Session X27: New Methods for Strongly Correlated Systems

Non-equilibrium DMFT - Polaritonics

Non-equilibrium physics recently really becomes important with the progress of ultrafast laser sciences. However in our understanding there is still a gap between equilibrium physics and the non-equilibrium, even though numerical methods have been advanced in recent years. We compare in this talk novel results at hand with equilibrium physics. The comparison will show that especially theoretical efforts are needed to explain many - so far - unresolved problems and to predict novel research on the basis of ab initio computing. We specifically discuss several non-equilibrium extensions of DMFT, numerical methods as well as semi-analytical solvers.   

Strange metals from quantum geometric fluctuations of interfaces

The emerging picture of strongly correlated electron systems is that they possess a multiplicity of nearly degenerate ground states. A general theoretical framework for such systems is lacking. Here we explore a new approach based on the observation that different ground states can coexist and fluctuate in real space. Specifying to systems near the Mott metal-insulator transition, we propose a real space picture as itinerant electrons functioning in the fluctuating geometries bounded by interfaces between metallic and insulating regions. The interface fluctuations give rise to non-Fermi liquid behavior for the itinerant electrons, and furthermore mediate Cooper pairing.   

A real-time impurity solver for DMFT

Dynamical mean-field theory (DMFT) offers a non-perturbative approach to problems with strongly correlated electrons. The method heavily relies on the ability to numerically solve an auxiliary Anderson-type impurity problem. While powerful Matsubara-frequency solvers have been developed over the past two decades to tackle equilibrium situations, the status of real-time impurity solvers that could compete with Matsubara-frequency solvers and be readily generalizable to non-equilibrium situations is still premature. We present a real-time solver which is based on a quantum Master equation description of the dissipative dynamics of the impurity and its exact diagonalization. As a benchmark, we illustrate the strengths of our solver in the context of the equilibrium Mott-insulator transition of the one-band Hubbard model and compare it with iterative perturbation theory (IPT) method. Finally, we discuss its direct application to a nonequilibrium situation.   

Constrained Path Monte Carlo with Matrix Product State trial wavefunctions

Constrained path Monte Carlo (CPMC) is a powerful method for simulating strongly correlated systems. By constraining the path with a trial wavefunction, CPMC circumvents the minus sign problem, but at the cost of introducing a bias. The Density Matrix Renormalization Group (DMRG) is an alternative simulation technique, which is immune to the minus sign problem, but which has an analogous "dimensionality problem" for two and three dimensions. Here we present a combination of these techniques, where we use a DMRG matrix product state as a trial wavefunction for CPMC. We demonstrate our method in two-dimensional Hubbard model, and show the comparison to DMRG alone and to CPMC with single-determinant trial functions.   

Efficient implementation of the parquet equations - role of the reducible vertex function and its kernel approximation

We present an efficient implementation of the parquet formalism which respects the asymptotic structure of the vertex functions at both single- and two-particle levels in momentum- and frequency-space. We identify the two-particle reducible vertex as the core function which is essential for the construction of the other vertex functions. This observation stimulates us to consider a two-level parameter-reduction for this function to simplify the solution of the parquet equations. The resulting functions, which depend on fewer arguments, are coined ``kernel functions". With the use of the ``kernel functions", the open boundary of various vertex functions in the Matsubara-frequency space can be faithfully satisfied. We justify our implementation by accurately reproducing the dynamical mean-field theory results from momentum-independent parquet calculations. The high-frequency asymptotics of the single-particle self-energy and the two-particle vertex are correctly reproduced, which turns out to be essential for the self-consistent determination of the parquet solutions. The current implementation is also feasible for the dynamical vertex approximation.   

Exact Real-time Dynamics with non-equilibrium QMC

We present an overview of recent methodological progress for non-equilibrium hybridization expansion diagrammatic Monte Carlo impurity solver and we examine the real-time dynamics of a correlated quantum dot in the mixed valence regime. We perform numerically exact calculations of currents and magnetic susceptibilities after a quantum quench from equilibrium by rapidly applying a bias voltage in a wide range of initial temperatures. We observe Kondo signatures both in transient regimes and in the steady state.   

