Questions tagged [goldstone-mode]

The Goldstone-mode is a massless quantum excitation arising in systems with spontaneous breaking of continuous symmetry. That is, the Noether symmetry currents are conserved, but the vacuum is not invariant under the symmetry, so the symmetry is not immediately apparent, realized non-linearly.

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Identification of physical pion states in chiral perturbation theory

In chiral perturbation theory, we introduce the parametrization of the Goldstone manifold $$ U = \exp\left( i \frac{1}{f_\pi} \pi_a \sigma_a\right), $$ where $\sigma_a$ are the Pauli matrices. Most ...
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Goldstone and longitudinal polarization equivalence between off-shell $W$s

If I understand correctly, the couplings of the longitudinal component of the $W$ and $Z$ bosons should be "equivalent" to the Goldstone bosons they ate after SSB. In practice, this makes ...
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Steps in getting an explicit expression for Goldstone bosons from two scalar fields

I have two scalar fields, which have the U(1) symmetry $\phi_1 \rightarrow e^{i\alpha}\phi_1$ and $\phi_2 \rightarrow e^{-i\beta}\phi_2$, $\alpha \neq \beta$. Then they both get their own vev in SSB, ...
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Factoring a the exponential form of a group element of a Lie group, using subgroups

I am working on non linear realization of Goldstone bosons, as is done by Weinberg in section 19.6 of Quantum theory of fields, volume II. We have a real, compact and connected Lie group $G$ with as ...
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Nambu-Goldstone mode without symmetry breaking

Superfluidity is often explained in terms of spontaneous breaking of global $U(1)$ symmetry. However, we know that in real, finite-size quantum systems, this symmetry can never be broken. Quantum ...
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Property of representations of Lie Groups in Weinberg's Quantum Theory of Fields

In section 19.6, page 212-213 in Weinberg's Quantum Theory of Fields Volume II, Weinberg shows that on can always eliminate the Goldstone Bosons from a field $\psi(x)$, by use of a space-dependent ...
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Vacuum Degeneracy for Massless Free Scalar Field

I am wondering how to explicitly see the vacuum degeneracy for a free massless scalar field, described by the action $$S = -\frac{1}{2}\int d^4x\,(\partial\phi)^2.$$ This action is invariant under ...
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SSB of Spacetime Symmetries in 2d

Colemans theorem states that there are no Goldstone Bosons in d=2 spacetime dimensions: https://inspirehep.net/literature/83738 because there is no spontaneous symmetry breaking. However i have read ...
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Spontaneous symmetry breaking in the Standard Model. What is "broken"?

I know this question has been asked other times, but I am looking for a confirmation of the following. When we say that the gauge group of the standard model is $G_{SM} = SU(3)_{c} \times SU(2)_{L} \...
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How does the wave nature of a Spin Wave arise?

I have been learning about Goldstone's theorem recently and I am a bit confused about show Goldstone excitations arise. Consider spin waves in a magnet. The Hamiltonian is given by (see: https://en....
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Spontaneous Chiral Symmetry Breaking in Schwartz

I am reading M. Schwartz's book on QFT, equation (28.24)/(28.22). They say that a set scalar fields will transform as, where $g_L$ belongs to $SU(2)_{L}$ and $g_R$ belongs to $SU(2)_R$: $$\Sigma\...
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Why is that if the broken symmetry is realized linearly we can not find rationale for the Lagrangian of the sigma model?

In the book The Quantum theory of fields Vol. 2 at page 193 Weinberg says $$ \mathscr{L}=-\frac{1}{2} \partial_{\mu} \phi_{n} \partial^{\mu} \phi_{n}-\frac{\mathscr{M}^{2}}{2} \phi_{n} \phi_{n}-\...
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Parametrization of the coset space using Goldstone boson

Suppose we have $n$ Goldstone bosons which is obtained from the fact that the ground state $\eta$ is invariant under a subgroup $H$ of $G$. Each of these Goldstone bosons will be described by an ...
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What does the absence of these Goldstone boson interactions mean physically?

