Research Highlights

Several quantum magnets comprise two sublattices of magnetic ions coupled by a geometrically frustrated exchange. This is for instance the case of a Heisenberg antiferromagnet on a body centered tetragonal lattice or a square lattice with nearest- and next-nearest-neighbor exchange interactions. In the regime of intersublattice coupling smaller than the intrasublattice exchange, it can be shown that the frustrated nature of the intersublattice exchange precludes a bilinear coupling between the order parameters of the two sublattices. The Hamiltonian symmetry only allows for an effective biquadratic coupling. The biquadratic interaction tends to align the order parameters of the sublattices in either parallel or antiparallel. This results in the Ising-like Z2 symmetry breaking in the ground state in addition to the usual spin-rotational symmetry breaking.

In Ref. 1, we investigated such a phase transition with the additional Ising-like symmetry breaking for the XY spin case, which is relevant for a spin-dimer compound BaCuSi2O6 [1]. More specifically, we performed Monte Carlo simulations to obtain a renormalization group flow diagram around the decoupled model with intersublattice interaction (∝ λ) being zero. Figure 1 is the obtained flow diagram. It shows several parameters of zero scaling dimensions and therefore at second order transitions the flow should converge to a stable fixed point. It can be clearly seen that the flow starting around the decoupled XY fixed point (indicated by a large filled circle) systematically deviates from it, evolving toward the region where a clear first order signature is obtained (λ = -2; see the inset where an energy histogram with the double-peak structure is shown).

Neither a separatrix nor a stable fixed point is found. It means that the biquadratic coupling is relevant at the decoupled XY fixed point (this can also be shown using a scaling argument [2]), and the resulting crossover behavior leads to a first-order transition.

So far there is no experimental report suggesting the first-order transition in BaCuSi2O6, and the experimental data such as specific heat can be fitted using the XY model. The reason is that BaCuSi2O6 is a quasi-two-dimensional system and thus the intersublattice biquadratic coupling which arises from second-order perturbation with respect to the ratio between the interlayer and the intralayer bilinear exchange coupling is extremely small in this case. Namely, the true discontinuous nature of the transition can be observed in a very narrow region near the transition point that could be easily beyond the standard experimental precision.

(by Yoshitomo Kamiya)

References

[1] Y. Kamiya, N. Kawashima, and C. D. Batista, J. Phys. Soc. Jpn. 78, 094008 (2009).
[2] Y. Kamiya, N. Kawashima, and C. D. Batista, Phys. Rev. B 82, 054426 (2010).

We study the Bose-Hubbard model with an external harmonic field, which is effective for modeling a cold atomic Bose gas trapped in an optical lattice. We modify the directed-loop algorithm to simulate large systems efficiently. As a demonstration we carry out the simulation of a system consisting of 1.8×10⁵ particles on a 64³ lattice. These numbers are comparable to those in the pioneering experimental work by Greiner et al. [Nature (London) 415, 39 (2002)]. Furthermore, we observe coherence between two superfluid spheres separated by a Mott insulator region in a “wedding-cake” structure.

(by Yasuyuki Kato)

Reference

Yasuyuki Kato and Naoki Kawashima, “Quantum Monte Carlo method for the Bose-Hubbard model with harmonic confining potential” Phys. Rev. E. 79, 021104 (2009).

We study the finite-temperature transition of the quasi-2D Bose gas in an uniaxially-compressed harmonic trap by numerically solving the projected Gross-Pitaevskii equation. Gradual emergence of superfluidity is confirmed by calculating the moment of inertia when a temperature decreases. By investigating the long-distance behavior of a phase correlation function, superfluid density gradually increases reflecting the development of the phase correlation around the center of the system. From these results, we obtain the evidence for the emergence of superfluidity in this system directly.

Reference

Toshihiro Sato, Takafumi Suzuki and Naoki Kawashima, J. Phys.: Conf. Ser. 150, 032094 (2009)

We study a finite-temperature phase transition in the two-dimensional classical Heisenberg model on a triangular lattice with a ferromagnetic nearest-neighbor interaction J₁ and an antiferromagnetic third-nearest-neighbor interaction J₃ using a Monte Carlo method. Apart from a trivial degeneracy corresponding to O(3) spin rotations, the ground state for J₃ ≠0 has a threefold degeneracy corresponding to 120° lattice rotations. We find that this model exhibits a first-order phase transition with the breaking of the threefold symmetry when the interaction ratio is J₃ / J₁ = -3.

