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.
81 questions
0
votes
0
answers
28
views
Spontaneous symmetry breaking - incorrect Peierl's argument in the continuous 3d $O(3)$ model
The following must be wrong but I don't see why:
Consider an $O(3)$ model in $d=2$ dimensions. Gapless Goldstone bosons (magnons) at $k\to 0$ are the lowest-energy excitations of the system with some ...
- 11
2
votes
1
answer
114
views
Parametrization of Spontaneous Symmetry Breaking
I am currently studying the spontaneous symmetry breaking part in Schwartz's Quantum Field Theory and the Standard Model. I am solving the Problem 28.1.
In the main text, Schwartz presents the example ...
- 134
0
votes
1
answer
82
views
In the Non-Abelian Higgs Mechanism, what is the definition of the resulting massive vector fields?
In the Abelian case of the Higgs mechanism, say $U(1)$, the way the Goldstone bosons that arise due to SSB are "eaten" by the gauge vectors is usually explained by going to the unitary gauge....
- 1
0
votes
1
answer
77
views
Goldstone Matrix for $SO(3) \longrightarrow SO(2)$ breaking
In these lecture notes (https://arxiv.org/abs/1506.01961) on composite Higgs models, on page 22, the authors calculate the Goldstone Matrix $U[\Pi]$ for an abelian composite Higgs scenario i.e. $SO(3)$...
- 311
1
vote
0
answers
40
views
What does stable/meta-stable vacuua actually mean in theories with spontaneously broken symmetries?
In SSB, there exists a scalar field has non-zero VEV. So we redefine the fields to have zero VEV for several reasons. But what is the vacuum used in calculation of the correlators in this field and ...
- 300
5
votes
1
answer
209
views
Confusion of the Proof of Goldstone Theorem in QFT
Context
I found many proof of the symmetry breaking is the following:
Let $\langle \delta \phi(x) \rangle$ be the corresponding symmetry transformation w.r.t. the charge $Q = \int d^Dx\; J^0(x)$.
$\...
- 170
1
vote
1
answer
139
views
Why Goldstone Bosons in QCD act as the effective long distance mediators?
QCD breaks axial generators spontaneously, as $m(u) < m(d) << Λ_{QCD}$ and for energies much less than $Λ_{QCD}$, the theory which predicts this spontaneous breaking (QCD) is already ...
- 149
1
vote
0
answers
53
views
Goldstone Theorem in Schwartz, follow-up
This is somewhat of a related question to
Goldstone theorem in Schwartz
and is related to equation 28.16 in Schwartz's QFT book.
One way to prove that
$$ \langle \Omega | J^\mu(x) | \pi(p) \rangle = i ...
- 29
0
votes
0
answers
45
views
Unitary Gauge Removing Goldstone Bosons
The Lagrangian in a spontaneously broken gauge theory at low energies looks like
$$ \frac{1}{2} m^2 ( \partial_\mu \theta - A_\mu )^2 $$
and the gauge transformations look like $\theta \rightarrow \...
- 29
5
votes
2
answers
446
views
Why are there no Goldstone modes in superconductor?
Usually, the absence of Goldstone modes in a superconductor is seen as an example of the Anderson-Higgs mechanism, related to the fact that there is gauge invariance due to the electromagnetic gauge ...
- 109
6
votes
1
answer
189
views
Goldstone theorem for classical and quantum potential
Consider a quantum theory $$\mathcal{L}(\phi^a) = \mathcal{L_{kin}}(\phi^a)-V(\phi^a),\tag{11.10}$$ depending on any type of fields $\phi^a$.
The VEV of this theory are constant fields $\phi_0^a$ such ...
- 421
2
votes
0
answers
96
views
Soliton and Goldstone boson
I'm learning Gross-Pitaevskii model. By spontaneous symmetry breaking one obtains Bogoljubov modes, which ensures Landau criterion. So those modes have two features, for one they are Goldstone bosons ...
- 136
2
votes
2
answers
244
views
Existence of low temperature phase in 2-Dimensional XY-Model
I'm reading the lecture notes of David Tong on Statistical Field Theory, specifically chapter $4.4$ on the Kosterlitz-Thouless Transition. He considers the XY model in $d=2$ dimensions and states the ...
3
votes
0
answers
49
views
Spectral function of superconductor in the BEC regime: how does Higgs mechanism affect the spectrum?
