

U/t

, $ $ is strongly reduced with respect to the saturation value 1/2 due to exchange fluctuations between nearest neighbors (NN) spins (

| |\simeq 0.35

for

U/t\to +\infty

). $ - ^2$ shows a maximum for

U/t=2.4

--2.9 which results from the interplay between the usual increase of $ $ with increasing

U/t

and the formation of important permanent moments $ $ at large

U/t

. NN sites show antiferromagnetic spin correlations which increase with increasing Coulomb repulsion. In contrast next NN sites are very weakly correlated over the whole range of

U/t

. The accuracy of the DMRG results is discussed by comparison with tight-binding exact results, independent DMRG calculations for the Heisenberg model and simple first-order perturbation estimates.

We study the spin-1 Heisenberg model on the square lattice with the antiferromagnetic nearest-neighbor

{J}_{1}

and the next-nearest-neighbor

{J}_{2}

couplings by using the infinite projected entangled-pair state (iPEPS) ansatz and the density matrix renormalization group (DMRG) calculation. The iPEPS simulation, which studies the model directly in the thermodynamic limit, finds a crossing of the ground state from the N\'eel magnetic state to the stripe magnetic state at ${J}_{2}/{J}_{1}\ensuremath{\simeq}0.549$, showing a direct phase transition. In the finite-size DMRG calculation on the cylinder geometry up to the cylinder width

{L}_{y}=10

, we find that around the same critical point the N\'eel and the stripe orders are strongly suppressed, which implies the absence of an intermediate phase. Both calculations identify that the stripe order comes with a first-order transition at ${J}_{2}/{J}_{1}\ensuremath{\simeq}0.549$. Our results indicate that unlike the spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ square model, quantum fluctuations in the spin-1 model may not be strong enough to stabilize an intermediate nonmagnetic phase.

A density matrix renormalization group (DMRG) algorithm is presented for the Bethe lattice with connectivity

Z=3

and antiferromagnetic exchange between nearest-neighbor spins

s=1/2

or 1 sites in successive generations

g

. The algorithm is accurate for

s=1

sites. The ground states are magnetic with spin

S(g)={2}^{g}s

, staggered magnetization that persists for large

g>20

, and short-range spin correlation functions that decrease exponentially. A finite energy gap to

S>S(g)

leads to a magnetization plateau in the extended lattice. Closely similar DMRG results for

s

=

1/2 and 1 are interpreted in terms of an analytical three-site model.

We study the phase diagram of the antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the pyrochlore lattice with

S=1

spins at zero and finite temperatures. We use a combination of complementary state-of-the-art quantum many-body approaches such as density matrix renormalization group (DMRG), density-matrix purification, and pseudo-Majorana functional renormalization group (PMFRG). We present an efficient approach to preserve the applicability of the PMFRG for spin-1 systems at finite temperatures despite the inevitable presence of unphysical spin states. The good performance of our methods is first demonstrated for the nearest-neighbor pyrochlore Heisenberg model where the finite-temperature behavior of the specific heat and uniform susceptibility show excellent agreement within PMFRG and density-matrix purification. Including an antiferromagnetic second-neighbor coupling we find that the nonmagnetic ground-state phase of the nearest-neighbor model extents up to ${J}_{2}/{J}_{1}\ensuremath{\sim}0.02$ within DMRG, beyond which magnetic $\mathbit{k}=0$ long-range order sets in. Our PMFRG calculations find the phase transition in a similar regime ${J}_{2}/{J}_{1}\ensuremath{\sim}0.035(8)$ which, together with the DMRG result, provides a strong argument for the existence of a small but finite nonmagnetic ground-state phase in the spin-1 pyrochlore Heisenberg model. We also discuss the origin of discrepancies between different versions of the functional renormalization group concerning the location of this phase transition.

Phase diagram of the antiferromagnetic

J_1

J_2

spin-

1

pyrochlore Heisenberg model

We study the phase diagram of the antiferromagnetic

J_1

J_2

Heisenberg model on the pyrochlore lattice with

S=1

\boldsymbol{k}=0

long-range order sets in. Our PMFRG calculations find the phase transition in a similar regime

J_2/J_1\sim 0.035(8)

which, together with the DMRG result, provides a strong argument for the existence of a small but finite non-magnetic ground-state phase in the spin-1 pyrochlore Heisenberg model. We also discuss the origin of discrepancies between different versions of the functional renormalization group concerning the location of this phase transition.