Superconductivity enhancement and particle-hole asymmetry: interplay with electron attraction in doped Hubbard model

arXiv:2402.11255 · cond-mat.str-el, cond-mat.supr-con · Submitted 2024-02-17 · 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 enhancement and particle-hole asymmetry".

Mira: Here's the gist of the paper. The gist NN electron attraction substantially bolsters superconductivity while concurrently mitigating CDW stripe order in the electron-doped (t′ > 0) region of the model where we find that a moderate V is sufficient to drive the system into a pronounced SC phase with dominant SC correlations<ref:2402.11255#pg2>.

Kai: Model and Method The study employs the density-matrix renormalization group to examine its effect in the lightly doped t-t′ - Hubbard model on six-leg square cylinders<ref:2402.11255#pg4>.

Mira: The Hamiltonian used is defined as H = − X ijσ t ij cˆ† iσcˆjσ + h.c.

Kai: + U X i nˆi↑nˆi↓ − V X⟨ij⟩ nˆinˆj<ref:2402.11255#pg4>.

Title and authors: Mira: So we're looking at this paper, "Superconductivity enhancement and particle-hole asymmetry: interplay with electron attraction in doped Hubbard model," and it really zeroes in on how adding that nearest-neighbor electron attraction, V, changes things when you have electron doping.

Kai: Right. It seems the main point here is that this attraction actually boosts superconductivity while at the same time it fights against those charge density wave stripes we usually see in these systems.

Lev: So what we're looking at is specifically in the electron-doped region, where t prime is positive, and they find that if you push V past a certain point, say Vc about zero point seven times t, you get a clear superconducting phase with strong SC correlations dominating everything <ref:2402.11255#pg1>.

Mira: That's interesting because earlier studies on similar models showed both superconductivity and CDW stripes existing together, but here the attraction seems to favor the superconducting side when it gets strong enough.

Kai: Exactly, and they show that this enhancement happens even up to a value of V = one point zero, which is pretty high for such an interaction strength in this setup <ref:2402.11255#pg1>.

Lev: What that means for us on real hardware is that if we were trying to engineer a system with these superconducting correlations, we'd need an attractive interaction strength near zero point seven times t to see those effects on the ground state calculations they’re doing here <ref:2402.11255#pg1>.

Mira: And it’s worth pointing out that in contrast, when they look at the hole-doped case, where t prime is negative, even with this attraction V up to one point zero, the system stays insulating and still has those charge density wave orders present <ref:2402.11255#pg1>.

Kai: So it's really an asymmetry in how this interaction plays out depending on whether you have electron doping or hole doping, which is a key finding here.

Lev: And from an engineering standpoint, if we were building a device based on this model, we need to be careful because the physics behaves so differently based on that sign of t prime.

Mira: Now they also point out some things they think are important improvements for the paper itself, focusing on how to make those results more robust or applicable.

Kai: They suggest looking at how the system handles longer cylinders, and they check their pair-pair correlation functions on longer cylinders to make sure the power-law decay holds up over longer distances.

Lev: That makes sense because in real simulations, you always have to worry about finite size effects, so checking that power-law behavior across different cylinder widths is a necessary step before trusting the long-distance correlations.

Mira: They also seem to emphasize that the d-wave pairing symmetry they saw earlier actually stays prevalent even when this electron attraction V is strong, up to V = one point zero, which supports the idea that it doesn't destroy the fundamental superconducting nature of the state <ref:2402.11255#pg1>.

Title and authors: Kai: That's a bit reassuring, knowing that even with a strong interaction like V = one point zero, the underlying pairing symmetry remains d-wave in this setup according to their analysis <ref:2402.11255#pg1>.

Lev: So for us building error correction codes or any kind of topological system, that stability under strong attraction is helpful information to have when thinking about parameter ranges.

Mira: And they also mention that the single particle gap stays small and largely unaffected by V across the entire range from zero point zero up to one point zero, which suggests the overall electronic band structure doesn't drastically change with this attraction <ref:2402.11255#pg2>.

Kai: So to sum up, this paper on "Superconductivity enhancement and particle-hole asymmetry: interplay with electron attraction in doped Hubbard model" shows that for electron doping, moderate attraction drives a pronounced SC phase by suppressing the CDW order, and they have suggestions on how to make the results more reliable by checking longer geometries.

Lev: And for me, it just confirms that the required interaction strength Vc about zero point seven times t is a specific threshold we'd need to tune into if we were trying to realize this superconducting behavior in a physical system <ref:2402.11255#pg1>.

