CritPt / Challenge_40_main / Effective field theory is a powerful tool used to construct phenomenological…

# Problem setup:
Effective field theory is a powerful tool used to construct phenomenological models via symmetries. The method has recently been extended to dissipative systems via the Schwinger-Keldysh formalism. In this problem we will study the dissipative effective field theory associated to spontaneous symmetry breaking of multipolar U(1) symmetries.


Consider a 1d system with a conserved density N=∫ρ, conserved dipole moments D=∫xρ, and a conserved quadrupole moment Q=∫x²ρ. Assume that these are the only conserved quantities in the system.

Suppose that the quadrupole Q generator is spontaneously broken, and the charge N and dipole D generators are unbroken.

# Main problem:

Compute the spectrum of hydrodynamic modes ω(k).

Let χ be the charge susceptibility, κ be the quadrupole superfluid stiffness, and σ be the coefficient of the leading order dissipative term. You may assume all other EFT coefficients are zero. Express your answer in terms of χ, κ, and σ.

# Problem setup:
Effective field theory is a powerful tool used to construct phenomenological models via symmetries. The method has recently been extended to dissipative systems via the Schwinger-Keldysh formalism. In this problem we will study the dissipative effective field theory associated to spontaneous symmetry breaking of multipolar $U(1)$ symmetries.


Consider a $1d$ system with a conserved density $N = \int \rho$, conserved dipole moments $D = \int x \rho$, and a conserved quadrupole moment $Q = \int x^2 \rho$. Assume that these are the only conserved quantities in the system.

Suppose that the quadrupole $Q$ generator is spontaneously broken, and the charge $N$ and dipole $D$ generators are unbroken.

# Main problem:

Compute the spectrum of hydrodynamic modes $\omega(k)$.

Let $\chi$ be the charge susceptibility, $\kappa$ be the quadrupole superfluid stiffness, and $\sigma$ be the coefficient of the leading order dissipative term. You may assume all other EFT coefficients are zero. Express your answer in terms of $\chi$, $\kappa$, and $\sigma$.