Density Matrix Renormalization Group Calculations Research Articles

Superconductivity in doped triangular Mott insulators: The roles of parent spin backgrounds and charge kinetic energy

We study the prerequisites for realizing superconductivity in doped triangular-lattice Mott insulators by considering three distinct parent spin backgrounds, i.e., ${120}^{\ensuremath{\circ}}$ antiferromagnets, quantum spin liquid, and stripy antiferromagnets, and all possible sign combinations $({\ensuremath{\tau}}_{1},{\ensuremath{\tau}}_{2})$ of nearest-neighbor hopping and next-nearest-neighbor hopping $({t}_{1},{t}_{2})$. Based on density matrix renormalization group calculations, we find that, with finite ${t}_{2}$ and specific sign combinations $({\ensuremath{\tau}}_{1},{\ensuremath{\tau}}_{2})$, the quasi-long-range superconductivity order can always be achieved, regardless of the nature of the parent spin backgrounds. Besides specific hopping signs $({\ensuremath{\tau}}_{1},{\ensuremath{\tau}}_{2})$, these superconductivity phases in triangular lattices are commonly characterized by short-ranged spin correlations and two charges per stripe. In the robust superconductivity phase realized at larger ${t}_{2}/{t}_{1}$, flipping the signs ${\ensuremath{\tau}}_{2}$ and ${\ensuremath{\tau}}_{1}$ gives rise to the stripe phase without strong pairing and a pseudogaplike phase without Cooper-pair phase coherence, respectively. Interestingly, the roles of the two hopping signs are switched at smaller ${t}_{2}/{t}_{1}$. Moreover, different sign combinations $({\ensuremath{\tau}}_{1},{\ensuremath{\tau}}_{2})$ would stabilize distinct phases including superconductivity, charge density waves, spin density waves, and pseudogaplike phases accordingly. Our findings suggest the important role of charge kinetic energy in realizing superconductivity in doped triangular-lattice Mott insulators.

Triangular lattice Hubbard model physics at intermediate temperatures

Moire systems offer an exciting playground to study many-body effects of strongly correlated electrons in regimes that are not easily accessible in conventional material settings. Motivated by a recent experiment on $\text{WSe}_2/\text{WS}_2$ moire bilayers [Y. Tang et al., Nature 579, 353-358 (2020)], which realizes a triangular superlattice with a small hopping (of approximately 10 Kelvin), with tunable density of holes, we explore the Hubbard model on the triangular lattice for intermediate temperatures $t \lesssim T < U$. Employing finite temperature Lanczos calculations, and closely following the fitting protocols used in the experiment, we recover the observed trends in the reported Curie-Weiss temperature $\Theta$ with filling, using the reported interaction strength $U/t=20$. We focus on the large increase of $|\Theta|$ on decreasing the density below half filling and the sign change of $\Theta$ at higher fillings, which signals the onset of ferromagnetism. The increase in $|\Theta|$ is also seen in the $t$-$J$ model (the low energy limit of the Hubbard model) in the intermediate temperature range, which we clarify is opposite to the trend in its high temperature limit. Differences between the low, intermediate and high temperature regimes are discussed. Our numerical calculations also capture the crossover between short-range antiferromagnetic to ferromagnetic order in the intermediate temperature regime, a result broadly consistent with the experimental findings. We find that this behavior is a finite-temperature remnant of the underlying zero temperature phase transition, which we explore with ground state density matrix renormalization group calculations. We provide evidence of ferromagnetism characterized by weak (but robust) correlations that explain the small $\Theta$ seen in the experiment.

Comparative study of charge order in undoped infinite-layer nickelate superconductors

To understand the microscopic mechanism of the charge order observed in the parent compound of the infinite-layer nickelate superconductors, we consider a minimal three-legged model consisting of a two-legged Hubbard ladder for the Ni $3d_{x^2-y^2}$ electrons and a free conduction electron chain from the rare-earths. With highly accurate density matrix renormalization group calculations, when the chemical potential difference is adjusted to make the Hubbard ladder with $1/3$ hole doping, we find a long-range charge order with period $3$ in the ground state of the model, while the spin excitation has a small energy gap. Moreover, the electron pair-pair correlation has a quasi-long-range behavior, indicating an instability of superconductivity even at half-filling. As a comparison, the same method is applied to a pure two-legged Hubbard model with $1/3$ hole doping in which the period-3 charge order is a quasi-long range one. The difference between them demonstrates that the free electron chain of the three-legged ladder plays the role of a charge reservoir and enhances the charge order in the undoped infinite-layer nickelates.

