Density Matrix Renormalization Group Technique Research Articles

Boundary quantum phase transitions in the spin- 12 Heisenberg chain with boundary magnetic fields

We consider the spin $\frac{1}{2}$ Heisenberg chain with boundary magnetic fields and analyze it using a combination of Bethe ansatz and density matrix renormalization group (DMRG) techniques. We show that the system exhibits several different ground states which depend on the orientation of the boundary magnetic fields. When both the boundary fields take equal values greater than a critical field strength, each edge in the ground state accumulates a fractional spin which saturates to spin $\frac{1}{4}$, which is similar to systems exhibiting symmetry protected topological phases (SPT). Unlike in SPT systems, the fractional boundary spin in the Heisenberg spin chain is not a genuine quantum number since the variance of the associated operator does not vanish, this is due to the absence of a bulk gap. The system exhibits high energy bound states when the boundary fields take values greater than the critical field. All the excitations in the system can be sorted out into towers whose number depends on the number of bound states exhibited by the system. As the boundary fields are varied, in addition to the ground state phase transition, we find that the system may undergo an eigenstate phase transition (EPT) where the number of towers of the Hilbert space changes. We further inquire how the EPT reflects itself on local ground state properties by computing the magnetization profile $\langle S^z_j \rangle$ using DMRG. We identify a clear qualitative change from low edge fields to high edge fields when crossing the critical field. We though are unable to conclude on the basis of our data that EPT corresponds to a genuine phase transition in the ground state.

Non-Abelian anyons with Rydberg atoms

We study the emergence of topological matter in two-dimensional systems of neutral Rydberg atoms in Ruby lattices. While Abelian anyons have been predicted in such systems, non-Abelian anyons, which would form a substrate for fault-tolerant quantum computing, have not been generated. To generate anyons with non-Abelian braiding statistics, we consider systems with mixed-boundary punctures. We obtain the topologically distinct ground states of the system numerically using the infinite Density Matrix Renormalization Group technique. We discuss how these topological states can be created using ancilla atoms of a different type. We show that a system with $2N+2$ punctures and an equal number of ancilla atoms leads to $N$ logical qubits whose Hilbert space is determined by a set of stabilizing conditions on the ancilla atoms. Quantum gates can be implemented using a set of gates acting on the ancilla atoms that commute with the stabilizers and realize the braiding group of non-Abelian Ising anyons.

Quantum criticality of a Z_{3}-symmetric spin chain with long-range interactions.

Based on large-scale density matrix renormalization group techniques, we investigate the critical behaviors of quantum three-state Potts chains with long-range interactions. Using fidelity susceptibility as an indicator, we obtain a complete phase diagram of the system. The results show that as the long-range interaction power α increases, the critical points f_{c}^{*} shift towards lower values. In addition, the critical threshold α_{c}(≈1.43) of the long-range interaction power is obtained for the first time by a nonperturbative numerical method. This indicates that the critical behavior of the system can be naturally divided into two distinct universality classes, namely the long-range (α<α_{c}) and short-range (α>α_{c}) universality classes, qualitatively consistent with the classical ϕ^{3} effective field theory. This work provides a useful reference for further research on phase transitions in quantum spin chains with long-range interaction.

Dynamical Simulations of Carotenoid Photoexcited States Using Density Matrix Renormalization Group Techniques.

We present a dynamical simulation scheme to model the highly correlated excited state dynamics of linear polyenes. We apply it to investigate the internal conversion processes of carotenoids following their photoexcitation. We use the extended Hubbard-Peierls model, , to describe the π-electronic system coupled to nuclear degrees of freedom. This is supplemented by a Hamiltonian, , that explicitly breaks both the particle-hole and two-fold rotation symmetries of idealized carotenoid structures. The electronic degrees of freedom are treated quantum mechanically by solving the time-dependent Schrödinger equation using the adaptive time-dependent DMRG (tDMRG) method, while nuclear dynamics are treated via the Ehrenfest equations of motion. By defining adiabatic excited states as the eigenstates of the full Hamiltonian, , and diabatic excited states as eigenstates of , we present a computational framework to monitor the internal conversion process from the initial photoexcited 11Bu+ state to the singlet triplet-pair states of carotenoids. We further incorporate Lanczos-DMRG to the tDMRG-Ehrenfest method to calculate transient absorption spectra from the evolving photoexcited state. We describe in detail the accuracy and convergence criteria for DMRG, and show that this method accurately describes the dynamical processes of carotenoid excited states. We also discuss the effect of the symmetry-breaking term, , on the internal conversion process, and show that its effect on the extent of internal conversion can be described by a Landau-Zener-type transition. This methodological paper is a companion to our more explanatory discussion of carotenoid excited state dynamics in Manawadu, D.; Georges, T. N.; Barford, W. Photoexcited State Dynamics and Singlet Fission in Carotenoids. J. Phys. Chem. A 2023, 127, 1342.

