Superconductivity in lightly doped Hubbard model on honeycomb lattice

arXiv:2303.12348 · cond-mat.str-el · Submitted 2023-03-22 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "Superconductivity in lightly doped Hubbard model on honeycomb lattice".

Mira: Here's the gist of the paper. The gist The ground state of lightly doped Hubbard model on long three- and four-leg cylinders suggests that it is consistent with a superconducting state with coexisting quasi-long-range superconducting and charge density wave orders, both decaying as a power law at long distances with corresponding exponents Ksc < 2 and Kc < 2 Model and Method The study utilizes large-scale density-matrix renormalization group (DMRG) simulations to investigate the ground state properties of the lightly doped single-band Hubbard model on the honeycomb lattice across long three- and four-leg cylinders.

Kai: The Hamiltonian considered is defined as H = - t X h i j i σ cˆ† i σ cˆj σ + h.c.

Mira: + U X i nˆi↑nˆi↓ (1).

Kai: The lattice geometry is cylindrical, with periodic boundary condition along the e2 = (1/2, √3/2) direction and open boundary condition along the e1 = (1, 0) direction.

Paper summary: Kai: So, to recap, this paper studies the lightly doped Hubbard model on long three- and four-leg cylinders using large-scale density-matrix renormalization group simulations. The main finding is that the ground state matches a superconducting state with coexisting quasi-long-range superconducting and charge density wave orders.

Mira: What's important about those orders is that they decay as power laws at long distances, specifically with exponents K sc and K c both being less than two <ref:2303.12348#pg1>.

Lev: So, the naive question a listener might have is why are these specific exponents less than two instead of just being exactly two <ref:2303.12348#pg1>?

Kai: Well, the paper finds that this power law decay for superconductivity implies a superconducting susceptibility that diverges as T approaches zero, specifically chi about T minus (two-K sc) > <ref:2303.12348#pg1>

Mira: That divergence is what makes it quasi-long-range rather than true long-range order. It's about the nature of the correlation itself.

Lev: For us on the ground, this means that if we were to try and build a real quantum computer using this physics, we'd have to deal with these algebraic decay rates when calculating things like susceptibility or correlation functions over large distances.

Kai: It matters because it connects the Hubbard model—which is believed to capture low-energy physics of cuprate superconductors

one–three: —to these specific types of emergent order <ref:2303.12348#pg1>.

Mira: And this result is interesting because the pairing symmetry they see is consistent with what you’d expect from a nematic d-wave, which reminds people of the plaquette d-wave seen on four-leg square cylinders.

Lev: But they point out that their specific setup, using only nearest-neighbor hopping terms for the Hubbard model, is different from studies on square or triangular lattices.

Kai: Exactly. So it’s this finding that quasi-long-range superconductivity can happen in the lightly doped Hubbard model with just nearest-neighbor electron hopping matrix.

Mira: And they are looking at this on geometries like three and four-leg cylinders, which is a specific way to probe these effects without needing infinite lattices.

Conclusion: Kai: So, looking at the full title, "Superconductivity in lightly doped Hubbard model on honeycomb lattice," it really sets up a picture of complex, competing orders coexisting.

Mira: The authors Cheng Peng, D. N. Sheng and Hong-Chen Jiang are using large-scale density-matrix renormalization group studies on long three- and four-leg cylinders to find this specific ground state structure.

Lev: From an engineering standpoint, the implication is that if we want to model real materials like cuprates, we can't just look for a simple phase transition; we have to account for these competing power-law decays in the superconducting and charge density wave sectors.

Kai: It means that when you tune parameters in the Hubbard model, especially doping delta, you are tuning not just whether superconductivity exists, but how it coexists with other structural orders like the charge density wave.

Mira: The main implication for me is that this confirms a specific type of emergent behavior in these strongly correlated systems where both pairing and charge modulation show algebraic decay at long scales.

Lev: For error correction research, this tells us that the low-energy physics we are modeling has these inherent correlation functions, and we have to design codes that respect those power-law decay rates instead of assuming simpler exponential behavior everywhere.

Kai: It's about realizing that the superconducting state in this model isn't just one thing; it has these other components—the CDW—that are intricately linked through these power-law interactions.

Mira: So, the paper shows a specific landscape for lightly doped Hubbard models where quasi-long-range superconductivity and charge density wave orders exist together under specific conditions.

