Effects of next-nearest neighbor hopping on the pairing and critical temperatures of the attractive Hubbard model on a square lattice

Using sign-problem-free determinant quantum Monte Carlo simulations, this study demonstrates that introducing next-nearest-neighbor hopping in the attractive Hubbard model on a square lattice can significantly enhance the critical temperature by up to 50% while simultaneously reducing the pseudogap region, offering a viable route to achieve experimentally accessible superconducting temperatures.

The Gist

Imagine a crowded dance floor where pairs of dancers (electrons) are trying to hold hands and waltz together. This is the essence of superconductivity: a state where electricity flows without any resistance because the electrons move in perfect, coordinated pairs.

Scientists have been trying to get these "dance floors" (materials) to work at higher temperatures, but so far, they need to be cooled down to near absolute zero, which is expensive and difficult. This paper asks a simple question: Can we change the layout of the dance floor to make the pairs dance better and stay together at warmer temperatures?

Here is the breakdown of their findings using everyday analogies:

1. The Setup: The Square Dance Floor

The researchers are studying a specific model called the Attractive Hubbard Model.

• The Dancers: Electrons.
• The Floor: A grid of squares (a lattice).
• The Rules: Electrons usually only hold hands with the person standing immediately next to them (North, South, East, West). This is called "nearest-neighbor hopping."
• The Problem: Even with these rules, the dance floor gets chaotic at higher temperatures, and the pairs break apart. The "Critical Temperature" ($T_c$) is the point where the dance floor freezes into a perfect, resistance-free state. Currently, this temperature is too low for practical use.

2. The New Idea: Adding Diagonal Paths

The authors propose a change to the dance floor. Imagine adding diagonal pathways so that dancers can also move to the corners of the square, not just the sides. In physics terms, this is Next-Nearest Neighbor (NNN) hopping (represented by $t'$).

Think of it like a city grid.

• Old Way: You can only walk North, South, East, or West. If a street is blocked, you are stuck.
• New Way: You can also cut diagonally across the blocks. If one path is crowded, you have a shortcut.

3. The Results: A 50% Temperature Boost

When the researchers simulated this new "diagonal" dance floor using powerful computers, they found something amazing:

• The Sweet Spot: By adjusting the strength of these diagonal paths, they could increase the temperature at which the electrons pair up by up to 50%.
• The Analogy: Imagine you have a group of people trying to form a line in a cold wind. If they can only move in a straight line, they might freeze and break the line. But if they can also move diagonally to huddle together more efficiently, they can stay warm and keep the line intact at a much higher temperature.

4. The Twist: Breaking the "Perfect Symmetry"

In the old model (without diagonals), there was a weird rule at exactly half-filling (when the dance floor is half-full). The electrons were so indecisive that they couldn't choose between forming pairs or forming a static pattern (like a checkerboard). This indecision killed the superconductivity, making the critical temperature zero.

Adding the diagonal paths broke this indecision. It forced the electrons to choose the "dance partner" option over the "static pattern" option.

• Result: Even on a half-full floor, superconductivity could now happen at a finite temperature. It's like adding a DJ who plays a specific beat that forces everyone to dance rather than stand still.

5. The Trade-off: Pre-formed Pairs vs. The Real Dance

The paper also looked at something called the "Pairing Temperature" ($T_p$).

• $T_p$ (The Pre-formed Pairs): This is the temperature where electrons start holding hands but haven't started dancing in sync yet. They are just "hanging out" as pairs. In many high-temperature superconductors, this happens way above the actual superconducting temperature.
• The Finding: As they added more diagonal paths, the "hanging out" phase ($T_p$) got shorter.
• The Metaphor: Imagine a wedding.
  • Old Model: The couple gets engaged (pairs form) a year before the wedding, but they don't actually get married (superconductivity) until the day of. There's a long, awkward waiting period.
  • New Model (with diagonals): The couple gets engaged and gets married almost at the same time. The "waiting period" disappears.
  • Why is this good? It means the system behaves more like a "classic" superconductor (BCS theory), which is more predictable and stable. The electrons don't waste energy waiting; they condense into the superconducting state more efficiently.