Dynamical simulations of strongly correlated electron materials

We present a formulation of quantum molecular dynamics that includes electron correlation effects via the Gutzwiller method. Our new scheme enables the study of the dynamical behavior of atoms and molecules with strong electron interactions. The Gutzwiller approach goes beyond the conventional mean-field treatment of the intra-atomic electron repulsion and captures crucial correlation effects such as band narrowing and electron localization. We use Gutzwiller quantum molecular dynamics to investigate the Mott transition in the liquid phase of a single-band metal and uncover intriguing structural and transport properties of the atoms.   

Ab Initio Dynamical Correlations from Auxiliary-field quantum Monte Carlo: applications in the Hubbard model

The possibility of calculating dynamical correlation functions from first principles provides a unique opportunity to explore the manifold of the excited states of a quantum many-body system. Such calculations allow us to predict interesting physical properties like spectral functions, excitation spectra and charge and spin gaps, which are more difficult to access from usual equilibrium calculations. We address the ab-initio calculation of dynamical Green functions and two-body correlation functions in the Auxiliary-field Quantum Monte Carlo method, using the two-dimensional Hubbard model as an example. When the sign problem is not present, an unbiased estimation of imaginary time correlation functions is obtained. We discuss in detail the complexity and the stability of the calculations. Moreover, we propose a new approach which is expected to be very useful when dealing with dilute systems, e.g. for cold gases, allowing calculations with a very favorable complexity in the system size.   

Bond Order Correlations in the 2D Hubbard Model

We use the dynamical cluster approximation to study the bond correlations in the Hubbard model with next nearest neighbor (nnn) hopping to explore the region of the phase diagram where the Fermi liquid phase is separated from the pseudogap phase by the Lifshitz line at zero temperature. We implement the Hirsch-Fye cluster solver that has the advantage of providing direct access to the computation of the bond operators via the decoupling field. In the pseudogap phase, the parallel bond order susceptibility is shown to persist at zero temperature while it vanishes for the Fermi liquid phase which allows the shape of the Lifshitz line to be mapped as a function of filling and nnn hopping. Our cluster solver implements NVIDIA's CUDA language to accelerate the linear algebra of the Quantum Monte Carlo to help alleviate the sign problem by allowing for more Monte Carlo updates to be performed in a reasonable amount of computation time.   

Entanglement Entropy of U(1) Quantum Spin Liquids

We investigate the entanglement structure of the ground state of a (3+1)-dimensional U(1) quantum spin liquid, described by the deconfined phase of a compact U(1) gauge theory. The excitations of the system are a gapless photon and gapped electric/magnetic charges. The elements of the entanglement spectrum can be grouped according to the electric flux between the two regions, leading to an interpretation in terms of particles living on the boundary. The entanglement spectrum is given additional structure due to the presence of the gapless photon. Making use of the Bisognano-Wichmann theorem and a local thermal approximation, these two contributions are recast in terms of boundary and bulk contributions, respectively. Both pieces give rise to universal subleading logarithms in the entanglement entropy, as opposed to the subleading constant in gapped topologically ordered systems. The photon term arises from the low-energy conformal field theory and is essentially local in character. The particle term arises due to the constraint of closed electric loops and is shown to be the natural generalization of topological entanglement entropy to the U(1) spin liquid. This contribution to the entanglement entropy can be isolated by means of a special geometric construction.   

Optical Conductivity in Holography with Hyperscaling Violation and Massive Gravity

One of the long-standing enigmas in the field of strongly-interacting electron systems is the mid-frequency power law form of the optical conductivity in the cuprates, $|\sigma| \sim \omega^{-\alpha}$. Many efforts have been put forth to obtain this power law using the AdS/CFT correspondence while maintaining the experimentally observed Drude form of the conductivity at low frequency. Some models have obtained the power law form over a very narrow range but none have matched the robust form lasting over decades as in experimental observations. We expand on previous constructions by introducing the dynamical exponent $z$ and hyperscaling parameter $\theta$ in a theory that breaks translational invariance using massive gravity. We seek a form of the optical conductivity that reproduces the functional form of the cuprates over all frequency regimes.   