Ground-state phase diagram of the extended two-leg t−J ladder

Inspired by the observation of a robust $d$-wave superconducting phase driven by tuning the next-nearest-neighbor (NNN) electron hopping in recent density matrix renormalization group (DMRG) studies of six- and eight-leg $t$-$J$ model, we systematically study the phase diagram of the two-leg $t$-$J$ ladder with the NNN couplings (including NNN hopping and spin interaction) in a large region of doping level, by means of the DMRG calculations. Upon doping from half filling, we identify the Luther-Emery liquid (LEL) phase, which can be distinguished as the pairing-dominant and charge density-dominant regime by comparing the Luttinger parameter $K_{\rho}$. With the growing NNN couplings, pairing correlations are enhanced and correspondingly $K_{\rho}$ increases, driving the system from the density-dominant to the pairing-dominant regime. In the Tomonaga-Luttinger liquid (TLL) phase in the larger doping region, we identify two TLL regimes with different features of charge density correlation. At the quarter filling ($1/2$ doping level), we find that the strong dimer orders of bond energy in the open system actually decay algebraically and thus do not indicate a spontaneous translational symmetry breaking. Our results show that in the LEL phase of the two-leg ladder, the NNN couplings seem to play the similar role as that on the wider $t$-$J$ cylinder, and studies on this more accessible system can be helpful towards understanding the emergence of the remarkable $d$-wave superconducting phase on the wider system.

One-dimensional projection of two-dimensional systems using spiral boundary conditions

The spiral boundary conditions can project any two-dimensional or three-dimensional lattice onto a one-dimensional periodic chain with translational symmetry. It enables one to apply efficiently the existing one-dimensional techniques, such as density-matrix renormalization group (DMRG), bosonization, Jordan-Wigner transformation, etc., to studies of lattice systems in higher dimensions. Here, the authors demonstrate the utility of spiral boundary conditions by performing analytical and DMRG calculations for square and honeycomb lattice problems. Thus, the spiral boundary conditions emerge as a principal option in condensed matter physics.

Superconductivity of repulsive spinless fermions with sublattice potentials

We explore unconventional superconductivity of repulsive spinless fermions on square and honeycomb lattices with staggered sublattice potentials. The two lattices can exhibit staggered $d$-wave and $f$-wave pairing, respectively, at low doping stemming from an effective two-valley band structure. At higher doping, in particular, the square lattice displays a much richer phase diagram including topological $p+ip$ superconductivity which is induced by a qualitatively different mechanism compared to the $d$-wave pairing. We illuminate this from several complementary perspectives: We analytically perform sublattice projection to analyze the effective continuum low-energy description and we numerically calculate the binding energies for pair and larger bound states for few-body doping near half filling. Furthermore, for finite doping, we present phase diagrams based on extensive functional renormalization group and and density matrix renormalization group calculations.

Mutual information prediction for strongly correlated systems

We have trained a new machine-learning (ML) model which predicts mutual information (MI) for strongly correlated systems. This is a complex quantity, which is much more difficult to predict than one-site entropies, but carries important information about the correlation structure inside electronic systems. In this work, we replaced the expensive density matrix renormalization group (DMRG) calculations by newly trained ML model for prediction of the mutual information. We show the performance of the model on two important tasks: (a) to determine the correlation structure and (b) to determine ordering of orbitals for accurate DMRG calculations. The results are compared with the MI obtained from accurate DMRG calculations.

Unraveling the nature of spin excitations disentangled from charge contributions in a doped cuprate superconductor

The nature of the spin excitations in superconducting cuprates is a key question toward a unified understanding of the cuprate physics from long-range antiferromagnetism to superconductivity. The intense spin excitations up to the over-doped regime revealed by resonant inelastic X-ray scattering bring new insights as well as questions like how to understand their persistence or their relation to the collective excitations in ordered magnets (magnons). Here, we study the evolution of the spin excitations upon hole-doping the superconducting cuprate Bi2Sr2CaCu2O8+δ by disentangling the spin from the charge excitations in the experimental cross section. We compare our experimental results against density matrix renormalization group calculations for a t-J-like model on a square lattice. Our results unambiguously confirm the persistence of the spin excitations, which are closely connected to the persistence of short-range magnetic correlations up to high doping. This suggests that the spin excitations in hole-doped cuprates are related to magnons—albeit short-ranged.

