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We recall the reasons for the exponential failure of the DMRG algorithm in more than one-dimension and link this with the disconnected nature of the theory. A connected version of the RG is described which allows the efficient and accurate treatment of higher dimensional systems. In addition, a new renormalisation group algorithm, based on a numerical generalisation of the separation of variables, is introduced.
V.O.Cheranovskii and I.Ozkan
We studied the temperature dependency of internal energy and specific heat of 2D Ising models on square lattice with competing interactions like 2D axial and 2D biaxial next-nearest-neighbor models. They do not belong to the class of IRF models and, therefore, the application of DMRG scheme requires some modifications. We propose modified renormalization schemes with overlapping spin blocks, which are applicable to the above models.
Although there are still some difficulties in exploiting the performances of the DMRG near critical points, there is general consensus on the fact that the possible universality classes of critical 1D quantum systems are well described by conformal field theories (CFT's). Once that the essential features of the appropriate CFT (central charge, degeneracies, ...) have been identified one can fruitfully employ the finite-size scaling predictions of the theory to extract the scaling dimensions and the critical exponents with high precision. However, from a numerical point of view, the identification of the correct CFT requires a careful analysis of various levels of the low-lying part of the spectrum. Taking as a nontrivial reference a two-parameter class of spin-1 Hamiltonians with anisotropic interactions along the $z$-axis, we discuss first the critical spectra obtained with the help of a multi-target DMRG code whose Lanczos routine can target various excited states at once by means of the so-called Thick-restart algorithm. We then focus on the off-critical continuum theories that one finds near the borders of the Haldane phase of the model, where a change of topological (hidden) order takes place. In particular, near the Gaussian ($c=1$) line we expect the appropriate field theory to be a massive sine-Gordon model and the numerical estimates of the exponents associated with the string order parameter (SOP) along $z$ suggest that the corresponding correlator is related to the so-called twist operator. At the Ising-like ($c=1/2$) transition, instead, the longitudinal SOP remains finite while the transverse one vanishes, revealing some analogies with what happens in an isotropic spin-1 Heisenberg chain in a staggered magnetic field. Interestingly enough, spin systems like these, being characterised by a string-ordered ground state of valence-bond type, have recently been proposed as valid quantum channels for the high level of entanglement they can carry.
Andrzej Drzewinski and J.M.J. van Leeuwen
We investigate the Rubinstein-Duke model for a non-uniformly charged polymer driven by an electric field through a gel. This non-equilibrium problem is successfully studied by means of non-Hermitian density-matrix renormalization-group (DMRG) techniques. The method enables us to perform a detailed analysis of both global quantities as drift and curvilinear velocities and diffusion coefficient, but also of local average shapes of the polymer. Since for reptation large finite-size scaling effects are present, the DMRG method is very effective, as long chains can be studied; up to N=150 reptons for weak fields. In this limit, which is experimentally the most relevant, the probability distributions in the stationary state are found to be rigorously equivalent for different charge distributions. At strong field some profiles show a non-monotonic behaviour, which has a simple interpretation in terms of an interface model. The extensive DMRG calculations are also an excellent test for the usefulness of this method for stochastic processes.
The Density Matrix Renormalization Group (DMRG) was introduced by Steven White in 1992 as a method for accurately describing the properties of one-dimensional quantum lattices. Based on its enormous success in that domain, it was subsequently proposed that the DMRG could be adapted for use on finite Fermi systems, through the replacement of real-space lattice sites by an appropriately ordered set of single-particle levels. Since then, there has been an enormous amount of work on the subject, ranging from efforts to clarify the optimal means of implementing the algorithm to extensive applications in a variety of fields, like quantum chemistry, quantum Hall systems, small superconducting grains, nuclei, etc. I will briefly review these recent developments focusing on applications to ultrasmall superconducting grains and nuclear structure calculations.
Explanation of a numerical self-consistent algorithm for three-dimensional classical lattice models is presented. This algorithm, Tensor Product Variational Approach (TPVA), has been developed in order to extend applicability of DMRG to higher dimensions. The variational background of the TPVA is given. Efficiency and stability of the numerical algorithm is observed through its application to a complex frustrated model, the so-called Axial-Next-Nearest-Neighbor Ising (ANNNI) model. Global phase diagram is constructed for this model. It contains numerous commensurate and incommensurate magnetic structures in a spin modulated phase. The devil's stairs behavior for the model is confirmed. Wavelength in the modulated phase increases to infinity if the boundary with ferromagnetic phase is approached. New properties of the ANNNI model will be presented.
