Density Matrix Renormalization Group Calculations Research Articles

Modeling the spin-Peierls transition of spin- 12 chains with correlated states: J1−J2 model, CuGeO3 , and TTF–CuS4C4(CF3)4

The spin-Peierls transition at $T_{SP}$ of spin-$1/2$ chains with isotropic exchange interactions has previously been modeled as correlated for $T > T_{SP}$ and mean field for $T < T_{SP}$. We use correlated states throughout in the $J_1-J_2$ model with antiferromagnetic exchange $J_1$ and $J_2 = \alpha J_1$ between first and second neighbors, respectively, and variable frustration $0 \leq \alpha \leq 0.50$. The thermodynamic limit is reached at high $T$ by exact diagonalization of short chains and at low $T$ by density matrix renormalization group calculations of progressively longer chains. In contrast to mean field results, correlated states of 1D models with linear spin-phonon coupling and a harmonic adiabatic lattice provide an internally consistent description in which the parameter $T_{SP}$ yields both the stiffness and the lattice dimerization $\delta(T)$. The relation between $T_{SP}$ and $\Delta(\delta,\alpha)$, the $T = 0$ gap induced by dimerization, depends strongly on $\alpha$ and deviates from the BCS gap relation that holds in uncorrelated spin chains. Correlated states account quantitatively for the magnetic susceptibility of TTF-CuS$_4$C$_4$(CF$_3$)$_4$ crystals ($J_1 = 79$ K, $\alpha = 0$, $T_{SP} = 12$ K) and CuGeO$_3$ crystals ($J_1 = 160$ K, $\alpha = 0.35$, $T_{SP} = 14$ K). The same parameters describe the specific heat anomaly of CuGeO$_3$ and inelastic neutron scattering. Modeling the spin-Peierls transition with correlated states exploits the fact that $\delta(0)$ limits the range of spin correlations at $T = 0$ while $T > 0$ limits the range at $\delta= 0$.

Continuous phase transition between bosonic integer quantum Hall liquid and a trivial insulator: Evidence for deconfined quantum criticality

The deconfined quantum critical point, a prototype Landau-forbidden transition, could exist, in principle, in the phase transitions involving a symmetry-protected topological phase; however, examples of such kinds of transitions in physical systems are rare beyond one-dimensional systems. Here, using a density-matrix renormalization-group calculation, we unveil a bosonic integer quantum Hall phase in a two-dimensional correlated honeycomb lattice, by full identification of its internal structure from the topological $\mathbf{K}$ matrix. Moreover, we demonstrate that imbalanced periodic chemical potentials can destroy the bosonic integer quantum Hall state and drive it into a featureless trivial (Mott) insulator, where all physical observables evolve smoothly across the critical point. At the critical point, the entanglement entropy reveals a characteristic scaling behavior, which is consistent with the critical field theory as an emergent ${\mathrm{QED}}_{3}$ with two flavors of Dirac fermions.

Density matrix renormalization group study of nematicity in two dimensions: Application to a spin-1 bilinear-biquadratic model on the square lattice

Nematic order is an exotic property observed in several strongly correlated systems, such as the iron-based superconductors. Using large-scale density matrix renormalization group (DMRG) techniques, we study at zero-temperature the nematic spin liquid that competes with spin dipolar and quadrupolar orders. We use these nematic orders to characterize different quantum phases and quantum phase transitions. More specifically, we study a spin-$1$ bilinear-biquadratic Heisenberg model on the square lattice with couplings beyond nearest neighbors. We focus on parameter regions around the highly symmetric $SU(3)$ point where the bilinear and biquadratic interactions are equal. With growing further-neighbor biquadratic interactions, we identify different spin dipolar and quadrupolar orders. We find that the DMRG results on cylindrical geometries correctly detect nematicity in different quantum states and accurately characterize the quantum phase transitions among them. Therefore, spin-driven nematicity -- here defined as the spontaneous breaking of the lattice invariance under a 90$^o$ rotation -- is an order parameter which can be studied directly in DMRG calculations in two dimensions in different quantum states.