Probing Critical Surfaces in Momentum Space Using Real-Space Entanglement Entropy: Bose versus Fermi

A co-dimension one critical surface in the momentum space can be either a familiar Fermi surface, which separates occupied states from empty ones in the non-interacting fermion case, or a novel Bose surface, where gapless bosonic excitations are anchored. Their presence gives rise to logarithmic violation of entanglement entropy area law. When they are \textit{convex}, we show that the shape of these critical surfaces can be determined by inspecting the leading logarithmic term of real space entanglement entropy. The fundamental difference between a Fermi surface and a Bose surface is revealed by the fact that the logarithmic terms in entanglement entropies differ by a factor of two: $S^{Bose}_{log} = 2 S^{Fermi}_{log}$, even when they have identical geometry. Our method has remarkable similarity with determining Fermi surface shape using quantum oscillation. We also discuss possible probes of \textit{concave} critical surfaces in momentum space.   

Entanglement spectrum and entangled modes of highly excited states in random XX spin chains

We examine the newly developed real space renormalization group method of finding excited eigenstate (RSRG-X) of the XX spin-1/2 chain, from entanglement perspectives. Eigenmodes of the entanglement Hamiltonian, especially the maximally entangled mode (that contributes the most to the entanglement entropy) and corresponding entanglement energies are studied and compared with predictions of RSRG-X. Our numerical results demonstrate the accuracy of the RSRG-X method in the strong disorder limit, and quantify its error when applied to weak disorder regime. Overall, our results validate the RSRG-X method qualitatively, but as in the case of real space renormalization group method for the ground state (RSRG) there are quantitative errors for weaker randomness, and also such error grows with increasing temperature/excitation energy density.   

Two-component Structure in the Entanglement Spectrum of Highly Excited States

We study the entanglement spectrum of highly excited eigenstates of two known models which exhibit a many-body localization transition, namely the one-dimensional random-field Heisenberg model and the quantum random energy model. Our results indicate that the entanglement spectrum shows a ``two-component'' structure: a universal part that is associated to Random Matrix Theory, and a non-universal part that is model dependent. The non-universal part manifests the deviation of the highly excited eigenstate from a true random state even in the thermalized phase where the Eigenstate Thermalization Hypothesis holds. The fraction of the spectrum containing the universal part decreases continuously as one approaches the critical point and vanishes in the localized phase in the thermodynamic limit. We use the universal part fraction to construct a new order parameter for the many-body delocalized-to-localized transition. Two toy models based on Rokhsar-Kivelson type wavefunctions are constructed and their entanglement spectra are shown to exhibit the same structure.   

Matrix-product-state method with local basis optimization for nonequilibrium electron-phonon systems

We present a method for simulating the time evolution of quasi-one-dimensional correlated systems with strongly fluctuating bosonic degrees of freedom (e.g., phonons) using matrix product states [1]. For this purpose we combine the time-evolving block decimation (TEBD) algorithm with a local basis optimization (LBO) approach. We discuss the performance of our approach in comparison to TEBD with a bare boson basis, exact diagonalization, and diagonalization in a limited functional space. TEBD with LBO can reduce the computational cost by orders of magnitude when boson fluctuations are large and thus it allows one to investigate problems that are out of reach of other approaches. First, we test our method on the non-equilibrium dynamics of a Holstein polaron [2] and show that it allows us to study the regime of strong electron-phonon coupling. Second, the method is applied to the scattering of an electronic wave packet off a region with electron-phonon coupling. Our study reveals a rich physics including transient self-trapping and dissipation.\\ \noindent [1] C. Brockt, F. Dorfner, L. Vidmar, F. Heidrich-Meisner, and E. Jeckelmann, arXiv:1508.00694\\ \noindent [2] F. Dorfner, L. Vidmar, C. Brockt, E. Jeckelmann, F. Heidrich-Meisner, Phys. Rev. B 91, 104302 (2015)