Symmetry-Enforced Fermi Surfaces

Imagine you are building a complex machine out of tiny, invisible Lego blocks. In the world of quantum physics, these blocks are electrons (or "fermions") sitting on a grid. Usually, when you build these machines, you have to be very careful. If you arrange the blocks just right, the machine might stop working entirely (becoming an "insulator" or a "gapped phase"). If you arrange them differently, it might hum with energy (becoming a "metal" or a "gapless phase").

Physicists have long known how to force a machine to stop (create a gap), but it is much harder to force a machine to keep humming. Usually, a machine only hums if you tune it perfectly. If you tweak a single screw, the humming stops.

This paper introduces a new, incredibly powerful "rule of the universe" (a symmetry) that acts like a magical safety net. No matter how you build your machine, as long as you follow this rule, the machine must hum. It cannot stop. Specifically, it forces the machine to have a "Fermi surface."

What is a "Fermi Surface"?

Think of a Fermi surface not as a physical surface, but as a boundary line in a map of possibilities.

Imagine a crowded dance floor where every dancer represents a possible energy state.

Gapped Phase (Insulator): The dance floor is empty in the middle. There is a clear, wide gap between the dancers who are dancing (occupied states) and the empty space where no one can dance.
Gapless Phase (Metal): The dancers are packed right up to the edge of the floor.
Fermi Surface: This is the exact line where the dancers end and the empty space begins. It is a "coastline" of energy.

In a normal metal, this coastline can be messy or disappear if you change the temperature or add impurities. But the symmetry discovered in this paper acts like a magnetic fence that forces this coastline to exist, no matter what.

The Two "Magic Rules"

The authors found that to force this coastline to exist, you need two specific rules working together:

The "Counting" Rule (U(1) Symmetry): You must be able to count the total number of dancers (fermions) and keep that number constant. You can't create or destroy dancers out of thin air.
The "Mirror-Walk" Rule (Majorana Translation): This is the tricky one. Imagine the dancers are made of two halves, a "left shoe" (Majorana $a$ $a$ ) and a "right shoe" (Majorana $b$ $b$ ).
- Normally, if you tell the whole dancer to walk one step to the right, both shoes move.
- This new rule says: The left shoes stay exactly where they are, but the right shoes move one step to the right.
- It's like a dance where one half of your body stays frozen while the other half walks.

When you combine "Counting" with this weird "Half-Walk" rule, the laws of physics get so twisted that the system cannot settle into a quiet, empty state. It is forced to have a boundary line (the Fermi surface) where the energy is zero.

The Shape of the Coastline

The paper also looks at what this forced coastline looks like.

It's not random: The coastline must be perfectly symmetrical. If you draw a line through the center of the map and flip the map over, the coastline must look exactly the same.
It's never a simple loop: In a normal world, you might draw a circle on a map. But because of the "Half-Walk" rule, this coastline cannot be a simple circle that you can shrink down to a dot.
The "Open Road" Analogy: The paper proves that this coastline must have at least two parts that are "open roads" stretching all the way across the map. You can't close them up. They are like roads that go off the edge of the world and wrap around to the other side.

Why is this a Big Deal?

Usually, to get a metal to act like a metal, you need to tune it carefully. If you add a little bit of "chemical potential" (like adding a chemical to the dance floor), the coastline disappears, and the machine stops humming.

But with this new symmetry, you cannot kill the humming. Even if you try to add terms to the machine that usually stop the flow, this symmetry forbids them. It is a "strong" enforcement. The machine is guaranteed to have a Fermi surface.

The "Onsager" Connection

The authors mention that the group of rules they found is related to something called the Onsager algebra. Think of this as a very complex, infinite library of instructions.

In normal physics, symmetries are like simple switches (On/Off).
Here, the symmetry is like a library with infinite books. The "Half-Walk" rule allows the system to access a massive, infinite-dimensional group of symmetries.
This massive symmetry is what holds the "coastline" in place. It's so powerful that it forces the system to behave like a "free fermion" system (a simple, non-interacting gas of particles) even if you tried to make the particles interact.

