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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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Unitary gauge for spontaneous symmetry breaking

I'm given a lagrangian $$ \mathcal{L} = \partial_{\mu} \Phi^{\dagger} \partial^{\mu} \Phi + m^2 \Phi^{\dagger} \Phi - \lambda (\Phi^{\dagger} \Phi)^2 $$ where $m^2 > 0, \lambda > 0$. This ...
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Physical interpretation of Bogolyubov sound with zero momentum in BEC

The point with zero momentum in the spectrum of acoustic phonon corresponds to a translation of the crystal. Is there an interpretation like this of the Bogolyubov sound in BEC? I have read that in ...
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Collective modes of charge density wave

The question is about collective modes of charge density waves, i.e., amplitude and phase fluctuations $\delta,\phi$ of the order parameter $\Delta(x,t)=(\Delta_0+\delta)e^{i\phi}$. I read on p.1 of ...
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How can I simulate a ground state degenerate system numerically?

I'm using numerical method like DMRG to simulate ground state of correlated systems. But the degeneracy of the ground state has long bothered me: When degeneracy exists the ground state isn't unique. ...
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What is the standard definition of quantum spontaneous symmetry breaking?

i found many answers about spontaneous symmetry breaking here but i am not sure to see what is the standard definition of SSB. i am interested in the BCS theory and i would like to know how the ...
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$W$, $Z$ bozons and fermions gained their masses WHEN?

I have read these questions: How does the Higgs mechanism work? How does the Higgs boson give mass to other elementary particles like electrons? What is the difference between the Higgs Boson ...
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What is the 't Hooft determinant?

The 't Hooft vertex/determinant is somehow generated by instantons and is responsible for the generation of mass gap in pseudo-Goldstone bosons, such as an axion. For example, the complex Peccei-...
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How do we determine who eats who (or which field gets massive) in a Higgs phase transition?

Consider the following theory in which a $2$-form field $B_{\mu\nu}$ with field strength $P_{\alpha\mu\nu}=\partial_{[\alpha}B_{\mu\nu]}$ is coupled to a $3$-form gauge field $C_{\alpha\beta\gamma}$ ...
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Why do we have constraints on the baryon asymmetry of the universe and not the lepton asymmetry?

What is it about baryons that means that CMB observations can determine their asymmetry without also including a charged leptonic component?
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Where is the gauge symmetry in an ideal Bose gas?

It seems in the literature that there is a certain notion of a “macroscopic wavefunction” associated with a Bose-Einstein system (see this PSE answer) which exhibits a global $U(1)$-phase symmetry. ...
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Understanding Ising Model in Statistical Mechanics

A section on the Ising model in the text "Introduction to modern statistical mechanics" by Chandler states the following: "We consider a system of $N$ spins arranged on a lattice. In the ...
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Obtaining an expression for spontaneous magnetization in 1D Ising model with $H=0$ from the beginning

The usual trick to find the spontaneous magnetization for the 1D Ising model is to calculate the partition function $Z$ with the Hamiltonian $$\mathscr{H}=-J\sum\limits_{i}S_iS_{i+1}-H\sum\limits_{i}...
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Order parameter and Bose-Einstein condensation

I want to study about order parameter and symmetry breaking related to bose einstein condensation in interacted system. Which book i should read? also i want to learn this in second quantization ...
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What happens to the massive gauge bosons in a simple Little Higgs model?

I'm trying to understand a simple Little Higgs (toy) model where the Higgs doublet is made of pseudo Nambu-Goldstone Bosons generated by breaking the symmetry from $SU(3)_L\times U(1)_Y$ to $SU(2)_L\...
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Mermin Wagner theorem proof, what does the K stand for ?

I've been reading about the Mermin-Wagner theorem recently. I think I understand pretty much every computation need to derive its result from the Bogoliub inequality, but there is one thing I don't ...
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Goldstone theorem in Weinberg vol 2

I was reading the proof of Goldstone's theorem (the operator proof starting on page 170) in Weinberg's book on QFT (Volume II) and got confused. I am able to follow each line of the proof, but as a ...
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Leptons during Electroweak Epoch

Prior to the electroweak force splitting, did charged leptons still carry a charge of $e$? I was under the impression that, since electromagnetism didn't exist in its current state prior to ...
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Why did David Tong say that the global topological $U(1)$ symmetry is unbroken in Higgs phase?

