Author: Natsuhiko Sato

Based on the worm algorithm in the path-integral representation, we propose a general quantum Monte Carlo algorithm suitable for parallelizing on a distributed-memory computer by domain decomposition. Of particular importance is its application to large lattice systems of bosons and spins. A large number of worms are introduced and its population is controlled by a fictitious transverse field. For a benchmark, we study the size dependence of the Bose-condensation order parameter of the hard-core Bose-Hubbard model with L×L×βt=10240×10240×16, using 3200 computing cores, which shows good parallelization efficiency.

Reference

[1] N. Prokof’ev, B. Svistunov and I. Tupitsyn, Sov. Phys. JETP 87, 310 (1998) .
[2] O. F. Syljuasen and A. W. Sandvik, Phys. Rev. E 66, 046701 (2012).
[3] A. Masaki-Kato, T. Suzuki, K. Harada, S. Todo and N. Kawashima, Phys. Rev. Lett. 112, 140603 (2014).

We study quantum entanglement in the ground state of the Affleck-Kennedy-Lieb-Tasaki (AKLT) model defined on two-dimensional graphs with reflection and/or inversion symmetry. The ground state of this spin model is known as the valence-bond-solid state. We investigate the properties of reduced density matrix of a subsystem which is a mirror image of the other one. Thanks to the reflection symmetry, the eigenvalues of the reduced density matrix can be obtained by numerically diagonalizing a real symmetric matrix whose elements are calculated by Monte Carlo integration. We calculate the von Neumann entropy of the reduced density matrix. The obtained results indicate that there is some deviation from the naive expectation that the von Neumann entropy per valence bond on the boundary between the subsystems is ln2. This deviation is interpreted in terms of the hidden spin chain along the boundary between the subsystems. In some cases where graphs are on ladders, the numerical results are analytically or algebraically confirmed.

(by Shu Tanaka)

Reference

[1] Hosho Katsura, Naoki Kawashima, Anatol N. Kirillov, Vladimir E. Korepin and Shu Tanaka, J. Phys. A: Math. Theor. 43 (2010) 255303.

 

Comprehensive experimental studies by magnetic, thermal and neutron measurements have clarified that Rb₄Mn(MoO₄)₃ is a model system of a quasi-2D triangular Heisenberg antiferromagnet with an easy-axis anisotropy, exhibiting successive transitions across an intermediate collinear phase. As a rare case for geometrically frustrated magnetism, quantitative agreement between experiment and theory is found for complete, anisotropic phase diagrams as well as magnetic properties.

(by Shu Tanaka)

Reference

[1]Rieko Ishii, Shu Tanaka, Keisuke Onuma, Yusuke Nambu, Masashi Tokunaga, Toshiro Sakakibara, Naoki Kawashima, Yoshiteru Maeno, Collin Broholm, Dixie P. Gautreaux, Julia Y. Chan, and Satoru Nakatsuji, Europhysics Letters 94, 17001 (2011).

The phase transition from a superfluid to a Mott insulator in bosonic systems has been intensively investigated since it was observed in an ultra cold Bose gas trapped on an optical lattice. [1] The atomic gas was trapped in a harmonic potential and a periodic lattice potential generated by laser beams. This system can be well described by Bose-Hubbard model (BHM)

where bi (bi^†) is the boson destruction (creation) operator at a site i and Z=6 is the coordination number in the cubic lattice. In the optical lattice system, the ratio of the transfer integral t and the repulsive interaction U can be controlled simply by tuning the intensity of the laser beams, which makes this system an ideal laboratory for investigating quantum many-body problems. In order to first discuss the thermodynamic properties, we neglect the gradient of the chemical potential corresponding to the trapping potential. We also choose μ/U=1/2 in order to study a typical case of the phase transition.

Here, we report the results of the quantum Monte Carlo (QMC) simulation of BHM using a modified directed-loop algorithm [2]. The directed-loop algorithm is one of the most widely applicable algorithms for QMC based on the Feynman path integral. The modification makes the algorithm very efficient in particular for the bosonic systems. We show the finite temperature phase diagram in Fig. 1. For large t/U at fixed temperature, the system is in the superfluid phase. When t/U is decreased and reaches a certain value, a phase transition to a normal gas phase occurs. As a function of the transfer integral, the transition temperature decreases as t/U decreases, and eventually it vanishes at the quantum critical point. Beyond this point, the system at T=0 is in the Mott insurating phase, with an excitation gap increasing with decreasing t/U.

In the time-of-flight experiment, a column-integrated momentum distribution of the boson is observed as an interference pattern. The corresponding quantity in the QMC is

where W(k) is the Fourier transformed Wannier function and n(k)= Σ_i,j < b_i^† bj > exp{ik⋅ (r_i –r_j)}. In Fig. 1, three typical cases, i.e., the superfluid, Mott insulator, and critical, are shown. As is naturally expected, we see sharp peaks in the pattern in the superfluid region, while they do not appear in the Mott insulator region. The sharp peaks, therefore, have been regarded as a clear indicator of the superfluidity, and their disappearance have been thought to occur at the quantum phase transition to the Mott insulator phase. However, in Fig. 2 we see rather sharply peaked interference pattern near the critical temperature (t/U=0.25 and T=1.1Tc ) but still clearly in the normal gas phase. In this region, the superfluid density is zero. Nevertheless, the sharp peaks are present. The width of the sharp peaks in the normal region corresponds to the inverse of the correlation length. Near the critical temperature, the correlation length is large and the peaks, therefore, tends to be so sharp that it cannot be distinguished from delta-function peaks that truely signify the superfluidity. Thus, our results show that the interference pattern with sharp peaks is not necessarily a reliable indicator of superfluidity.

（by Y. Kato）

Reference

[1] M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39-44 (2002).
[2] Y. Kato, T. Suzuki, and N. Kawashima, Phys. Rev. E 75, 066703 (2007).
[3] Y. Kato, Q. Zhou, N. Kawashima, and N. Trivedi, Nature Physics 4, 617 (2008).