Ground state phase diagram and superconductivity of the doped Hubbard model on six-leg square cylinders

arXiv:2303.15541 · cond-mat.str-el · Submitted 2023-03-27 · 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: Hi, I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: We're looking at the paper: "Ground state phase diagram and superconductivity of the doped Hubbard model on six-leg square cylinders".

Mira: Here's the gist of the paper. The gist: The ground state phase diagram of the doped Hubbard model on six-leg square cylinders is surprisingly sensitive to both next-nearest-neighbor electron hopping t0 and hole doping concentration δ, revealing distinct phases including d-wave superconductivity, charge density waves, and a novel holon Wigner crystal depending on the sign of t0 <ref:2303.15541#pg3>.

Kai: How it works The study employs density-matrix renormalization group (DMRG) to investigate the ground state properties of the lightly doped Hubbard model on square lattice defined by a specific Hamiltonian including nearest-neighbor (NN) and next-nearest-neighbor (NNN) electron hopping t0, and on six-leg square cylinders with periodic boundary conditions in one direction and open boundary conditions in the other <ref:2303.15541#pg3>.

Mira: The system is studied for hole doping concentrations of 5.55% ≤ δ ≤ 12.5% and next-nearest-neighbor hopping amplitudes of −0.4 ≤ t0/t ≤ 0.5 at U=12 <ref:2303.15541#pg3>.

Kai: The phase diagram is highly sensitive to the parameters, with the blue region around t0 ∼ 0 identified as an insulating phase characterized by a unidirectional “2/3-filled” charge stripes and mutually commensurate spin stripes, but short-range SC correlations <ref:2303.15541#pg3>.

Title and authors: Kai: So we're looking at the paper titled "Ground state phase diagram and superconductivity of the doped Hubbard model on six-leg square cylinders". Mira, what do you think about that title and who wrote it?

Mira: It sounds like it's mapping out how different physical states exist in this specific material setup. The Hubbard model is a fundamental thing in physics, but adding those "six-leg square cylinders" makes the geometry very particular.

Lev: From my side, I wonder what kind of fidelity we’re looking at when you try to simulate these long systems with DMRG. The structure of the problem dictates how complex the state space gets before you hit those computational limits.

Kai: Exactly, Lev, it's about building something concrete. These authors are Jiang and Devereaux and Hong-Chen Jiang, who are working in materials science and physics institutions at Stanford and ShanghaiTech University. They’re the ones doing the heavy lifting here with density-matrix renormalization group simulations on these systems.

Mira: Yeah, so when you see this title, you should think it's about finding a map of all the possible ground states for this doped Hubbard model on these specific shapes. It suggests they're using that geometry to probe things that might be hard to see in simpler setups.

Lev: And I mean, the implication is that whatever they find, it gives us a better picture of how strong correlations actually manifest in systems like cuprates or other strongly correlated materials.

The paper's summary: Kai: Okay, so the core of this study looks at how sensitive the ground state is to two things: next-nearest-neighbor electron hopping, which they call t zero and the hole doping concentration delta <ref:2303.15541#pg1,next-nearest-neighbor electron hopping>.

Mira: Right. And they find that this sensitivity is really key because it dictates what kind of physics you see emerging, like d-wave superconductivity or charge density waves. They’re looking at hole doping concentrations between five point five five percent and twelve point five percent on these six-leg cylinders with interaction strength U set to twelve <ref:2303.15541#pg1>.

Lev: So, they've mapped out a phase diagram where you can see distinct regions, and the blue area right around t zero equals zero is identified as an insulating phase characterized by unidirectional charge stripes and spin stripes <ref:2303.15541#pg1>.

Kai: That's the crucial part. It shows that even without a specific next-nearest-neighbor hopping term, you still get this stripe order with short-range superconducting correlations instead of something else entirely.

Mira: And then on the positive t zero side, they find a robust d-wave superconducting phase that has these charge density waves coexisting with it <ref:2303.15541#pg1,find a robust d-wave superconducting>. It’s interesting because that SC phase has a plaquette-type symmetry when electron doping occurs with positive t zero <ref:2303.15541#pg1>.

Lev: That plaquette symmetry detail is important for the experimentalists because it gives you a specific symmetry to look for in your measurements, which is something you can't get from just looking at the general d-wave correlation.