Summary

The paper says: "If you build a quantum system on a grid and enforce these two specific, slightly weird rules (counting particles and a 'half-step' translation), the system is forced to have a Fermi surface. It cannot become an insulator. The boundary between energy states will always exist, and it will always have a specific, non-trivial shape with open paths that wrap around the system."

It's a discovery of a new "law of nature" that guarantees a specific type of metallic behavior, regardless of how you try to build the machine.

1. Problem Statement

The paper addresses a fundamental gap in the classification of quantum phases of matter. While gapped phases (topological, symmetry-protected, etc.) and certain gapless phases (Goldstone modes, conformal field theories) are well-understood, the landscape of gapless phases with an infinite number of gapless excitations (specifically metals with Fermi surfaces) is less controlled.

Existing mechanisms for "Symmetry-Enforced Gaplessness" (SEG) typically enforce a finite number of gapless modes or rely on mixed UV/IR constraints (e.g., requiring compressibility). The authors ask: Is there a purely ultraviolet (UV) symmetry that strictly enforces the existence of a Fermi surface (a codimension-1 locus of gapless excitations) in any local lattice model, regardless of interactions?

2. Methodology

The authors construct a specific symmetry group acting on quantum lattice fermion models defined on a $d$ -dimensional Bravais lattice $\Lambda$ with one site per unit cell.

Model Setup:
- Hilbert space: One complex fermion $c_r$ per site $r$ .
- Hamiltonian: Local, Hermitian, and commuting with the symmetry group.
- Constraints: The system must conserve total fermion number and possess specific translation symmetries.
Symmetry Construction:
The enforcing symmetry is generated by $d+1$ locality-preserving unitary operators:
1. On-site $U(1)$ Fermion Number Symmetry: Generated by the charge operator $Q = \sum_r (c_r^\dagger c_r - 1/2)$ .
2. Non-on-site Majorana Translation Symmetry: Generated by operators $T^{(b)}_v$ $T_{v}^{(b)}$ acting on the lattice vector $v$ $v$ . These operators act differently on the two Majorana components of the fermion:
  - $a_r = c_r^\dagger + c_r$ (invariant under translation).
  - $b_r = i(c_r^\dagger - c_r)$ (shifted by $v$ ).
  - Specifically, $T^{(b)}_v a_r (T^{(b)}_v)^{-1} = a_r$ and $T^{(b)}_v b_r (T^{(b)}_v)^{-1} = b_{r+v}$ .
Algebraic Analysis:
The authors analyze the commutation relations between the charge operator $Q$ and the translation operators. They define shifted charges $Q_v = T^{(b)}_v Q (T^{(b)}_v)^{-1}$ and derive the Lie algebra they satisfy. This algebra is identified as a generalization of the Onsager algebra (specifically denoted as $\text{Ons}_d$ ), which is a non-compact, infinite-dimensional Lie group in the thermodynamic limit.

3. Key Contributions and Results

A. Enforcing the Fermi Surface

The authors prove that any local Hamiltonian commuting with both the $U(1)$ charge $Q$ and the Majorana translations $T^{(b)}_v$ must be a free fermion Hamiltonian of a specific form:
$H = i \sum_{r_1, r_2} g_{r_2-r_1} c_{r_1}^\dagger c_{r_2}$
where the hopping amplitudes satisfy $g_{-r} = -g_r$ (antisymmetric) and $g_0 = 0$ .

Consequence: The chemical potential term ( $\mu \sum c^\dagger c$ ) and pairing terms ( $c c + c^\dagger c^\dagger$ ) are strictly forbidden by this symmetry.
Dispersion: The single-particle dispersion is $\epsilon_k = -2 \sum_v g_v \sin(k \cdot v)$ .
Gaplessness: Since $\epsilon_{-k} = -\epsilon_k$ , the dispersion is odd under inversion. By the intermediate value theorem, $\epsilon_k$ must vanish on a codimension-1 locus (the Fermi surface) in the Brillouin zone. Thus, the symmetry enforces a Fermi surface.
Filling Factor: The symmetry enforces the ground state to be exactly half-filled (1/2 fermion per unit cell).

