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.


Introduction
The recent discovery of the Higgs particle [1] in the particle physics [2] [3] has stimulated study of the Higgs mode in the condensed matter physics [4].One can find many experimental reports on the existence of this mode [5]- [11].Among them we note that the experiment of superconducting films [12] close to the quantum phase transition gives us the strong evidence for the Higgs mode.In addition, theoretical study based on the sigma model, the spin wave theory and other effective models has been active.The purpose of the study is to find experimental conditions to observe the Higgs mode clearly [13]- [22].Another purpose is to understand the role of this mode near the critical point of the quantum phase transition [23]- [28].
In a previous study [29] we have studied the Higgs mode in the spin-1/2 Heisenberg antiferromagnet on the square lattice at zero temperature.It is well known that many materials realize the Heisenberg antiferromagnet because of its quite simple Hamiltonian.This system has been studied extensively by several numerical methods [30] [31] [32] as well as by the spin wave theory [33].One motivation of the previous study is to find directly the numerical evidence for the Higgs mode in the quantum antiferromagnet by the reliable method.Another motivation is to investigate how the Higgs mode is induced from the fundamental Hamiltonian.Since the Higgs mode is the resonance state and can couple with two Goldstone-Nambu modes, we have calculated the dynamical correlations of the two spin operators on the finite lattice using the exact diagonalization approach.On the finite lattice we cannot find the resonance itself, but we can find several excited states instead.Taking this into account we have proposed four procedures to find the evidences for the resonance.The results have showed that we can find the Higgs mode in the dynamical correlations of the two spin operators.Also we have clarified differences between the Higgs modes in the SU(2) symmetry and those in the U(1) symmetry through the study of the XXZ model [29].
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) [34]- [39].
When we apply the FTLM to calculations of the dynamical correlations of the operators at finite temperature, we use the set of the states generated by the Lanczos method to calculate the matrix elements of the operator between these Lanczos states.In the calculation at zero temperature the matrix elements are calculated between the Lanczos states and the ground state.In these calculations we need to keep the ground state, but we do not need to keep the Lanczos states after we calculate the matrix elements.In the calculations at finite temperature, on the other hand, we generate two kinds of sets of the Lanczos states.We then calculate the matrix elements between the Lanczos states of one set and the Lanczos states of another set.For these calculations we have to keep two kinds of sets of the Lanczos states on computer memory.For this reason we need the more memory than that of the calculations at zero temperature, or that of the calculations of the specific heat at finite temperature [39].
Here we propose a use of the stochastic state selection (SSS) method [40] for calculations of the matrix elements in order to weaken the limitation on the computer memory.This method has been proposed and developed by T. Munehisa and Y. Munehisa thirteen years ago [40]- [47].For the spin-1/2 antiferromagnet, the number of the basis states amounts to 2 N with the lattice size N , i.e. we need 2 N coefficients for one state.We select coefficients stochastically so that we force some to be zero and replace others by some finite values.The result from one sampling is not correct, but we can obtain the correct value after making the statistical average.Applying the SSS method we can drastically reduce the number of the basis states with non-zero coefficients.Then the limitation from the memory is relaxed.By this method it is possible to calculate the dynamical correlations on the 32 N = lattice using the moderate computer whose memory is 64 GB.Note that there is no other method so far to precisely calculate the matrix elements by small portions of the whole states.
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 20 N = to 32 N = show that the Higgs mode exists at low temperature and it disappears gradually when the temperature becomes large from the lower bound to the higher bound.
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.

Dynamical Correlations and FTLM
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 Â and B is defined by

G T dt tr e A t B e Z dt tr e e A e B e Z
β ω 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 ( )

2
. Therefore we approximate the trace calculation using the random state r R [48] [49], ( ) We therefore calculate the following ( ) , ; where ( ) 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 Ĥ . In the Lanczos method of the FLTM, it is important to choose an appropriate initial state for a good approximation.Therefore in calculations of ( ) given by (4), the first and second exponents of the Hamiltoniant are separately approximated by the suitable sets.As a result we need two sets of the Lanczos states, which are denoted by { } k r where r C is a normalization factor, where Br C is a normalization factor, ( ) ( ) Let us denote the i-th eigen value of the Hamiltonian ( ) E ψ and the eigen state by Similarly the second exponential operator When we use these eigenvalues and eigen states, we obtain the following expression for ( ) We cannot calculate the δ -function on the finite lattice because of the dis- crete eigenvalues.Instead of this singular function, therefore, we use a regular function with a parameter ε , ( ) Based on the discussions in [29], we make ε a moderate value in order to examine peaks of ( ) ω that are made by several eigen states.Then we obtain the following expression

