UV complete local field theory of persistent symmetry breaking in 2+1 dimensions (2409.10606v2)

Abstract: Spontaneous symmetry breaking can persist at all temperatures in certain biconical $\mathrm{O}(N)\times \mathbb{Z}_2$ vector models when the underlying field theories are ultraviolet complete. So far, the existence of such theories has been established in fractional dimensions for local but nonunitary models or in 2+1 dimensions but for nonlocal models. Here, we study local models at zero and finite temperature directly in 2+1 dimensions employing functional methods. At zero temperature, we establish that our approach describes the quantum critical behaviour with high accuracy for all $N\geq 2$. We then exhibit the mechanism of discrete symmetry breaking from $\mathrm{O}(N)\times \mathbb{Z}_2\to \mathrm{O}(N)$ for increasing temperature near the biconical critical point when $N$ is finite but large. We calculate the corresponding finite-temperature phase diagram and further show that the Hohenberg-Mermin-Wagner theorem is fully respected within this approach, i.e., symmetry breaking only occurs in the $\mathbb{Z}_2$ sector. Finally, we determine the critical $N$ above which this phenomenon can be observed to be $N_c \approx 15$.

• The paper demonstrates that persistent symmetry breaking occurs in O(N)×Z₂ models in 2+1 dimensions by employing the functional renormalization group method.
• It reveals a temperature-driven transition from O(N)×Z₂ to O(N) symmetry with a critical threshold N₍c₎ ≈ 15, consistent with the Hohenberg-Mermin-Wagner theorem.
• Extensive computational analyses validate the theoretical framework, offering significant insights for advancing quantum field theory and phase transition studies.

Overview of UV Complete Local Field Theory of Persistent Symmetry Breaking in 2+1 Dimensions

The manuscript discusses the theoretical foundation of spontaneous symmetry breaking in certain biconical vector models with $\mathrm{O}(N) \times \mathbb{Z}_2$ symmetries in 2+1 dimensions. It particularly focuses on establishing the conditions under which symmetry breaking persists across all temperatures, directly addressing the issue of ultraviolet (UV) completeness in local field theories.

The authors critically analyze these models using functional methods, notably the functional renormalization group (FRG), to explore their behavior at both zero and finite temperatures. With their approach, they aim to demonstrate that the quantum critical behavior is accurately captured for any finite but sufficiently large $N$.

Numerical and Functional Insights

The paper provides a computational exploration of the phase diagram and the symmetry-breaking conditions. Crucially, they have found that symmetry breaking transitions from $\mathrm{O}(N) \times \mathbb{Z}_2$ to $\mathrm{O}(N)$ as temperature increases, but only within the discrete $\mathbb{Z}_2$ sector due to the Hohenberg-Mermin-Wagner theorem.

The critical insight from their findings is the identification of a critical value, $N_c \approx 15$, above which persistent symmetry breaking can be observed. The authors demonstrate the high degree of accuracy in their predictions through extensive computational models.

Theoretical Implications

The theoretical implications lie in understanding how symmetry behaviors manifest at different temperatures and dimensional configurations. By resolving the critical phenomena through a FRG approach, which respects the constraints of the CHMW theorem, the paper fortifies the understanding of phase transitions in non-integer dimensions, offering an alternative to $D=4-\epsilon$ extrapolations typically marred by unitarity issues.

Future Directions

The paper opens potential avenues for further exploration in quantum field theories and statistical mechanics, particularly concerning the robust realization of symmetry-breaking across various dimensions. Theoretical advancements, such as coupling to additional matter or enlarging the discrete symmetry beyond $\mathbb{Z}_2$, are potential next steps for expanding upon these findings.

Conclusion

This work contributes significantly to the field of quantum field theory by rigorously assessing the conditions under which symmetry breaking persists in a UV-complete framework. The paper's findings are applicable to theoretical models that view temperature as an arbitrary scale, thus impacting how phase transitions are approached and conceptualized in complex systems geometries.