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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Mass parameters in the spontaneous symmetry breaking

In Ashok: Lectures on Quantum Field Theory Sec. 7.5 p.279 he discusses the mass parameters for the potential $$ V = -\frac{m^2}{2}(\sigma^2+\xi^2) + \frac{\lambda}{16}(\sigma^2 + \xi^2)^2. $$ My (...
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Anomaly vs spontaneous symmetry breaking

I was trying to gain a basic understanding of anomalies. In the case of anomalies, certain correlations which should have been zero based on symmetry considerations of the action, instead turn out to ...
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Cosmic strings and ground state degeneracy in SSB

There are two conflicting perceptions that I have regarding the notion of ground state post-SSB. Consider the Higgs mechanism e.g. for the electroweak theory. On the one hand, we say that the vacuum ...
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Higgs critical temperature?

Higgs mechanism is defined to operate (give spontaneous symmetry breaking) below some critical temperature. As mentioned, for ex. in Wikipedia: Below some extremely high temperature, the field ...
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What is the effect of the global $U(1)$ spontaneous symmetry breaking on the physical observables in superconductors?

I have been taught that in Ginzburg-Landau model for superconductors the transition to the superconductive states lead to the spontaneous symmetry breaking of global $U(1)$ symmetry. While I have ...
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Does pseudospin break the crystal symmetry?

Pseudospin is a concept to describe a superposition of two quantum states. Sometimes, I see a pseudospin texture in the momentum space which breaks a crystal symmetry. A simple example is the ...
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All infinite volume matrix product states are in different super-selection sectors?

Consider two matrix product states $\Psi_1,\Psi_2$, i.e. let them be described (schematically) by $$ \Psi_\alpha = \sum_{...i_n ... } \left(\ell_{\alpha}, \left[\prod_{i \in \mathbb{Z}} E_\alpha(i) \...
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Fermi liquid/gas and Goldstones

Often, we say that the low-energy excitations of a quantum system that spontaneously breaks certain symmetries is described by Goldstone bosons. It is also well-known that Fermi liquids and gases are ...
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Are sound waves in fluids also Goldstone modes?

One usually says that sound waves in a crystal, ie. phonons, are Goldstone modes of the broken translation symmetries. However, fluids have sound waves and also translation invariance. In fact, it's ...
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Does technical naturalness hold only for global symmetries, or also gauge symmetries?

Suppose you have an action $S(\epsilon) = S_1 + S_2 + \epsilon\, S_\mathrm{int}$. Assume that $S_1$ is gauge invariant under the action of the group $G$ and $S_2$ is gauge invariant under the action ...
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Symmetry properties of Goldstone mode dispersions

I've recently worked on a lattice model relevant to condensed matter physics. The model features an ordered phase with a broken $U(1)$ symmetry, so I anticipated the presence of a Goldstone mode. One ...
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Is the vacuum of a local ${\rm U(1)}$ gauge theory unique?

Consider a spontaneously broken scalar field theory with a global ${\rm U(1)}$ symmetry described by the Lagrangian $$\mathscr{L}=(\partial_\mu\phi^*)(\partial^\mu\phi)-\mathcal{V}(\phi),\\ \mathcal{V}...
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Coset construction for Goldstones in Weinberg Vol II

In section 19.6 of Weinberg Vol II, there is a discussion of the coset construction for effective actions of Goldstones arising from spontaneous symmetry breaking of internal symmetries. In this ...
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Bare mass versus the mass form spontaneous symmetry breaking

Consider renormalization in $\phi^4$ theory $$\mathscr{L}=(\partial\phi)^2-\frac{1}{2}m^2\phi^2+\frac{\lambda}{4}\phi^4$$ where $m$ and $\lambda$ are respectively the unobservable bare mass and bare ...
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Pion mass in theta vacuum

Does the mass of the charged pion depend on the QCD vacuum angle? I've seen it said---e.g., in these TASI lectures---that when the quark mass matrix is real and there is a nonzero QCD vacuum angle $\...
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Higgs amplitude mode vs. “spontaneous breaking” of local gauge symmetry

It seems that there are two definitions of the Higgs mode; one primarily used in particle theory, and one used in condensed matter. In the former, we can consider a simple Lagrangian of the form $$ \...
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How can the Higgs field “sense” that the temperature is high enough to restore the broken symmetry above this temperature?