Characterizing the phase diagram of finite-size dipolar Bose-Hubbard systems

We use state-of-the-art density matrix renormalization group calculations in the canonical ensemble to determine the phase diagram of the dipolar Bose-Hubbard model on a finite cylinder. We consider several observables that are accessible in typical optical lattice setups and assess how well these quantities perform as order parameters. We find that, especially for small systems, the occupation imbalance is less susceptible to boundary effects than the structure factor in uncovering the presence of a periodic density modulation. By analysing the non-local correlations, we find that the appearance of supersolid order is very sensitive to boundary effects, which may render it difficult to observe in quantum gas lattice experiments with a few tens of particles. Finally, we show how density measurements readily obtainable on a quantum gas microscope allow distinguishing between superfluid and solid phases using unsupervised machine-learning techniques.

Measurement-induced phase transition: A case study in the nonintegrable model by density-matrix renormalization group calculations

We study the effect of local projective measurements on the quantum quench dynamics. As a concrete example, a one-dimensional Bose-Hubbard model is simulated by the matrix product state and time-evolving block decimation. We map out a global phase diagram in terms of the measurement rate in spatial space and time domain, which demonstrates a volume-to-area law entanglement phase transition. When the measurement rate reaches the critical value, we observe a logarithmic growth of entanglement entropy as the subsystem size or evolved time increases. Moreover, we find that the probability distribution of the single-site entanglement entropy distinguishes the volume and area law phases, similar to the case of disorder-induced many-body localization. We also investigate the scaling behavior of entanglement entropy and mutual information between two separated sites, which is indicative of a single universality class and thus suggests a possible unified description of this transition.

Induced pairing of fermionic impurities in a one-dimensional strongly correlated Bose gas

We investigate numerically the problem of few (one, two) noninteracting spin$-1/2$ fermions in a shallow harmonic trap coupled via contact repulsive interactions to a uniform one-dimensional bath of lattice bosons, described by the Bose-Hubbard model. Through extensive density-matrix renormalization group calculations, we extract the binding energy and the effective mass of quasiparticles, including dressed impurities (polarons) and their two-body bound states (bipolarons), emerging from the effective non-local Casimir interaction between the impurities. We show that the mixture exhibits rather different pairing behaviors depending on the singlet \textit{vs}.~triplet spin state configurations of the two fermions. For opposite spin states, bipolarons are found for any finite value of the impurity-bath coupling. In particular, in the strong coupling regime their binding energy reduces to that of a single polaron, provided the boson-boson repulsion is not too weak. For equal spin states, we show that bipolarons emerge only beyond a critical strength of the Bose-Fermi interaction and their effective mass grows rapidly approaching the strong coupling regime.

Chiral spin liquid with spinon Fermi surfaces in the spin- 12 triangular Heisenberg model

We study the interplay of competing interactions in spin-$1/2$ triangular Heisenberg model through tuning the first- ($J_1$), second- ($J_2$), and third-neighbor ($J_3$) couplings. Based on large-scale density matrix renormalization group calculation, we identify a quantum phase diagram of the system and discover a new {\it gapless} chiral spin liquid (CSL) phase in the intermediate $J_2$ and $J_3$ regime. This CSL state spontaneously breaks time-reversal symmetry with finite scalar chiral order, and it has gapless excitations implied by a vanishing spin triplet gap and a finite central charge on the cylinder. Moreover, the central charge grows rapidly with the cylinder circumference, indicating emergent spinon Fermi surfaces. To understand the numerical results we propose a parton mean-field spin liquid state, the $U(1)$ staggered flux state, which breaks time-reversal symmetry with chiral edge modes by adding a Chern insulator mass to Dirac spinons in the $U(1)$ Dirac spin liquid. This state also breaks lattice rotational symmetries and possesses two spinon Fermi surfaces driven by nonzero $J_2$ and $J_3$, which naturally explains the numerical results. To our knowledge, this is the first example of a gapless CSL state with coexisting spinon Fermi surfaces and chiral edge states, demonstrating the rich family of novel phases emergent from competing interactions in triangular-lattice magnets.

Quantum Adiabatic Doping with Incommensurate Optical Lattices.

Quantum simulations of Fermi-Hubbard models have been attracting considerable effort in the optical lattice research, with the ultracold antiferromagnetic atomic phase reached at half filling in recent years. An unresolved issue is to dope the system while maintaining the low thermal entropy. Here we propose to achieve the low temperature phase of the doped Fermi-Hubbard model using incommensurate optical lattices through adiabatic quantum evolution. In this theoretical proposal, we find that one major problem about the adiabatic doping is atomic localization in the incommensurate lattice, potentially causing an exponential slowing down of the adiabatic procedure. We study both one- and two-dimensional incommensurate optical lattices, and find that the localization prevents efficient adiabatic doping in the strong lattice regime for both cases. With density matrix renormalization group calculation, we further show that the slowing down problem in one dimension can be circumvented by considering interaction induced many-body delocalization, which is experimentally feasible using Feshbach resonance techniques. This protocol is expected to be efficient as well in two dimensions where the localization phenomenon is less stable.

