Ground-state phase diagram of the t-t'-J model

Computational demonstrations of density wave of Cooper pairs and paired-electron liquid in the quarter-filled band-A brief review

There has been strong interest recently in the so-called Cooper pair density wave, subsequent to the proposition that such a state occurs in the hole-doped cuprate superconductors. As of now, there is no convincing demonstration of such a state in the cuprate theoretical literature. We present here a brief but complete review of our theoretical and computational work on the paired-electron crystal (PEC), which has also been experimentally seen in the insulating phase proximate to superconductivity (SC) in organic charge-transfer solid (CTS) superconductors. Within our theory, SC in the CTS does indeed evolve from the PEC. A crucial requirement for the finding of the PEC is that the proper carrier density of one charge carrier per two sites is taken into consideration at the outset. Following the discussion of CTS superconductors, we briefly discuss how the theory can be extended to understand the phase diagram of the cuprate superconductors that has remained mysterious after nearly four decades of the discovery of SC in this family.

Emergent Superconductivity and Competing Charge Orders in Hole-Doped Square-Lattice t-J Model

The square-lattice Hubbard and closely related t-J models are considered as basic paradigms for understanding strong correlation effects and unconventional superconductivity (SC). Recent large-scale density matrix renormalization group simulations on the extended t-J model have identified d-wave SC on the electron-doped side (with the next-nearest-neighbor hopping t_{2}>0) but a dominant charge density wave (CDW) order on the hole-doped side (t_{2}<0), which is inconsistent with the SC of hole-doped cuprate compounds. We re-examine the ground-state phase diagram of the extended t-J model by employing the state-of-the-art density matrix renormalization group calculations with much enhanced bond dimensions, allowing more accurate determination of the ground state. On six-leg cylinders, while different CDW phases are identified on the hole-doped side for the doping range δ=1/16-1/8, a SC phase emerges at a lower doping regime, with algebraically decaying pairing correlations and d-wave symmetry. On the wider eight-leg systems, the d-wave SC also emerges on the hole-doped side at the optimal 1/8 doping, demonstrating the winning of SC over CDW by increasing the system width. Our results not only suggest a new path to SC in general t-J model through weakening the competing charge orders, but also provide a unified understanding on the SC of both hole- and electron-doped cuprate superconductors.

Extracting the Speed of Light from Matrix Product States

We provide evidence that the spectrum of the local effective Hamiltonian and the transfer operator in infinite-system matrix product state simulations are identical up to a global rescaling factor, i.e., the speed of light of the system, when the underlying system is described by a 1+1 dimensional CFT. We provide arguments for this correspondence based on a path integral point of view. This observation turns out to yield very precise estimates for the speed of light in practice, confirming exact results to high precision where available, but also allowing us to finally determine the speed of light of the non-integrable, critical SU(2) Heisenberg chains with half-integer spin S>1/2 with unprecedented accuracy. We also show that the same technology applied to doped Hubbard ladders provides highly accurate velocities for a range of dopings. Combined with measurements of compressibilities we present new results for the Luttinger liquid parameter in the Luther-Emery regime of doped Hubbard ladders, outperforming earlier approaches based on the fitting of real-space correlation functions.

Controlled Bond Expansion for Density Matrix Renormalization Group Ground State Search at Single-Site Costs

DMRG ground state search algorithms employing symmetries must be able to expand virtual bond spaces by adding or changing symmetry sectors if these lower the energy. Traditional single-site DMRG does not allow bond expansion; two-site DMRG does, but at much higher computational costs. We present a controlled bond expansion (CBE) algorithm that yields two-site accuracy and convergence per sweep, at single-site costs. Given a matrix product state Ψ defining a variational space, CBE identifies parts of the orthogonal space carrying significant weight in HΨ and expands bonds to include only these. CBE-DMRG uses no mixing parameters and is fully variational. Using CBE-DMRG, we show that the Kondo-Heisenberg model on a width 4 cylinder features two distinct phases differing in their Fermi surface volumes.

Tangent Space Approach for Thermal Tensor Network Simulations of the 2D Hubbard Model

Accurate simulations of the two-dimensional (2D) Hubbard model constitute one of the most challenging problems in condensed matter and quantum physics. Here we develop a tangent space tensor renormalization group (tanTRG) approach for the calculations of the 2D Hubbard model at finite temperature. An optimal evolution of the density operator is achieved in tanTRG with a mild O(D^{3}) complexity, where the bond dimension D controls the accuracy. With the tanTRG approach we boost the low-temperature calculations of large-scale 2D Hubbard systems on up to a width-8 cylinder and 10×10 square lattice. For the half-filled Hubbard model, the obtained results are in excellent agreement with those of determinant quantum Monte Carlo (DQMC). Moreover, tanTRG can be used to explore the low-temperature, finite-doping regime inaccessible for DQMC. The calculated charge compressibility and Matsubara Green's function are found to reflect the strange metal and pseudogap behaviors, respectively. The superconductive pairing susceptibility is computed down to a low temperature of approximately 1/24 of the hopping energy, where we find d-wave pairing responses are most significant near the optimal doping. Equipped with the tangent-space technique, tanTRG constitutes a well-controlled, highly efficient and accurate tensor network method for strongly correlated 2D lattice models at finite temperature.

