Pair density wave and superconductivity in a kinetically frustrated doped Emery model on a square lattice

arXiv:2309.11786 · cond-mat.str-el, cond-mat.supr-con · Submitted 2023-09-21 · 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: "Pair density wave and superconductivity in a kinetically frustrated doped Emery model on a square lattice".

Mira: Here's the gist of the paper. The gist: The study investigates whether a pair density wave (PDW) state or distinct superconducting (SC) state emerges upon doping in the Emery model, revealing that for negative oxygen-oxygen hopping, SC pairing is dominant between neighboring oxygen sites with a dxy-wave symmetry rather than the expected dx2−y2 wave Model and Method The research employs a density-matrix renormalization group (DMRG) study to examine the Emery model on two-leg square cylinders to determine if the same PDW state or distinct SC state emerges upon doping and what the associated pairing symmetry is.

Kai: The model, also known as the three-band Hubbard model, considers a square lattice of copper and oxygen atoms where copper sites are represented by a single 3dx2−y2 orbital and each oxygen site has two active 2p orbitals.

Mira: Kinetic frustration is introduced by changing the sign of oxygen-oxygen hopping term tpp, which leads to a much reduced Cu-Cu antiferromagnetic exchange along with an enlarged charge transfer energy that changes the local properties of the model.

Kai: The study focuses on two-leg cylinders with width Ly = 2 and length up to Lx=96, considering hole densities of δ = 1/12 and 1/8.

Title and authors: Kai: So we're looking at a paper called "Pair density wave and superconductivity in a kinetically frustrated doped Emery model on a square lattice," right? It’s by Hong-Chen Jiang and Thomas Peter Devereaux.

Mira: Yeah, that title tells you the main thing right away: they are investigating whether a pair density wave state or some distinct superconducting state shows up when you dope this specific Emery model.

Kai: It’s really about seeing if this system develops that spatially modulated order parameter we call a pair density wave, which is something we haven't seen conclusively in cuprates yet.

Mira: And the authors are using a density-matrix renormalization group study to see if they get that PDW state or just some other kind of superconducting state and what the pairing symmetry looks like.

Lev: From an engineering standpoint, a DMRG study on two-leg square cylinders tells us about the complexity of simulating these higher dimensional systems without introducing too much error in the hardware setup.

Kai: Right, so they’re using that DMRG method to probe this Emery model on two-leg square cylinders with negative electron hopping term tpp between adjacent oxygen sites.

Mira: The implication is that by changing the sign of that oxygen-oxygen hopping, which introduces kinetic frustration, they are looking at how that affects the magnetic exchange and charge transfer energy.

Lev: That change in kinetic frustration is what allows them to look at a system where the Cu-Cu antiferromagnetic exchange gets much reduced, which is interesting for how we model those strong correlations.

The paper's summary: Kai: Okay, so the paper summarizes their findings on this pair density wave and superconductivity in a kinetically frustrated doped Emery model on a square lattice.

Mira: They found that when you look at the ground state phase diagram at doping delta equals one/eight if you keep the oxygen-oxygen hopping Vpp not strongly attractive, the system is consistent with a PDW state <ref:2309.11786#pg3>.

Kai: And in this PDW phase, they see power-law and mutually commensurate correlations for both charge density wave and spin density wave alongside that superconducting correlation.

Mira: The key thing they pointed out is that the SC correlations oscillate periodically in real space, but their spatial average vanishes, meaning it’s not a simple uniform state.

Lev: That oscillating correlation pattern suggests quasi-long-range order for the charge and spin channels because of those power-law decays they measure.

Kai: And regarding the superconductivity itself, they found that Cooper pairing prominently emerges between adjacent oxygen sites rather than between neighboring Cu sites.

Mira: This is a big finding because it means the pairing symmetry is consistent with dxy rather than the dxtwo-y2 wave we usually see in cuprates.

Lev: If we were building hardware to realize this, that means our coupling scheme has to strongly favor those oxygen-oxygen interactions over the direct copper-copper ones.

The paper's improvements: Kai: Now for the improvements they suggest with this work on the pair density wave and superconductivity in a kinetically frustrated doped Emery model on a square lattice.

Mira: They suggest that when you incorporate moderate near-neighbor interactions, specifically an attractive Vpp between adjacent oxygen sites, the SC correlations become quasi-long-ranged.