6. The "Flat Band" Effect

The researchers also noticed that adding these diagonal paths made the "energy landscape" flatter.

• Analogy: Imagine a roller coaster.
  • Steep Hills: Electrons zoom past quickly and don't stick together.
  • Flat Tracks: Electrons slow down and spend more time in the same spot, making it much easier for them to find a partner and stick together.
  • The diagonal paths created these "flat tracks," enhancing the ability of electrons to pair up.

The Bottom Line

This paper suggests that by simply tweaking the geometry of how electrons move (adding diagonal shortcuts), we can significantly boost the temperature at which superconductivity occurs.

Why does this matter?
If we can build materials or optical lattices (using lasers to trap atoms) that mimic this "diagonal hopping," we might be able to create superconductors that work at temperatures achievable with standard cooling, rather than requiring extreme, expensive cryogenics. It's a roadmap for turning a theoretical physics trick into a real-world energy revolution.

Technical Summary

Here is a detailed technical summary of the paper "Effects of next-nearest neighbor hopping on the pairing and critical temperatures of the attractive Hubbard model on a square lattice."

1. Problem Statement

The attractive Hubbard model (AHM) is a paradigmatic model for studying superconductivity and superfluidity, particularly in the context of ultracold atom experiments on optical lattices. While recent experiments have achieved unprecedentedly low temperatures ($T \sim 0.05t/k_B$), the critical temperatures ($T_c$) predicted by theory for the standard 2D square lattice (with only nearest-neighbor hopping, $t$) remain lower than the experimentally accessible range in many regimes. Specifically, for the AHM, the maximum $T_c$ is approximately $0.15t$at a filling of$\langle n \rangle \approx 0.87$.

The authors investigate whether introducing next-nearest-neighbor (NNN) hopping ($t'$), allowing fermions to move along the diagonals of the square lattice, can enhance $T_c$. This mechanism is motivated by:

• Bandwidth and Density of States (DOS): According to BCS theory ($T_c \propto \exp[-1/D(0)|U|]$), increasing the DOS at the Fermi level ($D(0)$) should increase $T_c$. NNN hopping can flatten bands or shift the Fermi level relative to van Hove singularities, potentially increasing $D(0)$.
• Symmetry Breaking: In the standard model ($t'=0$) at half-filling, charge density wave (CDW) and superconducting (SS) orders are degenerate, leading to $T_c=0$ due to the Mermin-Wagner theorem. Introducing $t'$ breaks this pseudo-spin SU(2) symmetry, potentially allowing a finite-temperature transition even at half-filling.
• Experimental Feasibility: NNN hopping can be engineered in optical lattices by adding specific laser beam configurations (triangular lattice geometries).

2. Methodology

The authors employ Determinant Quantum Monte Carlo (DQMC) simulations, a sign-problem-free numerical method suitable for the attractive Hubbard model.

• Model: The Hamiltonian includes nearest-neighbor hopping ($t$), next-nearest-neighbor hopping ($t'$), chemical potential ($\mu$), and on-site attractive interaction ($-|U|$).
• Simulation Parameters:
  • Lattice sizes: $L \times L$ with $L \leq 16$.
  • Temperatures: Sufficiently low to probe phase transitions.
  • Interaction strengths: $|U|/t$ ranging from 3.0 to 5.0.
  • Fillings: Half-filling ($\langle n \rangle = 1.0$) and doped cases ($\langle n \rangle = 0.87, 0.5$).
• Observables:
  • Critical Temperature ($T_c$): Determined via the superfluid density ($\rho_s$) using the universal jump relation for the Berezinskii-Kosterlitz-Thouless (BKT) transition: $T_c = \frac{\pi}{2} \rho_s(T_c)$. The intersection of the $\rho_s(T)$ curve with the line $2T/\pi$yields$T_c$.
  • Pairing Temperature ($T_p$): Identified by the downturn in the uniform spin susceptibility ($\chi_s$), signaling the formation of preformed pairs (pseudogap regime) above $T_c$.
  • Structure Factors: $s$-wave pairing ($P_s$) and charge density wave ($P_{CDW}$) to analyze competing orders.
  • Density of States (DOS): Calculated via the maximum-entropy method applied to the imaginary-time Green's function to distinguish between pseudogap and fully gapped superconducting states.

