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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.

The recent discovery of the Higgs particle [

In a previous study [

In order to confirm that the Higgs mode exists at finite temperature, we would like to study the Higgs mode in the spin-1/2 Heisenberg antiferromagnet on the square lattice using the finite temperature Lanczos method (FTLM) [

Here we propose a use of the stochastic state selection (SSS) method [

After numerical examinations of the SSS method in the FTLM, we present results about the Higgs mode at finite temperature. At low temperature we find the Higgs mode, whereas at high temperature we do not find this mode. We estimate two bounds of temperature by the strict and the loose conditions, under which we can find the Higgs mode. Our results on the lattices from

Contents of this paper are as follows. In the next section we present a brief description of calculations in the FTLM which will show the reason why a large number of the Lanczos states are necessary. Sections 3 and 4 are devoted to the SSS method. After explaining the SSS method in Section 3, we present numerical examinations of the SSS method in the calculations of the dynamical correlations in Section 4. Then in Section 5 we calculate the dynamical correlations in the spin-1/2 Heisenberg antiferromagnet on the square lattice in order to find the resonance that is associated with the Higgs mode. The final section is for summary and discussion of this work.

In this section we give a brief description of the dynamical correlation and of the FTLM we use in our calculations. The dynamical correlation of the operator

Here

It is not possible to calculate every eigen value and every eigen state on the lattice whose size is more than 15 because the number of the matrix element is more than

We therefore calculate the following

where

When

In the FTLM, instead of the exact values and the exact states, we use the eigenvalues and the eigen states that are calculated by the set of the Lanczos states. A set of the Lanczos states

When M is of order of 100, it is easy to obtain the eigenvalues of

In the Lanczos method of the FLTM, it is important to choose an appropriate initial state for a good approximation. Therefore in calculations of

where

Similarly the first state

where

Let us denote the i-th eigen value of the Hamiltonian

Similarly the second exponential operator

When we use these eigenvalues and eigen states, we obtain the following expression for

We cannot calculate the

Based on the discussions in [

In this section we briefly describe the stochastic state selection method [

Let us consider a probability variable

where

The statistical average of number

Next we consider an inner product

where

If we make the statistical average of the inner product we obtain the correct value,

In our calculation the statistical average

where

When

For quantitative discussions let us assume that

and

If

In this section we present numerical examinations of the SSS method in the calculations of the dynamical correlations. Our model is the spin-1/2 Heisenberg antiferromagnet on the square lattice. The Hamiltonian is given by

where

denoting the location of site i by

In the calculations of the dynamical correlations at finite temperature, we apply the SSS method to two sets of the Lanczos states,

Let us examine the accuracy by the SSS method then. For this examination we use a state

instead of

where

Note that, from

We apply the SSS method to

(36)

Here

SSS sampling. When

As discussed in section 3, we have the parameter

The error bar is the statistical error of

As for the accuracy of sample average, we have discussed that it would be proportional to

In order to examine

Note that here we set

Now we would like to examine the difference between

where

The most important purpose of this paper is the numerical verification of the Higgs mode in the quantum spin systems at finite temperature. In this section we would like to show it by calculating the dynamical correlations in the spin- 1/2 Heisenberg antiferromagnet on the square lattice. Since the Higgs mode is the excited state and couples to the two Goldstone-Nambu modes, we have to calculate the dynamical correlations of the two operators that contain the Goldstone-Nambu modes. In the Heisenberg antiferromagnet the spin operators con- tain these modes. Therefore we calculate the following dynamical correlation,

In order to obtain stable results at any temperature, we employ the Chebyshev polynomial expansion [

with the k-th Chebyshev polynomial

Before presenting our results we comment on parameters in our calculations, which we summarize in

0.80 -10.0 | |
---|---|

0.0 - 8.0 | |

20 - 32 | |

Minimum wave vector on the lattice | |

20 -100 | |

15 - 25 | |

0.5 | |

40-50 | |

0.01 - 0.04 | |

20 - 160 |

The inverse temperature

The lattice size

In our work, we calculate on the lattices of the size

The wave vector

The parameters

On the parameter

As for the number

We apply the SSS method to calculations for

The sampling number of the random states

In

What we are interested in is the shape of the dynamical correlation around the broad peak. Since the absolute value of the correlation strongly depends on

in the range

We would like to determine a boundary of

On other lattices we can determine

Such difference between the results on the odd-size lattices and those on the even-size lattices is ascribable to the behavior of the correlations at

size lattices than those on the odd-size lattices. Especially we observe that

The behavior of the dynamical correlations on the

In this research we have calculated, in order to find the Higgs mode, the dynamical correlations of the two spin operators in the spin-1/2 Heisenberg antiferromagnet on the square lattice at finite temperature. We have proposed an improved finite temperature Lanczos method using the stochastic state selection method for calculations on the lattices of up to 32 sites.

In the standard finite temperature Lanczos method we generate Lanczos states, calculate the eigenvalues and calculate matrix elements that are the values of the operator between two Lanczos states. In 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 for which we can calculate the matrix elements. Here we have proposed the 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 states. Only by the selected basis states we calculate the inner product. After we make the statistical average, we can obtain the correct value of the inner product. By the stochastic state selection method we can drastically reduce the number of the basis states for the calculations.

In order to study the Higgs mode at finite temperature, we have calculated 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. In calculations on the

A few comments are in order. The first comment is about the parameter

The second comment is about

The following three comments are about subjects for future study to be pursued. Since we employ the exact diagonalization approaches, the lattice size is severely limited even in the improved FTLM. Therefore it is desirable to make further study by other calculation methods, the high temperature expansions for example, which do not depend on the lattice size.

The results of this work and the previous work [

Another subject is about universality of the Higgs mode in the quantum spin systems. On the universality study the system on the triangle lattice is quite interesting because this system has the three kinds of the Nambu-Goldstone modes, whereas the system on the square lattice has the two kinds. It means that there must be an essential difference between both systems. Therefore it is very important to ask about what is the difference between the Higgs modes in these systems.

T. M. would like to thank Dr. Yasuko Munehisa for critical reading of the manuscript and for useful discussions.

Munehisa, T. (2017) Finite Temperature Lanczos Method with the Stochastic State Selection and Its Application to Study of the Higgs Mode in the Antiferromagnet at Finite Temperature. World Journal of Condensed Matter Physics, 7, 11-30. https://doi.org/10.4236/wjcmp.2017.71002