Study of Staggered Magnetization in the Spin-S Square-Lattice Heisenberg Model Using Spiral Boundary Conditions

We propose an efficient numerical method to obtain local order parameter in two-dimensional systems using spiral boundary conditions. As a benchmark, we first estimate the magnitude of staggered magnetization for the $S=1/2$ XXZ Heisenberg model on a square lattice in the whole range of the XXZ anisotropy by density-matrix renormalization group technique. The validity of our method is confirmed by comparing our results with the previous analytical and numerical studies. Then, as further demonstration, we apply our method to obtain the staggered magnetization for the higher-spin cases from $S=1$ to $S=6$. The accuracy of the obtained results is validated using the series expansion and the spin-wave theory.

Magnetic phases of spatially modulated spin-1 chains in Rydberg excitons: Classical and quantum simulations.

In this work, we study the magnetic phases of a spatially modulated chain of spin-1 Rydberg excitons. Using the Density Matrix Renormalization Group (DMRG) technique, we study various magnetic and topologically nontrivial phases using both single-particle properties, such as local magnetization and quantum entropy, and many-body ones, such as pair-wise Néel and long-range string correlations. In particular, we investigate the emergence and robustness of the Haldane phase, a topological phase of anti-ferromagnetic spin-1 chains. Furthermore, we devise a hybrid quantum algorithm employing restricted Boltzmann machine to simulate the ground state of such a system that shows very good agreement with the results of exact diagonalization and DMRG.

Low-lying excited states of Diphenylpolyenes and its derivatives in singlet fission: A Density Matrix Renormalization Group study

Singlet fission (SF) is a promising mechanism by which the power conversion efficiency (PCE) of the solar cells can be increased. α, ω-Diphenylpolyenes are known to draw research attention due to their fascinating optical and photophysical properties. The low-lying excited states of 1,6-diphenyl-1,3,5-hexatriene (DPH), 1,8-diphenyl-1,3,5,7-octatetraene (DPO), 1,10-diphenyl-1,3,5,7-decapentaene (DPD) molecules and their derivatives are calculated within Pariser–Parr–Pople(PPP) model Hamiltonian implementing Density Matrix Renormalization Group (DMRG) technique. The calculations reveal that all three molecules are promising candidates for SF mechanism. Excited state energies in singlet subspace have been calculated for 1,6-diphenyl-1,3,5-hexatriene (DPH) comprising 18 carbon atoms within exact diagonalization method which are in excellent agreement with results obtained in DMRG technique. Moreover, calculations on α, ω-diphenyl-μ, ν-dicyano oligoenes (DPDCs) show with increasing chain length, these derivatives support endoergic to isoergic SF

Phase diagram of a bilinear-biquadratic spin-1 model on the triangular lattice from density matrix renormalization group simulations

We investigate a highly frustrated spin-1 model on the triangular lattice, with nearest- and next-nearest-neighbor antiferromagnetic $S\ifmmode\cdot\else\textperiodcentered\fi{}S$ interactions and nearest-neighbor ${(S\ifmmode\cdot\else\textperiodcentered\fi{}S)}^{2}$ interactions. Using the density matrix renormalization group technique, we find three magnetically ordered phases, namely ${120}^{\ensuremath{\circ}}$ spiral order, stripe order, and tetrahedral order, as well as two spin nematic phases: ferroquadrupolar and antiferroquadrupolar. While our data could be consistent with a spin liquid phase between the ${120}^{\ensuremath{\circ}}$ spiral and antiferroquadrupolar orders, the more likely scenario is a direct continuous transition between these two orders.

Photoinduced metallization of excitonic insulators

Utilizing the time-dependent density-matrix renormalization group technique, we numerically prove photoinduced pairing states in the extended Falicov-Kimball model (EFKM) at half filling, both with and without internal SU(2) symmetry. In the time-dependent photoemission spectra an extra band appears above the Fermi energy after pulse irradiation, indicating an insulator-to-metal transition. Even in the absence of the SU(2) structure, the pair correlations are enhanced during the pump, and afterwards decrease over time. This implies the possible metallization of Ta$_2$NiSe$_5$, a strong candidate for an excitonic insulator material, for which the EFKM is considered to be the minimal theoretical model. Simulating the photoemission with optimized pulse parameters, we demonstrate a photoinduced quantum phase transition, in accord with recent findings in time- and angle-resolved photoemission spectroscopy experiments on Ta$_2$NiSe$_5$.