B. Topology of Symmetry-Enforced Fermi Surfaces

The paper provides a rigorous topological classification of these enforced Fermi surfaces ( $F$ ):

Inversion Symmetry: The Fermi surface is invariant under $k \to -k$ and must pass through all inversion-invariant points (e.g., $k=0$ ).
Non-contractible Components: A generic symmetry-enforced Fermi surface always possesses at least two non-contractible components (open orbits) in the Brillouin torus.
Contractible Components: Any contractible components must appear in even numbers and cannot pass through inversion-invariant points.
Proof: The authors prove that a contractible component passing through an inversion-symmetric point would necessarily self-intersect, violating the "generic" condition (smoothness/non-singularity) of the dispersion.

C. Emergent IR Symmetries

The authors investigate the infrared (IR) limit of the UV symmetry group $\text{Ons}_d \rtimes \prod Z_{L_i}$ :

Emanant Symmetry: The UV symmetry maps to a subgroup of the infinite-dimensional Ersatz Fermi Liquid symmetry group, denoted $L_F U(1)$ .
Structure: The IR symmetry is isomorphic to a quotient of the UV group. It contains a non-abelian, infinite-dimensional subgroup.
Relation to $L_F U(1)$ : The UV symmetry includes a specific subgroup of $L_F U(1)$ generated by even functions $f(k) = f(-k)$ on the Fermi surface. This is denoted $L_e F U(1)$ .
Anomaly Matching: Unlike standard Fermi liquid theories where the $L_F U(1)$ anomaly is emergent, the UV symmetry constructed here is anomaly-free (it allows gapped phases if the symmetry is broken). The IR anomaly of the full $L_F U(1)$ is not matched by this specific UV symmetry, suggesting the UV symmetry is a "stronger" constraint that forces the system into a specific free-fermion-like state.

4. Significance

Strong Symmetry-Enforced Gaplessness: This work identifies the first known example of Strong SEG that enforces an infinite number of gapless modes (a Fermi surface) rather than just a finite set. This distinguishes it from previous SEG examples which typically allow for gapped phases with broken symmetry.
New Symmetry Class: The construction introduces a non-compact, infinite-dimensional Lie group symmetry ( $\text{Ons}_d$ ) generated by non-on-site Majorana translations, expanding the known landscape of lattice symmetries beyond standard global symmetries and lattice translations.
Topological Constraints: The paper establishes that symmetry-enforced Fermi surfaces are not arbitrary; they possess rigid topological features (mandatory non-contractible loops) that distinguish them from generic metallic states.
Theoretical Framework: It provides a rigorous UV definition for systems that behave like "perfect" Fermi liquids without interactions, offering a new perspective on the stability of metallic phases against gapping perturbations.

5. Outlook and Future Directions

The authors suggest several avenues for future research:

Higher Codimension: Investigating if similar symmetries can enforce codimension- $p$ Fermi surfaces (e.g., nodal lines in 3D).
Physical Consequences: Exploring the physical implications of the non-trivial topology of these Fermi surfaces.
Gauge Fields: Studying the behavior of these symmetries in the presence of background gauge fields (e.g., uniform magnetic fields), noting preliminary results for $\pi$ -flux models.
Enhanced IR Symmetry: Determining if symmetric Fermi surfaces generally lead to enhanced emergent symmetries larger than the standard $L_F U(1)$ .

In summary, the paper demonstrates that a specific combination of on-site $U(1)$ and non-on-site Majorana translation symmetries creates a "no-go" theorem for gapped phases, forcing any local, symmetric lattice model to realize a topologically constrained Fermi surface.