Research Highlights

We considered a Bose-Hubbard model on a two-dimensional lattice. We showed that the model parameters (the lattice structure, the hopping constants, and the chemical potential) can be tuned in such a way that the ground state is generated by applying, to the vacuum, N creation operators of locally-defined quasi-particles, where N is proportional to the total number of lattice points. The resulting model has a free parameter β that controls the degree of quantum fluctuation; β=0 for no fluctuation and β=1/2 for the maximum fluctuation. We carried out Monte Carlo simulation of this model at zero temperature, and discovered that the model undergoes a quantum phase transition when we change this parameter. While the system is 2+1 dimensional, the new phase transition exhibits the characteristics of 2 dimensions, i.e., the Kosterlitz-Thouless phase transition analogous to the classical XY model in 2 dimensions. Accordingly, the large β phase is the quasi Bose-Einstein condensation phase.

The above-mentioned ground state is expressed exactly as a tensor network with the bond dimension 2. Tracing out this tensor network is equivalent to calculating the partition function of some classical loop gas model. This is why the quantum criticality exhibits the purely 2-dimensional characteristics. It also allows us to construct a Monte Carlo algorithm for efficient sampling from the space of classical occupation-number configurations. In other words, we can stochastically trace out the tensor network to obtain various physical quantities of interest, such as the static structure factors and the helicity modulus.

The lattice is generated from the square lattice by duplicating each bond into p copies and introducing an additional site in the middle of each copy. (Fig. 1) The direct hopping is allowed only within local clusters labelled by either x or u. While bosons interact with each other, the model is constructed in such a way that the dispersion of a single particle has a flat band if influence of the other paricles could be neglected. The result of the Monte Carlo simulation are shown in Fig.2. The static structure factor diverges around β=0.2 and, at the same time, the helicity modulus jumps from 0 to 2/π, the universal value characteristic to the KT transition.

[1] H. Katsura, N. Kawashima, S. Morita, A. Tanaka and H. Tasaki: Physical Review Research, 3 033190 (2021)

It is widely known that there is no spontaneous symmetry breaking in 1d classical systems at finite temperature, which suggests that 1d classical many body problems are almost trivial and boring.
Then, let us consider 1d classical `edges’ in 2d systems, or 1d classical systems attached to a 2d bulk.
Does something interesting happen in these 1d edges?
In fact, the situation is a bit different from the simple 1d cases.

When the 2d bulk is at criticality, in particular, the strong correlation at the bulk helps the edge ordering, which results in a variety of phase transitions on the 1d edges.
For example, the 2d classical tricritical Ising model exhibits a finite-surface-temperature symmetry breaking on its 1d edge, when the bulk is fine-tuned at the tricritical point [1].
It is also known that another surface transition occurs driven by the surface external fields.

As a theoretical way of studying surface criticality, boundary conformal field theory (BCFT) is very powerful, particularly for 2d classical / 1+1d quantum systems at criticality.
It could be essential to understand surface critical behavior precisely, since the restriction of the conformal invariance may enable us not only to classify the conformal boundary conditions but also to investigate stability of boundary fixed points and exact values of scaling dimensions.

Now, let us consider an extension of the above classical tricritical Ising model in 2d, called the tricritical 3-state Potts model, whose symmetry of the spin is \(S_3\) rather than \(Z_2\).
The Monte Carlo simulation of this model suggests, similarly to the tricritical Ising, rich surface phase transitions when the bulk is at tricriticality [2].
However, the precise study from the view point of BCFT has been missing for a long time.

In order to investigate the more detailed surface phase diagram of this model, we utilize the \(ADE\) classification of the minimal-series BCFTs and perform numerical simulation with tensor network renormaization (TNR) [3,4,5].
The tricritical 3-state Potts model can be described by the \(D\)-type minimal CFT with \(c=6/7\), whose modular invariant partition function is labeled as the pair of Lie algebra \((D_4, A_6)\) in the \(ADE\) classification.
Using the complete classification of the conformal boundary states in Ref. [3], one can derive the twelve conformal boundary conditions labeled by the nodes of the Dynkin diagrams \(D_4\) and \(A_6\).
The triality of \(D_4\) implies that those can be classified into three \(Z_3\)-symmetric boundary states and nine \(Z_3\)-broken ones.

To understand the physical picture of the above twelve boundary fixed points, we simulate the 3-state dilute Potts model on lattices with the TNR technique.
The extended TNR method suited for open-boundary systems enables us to extract accurate conformal spectrum from lattice models, by which one can study the correspondence between the boundary fixed points from BCFT and the surface phases in the phase diagram of the tricritical 3-state Potts model.
Our numerical analysis reveals that the eleven boundary fixed points among the twelve can be realized on the lattice by controlling the external field and coupling strength at the boundary.
The last unfound fixed point would be out of the physically sound region in the parameter space, and we leave it an open question to consider the realization of that boundary condition on the lattice model.
(by Shumpei Iino)

References:
[1] I. Affleck, J. Phys. A 33(37), 6473 (2000).
[2] Y. Deng and H. W. J. Blöte, Phys. Rev. E 70, 035107 (2004); Phys. Rev. E 71, 026109 (2005).
[3] R. E. Behrend, P. A. Pearce, V. B. Petkova, and J.-B. Zuber, Nucl. Phys. B 579(3), 707 (2000).
[4] G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015); S. Iino, S. Morita, and N. Kawashima, Phys. Rev. B 101, 155418 (2020).
[5] S. Iino, arXiv:2007.03182; J. Stat. Phys. 182, 56 (2021).

