We review recently introduced numerical methods for the unbiased detection of the order parameter and/or dominant correlations in many-body interacting systems, by using reduced density matrices. Most of the paper is devoted to the ‘quasi-degenerate density matrix’ (QDDM) which is rooted in Anderson's observation that the degenerate symmetry-broken states valid in the thermodynamic limit are manifested in finite systems as a set of low-energy ‘quasi-degenerate’ states (in addition to the ground state). This method, its original form due to Furukawa et al (2006 Phys. Rev. Lett. 96 047211 ), is given a number of improvements here, above all the extension from two-fold symmetry breaking to arbitrary cases. This is applied to two test cases: (1) interacting spinless hardcore bosons on the triangular lattice and (2) a spin-1/2 antiferromagnetic system at the percolation threshold. In addition, we survey a different method called the ‘correlation density matrix’, which detects (possibly long-range) correlations only from the ground state, but using the reduced density matrix from a cluster consisting of two spatially separated regions.
We review recently introduced numerical methods for the unbiased detection of the order parameter and/or dominant correlations in many-body interacting systems, by using reduced density matrices. Most of the paper is devoted to the ‘quasi-degenerate density matrix’ (QDDM) which is rooted in Anderson's observation that the degenerate symmetry-broken states valid in the thermodynamic limit are manifested in finite systems as a set of low-energy ‘quasi-degenerate’ states (in addition to the ground state). This method, its original form due to Furukawa et al (2006 Phys. Rev. Lett. 96 047211 ), is given a number of improvements here, above all the extension from two-fold symmetry breaking to arbitrary cases. This is applied to two test cases: (1) interacting spinless hardcore bosons on the triangular lattice and (2) a spin-1/2 antiferromagnetic system at the percolation threshold. In addition, we survey a different method called the ‘correlation density matrix’, which detects (possibly long-range) correlations only from the ground state, but using the reduced density matrix from a cluster consisting of two spatially separated regions.
A useful concept for finding numerically the dominant correlations of a given ground state in an interacting quantum lattice system in an unbiased way is the correlation density matrix (CDM). For two disjoint, separated clusters, it is defined to be the density matrix of their union minus the direct product of their individual density matrices and contains all the correlations between the two clusters. We show how to extract from the CDM a survey of the relative strengths of the system's correlations in different symmetry sectors and the nature of their decay with distance (power law or exponential), as well as detailed information on the operators carrying long-range correlations and the spatial dependence of their correlation functions. To achieve this goal, we introduce a new method of analysing the CDM, termed the dominant operator basis (DOB) method, which identifies in an unbiased fashion a small set of operators for each cluster that serve as a basis for the dominant correlations of the system. We illustrate this method by analysing the CDM for a spinless extended Hubbard model that features a competition between charge density correlations and pairing correlations, and show that the DOB method successfully identifies their relative strengths and dominant correlators. To calculate the ground state of this model, we use the density matrix renormalization group, formulated in terms of a variational matrix product state (MPS) approach within which subsequent determination of the CDM is very straightforward. In an extended appendix, we give a detailed tutorial introduction to our variational MPS approach for ground state calculations for one-dimensional quantum chain models. We present in detail how MPSs overcome the problem of large Hilbert space dimensions in these models and describe all the techniques needed for handling them in practice.
The reduced density matrix of an interacting system can be used as the basis for a truncation scheme, or in an unbiased method to discover the strongest kind of correlation in the ground state. In this paper, we investigate the structure of the many-body fermion density matrix of a small cluster in a square lattice. The cluster density matrix is evaluated numerically over a set of finite systems, subject to nonsquare periodic boundary conditions given by the lattice vectors ${\mathbf{R}}_{1}\ensuremath{\equiv}({R}_{1x},{R}_{1y})$ and ${\mathbf{R}}_{2}\ensuremath{\equiv}({R}_{2x},{R}_{2y})$. We then approximate the infinite-system cluster density-matrix spectrum by averaging the finite-system cluster density matrix (i) over degeneracies in the ground state, and orientations of the system relative to the cluster, to ensure it has the proper point-group symmetry; and (ii) over various twist boundary conditions to reduce finite size effects. We then compare the eigenvalue structure of the averaged cluster density matrix for noninteracting and strongly interacting spinless fermions, as a function of the filling fraction n, and discuss whether it can be approximated as being built up from a truncated set of single-particle operators.
This work is devoted to the study, the development, and the application of a new systematic method yielding the dominant correlations that govern a quantum many-body state in an unbiased way. The dominant correlations between any two disjoint blocks of a system are extracted by performing a singular value decomposition of the correlation density matrix (CDM) between those blocks. We determine several mathematical properties and features of this method, in particular the consequences of the lattice symmetries or the symmetries intrinsic to the studied state on the singular values spectrum. We investigate the relation between the norm of the CDM – providing a natural measure of the total correlation between the two blocks – and the so-called mutual information, a quantity originally introduced in quantum information theory. This novel tool is utilized for sheding new light on the zero temperature physics of the spin-1/2 frustrated ferromagnetic J1–J2 Heisenberg chain in a magnetic field as well as on the low-energy physics of the spin-1/2 antiferromagnetic Heisenberg model on the two-dimensional kagomé lattice. The states are computed using exact diagonalization and the density matrix renormalization group procedure in the first case, and exact diagonalization only in the second case. This work is introduced in Chapter 1. The first model is then presented in Chapter 2. Chapter 3 introduces the CDM method, and Chapter 4 is devoted to the study of the kagomé antiferromagnet. In the J1–J2 chain, we reveal a vector chiral phase at low magnetic field and a sequence of multipolar Luttinger liquid phases at high field. We explicitly show that these multipolar phases result from the destabilization – driven by a locking mechanism – of the classical spiral ground state in the absence of magnetic field. This point of view is completely new: multipolar phases were known to be a possible destabilization of ferromagnetic phases, but they have never been reported as a destabilization of spiral states yet. Regarding the kagomé antiferromagnet, we address for the first time the question of the nature of the singlet states forming its quite dense singlet spectrum above the ground state. We show that some of these low-lying singlet states have large dimer correlations which do not seem to significantly decrease with the distance, moreover our CDM studies confirm that the dominant correlations in those singlet states are of the dimer-dimer type. Studies of Von Neumann block entropies reveal a very short correlation length on the one hand, and entropies that are roughly independent on the energy of the state under consideration on the other hand. The scenario of a valence bond crystal phase is investigated and the relevance of different kinds of crystals (from the literature or ad hoc) for reproducing the dimer correlations in the 36-site sample is probed.