In the picture below of the potential of the Higgs field, we can see that the field has its highest energy if the field is zero, i.e. if no Higgs particles are present. The symmetry is restored at ...
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Implications of Instanton Corrections (to Degenerate Vacuua) for Spontaneous Symmetry Breaking

We consider that if the classical vacuua of a theory are degenerate then each of them can be non-invariant under one or more of the symmetries of the Lagrangian. We can choose one of the vacuua and ...
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Spontaneous breaking of supersymmetry

I am a beginner in SUSY Quantum mechanics. I had read that the symmetry is spontaneously broken if $A \left|\psi \right>_n^{(1)}\neq 0 $ and symmetric if $A \left|\psi \right>_n^{(1)}= 0. $ But ...
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Anderson-Higgs Mechanism

Consider an abelian gauge field coupled with a complex field: $$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+(D\varphi)^\dagger D\varphi+\mu^2 \varphi^\dagger\varphi-\lambda(\varphi^\dagger\varphi)^2.$...
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Sufficient and Necessary Conditions for Chiral Symmetry Breaking

In their 2005 paper, the authors write (just below eq. 3.19) we see that a non-zero value of $F_0$ is a necessary and sufficient criterion for spontaneous chiral symmetry breaking. On the other ...
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Higgs Field before or after Cosmological Inflation?

Question: When exactly did the Higgs Field begin to exist (before cosmological inflation or after cosmological inflation)?
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Are there Goldstone bosons in 1D or 2D?

The Mermin-Wagner theorem states that continuous symmetries cannot be spontaneously broken at finite temperature in systems with short-range interactions in dimensions d ≤ 2. And Goldstone bosons ...
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Why are Goldstone bosons associated to translation symmetry breaking corresponding to the coordinates transverse to the D-brane?

In some lecture notes, in the presence of a Dp-brane, translational symmetry is broken in the direction transverse to the brane. However, they identify the coordinates, $$\Phi^a,\quad a=p+1,\cdots,D-1$...
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SUSY Breaking in the Vacuum

Under what conditions is supersymmetry preserved in the vacuum state? In particular, suppose I have some super potential $W(x)$ which does not permit normalizable ground-state wave functions (such as $...
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Masslessness of Goldstone Bosons

I have a question about the origin of the massless nature of Goldstone bosons. This is what I understood: we know thanks to Goldstone's theorem that from the breaking of continuous symmetries emerge ...
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Why is the relation $M_W=M_Z\cos\theta_W$ true only at tree-level?

In Glashow-Weinberg-Salam electroweak theory, the relation $$M_W=M_Z\cos\theta_W\tag{1}$$ is said to be remain true only at the tree-level; it receives corrections from the loop diagrams. See here. ...
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Graph dimension and phase transitions

In this question I am mainly concerned with phase transitions associated with symmetry breaking. For phase transitions of this type, there are a number of results describing the relationship between ...
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Misconceptions in spontaneous symmetry breaking

Spontaneous symmetry breaking occurs when we have a potential like a mexican hat as shown in figure (right) and is unbroken for the potential shape as shown in left figure. Under the Symmetry ...
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Polar representation of complex scalar field in spontaneous symmetry breaking

In Rubakov's "Classical Theory of Gauge Fields", in the chapter on Higgs mechanism, he mentions that you can switch to the polar form of the complex scalar field only for non-zero vacuum value of the ...
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Shift symmetric scalar field

Suppose I study a quantum field theory in which among other fields a shift symmetric scalar field appear: $$\phi\rightarrow\phi+c$$ with $c$ a real constant. Can this always be interpreted as a Nambu-...
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Spontaneous symmetry breaking: proving the equivalence of two definitions

This question can be posed for both quantum and classical set-ups. For concreteness, let me consider a local, classical Hamiltonian $H$. The expectation values I consider are with respect to the usual ...
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Spinwaves, Mermin-Wagner theorem, Two-point correlation function and Heisenberg model

I was looking at the Mermin-Wagner theorem (as following the previous question) and the Heisenberg model seems to be presented, and they split the Hamiltonian $H$ in the matrix or vector n-components ...
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What happens to the (Wilsonian) effective action if a symmetry is spontaneously broken?

