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. Goldstone Modes are found throughout physics, with some celebrated examples stemming from the Higgs Mechanism.

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Existence of low temperature phase in 2-Dimensional XY-Model

I'm reading the lecture notes of David Tong on Statistical Field Theory, specifically chapter $4.4$ on the Kosterlitz-Thouless Transition. He considers the XY model in $d=2$ dimensions and states the ...
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Spectral function of superconductor in the BEC regime: how does Higgs mechanism affect the spectrum?

Consider the standard BCS theory but assume that the interaction energy $U$ that enters the definition of gap parameter (i.e. $\Delta = (U/N)\sum_k \langle c_{-k\downarrow} c_{k\uparrow} \rangle$, ...
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Goldstone bosons and spontaneous symmetry breaking in the complex triplet model

In the Standard model electroweak theory, the Higgs field is a complex doublet field, which couples to the $SU(2)$ gauge field. Suppose we replace the complex doublet with a complex triplet $\Sigma$: $...
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Infinite conductivity, goldstone bosons and generalized voltage

Weinberg [1] has put forth an elegant argument (summarized in [2, Section 3.3] and at the bottom of the question), which suggests that the broken symmetry phase of any $U(1)$ gauge theory will be ...
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An issue with spectral representation in Coleman's proof of Goldstone's theorem in his Lectures on QFT

I am working through the proof of Goldstone's theorem in Coleman's Lectures on QFT in chapter 43.4. He uses a Källén-Lehmann-like spectral representation of $$\langle 0 | j_{\mu}\left(x\right)\phi\...
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Goldstone bosons in 2 and 3 quark flavor symmetries [closed]

In my (undergraduate) advanced elementary particles class last semester, we learnt that for a 2 quark (u/d) model the symmetry of the Lagrangian is (and breaks as) $$ U(2)_L \otimes U(2)_R = SU(2)_L \...
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Explanation of massive Goldstone modes

I'm solving this exercise with a Heisenberg Hamiltonean in linear spin-wave theory and at some point we are asked to compute the dispersion relation at $k=0$, which leads me to finding two different ...
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Global symmetries QCD goldstone bosons

Beside the local $SU(3)$-Color-symmetrie The QCD Lagrangian also has global symmetries: $$L_{QCD}=\sum_{f,c}\bar{q_{fc}}(i\gamma^\mu D_\mu - m ) q_{fc} - \frac{1}{4}F^a_{\mu \nu} F^{a \mu \nu} $$ $SU(...
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liquid-vapor interface goldstone modes

The free energy of a liquid-vapor interface (far from critical point) can be approximates as $$ \mathcal{F}=γ A+\frac{γ}{2}\int dxdy|\nabla h(x,y)|^2 $$ where $h(x,y)$ is the height of the interface ...
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What is meant by the statement that Wigner-Weyl is linear realization and Nambu Goldstone is nonlinear realization?

In a paper, "Nonlinear Realization of Global Symmetries" by Zhong-Zhi Xianyu, I see the line "There are basically two ways of symmetry realization in Hilbert space. One is such that ...
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Goldstone modes/momentum generation from the vacuum

I'm quite confused by two short paragraphs in Schwarz 28. He proves that $$ Q = \int d^3 x J_0(x) = \int d^3 x \sum_m \frac{\partial L}{\partial \dot \phi_m} \frac{\delta \phi_m}{\delta \alpha} \tag{...
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Goldstone boson in Higgs Mechanism

On P&S's QFT book, page 693, the book discussed "Systematics of the Higgs Mechanism". For the kinetic part of Lagrangian, $$ \frac{1}{2}\left(D_\mu \phi_i\right)^2=\frac{1}{2}\left(\...
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Goldstone Bosons and Criticality

In the presence of spontaneous symmetry breaking of a continuous symmetry, there are massless goldstone bosons. However, they are not treated in the discussions of critical phenomena. Naively I would ...
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How to construct Goldstone bosons from a conserved charge?

The Goldstone boson associated with a conserved charge is described by $$ |\pi(\vec p)\rangle\propto \int d^3x~e^{i\vec p\cdot\vec x}j_0(x)|0\rangle. $$ How can I see that this is a state with energy ...
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Why magnon band is half filled in XY model?

This is my first contact with that field so I hope that I describe my problem clearly. In the notes I follow we consider a 1-dimensional XXZ problem with \begin{equation} \hat{H}_{XXZ}=-J_{\perp}\sum_{...
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Spontaneous Symmetry Breaking / Explicit Symmetry Breaking [closed]

I'm trying to understand the SSB and explicit symmetry breaking. If we have a complex scalar field theory with Mexican hat potential, $$\phi(x)=\frac{\sigma(x)}{\sqrt{2}}e^{i\Pi(x)/v}$$ choice of the ...
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Quantum numbers of a symmetry generator?

I am trying to understand the italicized part: a broken symmetry leads to a massless Goldstone boson with same quantum numbers as broken symmetry generator. Weinberg talks about it in Chapter 19, if ...
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Axions as goldstone bosons of anomalous $U(1)$ symmetry

In the $m_q \rightarrow 0$ limit the QCD lagrangian has the symmetry $U(N)_V \times U(N)_A$. Including just the two lightest quarks, $N=2$, and looking at the $U(2)_A=SU(2)_A \times U(1)_A$ part, we ...
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How could negative $W$ "eat" the positive Goldstone bosons $\phi_1$ and $\phi_2$?

In the Standard Model, the Higgs field is $\Phi=\left(\begin{array}{c}\phi^+\\\phi^0\\\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}{c}\phi_1+i\phi_2\\\phi_3+i\phi_4\\\end{array}\right)$. ...
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Goldstone Modes, Galilean Symmetry, and Negative Excitations in Fermi Gas

Considering the centrality of Goldstone quasiparticles in condensed matter theories, I was wondering if the converse of the theorem might also be true: Does the existence of a gapless excitation imply ...
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Identifying mesons with Goldstone modes

In QCD, due to the work of ‘t Hooft and Vafa-Witten, we know that confinement implies chiral symmetry breaking (David Tong’s gauge theory notes have a clear discussion of this). It is then said that ...
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Coset construction for space-time symmetry breaking: how is the covariant derivative built?

I am currently learning how to build effective field theories for Nambu-Goldstone modes (NG modes) by using the coset construction formalism. I essentially follow 2 reviews: For the breaking of ...
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How does the Nambu-Goldstone mode explain the absence of parity doubling?

I've been doing some reading about chiral symmetry breaking since it was not touched in my particle physics course I found these slides As explained in the above link, if we take $|\psi \rangle$ as ...
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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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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 ...
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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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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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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 ...
amilton moreira's user avatar
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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 ...
amilton moreira's user avatar
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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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