An Equilibrium for Frustrated Quantum Spin Systems in the Stochastic State Selection Method

We develop a new method to calculate eigenvalues in frustrated quantum spin models. It is based on the stochastic state selection (SSS) method, which is an unconventional Monte Carlo technique we have investigated in recent years. We observe that a kind of equilibrium is realized under some conditions when we repeatedly operate a Hamiltonian and a random choice operator, which is defined by stochastic variables in the SSS method, to a trial state. In this equilibrium, which we call the SSS equilibrium, we can evaluate the lowest eigenvalue of the Hamiltonian using the statistical average of the normalization factor of the generated state. The SSS equilibrium itself has been already observed in un-frustrated models. Our study in this paper shows that we can also see the equilibrium in frustrated models, with some restriction on values of a parameter introduced in the SSS method. As a concrete example, we employ the spin-1/2 frustrated J1-J2 Heisenberg model on the square lattice. We present numerical results on the 20-, 32-, 36-site systems, which demonstrate that statistical averages of the normalization factors reproduce the known exact eigenvalue in good precision. Finally we apply the method to the 40-site system. Then we obtain the value of the lowest energy eigenvalue with an error less than 0.2%.Comment: 15 pages, 12 figure

A constrained stochastic state selection method applied to quantum spin systems

We describe a further development of the stochastic state selection method, which is a kind of Monte Carlo method we have proposed in order to numerically study large quantum spin systems. In the stochastic state selection method we make a sampling which is simultaneous for many states. This feature enables us to modify the method so that a number of given constraints are satisfied in each sampling. In this paper we discuss this modified stochastic state selection method that will be called the constrained stochastic state selection method in distinction from the previously proposed one (the conventional stochastic state selection method) in this paper. We argue that in virtue of the constrained sampling some quantities obtained in each sampling become more reliable, i.e. their statistical fluctuations are less than those from the conventional stochastic state selection method. In numerical calculations of the spin-1/2 quantum Heisenberg antiferromagnet on a 36-site triangular lattice we explicitly show that data errors in our estimation of the ground state energy are reduced. Then we successfully evaluate several low-lying energy eigenvalues of the model on a 48-site lattice. Our results support that this system can be described by the theory based on the spontaneous symmetry breaking in the semiclassical Neel ordered antiferromagnet.Comment: 15 pgaes, 5 figure

The Stochastic State Selection Method Combined with the Lanczos Approach to Eigenvalues in Quantum Spin Systems

We describe a further development of the stochastic state selection method, a new Monte Carlo method we have proposed recently to make numerical calculations in large quantum spin systems. Making recursive use of the stochastic state selection technique in the Lanczos approach, we estimate the ground state energy of the spin-1/2 quantum Heisenberg antiferromagnet on a 48-site triangular lattice. Our result for the upper bound of the ground state energy is -0.1833 +/- 0.0003 per bond. This value, being compatible with values from other work, indicates that our method is efficient in calculating energy eigenvalues of frustrated quantum spin systems on large lattices.Comment: 11 page

An Attempt to Calculate Energy Eigenvalues in Quantum Systems of Large Sizes

We report an attempt to calculate energy eigenvalues of large quantum systems by the diagonalization of an effectively truncated Hamiltonian matrix. For this purpose we employ a specific way to systematically make a set of orthogonal states from a trial wavefunction and the Hamiltonian. In comparison with the Lanczos method, which is quite powerful if the size of the system is within the memory capacity of computers, our method requires much less memory resources at the cost of the extreme accuracy. In this paper we demonstrate that our method works well in the systems of one-dimensional frustrated spins up to 48 sites, of bosons on a chain up to 32 sites and of fermions on a ladder up to 28 sites. We will see this method enables us to study eigenvalues of these quantum systems within reasonable accuracy.Comment: 17pages, 4figures(eps-files