The paper's improvements: Kai: The authors point out that they’ve found several things that suggest ways to push the understanding further, especially concerning how we predict superconductivity. They suggest that in the electron-doped case with positive t zero long-range superconductivity might actually be possible in two dimensions <ref:2303.15541#pg1>.

Mira: That’s a big claim because most 2D models struggle with robust long-range pairing unless you have very specific conditions, so this opens up some thinking about what exactly suppresses that pairing in other systems <ref:2303.15541#pg1>.

Lev: From an error correction standpoint, if they show a path toward long-range SC in 2D under these conditions, it tells us the constraints on the required entanglement structure for a true superconducting state might be less severe than we currently assume <ref:2303.15541#pg1>.

Kai: And then they discuss the hole-doped case with negative t zero suggesting that long-range charge density wave order is present in six-leg cylinders even when t zero is negative <ref:2303.15541#pg1>. That connects back to those t-J models on both six and eight-leg cylinders with negative t zero <ref:2303.15541#pg1>.

Mira: They also highlight a novel insulating CDW phase, which they call the WC*, which appears at lower doping levels and larger negative t zero <ref:2303.15541#pg1>. This state is different from doped holes or hole pairs; it’s a Wigner crystal of holons.

Lev: That Wigner crystal of holons concept is interesting because it suggests a new way to understand charge ordering, moving away from just thinking about simple localized charge carriers.

Conclusion: Kai: So, to wrap up the "Ground state phase diagram and superconductivity of the doped Hubbard model on six-leg square cylinders," we see that the nature of the ground state is incredibly sensitive to t zero and delta <ref:2303.15541#pg1,Ground state phase diagram and superconductivity of the doped Hubbard model on>. The study clearly distinguishes between a robust d-wave SC phase with charge density waves in one regime, and an insulating stripe phase with long-range CDW order in another.

Mira: And they also found that this sensitivity leads to the discovery of a new insulating CDW phase called the WC*, which is a Wigner crystal of holons under certain conditions with negative t zero <ref:2303.15541#pg1>. This suggests doping can induce fractional phases that are structurally different from what we usually expect.

Lev: From an engineering viewpoint, this means that when we try to build hardware, whether it's for superconducting circuits or quantum simulators, we need to keep this phase diagram in mind because the ground state isn't just one thing; it depends on the parameters you tune.

Kai: It’s a reminder that even small changes in the model parameters, like adding t zero can fundamentally change what kind of correlations you observe in your system <ref:2303.15541#pg1>.

Mira: So, for anyone listening who isn't deep into this specific paper, it shows how complex the interplay between doping and hopping is when you're dealing with these specific lattice geometries.

Lev: And we’ll keep an eye on this work because understanding how these phases form could help us design better error correction codes or materials for stable superconducting devices.

School of Physical Science and Technology, ShanghaiTech University · Stanford Institute for Materials and Energy Sciences SLAC National Accelerator Laboratory Stanford University Department of Materials Science and Engineering Stanford University

cond-mat.str-el

Submitted: 2023-03-27

Updated: 2023-03-27

Comments: 8 pages, 6 figures

Journal ref: Phys. Rev. B 109, 085121 (2024)

DOI: 10.1103/PhysRevB.109.085121

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

Importance score: 83/100

The gist: The gist: The ground state phase diagram of the doped Hubbard model on six-leg square cylinders is surprisingly sensitive to both next-nearest-neighbor electron hopping t0 and hole doping

Key concepts

Doped Hubbard Model: This is a theoretical model used in physics to describe electrons moving on a lattice with interactions. 'Doped' means some holes (missing electrons) have been introduced into the material, which changes how the system behaves, potentially leading to superconductivity or other ordered states.
d-wave Superconductivity: This is a specific type of superconducting state where electron pairs form in a way that has a 'd-wave' symmetry. It is characterized by specific patterns in the superconducting correlations and is found when the next-nearest-neighbor hopping ($t_0$) is positive, indicating strong pairing tendencies.
Holon Wigner Crystal (WC*): This is a novel insulating phase where the charge carriers (holons) arrange themselves into a highly ordered, crystalline structure, similar to how classical ions form a crystal. It occurs under specific negative $t_0$ and doping levels and shows clear spin-charge separation signatures.
Spin-Charge Separation: This phenomenon describes a situation where the electron's properties can be separated into two independent entities: a 'holon' that carries the charge but no spin, and a 'spinon' that carries the spin but no charge. The paper shows this separation in the WC* phase through specific patterns in spin density profiles.