Traces of electron-phonon coupling in one-dimensional cuprates

The appearance of certain spectral features in one-dimensional (1D) cuprate materials has been attributed to a strong, extended attractive coupling between electrons. Here, using time-dependent density matrix renormalization group methods on a Hubbard-extended Holstein model, we show that extended electron-phonon (e-ph) coupling presents an obvious choice to produce such an attractive interaction that reproduces the observed spectral features and doping dependence seen in angle-resolved photoemission experiments: diminished 3k_F spectral weight, prominent spectral intensity of a holon-folding branch, and the correct holon band width. While extended e-ph coupling does not qualitatively alter the ground state of the 1D system compared to the Hubbard model, it quantitatively enhances the long-range superconducting correlations and suppresses spin correlations. Such an extended e-ph interaction may be an important missing ingredient in describing the physics of the structurally similar two-dimensional high-temperature superconducting layered cuprates, which may tip the balance between intertwined orders in favor of uniform d-wave superconductivity.

Robust charge-density-wave correlations in the electron-doped single-band Hubbard model

There is growing evidence that the hole-doped single-band Hubbard and t - J models do not have a superconducting ground state reflective of the high-temperature cuprate superconductors but instead have striped spin- and charge-ordered ground states. Nevertheless, it is proposed that these models may still provide an effective low-energy model for electron-doped materials. Here we study the finite temperature spin and charge correlations in the electron-doped Hubbard model using quantum Monte Carlo dynamical cluster approximation calculations and contrast their behavior with those found on the hole-doped side of the phase diagram. We find evidence for a charge modulation with both checkerboard and unidirectional components decoupled from any spin-density modulations. These correlations are inconsistent with a weak-coupling description based on Fermi surface nesting, and their doping dependence agrees qualitatively with resonant inelastic x-ray scattering measurements. Our results provide evidence that the single-band Hubbard model describes the electron-doped cuprates.

Quantum Phase Diagram and Spontaneously Emergent Topological Chiral Superconductivity in Doped Triangular-Lattice Mott Insulators

The topological superconducting state is a highly sought-after quantum state hosting topological order and Majorana excitations. In this Letter, we explore the mechanism to realize the topological superconductivity (TSC) in the doped Mott insulators with time-reversal symmetry (TRS). Through large-scale density matrix renormalization group study of an extended triangular-lattice t-J model on the six- and eight-leg cylinders, we identify a d+id-wave chiral TSC with spontaneous TRS breaking, which is characterized by a Chern number C=2 and quasi-long-range superconducting order. We map out the quantum phase diagram with by tuning the next-nearest-neighbor (NNN) electron hopping and spin interaction. In the weaker NNN-coupling regime, we identify a pseudogaplike phase with a charge stripe order coexisting with fluctuating superconductivity, which can be tuned into d-wave superconductivity by increasing the doping level and system width. The TSC emerges in the intermediate-coupling regime, which has a transition to a d-wave superconducting phase with larger NNN couplings. The emergence of the TSC is driven by geometrical frustrations and hole dynamics which suppress spin correlation and charge order, leading to a topological quantum phase transition.

Fragmented Cooper Pair Condensation in Striped Superconductors

Condensation of bosons in Bose-Einstein condensates or Cooper pairs in superconductors refers to a macroscopic occupation of a few single- or two-particle states. A condensate is called "fragmented" if not a single, but multiple states are macroscopically occupied. While fragmentation is known to occur in particular Bose-Einstein condensates, we propose that fragmentation naturally takes place in striped superconductors. To this end, we investigate the nature of the superconducting ground state realized in the two-dimensional t-t^{'}-J model. In the presence of charge density modulations, the condensate is shown to be fragmented and composed of partial condensates located on the stripes. The fragments of the condensates hybridize to form an extended macroscopic wave function across the system. The results are obtained from evaluating the singlet-pairing two-particle density matrix of the ground state on finite cylinders computed via the density matrix renormalization group method. Our results shed light on the intricate relation between stripe order and superconductivity in systems of strongly correlated electrons.

Stripe order enhanced superconductivity in the Hubbard model

The Hubbard model plays a central role in the theory of highly correlated systems. Its simplicity allows conceptual issues—which are generally complicated in the context of experiments on interesting materials—to be sharply posed and definitively answered. Recently, a variety of numerical studies have led to the conclusion that the “pure” Hubbard model on the square lattice at intermediate coupling, U, is not superconducting in the range of electron densities in which many previous approximate treatments had inferred high-temperature superconductivity. Here, using controlled density matrix renormalization group methods, we show that superconductivity is spectacularly enhanced if the hopping matrix elements are periodically modulated in a stripe-like pattern, with important (if suggestive) implications concerning the mechanism of unconventional superconductivity.

Unidirectional (“stripe”) charge density wave order has now been established as a ubiquitous feature in the phase diagram of the cuprate high-temperature superconductors, where it generally competes with superconductivity. Nonetheless, on theoretical grounds it has been conjectured that stripe order (or other forms of “optimal” inhomogeneity) may play an essential positive role in the mechanism of high-temperature superconductivity. Here, we report density matrix renormalization group studies of the Hubbard model on long four- and six-leg cylinders, where the hopping matrix elements transverse to the long direction are periodically modulated—mimicking the effect of putative period 2 stripe order. We find that even modest amplitude modulations can enhance the long-distance superconducting correlations by many orders of magnitude and drive the system into a phase with a substantial spin gap and superconducting quasi–long-range order with a Luttinger exponent, Ksc∼1.