Consider the standard BCS theory but assume that the interaction energy $U$ that enters the definition of gap parameter (i.e. $\Delta = (U/N)\sum_k \langle c_{-k\downarrow} c_{k\uparrow} \rangle$, ...
- 3,089
1
vote
1
answer
273
views
Goldstone bosons and spontaneous symmetry breaking in the complex triplet model
In the Standard model electroweak theory, the Higgs field is a complex doublet field, which couples to the $SU(2)$ gauge field. Suppose we replace the complex doublet with a complex triplet $\Sigma$:
$...
- 1,461
0
votes
0
answers
80
views
Infinite conductivity, goldstone bosons and generalized voltage
Weinberg [1] has put forth an elegant argument (summarized in [2, Section 3.3] and at the bottom of the question), which suggests that the broken symmetry phase of any $U(1)$ gauge theory will be ...
- 469
3
votes
0
answers
87
views
An issue with spectral representation in Coleman's proof of Goldstone's theorem in his Lectures on QFT
I am working through the proof of Goldstone's theorem in Coleman's Lectures on QFT in chapter 43.4.
He uses a Källén-Lehmann-like spectral representation of
$$\langle 0 | j_{\mu}\left(x\right)\phi\...
- 130
2
votes
0
answers
79
views
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)
$$
U(2)_L \otimes U(2)_R = SU(2)_L \...
- 23
3
votes
2
answers
398
views
Explanation of massive Goldstone modes
I'm solving this exercise with a Heisenberg Hamiltonean in linear spin-wave theory and at some point we are asked to compute the dispersion relation at $k=0$, which leads me to finding two different ...
- 548
4
votes
1
answer
351
views
Global symmetries QCD goldstone bosons
Beside the local $SU(3)$-Color-symmetrie The QCD Lagrangian also has global symmetries:
$$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} $$
$SU(...
- 63
1
vote
1
answer
40
views
liquid-vapor interface goldstone modes
The free energy of a liquid-vapor interface (far from critical point) can be approximates as
$$
\mathcal{F}=γ A+\frac{γ}{2}\int dxdy|\nabla h(x,y)|^2
$$
where $h(x,y)$ is the height of the interface ...
- 129
4
votes
2
answers
369
views
What is meant by the statement that Wigner-Weyl is linear realization and Nambu Goldstone is nonlinear realization?
In a paper, "Nonlinear Realization of Global Symmetries" by Zhong-Zhi Xianyu, I see the line
"There are basically two ways of symmetry realization in Hilbert space. One is such that ...
2
votes
1
answer
93
views
Goldstone modes/momentum generation from the vacuum
I'm quite confused by two short paragraphs in Schwarz 28. He proves that $$ 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{...
- 182
2
votes
1
answer
260
views
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,
$$ \frac{1}{2}\left(D_\mu \phi_i\right)^2=\frac{1}{2}\left(\...
- 1,461
2
votes
0
answers
117
views
Goldstone Bosons and Criticality
In the presence of spontaneous symmetry breaking of a continuous symmetry, there are massless goldstone bosons. However, they are not treated in the discussions of critical phenomena. Naively I would ...
- 71
1
vote
0
answers
97
views
How to construct Goldstone bosons from a conserved charge?
The Goldstone boson associated with a conserved charge is described by
$$
|\pi(\vec p)\rangle\propto \int d^3x~e^{i\vec p\cdot\vec x}j_0(x)|0\rangle.
$$
How can I see that this is a state with energy ...
- 19
2
votes
0
answers
108
views
Why magnon band is half filled in XY model?
This is my first contact with that field so I hope that I describe my problem clearly.
In the notes I follow we consider a 1-dimensional XXZ problem with
\begin{equation}
\hat{H}_{XXZ}=-J_{\perp}\sum_{...
- 175
1
vote
0
answers
359
views
Spontaneous Symmetry Breaking / Explicit Symmetry Breaking [closed]
I'm trying to understand the SSB and explicit symmetry breaking.
If we have a complex scalar field theory with Mexican hat potential,
$$\phi(x)=\frac{\sigma(x)}{\sqrt{2}}e^{i\Pi(x)/v}$$
choice of the ...
- 3,494
1
vote
0
answers
39
views
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 ...
- 453
1
vote
0
answers
263
views
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 ...
- 11
0
votes
1
answer
268
views
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)$.
...
3
votes
0
answers
214
views
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 ...
- 1,370
0
votes
1
answer
160
views
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 ...