We will review the different ways to calculate zero temperature dynamical properties of strongly correlated systems and present a number of successful applications. In addition we will show how the dynamical DMRG can be used together with the Dynamical Mean Field Theory (DMFT) to solve the associated impurity problem in the infinite-dimensional Hubbard model. This method is used to obtain the density of states of strongly correlated systems and, in particular, the insulator-metal transition. With this algorithm, more complex problems having a larger number of degrees of freedom can be considered and finite-size effects can be minimized.
I will present a density-matrix renormalization-group (DMRG) method for calculating spectral functions S(k,\omega) in lattice quantum many-body systems. The method is based on an exact variational principle for dynamical correlation functions [1,2]. With this dynamical version of DMRG one can calculate dynamical properties of large systems for all frequencies very accurately. I will also discuss the finite-size scaling of frequency-dependent spectra and describe methods for analyzing the scaling of dense spectra [2]. Combined with a finite-size scaling analysis the dynamical DMRG allows one to determine spectral properties in the thermodynamic limit (for instance, the shape of continuous bands). Moreover, I will present a simple technique based on quasi-momenta k for calculating momentum-dependent quantities in finite open chains [3]. With these methods one can compute various spectral functions S(k,\omega) in correlated electron models [3,4]. Comparisons with analytical results demonstrate the accuracy of this approach and show that the properties of infinite one-dimensional systems can be determined numerically.
[1] E. Jeckelmann, F. Gebhard, and F. Essler, Phys. Rev. Lett. 85, pp.
3910-3913 (2000).
[2] E. Jeckelmann, Phys. Rev. B 66, 045114 (2002).
[3] H. Benthien, F. Gebhard, and E. Jeckelmann, Phys. Rev. Lett. (2004),
in press; e-print cond-mat/0402664.
[4] Y.-J. Kim, J.P. Hill, H. Benthien, F.H.L. Essler, E. Jeckelmann, et al.,
Phys. Rev. Lett. 92, 137402 (2004).
In order to optimize the ordering of the lattice sites in the momentum space and quantum chemistry versions of the density matrix renormalization group (DMRG) method we have studied the separability and entanglement of the target state for the 1-D Hubbard model and various molecules. By analyzing the behavior of von Neumann entropy we have found criteria that help to fasten convergence. A new initialization procedure has been developed which maximizes the Kullback-Leibler entropy and extends the active space (AS) in a dynamical fashion. The dynamically extended active space (DEAS) procedure reduces significantly the effective system size during the first half sweep and accelerates the speed of convergence of momentum space DMRG and quantum chemistry DMRG to a great extent.
The effect of lattice site ordering on the number of block states to be kept during the RG procedure is also investigated and has been related to the field of quantum data compression. Therefore, we have also studied quantum data compression for finite quantum systems where the site density matrices are not independent, i.e., the density-matrix can not be given as direct product of site density matrices and the von Neumann entropy is not equal to the sum of site entropies. Using the density matrix renormalization group (DMRG) method for the 1-d Hubbard model, we have shown that a simple relationship exists between the entropy of the left or right block and dimension of the Hilbert space of that block as well as of the superblock for any fixed fidelity. The information loss during the RG procedure has been investigated and a more rigorous control of the relative error has been proposed based on Kholevo's theory.
I will describe how to apply the techniques of quantum angular momentum and other algebras (including discrete lattice symmetries) to DMRG. This scheme extends naturally to variants of DMRG, such as higher dimensional matrix product states and time evolution. Examples will be given using a new high performance C++ toolkit to be released with source code under a 'scientific' software license.
A dynamical density-matrix renormalization group approach to the spectral properties of quantum impurity problems is presented. The method is demonstrated on the flat-band symmetric single-impurity Anderson Model. This approach can provide the impurity spectral density for all frequencies and coupling strengths. Moreover, we show the application of our method to the dynamical mean-field theory.
Tomotoshi Nishino, Yukinobu Nishio, Andrej Gendiar
We consider a two-dimensional product of local tensors, which is a generalization of the matrix product state to higher dimension. Such a state can be employed as a trial state for 3D classical and 2D quantum lattice models, where variational (free) energy is easily obtained by use of DMRG or its variant, the corner transfer matrix renormalization group (CTMRG). We optimize the local tensor by solving a generalized eigenvalue problem self consistently. The optimization process can be regarded as a renormalization process to vertical direction, and it should be noted that the optimal block spin transformation cannot be obtained through the diagonalization of the density matrix.