Entanglement entropy and negativity in the Ising model with defects

Defects in two-dimensional conformal field theories (CFTs) contain signatures of their characteristics. In this work, we analyze entanglement properties of subsystems in the presence of energy and duality defects in the Ising CFT using the density matrix renormalization group (DMRG) technique. In particular, we compute the entanglement entropy (EE) and the entanglement negativity (EN) in the presence of defects. For the EE, we consider the cases when the defect lies within the subsystem and at the edge of the subsystem. We show that the EE for the duality defect exhibits fundamentally different characteristics compared to the energy defect due to the existence of localized and delocalized zero energy modes. Of special interest is the nontrivial ‘finite-size correction’ in the EE obtained recently using free fermion computations [1]. These corrections arise when the subsystem size is appreciable compared to the total system size and lead to a deviation from the usual logarithmic scaling characteristic of one-dimensional quantum-critical systems. Using matrix product states with open and infinite boundary conditions, we numerically demonstrate the disappearance of the zero mode contribution for finite subsystem sizes in the thermodynamic limit. Our results provide further support to the recent free fermion computations, but clearly contradict earlier analytical field theory calculations based on twisted torus partition functions. Subsequently, we compute the logarithm of the EN (log-EN) between two disjoint subsystems separated by a defect. We show that the log-EN scales logarithmically with the separation of the subsystems. However, the coefficient of this logarithmic scaling yields a continuously-varying effective central charge that is different from that obtained from analogous computations of the EE. The defects leave their fingerprints in the subleading term of the scaling of the log-EN. Furthermore, the log-EN receives similar ‘finite size corrections’ like the EE which leads to deviations from its characteristic logarithmic scaling.

Entanglement renormalization in interacting fermionic systems using single-particle transformations

One of the most powerful and popular quantum entanglement-based computation methods for the simulation of one-dimensional and quasi-two-dimensional strongly correlated materials is the Density Matrix Renormalization Group (DMRG) technique. A lower quantum entanglement between the system’s degrees of freedom will result in a higher accuracy of the method. However, quantum entanglement between two complementary subsystems of a closed system depends on how the degrees of freedom have been distributed between them, and therefore, it is not in general invariant under arbitrary unitary transformations. Entanglement is invariant only under those unitary transformations which do not mix the degrees of freedom associated with the two subsystems. Consequently, a fundamental question is that if it is possible to construct, algorithmically, unitary transformations which minimize entanglement between degrees of freedom of the system and therefore maximize the DMRG technique? A general solution to this problem is highly nontrivial in general. In this paper, we study interacting fermionic models and by considering single-particle unitary transformations only. We define a novel and efficient method to find the optimal single-particle unitary transformations. In this framework, we show that using our algorithm, the accuracy of the DMRG method at a given bond dimension is increased and the ground-state energy is decreased and approaches to its exact value. Our method paves the way toward finding more general (many-particle) optimized unitary transformations and helps us to better understand the behavior of fermions in strongly correlated materials.

Kinetic formation of trimers and multimers in a spinless fermionic chain

We study a chain of spinless fermions with a multimer hopping term which kinematically favors the formation of multimers and competes with single-particle hopping. We argue that this model generically stabilizes two different multimer phases, as well as intermediate phases where the free-fermion and multimer fluids coexist and do not spatially separate. Using density-matrix renormalization group techniques, we establish the phase diagram of the model in the case of trimers. For one of the intermediate phases, hybridization between the fermions and trimers liquids does occur and the onset of their correlations is well captured by a generalized BCS-like ansatz. The case of tetramers is finally addressed.

