Stochastic State Selection Method Research Articles

Finite Temperature Lanczos Method with the Stochastic State Selection and Its Application to Study of the Higgs Mode in the Antiferromagnet at Finite Temperature

We propose an improved finite temperature Lanczos method using the stochastic state selection method. In the finite temperature Lanczos method, we generate Lanczos states and calculate the eigenvalues. In addition we have to calculate matrix elements that are the values of an operator between two Lanczos states. In the calculations of the matrix elements we have to keep the set of Lanczos states on the computer memory. Therefore the memory limits the system size in the calculations. Here we propose an application of the stochastic state selection method in order to weaken this limitation. This method is to select some parts of basis states stochastically and to abandon other basis state. Only by the selected basis states we calculate the inner product. After making the statistical average, we can obtain the correct value of the inner product. By the stochastic state selection method we can reduce the number of the basis states for calculations. As a result we can relax the limitation on the computer memory. In order to study the Higgs mode at finite temperature, we calculate the dynamical correlations of the two spin operators in the spin-1/2 Heisenberg antiferromagnet on the square lattice using the improved finite temperature Lanczos method. Our results on the lattices of up to 32 sites show that the Higgs mode exists at low temperature and it disappears gradually when the temperature becomes large. At high temperature we do not find this mode in the dynamical correlations.

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 kindof Monte Carlo method we have proposed in order to numerically study largequantum spin systems. In the stochastic state selection method we make a samplingwhich is simultaneous for many states. This feature enables us to modify themethod so that a number of given constraints are satisfied in each sampling.In this paper we discuss this modified stochastic state selection method thatwill be called the constrained stochastic state selection method in distinctionfrom the previously proposed one (the conventional stochastic state selectionmethod) in this paper. We argue that, by virtue of the constrained sampling, somequantities obtained in each sampling become more reliable, i.e. their statisticalfluctuations 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 thatdata errors in our estimation of the ground state energy are reduced. Then we successfullyevaluate several low-lying energy eigenvalues of the model on a 48-site lattice. Ourresults support that this system can be described by the theory based on thespontaneous symmetry breaking in the semiclassical Néel ordered antiferromagnet.

An equilibrium for frustrated quantum spin systems in the stochastic state selectionmethod

We develop a new method to calculate eigenvalues in frustrated quantum spin models. It isbased on the stochastic state selection (SSS) method, which is an unconventionalMonte Carlo technique that we have investigated in recent years. We observethat a kind of equilibrium is realized under some conditions when we repeatedlyoperate a Hamiltonian and a random choice operator, which is defined by stochasticvariables in the SSS method, to a trial state. In this equilibrium, which we callthe SSS equilibrium, we can evaluate the lowest eigenvalue of the Hamiltonianusing the statistical average of the normalization factor of the generated state.The SSS equilibrium itself has already been observed in unfrustrated models. Our study in thispaper shows that we can also see the equilibrium in frustrated models, with some restriction onvalues of a parameter introduced in the SSS method. As a concrete example, we employ thespin-1/2 frustrated J1–J2 Heisenberg model on the square lattice. We present numerical results on the 20-, 32-, and 36-sitesystems, which demonstrate that statistical averages of the normalization factors reproducethe known exact eigenvalue to good precision. Finally, we apply the method to the 40-sitesystem. Then we obtain the value of the lowest energy eigenvalue with an error of less than0.2%.

The stochastic state selection method combined with the Lanczos approach to eigenvaluesin quantum spin systems

We describe a further development of the stochastic state selection method, a new MonteCarlo method we have proposed recently to make numerical calculations in largequantum spin systems. Making recursive use of the stochastic state selectiontechnique in the Lanczos approach, we estimate the ground state energy of thespin-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 ourmethod is efficient in calculating energy eigenvalues of frustrated quantum spin systems onlarge lattices.

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 ×10 7 . Employing a translationally invariant initial trial state, we evaluate the lowest energy eigenvalue within 0.5 % of the statistical errors.

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.