Mira: It’s interesting how they handle the distinction between the electron-doped and hole-doped regimes, showing that while both have SC correlations boosted by V, the outcome is very different regarding CDW stripes.

Kai: This asymmetry is what makes this paper compelling; it tells us exactly where in parameter space we should look for superconductivity versus charge ordering.

Lev: So the real value here is identifying that critical point Vc about zero point seven times t, because that's where the physics switches from one kind of order to another in the electron-doped side <ref:2402.11255#pg1>.

Mira: Exactly, and it sets a benchmark for how we might approach these correlated models in future theoretical work.

Kai: So this paper on "Superconductivity enhancement and particle-hole asymmetry: interplay with electron attraction in doped Hubbard model" gives us a clear picture of when attraction helps superconductivity and when it just enhances density wave order.

Lev: And for the next step, I think we need to look at how those specific numerical results translate into the actual complexity of running these simulations on real quantum hardware.

Mira: Definitely, because theoretically, this shows a clear mechanism—attraction fighting CDW—that we can use as a guide when designing experiments or looking at different material parameters.

Kai: So that's what we've got for this one; the paper on "Superconductivity enhancement and particle-hole asymmetry: interplay with electron attraction in doped Hubbard model" tells us about the role of nearest-neighbor interaction in driving superconductivity in electron-doped systems.

The paper's summary: Kai: So, to recap, this paper looks at how adding nearest-neighbor electron attraction changes the game when you have an electron-doped Hubbard model, and it finds that this attraction actually helps superconductivity while simultaneously messing with those charge density wave stripes we usually see.

Mira: Right. The core of it is that in the electron-doped side, where t prime is positive, a moderate level of this electron attraction V pushes the system into a real superconducting phase where those SC correlations are dominant over everything else, including the CDW stripe order.

Lev: So what we're hearing here is that this interaction isn't just adding noise; it has a structural effect on the ground state itself, shifting it away from an insulating charge ordered state toward a superconducting one.

Kai: And they pinpoint this switch happening around V equals zero point seven times t, which is a specific value for the attraction strength where superconductivity takes over as the main feature.

Mira: That threshold is key because it shows that you can tune the system into a superconducting state by controlling this interaction strength, which is something we've been trying to model in other contexts.

Lev: If we were trying to build a real material, that V equals zero point seven times t becomes our target parameter for tuning the electron-electron interactions.

Kai: It’s also interesting how they show that this enhancement of SC correlations keeps going even when the attraction V is pushed up to one point ten.

Mira: That's because even at strong attraction, you still maintain that d-wave pairing symmetry throughout the range from zero up to one point ten, which is a pretty stable feature for the superconducting part of the physics.

Lev: So from an engineering standpoint, that stability in the pairing symmetry is good news for any attempt to design a robust superconducting material using these interaction models.

Kai: Now, they also contrast this with the hole-doped side—where t prime is negative—showing that while you still get boosted SC correlations when V increases, the system stays insulating and shows very clear charge density wave order.

Mira: That's a big difference because it highlights that this electron attraction effect isn't universal; it depends heavily on whether you have electron doping or hole doping in your model.

Lev: So what this tells us is that we can’t just apply one interaction strategy to all doped systems; the physics of the interaction has to be considered relative to the sign of t prime.

Kai: It basically sets a benchmark for how we should approach tuning these models based on their doping regime.

Mira: And before we go deeper into those specific numbers, Lev, what does that critical Vc value actually mean in terms of the complexity of running these simulations?

Lev: Well, hitting that threshold means you've found the sweet spot where the SC correlations start to dominate, so for us on hardware, it means we need to ensure our error correction overhead can handle those strong superconducting correlations without losing fidelity.

Kai: That makes sense. It’s not just about finding the phase; it’s about being able to measure that phase reliably when you push the interaction up that far.

Mira: Exactly, and this asymmetry in how V affects the phases is a huge piece of context for understanding how these materials might behave in real life.

Lev: So we've seen where the theoretical boundary lies, now we need to figure out what kind of physical measurement would actually confirm if we are seeing that zero point seven times t behavior on a real experimental system.

The paper's improvements: Kai: So, to wrap up on that last bit, this paper isn't just stopping there; they actually lay out some stuff they think would help future researchers improve their own models and simulations of these systems.

Mira: They suggest focusing more on the longer cylinders, which is a practical thing because you can't trust short-distance calculations if you want to know about long-range physics.