Superuniversality of dimerized SU(N+M) spin chains

We explore the physics of the quantum Hall effect using the Haldane mapping of dimerised $SU(N+M)$ spin chains, the large $N$ expansion and the density matrix renormalization group technique. We show that while the transition is first order for $N+M >2$, the system at zero temperature nevertheless displays a continuously diverging length scale $\xi$ (correlation length). The numerical results for $(M, N) = (1,3), ~ (2, 2),~(1, 5)$ and $(1, 7)$ indicate that $\xi$ is a directly observable physical quantity, namely the spatial width of the edge states. We relate the physical observables of the quantum spin chain to those of the quantum Hall system (and, hence, the $\vartheta$ vacuum concept in quantum field theory). Our numerical investigations provide strong evidence for the conjecture of super universality which says the dimerised spin chain quite generally displays all the basic features of the quantum Hall effect, independent of the specific values of $M$ and $N$. For the cases at hand we show that the singularity structure of the quantum Hall plateau transitions involves a universal function with two scale parameters that may in general depend on $M$ and $N$. This includes not only the Hall conductance but also the ground state energy as well as the correlation length $\xi$ with varying values of $\vartheta \sim \pi$.

Quantum phases of Rydberg atoms on a frustrated triangular-lattice array.

The neutral atoms coupled to a highly excited Rydberg state on a two-dimensional triangular lattice are investigated by employing the density matrix renormalization group technique in the matrix product state form. The full ground-state phase diagram as a function of blockade radius and the detuning of the exciting laser is determined by the behavior of entanglement entropy. We find several quantum phases including stripe-ordered and symmetry-breaking density-wave-ordered phases featured with regular excitation patterns of different excitation densities ρ = 1/3, 1/4, and 1/7. In addition, a ρ = 2/3 ordered phase and an interesting "order-by-disorder" phase, which has been prepared experimentally, are also observed in this work. Our work provides an exploration of the possible quantum phases that can occur in a triangularly arrayed Rydberg system, and thus could be a faithful theoretical guide for further experimental research.

Spin- S impurities with XXZ anisotropy in a spin- 12 Heisenberg chain

We study the effect of anisotropy of spin-$S$ impurities on an antiferromagnetic SU(2) Heisenberg chain. The magnetic impurities are assumed to have an XXZ-type anisotropic exchange with its neighboring sites. First, using density-matrix renormalization group technique. we examine spin-spin correlation function, instability of N\'eel order, and local spin susceptibility in the presence of a single spin-$S$ impurity. Based on the results, we find that the types of spin-$S$ impurities are classified into two groups: $\rm(\hspace{.18em}i\hspace{.18em})$ nonmagnetic and $S=1$ impurities enhance only short-range antiferromagnetic correlation and $\rm(\hspace{.08em}ii\hspace{.08em})$ $S=1/2$ and $S>1$ impurities can, in contrast, stabilize a long-range N\'eel order in the disordered SU(2) Heisenberg chain. Then, we focus on the case of $S=1/2$ impurity as a representative of $\rm(\hspace{.08em}ii\hspace{.08em})$ and investigate the evolution of some experimentally observable quantities such as magnetization, specific heat, and magnetic susceptibility, as a function of concentration and XXZ anisotropy strength of the impurity. We confirm that the N\'eel order is induced in the bulk spin chain in the presence of finite amount of easy-axis XXZ $S=1/2$ impurities. Furthermore, we recover some of the aforementioned features using cluster mean-field theory, which allows us to present results on experimentally accessible quantities at finite temperatures. Interestingly, in the presence of uniform magnetic field, the total magnetization exhibits a pseudo-gap behavior for low values of applied field. We also discuss the dependence of NMR spectrum on various XXZ impurities and identify that the spin state of Co impurity in SrCu$_{0.99}$Co$_{0.01}$O$_2$ is $S = 3/2$.

Self-energy method for time-dependent spectral functions of the Anderson impurity model within the time-dependent numerical renormalization group approach

The self-energy method for quantum impurity models expresses the correlation part of the self-energy in terms of the ratio of two Green's functions and allows for a more accurate calculation of equilibrium spectral functions than is possible directly from the one-particle Green's function [Bulla et al., J. Phys.: Condens. Matter 10, 8365 (1998)], for example, within the numerical renormalization group method. In addition, the self-energy itself is a central quantity required in the dynamical mean field theory of strongly correlated lattice models. Here, we show how to generalize the self-energy method to the time-dependent situation for the prototype model of strong correlations, the Anderson impurity model. We use the equation of motion method to obtain closed expressions for the local Green's function in terms of a time-dependent correlation self-energy, with the latter being given as a ratio of a one-particle time-dependent Green's function and a higher-order correlation function. We benchmark this self-energy approach to time-dependent spectral functions against the direct approach within the time-dependent numerical renormalization group method. The self-energy approach improves the accuracy of time-dependent spectral function calculations, and the closed-form expressions for the Green's function allow for a clear picture of the time-evolution of spectral features at the different characteristic time-scales. The self-energy approach is of potential interest also for other quantum impurity solvers for real-time evolution, including time-dependent density matrix renormalization group and continuous-time quantum Monte Carlo techniques.