References:
I will discuss means of characterizing metallic and insulating phases as well as transitions between such phases within the DMRG. Gaps, correlation functions, order parameters, and susceptibilities appropriate to the phases and the transitions are best used in combination in order to completely characterize the behavior. I will discuss application to the Hubbard model with an ionic potential and with a next-nearest-neighbor hopping. I will make contact with both field-theoretic treatments and strong-coupling expansions and point out some anomalies discovered by the numerical calculations.
The properties of the reduced density matrices used in DMRG calculations can be obtained explicitly for a number of solvable quantum chains. In the talk this will be discussed with emphasis on the tight-binding model, which is a simple example of a critical lattice system. Some connections to other physical problems where the same reduced density matrices appear, will also be pointed out. This includes calculations of entanglement entropies and treatments of integrable one- and two-dimensional models.
We study the spectral properties of an Anderson impurity hosted in a chain with even electron hopping. We use density matrix renormalization group method to calculate the dynamical correlation function for the impurity[1]. The single particle energy levels are evenly distributed at the Fermi level for the conducting band in an even hopping chain. We find that the Kondo resonance can not be well resolved[2] at zero energy and so it is not good for self-consistent calculation in dynamical mean field study[3,4]. Away from the zero energy Kondo resonance, the impurity spectrum is calculated upto the energy of the order of the electron hopping in the chain. We find this intermediate energy spectrum is in correspondence to the spectrum of the conduction electrons. Our study in the intermediate energy scale is then discussed in related to the size of Kondo clouds[5] and Kondo clouds in the finite size rings[6].
Carsten Raas and Götz S. Uhrig
The quantitative determination of the dynamic correlations of single impurity models is essential in many active fields. We analyze the local propagator of a fermionic impurity in a bath at constant energy resolution by dynamical density-matrix renormalization (D-DMRG). This approach is particularly useful for higher-lying excitations. In contrast to other approaches, sharp dominant resonances at high energies are found. We analyze their line shapes and discuss their origin and importance. The Kondo energy scale is also analyzed. Technically, we address the following important points: (i) Efficient inverting of the linear equation $[\omega+i\eta-(\mathcal{H}-E_0)] \|\xi\rangle = d^{\dagger} |0\rangle$ for the calculation of the correction vector. (ii) Reliable deconvolution schemes to extract the non-negative spectral densities from the broadened DMRG raw data. (iii) Treatment of problems inherent to spectral densities with energy gap. (iv) Computation of the irreducible self-energy $\Sigma(z)$ via reducible self-energy $Q(z)$ for increased accuracy. Our findings provide a well-controlled numerical approach to impurity problems as they arise in dynamical mean field theory (DMFT) as well as for more complex impurities like quantum dots, molecules, or the effective problems of extended DMFT schemes.
Reference: Carsten Raas, Götz S. Uhrig, and Frithjof B. Anders: High Energy Dynamics of the Single Impurity Anderson Model, Physical Review B 69(4), 041102(R) (2004)
In this talk, a brief introduction to the models employed in the study of the electronic states of conjugated polymers will be presented. The nature of low-lying excitations and the need for exploiting symmetries of the Hamiltonian to target important excited states will be discussed. The technique for accurately building a large number of unit cells of topologically nonlinear polymers such as polyacenes, poly para phenylynes and poly para phenylene vinylenes will be described. The method of obtaining frequency dependent response coefficients, in particular the dynamic nonlinear optic coefficients will be dealt with in some detail. Applications of the DMRG method to study possible Peierls' like distortion in polyacenes, and computing properties such as exciton binding energies, equilibrium geometries in excited states and the ordering of the low-lying excitations in various conjugated polymers will be discussed.
The density-matrix renormalization group (DMRG) is employed to calculate optical properties of the half-filled Hubbard model with nearest-neighbor interactions. In order to model the optical excitations of oligoenes, a Peierls dimerization is included whose strength for the single bonds may fluctuate. Systems with up to 100 electrons are investigated, their wave functions are analyzed, and relevant length-scales for the low-lying optical excitations are identified. The presented approach provides a concise picture for the size dependence of the optical absorption in oligoenes.
We study Heisenberg models on diagonal ladders as an alternative route to approach the physics of two-dimensional lattices for AF Heisenberg and eventually t-J models. Diagonal ladders are defined by the number of transversal plaquettes N_p, much like standard square ladders are classified by their number los legs. For diagonal ladders with an odd number N_p we know that the system shows Ferrimagnetism. Here we focus on the properties of N_p even diagonal ladders and show that they do not exhibit Ferrimagnetism, and moreover, they show gapped behaviour with a singlet state for several range of couplings, including the isotropic case. We employ numerical tools like DMRG and Lanczos to do these studies and complement them with some analytical tools. Therefore, we may conclude that diagonal ladders fall into two universality classes depending on their even/odd character.