Time- and momentum-resolved tunneling spectroscopy of pump-driven nonthermal excitations in Mott insulators

We present a computational technique to calculate time and momentum resolved non-equilibrium spectral density of correlated systems using a tunneling approach akin scanning tunneling spectroscopy. The important difference is that our probe is extended, basically a copy of the sample, allowing one to extract the momentum information of the excitations. We illustrate the method by measuring the spectrum of a Mott-insulating extended Hubbard chain after a sudden quench with the aid of time-dependent density matrix renormalization group (tDMRG) calculations. We demonstrate that the system realizes a non-thermal state that is an admixture of spin and charge density wave states, with corresponding signatures that are recognizable as in-gap sub-bands. In particular, we identify a band of excitons and one of stable anti-bound states at high energies that gains enhanced visibility after the pump. We do not appreciate noticeable relaxation within the time-scales considered, which is attributed to the lack of decay channels due to spin-charge separation. These ideas can be readily applied to study transient dynamics and spectral signatures of correlation-driven non-equilibrium processes.

Nematic spin liquid phase in a frustrated spin-1 system on the square lattice

Frustration in quantum spin systems promote a variety of novel quantum phases. An important example is the frustrated spin-$1$ model on the square lattice with the nearest-neighbor bilinear ($J_1$) and biquadratic ($K_1$) interactions. We provide strong evidence for a nematic spin liquid phase in a range of $K_1/J_1$ near the SU(3)-symmetric point ($J_1 = K_1$), based on the linear flavor-wave theory and extensive density matrix renormalization group calculation. This phase displays no spin dipolar or quadrupolar order, preserves translational symmetry but spontaneously breaks $C_4$ lattice rotational symmetry, and possesses fluctuations peaked at the wavevector $(\pi, 2\pi/3)$. The spin excitation gap drops rapidly with system size and appears to be gapless, and the nematic order is attributed to the dominant $(\pi, 2\pi/3)$ fluctuations. Our results provide a novel mechanism for electronic nematic order and, more generally, open up a new avenue to explore frustration-induced exotic ground states.

Disorder and interaction in chiral chains: Majoranas versus complex fermions

We study the low-energy physics of a chain of Majorana fermions in the presence of interaction and disorder, emphasizing the difference between Majoranas and conventional (complex) fermions. While in the non-interacting limit both models are equivalent (in particular, belong to the same symmetry class BDI and flow towards the same infinite-randomness critical fixed point), their behavior differs drastically once interaction is added. Our density-matrix renormalization group calculations show that the complex-fermion chain remains at the non-interacting fixed point. On the other hand, the Majorana fermion chain experiences a spontaneous symmetry breaking and localizes for repulsive interaction. To explain the instability of the critical Majorana chain with respect to a combined effect of interaction and disorder, we consider interaction as perturbation to the infinite-randomness fixed point and calculate numerically two-wavefunction correlation functions that enter interaction matrix elements. The numerical results supported by analytical arguments exhibit a rich structure of critical eigenstate correlations. This allows us to identify a relevant interaction operator that drives the Majorana chain away from the infinite randomness fixed point. For the case of complex fermions, the interaction is irrelevant.

Orbital Entanglement Analysis of Exchange-Coupled Systems.

A new tool for the interpretation of multiconfigurational wave functions representing the spin states of exchange-coupled transition metal complexes is introduced. Based on orbital entanglement measures, herein derived from multiconfigurational density matrix renormalization group calculations, the complexity of the wave function is reduced, thus facilitating a connection with established concepts for the interpretation of magnetically coupled systems. We show that the entanglement of localized orbitals with a small basis set is a good representation of the magnetic coupling topology and that it is sensitive to chemical changes in homologous complexes. Furthermore, we introduce a measure for the magnetic relevance of orbitals in the active subspace and a concept for the quantitative comparison of different chemical species. The approach presented here will be easily applicable to higher nuclearity clusters, providing a direct insight into all states of the Heisenberg spin ladder for systems previously accessible only by single-configurational methods.