A Recursive Method of the Stochastic State Selection for Quantum Spin Systems

In this paper we propose the recursive stochastic state selection method, an extension of the recently developed stochastic state selection method in Monte Carlo calculations for quantum spin systems. In this recursive method we use intermediate states to define probability functions for stochastic state selections. Then we can diminish variances of samplings when we calculate expectation values of the powers of the Hamiltonian. In order to show the improvement we perform numerical calculations of the spin-1/2 anti-ferromagnetic Heisenberg model on the triangular lattice. Examining results on the ground state of the 21-site system we confide this method in its effectiveness. We also calculate the lowest and the excited energy eigenvalues as well as the static structure factor for the 36-site system. The maximum number of basis states kept in a computer memory for this system is about 3.6 x 10**7. Employing a translationally invariant initial trial state, we evaluate the lowest energy eigenvalue within 0.5 % of the statistical errors.Comment: 14 pages, 1 figur

Numerical Study of Excited States in the Shastry-Sutherland Model

We investigate excited states of the Shastry-Sutherland model using a kind of variational method. Starting from various trial states which include one or two triplet dimers, we numerically pursue the best evaluation of the energy for each set of quantum numbers. We present the energy difference as a function of either the coupling ratio or the momentum and compare them with the perturbative calculations. Our data suggest that the helical order phase exists between the singlet dimer phase and the magnetically ordered phase. In comparison with the experimental data we can estimate the intra-dimer coupling J and the inter-dimer coupling J' for SrCu2(BO3)2 : J'/J =0.65 and J = 87K.Comment: 15pages, 5figures to be published in JPS

A QED Shower Including the Next-to-leading Logarithm Correction in e+e- Annihilation

We develop an event generator, NLL-QEDPS, based on the QED shower including the next-to-leading logarithm correction in the e^+e^- annihilation. The shower model is the Monte Carlo technique to solve the renormalization group equation so that they can calculate contributions of alpha^m log^n(S/m_e^2) for any m and n systematically. Here alpha is the QED coupling, m_e is the mass of electron and S is the square of the total energy in the e^+e^- system. While the previous QEDPS is limited to the leading logarithm approximation which includes only contributions of (alpha log(S/m_e^2))^n, the model developed here contains terms of alpha(alpha log(S/m_e^2))^n, the the next-to-leading logarithm correction. The shower model is formulated for the initial radiation in the e^+e^- annihilation. The generator based on it gives us events with q^2, which is a virtual mass squared of the virtual photon and/or Z-boson, in accuracy of 0.04%, except for small q^2/S.Comment: 35 pages, 1 figure(eps-file

Test of QEDPS: A Monte Carlo for the hard photon distributions in e+ e- annihilation proecss

The validity of a photon shower generator QEDPS has been examined in detail. This is formulated based on the leading-logarithmic renormalization equation for the electron structure function and it provides a photon shower along the initial e+-. The main interest in the present work is to test the reliability of the generator to describe a process accompanying hard photons which are detected. For this purpose, by taking the HZ production as the basic reaction, the total cross section and some distributions of the hard photons are compared between two cases that these photons come from either those generated by QEDPS or the hard process e+e- -> H Z gamma gamma. The comparison performed for the single and the double hard photon has shown a satisfactory agreement which demonstrated that the model is self-consistent.Comment: 22 pages, 4 Postscript figures, LaTeX, uses epsf.te

The Stochastic State Selection Method for Energy Eigenvalues in the Shastry-Sutherland Model

We apply a recently developed stochastic method to the Shastry-Sutherland model on 4x4 and 8x8 lattices. This method, which we call the Stochastic State Selection Method here, enables us to evaluate expectation values of powers of the Hamiltonian with very limited number of states. In this paper we first apply it to the 4x4 Shastry-Sutherland system, where one can easily obtain the exact ground state, in order to demonstrate that the method works well for this frustrated system. We numerically show that errors of the evaluations depend much on representations of the states and that the restructured representation is better than the normal one for this model. Then we study the 8x8 system to estimate energy eigenvalues of the lowest S=1 state as well as of the lowest excited S=0 state, where S denotes the total spin of the system. The results, which are in good accordance with our previous data obtained by the Operator Variational method, support that an intermediate spin-gap phase exists between the singlet dimer phase and the magnetically ordered phase. Estimates of the critical coupling and of the spin gap for the transition from the dimer phase to the intermediate phase are also presented.Comment: 17 pages, 4 figure