Bad metal behavior and Lifshitz transition of a Nagaoka ferromagnet

Using an extended fermionic functional renormalization group method, this study maps the phase diagram of the infinite-U Hubbard model on a square lattice, revealing a progression from paramagnetic Fermi liquid to antiferromagnetic stripes and finally to an incoherent non-Fermi liquid Nagaoka ferromagnet that undergoes a Lifshitz transition between two distinct ferromagnetic regimes.

The Gist

Imagine a crowded dance floor where everyone is trying to move, but there's a strict rule: no two people can ever stand on the same spot at the same time. This is the world of the "Hubbard model" at infinite repulsion, a famous puzzle in physics that describes how electrons behave when they are extremely crowded and hate each other.

In this paper, three physicists (Jonas, Peter, and Andreas) used a powerful new mathematical tool to figure out exactly what happens on this dance floor as you add more and more dancers (electrons). Here is what they found, explained simply.

1. The Three Acts of the Dance

As they increased the number of dancers on the square dance floor, the crowd went through three distinct phases, like a play with three acts:

• Act 1: The Chill Crowd (Low Density). When there are few dancers, everyone moves around freely and politely. They bump into each other occasionally but keep their rhythm. In physics terms, this is a Fermi Liquid—a normal, well-behaved metal where electrons act like individual particles.
• Act 2: The Stalled Traffic (Medium Density). As the floor gets more crowded, the dancers start to get stuck in a gridlock. They can't move freely anymore. Instead of moving randomly, they start forming rigid lines and stripes, refusing to move unless everyone in the line moves together. This is a state of Antiferromagnetic Stripe Order—a frozen, organized chaos.
• Act 3: The Super-Group (High Density). When the floor is almost packed, something magical happens. The dancers suddenly decide to all face the same direction and move in perfect unison. This is the Nagaoka Ferromagnet. It's like a military parade where everyone marches in lockstep. This state is special because it's the only way a single "hole" (an empty spot) can move through the crowd without causing a traffic jam.

2. The "Bad Metal" Mystery

Usually, when things move in a coordinated way (like a parade), they move smoothly. But the authors found something weird about this high-density parade.

They looked at the "spectral function," which is like a camera recording the speed and energy of the dancers. In a normal metal, you see sharp, clear lines (like distinct footsteps). In this high-density state, the picture is blurry and flat.

• The Analogy: Imagine trying to run through a crowd where everyone is holding hands and moving as one giant blob. You can't tell where one person ends and another begins. The energy levels become a "flat, broad band."
• The Result: This creates what physicists call a "Bad Metal." It conducts electricity, but it does so in a messy, incoherent way. The electrons have lost their individual identities and are behaving like a chaotic, correlated soup.

3. The "Lifshitz Transition" (The Shape-Shifting Moment)

Here is the most exciting discovery. The authors found that the high-density "Super-Group" phase isn't just one thing; it actually splits into two different versions separated by a sudden change called a Lifshitz transition.

• The Analogy: Imagine the shape of the dance floor itself changing.
  • Phase 1 (FM1): The "dance floor" (Fermi surface) looks like a circle. The dancers are moving in a way that resembles particles.
  • The Transition: Suddenly, at a critical density, the floor cracks.
  • Phase 2 (FM2): The floor breaks into four separate islands (arcs). The dancers are now moving like "holes" (empty spaces) rather than particles.
• Why it matters: This change in shape is accompanied by the band becoming completely flat. It's as if the dancers stop moving forward entirely and just vibrate in place. This suggests that at the highest densities, the electrons are so correlated that they effectively "condense" into a state where they have almost no kinetic energy left to move.

4. How They Did It (The New Tool)

Why hasn't anyone solved this before? Because the math is incredibly hard. When electrons can't double up, the usual math tools (which assume you can treat them as slightly interacting particles) break down.

The authors used a new method called X-FRG.