In the paper A Duality Web in 2+1 Dimensions and Condensed Matter Physics by Seiberg, Senthil, Wang, and Witten, they studied the particle-vortex dualities in $2+1$ dimensions. On page 20, section 3....
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Anomaly is due to the noninvariance of the path-integral under a symmetry. Is the noninvariance reflected on 1PI effective action?

When a symmetry is anomalous, the path integral $Z=\int\mathcal{D}\phi e^{iS[\phi]}$ is not invariant under that group of symmetry transformations $G$. This is because though the classical action $S[\...
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How do know that Goldstone Boson actually become the longitude degrees of freedom in W+,W- and Z boson?

In many Quantum Field Theory text books they says these about Spontaneous Breaking and Higgs mechanism like this In unitary gauge, the Goldstone Bosons are eaten by $W^\pm$ and Z and become their ...
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Does a symmetry operator $U$ and its generator $Q$ acting on a vacuum $|0\rangle$ both represent new degenerate vacuum?

After the spontaneous breakdown of a symmetry characterized by $$\hat{U}=e^{i\hat{Q}\theta},$$ the commutation relation $[H,Q]=0$ continues to hold. Let us consider two states after this symmetry ...
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Symmetry breaking with Kinetic term dominating the Potential on a Lagrangian

Suppose a Lagrangian $L$ for a scalar field $\phi$, consists of a kinetic term and a Mexican-hat type potential. I am aware that if the vacuum has symmetry $H \subset G$ while the Lagrangian has ...
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Symmetry-breaking matrix/operator deformations: uniquely splitting eigenspaces into smaller ones?

Operators used in quantum mechanics, like Hamiltonian or angular momentum operator, usually have huge degeneracy of eigenspaces (symmetry inside them) - bringing a question of possibility to uniquely ...
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Can any body be uniform in the universe?

If I take any body in the shape of a rod and stretch that, after it reaches breaking stress it breaks at one point. Even though we apply the same the stress on each and every part of the rod it broke ...
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Doesn't the “Mexican hat” potential give a misleading impression that the barrier height between two vacua is finite?

For $\mathbb{Z}_2$ symmetry breaking in a Classical Field Theory described by a potential $$V(\phi)=\lambda(\phi^2-v^2)^2,\tag{1}$$ there is a finite energy barrier of height $\epsilon=\lambda v^4$ ...
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How is it possible to take the inner product of states which belong to two different Hilbert spaces?

Question 1 In case of spontaneous breakdown of a continuous symmetry e.g. the ${\rm U(1)}$ symmetry, two different vacua can be labelled as $|\theta\rangle$ and $|\theta^\prime\rangle$, and they ...
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Physically, what is the Fabri-Picasso theorem really trying to say?

The Fabri-Picasso (FP) theorem states that if a symmetry is spontaneously broken the corresponding conserved charge operator $Q$ does not exist in the Hilbert space. The state $Q|0\rangle$ will have ...
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Fabri-Picasso theorem: Why can we assume that the charge commutes with momentum?

To prove the Fabri-Picasso theorem, we assume that the charge $Q$ is translationally invariant, i.e. it commutes with the momentum operator: $$ [Q,P^\mu]=0 $$ Why can we assume this?
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Spontaneous symmetry breaking in fluids

BACKGROUND: One can think of solids as spontaneously breaking translational symmetries in the sense that each atom in a lattice has to pick a particular position. Yet, as with everything in our ...
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Counting degrees of freedom in the Higgs mechanism for different gauges

I am wondering how to count the degrees of freedom (dof) for a massive gauge field in different gauges. I've been reading some other answers, but haven't found a solution yet. I am looking at the ...
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Recovering nonrelativistic quantum mechanics from quantum field theory

In quantum field theory -- specially when applied to high energy physics -- we see that the requirements of Lorentz invariance, gauge invariance, and renormalizability strongly limit the kinds of ...
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How many connected components are there in the vacuum manifold of $\phi^4$ theory?

Consider the following theory in $1+1$ dimensions $$\mathcal L = \frac12(\partial\phi)^2 - \frac\lambda4 (\phi^2 - v^2)^2 \,,$$ which exhibits a $\mathbb Z_2 = \{0,1\}$ symmetry, $\phi \to -\phi$, ...
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Reference request for symmetry breaking Hartree-Fock

Stationary mean-field solutions break symmetries of the many-body Hamiltonian in favour of lowering the energy, e.g. translational or rotational symmetry, despite $[H,P]=0$, or $[H,L_z]=0$, ...
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Group structure of QCD‘s chiral symmetry (breaking)

With $3$ flavors of massless quarks, the QCD Lagrangian is invariant under flavor transformations$$SU(3)_V\ \otimes\ SU(3)_A\ \otimes\ U(1)_V\ \otimes\ U(1)_A.$$ Now, this is equivalent to $$SU(3)_R\ \...
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What is dynamical symmetry breaking, exactly?

