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arXiv:2107.12127 · Physical review. D/Physical review. D.

Operator spectrum of nonrelativistic CFTs at large charge

Vito Pellizzani

Published Jun 24, 2022• 10 citations
arXivHTMLPDF

Abstract

We extend and clarify the large-charge expansion of the conformal dimension ${\mathrm{\ensuremath{\Delta}}}_{Q}$ of the lowest operator of charge $Q$ in nonrelativistic conformal field theories using the state-operator correspondence. The latter requires coupling the theory to an external harmonic trap that confines the particles to a spherical cloud, at the edge of which the effective theory breaks down and leads to divergences. Only recently has this issue been overcome by constructing appropriate counterterms at the edge of the cloud [S. Hellerman and I. Swanson, arXiv:2010.07967]. In this paper, we extend these results by systematically analyzing the degree of divergence of operators in the effective action and show that there always exist appropriate edge counterterms that make the final contributions to ${\mathrm{\ensuremath{\Delta}}}_{Q}$ finite. On the other side of the correspondence, this also provides new corrections to the Thomas-Fermi approximation of the unitary Fermi gas, and we comment on their relevance for ultracold atom physics.

DOI
10.1103/physrevd.105.125018

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We extend and clarify the large-charge expansion of the conformal dimension \ensuremathΔQ{\mathrm{\ensuremath{\Delta}}}_{Q}\ensuremathΔQ​ of the lowest operator of charge QQQ in nonrelativistic conformal field theories using the state-operator correspondence. The latter requires coupling the theory to an external harmonic trap that confines the particles to a spherical cloud, at the edge of which the effective theory breaks down and leads to divergences. Only recently has this issue been overcome by constructing appropriate counterterms at the edge of the cloud [S. Hellerman and I. Swanson, arXiv:2010.07967]. In this paper, we extend these results by systematically analyzing the degree of divergence of operators in the effective action and show that there always exist appropriate edge counterterms that make the final contributions to \ensuremathΔQ{\mathrm{\ensuremath{\Delta}}}_{Q}\ensuremathΔQ​ finite. On the other side of the correspondence, this also provides new corrections to the Thomas-Fermi approximation of the unitary Fermi gas, and we comment on their relevance for ultracold atom physics.
We consider the large-charge expansion of the charged ground state of a Schrodinger-invariant, nonrelativistic conformal field theory in a harmonic trap, in general dimension d. In the existing literature, the energy in the trap has been computed to next-to-leading order (NLO) at large charge Q, which comes from the classical contribution of two higher-derivative terms in the effective field theory. In this note, we explain the structure of operators localized at the edge of the droplet, where the density drops to zero. We list all operators contributing to the ground-state energy with nonnegative powers of Q in the large-Q expansion. As a test, we use dimensional regularization to reproduce the calculation of the NLO ground state energy by Kravec and Pal , and we recover the same universal coefficient for the logarithmic term as in that work. We refine the derivation by presenting a systematic operator analysis of the possible edge counterterms, showing that different choices of cutoff procedures must yield the same renormalized result up to an enumerable list of Wilson coefficients for conformally invariant local counterterms at the droplet edge. We also demonstrate the existence of a previously unnoticed edge contribution to the ground-state operator dimension of order Q^{{2\over 3} - {1\over d}} in d spatial dimensions. Finally, we show there is no bulk or edge counterterm scaling as Q^0 in two spatial dimensions, which establishes the universality of the order Q^0 term in large-Q expansion of the lowest charged operator dimension in d=2.
Abstract The unitary Fermi gas, by virtue of its description as a nonrelativistic conformal field theory, has proven an interesting system by which the quantum properties of CFTs can be held to experimental verification. Here, we examine the structure of conformal dimensions of charge-Q operators in nonrelativistic CFTs, in the large-Q regime, from the non-linear sigma model perspective. We discuss in detail the renormalization of edge divergences using dimensional regularization, elucidating the presence of log(Q) terms in the large-charge expansion. Finally, we use dimensional regularization to compute the universal one-loop Q0 log Q contribution to the ground-state energy in d = 3 spatial dimensions, with the result Δ(Q)Q0=133log⁡(Q)+ \Delta {(Q)}_{Q^0}=\frac{1}{3\sqrt{3}}\log (Q)+ Δ(Q)Q0​=33​1​log(Q)+ Δ Q Q 0 = 1 3 3 log Q + const.