3. Key Contributions and Results

A. Enhancement of Critical Temperature ($T_c$)

• Significant Increase: The introduction of NNN hopping ($t'$) can increase $T_c$ by up to 50% compared to the $t'=0$ case.
• Optimal Parameters: The maximum enhancement occurs at a ratio of $|t'/t| \approx 1.2$.
  • For $\langle n \rangle = 0.87$ and $|U|/t = 5.0$, $T_c$ rises from $\approx 0.16t$ ($t'=0$) to $\approx 0.25t$ ($t'=-1.2t$).
  • Similar enhancements are observed at quarter-filling ($\langle n \rangle = 0.5$).
• Mechanism: The enhancement is linked to the evolution of the Fermi surface and DOS. As $|t'|$ increases, the Fermi surface transitions from electron-like to hole-like. For specific ranges ($0.7 \lesssim |t'/t| \leq 1.1$), the Fermi level aligns with a van Hove singularity, maximizing$D(0)$and thus$T_c$.

B. Finite $T_c$ at Half-Filling

• Symmetry Breaking: At half-filling ($\langle n \rangle = 1$), the standard model ($t'=0$) has $T_c=0$ due to the degeneracy between CDW and SS orders.
• Emergence of Order: Introducing $t' \neq 0$ breaks the pseudo-spin SU(2) symmetry, frustrating CDW order and favoring superconductivity.
• Result: A finite-temperature phase transition emerges at half-filling. Remarkably, for large $|t'|$, the $T_c$ at half-filling becomes the highest among all fillings studied, surpassing even the doped cases.

C. Suppression of the Pseudogap ($T_p$)

• Opposite Trend: While $T_c$ increases with $|t'|$, the pairing temperature $T_p$ (the scale for preformed pairs) decreases.
• Implication: The NNN hopping channel hinders the formation of local pairs (likely because unpaired fermions also gain more paths to move) but favors their condensation into a coherent superconducting state.
• BCS Limit: The convergence of $T_p$ and $T_c$ at large $|t'|$ suggests a transition toward a more BCS-like behavior, where pairing and condensation occur simultaneously, even at intermediate coupling strengths where a pseudogap usually dominates.

D. Evolution of the Density of States

• Pseudogap to Gap: Analysis of the interacting DOS ($D(\omega)$) confirms the transition from a pseudogap regime (suppressed states at the Fermi level but not zero) to a fully gapped superconducting state.
• Metric for $T_c$: The authors introduce $n_0$ (fermion density within a narrow energy window around the Fermi level). $n_0$ serves as a quantitative measure of the distance from $T_c$: it vanishes only when $T < T_c$ (full gap opening) and remains finite in the pseudogap regime.

4. Significance

• Experimental Relevance: The findings provide a concrete theoretical route to achieve higher critical temperatures in ultracold atom experiments. By tuning NNN hopping (via laser geometry), experimentalists could potentially reach $T_c$ values ($\sim 0.25t$) that are more accessible with current cryogenic optical lattice technologies.
• Theoretical Insight: The work clarifies the interplay between flat bands, DOS enhancement, and symmetry breaking in 2D superconductors. It demonstrates that introducing geometric frustration (via $t'$) can decouple the pairing scale from the condensation scale, potentially optimizing the superconducting state.
• Material Analogues: The results are relevant for understanding unconventional superconductivity in materials with complex hopping geometries, such as twisted bilayer graphene, where flat bands and enhanced DOS play a crucial role.

In conclusion, the paper establishes that next-nearest-neighbor hopping is a viable and powerful mechanism to significantly boost the critical temperature of the attractive Hubbard model, while simultaneously reducing the pseudogap region, thereby offering a promising pathway for experimental realization of high-$T_c$ superfluidity in optical lattices.