Quantum Kibble-Zurek physics in long-range transverse-field Ising models

We analyze the quantum phase transitions taking place in a one-dimensional transverse field Ising model with long-range couplings that decay algebraically with distance. We are interested in the Kibble-Zurek universal scaling laws emerging in non-equilibrium dynamics and in the potential for the unambiguous observation of such behavior in a realistic experimental setup based on trapped ions. To this end, we determine the phase diagram of the model and the critical exponents characterizing its quantum phase transitions by means of density-matrix renormalization group calculations and finite-size scaling theory, which allows us to obtain good estimates for different range of ferro- and antiferromagnetic interactions. Beyond critical equilibrium properties, we tackle a non-equilibrium scenario in which quantum Kibble-Zurek scaling laws may be retrieved. Here it is found that the predicted non-equilibrium universal behavior, i.e. the scaling laws as a function of the quench rate and critical exponents, can be observed in systems comprising an experimentally feasible number of spins. Finally, a scheme is introduced to simulate the algebraically decaying couplings accurately by means of a digital quantum simulation with trapped ions. Our results suggest that quantum Kibble-Zurek physics can be explored and observed in state-of-the-art experiments with trapped ions realizing long-range Ising models.

Natural-orbital impurity solver and projection approach for Green's functions

We extend a previously proposed rotation and truncation scheme to optimize quantum Anderson impurity calculations with exact diagonalization [PRB 90, 085102 (2014)] to density-matrix renormalization group (DMRG) calculations. The method reduces the solution of a full impurity problem with virtually unlimited bath sites to that of a small subsystem based on a natural impurity orbital basis set. The later is solved by DMRG in combination with a restricted-active-space truncation scheme. The method allows one to compute Green's functions directly on the real frequency or time axis. We critically test the convergence of the truncation scheme using a one-band Hubbard model solved in the dynamical mean-field theory. The projection is exact in the limit of both infinitely large and small Coulomb interactions. For all parameter ranges the accuracy of the projected solution converges exponentially to the exact solution with increasing subsystem size.

Out-of-equilibrium transport in the interacting resonant level model: Surprising relevance of the boundary sine-Gordon model

Using time-dependent density matrix renormalization group calculations we study the transport properties ($I-V$ curves and shot noise) of the interacting resonant level model (IRLM) in a large range of the interaction parameter $U$, in the scaling limit. We find that these properties can be described remarkably well by those of the Boundary sine-Gordon model (BSG), which are known analytically (Fendley, Ludwig and Saleur, 1995). We argue that the two models are nevertheless in different universality classes out of equilibrium: this requires a delicate discussion of their infra-red (IR) properties ({\it i.e.} at low voltage), where we prove in particular that the effective tunneling charge is $e$ in the infra-red regime of the IRLM (except at the self-dual point where it jumps to $2e$), while it is known to be a continuously varying function of $U$ in the BSG. This behavior is confirmed by careful analysis of the numerical data in the IR. The remarkable agreement of the transport properties, especially in the crossover region, remains however unexplained.

Quantification of electron correlation effects: Quantum Information Theory vs Method of Increments

AbstractUnderstanding electron correlation is crucial for developing new concepts in electronic structure theory, especially for strongly correlated electrons. We compare and apply two different approaches to quantify correlation contributions of orbitals: Quantum Information Theory (QIT) based on a Density Matrix Renormalization Group (DMRG) calculation and the Method of Increments (MoI). Although both approaches define very different correlation measures, we show that they exhibit very similar patterns when being applied to a polyacetelene model system. These results suggest one may deduce from one to the other, allowing the MoI to leverage from QIT results by screening correlation contributions with a cheap (“sloppy”) DMRG with a reduced number of block states. Or the other way around, one may select the active space in DMRG from cheap one‐body MoI calculations.

Hybrid exact diagonalization and density matrix renormalization group approach to the thermodynamics of one-dimensional quantum models

Exact diagonalization of small model systems gives the thermodynamics of spin chains or quantum cell models at high temperatures $T$. Density matrix renormalization group calculations of progressively larger systems are used to obtain excitations up to a cutoff ${W}_{C}$ and the low-$T$ thermodynamics. The hybrid approach is applied to the magnetic susceptibility $\ensuremath{\chi}(T)$ and specific heat $C(T)$ of spin-$1/2$ chains with isotropic exchange such as the linear Heisenberg antiferromagnet and the frustrated ${J}_{1}\ensuremath{-}{J}_{2}$ model with ferromagnetic ${J}_{1}l0$ and antiferromagnetic ${J}_{2}g0$. The hybrid approach is fully validated by comparison with Heisenberg antiferromagnet results. It extends ${J}_{1}\ensuremath{-}{J}_{2}$ thermodynamics down to $T\ensuremath{\sim}0.01|{J}_{1}|$ for ${J}_{2}/|{J}_{1}|\ensuremath{\ge}{\ensuremath{\alpha}}_{c}=1/4$ and is consistent with other methods. The criterion for the cutoff ${W}_{C}(N)$ in systems of $N$ spins is discussed. The cutoff leads to bounds for the thermodynamic limit that are best satisfied at a specific $T(N)$ at system size $N$.

