Fermi Arcs

In conventional metals, electrons occupy states up to the Fermi energy, forming a boundary known as the Fermi surface, which is invariably a closed loop or surface. This fundamental picture is challenged by the discovery of materials hosting "Fermi arcs"—seemingly impossible electronic states that trace open lines on a material's surface, like a coastline without a coast. This discrepancy raises a profound question: how can such an incomplete boundary exist and what principle protects it? This article delves into the fascinating physics of Fermi arcs, explaining the topological secrets that allow them to defy conventional wisdom.

The following sections will guide you through this topological landscape. In "Principles and Mechanisms," we will explore the deep connection between surface Fermi arcs and the material's bulk properties, introducing the concept of Weyl points as magnetic monopoles in momentum space. Subsequently, "Applications and Interdisciplinary Connections" will reveal how these exotic states are experimentally observed and how they lead to remarkable new physical phenomena and potential technologies, from nonlocal transport to new paradigms in photonics and quantum information.

In the world of metals, the motions of electrons are not entirely chaotic. Their allowed states in momentum space form a landscape of hills and valleys, and at zero temperature, the electrons fill all available states up to a certain energy level, the ​​Fermi energy​​. The boundary between the filled "lake" of states and the empty "land" of states is called the ​​Fermi surface​​. For any ordinary metal you’ve ever encountered, from the copper in your wires to the aluminum in your soda cans, this boundary is always like a coastline of an island or a continent—it is always a closed loop or a closed surface. It has to be. How could a boundary not be closed?

Now, imagine you are an explorer of the quantum world. You use a powerful technique, Angle-Resolved Photoemission Spectroscopy (ARPES), to map out the "coastline" on the surface of a newly discovered crystal. On one material, you see exactly what you expect: a neat, closed loop. But on another, you see something that seems impossible: the coastline is just a line segment. It starts at one point, runs for a while, and then just... stops. This is a ​​Fermi arc​​.

To check if you are seeing things, you decide to poke at it. You carefully deposit a thin, insulating film over the surfaces of both crystals. When you look again, the closed loop on the first material has vanished, its states washed away by the disturbance. But the open arc on the second material is still there, defiantly shimmering in your detector, albeit slightly shifted.

This little thought experiment, based on real observations, tells us something profound. The Fermi arc is not some fragile, accidental feature of the surface. Its existence is robust, protected by a deep physical principle. The material with the fragile loop is just a trivial metal with an ordinary surface state. The material with the indomitable arc is something new, something topological: a ​​Weyl semimetal​​. The secret to this impossible coastline lies not on the surface, but deep within the bulk of the crystal.

To understand the Fermi arc, we must venture into the material's bulk electronic structure. Here, we find special points in momentum space where the band of electron-filled states (the valence band) and the band of empty states (the conduction band) touch. These touching points are called ​​Weyl points​​. They are not just incidental crossings; they are points of immense topological significance.

A Weyl point acts like a ​​magnetic monopole​​, but in the abstract world of momentum space. You know that if you have a magnetic monopole (a hypothetical particle that is just a "north" or "south" pole), lines of magnetic field would emanate from it or converge on it. Similarly, a Weyl point is a source or a sink for a quantity called ​​Berry curvature​​, an "effective magnetic field" that lives in momentum space and governs the quantum geometry of the electron wavefunctions.

Each Weyl point is assigned a topological charge, an integer called ​​chirality​​ ($\chi$), which is either $+1$ (a source) or $-1$ (a sink). For a simple Weyl node, this charge can be found from its low-energy Hamiltonian, which has the form $H(\mathbf{q}) = v_x q_x \sigma_x + v_y q_y \sigma_y + v_z q_z \sigma_z$. The chirality is simply given by the sign of the product of the velocities: $\chi = \mathrm{sgn}(v_x v_y v_z)$.

Just as magnetic monopoles have never been found in isolation, Weyl points obey a strict rule known as the Nielsen-Ninomiya theorem: the total chirality summed over the entire momentum space must be zero. You cannot have a single source without a sink. Therefore, in any real material, Weyl points must come in pairs (or multiples of pairs) of opposite chirality. It is the existence of these separated "charges" in momentum space that sets the stage for the Fermi arc.