Anisotropic quantum Hall states in the presence of interactions with fourfold rotational symmetry

We study the effects of anisotropic interactions in the quantum Hall effect in the presence of a fourfold discrete rotational ($C_4$) symmetry. Employing the density matrix renormalization group technique on an infinite cylinder geometry (iDMRG), we calculate the anisotropy response of the Laughlin state at $\nu=1/3$ and the composite-Fermi liquid (CFL) state at $\nu=1/2$. We find that the anisotropy transferred from the interaction potential to the $\nu=1/3$ state is stronger when compared to the complementary case of an anisotropic band. Further, the strength of anisotropy reduces as the interaction is made shorter ranged. Quite surprisingly, at $\nu=1/2$, we find that the deformation in the CF Fermi-surface changes sign as the interaction range is reduced. Our results imply that the short-distance and long-distance parts of the interaction potential have opposite effects on the CFL state in the presence of $C_4$-symmetric anisotropy.

The quantum sine-Gordon model with quantum circuits

Analog quantum simulation has the potential to be an indispensable technique in the investigation of complex quantum systems. In this work, we numerically investigate a one-dimensional, faithful, analog, quantum electronic circuit simulator built out of Josephson junctions for one of the paradigmatic models of an integrable quantum field theory: the quantum sine-Gordon (qSG) model in 1+1 space-time dimensions. We analyze the lattice model using the density matrix renormalization group technique and benchmark our numerical results with existing Bethe ansatz computations. Furthermore, we perform analytical form-factor calculations for the two-point correlation function of vertex operators, which closely agree with our numerical computations. Finally, we compute the entanglement spectrum of the qSG model. We compare our results with those obtained using the integrable lattice-regularization based on the quantum XYZ chain and show that the quantum circuit model is less susceptible to corrections to scaling compared to the XYZ chain. We provide numerical evidence that the parameters required to realize the qSG model are accessible with modern-day superconducting circuit technology, thus providing additional credence towards the viability of the latter platform for simulating strongly interacting quantum field theories.

Performance of the rigorous renormalization group for first-order phase transitions and topological phases

Expanding and improving the repertoire of numerical methods for studying quantum lattice models is an ongoing focus in many-body physics. While the density matrix renormalization group (DMRG) has been established as a practically useful algorithm for finding the ground state in 1D systems, a provably efficient and accurate algorithm remained elusive until the introduction of the rigorous renormalization group (RRG) by Landau et al. [Nature Physics 11, 566 (2015)]. In this paper, we study the accuracy and performance of a numerical implementation of RRG at first order phase transitions and in symmetry protected topological phases. Our study is motived by the question of when RRG might provide a useful complement to the more established DMRG technique. In particular, despite its general utility, DMRG can give unreliable results near first order phase transitions and in topological phases, since its local update procedure can fail to adequately explore (near) degenerate manifolds. The rigorous theoretical underpinnings of RRG, meanwhile, suggest that it should not suffer from the same difficulties. We show this optimism is justified, and that RRG indeed determines accurate, well-ordered energies even when DMRG does not. Moreover, our performance analysis indicates that in certain circumstances seeding DMRG with states determined by coarse runs of RRG may provide an advantage over simply performing DMRG.

Finite-temperature photoemission in the extended Falicov-Kimball model: a case study for Ta$_2$NiSe$_5$

Utilizing the unbiased time-dependent density-matrix renormalization group technique, we examine the photoemission spectra in the extended Falicov-Kimball model at zero and finite temperatures, particularly with regard to the excitonic insulator state most likely observed in the quasi-one-dimensional material Ta_22NiSe_55. Working with infinite boundary conditions, we are able to simulate all dynamical correlation functions directly in the thermodynamic limit. For model parameters best suited for Ta_22NiSe_55 the photoemission spectra show a weak but clearly visible two-peak structure, around the Fermi momenta k\simeq\pm k_{F}k≃±kF, which suggests that Ta_22NiSe_55 develops an excitonic insulator of BCS-like type. At higher temperatures, the leakage of the conduction-electron band beyond the Fermi energy becomes distinct, which provides a possible explanation for the bare non-interacting band structure seen in time- and angle-resolved photoemission spectroscopy experiments.