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.
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.
In contrast to classical many-particle systems for which the number of degrees of freedom grows linearly with the system size, it grows exponentially for quantum many-particle systems. On the one hand, the resulting complexity of the quantum systems has a high potential for technological advances if it can be exploited directly. For example, in recent years, the fundamentals for the realization of quantum computers and their possible applications have been investigated intensely. On the other hand, the high complexity, especially in strongly correlated regimes, often makes the analytical or numerical analysis of underlying physical phenomena very difficult. This is why some systems of high technological relevance, like the high-temperature superconductors, are still unsatisfactorily understood. A main topic of this thesis is the development of efficient numerical methods for the simulation of strongly correlated quantum lattice models. For one-dimensional systems, the density-matrix renormalization-group (DMRG) is such a very successful method. The physical states of interest are approximated within a certain class of ansatz states. These ansatz states are designed in a way that the number of degrees of freedom are prevented from growing exponentially. With the DMRG method, ground states of interacting one-dimensional systems can often be determined with very high precision. This observation indicates already that ground states actually do not exhaust the available exponential number of degrees of freedom but can be described efficiently with much fewer parameters. The ansatz states used in DMRG are the so-called matrix product states. A very useful criterion for the construction of corresponding appropriate ansatz states for higher-dimensional systems is the quantum nonlocality as for example quantified by the entanglement entropy. The ansatz states should be able to reproduce the scaling properties of the quantum nonlocality in the physical states of interest with as few parameters as possible. The first part of the thesis, therefore, provides analytical and numerical analysis of the scaling of quantum nonlocality with the system size or time in different, physically relevant scenarios. For example, the scaling of Renyi entropies and their dependence on boundary conditions is derived within the 1+1-dimensional conformal field theory. Conjectures and analytical indications concerning the properties of entanglement entropy in critical fermionic and bosonic systems are confirmed numerically with high precision. For integrable models in the thermodynamic limit, general preconditions are derived under which subsystems converge to steady states. These steady states are non-thermal and retain information about the initial state. It is shown that the entanglement entropy in such steady states is extensive. For short times, the entanglement entropy grows typically linearly with time, causing an exponential increase in computation costs for the DMRG method. Consequently, time-dependent DMRG simulations usually have no direct access to the long time regime. The second part of the thesis focuses on the development and improvement of the above-mentioned numerical techniques. The time-dependent DMRG is complemented with an extrapolation technique for the evaluated observables. In this way, the problem of the entropy increase can be circumvented, allowing for a precise determination of spectral functions. The method is demonstrated using the example of the Heisenberg antiferromagnet and results are compared to Bethe-Ansatz data for T=0 and quantum Monte Carlo data for T>0. For the simulation of higher-dimensional systems, projected-entangled pair-states (PEPS) and the multiscale entanglement renormalization ansatz (MERA) turn out to be appropriate wavefunction classes. Initially, they could only be used for spin systems. Within this thesis, variants of those state classes are being suggested for the fermionic case. It is shown how the known algorithms for spin systems can be translated into corresponding algorithms for the fermionic systems. The occurring computation costs increase only by a marginal overhead. First numerical benchmarking tests proceeded successfully. Thus, new techniques are available which can for example be used to examine the two-dimensional Hubbard model. This model is a candidate for the description of high-temperature superconductivity.
The density matrix renormalization group (DMRG) method allows an efficient computation of the properties of interacting 1D quantum systems. Two-dimensional (2D) systems, capable of displaying much richer quantum behavior, generally lie beyond its reach except for very small system sizes. Many of the physical properties of 2D systems carry into the quasi-1D case, for which, unfortunately, the standard 2D DMRG algorithm fares little better. By finding the form of the entanglement entropy in quasi-1D systems, we directly identify the reason for this failure. Using this understanding, we explain why a modified algorithm, capable of cleverly exploiting this behavior of the entanglement entropy, can accurately reach much larger system sizes. We demonstrate the power of this method by accurately finding quantum critical points in frustration induced magnetic transitions, which remain inaccessible using the standard DMRG or the Monte Carlo methods. We contrast half-integer and integer spin cases.