We examine the performance of the density matrix embedding theory (DMET) recently proposed in Knizia and Chan [Phys. Rev. Lett. 109, 186404 (2012)]. The core of this method is to find a proper one-body potential that generates a good trial wave function for projecting a large-scale original Hamiltonian to a local subsystem with a small number of bases. The resultant ground state of the projected Hamiltonian can locally approximate the true ground state. However, the lack of the variational principle makes it difficult to judge the quality of the choice of the potential. Here we focus on the entanglement spectrum (ES) as a judging criterion; accurate evaluation of the ES guarantees that the corresponding reduced density matrix well reproduces all physical quantities on the local subsystem. We apply the DMET to the Hubbard model on the one-dimensional chain, zigzag chain, and triangular lattice, and test several variants of potentials and cost functions. It turns out that ES serves as a more sensitive quantity than the energy and double occupancy to probe the quality of the DMET outcomes. A symmetric potential reproduces the ES of the phase that continues from a noninteracting limit. The Mott transition as well as symmetry-breaking transitions can be detected by the singularities in the ES. However, the details of the ES in the strongly interacting parameter region depends much on these variants, meaning that the present DMET algorithm allowing for numerous variant is insufficient to fully characterize the particular phases that require characterization by the ES.
The recent development of bootstrap methods based on semidefinite relaxations of positivity constraints has enabled rigorous two-sided bounds on local observables directly in the thermodynamic limit. However, these bounds inevitably become loose in symmetry broken phases, where local constraints are insufficient to capture long-range order. In this work, we identify the origin of this looseness as order parameter defects which are difficult to remove using local operators. We introduce a defect bootstrap framework that resolves this limitation by embedding the system into an auxiliary defect model equipped with ancilla degrees of freedom. This construction effectively enables local operators to remove order parameter defects, yielding tighter bounds in phases with spontaneous symmetry breaking. This approach can be applied broadly to pairwise-interacting local lattice models with discrete or continuous internal symmetries that satisfy a property we call defect diamagnetism, which requires that the ground state energy does not decrease upon adding any finite number of symmetry defects. Applying the method to the transverse field Ising models in 1D and 2D, we obtain significantly improved bounds on energy densities and spin correlation functions throughout the symmetry broken phase in 1D and deep within the phase in 2D. Our results demonstrate that physically motivated constraint sets can dramatically enhance the power of bootstrap methods for quantum many-body systems.
This thesis contributes to the field of strongly correlated electron systems with studies in two distinct fields thereof: the specific nature of correlations between electrons in one dimension and quantum quenches in quantum impurity problems. In general, strongly correlated systems are characterized in that their physical behaviour needs to be described in terms of a many-body description, i.e. interactions correlate all particles in a complex way. The challenge is that the Hilbert space in a many-body theory is exponentially large in the number of particles. Thus, when no analytic solution is available - which is typically the case - it is necessary to find a way to somehow circumvent the problem of such huge Hilbert spaces. Therefore, the connection between the two studies comes from our numerical treatment: they are tackled by the density matrix renormalization group (DMRG) [1] and the numerical renormalization group (NRG) [2], respectively, both based on matrix product states. The first project presented in this thesis addresses the problem of numerically finding the dominant correlations in quantum lattice models in an unbiased way, i.e. without using prior knowledge of the model at hand. A useful concept for this task is the correlation density matrix (CDM) [3] which contains all correlations between two clusters of lattice sites. We show how to extract from the CDM, a survey of the relative strengths of the system’s correlations in different symmetry sectors as well as detailed information on the operators carrying long-range correlations and the spatial dependence of their correlation functions. We demonstrate this by a DMRG study of a one-dimensional spinless extended Hubbard model [4], while emphasizing that the proposed analysis of the CDM is not restricted to one dimension. The second project presented in this thesis is motivated by two phenomena under ongoing experimental and theoretical investigation in the context of quantum impurity models: optical absorption involving a Kondo exciton [5, 6, 7] and population switching in quantum dots [8, 9, 10, 11, 12, 13, 14, 15]. It turns out that both phenomena rely on the various manifestations of Anderson orthogonality (AO) [16], which describes the fact that the response of the Fermi sea to a quantum quench (i.e. an abrupt change of some property of the impurity or quantum dot) is a change of the scattering phase shifts of all the single-particle wave functions, therefore drastically changing the system. In this context, we demonstrate that NRG, a highly accurate method for quantum impurity models, allows for the calculation of all static and dynamic quantities related to AO and present an extensive NRG study for population switching in quantum dots.