(by Ryo TAMURA)

Reference:

R. Tamura and N. Kawashima,  J. Phys. Soc. Jpn. 77, 103002 (2008).

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).

The S=1/2 Ising-like XXZ model has been intensively studied as a good stage to clarify exotic phenomena, such as supersolid behavior, stabilization of the valence-bond-solid state on frustrated lattices, and spin-flop transition in magnetic fields [1]. In particular, the spin-flop transition has been a long standing topic in condensed matter physics and the nature has been clarified by several theoretical approaches [2]; the Neel state (z-Neel) with the antiferromagnetically ordered spins along the easy-axis direction shows a first-order transition to the spin-canted state (xy-Neel) at a low temperature. An interesting point on this spin-flop transition is that the first order transition accompanying a magnetization jump occurs irrespectively of whether the quantum or the classical spins in both two-dimensional (2D) and three-dimensional (3D) isotropic lattice cases. However, in quasi-one-dimensional limit, the system can be regarded as a 1D-quantum spin chain in a low temperature region, because the interchain couplings are suppressed by the thermal fluctuation. The S=1/2 Ising-like XXZ spin chain shows a critical behavior from the transition field Hc1 up to the saturation field Hc2, when the magnetic field H is applied along the easy axis at zero temperature T=0. The low energy excitation for Hc1 < H < Hc2 is described by the Tomonaga-Luttinger (TL) liquid, which is characterized by the TL exponents appearing in the long-distance behavior of the longitudinal and transverse spin-spin correlations. An important point arising from the Ising anisotropy is that the longitudinal incommensurate correlation becomes dominant for Hc1 < H < H* while the transverse staggered one survives at long distance for H* < H < Hc2. When the interchain interactions become relevant in the TL liquid state, the long-range-ordered state corresponding to the dominant spin fluctuation can be realized, accompanying the finite-temperature phase transition. Indeed, the existence of such incommensurate order is suggested in recent experiment on a quasi 1D S=1/2 XXZ antiferromagnet BaCo2V2O8 [3].

In the above two descriptions of the field-induced transition, the phase transition in the spin-flop transition and the q1D case seem to be incompatible with each other. Then, a natural but nontrivial question may arise; Is it possible to connect the 1D-based description with the spin-flop phase diagram in the pure 2D or 3D lattice case, varying the amplitude of the interchain couplings? Since the critical phenomena are strongly affected by the dimensionality of the system, the interchain-coupling dependence of the q1D system involves some essentially important physics. The enhancement of the incommensurate fluctuation which is not hold in the classical case makes the field-induced phase of the q1D XXZ model more complicated.

We performed quantum Monte Carlo simulations based on the directed loop algorithm for the S=1/2 Ising-like XXZ spin chains constructing the 2D square lattice and 3D cubic lattice through weak interchain couplings. Based on the obtained results, we illustrated the H-T phase diagrams in Figure 1. In the low magnetized region of the 2D case, there is no finite temperature transition because incommensurate correlations along both the spin-chain and the interchain directions suppress the development of a two-dimensional antiferromagnetic correlation. On the other hand, in the 3D case, the incommensurate fluctuation can be stabilized as a long-range-ordered state at a finite temperature [4].

(by Takafumi Suzuki)

References

[1] F. H’ebert, et al., Phys. Rev. B 65, 014513 (2001); M. Boninsegni and N. Proko’ev, Phys. Rev. Lett. 95, 237204 (2005); D. C. Cabra, et al., Phys. Rev. B 71, 144420 (2005); S. V. Isakov, et al., Phys. Rev. Lett. 97, 147202 (2005); A. Banerjee, et al., Phys. Rev. Lett. 100, 047208 (2008).
[2] A. van Otterlo, et al., Phys. Rev. B 52, 16176 (1995); M. Kohno and M. Takahashi, Phys. Rev. B 56, 3212 (1997); G. Schmid, et al., Phys. Rev. Lett. 88, 167208 (2005).
[3] S. Kimura, et al., Phys. Rev. Lett. 99, 087602 (2007).
[4] T. Suzuki and N. Kawashima, K. Okunishi, J. Phys. Soc. Jpn. 76, 123707 (2007); K. Okunishi and T. Suzuki, Phys. Rev. B 76, 224411 (2007).

