Imperfections in correlated materials can alter their ground state as well as finite-temperature properties in significant ways. Here, we develop a method based on numerical linked-cluster expansions for calculating exact finite-temperature properties of disordered lattice models directly in the thermodynamic limit. We show that a continuous distribution for disordered parameters can be achieved using a set of carefully chosen discrete modes in the distribution, which allows for the averaging of properties over all disorder realizations. We benchmark our results for thermodynamic properties of the square lattice Ising and quantum Heisenberg models with bond disorder against Monte Carlo simulations and study them as the strength of disorder changes. We also apply the method to the disordered Heisenberg model on the frustrated checkerboard lattice, which is closely connected to Sr$_2$Cu(Te$_{0.5}$W$_{0.5}$)O$_6$. Our method can be used to study finite-temperature properties of other disordered quantum lattice models including those for interacting lattice fermions.
The interplay of disorder and strong correlations in quantum many-body systems remains an open question. That is despite much progress made in recent years with ultracold atoms in optical lattices to better understand phenomena such as many-body localization or the effect of disorder on Mott metal-insulator transitions. Here, we utilize the numerical linked-cluster expansion technique, extended to treat disordered quantum lattice models, and study exact thermodynamic properties of the disordered Fermi-Hubbard model on the square and cubic geometries. We consider box distributions for the disor…
We introduce a white graph expansion for the method of perturbative continuous unitary transformations when implemented as a linked cluster expansion. The essential idea behind an expansion in white graphs is to perform an optimized bookkeeping during the calculation by exploiting the model-independent effective Hamiltonian in second quantization and the associated inherent cluster additivity. This appoach is shown to be especially well suited for microscopic models with many coupling constants, since the total number of relevant graphs is drastically reduced. The white graph expansion is exem…
We identify a fundamental challenge for non-perturbative linked cluster expansions (NLCEs) resulting from the reduced symmetry on graphs, most importantly the breaking of translational symmetry, when targeting the properties of excited states. A generalized notion of cluster additivity is introduced, which is used to formulate an optimized scheme of graph-based continuous unitary transformations (gCUTs) allowing to solve and to physically understand this fundamental challenge. Most importantly, it demands to go beyond the paradigm of using the exact eigenvectors on graphs.
For arbitrary space dimension $d$ we investigate the quantum phase transitions of two paradigmatic spin models defined on a hypercubic lattice, the coupled-dimer Heisenberg model and the transverse-field Ising model. To this end high-order linked-cluster expansions for the ground-state energy and the one-particle gap are performed up to order 9 about the decoupled-dimer and high-field limits, respectively. Extrapolations of the high-order series yield the location of the quantum phase transition and the correlation-length exponent $ν$ as a function of space dimension $d$. The results are compl…
We propose a hypergraph expansion which facilitates the direct treatment of quantum spin models with many-site interactions via perturbative linked cluster expansions. The main idea is to generate all relevant subclusters and sort them into equivalence classes essentially governed by hypergraph isomorphism. Concretely, a reduced König representation of the hypergraphs is used to make the equivalence relation accessible by graph isomorphism. During this procedure we determine the embedding factor for each equivalence class, which is used in the final resummation in order to obtain the final res…
We use non-perturbative linked-cluster expansions to determine the ground-state energy per site of the spin-one Heisenberg model on the kagome lattice. To this end, a parameter is introduced allowing to interpolate between a fully trimerized state and the isotropic model. The ground-state energy per site of the full graph decomposition up to graphs of six triangles (18 spins) displays a complex behaviour as a function of this parameter close to the isotropic model which we attribute to divergencies of partial series in the graph expansion of quasi-1d unfrustrated chain graphs. More concretely,…
We implement a highly efficient strong-coupling expansion for the Green's function of the Hubbard model. In the limit of extreme correlations, where the onsite interaction is infinite, the evaluation of diagrams simplifies dramatically enabling us to carry out the expansion to the eighth order in powers of the hopping amplitude. We compute the finite-temperature Green's function analytically in the momentum and Matsubara frequency space as a function of the electron density. Employing Pade approximations, we study the equation of state, Kelvin thermopower, momentum distribution function, quasi…
We introduce a generic scheme to perform non-perturbative linked cluster expansions in long-range ordered quantum phases. Clusters are considered to be surrounded by an ordered reference state leading to effective edge-fields in the exact diagonalization on clusters which break the associated symmetry of the ordered phase. Two approaches, based either on a self-consistent solution of the order parameter or on minimal sensitivity with respect to the ground-state energy per site, are formulated to find the optimal edge-field in each NLCE order. Furthermore, we investigate the scaling behavior of…
We use linked-cluster expansions to analyze the quantum phase transitions between symmetry unbroken trivial and topological Haldane phases in two different spin-one chains. The first model is the spin-one Heisenberg chain in the presence of a single-ion anisotropy while the second one is the dimerized spin-one Heisenberg chain. For both models we determine the ground-state energy and the one-particle gap inside the non-topological phase as a high-order series using perturbative continuous unitary transformations. Extrapolations of the gap series are applied to locate the quantum critical point…
Numerical linked-cluster expansions allow one to calculate finite-temperature properties of quantum lattice models directly in the thermodynamic limit through exact solutions of small clusters. However, full diagonalization is often the limiting factor for these calculations. Here, we show that a partial diagonalization of the largest clusters in the expansion using the Lanczos algorithm can be as useful as full diagonalization for the method while mitigating some of the time and memory issues. As a test case, we consider a frustrated Heisenberg model on the checkerboard lattice and find that …