The large-charge master field which generates all n -point correlation functions with an insertion of large charge Q in nonrelativistic conformal field theory is obtained. This field is used to compute Schrödinger-invariant n -point correlation functions of large-charge operators via a direct evaluation of the path integral. Conformal dimensions are found to agree with calculations based on the state-operator correspondence. The master field solution exhibits an emergent harmonic trap whose frequency is a function of the Euclidean time. The large-charge effective action with operator insertions describes a droplet of superfluid matter whose spatial size scales with the time separation of sources. The solution is used to compute Schrödinger symmetry breaking corrections in the large-charge effective field theory (EFT) due to a finite scattering length in the fundamental theory of fermions near unitarity. The scaling of these effects in the large-charge power counting scheme is established, and the size of the effects is quantified using input from quantum Monte Carlo simulations of the near-unitary gas, as well as from the large- N expansion at large charge. Published by the American Physical Society 2024
A bstract We study the unitary Fermi gas in a harmonic trapping potential starting from a microscopic theory in the limit of large charge and large number of fermion flavors N . In this regime, we present an algorithmic procedure for extracting data from perturbation theory, order-by-order, without the need for other assumptions. We perform a gradient expansion in the interior of the particle cloud, sufficiently far from the cloud edge where the particle density drops rapidly to zero. In this latter region we present the first microscopic computation characterizing the contribution of the edge terms. The microscopic theory reproduces the predictions of the superfluid eft , including the action, the form of the gap equation, and the energy of the system in a harmonic trap (which maps, via the non-relativistic state-operator correspondence, to the scaling dimension of the lowest operator of charge Q ). We additionally give the Wilsonian coefficients at leading order in N up to nnlo in the large-charge expansion.
We study the unitary Fermi gas in a harmonic trapping potential starting from a microscopic theory in the limit of large charge and large number of fermion flavors N. In this regime, we present an algorithmic procedure for extracting data from perturbation theory, order-by-order, without the need for other assumptions. We perform a gradient expansion in the interior of the particle cloud, sufficiently far from the cloud edge where the particle density drops rapidly to zero. In this latter region we present the first microscopic computation characterizing the contribution of the edge terms. The microscopic theory reproduces the predictions of the superfluid EFT, including the action, the form of the gap equation, and the energy of the system in a harmonic trap (which maps, via the non-relativistic state-operator correspondence, to the scaling dimension of the lowest operator of charge Q). We additionally give the Wilsonian coefficients at leading order in N up to NNLO in the large-charge expansion.
The field theory of nonrelativistic fermions interacting via contact interactions can be used to calculate the properties of few-body systems of cold atoms confined in harmonic traps. The state-operator correspondence of nonrelativistic conformal field theory (NRCFT) shows that the energy eigenvalues (in oscillator units) of NNN harmonically trapped fermions can be calculated from the scaling dimensions of NNN-fermion operators in the NRCFT. They are also in one-to-one correspondence with zero energy, scale-invariant solutions to the NNN-body problem in free space. We show that these two mappings of the trapped fermion problem to free space problems are related by an automorphism of the SL(2,R)SL(2,R)SL(2,R) algebra of the conformal symmetry of fermions at the unitary limit. This automorphism exchanges the internal Hamiltonian of the gas with the trapping potential and hence provides a method for deriving virial theorems for trapped Fermi gases at the unitary limit. We also show that the state-operator correspondence can be applied directly in three spatial dimensions by calculating the scaling dimensions of two- and three-fermion operators and finding agreement with known exact results for energy levels of two and three trapped fermions at the unitary limit.