Kai: And they also note that this attractive Vpp leads to a pronounced divergence in the PDW susceptibility, which is a very strong signal for that ordering.

Lev: From an error correction perspective, that divergence in susceptibility would mean the system is extremely sensitive to small perturbations, which makes running it on real hardware pretty challenging because you need very precise control.

Mira: Further increase in that attractive Vpp allows the system to give way to an unconventional d-wave superconductivity where those PDW signatures get much suppressed.

Kai: So they suggest a pathway where tuning the oxygen-oxygen interaction can drive the system from a PDW state into a conventional d-wave superconducting state.

Lev: That transition point, where the PDW signatures get suppressed, would be a crucial benchmark for testing any error correction protocol designed to handle these competing orders.

Conclusion: Kai: So to wrap up on this paper about pair density wave and superconductivity in a kinetically frustrated doped Emery model on a square lattice.

Mira: The main implication is that the system’s ground state is strongly aligned with the characteristics of a PDW state, showing quasi-long-range order and strong susceptibility.

Kai: And that Cooper pairing prominently emerges between adjacent oxygen sites, which points to a dxy symmetry instead of the dxtwo-y2 we usually expect.

Mira: This work also shows that there's both a gapless charge mode and a gapless spin mode present in this system, which is indicated by the central charge being approximately given by c equals one point nine five or c equals two point zero depending on the parameters <ref:2309.11786#pg1>.

Lev: For real hardware, that presence of both gapless modes means we’re dealing with complex dynamics that require careful modeling to ensure any error correction protocol can actually capture those excitations correctly and not just miss them in the simulation.

Kai: Exactly, so tuning the oxygen-oxygen interaction lets you switch between these different phases like PDW and d-wave superconductivity.

Mira: It’s a detailed look at how kinetic frustration can change the fundamental pairing symmetry of these correlated electron systems.

Lev: The whole point is to understand the real physical constraints on how we build those exotic materials and see what kind of excitations we have to deal with when trying to engineer them for quantum computation.

Stanford Institute for Materials and Energy Sciences · Stanford University

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

Submitted: 2023-09-21

Updated: 2023-09-21

Comments: 7 pages, 6 figures

Journal ref: Front. Electron. Mater. 3, 1323404 (2023)

DOI: 10.3389/femat.2023.1323404

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

Importance score: 89/100

The gist: The gist: The study investigates whether a pair density wave (PDW) state or distinct superconducting (SC) state emerges upon doping in the Emery model, revealing that for negative oxygen-oxygen

Key concepts

Emery Model: This is a specific theoretical model used to describe materials like cuprates. It models copper and oxygen atoms on a square lattice, considering their different electronic states. The study uses this model to understand how doping affects the material's properties and whether it becomes superconducting or forms other ordered states.
Pair Density Wave (PDW): This is an ordered state where Cooper pairs form in a specific pattern across the lattice. In this study, the PDW phase is found to be dominant, characterized by oscillating superconducting correlations that average to zero over space. This ordering wavevector Q is related to the doping level.
dxy-wave Symmetry: This refers to a specific type of pairing symmetry in superconductivity. The paper finds that when oxygen hopping is negative, the dominant SC pairing exhibits dxy symmetry instead of the more commonly expected dx2-y2 wave found in cuprates. This difference is crucial for understanding the material's electronic behavior.
Kinetic Frustration (Negative tpp): This term describes a situation where kinetic energy constraints make it difficult for electrons to move freely, which is introduced by changing the sign of oxygen-oxygen hopping (tpp). Negative tpp significantly reduces magnetic exchange between copper spins, altering the system's magnetic and electronic ground state.

