Abstract
In this note we search for the ground state of the D = 3 Wilson-Fisher conformal O(4) model, at nonzero values of the two independent charge densities ρ1,2, on the torus spatial slice. Using an effective theory valid on scales longer than the scale defined by the charge density, we show that the ground-state configuration is inhomogeneous for generic ratios ρ1/ρ2. This result confirms, within the context of a well-defined effective theory, a recent no-go result of [1]. We also show that any spatially periodic ground state solutions have an energetic preference towards longer periods, within some range of ρ1/ρ2 containing a neighborhood of zero. This suggests that the scale of variation of the ground state solution in finite volume will be the infrared scale, and that the use of the effective theory at large charge in finite volume is self-consistent. Note added: the statements in this paper are true for arbitrary ratio of ρ1/ρ2, which we proved after we uploaded this paper. See [2].
Highlights
In order to get started on such a calculation, one needs to know the structure of the large-charge effective Lagrangian, and the nature of the ground state carrying a given set of global charges
In the limit where the charge is taken to infinity, one can try to flatten out the sphere and consider the system in infinite flat space at fixed charge density ρ
We will find that there are no exactly homogeneous ground states, but a family of inhomogeneous, spatially periodic solutions of spatial period matching the larger cycle of the spatial torus, in some range of ratios of the two independent large charges, 0 < J1/J2 ≪ 1. (Note: the same holds for arbitrary ratio 0 < J1/J2, which was proven in [2] after this paper was completed.) In this range of charges, the system will be driven dynamically into a regime where the fields vary slowly compared with the scale of the charge density, and the large-charge effective Lagrangian is parametrically reliable
Summary
To answer these questions quantitatively, we must find a convenient way to express the charge density itself, as an element of the adjoint of SO(4), that is, a general 4×4 imaginary antisymmetric matrix Such a matrix has real eigenvalues that occur in pairs with equal magnitude and opposite sign. Rather than parametrizing the charge density directly by the two independent eigenvalues ρ ≪ 1, 2 of the charge matrix, we follow [1] in choosing a basis for the chemical potential, which is equivalent to diagonalizing the generator defining the symmetry of the helical solution. Choose a complex basis for the fundamental of U(2) ⊂ SO(4), and parametrize the charge generator by the two matrix elements ρ ≪ 1, 2 on the diagonal This will turn out to be equivalent: for helical solutions, the charge matrix commutes with the chemical potential, its off-diagonal terms always vanish, and ρ1/ρ2 is equal to ρ1/ρ2. We will comment in the Discussion section on the relevance to the ground state in finite volume
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