Conformal Field Theory Interpretation

• Conformal Field Theory Interpretation is a unified framework that defines how quantum field theories invariant under conformal transformations reveal low-energy excitations and scaling laws.
• It maps low-energy Fermi surface excitations to 1+1-dimensional chiral CFTs, enabling precise calculations of entanglement entropy and number fluctuations.
• The approach shows that universal properties in Fermi liquids depend solely on Fermi surface geometry, robustly predicting behavior even at finite temperature and weak interactions.

A conformal field theory (CFT) interpretation provides a rigorous and unifying framework for understanding quantum field theories that are invariant under conformal transformations, with wide-ranging consequences in both mathematical structure and physical applications. The CFT interpretation is especially notable for its central role in elucidating the structure of low-energy excitations, universal scaling properties, operator spectra, and correlation functions in diverse contexts ranging from condensed matter physics to string theory.

1. Low-Energy Degrees of Freedom as Chiral CFTs on the Fermi Surface

The central paradigm introduced in "Conformal Field Theory on the Fermi Surface" (Swingle, 2010) is the decomposition of the low-energy excitation space of a Fermi surface into a continuum of 1+1-dimensional chiral conformal field theories. For a free Fermi gas in $d_s$ spatial dimensions, low-energy degrees of freedom correspond to particle-hole excitations that are localized in momentum space near each point of the Fermi surface.

Let $\psi_\theta(k, t)$ denote the fermionic field in the patch labeled by $\theta$ (specifying a direction on the Fermi surface, with local Fermi momentum $k_F$ and velocity $v_F$). The effective action for these excitations is

$$S_{\psi} = \int d\theta \int dk\, dt\, \psi^+_\theta(k,t) [i\partial_t - v_F k]\psi_\theta(k,t),$$

so that each patch behaves as a chiral, massless 1+1-dimensional fermion. These chiral CFTs are parametrized continuously by the Fermi surface, and each contributes independently to observables at low energies. In this picture, the chiral nature (e.g., $c_L=1$, $c_R=0$) reflects directionality (outward or inward) relative to the Fermi surface normal.

This reformulation permits direct computation of entanglement and fluctuation quantities using well-established methods from 1+1-dimensional CFT.

2. Anomalous Ground-State Properties: Entanglement Entropy and Fluctuations

A distinguishing feature of gapless fermionic systems with a Fermi surface is the violation of the standard area law in entanglement entropy. Generic gapped or bosonic systems in $d_s$ dimensions obey $S_A \sim L^{d_s-1}$ for a region $A$ of linear size $L$. In contrast, CFT methods reveal that for free fermions,

$S_A \sim L^{d_s-1} \ln L.$

This scaling arises from summing the contributions of all chiral patches intersected by the boundary $\partial A$ of region $A$. The number of such modes is $N_{\mathrm{modes}} \sim (k_F L)^{d_s-1}$. The contribution from each 1+1 mode, for interval length $L$, is

$S = \frac{c}{6} \ln \frac{L}{\epsilon},$

where $\epsilon$ is a UV cutoff and $c = 1$ for a chiral branch. Integrating over the Fermi surface and real-space boundary, the Widom formula is obtained: $$S = \frac{1}{(2\pi)^{d_s-1}} \frac{\ln L}{12} \int dA_k \int dA_x |n_x \cdot n_k|,$$ where $n_x$ ($n_k$) are the normals to the region boundary and Fermi surface, respectively.

Particle-number fluctuations $\Delta N_A^2$ in the region $A$ also exhibit a logarithmic enhancement,

$\Delta N_A^2 \sim L^{d_s-1} \ln L,$

which is again derived via the chiral CFT picture: $$\Delta N_A^2 \sim \frac{1}{(2\pi)^{d_s-1}} \frac{\ln L}{4\pi^2} \int dA_k \int dA_x |n_x \cdot n_k|.$$

This framework directly and quantitatively explains the anomalous scaling of both entanglement and number fluctuations in metallic Fermi systems.

