Chiral non-Fermi liquids

A non-Fermi liquid state without time-reversal and parity symmetries arises when a chiral Fermi surface is coupled with a soft collective mode in two space dimensions. The full Fermi surface is described by a direct sum of chiral patch theories, which are decoupled from each other in the low-energy limit. Each patch includes low-energy excitations near a set of points on the Fermi surface with a common tangent vector. General patch theories are classified by the local shape of the Fermi surface, the dispersion of the critical boson, and the symmetry group, which form the data for distinct universality classes. We prove that a large class of chiral non-Fermi liquid states exists as stable critical states of matter. For this, we use a renormalization group scheme where low-energy excitations of the Fermi surface are interpreted as a collection of $(1+1)$-dimensional chiral fermions with a continuous flavor labeling the momentum along the Fermi surface. Due to chirality, the Wilsonian effective action is strictly UV finite. This allows one to extract the exact scaling exponents although the theories flow to strongly interacting field theories at low energies. In general, the low-energy effective theory of the full Fermi surface includes patch theories of more than one universality classes. As a result, physical responses include multiple universal components at low temperatures. We also point out that, in quantum field theories with extended Fermi surface, a noncommutative structure naturally emerges between a coordinate and a momentum which are orthogonal to each other. We show that the invalidity of patch description for Fermi liquid states is tied with the presence of UV/IR mixing associated with the emergent noncommutativity. On the other hand, UV/IR mixing is suppressed in non-Fermi liquid states due to UV insensitivity, and the patch description is valid.

Figure 1

(a) Two stacks of integer quantum Hall layers with $\nu =1$. (b) When the directions of magnetic field are opposite in the two stacks, the edge modes have same chirality near the region where two stacks are close to each other. The interface of the two stacks is described by a two-dimensional chiral metal with two flavors.

Figure 2

The chiral Fermi sea. Generically, there are two points on the Fermi surface which have a common tangent vector. For example, the points ${\mathcal{P}}_A$ and ${\mathcal{P}}_B$ have the same tangent vectors and they remain strongly coupled to each other in the low-energy limit. A minimal patch theory should include all the points with the common tangent vector.

Figure 3

The solid curve is the Fermi surface, and the dashed curves represent the Fermi surface shifted by $\pm {\Lambda }_x$ along the $k_x$ direction. (a) The two arrows represent the minimum and maximum $x$ momenta a particle-hole pair ${\psi }_{i,k+p}^{*}{\psi }_{j,k}$ with fixed $k$ and $p$ can have within the thin shell of width $2{\Lambda }_x$. (b) The $x$ momenta of two pairs of particle-hole excitations created by ${\psi }_{i,k+p}^{*}{\psi }_{j,k}$ and ${\psi }_{i,k+\Delta k+p}^{*}{\psi }_{j,k+\Delta k}$ cannot be resolved within the uncertainty $2{\Lambda }_x$ if $\Delta k$ is too small. This is because the allowed $x$ momenta for the first pair overlap with those for the second. (c) If $\Delta k$ is large enough, the $x$ momenta of the two pairs can be resolved because the largest possible $x$-momentum of the first particle-hole pair is smaller than the smallest possible momentum of the second pair.

Figure 4

One-loop vertex function.

Figure 5

The solid curve is the Fermi surface, and the dashed curves represents the Fermi surface shifted by $\pm {\Lambda }_x$ along the $k_x$ direction. The (red) arrows denote particle-hole pairs with momentum $\vec{p}$ created near the Fermi surface. The dotted horizontal lines represent the UV cutoff in the $y$ momentum $(\pm \Lambda )$, which is set by the size of the Fermi surface. (a) If the $y$ momentum of a particle-hole pair, $p$, is large, then the phase space available to the pair within energy ${\Lambda }_x$ is limited by $\frac{{\Lambda }_x}{\gamma p}$. Therefore, the arrows go outside the thin shell before they sense the full extension of the Fermi surface. (b) If $p$ is small, then the available phase space of the particle-hole pairs is only limited by the size of Fermi surface.

Figure 6

Corrections to the quadratic vertex due to normal ordering of the quartic vertex.

Figure 7

Any group of operators, represented by small dots in the top panel, where separations between nearest neighbors are less than $X_0$ in $x$ direction are fused into a composite operator which are denoted by big dots in the bottom panel.

