Questions tagged [symmetry-breaking]

In explicit symmetry breaking, the equations of motion of a physical system are variant under the broken symmetry; by contrast, for spontaneous symmetry breaking (SSB), these equations are invariant, but the entire system is not because its vacuum (background) is non-invariant. Further use for the SSB characteristic nonlinear realizations (Goldstone mode), and the group theoretical patterns involved.

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Is this asymmetrically frozen ice cream an example of symmetry breaking?

Today I noticed my ice cream had frozen in an asymmetric pattern because I left it resting at an angle on the fridge. I'm wondering if this is analogous to the concept of symmetry breaking as used in ...
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Frustrated classical field theory

The frustrated Ising model (see e.g. this answer) is an example of a system that shows no unique ground state and many metastable states (its "energy landscape" is extremely complex). ...
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Statistical physics of broken symmetry state?

If we are considering a system, like a paramagnet, above its critical temperature $T_c$ we can statistically describe it using a canonical ensemble: $$ \langle A \rangle = Tr\left[ \rho A \right]; \...
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How can the Higgs Field have low energy but high vacuum expectation value?

I'm not a math student, I'm just curious as to know why and how can a vacuum be more stable with high Vacuum expectation value than 0 vev. A similar question would be what does it mean when the EM and ...
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System of particles with spin & symmetry breaking

I was solving an exercise in which I had to find how the entropy of a system of N (assuming distinguishable) particles with spin changes when the temperature $T\rightarrow0$ and $T\rightarrow \infty$. ...
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In an $SO(10)$ GUT, how many gauge fields acquire masses that are above the electro-weak scale?

The $SO(10)$ group is spontaneously broken down into the Standard Model gauge group at around $10^{16}$ GeV and the electroweak scale is ~246 GeV. I think there should be 45 gauge fields predicted by ...
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Have magnetic monopoles been predicted to have been produced in other cases than those of symmetry breaking in GUTs?

Also, what other configurations are magnetic monopoles theorized to exist in? (such as black holes, monopole walls etc.)
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What related massive/massless degrees of freedom and the generators of a field?

A lecturer has stated that it is possible to determine the number of massive and massless degrees of freedom from the known generators of a $SO(3)$ field and a given v.e.v. In the given example, the ...
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Is the early universe cosmology phase transition thermal or quantum? WHY?

In this question How does SSB happen?, one of the answers assumes the nature of the early universe phase transition to be thermal. I need to know why can't such phase transition be a quantum phase ...
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Does the number of broken generators in SSB depend on the choice of VEV?

I take the Lagrangian, $$\mathcal{L}=\frac{1}{2}\partial_\mu \phi^T\,\partial^\mu\phi\,-\, \frac{1}{2}\mu^2\phi^T\phi-\frac{\lambda}{4}(\phi^T\phi)^2~,$$ where $\phi=(\phi_1,\,\phi_2,\,\phi_3)$ (real ...
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What determines a generator as broken or unbroken?

From my understanding of "generators"(e.g. below is $T^a$), one is broken if: $$T^a \phi \not= 0 \tag{1}$$ where $\phi$ is the v.e.v. of a field. But in an example given by a lecturer we ...
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What does breaking time translation symmetry mean? What's wrong with time crystal?

I was reading about a new state of matter which is called time crystal, this seemingly violates thermodynamic which had intrigue me. Someone told me that when you perform an experiment in the past, ...
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How does SSB happen?

I've just finished studying for an exam on the Standard Model (so electroweak theory and symmetry breaking) and I can't figure out how this question never crossed my mind. I'm now studying the QCD ...
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More rigorous explanation as to why $G/H = G'/H'$ for vacuum manifolds?

When studying topological defects, the parameter space (or vacuum manifold in QFT) is denoted as the coset space $G/H$, where $G$ corresponds to the symmetry group of the Lagrangian and $H$ the ...
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Why is the chiral condensate a negative quantity?

The chiral condensate serves as an order parameter for the chiral phase transition. Thus, it is a finite quantity in one phase and vanishes in the other phase. It is given as a vacuum expectation ...
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Homotopy group of $[SU(2) \times U(1)] / U(1)$ dependent on embedding of $U(1)$?

I have read, that the topology and thus the homotopy groups of $[SU(2) \times U(1)] / U(1)$ depend upon the embedding of $U(1)$ into $SU(2) \times U(1)$. For the Electroweak theory inside the standard ...
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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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Homopolar Motor and Rotational Symmetry

Take a battery, stick a magnet under it and then place a perfectly circular wire on top of the battery, in such a way to form a circuit. We now have an homopolar motor with a core that has perfect ...
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Lagrangian density: What assumption are we making and how could it change?

I'm having a hard time picturing the physical system under consideration in QFT. Scattering and propagators are understandable since we a priori assume we have certain particles given by the ...
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How an applied magnetic field breaks the inversion symmetry in a centrosymmetric system?