We extended the Density Matrix Renormalization Group method to study the real time dynamics of interacting one dimensional spinless Fermi systems by applying the full time evolution operator to an initial state. As an example we describe the propagation of a density excitation in an interacting clean system and the transport through an interacting nano structure.
I will discuss the recently introduced adaptive time-dependent DMRG with special emphasis on its algorithmic root in quantum information theory and the theory of matrix product ansatzes. I will discuss applications related to driven quantum phase transitions from superfluid to Mott insulating phases in Bose-Hubbard models as realized in cold atomic gases in optical lattices.
In a high perpendicular magnetic field, the kinetic energy of electrons in a two-dimensional system is completely quenched and macroscopic degeneracy appears in each Landau level. In ideal systems the degeneracy is lifted only by Coulomb interaction, and various interesting ground states -- including compressible liquids, incompressible liquids, stripes, bubbles and the Wigner crystal -- are realized.
To determine the ground state of this system, we use the density matrix renormalization group (DMRG) method. Although the DMRG was developed for one-dimensional systems, we can apply this method to two-dimensional systems with the use of the eigenstates of free electrons in Landau gauge. Since each eigenstate is uniquely identified by the x-component of the guiding center coordinates, we can map the system onto an effective one-dimensional system. The truncation error estimated from the eigenvalues of the density matrix is usually smaller than the order of $10^{-4}$ with keeping 200 basis states.
We have calculated the ground state wave function and the pair correlation functions at various fillings in the lowest three Landau levels. The determined ground state phase diagrams consist of compressible liquids, incompressible liquids, stripes, bubbles and the Wigner crystal, which shows that diverse electronic states are realized only by changing magnetic field.
We discuss DMRG from the perspective of Quantum Information Theory, and introduce extensions to periodic boundary conditions, finite-T and higher dimensions.
I will review recent work on obtaining real time dynamics within DMRG, including Vidal's quantum information approach to DMRG and Trotter approximation based approaches to time evolution. I will also discuss our recent progress on time-dependent methods for systems with non-nearest-neighbor couplings, including ladder systems.
In this work, we will explore some parts of the phase diagram of the N-leg Kondo ladders calculated by the density matrix renormalization group (DMRG) technique (keeping up m=2000 states per block). The N-leg Kondo ladders consist of $N$ Kondo chains of length $L$ coupled by the hopping term. The 2D system is obtained by taking both $N$ and $L$ to infinity. As far as we know this is the first study of the $N$-leg Kondo ladders.
Firstly, we are going to focus in the "1-leg Kondo Ladders" (1D model) at quarter-filling. We will show a new phase found recently at this filling (PRL 90, 247204 (2003)). We will report the presence of spin dimerization in the ground state. We will see that RKKY-interaction induces this dimerized phase.
We also will investigate the spin and charge gaps of the half-filled N-leg Kondo ladders (PRB 68, 134422 (2003)). Based on the behavior of the spin gap of the 1- to 3-leg Kondo ladders, we expect non-zero charge and spin gaps in the N-leg Kondo ladders for any $J>0$. We will also show evidence of the existence of a QCP in the 2D Kondo model at half-filling in agreement with previous studies. Moreover, our results for the binding energy of two holes do not show any evidence for an effective hole-hole attraction in the 2D KLM for any Kondo coupling.
Finally, we are going to explore the whole phase diagram of the two-leg Kondo ladder. Our results show that the ferromagnetic phase is present only for small density and large Kondo coupling $J$, while for the densities $n\gtrsim0.4$ and any Kondo coupling $J>0$ we found a paramagnetic phase. We also report the existence of dimerization along the legs for the densities $n=1/4$ and $n=1/2$. The results of the charge structure factor suggest that the charges behave as free particle with spin-1/2 (spinless) for small (large) values of $J$.
We present an algorithm to study finite-temperature dynamics in one-dimensional quantum many-body systems, such as spin chains. This algorithm, a generalization of that of [G. Vidal, quant-ph/0310089, to appear in Phys. Rev. Lett.] for pure-state Hamiltonian dynamics, can be used to construct thermal states and to simulate real time evolution as given by a generic master equation. Thus, it can be applied to the study of transport properties, finite temperature effects and dissipation in quantum systems far from equilibrium, The efficiency of the simulation depends on the amount of correlations between subsystems. The algorithm can be incorporated into standard DMRG implementations.