Lev: It’s a solid piece of physics showing how complex the ground state can be even in this relatively simple effective model.

Stanford Institute for Materials and Energy Sciences · California State University

cond-mat.str-el

Submitted: 2023-03-22

Updated: 2023-03-22

Comments: 7 pages, 8 figures

Journal ref: Phys. Rev. B 111, 085108 (2025)

DOI: 10.1103/PhysRevB.111.085108

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 83/100

The gist: The gist The ground state of lightly doped Hubbard model on long three- and four-leg cylinders suggests that it is consistent with a superconducting state with coexisting quasi-long-range

Key concepts

Density-Matrix Renormalization Group (DMRG): This is a powerful computational method used to find the lowest energy state of quantum systems like the Hubbard model. It works by systematically reducing the size of the system while keeping track of its essential properties, allowing researchers to study complex ground states on large lattices.
Charge Density Wave (CDW): This refers to a periodic modulation in electron density across the lattice. In this study, it manifests as 'partially filled' charge stripes where holes are arranged in a specific pattern, with the stripe wavelength ($\lambda_c$) depending on the doping concentration.
Superconducting Correlations: These correlations describe how superconducting pairs are linked across different parts of the material. The study found quasi-long-range superconductivity characterized by a power-law decay exponent ($K_{sc} < 2$), suggesting the superconducting order is present but not truly long-range.
Luttinger Exponent (Kc, Ksc): These exponents quantify the rate at which correlations (like charge density or superconducting pairing) decay with distance. A power-law decay with an exponent less than two ($K < 2$) signifies quasi-long-range order, meaning the correlations persist over long distances but do not decay as fast as in a true ordered state.

Terminology

Hubbard Model — 2× in this paper · explained in 4 episodes

Summary

The gist

The ground state of lightly doped Hubbard model on long three- and four-leg cylinders suggests that it is consistent with a superconducting state with coexisting quasi-long-range superconducting and charge density wave orders, both decaying as a power law at long distances with corresponding exponents Ksc < 2 and Kc < 2

Model and Method

The study utilizes large-scale density-matrix renormalization group (DMRG) simulations to investigate the ground state properties of the lightly doped single-band Hubbard model on the honeycomb lattice across long three- and four-leg cylinders. The Hamiltonian considered is defined as H = - t X h i j i σ cˆ† i σ cˆj σ + h.c. + U X i nˆi↑nˆi↓ (1). The lattice geometry is cylindrical, with periodic boundary condition along the e2 = (1/2, √3/2) direction and open boundary condition along the e1 = (1, 0) direction. For three-leg cylinders Ly = 3, the total number of sites is N = Lx × Ly × 2 = 2Nu, where Nu denotes the number of unit cells. The doping concentration away from half-filling is defined as δ = Nh/(2Nu). The simulation parameters include setting t = 1 as an energy unit and considering U = 8 and U = 12. More than 100 sweeps are performed, keeping up to m = 40,000 in each DMRG block with a typical truncation error of ∼ 10−6.

Charge Density Wave Order

The charge density profile n(x) is calculated as n(x, y) = hnˆ(x, y)i on-site i = (x, y), and the rung average is n(x) = PLy / y=1 n(x, y)/Ly. The results indicate that the system forms “partially filled” charge stripes with two doped holes in each CDW unit cell (to sum the hole density of all legs). Specifically, the wavelength of charge stripes is λc = 1/3δ. For four-leg cylinders, the CDW wavelength is λc = 1/4δ. At long distances, the spatial decay of the CDW correlations is dominated by a power law with a Luttinger exponent Kc. Numerically, the exponent Kc can be obtained by fitting charge density oscillations induced by open boundaries of the cylinder.

Superconducting Correlations

The equal-time spin-singlet SC correlation is defined as Φαβ(r) = h∆† α(x0, y)∆β(x0 + r, y)i. The pairing symmetry of SC correlations is consistent with that of a nematic dwave, which is reminiscent of the plaquette d-wave of the lightly doped t-J and Hubbard models on four-leg square cylinders. At long distances, the dominant SC correlation Φ(r), e.g., Φbb(r) and Φcc(r), is characterized by a power law with an appropriate Luttinger exponent Ksc which is defined as Φ(r) ∼ r −Ksc. A slow decay of the SC correlation with an exponent Ksc < 2 implies a SC susceptibility that diverges as χ ∼ T −(2−Ksc) as the temperature T → 0.