Suppose we want to generate a bit string with length \(N\) in which each bit is set with arbitrary probability \(p\). If we adopt the simple algorithm, which repeats check for each bit, we need to generate \(N\) random numbers. However, we can reduce the number of random number generations by adopting more sophisticated algorithms [1]. We proposed three algorithm, the Binomial-Shuffle, the Poisson-OR, and the finite-digit algorithms.

The Binomial-Shuffle algorithm is an algorithm that first calculates how many bits are set and then shuffles them. Adopting Walker’s alias method and Floyd’s sampling method, it is possible to generate a random bit string with \(pN+1\) random number generations. The Poisson-OR algorithm generates \(k\) bit strings with one of the bits is set randomly and takes the logical OR of them. This is a special case of the algorithm proposed by Todo and Suwa [2]. The average number of the random number generations is \(-N \log(1-p)+1\). The finite digit method is effective when the probability \(p\) can be represented by a finite digit in the binary notation. When the probability p can be expressed by \(n\) digits in the binary notation, we can generate a random bit string with that probability by generating \(n\) bit strings in which each bit is set with a probability \(1/2\) and properly taking the logical operations. Therefore, the finite-digit algorithm with n digits involves the random number generates \(n\) times.

While the Binomial-Shuffle and the Poisson-OR algorithms are effective when the value of probability \(p\) is small, the required number of random numbers increases as \(p\) increases. Meanwhile, the required number for the finite-digit method does not depend on \(p\), although the method cannot be applied directly unless p can be expressed by \(n\) digits in the binary notation. Therefore, we constructed a hybrid algorithm, i.e., which first generates a bit string with probability \(\tilde{p}\) close to \(p\) with the finite digit algorithm and corrects the difference \(\tilde{p}-p\) by the Binomial-Shuffle or Poisson-OR algorithms. The expected number of random numbers generated to generate a string in which each bit is set with arbitrary probability is at most 7 for the 32-bit case and 8 for 64-bit case.

Employing the developed algorithm, we applied the multispin coding technique to the one-dimensional bond-directed percolation and obtained a factor of 14 speed-up.

References

1. H. Watanabe, S. Morita, S. Todo, and N. Kawashima, J. Phys. Soc. Jpn. 88, 024004 (2019)
2. S. Todo and H. Suwa, J. Phys.: Conf. Ser. 473, 012013 (2013).

After the discovery of the high-temperature superconductors, their parent compounds are conjectured to be in a spin liquid phase which becomes superconducting when charge carriers are doped. Spin liquid is a highly entangled quantum disordered state. Due to its strong geometrical frustration, the Kagome quantum magnets are expected to host such novel phase. Searching for an ideal Kagome quantum magnet has led to recent discovery of new materials from structural polymorphs of herbertsmithite, i.e., Ca-kapellasite which is described in Fig.1(a) and (b).

However, according to exhaustive theoretical investigations, it may host many different types of spin liquid phase including RVB, Z₂ and Dirac spin liquids. Therefore, it is very important to find the elementary excitations expected to emerge in these unknown phases through experiments on ideal Kagome materials, though it has been remained as a very challenging problem. As an evidence of emergence of spin liquid phase and its excitation, the thermal Hall response, which is an analogue of Hall response in electronic system, has been suggested and, in fact, the finite thermal Hall conductivity was measured. However, its origin was not so clear since the phonon also can transport the heat.

In this study, a clear thermal Hall signal was observed in the spin-liquid phase of the S=1/2 kagome antiferromagnet Ca-kapellasite and provide a strong evidence suggesting the heat is carried by the spinon excitation on top of the spin liquid state. In order to obtain a theoretical estimation of thermal Hall conductivity, we employ the Schwinger-Boson mean field theory (SBMFT) and the U(1) spin liquid ansatz. The thermal Hall conductivity can be measured by the following formula

Where c₂ is a distribution function of the Schwinger boson, and n_B the Bose-Einstein distribution function, E is the Schwinger boson band, and Ω is the corresponding Berry curvature.  Remarkably, the (dimensionless) thermal Hall conductivity obtained from the SBMFT is perfectly fitted by the ones obtained from the experiment as shown in Fig. 2.

Also, we find all observed thermal Hall conductivity curves from many samples converge onto a single curve by adjusting only the exchange interaction and Dzyloshinkii-Moriya interaction, which suggests that it has a common temperature dependence on the Kagome quantum magnet. This excellent agreement, both qualitative and quantitative, demonstrates that the thermal Hall effect in these kagome antiferromagnets derives from spins in the spin-liquid phase.