Lev: Yeah, that makes sense from an error correction viewpoint; if you want to run a real simulation, you need to know how those correlations decay over larger distances so the overhead doesn't blow up.

Kai: They also point out checking the pair-pair correlation functions on longer cylinders to ensure that power-law decay keeps holding true across different cylinder widths.

Mira: That addresses one of the biggest assumptions they make when predicting long-range behavior in these systems, which is how they're going to check those results against reality.

Lev: It’s good that they are being rigorous about checking those finite size effects, because if you get a false signal on a small cylinder, it’s just bad data for any hardware implementation.

Kai: And they emphasize the importance of looking at how the attraction V affects other phases besides the superconducting one, like the charge stripe phase.

Mira: Right, because while this attraction boosts SC correlations there too, they make sure to state clearly that this enhancement doesn't actually transition that system into a full superconducting phase.

Lev: It’s important for us to know where the limits are; you can boost one thing without achieving the main goal of getting superconductivity on its own.

Kai: So the paper is essentially saying, "Here's how we found it, but here's what you need to do next to make it more trustworthy."

Mira: And that focus on testing those long-distance correlations and checking the other phases really shows they are thinking about applying this to actual materials science experiments.

Lev: If we can get those simulation results validated with longer geometries, then we can start talking about how this interaction strength V equals zero point seven times t translates into something tangible for designing superconducting circuits.

Conclusion: Tom: So we're wrapping up on this paper titled "Superconductivity enhancement and particle-hole asymmetry: interplay with electron attraction in doped Hubbard model," which is really about how that electron attraction, V, changes things when you have electron doping.

Kai: Basically, they found that in the electron-doped regime, a moderate V pushes the system into a clear superconducting phase by actively suppressing those charge density wave stripes we see otherwise.

Mira: And they established this critical point where SC correlations start to dominate over CDW order at around V equals zero point seven times t, which is a really specific number for tuning interactions.

Lev: From an error correction standpoint, that means if we were building hardware based on this model, we'd need to make sure our system can handle those strong superconducting correlations without losing fidelity during the measurements.

Kai: Right, so it’s not just a theoretical finding; it tells us exactly what kind of interaction strength to aim for in a physical material.

Mira: It also highlighted that while this attraction boosts superconductivity even in the charge stripe phase on the hole-doped side, it doesn't actually cause a transition into a full superconducting state there.

Lev: That asymmetry is crucial because it shows how sensitive the system is to whether you have electron doping or hole doping, which tells us we can’t use one single interaction strategy for everything.

Kai: So this paper gives us a clear roadmap for where to look in the parameter space when trying to find high-temperature superconductivity in these Hubbard models.

Mira: It sets a benchmark for how we can model and predict phase transitions based on electron-electron interactions, which is what the theory needs to be grounded in.

Lev: If we can use this threshold Vc about zero point seven times t, it gives us a concrete target to aim for when designing experiments or looking at different material parameters.

Kai: It’s interesting how they handle the distinction between the electron-doped and hole-doped regimes, showing that while both have SC correlations boosted by V, the outcome is very different regarding CDW stripes.

Mira: That asymmetry is what makes this paper compelling; it tells us exactly where in parameter space we should look for superconductivity versus charge ordering.

Kai: I think this whole study on "Superconductivity enhancement and particle-hole asymmetry: interplay with electron attraction in doped Hubbard model" is a good example of how subtle interactions can have massive effects on the final state.

Mira: It's a solid piece of work that gives us concrete numbers to test against our broader theories about correlated electron systems.

Lev: I'm looking forward to seeing what others build on this threshold Vc value when they try to map it onto actual quantum hardware architectures.

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

cond-mat.str-el, cond-mat.supr-con

Submitted: 2024-02-17

Updated: 2024-02-17

Comments: 4 pages, 4 figures

Journal ref: SciPost Phys. 19, 137 (2025)

DOI: 10.21468/SciPostPhys.19.5.137

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

Importance score: 75/100

The gist: The gist NN electron attraction substantially bolsters superconductivity while concurrently mitigating CDW stripe order in the electron-doped (t′ > 0) region of the model where we find that a

Key concepts

Electron Attraction (V): This is an added interaction term in the model that represents an attractive force between electrons. It's introduced into the Hamiltonian and has a significant, asymmetric effect depending on whether the system has more holes or electrons (electron-doped vs. hole-doped).
Charge Density Wave (CDW): This refers to a specific ordering where electron density in the material forms periodic stripes. The paper observes that adding attraction can weaken this CDW order, especially in electron-doped systems, suggesting that superconductivity can compete with or suppress stripe formation.
Superconducting Correlation (Ksc): This measures how strongly Cooper pairs are correlated across the material. It is quantified by a Luttinger exponent (Ksc). The study shows that increasing the attractive interaction V consistently decreases Ksc, indicating that stronger attraction enhances the tendency toward superconductivity as temperature approaches zero.

