# Spontaneous Symmetry Breaking and Nambu–Goldstone Bosons in Quantum Many-Body Systems

^{1}

^{2}

## Abstract

**:**

**PACS**11.30.Qc

## 1. Introduction

## 2. Basic Properties of Spontaneously Broken Symmetries

#### 2.1. Ground state and finite symmetry transformations

#### 2.2. Explicit symmetry breaking and the choice of the ground state

## 3. Example: Free Nonrelativistic Particle

#### 3.1. Hilbert space and inequivalent ground states

#### 3.2. Spontaneous symmetry breaking and the Nambu–Goldstone boson

## 4. Achieving Spontaneous Symmetry Breaking: Minimization of Higgs Potentials

#### 4.1. Group action on the order parameter space

**Theorem 1**

**(Michel)**

#### 4.2. Minimization of Higgs potentials

#### 4.3. Example: spin-one color superconductor

## 5. Goldstone Theorem and The Counting of Nambu–Goldstone Bosons

#### 5.1. Goldstone theorem

**Theorem 2**

**(Goldstone)**

#### 5.2. Goldstone boson counting: Dispersion relations

^{3}He or the spin-one color superconductor analyzed in Section 4.3., leaving unbroken $\mathrm{SO}(2)$ and $\mathrm{U}(1)$ subgroups which are isomorphic, but realized differently due to a different structure of the ground state. (The respective values of the order parameter lie in different strata.) The three broken generators give rise to three NG bosons with linear dispersions in the polar phase, and to one linear and one quadratic NG mode in the A phase. Similar conclusions have recently been reported for spin-two Bose–Einstein condensates [46].

**Theorem 3**

**(Nielsen**

**and**

**Chadha)**

#### 5.3. Goldstone boson counting: Charge densities

**Theorem**

**4 (Schäfer et al.)**

#### 5.4. Linear sigma model

**Theorem**

**5**

## 6. Further Examples

#### 6.1. Nonrelativistic Boulware–Gilbert model

#### 6.2. Heisenberg ferromagnet

#### 6.3. Linear sigma model

## 7. Low-Energy Effective Field Theory for NG Bosons

#### 7.1. Coset construction of effective Lagrangians

#### 7.2. Geometric interpretation

#### 7.3. Nonrelativistic effective Lagrangians

#### 7.4. Applications of effective field theory

## 8. Conclusions

^{3}He [82,83]. In the appropriate approximation, the system then exhibits more gapless states than would correspond to the symmetry of the full theory, some of them stemming from the extended symmetry of the part of the Lagrangian. These spurious NG bosons receive a gap once quantum corrections are taken into account. This mechanism can be responsible for the presence of naturally light states in the spectrum.

^{3}He (see [25] for an extensive review), spin-one color superconductors [32], or even imbalanced spin-zero color superconductors [84], and in Bose–Einstein condensates of relativistic vector fields [85,86], as well as inhomogeneous states such as crystalline solids [87] or superconductors with inhomogeneous pairing [88]. Accordingly, the behavior of NG bosons may be highly nontrivial. For example, in helical ferromagnets [89,90] the local magnetization field forms a spiral structure. Along the axis of the helix, the average magnetization is zero and the NG boson is type-I. In the transverse directions, the NG boson feels the uniform magnetization and is type-II like in ordinary ferromagnets.

## Acknowledgments

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**Figure 1.**Geometric minimization of the quantity $\Xi ({\mathcal{A}}_{\alpha})\equiv {\sum}_{\alpha \ne 1}{\lambda}_{\alpha}{\mathcal{A}}_{\alpha}$. The region of allowed values of ${\mathcal{A}}_{\alpha}$ is indicated by gray shading. The lines of constant $\Xi ({\mathcal{A}}_{\alpha})$, having the common normal vector $({\lambda}_{2},{\lambda}_{3})$, are in red. The red dot denotes the position of the absolute minimum of the potential.

**Figure 2.**Left panel: target space of a spin-one color superconductor. The dashed lines are given by various bounding inequalities as described in the text. The $AX$ segment of the boundary is occupied by the non-inert $\epsilon $ phase. Right panel: phase diagram of the spin-one color superconductor as a function of the quartic couplings. Thick solid and dashed lines stand for first and second order phase transitions, respectively.