Numerical Block Diagonalization and Linked-Cluster Expansion for Deriving Effective Hamiltonians: Applications to Spin Excitations
RIKEN Center for Emergent Matter Science（理化学研究所物质结构研究中心）；School of Physical and Chemical Sciences, Queen Mary University of London（伦敦大学玛丽女王学院物理与化学科学学院）
We develop a numerical linked cluster expansion (NLCE) method that can be applied directly to inhomogeneous systems, for example Hamiltonians with disorder and dynamics initiated from inhomogeneous initial states. We demonstrate the method by calculating dynamics for single-spin expectations and spin correlations in two-dimensional spin models on a square lattice, starting from a checkerboard state. We show that NLCE can give moderate to dramatic improvement over an exact diagonalization of comparable computational cost, and that the advantage in computational resources grows exponentially as …
We study quantum quenches in the transverse-field Ising model defined on different lattice geometries such as chains, two- and three-leg ladders, and two-dimensional square lattices. Starting from fully polarized initial states, we consider the dynamics of the transverse and the longitudinal magnetization for quenches to weak, strong, and critical values of the transverse field. To this end, we rely on an efficient combination of numerical linked cluster expansions (NLCEs) and a forward propagation of pure states in real time. As a main result, we demonstrate that NLCEs comprising solely recta…
We demonstrate that numerical linked cluster expansions (NLCE) yield a powerful approach to calculate time-dependent correlation functions for quantum many-body systems in one dimension. As a paradigmatic example, we study the dynamics of the spin current in the spin-1/2 XXZ chain for different values of anisotropy, as well as in the presence of an integrability-breaking next-nearest neighbor interaction. For short to intermediate time scales, we unveil that NLCE yields a convergence towards the thermodynamic limit already for small cluster sizes, which is much faster than in direct calculatio…
We present a short review of our studies of disorder influence upon Ginzburg - Landau expansion coefficients in Anderson - Hubbard model with attraction in the framework of the generalized DMFT+$Σ$ approximation. A wide range of attractive potentials $U$ is considered - from weak coupling limit, where superconductivity is described by BCS model, to the limit of very strong coupling, where superconducting transition is related to Bose - Einstein condensation (BEC) of compact Cooper pairs, which are formed at temperatures significantly higher than the temperature of superconducting transition, a…
We study the Heisenberg model on the pinwheel distorted kagome lattice as observed in the material Rb_2Cu_3SnF_12. Experimentally relevant thermodynamic properties at finite temperatures are computed utilizing numerical linked-cluster expansions. We also develop a Lanczos-based, zero-temperature, numerical linked cluster expansion to study the approach of the pinwheel distorted lattice to the uniform kagome-lattice Heisenberg model. We find strong evidence for a phase transition before the uniform limit is reached, implying that the ground state of the kagome-lattice Heisenberg model is likely…
Non-equilibrium quantum phenomena are ubiquitous in nature. Yet, theoretical predictions on the real-time dynamics of many-body quantum systems remain formidably challenging, especially for high dimensions, strong interactions or disordered samples. Here we consider a notable paradigm of strongly correlated Fermi systems, the Mott phase of the Hubbard model, in a setup resembling ultracold-gases experiments. We study the three-dimensional expansion of a cloud into an optical lattice after removing the confining potential. We use time-dependent density-functional theory combined with dynamical …
We study the effect of quenched disorder on the semimetal-superconductor quantum phase transition in a model of two-dimensional Dirac semimetal with $N$ flavors of two-component Dirac fermions, using perturbative renormalization group methods at one-loop order in a double epsilon expansion. For $N\geq 2$ we find that the Harris-stable clean critical behavior gives way, past a certain critical disorder strength, to a finite-disorder critical point characterized by non-Gaussian critical exponents, a noninteger dynamic critical exponent $z>1$, and a finite Yukawa coupling between Dirac fermions a…
Motivated by Weyl semimetals and weakly doped semiconductors, we study transport in a weakly disordered semiconductor with a power-law quasiparticle dispersion $ξ_{\bf k}\propto k^α$. We show, that in $2α$ dimensions short-correlated disorder experiences logarithmic renormalisation from all energies in the band. We study the case of a general dimension $d$ using a renormalisation group, controlled by an $\varepsilon=2α-d$-expansion. Above the critical dimensions, conduction exhibits a localisation-delocalisation phase transition or a sharp crossover (depending on the symmetries of the Hamilton…
We study the effect of uncorrelated random disorder on the temperature dependence of the superfluid stiffness in the two-dimensional classical XY model. By means of a perturbative expansion in the disorder potential, equivalent to the T-matrix approximation, we provide an extension of the effective-medium-theory result able to describe the low-temperature stiffness, and its separate diamagnetic and paramagnetic contributions. These analytical results provide an excellent description of the Monte Carlo simulations for two prototype examples of uncorrelated disorder. Our findings offer an intere…
Superconducting instability can occur in three-dimensional quadratic band crossing semimetals only at a finite coupling strength, due to the vanishing of density of states at the quadratic band touching point. Since realistic materials are always disordered to some extent, we study the effect of short-ranged correlated disorder on this superconducting quantum critical point, using a controlled loop-expansion applying dimensional regularization. The renormalization group (RG) scheme allows us to determine the RG flows of the various interaction strengths, and shows that disorder destroys the su…
Disordered quantum systems undergoing a many-body localization (MBL) transition fail to reach thermal equilibrium under their own dynamics. Distinguishing between asymptotically localized or delocalized dynamics based on numerical results is however nontrivial due to finite-size effects. Numerical linked cluster expansions (NLCE) provide a means to tackle quantum systems directly in the thermodynamic limit, but are challenging for models without translational invariance. Here, we demonstrate that NLCE provide a powerful tool to explore MBL by simulating quench dynamics in disordered spin-$1/2$…
