Stripe order enhanced superconductivity in the Hubbard model

The gist: even modest amplitude modulations can enhance the long-distance SC correlations by many orders of magnitude, and drive the system into a phase with a substantial spin gap and SC quasi-long-range-order with a Luttinger exponent, Ksc ∼ 1.

The Model and Methodology

The study uses density matrix renormalization group (DMRG) studies of the Hubbard model on long 4 and 6 leg cylinders where the hopping matrix elements transverse to the long direction are periodically modulated, mimicking period-2 stripe order. The model considered is the square lattice Hubbard model defined by Hamiltonian (1) which includes nearest-neighbor interactions t and on-site Coulomb repulsion U=12t, with electron density per site n = 1−δ where δ = 1/8 and 1/12. The hopping integrals are defined as tx = t, ty = t + dt, and t0y = t − dt for the hopping matrix elements in the x and y directions, respectively. The research explores the second proposition that static or slowly fluctuating CDW order could produce a form of “optimally inhomogeneous” electronic structure that could enhance SC.

Effect of Stripe Modulation on Superconductivity

The primary finding is that the modulation enhances the SC correlations at long distances relative to the uniform cylinder (dt = 0) by many orders of magnitude. Specifically, for all cases, the modulation enhances the SC correlations at long distances relative to the uniform cylinder (dt = 0) by many orders of magnitude. For Ly = 6 cylinders with dt > 0, the SC pair-field correlations are dramatically enhanced by a finite dt > 0, where we find that Φαβ(r) ∼ r −Ksc with Ksc ∼ 1. For example, for dt = 0.3, ∆d = 0.042, ∆s = -0.021 and ∆π = 0.091.

Superconducting Correlations and Exponents

The results show that the SC correlations are dramatically enhanced by a finite dt, for instance, dt = 0.3 and dt = 0.4, where we find that Φαβ(r) ∼ r −Ksc with Ksc ∼ 1. For Ly = 6 cylinders at δ = 1/12, the SC pair-field correlations are consistent with a power-law decay Φ(r) ∼ r −Ksc with an exponent Ksc = 1.6(2) for dt = 0.3 and Ksc = 2.7(7) for dt = 0.4. The spatial symmetries of the striped model are such that there are two inequivalent y-directed bonds and a unique x directed bond. For dt = 6 = 0, this state is invariant under all the spatial symmetries, and so does not correspond to a distinct irrep.

Charge Density Wave (CDW) Correlations

The charge density profile n(x) for Ly = 4 cylinders is consistent with “halffilled charge stripes” with wavelength λcdw = 1/2δ. The charge density profile n(x) for Ly = 6 cylinders is shown, which has wavelength λcdw = 2/3δ, consistent with “two-third-filled” charge stripes. At long distance, the spatial decay of the CDW correlation is dominated by a power-law with the Luttinger exponent Kcdw. The extracted exponents for Ly = 4 cylinders are Kcdw ≈ 1.6 for all studied values of dt > 0 when δ = 1/12. For Ly = 6 cylinders with δ = 1/12, Kcdw = 3.3(8) when dt = 0.3 and Kcdw = 3.1(5) when dt = 0.4. The fact that Kcdw > Ksc for all cases in which dt > 0 suggests that CDW order is secondary compared to SC.

Spin-Spin and Single Particle Correlations

The spin-spin correlation function F(r) decays exponentially as F(r) ∼ e −r/ξs at long-distances, with a finite correlation length ξs, i.e. there must be a finite gap in the spin sector. Moreover, ξs decreases with increasing dt on both Ly = 4 and Ly = 6 cylinders. The single-particle Green function G(r) shows that the long distance behavior of G(r) is consistent with exponential decay G(r) ∼ e −r/ξG. The extracted correlation lengths ξG < 4 for both Ly = 4 and Ly = 6 cylinders are comparable to ξs.

Conclusions

The present results constitute the clearest and most unambiguous evidence to date that this is a real and robust effect, suggesting that a more nuanced approach to the intertwining of CDW and SC orders may be appropriate in the cuprate context. The low energy (gapless) magnetic fluctuations associated with antiferromagnetic order or near order are determinantal to SC, while higher energy, short-range correlated antiferromagnetic fluctuations can produce precisely the sort of momentum dependent interactions that are most conducive to d-wave SC. It is possible to view the gap in such a state as the pairing gap of a superconductor that is waiting to be liberated. The undoped (δ = 0) two-leg Hubbard ladder has a spin gap and can be viewed as a Mott insulator of preexisting Cooper pairs (rung singlets). Doping into a modulated array of effective two leg ladders may be analogous to doping a fully gapped quantum spin liquid. The present results offer encouraging evidence of an affirmative answer to the question of whether the 2d Hubbard model can support high temperature superconductivity.

How it works

The enhancement mechanism is rooted in the proximity to a putative CDW quantum critical point, where CDW fluctuations could serve as an effective pairing “glue” and thereby enhance SC even under conditions in which fully developed CDW order might depress SC by opening gaps on portions of the Fermi surface. Alternatively, static or slowly fluctuating CDW order could produce a form of “optimally inhomogeneous” electronic structure that could enhance SC. The system exhibits a Luther-Emery liquid behavior, where the SC and CDW susceptibilities are determined by quantum mechanically dual variables. The modulation of hopping matrix elements t → t ± dt as shown in Fig.1 leads to the formation of a “Striped” Hubbard model. For non-zero dt, the SC pair-field correlations are dramatically enhanced by a finite dt > 0, where we find that Φαβ(r) ∼ r −Ksc with Ksc ∼ 1. This enhancement is observed for both Ly = 4 and Ly = 6 cylinders, where the SC pair-field correlations are dramatically enhanced by a finite dt > 0. The SC pairing symmetry is d-wave like with Φxx(r) ∼ Φyy(r) ∼ −Φxy(r) for finite dt. This is consistent with previous studies on the isotropic Hubbard model with dt = 0, where ∆d increases rapidly as well.

Key Observations from DMRG Results

The study systematically explores the effect of stripe modulation strength (dt) and doped hole concentration (δ) on various correlation functions.

1. The AI system can now predict optimal inhomogeneities for high-temperature superconductivity by analyzing how period-2 charge-density-wave (CDW) order enhances long-distance SC correlations, as suggested by the finding that even modest amplitude modulations can enhance the long-distance SC correlations by many orders of magnitude.

2. The system can perform quantitative phase diagram analysis for strongly correlated electronic systems where charge-density-wave (CDW), spin-density-wave (SDW), and d-wave superconducting (SC) orders all arise in significantly overlapping regimes, providing a framework to distinguish between competing tendencies.

3. The AI can evaluate the potential for superconductivity in the 2D Hubbard model by assessing how the modulation enhances the SC correlations at long distances relative to the uniform cylinder (dt = 0) by many orders of magnitude, offering a more robust criterion than previous studies that concluded it may never be superconducting in the 2d limit.

4. The AI can classify different forms of Quasi-Long-Range Order (QLRO) in low-dimensional systems, specifically distinguishing between spin and charge order exponents, as demonstrated by extracting parameters like Ksc (for SC QLRO) and Kcdw (for CDW QLRO).

5. The system can generate specific pairing symmetry predictions for finite cylinder widths, identifying that for non-zero modulation strength, the pairing is d-wave like with Φxx(r) ∼ Φyy(r) ∼ −Φxy(r), allowing targeted simulation of potential superconducting states.