Terminology

Frustration — 1× in this paper · explained in 3 episodes
Hubbard Model — 1× in this paper · explained in 5 episodes

Summary

The gist: The study investigates whether a pair density wave (PDW) state or distinct superconducting (SC) state emerges upon doping in the Emery model, revealing that for negative oxygen-oxygen hopping, SC pairing is dominant between neighboring oxygen sites with a dxy-wave symmetry rather than the expected dx2−y2 wave

Model and Method

The research employs a density-matrix renormalization group (DMRG) study to examine the Emery model on two-leg square cylinders to determine if the same PDW state or distinct SC state emerges upon doping and what the associated pairing symmetry is. The model, also known as the three-band Hubbard model, considers a square lattice of copper and oxygen atoms where copper sites are represented by a single 3dx2−y2 orbital and each oxygen site has two active 2p orbitals. Kinetic frustration is introduced by changing the sign of oxygen-oxygen hopping term tpp, which leads to a much reduced Cu-Cu antiferromagnetic exchange along with an enlarged charge transfer energy that changes the local properties of the model. The study focuses on two-leg cylinders with width Ly = 2 and length up to Lx=96, considering hole densities of δ = 1/12 and 1/8.

Phase Diagram

The ground state phase diagram for the Emery model on two-leg square cylinders at δ = 1/8 shows that when Vpp between oxygen sites are not strongly attractive, the ground state is consistent with a PDW state with power-law and mutually commensurate SC, charge-density wave (CDW) and spin-densitywave (SDW) correlations. The SC correlations oscillate periodically in real space such that its spatial average vanishes and the PDW ordering wavevector Q ≈ 2πδ is incommensurate. Contrary to the positive tpp case, the results show that the SC pairing is dominant between adjacent oxygen sites instead of Cu sites. When moderate near-neighbor interactions, particularly an attractive Vpp between adjacent oxygen sites, are incorporated, SC correlations become quasi-long-ranged accompanied by a pronounced divergence in the PDW susceptibility. When further increase in the attractive Vpp occurs, the system gives ways to an unconventional d-wave superconductivity where PDW signatures get much suppressed.

Pair Density Wave Phase

The majority of the ground state phase diagram is occupied by the PDW phase, where SC correlations decay as a power-law at long distances and oscillate periodically in real space in such a way that its spatial average vanishes. In this PDW phase, Cooper pairing prominently emerges between adjacent oxygen sites rather than between neighboring Cu sites. The pairing symmetry is consistent with dxy rather than dx2−y2 wave. The spatial oscillation of the SC correlations Φdd(r) is captured by the normalized function ϕdd(r), which aligns well with the fitting function ϕdd(r) ∼ sin(Qr + θ). The PDW ordering wavevector appears to be incommensurate as Q ≈ 2πδ with a corresponding wavelength λsc ≈ 1/δ.

Superconducting Correlations

The equal-time spin-singlet SC pair-pair correlations defined as Φαβ(r) are analyzed, and the strongest SC correlations are prominently exhibited in Φdd(r) and Φd¯d¯(r). This suggests that the pairing is more dominant between neighboring oxygen sites rather than Cu sites. Although the pairing symmetry aligns with the d-wave, our findings indicate a shift to the dxy-wave symmetry in this particular scenario, diverging from the previously understood dx2−y2 wave. The spatial oscillation of SC correlations Φdd(r) is captured by ϕdd(r) ∼ f(r)ϕdd(r), where f(r) acts as the envelope and gives rise to the spatial oscillation. For longer distances, the envelope function f(r) adheres to a power-law decay f(r) = A∗r−Ksc. The exponent Ksc is found to be ≈ 2.4 at Vpd = 1.0 and Vpp = 0.75.

Charge Density Wave

The charge density profile nα(x, y) is calculated, and its spatial oscillation is characterized by two ordering wavevectors at Q and 2Q, corresponding to wavelengths λQ ≈ 1/δ and λ2Q ≈ 1/2δ. The long-distance spatial decay of the CDW correlation is dominated by a power-law with an exponent Kc, which can be obtained by fitting the charge density oscillations induced by the cylinder boundaries. For δ = 1/8, Kc(Cu)≈ 1.6 and Kc(Oy)≈ 1.7 for Vpd = 1 and Vpp = 0.75.

Spin-Spin Correlations

The spin-spin correlation functions Fα(r) are examined, and the findings highlight a dominant correlation between oxygen sites in the kinetically frustrated case. This decay is described by F(r) ∼ r−Ks over extended distances. The associated Luttinger exponent Ks(Oy)≈ 1.1 for Vpd = Vpp = 0 and Ks(Oy) ≈ 1.3 for Vpd = −0.8 and Vpp = −0.2.

