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AIP conference proceedings

Modelling and simulating strongly correlated fermions

Didier Poilblanc

Published Jan 1, 2007• 0 citations
Publisher

Abstract

I shall briefly review here some aspects of the Exact Diagonalization technique and put it in perspective with other numerical techniques such as Density Matrix Renormalization Group (DMRG) or Quantum Monte Carlo methods (partially) covered in this AIP Series by Reinhard Noack and Richard Scalettar respectively. I will then illustrate this lecture in discussing some simple applications, mainly for one dimensional systems.

DOI
10.1063/1.2751990

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I shall briefly review here some aspects of the Exact Diagonalization technique and put it in perspective with other numerical techniques such as Density Matrix Renormalization Group (DMRG) or Quantum Monte Carlo methods (partially) covered in this AIP Series by Reinhard Noack and Richard Scalettar respectively. I will then illustrate this lecture in discussing some simple applications, mainly for one dimensional systems.
In these lecture notes, we present a pedagogical review of a number of related numerically exact approaches to quantum many‐body problems. In particular, we focus on methods based on the exact diagonalization of the Hamiltonian matrix and on methods extending exact diagonalization using renormalization group ideas, i.e., Wilson’s Numerical Renormalization Group (NRG) and White’s Density Matrix Renormalization Group (DMRG). These methods are standard tools for the investigation of a variety of interacting quantum systems, especially low‐dimensional quantum lattice models. We also survey extensions to the methods to calculate properties such as dynamical quantities and behavior at finite temperature, and discuss generalizations of the DMRG method to a wider variety of systems, such as classical models and quantum chemical problems. Finally, we briefly review some recent developments for obtaining a more general formulation of the DMRG in the context of matrix product states as well as recent progress in calculating the time evolution of quantum systems using the DMRG and the relationship of the foundations of the method with quantum information theory.
One of the most intriguing areas of physics is the study of strongly correlated many-body systems. In these kinds of systems, a large number of particles interacts via interactions that are strong enough to play a major role in determining the properties of the system, which is particularly interesting when multiple interaction terms compete and cannot be satisfied simultaneously. Examples of phases they realize range from simple magnetism to puzzling phenomena such as superconductivity or topological order. Of special interest in condensed matter are fermionic models which have an extra layer of complexity added by the possibly intricate sign structure of the wave function. \nHowever, exactly solvable models that exhibit the sought-after phenomena are scarce and Hamiltonians that are supposed to model experiments often elude analytical solutions precisely due to the strong interactions. Numerical methods are thus essential to gain insight into these systems and provide an important link between theoretical models and real world experiments. There are many different methods that in principle allow solving a problem numerically exact, most prominently exact diagonalization, tensor network based methods such as DMRG and (quantum) Monte Carlo. Exact diagonalization is certainly the most straightforward technique which allows measuring any observable of interest but its application to interacting models is severely limited by the exponential growth of the Hilbert space with the system size which only allows studying rather small systems. Tensor network methods excel in the study of gapped, one-dimensional systems and while they have shown some impressive results for two-dimensional systems, they need a lot of fine tuning and careful analysis to be considered reliable in this very important domain. Quantum Monte Carlo, on the other hand, is in principle neither limited by the size of the Hilbert space nor by the strength of the interactions and has thus been a tremendously important tool for studying condensed matter systems which is why it is the focus of this thesis. \nIt is necessary to continuously advance existing and develop new techniques to keep up with theoretical as well as with experimental progress. In this thesis, new methods to identify and characterize conventional and novel phases of matter within quantum Monte Carlo are developed and tested on archetypical models of condensed matter. \nIn the first part of this thesis, a method to study entanglement properties of strongly interacting fermionic models in quantum Monte Carlo is presented. Measurements of this kind are needed because some characteristics of a system remain hidden from conventional approaches based on the calculation of correlation functions. Prime examples are systems that realize so-called topological order entailing long-range entanglement that can be positively diagnosed using entanglement techniques. Realizing such measurements in a concrete algorithm is not straightforward for fermions. A large section of this part is thus devoted to developing a solution to the problem of implementing entanglement measurements and benchmarking the results against known data. \nThe second part is concerned with a truly novel and recent approach to the many-body problem, namely the symbiosis of machine learning and quantum Monte Carlo techniques. Out of the many different possibilities to combine the two, this part is concerne with exploring ways in which machine learning can help to effortlessly explore phase diagrams of hitherto unknown Hamiltonians. \nDespite the power and versatility of quantum Monte Carlo techniques, it does suffer from one significant weakness called the fermion sign problem. At its core related to the peculiar exchange statistics of fermions, it is in principle merely a technical problem of the Monte Carlo procedure but one that actually prohibits the use of Monte Carlo methods for a large number of interesting problems. The sign problem has been known for a long time but remains unsolved to this day. Thus, it is time to approach the problem from a different perspective with the hope of learning something new that can either help to better understand the problem itself or the properties of the models affected by it. Both of the two major research areas, entanglement measures and machine learning, are capable of providing such new perspectives and the sign problem will be revisited in more detail in the context of each of them.
The Density Matrix Renormalization Group (DMRG) has become a powerful numerical method that can be applied to nanoscopic and low-dimensional strongly correlated fermionic and bosonic systems. It allows for a very precise calculation of static, dynamic and thermodynamic properties. Its field of applicability has now extended beyond Condensed Matter, and is successfully used in Quantum Chemistry, Statistical Mechanics, Quantum Information Theory, Nuclear and High Energy Physics as well. In this article, we briefly review the main aspects of the method and present some of the most relevant applications. The recent quantum information interpretation, the development of highly accurate time-dependent algorithms and the possibility of using the DMRG as the impurity-solver of the Dynamical Mean Field Method (DMFT) give new insights into its present and potential uses. We review the numerous very recent applications of these techniques whe the DMRG has shown to be one of the most reliable and versatile methods in modern computational physics.