I have read that in several statistical models exhibiting spontaneous symmetry breaking, the resulting Goldstone bosons do not interact with each other via $\theta^{2n}$ terms — only via derivative ...
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Skin depth and Mermin-Wagner theorem

I recently became aware of the Coleman-Mermin-Wagner theorem presented in [1802.07747] for higher-form symmetries and I have a question about how it might be applied to electromagnetism. The theorem ...
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About Itzykson and Zuber's proof of Goldstone's theorem

In chapter 11-2-2, I&Z discuss Goldstone's theorem. They start by claiming that if an operator $A$ exists, such that $$ \delta a(t) \equiv \langle 0| [Q(t),A]|0\rangle \neq 0 \tag{11-30} $$ the ...
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How scalars appear in Goldstone's theorem

Let $G$ be a lie group and $G_i$ its generators. Suppose for a set of fields $\chi_{\alpha}(x)$ we have $$ \left[G_{i}, \chi_{\alpha}(x)\right]=-\left(g_{i}\right)_{\alpha \beta} \chi_{\beta}(x) $$ If ...
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Fetter and Walecka's derivation of Goldstone's Theorem

I've been learning Green's Functions (Zero-Temprature) from Fetter and Walecka's Quantum Theory of Many-Particle Systems, and find it hard to understand the proof of Goldstone's Theorem, in page 111, ...
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Question about the Weinberg proof of the Goldstone's theorem

When proving Goldstone's theorem in Vol. 2, p. 171, Weinberg takes the following: $$\langle \{J^\lambda(y),\phi_n(x)\}\rangle= \frac{\partial}{\partial y_\lambda}\int d\mu^2 \rho_n(\mu^2) \left[ \...
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Can be Goldstone bosons measured?

I'm reading about the Higgs mechanism considering the Lagrangian: $$ \mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu} + \left(D_\mu\phi\right)\left(D^\mu\phi\right)^\dagger-\mu^2\left(\phi\phi\right)^\...
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Why phonons are Goldstone modes?

I read this in the lecture notes by David Tong: "Gapless excitations often dominate the low-temperature behaviour of a system, where they are the only excitations that are not Boltzmann ...
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Do Goldstone modes, by definition, have to be low-energy or long wavelength?

For a spontaneously broken continuous global symmetry, there exist Goldstone modes for which the dispersion relation is such that as the momentum or the wavenumber goes to zero the energy or frequency ...
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Issue with understanding the Weinberg's proof of the Goldstone boson theorem

To prove Goldstone's theorem Weinberg (in volume II, page 170, equation 19.2.18) starts with the expectation value over the vacuum of the commutator between one field $\phi_n(x)$ and the conserved ...
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Methods for deriving goldstone theorem

I am reading chapter 19 of Weinberg Vol. 2, and in section 19.2 he gives two proof of goldstone theorems - one uses functional method involving quantum effective action and other is operator method ...
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Confusion about matrix element of conserved current between vacuum and one-particle spin-zero boson in Weinberg 19.2

I am going through Weinberg's second proof of Goldstone's theorem. I am able to follow it up to Eqn. (19.2.34), where he asserts that the following relation holds by Lorentz invariance $$\langle \...
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Dispersion for goldstone bosons

In particle and condensed matter physics, Goldstone bosons or Nambu–Goldstone bosons (NGBs) are bosons that appear necessarily in models exhibiting spontaneous breakdown of continuous symmetries. ...
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Are Spring Oscillations Goldstone Modes?

When considering a spring-like object, it seems almost intuitive that it would be "springy", i.e., harmonically oscillate. However, these oscillations occur at a macroscopic scale, unlike ...
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Realization of $\operatorname{SU}(N) \times \operatorname{SU}(N)$ in chiral perturbation theory

Suppose we have $n$ Goldstone bosons which is obtained from the fact that the ground state $\eta$ is invariant under a subgroup $H$ of $G$. Each of these Goldstone bosons will be described by an ...
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Matrix element of the currents associated with the broken generators between the vacuum and Goldstone's bosons

Let $G$ be a Lie group and $L^i$ the generators of this group. Suppose we have $L^{j}|0\rangle \neq 0$ where $|0\rangle \neq 0$ denotes the vacuum. If $G$ is associated with a symmetry of the ...
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Goldstone bosons in soft limit

There's interesting statement about amplitude of Goldstone bosons: I wanna to understand simple argument for statement of vanishing of amplitudes in the single scalar soft limit. In principle, it is ...
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Why there is an isomorphism between the coset group and the Goldstone bosons?