Stochastic State Selection Method
In this section we briefly describe the stochastic state selection method [40].A state ψ is given by a set of a basis states { }( Let us consider a probability variable η for a parameter 1 a ≥ , which is de- fined by a η = with the probability ( ) 1 P a a η = = and 0 η = with the pro- bability ( ) . The average of this variable is one, i.e.
( ) ( ) ( ) where , c i η is a probability variable generated with ( ) Here S ε is a parameter to control the accuracy of η ψ and the number of the selected states in the SSS method.When we make the statistical average of η ψ , we obtain the correct state, The statistical average of number N ψ of basis states with non-zero coeffi- cients in the sampling is given by Next we consider an inner product of two states ψ and ( ) If we make the statistical average of the inner product we obtain the correct value, In our calculation the statistical average ⋅ is replaced by a sample average with a sample number sm N , where k A is a value of A in one sampling.When sm N becomes infinitely lar- ge, the sample average agrees with the statistical average.When S ε becomes large the number of the non-zero η decreases, but the larger sm N is necessary for more accurate value.In order to estimate the re- quired sm N , we calculate a variance 2 σ of the inner product, because the accu- racy of the sample average is given by For quantitative discussions let us assume that 1 for all i.Then we obtain { } ( ) If

Numerical Examinations of SSS Method
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 is a spin operator on a site ( )  [29], 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, { } i r In the calculations of ficients on the lattice of the size N without the SSS method.When we apply the SSS method, on the other hand, we need to keep 2 M N ψ ∼ × × coefficients instead.As we have discussed in section 3 we can impose N ψ which is much smaller than 2 N .Let us examine the accuracy by the SSS method then.For this examination we use a state F instead of ( ) and calculate a correlation.
instead of ( ) 4).Following the procedure of having ( 15) from (4) in the section 2, we obtain , 0 where F C and BF C are normalization factors, ( ) ( ) Note that, from , we have We apply the SSS method to .Using these states we define ( ) Here ( ) When S ε is small the accuracy is high, but the number of the basis states to be kept is large.We would like to examine the accuracy and the cost in calcu- The error bar is the statistical error of ( ) , , ; ,  In order to examine ( ) Note that here we set 1 sm N = because one sampling for each random set will be enough when R N is large.Further discussion on this point will be given in the final section.the same order for any value of S ε we employed.This fact implies that in calculations of the dynamical correlations using the SSS method we do not need more number of the sampling compared to that without the SSS.By these examinations we conclude that we can apply the SSS method to calculations of the dynamical correlations.