**Figure 3.**Hierarchy of the little groups in a spin-one color superconductor. Arrows indicate successive breaking into smaller and smaller subgroups. The relation ${\mathrm{G}}_{1}\to {\mathrm{G}}_{2}$ means that the group ${\mathrm{G}}_{2}$ is conjugate to a subgroup of ${\mathrm{G}}_{1}$, or in other words, the orientation of the order parameter in the two phases can be chosen so that ${\mathrm{G}}_{2}\subset {\mathrm{G}}_{1}$.

**Figure 4.**Characteristics of NG spectrum and their interconnections. Nielsen and Chadha [3] provided a general relation between the number of NG bosons and their dispersion relations. The connection of NG dispersion relations and the presence of charge densities was clarified by Leutwyler [4] using low-energy effective field theory. Partial results are indicated by dashed lines.

**Table 1.**Classification of different ground states of a spin-one color superconductor, based on the pattern of spontaneous breaking of continuous symmetries. First line: name of the phase; second line: unbroken continuous symmetry; third line: representative element of the stratum. Lower indices ${}_{\mathrm{L},\mathrm{R}}$ denote subgroups of $\mathrm{U}{(3)}_{\mathrm{L}}$ and $\mathrm{SO}{(3)}_{\mathrm{R}}$, while ${}_{\mathrm{V}}$ stands for a “diagonal” subgroup, mixing transformations from the two.

Oblate | Cylindrical | $\epsilon $ | A |

$\mathrm{SO}{(2)}_{\mathrm{V}}$ | $\mathrm{SO}{(2)}_{\mathrm{V}}\times \mathrm{U}{(1)}_{\mathrm{L}}$ | $\mathrm{SO}{(2)}_{\mathrm{V}}\times \mathrm{U}{(1)}_{\mathrm{L}}$ | $\mathrm{SU}{(2)}_{\mathrm{L}}\times \mathrm{SO}{(2)}_{\mathrm{V}}\times \mathrm{U}{(1)}_{\mathrm{L}}$ |

$\left(\begin{array}{ccc}{\Delta}_{1}& +\mathrm{i}a& 0\\ -\mathrm{i}a& {\Delta}_{1}& 0\\ 0& 0& {\Delta}_{2}\end{array}\right)$ | $\left(\begin{array}{ccc}\Delta & +\mathrm{i}a& 0\\ -\mathrm{i}a& \Delta & 0\\ 0& 0& 0\end{array}\right)$ | $\left(\begin{array}{ccc}{\Delta}_{1}& +\mathrm{i}{\Delta}_{1}& 0\\ -\mathrm{i}{\Delta}_{1}& {\Delta}_{1}& 0\\ 0& 0& {\Delta}_{2}\end{array}\right)$ | $\left(\begin{array}{ccc}1& +\mathrm{i}& 0\\ -\mathrm{i}& 1& 0\\ 0& 0& 0\end{array}\right)$ |

CSL | Polar | ${\mathrm{N}}_{1}$ | ${\mathrm{N}}_{1}$ |

$\mathrm{SO}{(3)}_{\mathrm{V}}$ | $\mathrm{SU}{(2)}_{\mathrm{L}}\times \mathrm{SO}{(2)}_{\mathrm{R}}\times \mathrm{U}{(1)}_{\mathrm{L}}$ | $\mathrm{U}{(1)}_{\mathrm{L}}$ | $\mathrm{SU}{(2)}_{\mathrm{L}}\times \mathrm{U}{(1)}_{\mathrm{L}}$ |

$\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$ | $\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right)$ | $\left(\begin{array}{ccc}0& 0& 0\\ {z}_{1}& {z}_{2}& {z}_{3}\\ {z}_{4}& {z}_{5}& {z}_{6}\end{array}\right)$ | $\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ {z}_{1}& {z}_{2}& {z}_{3}\end{array}\right)$ |

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**MDPI and ACS Style**

Brauner, T.
Spontaneous Symmetry Breaking and Nambu–Goldstone Bosons in Quantum Many-Body Systems. *Symmetry* **2010**, *2*, 609-657.
https://doi.org/10.3390/sym2020609

**AMA Style**

Brauner T.
Spontaneous Symmetry Breaking and Nambu–Goldstone Bosons in Quantum Many-Body Systems. *Symmetry*. 2010; 2(2):609-657.
https://doi.org/10.3390/sym2020609

**Chicago/Turabian Style**

Brauner, Tomáš.
2010. "Spontaneous Symmetry Breaking and Nambu–Goldstone Bosons in Quantum Many-Body Systems" *Symmetry* 2, no. 2: 609-657.
https://doi.org/10.3390/sym2020609