Suppose we have $n$ Goldstone bosons which is obtained from the fact that the ground state $\eta$ is invariant under a subgroup $H$ of $G$. Each of these Goldstone bosons will be described by an ...
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On Goldstone fermions / goldstino: SUSY breaking

There are statements about Goldstone fermions, or goldstino, seem confusing to me. (1) Goldstone boson requires a continuous symmetry spontaneously broken. Does Goldstone fermion imply continuous SUSY ...
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Supersymmetry non-breaking $\iff$ no "Goldstone fermion"?

In Supersymmetry and Morse Theory (1982) by E Witten, Concern whether the supersymmetry is broken by checking whether $$ Q | 0 \rangle=0 $$ exists or not --- Witten said: SUSY breaking: A solution ...
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Goldstone counting from symmetries and from the expansion in the Lagrangian

Goldstone theorem states that when a continuous symmetry is broken there is a massless mode for each broken generator. To exemplify the theorem, many references consider the complex scalar theory with ...
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Confusion on the proof of Goldstone’s theorem

I amd reading a proof Goldstone’s theorem in Zee's QFT book. On page $228$, Zee presents the proof as follows. The conserved charge $Q$ is given by \begin{equation} Q=\int d^D\vec{x}J^0(t,\vec{x}). \...
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Does Goldstone-boson absorption by $W$ and $Z$ bosons mean that their masses are stochastic?

I have heard it explained that a complex doublet scalar field produces the Higgs boson, and also three Goldstone bosons which are absorbed by $W$ and $Z$ bosons, giving them mass. Does this mean that ...
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Confusions on $U(1)_{A}$ problem

In Cheng&Li's book "Gauge theory of elementary particle physics" section 16.3, the $U(1)$ problem is presented. Here is the story: Suppose we have a theoy which contains only two quarks $...
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Schwartz's and Zee's proof of Goldstone theorem

In Refs. 1 & 2 the Goldstone theorem is proven with a rather short proof which I paraphrase as follows. Proof: Let $Q$ be a generator of the symmetry. Then $[H, Q] = 0$ and we want to consider ...
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Number of Goldstone Modes in the Heisenberg Model?

I am getting confused about the number of Goldstone modes present in the Heisenberg model. After a Holstein-Primakoff transformation the energy can be written a: $$H=-JS^2 Nz+ \sum_{\vec k} \...
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Measuring and calculating free quark masses

Particle data book contains masses of the free quarks. I wonder, how do experimentalists determine the masses of the free quarks even though they are trapped inside hadrons (except perhaps in quark-...
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Goldstone's theorem, symmetry breaking and the Heisenberg model

I'm currently researching symmetry breaking and Goldstone's theorem for a project in my third year of my theoretical physics degree. So my knowledge isn't from a formal teaching but my own research. I ...
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Does the existence of soft Goldstone mode makes a topological superfluid unstable?

When people talk about topological order and Majorana fermions in a BCS superfluid, do they admit the existence of a massless Goldstone mode? (As I've heard that soft Goldstone mode always exists in ...
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Are acoustic phonon frequencies always linear in $k$?

I'm confused by a discussion in Ashcroft and Mermin's textbook on pg. 512-513. They say that if we have a bunch of ions in a solid and neglect the effect of the conduction electrons, then waves will ...
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In what sense do Goldstone bosons live in the coset?

Goldstone's theorem says that if a group, $G$, is broken into its subgroup, $H$, then massless particles will appear. The number of massless particles are given by the dimension of the coset, $G/H$. ...
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Unified Description of Nambu-Goldstone Bosons without Lorentz Invariance

I am reading an article Unified Description of Nambu-Goldstone Bosons without Lorentz Invariance, arXiv:1203.0609, by Watanabe & Murayama. It gives a proof on the counting of Nambu–Goldstone ...