- 163
2
votes
0
answers
298
views
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 ...
0
votes
1
answer
207
views
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 ...
- 3,494
0
votes
1
answer
132
views
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),
$$
where $\sigma_a$ are the Pauli matrices. Most ...
- 577
1
vote
0
answers
53
views
Goldstone and longitudinal polarization equivalence between off-shell $W$s
If I understand correctly, the couplings of the longitudinal component of the $W$ and $Z$ bosons should be "equivalent" to the Goldstone bosons they ate after SSB.
In practice, this makes ...
- 41
6
votes
1
answer
154
views
Factoring a the exponential form of a group element of a Lie group, using subgroups
I am working on non linear realization of Goldstone bosons, as is done by Weinberg in section 19.6 of Quantum theory of fields, volume II.
We have a real, compact and connected Lie group $G$ with as ...
- 577
2
votes
0
answers
105
views
Nambu-Goldstone mode without symmetry breaking
Superfluidity is often explained in terms of spontaneous breaking of global $U(1)$ symmetry. However, we know that in real, finite-size quantum systems, this symmetry can never be broken. Quantum ...
- 8,517
2
votes
0
answers
59
views
Property of representations of Lie Groups in Weinberg's Quantum Theory of Fields
In section 19.6, page 212-213 in Weinberg's Quantum Theory of Fields Volume II, Weinberg shows that on can always eliminate the Goldstone Bosons from a field $\psi(x)$, by use of a space-dependent ...
- 577
10
votes
2
answers
400
views
Vacuum Degeneracy for Massless Free Scalar Field
I am wondering how to explicitly see the vacuum degeneracy for a free massless scalar field, described by the action $$S = -\frac{1}{2}\int d^4x\,(\partial\phi)^2.$$ This action is invariant under ...
- 739
2
votes
0
answers
55
views
SSB of Spacetime Symmetries in 2d
Colemans theorem states that there are no Goldstone Bosons in $d=2$ spacetime dimensions : https://inspirehep.net/literature/83738 because there is no spontaneous symmetry breaking.
However I have ...
- 149
4
votes
1
answer
527
views
Spontaneous symmetry breaking in the Standard Model. What is "broken"?
I know this question has been asked other times, but I am looking for a confirmation of the following.
When we say that the gauge group of the standard model is $G_{SM} = SU(3)_{c} \times SU(2)_{L} \...
- 1,084
2
votes
2
answers
223
views
Spontaneous Chiral Symmetry Breaking in Schwartz
I am reading M. Schwartz's book on QFT, equation (28.24)/(28.22). They say that a set scalar fields will transform as, where $g_L$ belongs to $SU(2)_{L}$ and $g_R$ belongs to $SU(2)_R$:
$$\Sigma\...
- 462
1
vote
0
answers
35
views
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
$$
\mathscr{L}=-\frac{1}{2} \partial_{\mu} \phi_{n} \partial^{\mu} \phi_{n}-\frac{\mathscr{M}^{2}}{2} \phi_{n} \phi_{n}-\...
- 485
2
votes
1
answer
108
views
What does the absence of these Goldstone boson interactions mean physically?
I have read that in several statistical models exhibiting spontaneous symmetry breaking, the resulting Goldstone bosons do not interact with each other via $\theta^{2n}$ terms — only via derivative ...
- 61
9
votes
1
answer
164
views
Skin depth and Mermin-Wagner theorem
I recently became aware of the Coleman-Mermin-Wagner theorem presented in [1802.07747] for higher-form symmetries and I have a question about how it might be applied to electromagnetism.
The theorem ...
- 1,113
9
votes
1
answer
352
views
About Itzykson and Zuber's proof of Goldstone's theorem
In chapter 11-2-2, I&Z discuss Goldstone's theorem. They start by claiming that if an operator $A$ exists, such that
$$ \delta a(t) \equiv \langle 0| [Q(t),A]|0\rangle \neq 0 \tag{11-30} $$
the ...
- 2,678
1
vote
1
answer
148
views
How scalars appear in Goldstone's theorem
Let $G$ be a lie group and $G_i$ its generators. Suppose for a set of fields $\chi_{\alpha}(x)$ we have
$$
\left[G_{i}, \chi_{\alpha}(x)\right]=-\left(g_{i}\right)_{\alpha \beta} \chi_{\beta}(x)
$$
If ...
- 485
2
votes
0
answers
63
views
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, ...
- 143