Gradient corrections to the local-density approximation in the one-dimensional Bose gas

The local density approximation (LDA) is the central technical tool in the modeling of quantum gases in trapping potentials. It consists in treating the gas as an assembly of independent mesoscopic fluid cells at equilibrium with a local chemical potential, and it is justified when the correlation length is larger than the size of the cells. The LDA is often regarded as a crude approximation, particularly in the ground state of the one-dimensional (1D) Bose gas, { where the correlation length is "therefore said to be" infinite (in the sense that correlation functions decay as a power law).} Here we take another look at the LDA. The local density $\rho(x)$ is viewed as a functional of the trapping potential $V(x)$, to which one applies a gradient expansion. The zeroth order in that expansion is the LDA. The first-order correction in the gradient expansion vanishes due to reflection symmetry. At second order, there are two corrections proportional to $d^2V/dx^2$ and $(dV/dx)^2$, and we propose a method to determine the corresponding coefficients by a perturbative calculation in the Lieb-Liniger model. This leads to an expression for the coefficients in terms of matrix elements of the density operator, which can in principle be evaluated numerically for an arbitrary coupling constant; here we show how to efficiently evaluate the coefficient associated to the curvature of the potential $d^2V/dx^2$, which dominates the deviation to LDA near local minima or maxima of the trapping potential. Both coefficients are evaluated analytically in the limits of infinite repulsion (hard-core bosons) and small repulsion (quasi-condensate).} The corrected LDA density profiles are compared to DMRG calculations, with significant improvement compared to zeroth-order LDA.

Emergent many-body composite excitations of interacting spin-1/2 trimers

Understanding exotic forms of magnetism in quantum spin systems is an emergent topic of modern condensed matter physics. Quantum dynamics can be described by particle-like carriers of information, known-as quasiparticles that appear from the collective behaviour of the underlying system. Spinon excitations, governing the excitations of quantum spin-systems, have been accurately calculated and precisely verified experimentally for the antiferromagnetic chain model. However, identification and characterization of novel quasiparticles emerging from the topological excitations of the spin system having periodic exchange interactions are yet to be obtained. Here, we report the identification of emergent composite excitations of the novel quasiparticles doublons and quartons in spin-1/2 trimer-chain antiferromagnet Na2Cu3Ge4O12 (having periodic intrachain exchange interactions J1-J1-J2) and its topologically protected quantum 1/3 magnetization-plateau state. The characteristic energies, dispersion relations, and dynamical structure factor of neutron scattering as well as macroscopic quantum 1/3 magnetization-plateau state are in good agreement with the state-of-the-art dynamical density matrix renormalization group calculations.

Fermionic interpretation of the quantum phase transition in XXZ magnets

Quantum phase transitions in quantum magnets constitute an important field that has attracted great interests. Although a variety of analytic and numerical methods have been introduced in this direction, faithful fermionic descriptions are still desired because they can transform the spin models to a form tractable by conventional many-body techniques, yielding more transparent physical pictures. We generalize the Chern-Simons fermionization approach and apply it to the XXZ quantum magnets. After fermionzation, the Ising interaction of the XXZ spin model leads to a fermion-fermion interaction. We show that the additional fermion-fermion interaction generates an interesting phase transition between two fermionic ordered states, i.e., the Chern-Simons superconductor and the Chern-Simons exciton insulator. We also demonstrate that this transition in the fermionic language essentially describes the quantum phase transition between the planar and out-of-plane N\'eel orders in the spin picture. The fermionic mean-field theory further leads to a nonlinear $\ensuremath{\sigma}$ model that describes the quantum phase transition, which is further supported by our density matrix renormalization group calculations in the original spin model. Our work introduces a fermionic interpretation of quantum phase transitions and advances the existing knowledge of quantum magnets.

One-particle entanglement for one-dimensional spinless fermions after an interaction quantum quench

Particle entanglement provides information on quantum correlations in systems of indistinguishable particles. Here, we study the one particle entanglement entropy for an integrable model of spinless, interacting fermions both at equilibrium and after an interaction quantum quench. Using both large scale exact diagonalization and time dependent density matrix renormalization group calculations, we numerically compute the one body reduced density matrix for the J-V model, as well as its post-quench dynamics. We include an analysis of the fermionic momentum distribution, showcasing its time evolution after a quantum quench. Our numerical results, extrapolated to the thermodynamic limit, can be compared with field theoretic bosonization in the Tomonaga-Luttinger liquid regime. Excellent agreement is obtained using an interaction cutoff that can be determined uniquely in the ground state.