Terminology

Hubbard Model — 3× in this paper · explained in 4 episodes
Particle-hole asymmetry — 1× in this paper · explained in 2 episodes

Summary

The gist

NN electron attraction substantially bolsters superconductivity while concurrently mitigating CDW stripe order in the electron-doped (t′ > 0) region of the model where we find that a moderate V is sufficient to drive the system into a pronounced SC phase with dominant SC correlations<ref:2402.11255#pg2>.

Model and Method

The study employs the density-matrix renormalization group to examine its effect in the lightly doped t-t′ - Hubbard model on six-leg square cylinders<ref:2402.11255#pg4>. The Hamiltonian used is defined as H = − X ijσ t ij cˆ† iσcˆjσ + h.c. + U X i nˆi↑nˆi↓ − V X⟨ij⟩ nˆinˆj<ref:2402.11255#pg4>. The lattice geometry considered is cylindrical with periodic boundary condition in the ˆy = (0, 1) direction and open boundary condition in the ˆx = (1, 0) direction<ref:2402.11255#pg4>. For this study, they focus on six-leg square cylinders with width Ly = 6 and fix the hole doping concentration δ = 1/12<ref:2402.11255#pg4>. The simulation parameters include t = 1 as the unit of energy and U = 12<ref:2402.11255#pg4>. They perform up to 130 sweeps and keep up to m = 43,000 states in each DMRG block to reach a typical truncation error of ε ≲ 3 × 10−6<ref:2402.11255#pg4>.

Enhanced SC Correlation in d-Wave Phase

Previous DMRG studies have observed the simultaneous presence of quasilong-range d-wave SC and CDW orders in the electrondoped (t′ > 0) region on both four-leg and six-leg square cylinders<ref:2402.11255#pg4>. The Luttinger exponents Ksc and Kcdw derived from the decay rate of the correlation functions for SC and CDW orders were found to be comparable<ref:2402.11255#pg4>. However, within a specific region of the phase diagram, the SC correlation was weaker than the CDW order, characterized by Ksc > Kcdw<ref:2402.11255#pg4>. The equaltime spin-singlet SC correlation function is defined as Φαβ (r) = D ∆† α,(x0,y0)∆β,(x0+r,y0)E, where ∆ˆ † α,(x,y) = √ 1/2 c† ↑,(x,y) c† ↓,(x,y)+α − c† ↓,(x,y) c† ↑,(x,y)+α<ref:2402.11255#pg4>. As V increases from 0.0 to 1.0, the exponent Ksc consistently decreased, reaching down to Ksc = 1.0(1)<ref:2402.11255#pg4>. This trend suggests that as temperature T approaches 0, the SC susceptibility χsc ∼ T − (2−Ksc) becomes increasingly divergent for stronger electron attraction<ref:2402.11255#pg4>.

Charge Density Wave

The charge density profile n(x), depicted in Fig.2(a), shows a pattern of clear spatial oscillations which have a wavelength of approximately λ ≈ 1/3δ = 4 in the ˆx direction, indicative of a stripy CDW characterized by an ordering vector Q ≈ 6πδ<ref:2402.11255#pg4>. When V = 0, the value of the exponent is Kcdw = 0.8(4), which is found to be larger than the exponent for the SC correlation, Ksc = 1.5(2)<ref:2402.11255#pg4>. Adding a small attractive V slightly raises the average electron density in the center, as illustrated in Fig.2(a), as this attraction energetically prefers fewer electrons at the edges<ref:2402.11255#pg4>. When the attraction strength exceeds approximately V ≈ 0.4, there’s a noticeable increase in the exponent Kcdw, corresponding to a clear weakening of the CDW order<ref:2402.11255#pg4>. A clear observation is that SC order starts to dominate when the attraction strength surpasses the intersection point, around Vc ≈ 0.7, of the two trend lines<ref:2402.11255#pg4>.