Entanglement Entropy

The presence of multiple gapless modes is indicated by the central charge, c, which is approximately given by c ≈ 2. Specifically, a central charge of c ≈ 1.95 at Vpd = Vpp = 0 and c ≈ 2.0 at Vpd = −0.8 and Vpp = −0.2 at δ = 1/8. These results point to the presence of both a gapless charge mode and a gapless spin mode.

d-wave Superconductivity

When further increase in the attractive Vpp, the system evolves into a d-wave SC phase where Φdd(r) and Φd¯d¯(r) exhibit the strongest SC correlations. The pairing symmetry is consistent with the dxy-wave symmetry characterized by the fact that Φdd(r) ∼ Φd¯d¯(r) ∼ −Φdd¯(r). In this d-wave SC phase, the SC correlations maintain a consistent sign in real space, and the spin-spin correlation functions are short-ranged and undergo exponential decay. A singular gapless mode with c ≈ 1 is evident in this phase. The ground state of the d-wave SC phase aligns with the characteristics of a Luther-Emery liquid.

Summary and Discussion

The results strongly suggest that the system’s ground state is aligned with the characteristics of a PDW state, showcasing quasi-long-range PDW order and pronounced susceptibility. A key finding is that Cooper pairing prominently emerges between adjacent oxygen sites rather than between neighboring Cu sites. This manifests as a dxy symmetry instead of the expected dx2−y2 pairing symmetry characteristic of cuprates. The effective spin exchange between Cu spins is greatly reduced for negative tpp, and a negative Vpd increases the magnetic exchange for the two-hole ground state configuration. The predominance of dxy pairing observed in the phase diagram may be related to this tendency to bind neighboring O holes. The system exhibits both a gapless charge mode and a gapless spin mode. This work affirms that the ground state of the d-wave SC phase aligns with the characteristics of a Luther-Emery liquid.

TABLE I

The effective single band exchange parameter J and NN hopping t determined from Cu2O7 clusters for different values of tpp as indicated. The exchange energy J = 0.0179 for tpp = −0.5, compared to 0.165 if the sign of tpp is reversed. The effective spin exchange between Cu spins is greatly reduced for negative tpp. A negative Vpd increases the magnetic exchange for the two-hole ground state configuration.

FIG. 2

Ground state phase diagram of the Emery model on two-leg square cylinders at δ = 1/8. When Vpp between oxygen sites are not strongly attractive, we find the the ground state of the system is consistent with that of a PDW state with power-law and mutually commensurate SC, charge-density wave (CDW) and spin-densitywave (SDW) correlations. The SC correlations oscillate periodically in real space in such a way that its spatial average vanishes and the PDW ordering wavevector Q ≈ 2πδ is incommensurate. Contrary to the positive tpp case, our results show that the SC pairing is dominant between adjacent oxygen sites instead of Cu sites.

Improvements for AI systems

The AI system can accurately predict whether a given parameter set in an Emery model on two-leg square cylinders will yield a Pair Density Wave (PDW) state or a distinct superconducting (SC) state by analyzing the ground state phase diagram shown in Fig. 2.
The system can determine the dominant SC pairing symmetry by analyzing the Cooper pairing correlations, specifically noting that our results differ significantly for the negative tpp case and that the corresponding pairing symmetry is consistent with dxy rather than dx2−y2 wave.
The AI can predict the nature of long-range correlations in both charge and spin channels by calculating Luttinger exponents, such as determining if the system exhibits quasi-long-range PDW order based on the exponent "Ksc < 2."
The system can identify the dominant pairing channel by analyzing equal-time spin-singlet SC pair-pair correlations, specifically noting that in the negative tpp case, the strongest SC correlations are prominently exhibited in Φdd(r) and Φd¯d¯(r), suggesting pairing is more dominant between neighboring oxygen sites, rather than Cu sites.
The AI can distinguish between the PDW phase and a conventional d-wave SC phase by analyzing entanglement entropy, noting that in the d-wave SC phase, the SC correlations Φαβ(r) maintain a consistent sign in real space while in the PDW state they oscillate periodically in real space in such a way that its spatial average vanishes.
The AI can infer the presence of gapless excitations by characterizing the system's central charge, noting that Our findings reveal a central charge approximately given by c ≈ 2, pointing to the presence of both a gapless charge mode and a gapless spin mode.