Excitation spectra of fully correlated donor-acceptor complexes by density matrix renormalisation group

In this paper, we investigate the convergence properties of density matrix renormalisation group (DMRG) calculations for vertical electronic excitations performed on three bimolecular complexes motivated by Szalay's recent benchmark results [JCTC. 2020;16:7]. Besides the high-level coupled-cluster reference spectrum, the extrapolated truncation-free DMRG results are compared against several alternative solutions based on multireference configuration interaction and coupled-cluster theories. By taking advantage of the error cancellation effects, which are inherent in the state-averaged computational scheme, we demonstrate that the extrapolated solution based on only low bond-dimensional DMRG data could already provide quantitative predictions even for large active spaces. We also show that orbital optimisation improves further the accuracy of the DMRG results which systematically approach the excitation energies of coupled cluster single-double-triple at similar computational costs.

Designer flat bands: Topology and enhancement of superconductivity

We construct quasi one-dimensional topological and non-topological three-band lattices with tunable band gap and winding number of the flat band. Using mean field (MF) and exact density matrix renormalization group (DMRG) calculations, we show explicitly how the band gap affects pairing and superconductivity (SC) in a Hubbard model with attractive interactions. We show excellent agreement between MF and DMRG. When a phase twist is applied on the system, a phase difference appears between pairing order parameters on different sublattices, and this plays a very important role in the SC density. The SC weight, $D_s$, on the gapped topological, $W\neq0$, flat band increases linearly with interaction strength, $U$, for low values, and with a slope that depends on the details of the compact localized state at $U=0$. As $U\to 0$ for the gapped non-topological flat band ($W=0$), $D_s$ decays with a power law faster than quadratic but slower than exponential. This indicates that isolated non-topological flat bands are less beneficial to SC. In the gapless case (flat band touching the band above it), we find at low $U$ (both for $W=0$ and $W\neq 0$) that $D_s\propto U^\varphi$ with $\varphi<1$ contrary to the $U{\rm ln}\, ({\rm const.}/U)$ behavior reported in the literature. In other words, $D_s$ increases faster than linearly for low $U$ thus favoring SC at weak interaction more than the gapped case. For systems with touching bands, we observe that the one-body correlation length, $\xi$, diverges as a power law as $U\rightarrow0$, while for the isolated flat band $\xi(U\to 0)$ is a constant smaller than one lattice spacing. Both behaviors are distinct from the exponential divergence of $\xi$ in the dispersive case. Our results re-establish that the BCS mean field and quantum metric alone are insufficient to characterize SC at weak coupling.

Crossover of high-energy spin fluctuations from collective triplons to localized magnetic excitations in Sr14−xCaxCu24O41 ladders

We studied the magnetic excitations in the quasi-one-dimensional (q-1D) ladder subsystem of Sr14−xCaxCu24O41 (SCCO) using Cu L3-edge resonant inelastic X-ray scattering (RIXS). By comparing momentum-resolved RIXS spectra with high (x = 12.2) and without (x = 0) Ca content, we track the evolution of the magnetic excitations from collective two-triplon (2 T) excitations (x = 0) to weakly-dispersive gapped modes at an energy of 280 meV (x = 12.2). Density matrix renormalization group (DMRG) calculations of the RIXS response in the doped ladders suggest that the flat magnetic dispersion and damped excitation profile observed at x = 12.2 originates from enhanced hole localization. This interpretation is supported by polarization-dependent RIXS measurements, where we disentangle the spin-conserving ΔS = 0 scattering from the predominant ΔS = 1 spin-flip signal in the RIXS spectra. The results show that the low-energy weight in the ΔS = 0 channel is depleted when Sr is replaced by Ca, consistent with a reduced carrier mobility. Our results demonstrate that off-ladder impurities can affect both the low-energy magnetic excitations and superconducting correlations in the CuO4 plaquettes. Finally, our study characterizes the magnetic and charge fluctuations in the phase from which superconductivity emerges in SCCO at elevated pressures.

Observation of competing, correlated ground states in the flat band of rhombohedral graphite.

In crystalline solids, the interactions of charge and spin can result in a variety of emergent quantum ground states, especially in partially filled, topological flat bands such as Landau levels or in "magic angle" graphene layers. Much less explored is rhombohedral graphite (RG), perhaps the simplest and structurally most perfect condensed matter system to host a flat band protected by symmetry. By scanning tunneling microscopy, we map the flat band charge density of 8, 10, 14, and 17 layers and identify a domain structure emerging from a competition between a sublattice antiferromagnetic insulator and a gapless correlated paramagnet. Our density matrix renormalization group calculations explain the observed features and demonstrate that the correlations are fundamentally different from graphene-based magnetism identified until now, forming the ground state of a quantum magnet. Our work establishes RG as a platform to study many-body interactions beyond the mean-field approach, where quantum fluctuations and entanglement dominate.