The connection between the bulk Weyl points and the surface Fermi arcs is one of the most beautiful examples of the ​​bulk-boundary correspondence​​ in physics. Let's see how it works.

Imagine our 3D momentum space is a stack of 2D parallel slices, like pages in a book. Let's say we have a pair of Weyl points, one with $\chi=+1$ and one with $\chi=-1$, separated along the $k_z$ axis. So, one node is at $k_z = -k_0$ and the other is at $k_z = +k_0$.

Now, let's pull out one of the 2D slices at a fixed $k_z$ that lies between the two nodes (say, $k_z = 0$). This 2D slice "encloses" one of the monopoles but not the other. Just as Gauss's law tells you that a surface enclosing a charge has a net electric flux, our 2D slice of momentum space enclosing a Weyl point has a net flux of Berry curvature. This net flux is a quantized topological invariant called the ​​Chern number​​, $C$. For any slice with $-k_0 k_z k_0$, the Chern number is non-zero (e.g., $C(k_z)=1$), while for slices outside this region ($|k_z| > k_0$), the Chern number is zero.

Here comes the magic. It is a fundamental result that a 2D system with a non-zero Chern number is a ​​topological insulator​​, and it must host conducting states at its one-dimensional edges. These are not just any conducting states; they are ​​chiral edge states​​, meaning electrons can only travel in one direction along the edge—they are like one-way electronic highways.

Now, back in our 3D Weyl semimetal slab, the "edge" of each of these 2D slices is simply the surface of the slab. Therefore, for every single $k_z$ value between $-k_0$ and $+k_0$, our slab's surface must host a chiral state. If we plot the position of these required states in the 2D surface momentum space (spanned by $k_y$ and $k_z$), they trace out a continuous line. This line is precisely the Fermi arc! It exists only for the range of $k_z$ between the Weyl points, and thus it gracefully connects the projection of one Weyl point to the projection of the other. One can even write down a simple model where the energy of the surface state is directly tied to this bulk invariant: $E_s(k_y, k_z) \propto C(k_z)$, beautifully demonstrating this connection.

This picture also resolves the paradox of an "open" coastline. An electron traveling along a Fermi arc on the top surface reaches the end—the projection of a Weyl point. At this point, it can "dive" into the bulk and travel to the other Weyl point, where it can emerge on the bottom surface of the slab. On the bottom surface, another Fermi arc will guide it back to the projection of the first Weyl node. The full circuit is complete: top arc, bulk, bottom arc, bulk. The system is perfectly self-contained, and particle number is conserved.

What is it like to be an electron living on a Fermi arc? It's a strange and wonderful existence, governed by the underlying topology.

First, as we saw, this existence is ​​robust​​. The arc is a consequence of the bulk topological charge, which cannot be changed by small local perturbations on the surface. You can't just wish it away.

Second, these states are truly surface-dwellers. Their wavefunction is peaked at the crystal's boundary and decays exponentially as one goes deeper into the bulk. The characteristic length of this decay is the ​​penetration depth​​.

Third, these states can exhibit unique transport properties. For a simple, straight Fermi arc, the density of states per unit energy can be a constant, independent of energy. This is highly unusual—in most metals, the density of states varies with energy. This constant density of states implies that the material's thermodynamic response (like its heat capacity) will have a peculiar character.

Finally, the arc carries a "hidden" topological fingerprint. If an electron were to travel from one end of the arc to the other, its quantum mechanical phase would change in a very specific way. The integral of the Berry connection along the arc, a quantity called the ​​Zak phase​​, is quantized to be $\pi$ (modulo $2\pi$). This is another subtle, yet profound, consequence of a path connecting two points of opposite topological charge.

The story of the Fermi arc is a perfect illustration of how topology reshapes our understanding of materials. But nature is ever inventive, and it's worth noting that the term "Fermi arc" has appeared in another, very different, corner of physics: the study of high-temperature copper-oxide superconductors.

In these materials, ARPES experiments also reveal what look like disconnected Fermi arcs in their strange "pseudogap" phase. However, these arcs are not believed to originate from bulk Weyl points. Instead, they are thought to be remnants of a full Fermi surface that has been partially obliterated by incredibly strong electron-electron repulsive forces—a phenomenon rooted in ​​Mott physics​​. In this picture, the interactions become so strong in certain directions in momentum space that they destroy the electronic states, leaving only the more resilient "nodal" regions as conducting arcs.