Other Properties and Longer Cylinders

The single particle Green’s function decays exponentially at long distances and can be well fitted by an exponentially decaying function G(r) ∼ e − r/ξG, where ξG is the single particle correlation length<ref:2402.11255#pg4>. For the t′ = 0.4 case, a long but finite correlation length, ξG ≈ 20, is observed across a range of attraction strengths V from 0.0 to 1.0<ref:2402.11255#pg4>. To ensure the accuracy of our results over long distances, we have carefully examined the potential impact of finite size effects by analyzing the SC pair-pair correlation functions on longer cylinders<ref:2402.11255#pg4>. Consistent with the results for Lx = 36 shown in Fig.1, Φyy for Lx = 48 also exhibits similar power-law decay at long distances across all cases examined<ref:2402.11255#pg4>.

Effect of Attraction in Other Phases

Although the presence of electron attraction V can also significantly boost SC correlations in the charge stripe phase which is similar with the electron doped side, this enhancement doesn’t transition the system into a SC phase, evidenced by the exponential decaying SC correlations<ref:2402.11255#pg5>. In the WC phase (t′ = −0.4), the impact of V is minimal due to the absence of Cooper pairs, with SC correlations at V = 0.0 and V = 0.6 nearly identical, as seen in Fig.4(b)<ref:2402.11255#pg5>.

Summary and Conclusion

The study has shown that incrementing the attractive V systematically amplifies SC correlations in both the SC and charge stripe phases<ref:2402.11255#pg7>. For electron-doped (positive t′) case, an increase in V markedly enhances SC correlations while simultaneously diminishing CDW ordering, underscoring the pivotal role of electron attraction in augmenting superconductivity in the doped Hubbard model<ref:2402.11255#pg7>. The threshold attraction strength Vc, at which SC correlations become dominant over CDW correlations, is found to be around Vc ≈ 0.7, in agreement with experimental observations on cuprates<ref:2402.11255#pg7>. Future research delving into mechanisms that could boost SC ordering in the negative t-t′ - Hubbard model will be crucial<ref:2402.11255#pg7>.

How it works

The attractive interaction V is introduced into the Hamiltonian as − V X⟨ij⟩ nˆinˆj<ref:2402.11255#pg4>. This attraction has a significant interplay with the electron hopping amplitudes t and t′<ref:2402.11255#pg4>. The effect is asymmetric between the electron-doped and hole-doped regimes<ref:2402.11255#pg4>.

Key Findings

In the electron-doped (t′ > 0) region, a moderate V is sufficient to drive the system into a pronounced SC phase with dominant SC correlations<ref:2402.11255#pg4>.
The critical value of Vc needed to enhance SC order in six-leg square cylinders is noticeably lower than that for four-leg square cylinders<ref:2402.11255#pg7>.
In the hole-doped (t′ < 0) regime, while heightened superconducting correlations have also been observed in the charge stripe phase, the systems remain insulating with pronounced charge density wave order<ref:2402.11255#pg2>.
The single particle gap remains small but largely unaffected by V across a range of V from 0.0 to 1.0<ref:2402.11255#pg8>.

Effect of Attraction in Other Phases

The attraction V can also significantly boost SC correlations in the charge stripe phase, but this enhancement doesn’t transition the system into a SC phase<ref:2402.11255#pg5>.

Improvements for AI systems

textbf Stronger SC Correlation Prediction in Electron-Doped Systems (t' > 0): This improved system can predict that "NN electron attraction substantially bolsters superconductivity while concurrently mitigating CDW stripe order in the electron-doped (t′ > 0) region of the model where we find that a moderate V is sufficient to drive the system into a pronounced SC phase with dominant SC correlations."
textbf Critical Attraction Threshold Determination: The AI can now accurately determine the critical value of Vc needed to enhance SC order in six-leg square cylinders is noticeably lower than that for four-leg square cylinders [27], enabling more precise parameter tuning for materials science simulations.
textbf Phase Discrimination in Hole-Doped Regimes (t' < 0): The system can differentiate between phases by noting that while heightened superconducting correlations have also been observed in the charge stripe phase, the systems remain insulating with pronounced charge density wave order when V is increased on the hole-doped side.
textbf Robustness of Pairing Symmetry: The AI can verify that the d-wave pairing symmetry, identified in earlier research [25], continues to be prevalent even when the electron attraction is strong, up to a value of V = 1.0, ensuring that predicted superconducting states are topologically stable under varying interaction strengths.
textbf Single-Particle Gap Stability Assessment: The system can predict that the system’s single particle gap remains small but largely unaffected by V across the range of attraction strengths, providing reliable estimates for the electronic band structure stability in correlated materials.