Pair density wave in doped three-band Hubbard model on two-leg square cylinders

arXiv:2209.11381 · cond-mat.str-el · Submitted 2022-09-23 · 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: "Pair density wave in doped three-band Hubbard model on two-leg square cylinders".

Mira: Here's the gist of the paper. The gist: The ground state of hole-doped three-band Hubbard model on two-leg square cylinders is consistent with a Pair Density Wave (PDW) state featuring mutually commensurate and powerlaw SC, charge (CDW) and spin (SDW) density wave correlations.

Kai: Model and Setup The study employs the three-band Hubbard model, also known as the Emery model, on long two-leg square cylinders to determine if this minimal model supports PDW order The Hamiltonian is defined by hopping matrix elements like tpd and tpp, and Coulomb interactions such as Ud and Up The lattice geometry considered is cylindrical with periodic/open boundary condition in the e2/e1 direction The overall hole density of the system is defined as ρ = 1 + δ, where δ = Nh/Nu

39: Both δ = 1/12 and δ = 1/8 hole doping concentrations are considered The DMRG study performs up to 157 sweeps and keeps up to m = 25000 number of states with a typical truncation error ∼ 10−9.

Mira: Superconducting Correlation The equal-time SC pair-field correlations are calculated, focusing on the spin-singlet SC correlation defined as Φαβ(r)

40: The pairing symmetry is consistent with d-wave, which is characterized by the fact that Φuu(r) ∼ Φhh(r) ∼ −Φuh(r) The spatial distribution of SC correlations exhibits clear spatial oscillations which can be well fitted by φhh(r) ∼ sin(Qr + θ) The PDW ordering wavevector Q is incommensurate and hole doping dependent as Q ≈ 2πδ, corresponding to a wavelength λsc ≈ 1/δ The static PDW susceptibility diverges as χpdw ∼ T −(2−Ksc) since Ksc < 2 as T → 0.

Kai: Charge Density Wave Order The spatial oscillation of the charge density profile nα(x) in the three-band Hubbard model is characterized by two wavevectors at Q and 2Q, corresponding to wavelengths λQ ≈ 1/δ and λ2Q ≈ 1/2δ, respectively The spatial decay of the CDW correlation is dominated by a power-law with Luttinger exponent Kc The CDW strength can be notably suppressed by NN attraction evidenced by the monotonic increase of Kc with the increase of NN attraction Vpd in Fig.3B When the NN interaction Vpd ≤ −0.2 as shown in Fig.3D, it is clear that the CDW at wavevector Q (QCDW) is dominant while the CDW at 2Q (2Q-CDW) is secondary.

Paper summary: Kai: Looking at the full paper "Pair density wave in doped three-band Hubbard model on two-leg square cylinders," the authors are essentially showing that hole doping in this specific three-band Hubbard model leads to a PDW ground state.

Mira: They argue that this is because they find mutually commensurate and powerlaw correlations for superconductivity, charge density wave, and spin density wave orders all existing together at the same ordering wavevector Q.

Lev: The key implication here is the role of those near-neighbor interactions, particularly Vpd, which can enhance the superconducting correlations while suppressing the charge density wave correlations.

Kai: So if you only listen to this show, what does it change for you? It means that in these strongly correlated systems we study, we can expect all three types of order to be locked together in a specific pattern dictated by doping.

Mira: They point out that the uniform superconducting component might actually be absent or at least very small in the resulting ground state they found.

Lev: For those of us building error-corrected hardware, this suggests that trying to stabilize a simple uniform superconductor might not be the most straightforward path if you want to capture these features.

Kai: And what about the title itself? "Pair density wave in doped three-band Hubbard model on two-leg square cylinders." It sets up a very specific problem they tackle with this minimal model.

Mira: The paper addresses a challenge because much is known about PDW properties, but realizing them in microscopic models with divergent susceptibility has been quite difficult before.

Lev: So the authors are providing one of those realizations in the three-band Hubbard model on long two-leg square cylinders, which is important for testing concepts in real materials.

Conclusion: Kai: So we just looked at this paper, "Pair density wave in doped three-band Hubbard model on two-leg square cylinders." It’s about finding a specific state of matter in cuprates that we call a Pair Density Wave, or PDW.

Mira: Yeah, the authors are using this three-band Hubbard model to see if their math actually supports this PDW thing they're talking about.

Lev: And they found that hole doping on these two-leg square cylinders leads to a ground state that has these coupled superconductivity and density wave orders all running at the same wavevector Q.

Kai: It means they aren't just seeing some random fluctuations; this is a structured pattern where everything is linked.

Mira: The authors highlight how certain near-neighbor interactions, specifically that Vpd interaction between copper and oxygen sites, can actually boost the superconducting part while slowing down the charge density wave.

Lev: From an error correction standpoint, if you had to simulate this on real hardware today, running a system with three coupled orders all at one Q vector would put a serious strain on your coherence.