Terminology

Hubbard Model — 3× in this paper · explained in 4 episodes
Charge Density Wave (CDW) — 1× in this paper · explained in 3 episodes

Summary

The gist: The ground state phase diagram of the doped Hubbard model on six-leg square cylinders is surprisingly sensitive to both next-nearest-neighbor electron hopping t0 and hole doping concentration δ, revealing distinct phases including d-wave superconductivity, charge density waves, and a novel holon Wigner crystal depending on the sign of t0 <ref:2303.15541#pg3>.

How it works

The study employs density-matrix renormalization group (DMRG) to investigate the ground state properties of the lightly doped Hubbard model on square lattice defined by a specific Hamiltonian including nearest-neighbor (NN) and next-nearest-neighbor (NNN) electron hopping t0, and on six-leg square cylinders with periodic boundary conditions in one direction and open boundary conditions in the other <ref:2303.15541#pg3>. The system is studied for hole doping concentrations of 5.55% ≤ δ ≤ 12.5% and next-nearest-neighbor hopping amplitudes of −0.4 ≤ t0/t ≤ 0.5 at U=12 <ref:2303.15541#pg3>.

The phase diagram is highly sensitive to the parameters, with the blue region around t0 ∼ 0 identified as an insulating phase characterized by a unidirectional “2/3-filled” charge stripes and mutually commensurate spin stripes, but short-range SC correlations <ref:2303.15541#pg3>.

Key Findings in Different Regimes

The nature of the ground state is remarkably sensitive to the presence of next-nearest-neighbor electron hopping t0 <ref:2303.15541#pg2>. On the positive t0 side, a robust d-wave superconducting (SC) phase is found, which is characterized by coexisting quasi-long-range SC and charge density wave (CDW) correlations <ref:2303.15541#pg3>. This SC phase exhibits a plaquette-type d-wave symmetry when electron doping occurs with positive t0 <ref:2303.15541#pg3>.

In contrast, without t0, the ground state forms an insulating stripe phase with long-range CDW order but short-range spin-spin and SC correlations <ref:2303.15541#pg3>. When t0 is negative, the doped Hubbard model on six-leg cylinders in the hole-doped case appears insulating with strong CDW order, where other correlations like SC and spin-spin correlations are short-ranged with fairly short correlation lengths <ref:2303.15541#pg3>.

Characterization of Phases

The paper identifies several distinct phases on the ground state phase diagram <ref:2303.15541#pg3>:

- d-wave SC phase:

This phase is found in the electron-doped side with positive t0 and is characterized by coexisting quasi-long-range SC and CDW orders, but short-range spin-spin correlations <ref:2303.15541#pg3>. Superconducting correlations decay as a power law with a Luttinger exponent Ksc < 2, establishing quasi-long-range SC for the lightly doped Hubbard model on six-leg cylinders with positive t0 <ref:2303.15541#pg4>.

- CDW-2/3 phase:

This insulating CDW phase appears in the blue region of the phase diagram, exhibiting strong unidirectional charge stripe order but short-range SC correlations <ref:2303.15541#pg5>. The spatial decay of charge density oscillations is nearly invisible for these lengths of ladders, consistent with long-range charge stripe order where Kc is close to 0 <ref:2303.15541#pg5>.

- Holon Wigner Crystal (WC

This second CDW phase appears in the left-bottom corner of the phase diagram at relatively lower doping levels and larger negative t0, and it is similar with the one reported for doped Kagome lattice spin liquids <ref:2303.15541#pg5>. This state is consistent with a Wigner crystal of holons, where the holon carries the charge of an electron but without its spin <ref:2303.15541#pg5>. The spin-charge separation is evident in the modified spin density profile, which shows clear antiphase domain-walls only in the x direction and maxima of |S z x,y| appear exactly at the minima of the hole density profile <ref:2303.15541#pg6>.