Magnetization process and ordering of the S=12 pyrochlore Heisenberg antiferromagnet in a magnetic field

We study the $S=\frac{1}{2}$ pyrochlore Heisenberg antiferromagnet in a magnetic field. Using large-scale density-matrix renormalization group calculations for clusters with up to 128 spins, we find indications of a finite triplet gap, causing a threshold field to nonzero magnetization in the magnetization curve. We obtain a robust saturation field consistent with a magnon crystal, although the corresponding $5/6$ magnetization plateau is very slim and possibly unstable. Most remarkably, there is a pronounced and apparently robust $1/2$ magnetization plateau where the ground state breaks the rotational symmetry of the lattice, exhibiting oppositely polarized spins on alternating kagome and triangular planes. Reminiscent of the kagome ice plateau of the pyrochlore Ising antiferromagnet known as spin ice, it arises via a much more subtle ``quantum order by disorder'' mechanism.

Theory of the dark state of polyenes and carotenoids

A theory is developed to describe the singlet dark state (usually labeled S1 or 2Ag) of polyenes and carotenoids. The theory assumes that in principle this state is a linear combination of a singlet triplet-pair and an odd-parity charge-transfer exciton. Crucially, these components only couple when the triplet-pair occupies neighboring dimers, such that an electron transfer between the triplets creates a nearest-neighbor charge-transfer excitation. This local coupling stabilises the 2Ag state and induces a nearest neighbor attraction between the triplets. In addition, because of the electron-hole attraction in the exciton, the increased probability that the electron-hole pair occupies neighboring dimers enhances the triplet-triplet attraction: the triplet pair is `slaved' to the charge-transfer exciton. The theory also predicts that as the Coulomb interaction is increased, the 2Ag state evolves from a predominately odd-parity charge-transfer exciton state with a small component of triplet-pair character to a state predominately composed of a triplet-pair with some exciton character. Above a critical Coulomb interaction there is a decoupling of the triplet-pair and charge-transfer exciton subspaces, such that the 2Ag state becomes entirely composed of an unbound spin-correlated triplet pair. The predictions of this theory are qualitatively consistent with high-level density matrix renormalization group calculations of the Pariser-Parr-Pople (or extended Hubbard) model.

Integer quantum Hall effect of two-component hard-core bosons in a topological triangular lattice

We study the many-body ground states of two-component hardcore bosons in topological triangular lattice models. Utilizing exact diagonalization and density-matrix renormalization group calculations, we demonstrate that at commensurate two-thirds filling per lattice site, two-component bosonic integer quantum hall (BIQH) effect emerges with the associated $\mathbf{K}=\begin{pmatrix} 0 & 1\\ 1 & 0\\ \end{pmatrix}$ matrix under strong intercomponent Hubbard repulsion. The topological nature is further elucidated by (i) a unique ground state degeneracy with a robust spectrum gap, (ii) a quantized topological Chern number matrix $\mathbf{C}=\mathbf{K}^{-1}$, and (iii) two counterpropagating edge branches. Moreover, with increasing nearest-neighbor repulsions, the ground state undergoes a first-order transition from a BIQH liquid to a commensurate solid order.

Exploring optimal multimode vibronic pathways in singlet fission of azaborine analogues of perylene.

The development of new singlet fission chromophores is a vibrant area of research to explore the possibility of efficient photovoltaic devices. Using high-level ab-initio density matrix renormalization group calculations, we present a systematic analysis of BN-doped perylenes for their potential application as singlet fission candidates. Four singlet fission chromophores are identified considering the monomer-based properties and their excitonic characters are further analyzed in a dimer configuration optimized in a six-dimensional space for local maxima of fission rates. Furthermore, a multistate multimode vibronic Hamiltonian is employed to identify intra- and interstate vibrational pathways for excitation energy modulation. Several photophysical properties such as Davydov splitting, activation energy and vibronic admixture of multiexcitonic and charge-transfer states are calculated for physically accessible dimers. The optimal dimer packing results in appropriate vibrational relaxation of singlet fission states and promotes significant population transfer which would be more attenuated without such couplings. This work not only identifies potential singlet fission systems with favorable electronic properties but also highlights the sensitivity of dimer packings with respect to the substitution patterns in singlet fission chromophores.