Single-particle and Spin-spin Correlations

The long-distance behavior of the single-particle correlation function Gσ(r) is consistent with an exponential decay Gσ(r) ∼ e −r/ξG. Similarly, the spin-spin correlation function F(r) decays exponentially as F(r) ∼ e −r/ξs at long distances. This suggests a finite gap in both the spin and single-particle sectors, evidenced by short-range spin-spin and single-particle Green correlations.

Summary and Discussion

The ground state of the system is consistent with that of an SC state where both the SC and CDW orders coexist which decay as a power law at long distances with corresponding exponents Ksc < 2 and Kc < 2. The pairing symmetry is consistent with that of a d-wave, its manifestation on finite honeycomb cylinders is different from that on finite triangular cylinders because one component of the SC correlations can be orders of magnitude weaker than other components at long distances. The fact that the quasi-long-range superconductivity can be realized in the lightly doped Hubbard model on the honeycomb lattice with only nearest-neighbor electron hopping matrix is notably different from the doped Hubbard model on both square and triangular lattices. The study focuses on the lightly doped Hubbard model with only nearest-neighbor electron hopping term, and it will be interesting to study the higher doping case as well as the effect of longer range electron hopping terms.

Appendix Results

For U = 8 with t = 1 at 1/12 hole doping concentration on the three-leg cylinder with Lx = 32, the central charge is c ∼ 3, which indicates that the ground state deviates from the CDW/superconducting dominant state to become a Luttinger liquid. For four-leg systems with U = 12 at 1/18 hole doping, the results are displayed in Fig.A2. The density-density correlation function decays in power-law with the same Luttinger exponent Kc as the power. The fitted Kc is 1.46(4) for U = 12 at δ = 1/12 hole doping, and 1.57(6) for U = 8 at δ = 1/16 hole doping. The density-density correlation function Kc is bigger than that fitted from the CDW; however, it still satisfies Kc < 2.

Appendix C Results

The spin-triplet superconducting correlation function Φtriplet αβ (r) shown in Supplementary Figure A4 are much weaker which decay faster for the three-leg cylinder at δ = 1/12 doping and four-leg cylinder δ = 1/16 doping. The results show that the spin-triplet SC correlations are much weaker than the spin-singlet SC correlations, as shown in the SM. The triplet correlation function decays faster for both cylinder types.

Table A1 Summary

Table A1 lists the lattice parameters, Luttinger exponents (Kc, Ksc), and correlation lengths (ξs, ξG) in the unit of unit cell at different on-site repulsion U, with t = 1 as an energy unit, hole doping concentrations δ, and the lattice size N. The table summarizes the numerical results obtained across different model parameters.

Improvements for AI systems

Rewriting Material Science Simulations for Strongly Correlated Electron Systems: A specific improvement is developing a density-matrix renormalization group (DMRG) based simulation framework capable of accurately capturing the coexistence of quasi-long-range superconducting and charge density wave orders, as found in the ground state of the lightly doped Hubbard model on honeycomb lattices. This improved AI system can predict whether a given material configuration, modeled by a Hubbard Hamiltonian, will exhibit superconductivity or be dominated by competing phases like charge density waves by analyzing the decay exponents (Ksc < 2 and Kc < 2).
Enhanced Phase Characterization for High-Temperature Superconductors: The AI can perform systematic analysis of superconducting correlation functions, specifically identifying pairing symmetries consistent with a nematic dwave based on bond correlations like Φbb(r) ∼ Φcc(r) Φaa(r). This allows the system to distinguish between different microscopic models (e.g., those with or without next-nearest-neighbor hopping terms) by analyzing how the SC correlation decays in finite geometries.
Predictive Modeling of Electronic Structure and Transport: The improved AI can predict single-particle and spin correlations using the exponential decay forms, such as Gσ(r) ∼ e −r/ξG and F(r) ∼ e −r/ξs. This enables the system to estimate key physical parameters like correlation lengths (ξs, ξG) directly from simulation data, helping researchers quickly assess the presence of a finite gap in both spin and single-particle sectors.