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The Higgs mode is expected to exist in any system with the spontaneous symmetry breaking of the continuous symmetry. We make numerical study about the Higgs mode in the Heisenberg antiferromagnet on the square lattice by the exact diagonalisation approach. Since the Higgs mode can couple with a pair of the Nambu-Goldstone modes, we calculate the dynamical correlation of the two spin operators, employing the finite temperature Lanczos method. Because the lattice size is severely limited, we make a careful discussion on procedures of finding evidences for the Higgs mode by numerical works. By the discussed procedures, we present numerical results for the dynamical correlation at zero temperature. Then we obtain clear evidences for the Higgs mode of the spin-1/2 Heisenberg antiferromagnet on the square lattice.

The spontaneous symmetry breaking (SSB) of the continuous symmetry has been one of the most important concepts and phenomena of the modern physics [

In the condensed matter physics the Higgs mode has been extensively studied recently [

On the other hand theoretical study has been active, especially using the sigma model and other effective models. Its purpose is to find experimental possibilities of observing the Higgs mode [

In this work we would like to study the Higgs mode in the spin-1/2 Heisenberg antiferromagnet on the square lattice [

The first motivation of our study is to find directly the Higgs mode in the quantum antiferromagnet. In this system the Hamiltonian is clearly defined and we have many materials that realize it. Therefore the numerical evidence for the Higgs mode should stimulate researchers to study experimentally as well as theoretically for this new degree of freedom in the condensed matter physics. Another motivation is to investigate how the Higgs mode is induced from the fundamental Hamiltonian. In the particle physics the Higgs particle is the object to constitute the Hamiltonian. By contrast, in the condensed matter physics the Higgs mode is not an object of the Hamiltonian, but the collective mode induced from it. Therefore we should be able to calculate the gap energy and other properties of the Higgs mode without any assumption. For this purpose the Hamiltonian of the Heisenberg antiferromagnet on the two-dimensional lattice is the most suitable because one can investigate it by various methods.

In our study on the Heisenberg antiferromagnet, the most important purpose is to find evidences for the Higgs mode on the finite lattice by the reliable method of the numerical calculations at zero temperature. Also we would like to clarify differences between the Higgs modes in the SU(2) symmetry and those in the U(1) symmetry through the study of the XXZ model [

Since the Higgs mode is an excited state, we calculate the dynamical spin correlation. Here we employ the finite temperature Lanczos method [

In the next section after a brief description of the Higgs mode, we discuss procedures of finding evidences for it in our calculation. Here we emphasize that the Higgs mode is the excited mode and it couples with a pair of the NG-modes. Therefore the Higgs mode should be a resonance in the dynamical correlation of two spin operators. On the infinitely large lattice we can easily judge signals for the resonance. On the finite lattice, however, the numerical study for the resonance is a non-trivial task, because we calculate not continuous, but discrete energy eigen values. By taking this discreteness into account, we suggest four procedures to find the evidences. Numerical results are presented in Section 3. This section is divided to four subsections. In each subsection, we show the evidence for Higgs mode by using each suggested procedure. The final section is devoted to a summary and discussion for future researches.

The spin-1/2 Heisenberg antiferromagnet on the square lattice is given by

Here

obtain eigen values of the

number. For each

cause the symmetry is not broken on a finite lattice system. In our representation the evidence of the SSB is given by the non-zero value of

Here we denote a location by

From the field theory on the SSB [

Here

and

Here f is the decay constant and Z is the renormalization factor. If we apply the above discussion to the Heisenberg antiferromagnet on the square lattice, we have the following correspondence, as discussed in [

Using the annihilation operator

Here

the creation operator

In the sigma model of the SSB, we have an interaction between the two NG modes and the Higgs mode. In order to obtain a state of the two NG modes, we apply two spin operators of

Here

and

with the infinitely small

We make

If only one resonance state exists as the Higgs mode in the large size limit, we have

Here

Here we assume that

about the Higgs mode on the finite lattice, we propose following four procedures to obtain numerical evidences on the existence of the Higgs mode.

