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arXiv:2105.04970

Dispersion Relations of Nambu-Goldstone Modes

Tohru Koma

Published May 11, 2021• 2 citations
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Abstract

We study the energy-momentum relations of the Nambu-Goldstone modes in quantum antiferromagnetic Heisenberg models on a hypercubic lattice. This work is a sequel to the previous series about the models. We prove that the Nambu-Goldstone modes show a linear dispersion relation for the small momenta, i.e., the low energies of the Nambu-Goldstone excitations are proportional to the momentum of the spin wave above the infinite-volume ground states with symmetry breaking. Our method relies on the upper bounds for the susceptibilities which are derived from the reflection positivity of the quantum Heisenberg antiferromagnets. The method is also applicable to other systems when the corresponding upper bounds for the susceptibilities are justified.

DOI
10.48550/arxiv.2105.04970

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We study a quantum antiferromagnetic Heisenberg model on a hypercubic lattice in three or higher dimensions d≥3d\ge 3d≥3. When a phase transition occurs with the continuous symmetry breaking, the nonvanishing spontaneous magnetization which is obtained by applying the infinitesimally weak symmetry breaking field is equal to the maximum spontaneous magnetization at zero or non-zero low temperatures. In addition, the transverse correlation in the infinite-volume limit exhibits a Nambu-Goldstone-type slow decay. In this paper, we assume that the transverse correlation decays by power law with distance. Under this assumption, we prove that the power is equal to 2−d2-d2−d at non-zero low temperatures, while it is equal to 1−d1-d1−d at zero temperature. The method is applied also to a quantum XY model and a classical Heisenberg model at non-zero low temperatures in three or higher dimensions. The resulting power is given by the same 2−d2-d2−d at non-zero low temperatures.
We introduce a lattice fermion model in one spatial dimension with supersymmetry (SUSY) but without particle number conservation. The Hamiltonian is defined as the anticommutator of two nilpotent supercharges QQQ and ${Q}^{\ifmmode\dagger\else\textdagger\fi{}}$. Each supercharge is built solely from spinless fermion operators and depends on a parameter ggg. The system is strongly interacting for small ggg, and in the extreme limit g=0g=0g=0, the number of zero-energy ground states grows exponentially with the system size. By contrast, in the large-ggg limit, the system is noninteracting and SUSY is broken spontaneously. We study the model for modest values of ggg and show that under certain conditions spontaneous SUSY breaking occurs in both finite and infinite chains. We analyze the low-energy excitations both analytically and numerically. Our analysis suggests that the Nambu-Goldstone fermions accompanying the spontaneous SUSY breaking have cubic dispersion at low energies.
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.
Abstract We discuss spontaneous symmetry breaking of open classical and quantum systems. When a continuous symmetry is spontaneously broken in an open system, a gapless excitation mode appears corresponding to the Nambu–Goldstone mode. Unlike isolated systems, the gapless mode is not always a propagation mode, but it is a diffusion one. Using the Ward–Takahashi identity and the effective action formalism, we establish the Nambu–Goldstone theorem in open systems, and derive the low-energy coefficients that determine the dispersion relation of Nambu–Goldstone modes. Using these coefficients, we classify the Nambu–Goldstone modes into four types: type-A propagation, type-A diffusion, type-B propagation, and type-B diffusion modes.