So, while they may look similar in an experiment, the Fermi arc of a Weyl semimetal and the Fermi arc of a cuprate are born from vastly different physics: one from the elegant topology of non-interacting bands, the other from the messy, complex brawl of strongly interacting electrons. It serves as a beautiful reminder that in physics, understanding the "why" is just as important as observing the "what". The journey to uncover the principles and mechanisms behind what we see is the true adventure.

In the last chapter, we were introduced to a rather strange beast: the Fermi arc. We learned that on the surface of a special kind of material called a Weyl semimetal, the electrons are not confined to the usual closed loops of a Fermi surface. Instead, they live on open line segments—like highways that seem to begin and end in the middle of nowhere. These arcs are the surface footprints of the bizarre topological dance happening in the bulk of the material.

Now, a reasonable person might ask: So what? Is this just a physicist's curiosity, a peculiar squiggle on a graph, or does it lead to something real and useful? It's a fair question. And the answer is a resounding "yes!" These strange highways on the edge of a crystal are not just a theoretical oddity; they are the gateways to a whole new world of physical phenomena and potential technologies. We are about to embark on a journey to see where these highways lead, from seeing them with our own "electron cameras" to watching them conduct electricity between parallel worlds.

Before we can use a new discovery, we must first be certain it's really there. How can we be sure that these Fermi arcs, these one-dimensional electronic roads, actually exist? You can't just look at a crystal and see them. We need a special kind of camera, one that can take a picture not of the atoms, but of the electrons' momentum and energy.

That camera is a technique called Angle-Resolved Photoemission Spectroscopy, or ARPES. In an ARPES experiment, we shine a bright light on the material's surface, which knocks electrons out. By measuring the direction and energy of these escaping electrons, we can reconstruct a "traffic map" of the electronic states on the surface. For a normal metal, this map shows what you might expect: closed loops, like roundabouts or racetracks, where electrons can happily travel in circles.

But when we point our ARPES camera at the surface of a Weyl semimetal, we see something that, by all conventional rules, should not exist. We find a bright, sharp line of states that is not a closed loop. It's an open arc. It’s a road that just starts at one point and stops at another, with no connection back to its beginning. This observation of an unclosed line segment of states is the definitive, smoking-gun signature of a Fermi arc. It's the first proof that we've found one of these topological highways.

Our confidence in this picture is bolstered by computer simulations. When we model a slab of a Weyl semimetal, these peculiar surface-bound states appear in the calculations exactly where the theory of topology predicts they should. If we run the same simulation for an ordinary material, they are nowhere to be found. This perfect agreement between deep theory, sophisticated computation, and precise experiment is what gives us the certainty to say that these arcs are real. The highways are open for business.

So, we have these highways. What happens when an electron starts moving on one? This is where the story gets truly fantastical. Imagine our slab of material is like two parallel universes: the top surface and the bottom surface. The Fermi arcs are strange, one-way streets within each of these universes. Now, what happens if we apply a magnetic field perpendicular to these surfaces? The magnetic field acts as a cosmic traffic controller, but it also does something much more magical: it opens up one-way teleporters between the two universes.

These "teleporters" are the special chiral Landau levels that exist in the bulk of the material, a direct consequence of the Weyl points. In a magnetic field, an electron moving along the Fermi arc on the top surface is pushed by the Lorentz force until it reaches the end of the arc—the projection of a bulk Weyl point. At this terminus, it can't go any further on the surface. Instead, it "jumps" into one of these chiral Landau levels and shoots straight through the bulk of the crystal to the bottom surface, like an express elevator.

Upon arriving at the bottom surface, it lands on the a Fermi arc there and travels along it until it reaches the other Weyl point projection. From there, it jumps into the other chiral Landau level (which goes in the opposite direction) and zips back to the top surface, completing a giant, spectacular loop. This hybrid trajectory—part surface travel, part bulk teleportation—is known as a "Weyl orbit."

This isn't just a fun story; it has stunning and measurable consequences.

First, it allows for ​​nonlocal transport​​. Because the Weyl orbit physically connects the top and bottom surfaces, you can inject an electrical current into the top surface and measure a voltage on the bottom surface. The Weyl orbit acts as a wire that traverses the crystal, a direct conduction path between two points that are, in ordinary circumstances, isolated from each other.