I want to understand why magnetic dipole transition breaks the inversion symmetry in a centrosymmetric system and gives rise to second-order nonlinearity.
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Restoration of symmetry explicitly broken by anomaly

What is the meaning of the restoration at finite temperature of a symmetry that is "broken" by the presence of an anomaly. If the symmetry is not there why is it restored at finite ...
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Order and Disorder Operators in QFT

On the wikipedia page, the 't Hooft loop operator is called a "disorder parameter," in contrast to the Wilson loop operator, which is an "order parameter." From my limited ...
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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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Local order parameter for confinement in gauge theories

I would like some help clarifying what Zinn Justin is saying in his book "Quantum Field Theory and Critical Phenomena" p.805 on detecting confinement of gauge theories. In particular, I ...
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Could the Higgs mechanism also cause the flow of time?

The Higgs mechanism gives mass to the standard model particles because it has a nonzero vacuum expectation value (vev). Now I read that there are models where the quadratic term in the Higgs potential ...
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How QCD vacuum is different from hadrons?

Although question's main focus is QCD, it actually addresses all kinds of quantum field(s) fluctuations, which are famously expressed as a constantly creating and annihilating virtual particle-...
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Long-range correlations in transverse field Ising model

The transverse field Ising model in 1+1d has two phases: a symmetric "disordered" phase and a symmetry-breaking "ordered" phase. Both of these phases have a finite excitation gap. ...
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Time Crystals in the Early Universe?

There is an extremely exciting footnote/remark from Prof. Frank Wilczek during the subsequent Question and Answer (Q&A) session (starting from 01 h 03 min 42 sec) of his recently held presentation:...
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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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Does the transverse Ising model with a quartic term have a first order phase transition?

Mean field theory applied to spin-chains, where the ground state is assumed a product state and the energy is variationally minimized, is one of the tools in my toolkit. The examples I give in this ...
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Mixing $SU(N)$ and $U(1)$ generators to form an unbroken $U(1)’$

I’m trying to understand some symmetry breaking patterns and have been reading David Tong’s Gauge Theory notes for an overview. I’m getting very confused about how one can mix $SU(N)$ and $U(1)$ ...
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How can one physically describe imaginary mass in the Higgs mechanism?

I studied the Higgs mechanism a couple of times now and one question that always comes to my mind is the imaginary part of the mass in the Higgs potential. The Higgs potential can be written as $$V = -...
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How spontaneous symmetry breaking plays with the gauging of a finite Abelian higher-form global symmetry

My question is about a claim which I suspect is true in some generality, but that I have not seen stated anywhere in the literature, and for which I do not know of any counter-examples. Most of the ...
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Hidden gauge symmetry in spontaneously broken phase?

Throughout this post I will be using Weinberg's notation. When a global symmetry group $G$ is spontaneously broken down to a subgroup $H$, it is often useful to reparametrize whatever fields that make ...
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Doesn't vector bosons enter in the formula for the calculation of beta function coefficients?

It is a well established result that, for a popular set of conventions, the lowest order $\beta$ function coefficient is calculated as $$ b=-\frac{11}{3}C_2(G)+\frac{4}{3}\kappa S_2(F)+\frac{1}{6}\eta ...
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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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Vacuum manifold and moduli space

Vacuum manifold is just another name for the manifold spanned by the ground states of quantum field theory. It is also called moduli space. According to https://en.wikipedia.org/wiki/Vacuum_manifold, ...
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Time Crystals - Question on the Hamiltonian for space crystals as in the seminal paper

I was going trough the paper "Classical Time Crystals" by A. Shapere and F. Wilczek, Physical Review Letters 109 (2012). At the beginning they state that it is easy to construct Hamiltonians ...
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Mermin-Wagner and superconductivity

Why can superconductivity exist in 2D since Mermin-Wagner should forbit it? This question was asked here before, but I don't think anyone gave a satisfactory answer, so let me revisit it. I have read ...
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How to calculate the amplitude of two degenerate vacuums?

I've seen in textbooks and heard from lecturers that spontaneous symmetry breaking (SSB) does not happen in quantum mechanics because of the tunnelling between two degenerate vacua (i.e amplitude of ...
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How does CPT symmetry hold while Charge, Parity and Time can all be violated?

To get it straight, Charge symmetry can be broken by U-236 atoms, Parity can be broken by K-one atoms and Time symmetry can be broken by quarks. So, how do they together hold true?
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Can the gauge boson respective of a spontaneously broken generator remain massless in the context of the Higgs Mechanism?

I'm studying a 3-3-1 model which is an extension of the standard model. The breaking $$SU(3)\times U(1) \to SU(2)\times U(1)'$$ occurs in a single step and through a single scalar VEV. The problem is ...
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Gauge invariance at the quantum level post SSB in the case of Abelian Gauge theory

On page 690 in Peskin & Schroeder's book, the Higgs mechanism is discussed in the context of an Abelian Gauge theory. After the SSB, among the many terms that appear, there is a mass term for the ...

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