The principal effect of frustration is cancellation among competing couplings. In quasi-two-dimensional systems, when the frustration exists in inter-layer couplings, it tends to enhance murual independence of layers. At finite temperature, however, this cancellation can never be perfect so as to make the system purely two-dimensional because thermal fluctuation generates effective net interlayer couplings. Even at zero temperature, the geometric frustration fails to decouple layers completely since the zero-point fluctuations restore the inter-layer coupling as discussed by Maltseva and Coleman. As a result, the critical phenomena in any real quasi-two-dimensional system belongs to the universality class of some three dimensional theory no matter how perfectly the frustration may seem to cancel the couplings. However, we find that this is not necessarily true at quantum critical points. [1]

The first experimental evidence of this phenomenon was found by measuring critical exponents of a field induced QCP in BaCuSi2O6. [2]  This is a spin dimer system whose highly symmetric crystal structure gives it unique advantages for tackling the fundamental role of dimensionality in quantum criticality. The material consists of layers stacked on top of each other forming a body-centered-cubic (BCT) lattice of dimers. By neglecting two high-energy triplet states of each dimer, the system can be represented by an S=1/2 XY model on BCT lattice. We examined the magnon excitations by using the spin-wave treatment..[1] The effective coupling between layers is proportional to the number of excited magnons, which means that the fluctuation responsible for the effective inter-layer coupling dies out as we approach zero temperature and the quantum critical point can possess purely two dimensional characteristics. Thermal or quantum fluctuations at finite temperature induce a crossover to d = 3 away from the QCP. The two-dimensional value of the exponent characterizes the phase boundary near the zero temperature, in agreement with the experiment [2].

The agreement between the pure two-dimensional behavior and the real experiment is not only qualitative, but it is quantitatively good as can be seen in Fig. 2. The upper panel shows the critical field measured in the experiment on BaCuSi2O6 plotted with the phase boundary obtained by quantum Monte Carlo simulation assuming that the inter-layer coupling is zero (filled circles) . Both the results show the two-dimensional quantum criticality that is characterized by the asymptotic linear dependence. The correspondence is further confirmed by other quantities: the field-dependence of the magnetization at zero (or very low) temperature  (lower left panel), and the temperature-dependence of the magnetization at critical magnetic field (lower right panel).

(by Naoki KAWASHIMA)

Reference:

[1] C. D. Batista,  J. Schmalian,  N. Kawashima, P. Sengupta, S. E. Sebastian, N. Harrison, M. Jaime and I. R. Fisher, Phys. Rev. Lett. 98, 257201 (2007).
[2] S. E. Sebastian, N. Harrison, C. D. Batista, L. Balicas, M. Jaime, P. A. Sharma, N. Kawashima and I. R. Fisher, Nature 441, 617 (2006).

We modify the directed-loop algorithm (DLA) to solve the problem that typically arises from large on-site interaction. The large on-site interaction is inevitable when one tries to simulate a Bose gas system in continuum by discretizing the space with small lattice spacings. While the efficiency of a straightforward application of DLA decreases as the mesh becomes finer, the performance of the new method does not depend on it except for the trivial factor due to the increase in the number of lattice points.
(Abstract of Ref. [1])

(By Yasuyuki Kato)

Reference

[1] Yasuyuki Kato, Takafumi Suzuki and Naoki Kawashima: “Modification of directed-loop algorithm for continuous space simulation of bosonic systems”, Phys. Rev. E 75 066703(1-8) (2007).

We investigate a supersolid state in hardcore boson models on the face-centered-cubic (fcc) lattice. The supersolid state is characterized by a coexistence of crystalline order and superfluidity. Using a quantum Monte Carlo method based on the directed-loop algorithm, we calculate static structure factors and superfluid density at finite temperature, from which we obtain the phase diagram. The supersolid phase exists at intermediate fillings between a three-quarter-filled solid phase and a half-filled solid phase. We also discuss the mechanism of the supersolid state on the fcc lattice.
(Abstract of Ref. [1])

(by Takafumi Suzuki)

Reference

[1] T. Suzuki and N. Kawashima, Phys. Rev. B 75, 180502(R) (2007).