On operator product expansion in the spin-orbit coupled bosonic system

This paper derives the operator product expansion for two operators in a spin-orbit coupled bosonic system, with a specific focus on determining the contact density term that governs the system's universal physics.

The Gist

Imagine a vast, ultra-cold dance floor filled with billions of tiny, invisible dancers (bosonic atoms). In a normal ballroom, these dancers might just bump into each other randomly. But in this specific experiment, scientists have set up a special "laser spotlight" that makes the dancers interact in a very specific, choreographed way. This setup is called a spin-orbit coupled system.

Depending on how the laser is tuned, the dancers can form different patterns:

• The Rabi case: They move in perfect sync, like a single, unified crowd (ferromagnetic phase).
• The Spin-Orbit case: They start moving in waves, forming stripes or even a "supersolid" state where they act like both a solid crystal and a flowing liquid at the same time.

The authors of this paper are like detectives trying to understand the secret handshake between these dancers when they get extremely close to each other.

The Detective's Tool: The "Operator Product Expansion" (OPE)

In the world of quantum physics, it's hard to look at two particles when they are right on top of each other because the math gets messy and infinite. To solve this, the authors use a tool called Operator Product Expansion (OPE).

Think of OPE like a magnifying glass with a special lens.

• Normally, if you look at two dancers standing side-by-side, you see two separate people.
• But if you zoom in extremely close (mathematically speaking, as the distance between them goes to zero), the OPE lens tells you that their interaction isn't just "two people." It reveals a hidden "contact density."
• This "contact density" is like a universal fingerprint. It doesn't matter if the dancers are in a slow waltz or a fast tango; this fingerprint tells you exactly how they interact when they touch. This fingerprint controls the "universal physics" of the whole system, specifically how the dancers behave at very high speeds (high momentum).

The Investigation: What Happens When They Touch?

The authors spent the paper calculating exactly what this "fingerprint" looks like for their specific laser-tuned dancers.

1. The Setup: They started with the basic rules of the dance floor (the Hamiltonian), which includes how the dancers move and how the laser pushes them.
2. The Complication: In the "Spin-Orbit" version, the laser breaks the symmetry of the room. It's like the dance floor has a strong wind blowing from one side. This makes the math much harder because the dancers' behavior depends on which way they are facing relative to the wind.
3. The Calculation: They used complex diagrams (Feynman diagrams) to simulate collisions between pairs of dancers. They looked at what happens when two dancers collide, scatter, and then collide again.
4. The Discovery: They found that even though the "wind" (the spin-orbit coupling) makes the dance complex, the fundamental rule of how they touch (the contact density) remains surprisingly simple and robust.

The Three Dance Floors (Phases)

The authors checked how this "touching rule" changes in three different dance styles (phases) the system can take:

• The Plane Wave Phase: The dancers are all moving in a specific direction, like a marching band. Here, the "touching rule" has a specific value based on how fast they are marching.
• The Zero Momentum Phase: The dancers are standing still in a calm, uniform crowd. Here, the rule is simple and symmetric.
• The Stripe Phase: The dancers form a pattern of stripes, like a zebra. They are moving back and forth, creating a wave of density.

The Big Finding:
The authors discovered that when the dancers switch from the "calm crowd" (Zero Momentum) to the "marching band" (Plane Wave), the "touching rule" changes smoothly, like a dimmer switch slowly turning down the light.

However, when they switch from the "marching band" to the "striped zebra" (Stripe phase), the "touching rule" jumps. It's like the dimmer switch suddenly snapping to a different setting. This jump tells the scientists that this specific phase transition is a "first-order" transition (a sudden, dramatic change) rather than a smooth one.

The Bottom Line

This paper doesn't invent a new machine or cure a disease. Instead, it provides a mathematical dictionary for understanding how ultra-cold atoms interact when they are squeezed together.

By deriving the "Operator Product Expansion," the authors gave us the exact formula for the contact density. This formula is crucial because it allows physicists to predict how these exotic quantum systems will behave at high energies, regardless of the complex dance moves they are performing. It confirms that even in a chaotic, wind-blown quantum dance floor, the fundamental rules of how particles "touch" remain consistent and calculable.

Technical Summary

Technical Summary: Operator Product Expansion in Spin-Orbit Coupled Bosonic Systems

Problem Statement
Ultra-cold bosonic systems, particularly those exhibiting spin-orbit coupling (SOC) induced by counter-propagating Raman lasers, display rich quantum phases such as zero-momentum, plane-wave, and supersolid states. While the Operator Product Expansion (OPE) has been extensively utilized in fermionic systems to derive universal relations (e.g., Tan's relations) and understand short-distance physics, its application to spin-orbit coupled and coherently coupled bosonic systems remains under-explored. The primary objective of this work is to derive the OPE for two local field operators, $\psi^\dagger_\sigma(\vec{R} - \frac{1}{2}\vec{r})$ and $\psi_{\sigma'}(\vec{R} + \frac{1}{2}\vec{r})$, in these bosonic systems. Specifically, the authors aim to isolate the "contact density" term—the leading non-analytic contribution in the separation distance $r$—which governs the asymptotic fall-off of the momentum distribution at large wave numbers ($k \to \infty$).

