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arXiv:1305.0352

Numerical Study on Spontaneous Symmetry Breaking in a XY Quantum Antiferromagnet on a Finite Triangular Lattice

Tomo Munehisa, Yasuko Munehisa

Published May 2, 2013• 0 citations
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Abstract

Motivated by recent experiments that require more complicated macroscopic wave functions in the condensed matters, we make numerical study on a XY quantum antiferromagnet on a finite triangular lattice using the variational Monte Carlo method and the stochastic state selection method. One of our purpose is a numerical confirmation on dominance of a Nambu-Goldstone boson in low energy excitation. For another purpose, we calculate energy, an expectation value of a symmetry breaking operator and structure functions of spin by fixing a quantum number of the symmetry. These calculations are made for states that become degenerate in an infinitely large lattice. By numerical calculations we confirm existence of a Nambu-Goldstone boson, and find dependence of a square of the quantum number for the above quantities. Using these results we can discuss on complicated macroscopic wave functions in quantum spin systems.

DOI
10.48550/arxiv.1305.0352

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Motivated by recent experiments that require more complicated macroscopic wave functions in the condensed matters, we make numerical study on a XY quantum antiferromagnet on a finite triangular lattice using the variational Monte Carlo method and the stochastic state selection method. One of our purpose is a numerical confirmation on dominance of a Nambu-Goldstone boson in low energy excitation. For another purpose, we calculate energy, an expectation value of a symmetry breaking operator and structure functions of spin by fixing a quantum number of the symmetry. These calculations are made for states that become degenerate in an infinitely large lattice. By numerical calculations we confirm existence of a Nambu-Goldstone boson, and find dependence of a square of the quantum number for the above quantities. Using these results we can discuss on complicated macroscopic wave functions in quantum spin systems.
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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 cluster property is one of fundamental properties in physics.This property means that there are no relations between two events that are sufficiently separated.Because the cluster property is directly connected with entanglement in quantum field theory and in many-body systems, theoretical and experimental progress on entanglement stimulates us to study this property deeply.In this paper we investigate the cluster property in the spin 1/2 XXZ antiferromagnet on the square lattice with an explicitly symmetry breaking interaction of strength g.In this model spontaneous symmetry breaking occurs when the lattice size N is infinitely large.On the other hand, we have to make g zero in order to obtain quantities in the XXZ model with no symmetry breaking interaction.Since some results depend on the sequence of limit operations -N → ∞ and 0 g → , it is difficult to draw a clear conclusion in these limits.Therefore we study the model with finite g on the finite lattice, whose size N is supposed to be 10 20 , for our quantitative calculations.Then we can obtain the concrete ground state.In order to study the cluster property we calculate the spin correlation function.It is known that the function due to Nambu-Goldstone mode (gapless mode), which is calculated using linear spin wave theory, satisfies this property.In this paper we show that almost degenerate states also induce the spin correlation.We assume that the spin correlation function in measurements is a sum of the function due to Nambu-Goldstone mode and one due to these degenerate states.Then we examine whether the additional correlation function violates the cluster property or not.Our conclusion is that this function is finite at any distance, which means the violation of the cluster property, and it is of order of ( )for the case of extremely small g, this violation is the fine effect.Therefore the correlation function due to the degenerate states can be observed only when it is larger than the spin correlation function due to Nambu-Goldstone mode.We show that g required for this condition depends on the distance between positions of two spin operators.
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. PACS number: 05.10.Ln, 02.70.Ss, 75.10.Jm 1
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.