• The Analogy: Imagine trying to predict the weather. Old methods tried to add up small changes in temperature. But when the atmosphere is a total storm (extreme correlation), you need a different approach.
• The X-FRG method starts with a "perfect" snapshot of the system (where no one is moving) and then slowly "turns on" the movement, tracking how the chaos builds up step-by-step without losing the rules of the game. This allowed them to see the "Bad Metal" and the "Shape-Shifting" that previous methods missed.

The Big Picture

This paper tells us that when you squeeze electrons together tightly enough:

1. They stop acting like individuals and start acting like a collective, chaotic blob (Bad Metal).
2. They organize into a ferromagnetic parade.
3. At a critical point, the very geometry of how they move changes abruptly (Lifshitz transition), turning them into a state of "flat" energy where movement almost stops.

This helps explain the strange behavior of "strange metals" found in high-temperature superconductors and other exotic materials, suggesting that extreme crowding creates a new kind of physics where the rules of normal movement no longer apply.

Technical Summary

1. Problem Statement

The paper addresses the long-standing challenge of characterizing the ground state and electronic spectral properties of the Hubbard model at infinite on-site repulsion ($U = \infty$) in dimensions greater than one.

• Theoretical Context: The Nagaoka theorem (1966) proves that a single hole doped into a half-filled bipartite lattice with $U=\infty$ results in a ferromagnetic ground state due to kinetic energy minimization via path interference. However, the stability of this "Nagaoka ferromagnetism" for finite hole doping (more than one hole) in the thermodynamic limit remains controversial and difficult to resolve.
• Methodological Gap: Standard perturbative methods (weak-coupling expansions) and conventional Functional Renormalization Group (FRG) approaches fail because the $U=\infty$ limit requires a projection onto a Hilbert space where double occupancy is forbidden. This projection is non-perturbative and cannot be implemented via standard expansions.
• Missing Physics: Previous studies have largely focused on phase stability (magnetic ordering) but have neglected the single-particle spectral function in the strongly correlated Nagaoka regime. Understanding the nature of quasiparticles (or the lack thereof) in this regime is crucial for connecting to "bad metal" phenomenology and experimental realizations in optical lattices.

2. Methodology: X-FRG

The authors employ a recently developed extension of the fermionic Functional Renormalization Group (FRG) known as X-FRG (Hubbard X-operator FRG).

• Core Concept: Instead of treating the projection perturbatively, the X-FRG incorporates the Hilbert space projection exactly by imposing a non-trivial initial condition on the FRG flow.
• Formalism:
  • The system is mapped to the $t$-model (projected kinetic energy), where the electron operators are replaced by Hubbard $X$-operators (holons) acting on a restricted Hilbert space (no double occupancy).
  • These operators obey non-canonical anticommutation relations, encoding the constraint $n_{i\uparrow} + n_{i\downarrow} \leq 1$.
  • The flow starts from the exactly solvable limit of isolated Hubbard atoms ($t=0$) and integrates out hopping effects ($t \to t\Lambda$) non-perturbatively.
• Implementation:
  • The flow equations are solved for the holon self-energy $\Sigma_\Lambda(K)$ and the chemical potential counter-term.
  • The two-body interaction vertex is decomposed into superconducting (SC), magnetic (M), and charge (C) channels.
  • Crucially, the authors retain all three channels because the initial conditions for the magnetic and charge channels are singular and finite, meaning single-channel truncations would miss essential physics of the projected space.
  • Numerical solutions are performed on a grid of 190 $k$-points and 100 Matsubara frequencies. Analytical continuation to real frequencies is achieved using Padé approximants to compute spectral functions.

3. Key Contributions

1. Methodological Breakthrough: Demonstrates that X-FRG can successfully handle the $U=\infty$ limit, transcending the limitations of perturbative FRG and providing access to spectral functions in the extremely correlated regime.
2. Complete Phase Diagram: Maps the ground state phase diagram of the square lattice $t$-model as a function of electron density $n$.
3. Spectral Characterization: Provides the first detailed calculation of the single-particle spectral function $A(k, \omega)$ and magnetic spectral function in the Nagaoka ferromagnetic regime, identifying it as an incoherent non-Fermi liquid.
4. Discovery of Lifshitz Transition: Identifies a topological transition in the Fermi surface within the ferromagnetic phase, separating two distinct ferromagnetic regimes.