The df-d Dative Bonding in a Uranium-Cobalt Heterobimetallic Complex for Efficient Nitrogen Fixation.

Single transition-metal site catalysts with s-, p-, or d-block atom anchor for nitrogen fixation have been extensively studied, and yet the studies of the f-block atom anchor are rarely reported. Thus, we investigate the feasibility of using a newly synthesized U-Co complex featuring a single CoI site coordinated by tetrakis(phophinoamide) and an UIV anchor for N2-to-NH3 conversion by theoretical modeling. We characterize the evolution of oxidation states of U and Co along the reaction pathways from ab initio density matrix renormalization group (DMRG) calculations, and we find that the variation of the Co → U dative bond is correlated with the changes of oxidation states. Both uranium and cobalt can serve as electron reservoirs to facilitate breaking the N-N bond. Our study demonstrates the viability of metal → metal dative bonds, particularly the df-d one, for the reduction of N2 to NH3, and thus, this opens up a new avenue to the rational design of efficient catalyst for nitrogen fixation.

Pair Density Wave in the Doped t-J Model with Ring Exchange on a Triangular Lattice.

In our previous work [Phys. Rev. Lett. 121, 046401 (2018)PRLTAO0031-900710.1103/PhysRevLett.121.046401], we found a quantum spin liquid phase with a spinon Fermi surface in the two dimensional spin-1/2 Heisenberg model with four-spin ring exchange on a triangular lattice. In this work we dope the spinon Fermi surface phase by studying the t-J model with four-spin ring exchange. We perform density matrix renormalization group calculations on four-leg cylinders of a triangular lattice and find that the dominant pair correlation function is that of a pair density wave; i.e., it is oscillatory while decaying with distance with a power law. The doping dependence of the period is studied. This is the first example where a pair density wave is the dominant pairing in a generic strongly interacting system where the pair density wave cannot be explained as a composite order and no special symmetry is required.

Kosterlitz-Thouless and Gaussian criticalities in a mixed spin-( 12, 52, 12 ) Heisenberg branched chain with exchange anisotropy

Density-matrix renormalization group calculations are used to determine the ground-state phase diagram of the mixed spin-(1/2, 5/2, 1/2) Heisenberg chain whose backbone consists of regularly alternating $s=1/2$ and $S=5/2$ spins, the latter of which are coupled to additional $s=1/2$ spins providing lateral branching. The proposed magnetic structure aims to describe the main characteristics of the heterotrimetallic coordination polymer $[{\mathrm{Cu}}^{\mathrm{II}}{\mathrm{Mn}}^{\mathrm{II}}({\mathrm{L}}^{1})][{\mathrm{Fe}}^{\mathrm{III}}(\mathrm{bpb}){(\mathrm{CN})}_{2}]\phantom{\rule{4pt}{0ex}}\ifmmode\cdot\else\textperiodcentered\fi{}\phantom{\rule{4pt}{0ex}}{\mathrm{ClO}}_{4}\phantom{\rule{4pt}{0ex}}\ifmmode\cdot\else\textperiodcentered\fi{}\phantom{\rule{4pt}{0ex}}{\mathrm{H}}_{2}\mathrm{O}$. The full ground-state phase diagram in the parameter space exchange anisotropy versus magnetic field unveils special critical points, at which the intermediate magnetization plateaus emergent at 3/7 and 5/7 of the saturation magnetization vanish. A detailed description of the zero-temperature magnetization process and pair entanglement entropy is accomplished with the aim to distinguish the main coupling regimes. A scaling analysis is performed to accurately locate Kosterlitz-Thouless and Gaussian critical points. In particular, we introduce a new finite-size scaling protocol to properly extract the critical parameters from the size dependence of the magnetic-field range corresponding to the magnetization plateau in the close vicinity of the Gaussian point at which two gapped phases meet.