Kai: And what does this mean for us in terms of understanding these high-temperature superconductors? It suggests that the stripe picture might be too simple for what's actually happening.

Mira: Exactly. They argue that having all three orders—PDW, charge density wave, and spin density wave—at the same Q vector is a very specific configuration that needs to be considered.

Lev: The paper notes that the uniform superconducting component might even be absent in this ground state they found, which is a bit surprising when you think about typical cuprate behavior.

Kai: It’s interesting because it pushes us away from just looking for a simple uniform superconductor and towards these more complex, modulated states.

Mira: So the real question is how these specific coupling mechanisms control the relative strengths of those three different orders in the system.

Lev: That leads us into what the next part of this research might look like, seeing if we can build a model that actually captures this level of complexity with manageable error scaling.

Stanford Institute for Materials and Energy Sciences · Stanford University · SLAC National Accelerator Laboratory

cond-mat.str-el

Submitted: 2022-09-23

Updated: 2023-02-21

Comments: 8 pages, 5 figures

Journal ref: Phys. Rev. B 107, 214504 (2023)

DOI: 10.1103/PhysRevB.107.214504

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

Importance score: 91/100

The gist: The gist: The ground state of hole-doped three-band Hubbard model on two-leg square cylinders is consistent with a Pair Density Wave (PDW) state featuring mutually commensurate and powerlaw SC,

Key concepts

Pair Density Wave (PDW): A superconducting state characterized by a spatially modulated order parameter. In this model, it means the superconducting pairing pattern is not uniform but repeats periodically in space, leading to a characteristic wavevector Q.
Three-band Hubbard Model (Emery Model): A minimal microscopic model used to describe cuprate materials. It includes three electronic bands and accounts for hopping between Cu and oxygen sites, as well as Coulomb interactions. This model is essential for studying the complex physics of high-temperature superconductors.
Doping ($ ho = 1 + ext{δ}$): The process of adding holes (removing electrons) to the system, represented by the doping concentration δ. The study examines two specific doping levels, $ ext{δ} = 1/12$ and $ ext{δ} = 1/8$, to see how hole concentration affects the resulting electronic order.
Charge Density Wave (CDW) Suppression: The tendency of electrons to arrange themselves into a periodic pattern that modulates the charge density. The study found that attractive interactions between neighboring Cu and oxygen sites significantly suppress this CDW order, allowing superconductivity to become more prominent.

Terminology

Hubbard Model — 2× in this paper · explained in 5 episodes

Summary

The gist: The ground state of hole-doped three-band Hubbard model on two-leg square cylinders is consistent with a Pair Density Wave (PDW) state featuring mutually commensurate and powerlaw SC, charge (CDW) and spin (SDW) density wave correlations.

Model and Setup

The study employs the three-band Hubbard model, also known as the Emery model, on long two-leg square cylinders to determine if this minimal model supports PDW order The Hamiltonian is defined by hopping matrix elements like tpd and tpp, and Coulomb interactions such as Ud and Up The lattice geometry considered is cylindrical with periodic/open boundary condition in the e2/e1 direction The overall hole density of the system is defined as ρ = 1 + δ, where δ = Nh/Nu [39] Both δ = 1/12 and δ = 1/8 hole doping concentrations are considered The DMRG study performs up to 157 sweeps and keeps up to m = 25000 number of states with a typical truncation error ∼ 10−9.

Superconducting Correlation

The equal-time SC pair-field correlations are calculated, focusing on the spin-singlet SC correlation defined as Φαβ(r) [40] The pairing symmetry is consistent with d-wave, which is characterized by the fact that Φuu(r) ∼ Φhh(r) ∼ −Φuh(r) The spatial distribution of SC correlations exhibits clear spatial oscillations which can be well fitted by φhh(r) ∼ sin(Qr + θ) The PDW ordering wavevector Q is incommensurate and hole doping dependent as Q ≈ 2πδ, corresponding to a wavelength λsc ≈ 1/δ The static PDW susceptibility diverges as χpdw ∼ T −(2−Ksc) since Ksc < 2 as T → 0.

Charge Density Wave Order

The spatial oscillation of the charge density profile nα(x) in the three-band Hubbard model is characterized by two wavevectors at Q and 2Q, corresponding to wavelengths λQ ≈ 1/δ and λ2Q ≈ 1/2δ, respectively The spatial decay of the CDW correlation is dominated by a power-law with Luttinger exponent Kc The CDW strength can be notably suppressed by NN attraction evidenced by the monotonic increase of Kc with the increase of NN attraction Vpd in Fig.3B When the NN interaction Vpd ≤ −0.2 as shown in Fig.3D, it is clear that the CDW at wavevector Q (QCDW) is dominant while the CDW at 2Q (2Q-CDW) is secondary.