4. Key Results

A. Ground State Phase Diagram

As electron density $n$ increases from zero to unity, the system undergoes three distinct phases:

1. Low Density ($n \lesssim 0.48$): Paramagnetic Fermi Liquid. The system exhibits well-defined quasiparticles and a Fermi surface.
2. Intermediate Density ($0.48 \lesssim n \lesssim 0.59$): Antiferromagnetic Stripe Order. The ground state transitions to a state with stripe order (ordering vector $q=(\pi, 0)$). This phase separates the paramagnetic and ferromagnetic regimes, a finding consistent with variational wavefunction studies.
3. High Density ($n \gtrsim 0.6$): Extended Nagaoka Ferromagnet. The system enters a ferromagnetic phase with ordering vector $q=(0,0)$.

B. Spectral Properties and "Bad Metal" Behavior

In the high-density ferromagnetic regime, the single-particle spectral function exhibits striking features distinct from a standard Fermi liquid:

• Incoherent Flat Band: The spectrum develops a flat but broad band near the Fermi energy.
• Absence of Quasiparticles: The significant broadening indicates the absence of well-defined quasiparticles, characteristic of an incoherent non-Fermi liquid or "bad metal."
• Asymmetry: There is a marked asymmetry between particle-like and hole-like excitations. Holes can lower kinetic energy by binding a ferromagnetic bubble, forming a continuum of Nagaoka polarons. This leads to a weak momentum dependence in the spectrum.
• Spectral Weight Shift: A significant shift of spectral weight to negative energies is observed.

C. Lifshitz Transition and Two Ferromagnetic Regimes

The authors identify a Lifshitz transition at a critical density $n_{c,3} \approx 0.85$, dividing the ferromagnetic phase into two regimes (FM1 and FM2):

• FM1 ($n < 0.85$): The Fermi surface is particle-like. The electronic bandwidth $W$ is strongly renormalized but remains finite as $T \to 0$.
• FM2 ($n \geq 0.85$): The Fermi surface topology changes discontinuously to become hole-like, consisting of four disconnected arcs in the first Brillouin zone.
  • In this regime, the bandwidth $W$ collapses drastically ($W \to 0$) as $T \to 0$, indicating the emergence of a truly flat band.
  • This transition coincides with a qualitative change in the spin stiffness $\rho_s(n)$, which shows two distinct peaks, and a shift from a partially polarized to a fully polarized ferromagnet.

5. Significance

• Validation of Strong Correlation Theories: The results support the "fermion condensation" scenario proposed by Khodel and Shaginyan, demonstrating that extremely correlated fermions can indeed support flat bands and incoherent metallic states.
• Resolution of Nagaoka Stability: The study confirms the stability of the Nagaoka ferromagnet at high densities but clarifies that it is separated from the low-density paramagnet by a stripe-ordered phase, resolving ambiguities in previous numerical studies.
• Experimental Relevance: The calculated spectral line shapes and the identification of "bad metal" behavior provide concrete predictions for future experiments using ultracold atoms in optical lattices, which can now realize the infinite-$U$ Hubbard model. The detection of Nagaoka polarons and flat bands offers specific targets for spectroscopic measurements.
• Methodological Impact: The X-FRG approach establishes a new pathway for studying strongly correlated systems where Hilbert space projections are essential, potentially applicable to the $t-J$ model and other realistic materials.

In summary, this work provides a comprehensive, non-perturbative description of the infinite-$U$ Hubbard model, revealing a rich phase diagram dominated by stripe order at intermediate densities and a complex, incoherent ferromagnetic state at high densities, characterized by a Lifshitz transition and bad metal behavior.