3. Generalization to Finite Temperature and Interacting Fermi Liquids

The 1+1-dimensional CFT description is robust against generalizations in temperature and (weak) interactions. At temperature $T = 1/\beta$, the entanglement entropy for a chiral interval is given by

$$S = \frac{c}{6} \ln\left[\frac{\beta v}{\pi \epsilon} \sinh\left(\frac{\pi L}{\beta v}\right)\right].$$

When considering a real-space region (e.g., a disk of radius $R$), the effective length $L_\mathrm{eff}$ for each patch labeled by $\theta$ is $2R |\cos\theta|$, so the final entropy involves an angular integration over the Fermi surface.

Forward scattering interactions in a Fermi liquid do not introduce new low-energy singularities but only renormalize parameters such as $v_F$ and Landau parameters. The topological features of the Fermi surface (total mode counting) are protected by Luttinger’s theorem. The universal form of entanglement and number fluctuations is preserved, with interacting $v_F$ replacing the bare value.

4. Geometric Universality: Dependence Solely on Fermi Surface Structure

A crucial outcome of the CFT perspective is the universality of low-energy entanglement structure for Fermi liquids:

• The Widom and fluctuation formulas depend only on the geometry of the Fermi surface (shape, area, and orientation), not on microscopic details or interaction parameters (beyond those renormalizing $v_F$).
• All prominent anomalous properties are encoded in the integral kernel

$\int dA_k \int dA_x |n_x \cdot n_k|$

which projects CFT contributions according to the intersection of boundary and Fermi surface orientations.

This universality allows predictive control over entanglement and number fluctuation scaling in any Fermi liquid once the Fermi surface is known.

5. Implementation: Calculational Strategies and Scaling Laws

Translating the CFT interpretation to calculations for observables involves the following steps:

1. Patch Decomposition: Decompose the Fermi surface into infinitesimal segments, each mapped to a 1+1 chiral CFT.
2. Mapping Spatial Region: For a real-space region $A$, determine the intersection properties (projected effective lengths $L_\mathrm{eff}$) for each patch.
3. Mode Counting: Integrate over the Fermi surface and the boundary of $A$, with the geometric weight $|n_x \cdot n_k|$.
4. Observable Calculation: Sum contributions of each patch, using known results from 1+1 CFT for entanglement entropy and number variance, to obtain aggregate quantities using Widom-type integrals.

This approach yields, for ground-state entanglement entropy,

$$S_A = \frac{1}{(2\pi)^{d_s-1}} \frac{\ln L}{12} \int dA_k \int dA_x |n_x \cdot n_k|,$$

and for number fluctuation variance,

$$\Delta N_A^2 = \frac{1}{(2\pi)^{d_s-1}} \frac{\ln L}{4\pi^2} \int dA_k \int dA_x |n_x \cdot n_k|.$$

6. Broader Significance and Theoretical Impact

The chiral CFT interpretation of Fermi surface physics is profound for several reasons:

• It links multi-dimensional metallic Fermi systems with the rich mathematical and physical theory of 1+1-dimensional CFT, facilitating the transfer of powerful analytic tools.
• Anomalies seen in metallic ground states (e.g., logarithmic corrections to entropy and fluctuations) are transparently reinterpreted as consequences of underlying chiral CFT structure.
• The universality rooted in Fermi surface geometry generalizes to situations involving finite temperature, interactions, and nontrivial real-space region boundaries.
• The framework provides a rigorous justification for the breakdown of strict area laws in higher-dimensional gapless fermion systems.
• This perspective clarifies the relationship between microscopic fermionic models and emergent low-energy quantum many-body phenomena.

Overall, the CFT interpretation on the Fermi surface serves as a unifying theoretical construct for understanding the low-energy physics and universal scaling of entanglement and fluctuations in Fermi liquids and metals, with predictive power determined by Fermi surface geometry rather than microscopic details (Swingle, 2010).