Reference

Hayato Doki, Masatoshi Akazawa, Hyun-Yong Lee, Jung Hoon Han, Kaori Sugii, Masaaki Shimozawa, Naoki Kawashima, Migaku Oda, Hiroyuki Yoshida, and Minoru Yamashita, Phys. Rev. Lett. 121, 097203 (2018)

Tensor network methods are becoming powerful tools in the study of many-body problems. Real-space renormalization by coarse-graining tensor networks, including tensor renormalization group (TRG) [1] and its derivatives, is an efficient numerical method for the contraction of tensor networks. Accuracy of approximation is improved by increasing the bond dimension. However the computational complexity of TRG is proportional to the sixth power of the bond dimension \(\chi\). Thus complexity reduction without loss of accuracy is desired.

We proposed a new algorithm of TRG based on a randomized algorithm for singular value decomposition (RSVD) [2]. Its computational complexity scales as \(O(\chi^5)\). The main operations in TRG are “decomposition” and “contraction”. Since the both have \(O(\chi^6)\) complexity, we need to reduce the complexity not only by replacing the full SVD with RSVD but also by avoiding the explicit creation of a four-leg tensor. An oversampling parameter tunes the statistical error of RSVD. We showed that the oversampling parameter larger than the bond dimension reproduced the same result as the full SVD in two-dimensional Ising model. [3]

Since “decomposition” and “contraction” are dominant operations in most tensor network schemes, the techniques presented here would be useful in improving most of the tensor network calculations in an essential way.

References

1. M. Levin and C. P. Nave, Phys. Rev. Lett. 99, 120601 (2007).
2. N. Halko, P. G. Martinsson, and J. A. Tropp, SIAM Rev. 53, 217 (2011).
3. S. Morita, R. Igarashi, H.-H. Zhao, and N. Kawashima, Phys. Rev. E 97, 033310 (2018).

We investigate the ground state properties of ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ based on numerical calculations of the recently proposed ab initio Hamiltonian represented by Kitaev and extended Heisenberg interactions. To overcome the limitation posed by small tractable system sizes in the exact diagonalization study employed in a previous study [Y. Yamaji et al., Phys. Rev. Lett. 113, 107201 (2014)], we apply a two-dimensional density matrix renormalization group and an infinite-size tensor-network method. By calculating at much larger system sizes, we critically test the validity of the exact diagonalization results. The results consistently indicate that the ground state of ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ is a magnetically ordered state with zigzag configuration in agreement with experimental observations and the previous diagonalization study. Applications of the two independent methods in addition to the exact diagonalization study further uncover a consistent and rich phase diagram near the zigzag phase beyond the accessibility of the exact diagonalization. For example, in the parameter space away from the ab initio value of ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ controlled by the trigonal distortion, we find three phases: (i) an ordered phase with the magnetic moment aligned mutually in 120 degrees orientation on every third hexagon, (ii) a magnetically ordered phase with a 16-site unit cell, and (iii) an ordered phase with presumably incommensurate periodicity of the moment. It suggests that potentially rich magnetic structures may appear in $A_2{\mathrm{IrO}}_3$ compounds for $A$ other than Na. The present results also serve to establish the accuracy of the first-principles approach in reproducing the available experimental results thereby further contributing to finding a route to realize the Kitaev spin liquid.

             

(by Naoki Kawashima)

[Reference]

[1] G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, 017205 (2009).

[2] Y. Yamaji, Y. Nomura, M. Kurita, R. Arita, and M. Imada, Phys. Rev. Lett. 113, 107201 (2014).

[3] Tsuyoshi Okubo, Kazuya Shinjo, Youhei Yamaji, Naoki Kawashima, Shigetoshi Sota, Takami Tohyama, and Masatoshi Imada, Phys. Rev. B 96, 054434 (2017)

Two-dimensional Ising models on the honeycomb lattice and the square lattice with striped random impurities are studied to obtain their phase diagrams. Assuming bimodal distributions of the random impurities where all the non-zero couplings have the same magnitude, exact critical values for the fraction \(p\) of ferromagnetic bonds at the zero-temperature \(T=0\) are obtained. The critical lines in the \(p-T\) plane are drawn by numerically evaluating the Lyapunov exponent of random matrix products.

 

 

Reference

[1] S. Morita and S. Suzuki: J. Stat. Phys. 162, 123 (2016).

The $\mathrm{SU}\left(N\right)$ symmetric antiferromagnetic Heisenberg model with multicolumn representations on the two-dimensional square lattice is investigated by quantum Monte Carlo simulations. For the representation of a Young diagram with two columns, we confirm that a valence-bond solid (VBS) order appears as soon as the Néel order disappears at $N=10$, indicating no intermediate phase. In the case of the representation with three columns, there is no evidence for either the Néel or the VBS ordering for $N\ge 15$. This is actually consistent with the large-$N$ theory, which predicts that the VBS state immediately follows the Néel state, because the expected spontaneous order is too weak to be detected.

(by Tsuyoshi Okubo)

Reference:

T. Okubo, K. Harada, J. Lou and N. Kawashima: Phys. Rev. B 92, 134404 (2015).