Universal one-dimensional atomic gases near odd-wave resonance

We show the renormalization of the contact interaction for odd-wave scattering in one-dimension (1D). Based on the renormalized interaction, we exactly solve the two-body problem in a harmonic trap and further explore the universal properties of spin-polarized fermions near odd-wave resonance by using the operator-product-expansion method. It is found that the high-momentum distribution behaves as $C/k^2$, with $C$ being the odd-wave contact. Various universal relations are derived. Our work suggests a universal system emergent in 1D with large odd-wave scattering length.

Figure 1

Diagrams for interaction renormalization. The solid (hollow) circle represents the effective (bare) interaction $T\left(U\right)$. The lines denotes the incident and outgoing scattering states for the relative motion of two particles, given that it is separable from the center-of-mass motion. In the continuum, the state can be characterized by wave vector $k$.

Figure 2

(a) Energy spectrum for the relative motion of two atoms in a harmonic trap with odd-wave ($E_o\text{−}{l}_o$, black solid) and even-wave ($E_e\text{−}{l}_e$, red dashed) interactions. $\omega$ and $d[=\sqrt{2/\left(m\omega \right)}]$ are the trap frequency and confinement length, respectively. Horizontal gray lines show the noninteracting odd-wave or the hard-core even-wave energy levels $(2l+3/2)\omega$ (here $l=0,1,...$). While the vertical gray line at $l_o=l_e=\infty$ crosses the spectrum with energies $(2l+1/2)\omega$. (b) Normalized wave function $\psi \left(x\right)$ for odd-wave scattering at three scattering lengths $l_o/d=-0.675$ (solid), $\infty$ (dashed), and 0.675 (dash-dotted), as marked by squares from left to right in panel (a). The corresponding linear fits (with same line style) at $x\to 0^{+}$ confirm the boundary condition [Eqs. (2) and (13)], i.e., $\psi (x\to 0^{+})\propto (x-l_o)$.

Figure 3

The odd- (black solid) and even-wave(red dashed) two-body spectra for the lowest two branches with finite effective ranges ${\xi }_o=0.1md,{\xi }_e=-0.1m{d}^3$ ($d$ is the confinement length). For comparison, gray lines show the spectra with zero range [same as in Fig. 2].

Figure 4

(a) Two-body scattering vertex $iA\left(E\right)q{q}^{\prime }$. Here $q$ ($q^{\prime }$) is the incident (outgoing) momentum of the relative motion; $E=q^2/m$. (b) Diagram producing nonanalyticity in $\langle {\psi }^{†}(R-\frac{x}{2})\psi (R+\frac{x}{2})\rangle$. The two open dots represent the operators ${\psi }^{†}(R-\frac{x}{2})$ and $\psi (R+\frac{x}{2})$, respectively. Panels (c1)–(c4) show four diagrams contributing to $\langle \mathcal{V}\left(R\right)\rangle$. Each open dot connecting with four lines represents the local operator $\mathcal{V}\left(R\right)$.

Figure 5

Diagrams of local operators: (a1) $\psi \left(R\right)$, (a2) ${\psi }^{†}\left(R\right)$; (b) one-body operators, such as ${\psi }^{†}\left(R\right)\psi \left(R\right)$ and its derivatives; (c) two-body operators, such as $\mathcal{V}\left(R\right)$.

Figure 6

Diagrams for matrix elements of bilocal operator ${\mathrm{\Psi }}^{†}(R-\frac{x}{2})\mathrm{\Psi }(R+\frac{x}{2})$ between the two-particle scattering states. Note that the diagram (d) is equivalent to Fig. 4 in the main text.

Figure 7

Diagrams for matrix elements of one-body local operators, such as ${\mathrm{\Psi }}^{†}\mathrm{\Psi }\left(R\right)$ and its derivatives, between the two-particle scattering states.