Higgs Mode
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 contain these modes.Therefore we calculate the following dynamical correlation, ( ) in (39) with the two spin operators In order to obtain stable results at any temperature, we employ the Chebyshev polynomial expansion [39] for the calculation of 2 with the k-th Chebyshev polynomial ( ) k T x and the k-th coefficient k p .
Before presenting our results we comment on parameters in our calculations, which we summarize in Table 1.
Table 1.Parameters of calculations; the symbol and the range in our calculations.The inverse temperature 1 T β = and the energy ω are the physical quan- tities.
The lattice size N is restricted because of the exact diagonalization ap- proach.
In our work, we calculate on the lattices of the size 20 32 N ≤ ≤ .For the pe- riodic boundary condition we have two edge vectors ( ) , l l and ( ) , l l .
Since we impose the π 2 rotational symmetry to the Hamiltonian, the edge ve- ctor ( ) , l l is given by ( ) , l l − and the lattice size is given by On the parameter ε in ( 14), we have presented the careful discussion in the previous work [29].This discussion has showed that 0.5 ε = is most suitable.Therefore we use this value for ε .
As for the number M of the Lanczos states we fix it to be 50 following the discussion of the previous work [29] and the preliminary study.In the The sampling number of the random states R N is determined by requiring that the relative precision of our calculations on the dynamical correlations is 5%.
In Figure 3 we present the dynamical correlations ( ) lattice as a function of ω .At the low temperatures 1 10.0, 6.4 T = and 3.2 we find the broad peaks clearly, as expected.These broad peaks could be the Higgs mode which has been found at 0 T = in the previous work [29].On the other hand, at the high temperature 1 1.2 T = we cannot find any peak that is relevant with the Higgs mode.In order to confirm that we find the broad peak at the low T in the contrast to no broad peak at high T , we plot the dynamical correlations at the low T and those at the high T on various lattices in Figure 4, where the correlations at   lattice as a function of ω for various values of 1 T between 1.12 and 10.0.We see that the broad peak gradually disappears as 1 T becomes small.For example we can clearly find the broad peak at 1 3.20 T = , while it is not easy to find the peak when 1 1.44 T = and there is no peak at 1 1.20 T = .
We would like to determine a boundary of 1 T below which the broad peak vanishes.It is, however, difficult to estimate such boundary temperature because the broad peak disappears gradually when 1 T decreases.Therefore we intro- duce two kinds of T , which we denote s T and l T .We can insist that the broad peak exists for s T T ≤ .On the other hand we admit that it is difficult to find the broad peak for l T T ≥ .In other words s T is a boundary by the strict condition for the broad peak, while l T is a boundary by the loose condition for it.On the On other lattices we can determine s T and l T in the same way.When the lattice size N is odd we observe that 1 s T and 1 l T scarcely depend on N .
In Figure 6    The behavior of the dynamical correlations on the 20 N = lattice is much the probability variables η by large sm N for each random state.But it is also possible to make the sampling on the probability variables η and the sampling on the random state at the same time.This sampling means that we make one sampling on the probability variables of η , i.e. was made in the study of the SSS method [43] extensively.
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 [29] suggest that one can find the Higgs mode in experiments of the quantum antiferromagnet on the square lattice if we measure the dynamical correlations of the two spin operators.
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.
is the inverse of the temperature T .Using the eigen value i E and the eigen state i Ψ of the Hamiltonian Ĥ , we obtain the inner product with satisfyingly small σ even ifV N N ψ .In this case we can obtain the accuracy of 0.01 sm N for the inner product ψ φ .

.
As discussed in section 3, we have the parameter S ε in the SSS method.
of ω .In this figure we plot the difference

.
zero.In comparison with the black data and the red data: are closer to zero.The average numbers of the basis states with non-zero coefficients in the SSS method are ~9500 with 0In section 3 we have argued that the average number of the basis states with non-zero coefficients is drastically reduced to be of order ofV SN ε by the SSS method.We see that the mea- sured values are a little less thanV SN ε , which are 12500 for accuracy of sample average, we have discussed that it would be 04 in the black data, ~ 0.01 in the red data and ~ 0.01 in the green data.)} = 18 0.04 0.01 ∼ , these results support the discussion in section 3.

Figure 1
Figure 1.The difference Now we would like to examine the difference between on this small lattice, however, it is difficult to obtain ( ), GT ω because we need all eigenvalues and eigen states for the correlations.We therefore use

Figure 2
Figure 2. The difference ( ) , R S D N ε defined in (40) in the SSS method on the 20 N =

10 −
edge vectors are defined uniquely for a given lattice size N ex- cept for an accidental case 25 N = .In this exceptional case we distinguish two different 25 N = lattices by 25a and 25b .For the lattices of and (4, 4), respectively.The wave vector k is the non-zero wave vector of the lowest magnitude on the each lattice.For the lattices 20, 25 , 25 , 26, 29 π 5 , 2π 5, 0 , 8π 25, 6π 25 , 5π 13, π 13 , 10π 29, 4π 29 and ( ) π 4, π 4 .The parameters c N and c h control the accuracy of the Chebyshev poly- nomial expansion.They are determined by the request that the precision is of order 12 .As a result they depend on values of N and T .
reduce a huge calculation time.We apply the SSS method to calculations for 25 N ≥ .The parameter S

Figure 4 .
Figure 4.The dynamical correlations with 1 3.2 T = and 1.2 for various lattices plotted as a function of ω .The error bars are the statistical errors.
which covers the area of the broad peak at 0 T = , so that we can easily compare our results for the different values of T .We employ 2 , using the central energy of the board peak c ω and the width Γ at 0 T =[29].In Figure5we plot

Figure 5 .
Figure 5.The normalized dynamical correlations for various values of T on the 25 N a =

Figure 6 .
Figure 6.The dynamical correlations for s T and l T on the odd-size lattices.The vertical dotted lines show the values of c ω .

Figure 7 .
Figure 7.The dynamical correlations for s T and l T on the even-size lattices.The vertical dotted lines show the values of c ω .

1 smN
= , for each random state R .Adopting this sampling method, we can obtain the correct values of denotes the nearest neighbor pair on the square lattice.The z-component ˆz i s