A spontaneously broken symmetry is a symmetry of the action which does not manifest itself in physical states. Since the action is still invariant under this symmetry, can we say the same about the ...
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Relating the Yang-Mills field-strength to the Maxwell tensor in $SU(2)$ gauge theory

I'm studying topological monopoles in a $SU(2)$ Yang-Mills theory with spontaneous symmetry breaking, through the book "Topological Solitons", by Manton and Sutcliffe. In section 8.2, the authors ...
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Mermin-Wagner and graphene

I have been told that the Mermin-Wagner theorem disallows the existence of the crystal of graphene. However, I don't have enough knowledge to understand the Mermin-Wagner theorem. If possible can ...
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Detecting the electroweak unification in data of quark gluon plasma

Electroweak unification is discussed in the Big Bang model and variant proposals, and there is a transition at energies of 100 GeV where the EW symmetry is unbroken and a quark gluon plasma phase ...
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Super GoldStone model and SuperHiggs theorem

I am currently studying SuperSymmetry and I have reached a problem which I have not found an answer to. I clearly understand how the Goldstone theorem works for the boson case (without any susy) ...
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If the gauge symmetry is not broken by spontaneous symmetry breaking, what symmetry is broken?

In this post, the answer by buzhidao showed that the $U(1)$ gauge symmetry is not broken by spontanous symmetry broken and Higgs mechanism. What role does "spontaneously symmetry breaking" ...
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Goldstone bosons when SSB potential has two fields

A theory consists of two complex scalar fields $φ_0$ and $φ _1$ with a symmetry-breaking potential $$V(|φ_0|^2 + |φ_1|^2).$$ How many Goldstone particles will there be in the theory?
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Mass splitting for leptons and down quarks in $SU(5)$ model?

In the minimal $SU(5)$ GUT model, the masses of the leptons and the corresponding down-type quarks must be equal. i.e. $m_e=m_d$, $m_\mu=m_s$, $m_\tau=m_b$. Whereas this is approximately correct for ...
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Axion CDM analogous to reheating?

In a paper on Axions Peccei writes: When the Universe goes through the PQ phase transition at $T ∼ f_a > > Λ_{QCD}$ the QCD anomaly is ineffective. As a result, early in the Universe $< ...
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Direction of chiral symmetry breaking

What happens if one breaks the SU(2) chiral symmetry in a non-real direction(a direction apart from the one that the small masses of hadron's constituents(quarks) dictate). Or equivalently We know ...
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Could electroweak interactions derive from higgsing of gauged chiral symmetry?

A recent question discussed the practice of introducing electroweak interactions into chiral perturbation theory by gauging part of the chiral symmetry. Gauging chiral symmetry is normally regarded as ...
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Embedding of $SU(2)_L \times U(1)_Y$ into $SU(2)_L \times SU(2)_R$ in electroweak chiral effective theories

In the context of chiral effective theories we usually deal with the pion field \begin{equation} U= ie^{\frac{\pi^a \sigma^a}{2f}} \end{equation} where $\pi^a=\big(\pi^1, \pi^2, \pi^3\big)$ are the ...
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Is Lorentz symmetry broken if SUSY is broken?

I have seemingly convinced myself that the entire Poincare group is spontaneously broken if one of the supersymmetric charges is spontaneously broken. We know that if one of the supersymmetric ...
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Kitaev's reason for looking at p-wave superconductors

I would really like to understand the thought process that lead Kitaev to proposing the Kitaev chain but most expositions really just start from the Hamiltonian and derive the Majoranas without any ...
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How to break softly a symmetry

If you start from a theory (non-SUSY) with whatever symmetry and you break it by adding a soft term. Do you have to add to your Lagrangian all possible soft terms that break the symmetry? If so, why? ...
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Simplest model in field theory which leads to a pseudo-Goldstone boson

What can be a simple (if not simplest) continuum field theory model that gives rise to a pseudo Goldstone boson (doesn't matter if it is a toy model)? For example, I would be very happy if one can ...
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Symmetry and symmetric vacuum in Quantum Field theory

In the start of section 28.2 of Schwartz's Quantum Field theory and the Standard Model, Schwartz states that for a conserved charge, $\hat{Q}$, which generates the corresponding symmetry ...