Second, because this is the quantum world, the entire Weyl orbit is quantized. Its allowed energy levels are discrete, like the rungs of a ladder. This quantization leads to beautiful quantum oscillations: as you vary the strength of the magnetic field, properties like electrical resistance wobble up and down in a regular pattern. Crucially, the frequency of these oscillations depends on the thickness of the crystal slab. This thickness dependence is the ultimate proof that the electrons are really making the full journey from top to bottom and back again. These quantized energy levels also mean that the system can absorb particles of light—microwaves, for instance—at a very specific resonant frequency. The absorption frequency turns out to have a beautifully simple relationship with the slab thickness, $d$, and the velocity of the electron on its chiral express journey, $v_F$. A hypothetical model suggests this frequency could be as simple as $\omega_{\text{res}} = \frac{\pi v_F}{d}$.

The unique properties of Fermi arcs make them fascinating building blocks for a new generation of electronic and quantum devices. Their behavior is unlike anything found in conventional materials.

For instance, what if we form a junction by joining a standard p-type semiconductor with the surface of a Weyl semimetal? We create a new type of electronic component. When a forward voltage $V_a$ is applied, electrons can tunnel from the semiconductor into the Fermi arc states. Because the Fermi arc has a very peculiar density of states (in the simplest models, it's constant), the way the current $J$ depends on the voltage is unique. Instead of the exponential behavior of a typical diode, we find a power-law relationship, something of the form $J \propto (eV_a - \delta_p)^{3/2}$ for a certain voltage range. This unique characteristic, dictated by the topological nature of the final states, could be exploited to design novel electronic circuits.

The influence of Fermi arcs extends into the quantum realm. Imagine placing a single artificial atom, a quantum dot, near the surface of a Weyl semimetal. The quantum dot and the Fermi arc can exchange electrons. The arc acts as a very special kind of environment, or "bath," for the quantum dot. Due to the linear, one-dimensional nature of the arc states, the way they perturb the energy level of the quantum dot is unique. The level broadening, $\Gamma(E)$, which is a measure of how quickly an electron can hop between the dot and the surface, becomes constant with energy—a direct fingerprint of the arc's unusual dispersion. This provides a new tool for controlling and probing single quantum systems, a crucial task for building quantum computers.

Perhaps the most profound aspect of a great idea in physics is its universality. The principles behind Fermi arcs are so fundamental that they are not limited to electrons in a crystal. The same ideas pop up in completely different physical systems, a beautiful testament to the unity of nature's laws.

Consider light. By fabricating materials with periodic structures on the scale of the wavelength of light—so-called photonic crystals—we can control the flow of photons in much the same way a semiconductor crystal controls the flow of electrons. It's possible to design a 3D photonic crystal that exactly mimics the physics of a Weyl semimetal. And what do we find on its surface? You guessed it: ​​photonic Fermi arcs​​. These are one-dimensional, topologically protected pathways for light. This is not just a clever analogy; it opens the door to creating optical devices, like waveguides, that are exceptionally robust against defects and imperfections. One could even consider the thermodynamics of a "gas" of photons living on these arcs, which would exhibit its own unique signature, like a specific heat that scales linearly with temperature, $c_A(T) \propto T$.

The connection gets even deeper, touching the very fabric of quantum mechanics: entanglement. Let’s take the ground state of a Weyl semimetal—the state of lowest energy where all the quantum particles are settled. Now, let’s mathematically divide the system into two halves and ask: "How much are these two halves quantum-mechanically entangled with each other?" The answer is encoded in a mathematical object called the entanglement Hamiltonian. Astonishingly, the spectrum of this entanglement Hamiltonian contains its own gapless modes that form a perfect mimic of the physical surface states—an ​​entanglement Fermi arc​​. This means the topology of the bulk material is not just reflected on its physical boundary, but also in the abstract, informational structure of its quantum entanglement. The highway on the edge of the crystal is a ghost, a shadow, of the deepest quantum correlations hidden within.

From a strange squiggle in an experiment to a blueprint for new technologies and a reflection of fundamental quantum information, the Fermi arc has taken us on a remarkable tour. It shows us, once again, that by looking carefully at the strange and unexpected corners of the universe, we often find the most beautiful and powerful ideas.