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Goldstone bosons across thermal phase transitions

Peter Lowdon, Owe Philipsen

TL;DR

This work extends the non-perturbative understanding of Goldstone's theorem to finite temperature by analyzing the $ ext{U}(1)$ complex scalar theory on the lattice. It shows that the Goldstone mode persists as a thermoparticle across the thermal transition, with its dissipative damping door-signaling the phase: weak damping in the broken phase and strong damping in the symmetry-restored phase. The lattice results extract explicit damping parameters and demonstrate a thermoparticle spectral structure above $T_c$, consistent with a dissipative, non-hydrodynamic thermal medium. This provides a robust, non-perturbative criterion for thermal phase transitions in QFTs and has implications for the behavior of Goldstone-like excitations in high-temperature environments ranging from condensed matter systems to QCD.

Abstract

Temperature has a significant effect on the properties of quantum field theories (QFTs) with a spontaneously broken symmetry, in particular on the massless Goldstone bosons that exist in the vacuum state. It has recently been shown using lattice calculations for a $\mathrm{U}(1)$ complex scalar field theory that the Goldstone mode persists even when the symmetry is restored above the critical temperature $T_{c}$, and has the properties of a screened excitation, a so-called thermoparticle. In this work, we continue the investigation of this theory by determining explicitly how the Goldstone mode evolves as the temperature is increased both below and above $T_{c}$. We find that the two phases of the theory are entirely characterised by the thermal dissipative effects experienced by the Goldstone mode, with the broken and symmetry-restored phases associated with weak and strong damping, respectively. These findings are consistent with the non-perturbative constraints imposed by spontaneous symmetry breaking, and provide a new way in which to characterise thermal phase transitions in QFTs.

Goldstone bosons across thermal phase transitions

TL;DR

This work extends the non-perturbative understanding of Goldstone's theorem to finite temperature by analyzing the complex scalar theory on the lattice. It shows that the Goldstone mode persists as a thermoparticle across the thermal transition, with its dissipative damping door-signaling the phase: weak damping in the broken phase and strong damping in the symmetry-restored phase. The lattice results extract explicit damping parameters and demonstrate a thermoparticle spectral structure above , consistent with a dissipative, non-hydrodynamic thermal medium. This provides a robust, non-perturbative criterion for thermal phase transitions in QFTs and has implications for the behavior of Goldstone-like excitations in high-temperature environments ranging from condensed matter systems to QCD.

Abstract

Temperature has a significant effect on the properties of quantum field theories (QFTs) with a spontaneously broken symmetry, in particular on the massless Goldstone bosons that exist in the vacuum state. It has recently been shown using lattice calculations for a complex scalar field theory that the Goldstone mode persists even when the symmetry is restored above the critical temperature , and has the properties of a screened excitation, a so-called thermoparticle. In this work, we continue the investigation of this theory by determining explicitly how the Goldstone mode evolves as the temperature is increased both below and above . We find that the two phases of the theory are entirely characterised by the thermal dissipative effects experienced by the Goldstone mode, with the broken and symmetry-restored phases associated with weak and strong damping, respectively. These findings are consistent with the non-perturbative constraints imposed by spontaneous symmetry breaking, and provide a new way in which to characterise thermal phase transitions in QFTs.

Paper Structure

This paper contains 14 sections, 31 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Goldstone's theorem for QFTs
  3. Zero temperature
  4. Finite-temperature generalisation
  5. Vacuum Goldstone mode
  6. Thermal Goldstone mode
  7. Goldstone evolution across the $\mathrm{U}(1)$ phase transition
  8. Thermal Goldstone correlators
  9. Lattice analysis of the $\mathrm{U}(1)$ theory
  10. Broken phase
  11. Symmetry-restored phase
  12. Spectral characteristics across the $\mathrm{U}(1)$ phase transition
  13. Contrast with existing studies
  14. Conclusions

Figures (5)

  • Figure 1: $C_{L}(0,z=L/2)$ as a function of $L/a$ for different values of $N_{\tau}$.
  • Figure 2: Infinite-volume extrapolation of $a^{2}C_{L}(0,L/2)$ for $N_{\tau}=32$ (top left), $N_{\tau}=16$ (top right), and $N_{\tau}=8$ (bottom).
  • Figure 3: Fit parameter values of $d_{L}/a$ and $am_{L}$ obtained using the fit ansatz in Eq. \ref{['Cz_L']} for $N_{\tau}=2,4, 6$ at different volumes (left), and the screening mass $m_{\text{scr}}$ at $N_{s}=160$ versus temperature normalised to the field expectation value $|v_{0}|$ on the coldest lattice $N_{\tau}=32$ (right). The error bars in the plot are smaller than the symbol size.
  • Figure 4: Thermal Goldstone temporal correlator prediction (blue dashed line) and data (black points) for $L/a=160$ at $N_{\tau}=6$ (left), $N_{\tau}=4$ (middle), and $N_{\tau}=2$ (right).
  • Figure 5: Evolution of the Goldstone spectral function $\rho_{G}(a\omega,a|\vec{p}|)/a^{2}$ with increasing temperature for $L/a=160$ at $N_{\tau}=6$ (top left), $N_{\tau}=4$ (top right), and $N_{\tau}=2$ (bottom).