Summary and Discussion

The study concludes that in the electron-doped case with positive t0, long-range SC could also be realized in two dimensions <ref:2303.15541#pg7>. In the hole-doped case with negative t0, the Hubbard model on six-leg cylinders possesses long-range CDW order, which is consistent with holedoped t-J models on both six and eight-leg square cylinders with negative t0 <ref:2303.15541#pg7>. Furthermore, a novel insulating CDW phase, the WC∗, appears at relatively lower doping level and larger negative t0, suggesting a doping induced fractional phase similar to those reported for doped Kagome lattice spin liquids <ref:2303.15541#pg7>. This novel WC∗ phase is a Wigner crystal of holons rather than doped holes or hole pairs <ref:2303.15541#pg7>. The findings suggest that the mechanism of high temperature superconductivity warrants further investigation, specifically whether phase separation can be suppressed by introducing additional terms like further-neighbor electron hopping and extended electron interaction <ref:2303.15541#pg7>.

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Superconducting pair-pair correlations in the d-wave SC phase

The pairing symmetry of SC correlations is consistent with 2D d-wave as demonstrated in Fig.S1(a) with Φxx(r) ∼ Φyy(r) ∼ −Φxy(r) <ref:2303.15541#pg8>.

Superconducting correlations decay exponentially as Φyy(r) ∼ e − r/ξsc with a short correlation length ξsc ≈ 2 <ref:2303.15541#pg8>. The pairing symmetry is consistent with that of a d-wave form evidenced by Φyy(r) ∼ −Φxy(r) ∼ Φxx(r) <ref:2303.15541#pg8>.

The spin-spin correlation length is ξs ≈ 3 which is shorter than the separation between two adjacent holons in the x direction <ref:2303.15541#pg8>. This is a clear signature of spin-charge separation which is consistent with a holon Wigner crystal <ref:2303.15541#pg8>.

The superconducting correlation length is ξsc ≈ 2 <ref:2303.15541#pg8>.

--- Page 8 ---

Results for the holon Wigner crystal phase

Fig.S2 shows the density profile n(x, y) and modified spin density profile (−1)x+y S z x,y of the U = 8 doped Hubbard model with t0 = −0.4 at δ = 1/18 on N = 48 × 6 cylinder with m = 16000 states. It is clear that n(x, y) breakstranslation symmetries along both x and y directions, and the number of the emergent CDW unit cells is equal to the number of doped holes.

The spin density profile shows a clear signature of spin-charge separation as discussed in the main text. Consistent with that of a WC∗, we find that spinspin correlations decay exponentially as F(r) ∼ e − r/ξs as shown in Fig.S2(b). The spin-spin correlation length is ξs ≈ 3 which is shorter than the separation between two adjacent holons in the x direction.

Similar density profiles have been observed in the WC∗ phase including the sets of parameters (t0, δ) = (−0.4, 1/12), (−0.5, 1/18) and (−0.25, 1/18).

--- Page 9 ---

The entire charge density profile appears to prefer a rectangular lattice with an emergent larger unit cell each containing one of the blue spots. The number of these emergent unit cells is equal to the number of doped holes.

Improvements for AI systems

Improved Phase Transition Prediction: AI systems can be improved to accurately predict whether a system will exhibit a robust d-wave superconducting (SC) phase based on parameters like next-nearest-neighbor electron hopping t0, as the paper shows that the nature of the ground state is remarkably sensitive to the presence of next-nearest-neighbor electron hopping t0.
Holon Crystal Characterization: AI models can be enhanced to distinguish between different insulating phases by analyzing charge density profiles; specifically, they can identify when a system forms a novel holon Wigner crystal (WC∗) characterized by the observation that the number of these emergent unit cells is equal to the number of doped holes.
Spin-Charge Separation Detection: AI systems can be trained to detect signatures of spin-charge separation in the WC∗ phase by analyzing spin density profiles, specifically looking for clear signature of spin-charge separation as discussed in the main text evidenced by the maxima of S z(x,y) appear exactly at the minima of the hole density profile.
Correlation Length Extrapolation: AI can perform automated extrapolation to zero truncation error for correlation functions, mimicking the DMRG technique described in the paper where they use a second order polynomial Φ = Φ0 + a1 + a2 to extract physical properties from finite-size simulations.