We present a tree-tensor-network-based method to study strongly correlated systems with nonlocal interactions in higher dimensions. Although the momentum-space and quantum-chemistry versions of the density-matrix renormalization group (DMRG) method have long been applied to such systems, the spatial topology of DMRG-based methods allows efficient optimizations to be carried out with respect to one spatial dimension only. Extending the matrix-product-state picture, we formulate a more general approach by allowing the local sites to be coupled to more than two neighboring auxiliary subspaces. Following [Y. Shi, L. Duan, and G. Vidal, Phys. Rev. A 74, 022320 (2006)], we treat a treelike network ansatz with arbitrary coordination number z, where the z=2 case corresponds to the one-dimensional (1D) scheme. For this ansatz, the long-range correlation deviates from the mean-field value polynomially with distance, in contrast to the matrix-product ansatz, which deviates exponentially. The computational cost of the tree-tensor-network method is significantly smaller than that of previous DMRG-based attempts, which renormalize several blocks into a single block. In addition, we investigate the effect of unitary transformations on the local basis states and present a method for optimizing such transformations. For the 1D interacting spinless fermion model, the optimized transformation interpolates smoothly between real space and momentum space. Calculations carried out on small quantum chemical systems support our approach.
Density matrix renormalization group (DMRG) and its extensions in the form of matrix product states are arguably the choice for the study of one-dimensional quantum systems in the last three decades. However, due to the limited entanglement encoded in the wave-function ansatz, to maintain the accuracy of DMRG with the increase of the system size in the study of two-dimensional systems, exponentially increased resources are required, which limits the applicability of DMRG to only narrow systems. We introduce a new ansatz in which DMRG is augmented with disentanglers to encode area-law-like entanglement entropy (entanglement entropy supported in the new ansatz scales as l for an l × l system). In the new method, the O(D 3) low computational cost of DMRG is kept (with an overhead of O(d 4) and d the dimension of the physical degrees of freedom). We perform benchmark calculations with this approach on the two-dimensional transverse Ising and Heisenberg models. This new ansatz extends the power of DMRG in the study of two-dimensional quantum systems.
Tensor network quantum states are powerful tools for strongly correlated systems, tailored to capture local correlations such as in ground states with entanglement area laws. When applying tensor network states to interacting fermionic systems, a proper choice of the basis or orbitals can reduce the bond dimension of tensors and provide physically relevant orbitals. We introduce such a change of basis with unitary gates obtained from compressing fermionic Gaussian states into quantum circuits corresponding to various tensor networks. These circuits can reduce the ground-state entanglement entropy and improve the performance of algorithms such as the density matrix renormalization group. We study the Anderson impurity model with one and two impurities to show the potential of the method for improving computational efficiency and interpreting impurity physics. Furthermore, fermionic Gaussian circuits can also suppress entanglement during the time evolution out of low-energy state. Last, we consider Gaussian multiscale entanglement renormalization ansatz (GMERA) circuits which compress fermionic Gaussian states hierarchically. The emergent coarse-grained physical models from these GMERA circuits are studied in terms of their entanglement properties and suitability for performing time evolution.
Renormalization is often described as the removal or ‘integrating out’ of high energy degrees of freedom. In the context of quantum matter, one might suspect that quantum entanglement provides a sharp way to characterize such a loss of degrees of freedom. Indeed, for quantum many-body systems with Lorentz invariance, such entanglement monotones have been proven to exist in one, two, and three spatial dimensions. In each dimension d , a certain term in the entanglement entropy of a d -ball decreases along renormalization group (RG) flows. Given that most quantum many-body systems available in the laboratory are not Lorentz invariant, it is important to generalize these results if possible. In this work we demonstrate the impossibility of a wide variety of such generalizations. We do this by exhibiting a series of counterexamples with understood renormalization group flows which violate entanglement RG monotonicity. We discuss bosons at finite density, fermions at finite density, and majorization in Lorentz invariant theories, among other results.