Spin-Spin Correlation

The spin-spin correlations between Cu sites are dominant evidenced by FCu(r) FOy(r) ∼ FOx(r) Contrary to the exponential decaying spin-spin correlation in the Luther-Emery liquid of the square lattice single-band Hubbard model [46–48, 50, 51], we find that F(r) in the lightly doped three-band Hubbard model decays as a power-law Fα(r) ∼ r −Ks(α) at long distances, with a Luttinger exponent Ks ∼ 1 Consistent with a PDW state, we find that F(r) exhibits clear spatial oscillation as shown in Fig.4D at the PDW ordering wavevector Q with a wavelength λs ≈ λsc ≈ 1/δ, i.e., λs ≈ 8 at δ = 1/8.

Entanglement Entropy

The von Neumann entanglement entropy S(x) suggests that there is one gapless charge mode and one gapless spin mode with a central charge c ≈ 2 This suggests that there is one gapless charge mode and one gapless spin mode. The results suggest that the uniform SC component may be absent or at least small.

In summary, the ground state of hole-doped three-band Hubbard model on two-leg square cylinders is consistent with that of a PDW state, with mutually commensurate and powerlaw SC, CDW and SDW correlations. The Cooper pairing is dominant between neighboring Cu sites with d-wave symmetry For modestly strong NN attraction, quasi-long-range PDW order with a divergent susceptibility emerges. When the NN attraction Vpd ≤ −0.2 as shown in Fig.3D, it is clear that the CDW at wavevector Q (QCDW) is dominant while the CDW at 2Q (2Q-CDW) is secondary. The uniform SC component may be absent or at least small. The underlying mechanism of our observations that all three orders (PDW, CDW and SDW) are all at the same Q can be very different from the stripe picture of PDW [4].

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This work was supported by the Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, under Contract DE-AC02-76SF00515. The paper is a densitymatrix renormalization group study of a three-band Hubbard model (also known as the Emery model) for cuprates on long two-leg square cylinders.

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The authors are grateful to Steven Kivelson, Thomas Devereaux, John Tranquada, Patrick Lee and Ziqiang Wang for insightful discussions and invaluable suggestions, and Patrick Lee for pointing out the importance of three orders at the same Q. This work was supported by the Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, under Contract DE-AC02-76SF00515.

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The paper discusses the physics of pair density waves in cuprate high-temperature superconductors and other strongly correlated systems. The first example of PDW is the Fulde-Ferrell-Larkin-Ovchinnikov state in which the PDW state is more stable than the spatially uniform SC state when the Fermi surfaces are split by an external magnetic field [5, 6]. The PDW state has been discussed as a leading candidate state to understand the physics of cuprate high-temperature superconductors and other strongly correlated systems, where it has been proposed that various phases, including the unconventional superconductivity, charge density wave (CDW) and spin density wave (SDW) orders, can emerge by partially melting the PDW state [2–4, 7]. Recent interest in the PDW state has emerged due to experimental observations in underdoped cuprate superconductors, where signatures of PDW states have been observed via local Cooper pair tunneling and scanning tunneling microscopy in underdoped Bi2Sr2CaCu2O8+x (BSCCO) [8–11] and the dynamical inter-layer decoupling in La1.875Ba0.125CuO4 (LBCO) [12–18]. While much is known about the properties of the PDW state [1–4, 16, 17, 21], there are very few microscopic models which are shown to have PDW ground states [22–29] and especially no PDW long-range order has been established in unbiased calculations in two or higher dimensions.

Improvements for AI systems

Modeling Complex Correlated Phases The improved AI system can accurately model ground states of strongly correlated systems like cuprates by utilizing density-matrix renormalization group (DMRG) studies on three-band Hubbard models, specifically determining whether the ground state is consistent with that of a PDW state.
Predicting Superconducting Order Parameters The system can predict the nature and spatial modulation of superconducting correlations by analyzing equal-time SC pair-field correlations, including identifying d-wave symmetry and fitting the spatial distribution with a function like Φhh(r) ∼ f(r)φhh(r).
Characterizing Competing Density Waves The AI can distinguish between different charge density wave (CDW) orders by analyzing spatial oscillations, noting that in the three-band model, the CDW is characterized by two wavevectors at Q and 2Q.
Assessing Interaction Effects on Order The system can predict how specific interactions modify ground state phases; for instance, it can determine that near-neighbor attractions, especially Vpd between neighboring Cu and O sites, can notably enhance the SC correlations while suppressing the CDW correlations.
Determining Critical Behavior of PDW States The AI can analyze the divergence of susceptibility by calculating "χpdw ∼ T − (2−Ksc) since Ksc < 2 as T → 0," providing a quantitative measure related to the proximity to a critical attraction.
Identifying Gapless Excitations and Entanglement Entropy The system can calculate entanglement entropy to suggest the existence of gapless modes, finding the extracted c ≈ 2 with examples shown in Fig.5, indicating one gapless charge mode and one gapless spin mode in the critical systems described by conformal field theory.