Methodology
The study employs a combination of quantum field theory techniques and diagrammatic analysis:

1. Hamiltonian and Basis Transformation: The authors start with the action of a spin-orbit coupled bosonic system involving two hyperfine states ($\psi_1, \psi_2$) coupled by Rabi frequency $\Omega$ and momentum transfer $\kappa_0$. They perform a unitary transformation to diagonalize the single-particle Hamiltonian, introducing "dressed" particle operators ($\alpha_\pm$). However, recognizing that interactions become momentum-dependent in this dressed basis, the scattering calculations are performed in the original field basis ($\psi_{1,2}$).
2. Scattering Amplitude Calculation: Using the LSZ reduction formula, the authors compute the $2 \to 2'$ scattering amplitudes. They calculate the Green's functions in momentum space, including the effects of the Raman coupling $\Omega$. The scattering amplitude is derived by solving a matrix equation involving one-loop integrals ($Z_i(E)$) that account for the self-energy and vertex corrections.
3. OPE Derivation via Expectation Values: The OPE is derived by evaluating the expectation value of the bilocal operator product in two-particle states. This is represented diagrammatically as a sum of contributions from:
   • No scattering (analytic in $\vec{r}$).
   • Single scattering (analytic in $\vec{r}$).
   • Double scattering (potentially non-analytic in $\vec{r}$).
     The authors focus on the double-scattering diagrams, as the integration over the momentum dual to $\vec{r}$ generates terms non-analytic in $\vec{r}$ (specifically proportional to $r$).
4. Coefficient Matching: The non-analytic terms derived from the double-scattering diagrams are matched against the expectation values of local four-operator contact density terms (e.g., $(\psi^\dagger \psi)^2$). This allows the determination of the Wilson coefficients for the OPE.
5. Phase Analysis: The derived contact densities are evaluated across the three distinct phases of the SOC system: the Plane Wave phase, the Zero Momentum phase, and the Stripe phase, using a variational coherent wavefunction ansatz.

Key Contributions and Results

• Derivation of OPEs: The paper provides explicit expressions for the OPE of field operators in both coherently coupled ($\kappa_0 = 0$) and spin-orbit coupled ($\kappa_0 \neq 0$) regimes. The leading non-analytic term (order $r$) for the diagonal operator $\psi^\dagger_1(\vec{R} - \frac{1}{2}\vec{r})\psi_1(\vec{R} + \frac{1}{2}\vec{r})$ is found to be:
  $$\psi^\dagger_1 \psi_1(\vec{R}) + \vec{r} \cdot \psi^\dagger_1 \overset{\leftrightarrow}{\nabla} \psi_1(\vec{R}) - \frac{r}{8\pi} \left[ g^2(\psi^\dagger_1 \psi_1)^2 + \lambda^2 \psi^\dagger_1 \psi_1 \psi^\dagger_2 \psi_2 \right](\vec{R}) + \dots$$
  Similar expressions are derived for off-diagonal operators and the $\psi_2$ component.
• Robustness against Spin-Orbit Coupling: A significant finding is that the form of the leading non-analytic OPE terms remains unchanged in the presence of the recoil momentum $\kappa_0$. The authors argue that while $\kappa_0$ breaks Galilean invariance and introduces momentum dependence in scattering amplitudes, the specific integrals responsible for the non-analytic $r$-dependence yield the same structural results as the coherently coupled case.
• Contact Density in Different Phases: The authors calculate the expectation values of the contact density operators in the three phases:
  • Zero Momentum Phase: Contact densities are constant and continuous across the phase boundary with the Plane Wave phase.
  • Plane Wave Phase: Densities depend on the coupling parameters and connect continuously to the Zero Momentum phase.
  • Stripe Phase: Densities exhibit spatial modulation ($\cos(2\kappa_1 x - \delta)$). Crucially, the authors find that at the phase boundary between the Stripe and Plane Wave phases, the contact densities are discontinuous. This discontinuity serves as a signature of the first-order nature of this specific phase transition.

Significance and Claims
The paper claims to fill a gap in the literature by explicitly deriving the OPE and contact densities for spin-orbit coupled and coherently coupled bosonic systems, extending concepts previously established primarily for fermions. The work demonstrates that the universal large-$k$ behavior of the momentum distribution in these bosonic systems is controlled by the derived contact densities.

The authors modestly frame their results as a foundational step. They explicitly note that they have excluded three-body physics (the Efimov effect), which is known to introduce additional contact densities in bosonic systems, and suggest this as a necessary direction for future investigation. Furthermore, they identify the OPE of conserved currents as an open question for future work to explore potential universal physics in current correlations. The study establishes that the nature of phase transitions (continuous vs. first-order) can be directly observed through the continuity or discontinuity of contact densities.