As far as I understand, a symmetry can be broken explicitly (either by manually putting symmetry-violating terms in the Lagrangian or via an anomaly) or spontaneously. I want to focus on the second ...
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Rest frame of massive photon in Meissner–Ochsenfeld effect

The Meissner–Ochsenfeld effect together with spontaneous symmetry breaking and the London Equation yields $(\Box+M^2)A^\mu=0$ and gives photon an effective mass $q\sqrt{\frac{n_c}{m_c}}$ Which is ...
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Two-nucleon coupling: symmetrization postulate

I have some trouble understanding isospin symmetry. If we consider a system of a proton and a neutron as two different iso-states of a nucleon and we wish to write the wave function, do we have to ...
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Phase Transitions from a Bayesian Perspective of Statistical Mechanics

I have been reading papers by E.T. Jaynes recently about viewing all of statistical mechanics as just Bayesian inference applied to physics. (For an introduction: https://journals.aps.org/pr/abstract/...
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How do right handed quark doublets transform under $SU(2)_R$ transforms?

In the context of proving that the SM custodial symmetry is broken by Yukawa coupling to fermions using a 4D Higgs vector representation, $$\underline{\Phi}=\begin{pmatrix}\Phi\\i\sigma_2\Phi\end{...
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Classification of $E_6$ symmetry breaking

Is there a good reference which details the various ways of breaking $E_6$ into smaller groups using Wilson lines? I'm looking for a list of discrete (Abelian) symmetries, e.g. $\mathbb{Z}_5$, and the ...
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CERN and electroweak interaction

I know that the EM and the weak force were described in the same mathematical formalization by GSW and that it was predicted that they would appear as one at energies of 200 GeV. I know that the $W^+$,...
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Suppressed emission of composite particles

If a composite (pseudo)Goldstone boson $\phi$ emerges in a spontaneous symmetry breaking (similar to the mesons of QCD), is the emission of the $\phi$ particle suppressed in high-energy processes, i.e....
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Parametrization of scalar multiplet for spontaneous symmetry breaking in the non-abelian case

Describing abelian symmetry breaking in his book on gauge theories, after favouring a vacuum (whose expectation value is $v$) from the symmetric continuum, Quigg parametrize the complex scalar as $$ ...
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Elitzur theorem and the Ising model

I was recently studying the Elitzur theorem and its application to the Ising model on Kogut: An introduction to lattice gauge theories and spin systems, chapter $5$C. I was wondering how he obtain $\...
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What does soft symmetry breaking physically mean?

A symmetry can be explicitly broken by adding terms in the Lagrangian that aren't compatible with the symmetry, and we say the symmetry is softly broken if all these terms have positive mass dimension....
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Is spontaneous symmetry breaking robust against weak perturbations to the Hamiltonian?

Setup Suppose we have some Hamiltonian $H$ which is known to exhibit spontaneous symmetry breaking (SSB), at least in some parameter regime. For simplicity, we might consider the 2D Ising model below ...
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Non-abelian Goldstone bosons

Suppose we have a theory with a non-abelian symmetry group $G$ that is spontaneously broken to the subgroup $H\subset G$--this is a global symmetry, $not$ a gauge redundancy. Let $X^a$ be the ...
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Metallic state in 2D

Mermin Wagner theorem prevents any spontaneous breaking of continuous symmetry for short ranged interactions for dimensions, D $\leq$2 for any finite temperature. So I understand that superconducting ...
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If spontaneous symmetry breaking only occurs in infinite systems, why do we observe similar effects in finite systems?

Background No SSB in finite systems Consider a system interacting with a heat bath at inverse temperature $\beta$, with the resultant dynamics of the system described by a Liouvillian superoperator $...
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Symmetry breaking of $SO(10)$ or $Spin(10)$

The Spin(10) grand unification has a symmetry breaking of SO(10), or Spin(10). In Wikipedia, it says, "The symmetry breaking of SO(10) is usually done with a combination of (( a $45_H$ OR a $54_H$...