(A) Since we generate the state with the pair of the NG modes from the ground state, which is

(B) On a lattice of the finite size N we have the lowest energy state

Therefore the Higgs mode must exist on each

(C) On the square lattice, there is the NG mode at

of the peak in

(D) We will consider the XXZ model [

where

In the next section we will show the above four evidences through numerical calculations of the dynamical correlation.

In this section we present numerical results of our calculations about the Higgs mode in the Heisenberg antiferromagnet on the finite square lattice. Before presenting results shown in subsections, we describe three steps in calculations of

Since the magnitude of the

where C is determined for the correlation to satisfy a following normalization,

The third step of the calculation is to apply the finite temperature Lanczos method [

In this subsection we examine several peaks of the dynamical correlation in the narrow energy region in order to obtain the evidence by the first procedure (A) discussed in the Section 2. First we show results of the dynamical correlation

to be sufficiently small in order that each peak corresponds to a single energy eigen state. By the procedure (A) of finding several peaks in the narrow energy region, we can obtain the first evidence for the Higgs mode.

We suppose that the Higgs mode appears as a resonance on the infinitely large lattice. On the finite lattice,

this resonance consists of the several energy eigen states. We would like to investigate the resonance which survives in the large N limit. For this purpose we make several peaks in

By taking both conditions into account, we determine

Here we make a comment on the sharp peak whose energy

As discussed in the previous section, if the broad peaks found in

In order to make quantitative discussion, we introduce definitions for the location of the peak and its broadness. First we find the

the value of

For a concrete example, see

In

Next we will discuss the width

Summarizing this subsection we conclude that the observation in the state for

The SSB of the SU(2) symmetry of the Heisenberg antiferromagnet on the square lattice implies two kinds of the NG modes, as discussed in the previous section. One is the excited state of the small wave vector

In

these peaks. Also the result on the N = 32 lattice shows the similar agreement between peaks of the correlation with the wave vector

In this subsection we will carefully examine the broad peaks in

As discussed in the previous subsection, the agreement between the peaks in these correlations reflects the symmetry of SU(2). If the symmetry of the model is U(1), we should not expect agreements between these peaks. Therefore we would like to study the correlation in the XXZ model which has only the U(1) symmetry. The Hamiltonian of the XXZ model is given by (15) in the Section 2. Since in this model the conserved charge is

In

In this work we have presented the numerical evidences for the Higgs mode of the spin-1/2 Heisenberg antiferromagnet on the square lattice, using the exact diagonalisation method and the finite temperature Lanczos method. Since the Higgs mode is the resonance which couples with the two Nambu-Goldstone (NG) modes, we calculate the dynamical correlation of the two spin operators with various wave vectors at zero temperature. For the dynamical correlation we have to calculate the inverse of the Hamiltonian operator. Since our calculations are carried out on the finite lattice, we need the smearing parameter

In this work we suggest four procedures (A)-(D) to find the Higgs mode in the calculation of the dynamical correlation. Using the first procedure (A) about peaks in the correlation with the small value of

Now we comment on the dip that is seen in the broad peak of the correlation of N = 36,

Since we have calculated the central energy

we obtain

Finally we would like to discuss about further study of the Higgs mode. If the broad peak is the resonance of the Higgs mode, we should find the same peak that has the same central energy

In this work our calculations are made at the zero temperature. The extensive study at the finite temperature is an important subject because the experiments are performed at the finite temperature. Also the Higgs mode in two dimensional systems near a quantum critical point has been a subject of debate [

M. T. thanks Dr. Yasuko Munehisa for every encouragement on his study.

Tomo Munehisa, (2015) Numerical Study of the Higgs Mode in the Heisenberg Antiferromagnet on the Square Lattice. World Journal of Condensed Matter Physics,05,261-274. doi: 10.4236/wjcmp.2015.54027