Figure 8

The origin represents the position of one operator in the $(1+1)$-dimensional real space. If the second operator is within the shaded region in the space of the relative coordinate $(x_{12},{\tau }_{12})$, two operators are fused into one normal ordered operator.

Figure 9

The space of relative coordinates between two operators. Each quartic vertex is drawn as an extended object, although there is no spatial extension of the wiggly line in the $x$ direction. The shaded region represents $|x_{12}|<X_0$ (see Fig. 8). (a) If the separation between two operators along the $x$ axis is greater than $X_0$, the two operators are considered as independent operators. (b) If the separation between the two operators along the $x$ axis is less than $X_0$, they are fused into one operator. Here we show a particular channel of fusion where two quartic operators are fused into one quartic operator. Each internal (dotted) line represents a pair of contracted fields, whereas external (solid) lines represent uncontracted fields.

Figure 10

In the Wilsonian effective action, “short-distance modes” between UV scales set by ${\Lambda }^2$, ${\eta }^{-2/(\theta -1)}$, and IR scale $X_0^{-1}$ are integrated out. In the chiral theory, the resulting Wilsonian effective action is regular in the $\Lambda \to \infty$ and $\eta \to 0$ limit. As a result, only the running cutoff scale $X_0$ enters as a scale of the effective action in the $\mu X_0^{1/2}\to 0$ limit.

Figure 11

A three-loop correction to the quadratic action. There are different ways of assigning internal frequencies, where internal frequencies are assigned to run along the dotted lines in (a) and (b). (a) All the fermion propagators in the loop of internal frequency ${\omega }_2$ are shared with other loops associated with the internal frequencies, ${\omega }_1$ and ${\omega }_3$. (b) Each loop denoted by the dotted lines has an exclusive fermion propagator that is not shared with other loops. (c) The propagators that are exclusive to each loop in (b) are represented by dashed lines. The numbers in the boxes indicate the indices for the internal frequencies that run through the exclusive propagators. (d) A tree diagram whose legs are contracted to yield (b). The legs with matching numbers are contracted to give the corresponding dashed propagator in (c).

Figure 12

A three-loop contribution to self-energy.

Figure 13

A fusion of two quartic vertices which results in an operator which is independent of $X_0$.

Figure 14

Fusion channels that give rise to $O\left(1\right)$ corrections to the Wilsonian effective action in the small $X_0$ limit.

Figure 15

A subleading correction to the quartic vertex. (a) With bare vertices and propagators, the resulting operator is IR divergent in the $\mu \to 0$ limit. (b) Once the vertices and propagators are dressed by all other $O\left(1\right)$ corrections, the resulting operator is finite as $\mu \to 0$.

Figure 16

A three-loop correction to the quadratic vertex. In the full quantum effective action, this is one of the infinite series of diagrams which is of the same order as the RPA corrections. In the Wilsonian effective action, this is subleading (see the text).

Figure 17

A plot of the function $f_2\left(s\right)$ in Eq. (78) for $\theta =2$.

Figure 18

The list of quantum corrections generated from the leading-order terms in the Wilsonian effective action at each step of coarse graining.

Figure 19

A six-fermion vertex generated from two quartic vertices.

Figure 20

Two inflections points on the chiral Fermi surface at which the quadratic curvature vanishes for the dispersion in Eq. (3).

Figure 21

A non-RPA contribution to the quartic vertex. The double wiggly line represents the RPA dressed quartic vertex.

Figure 22

A diagram where two quartic vertices (boson propagator) outside the box carry a same momentum. The double wiggly line represents the RPA dressed quartic vertex.

Figure 23

(a) The diagram in Fig. 22 is a part of the “one-loop” fermion self-energy. (b) The zigzag line in (a) represents the quartic vertex dressed with an infinite series of the three-loop boson self-energy drawn inside the box in Fig. 22.

Figure 24

Plot of ${{\rm Y}}_{\theta }(\alpha ,L)$ as a function of $\alpha$ with $L={10}^6$ and $\theta =0.2$.

Figure 25

Under the inversion along the $y$ direction the two patches are exchanged. Momentum defined away from the inversion points in each patch are also exchanged. For example, a particle-hole pair denoted as an arrow in patch ${\mathcal{P}}_{+}$ is mapped to the other arrow in patch ${\mathcal{P}}_{-}$.