MATERIALS SCIENCE AND CHEMISTRY

Introduction

Topology and correlations are notions that summarize one of the most active research field in the solid state. In this domain, our aim is to understand theoretical aspects of collective and emergent phenomena and their realization in nature. To achieve this goal large scale approximation free simulations are imperative, since one can only sharply define collective excitations and phase transitions in the thermodynamic limit. Clearly, this limit can never be taken, even experimentally. However, the inverse size of the system defines an energy scale, or length scale, above which one can observe such excitations and critical fluctuations. Thus, the bigger the system size, the more accurate and reliable the results. From the numerical point of view the above defines two challenges. i) developing new algorithms that are more efficient [1] and ii) optimal implementation of existing algorithms on supercomputers [2]. Below we will describe the numerical method used as well as a selection of recent results.

Results and Methods

Algorithms for lattice fermions – A general implementation of the auxiliary field Quantum Monte Carlo method.

All our projects are based on the so-called auxiliary field quantum Monte Carlo algorithm. We have been, and are, spending a lot of time and effort at developing an optimized and general OpenMP/MPI package that allows simulations of a number of models at limited programming cost. A first version of this open source project entitled the ALF (Algorithms for Lattice Fermions) can be found online and the documentation has been published in Ref. [2]. The ability to play with models of correlated electron systems at a minimal programming cost is crucial. It is a priori not clear that the models we design will deliver the collective emergent phenomena we wish to study: in many cases, simple mean-field type instabilities -- as opposed to highly entangled quantum states - will intervene. By the same token, access to computational resources that allow us to quickly and efficiently map out the nature of the phase diagram is imperative.

Correlations and frustration

Classically, frustration leads to a macroscopic degenerate ground state that violates the third law of thermodynamics. Turning on quantum effects is bound to generate new stable and potentially exotic states of matter. Typical examples of the above are quantum spin liquids, or the fractional quantum Hall effect. In Ref. [3], we have introduced a new class of coupled frustrated spin fermion models that can be simulated - free of the so-called negative sign problem - in the realm of the ALF [2]. As a case study we present in Fig. [1] results for a Kondo lattice model on the Honeycomb lattice, supplemented by frustrating couplings between the localized spins. We have shown in Ref. [3] that this coupling term generates so called partial Kondo screened states of matter, where Kondo screening becomes site dependent so as to alleviate frustration effects. To the best of our knowledge, these are the first approximation free numerical simulations that exhibit such a state of matter.

Competing orders in Dirac systems

In Ref. [4] we designed a model that allows for spontaneous antiferromagnetic (AFM) and Kekulé valence bond solid (KVBS) orders (See. Fig 2). We consider Dirac fermions, as realized by a tight binding model with matrix element t on the honeycomb lattice, supplemented by a Hubbard U term. This term triggers an O(3) Gross-Neveu transition between the semi-metal and an AFM. To account for a competing ordered state, we couple our model to an Ising degree of freedom in a transverse field of magnitude h. The coupling between the Ising and fermionic degrees of freedom is chosen such that when the Ising field orders it triggers a KVBS.

The main result of the paper, is the observation of a direct and continuous transition between the AFM and KVBS with emergent SO(4) symmetry. This transition cannot be understood in the realm of mean-field type Ginzburg-Landau transitions, and belongs to the category of so called deconfined quantum critical points (DQCP). Here, the emergent SO(4) symmetry allows for a topological Θ-term at Θ=π in the low energy effective field theory. As a consequence the domain wall defects of the KVBS order harbor spin ½ chains. At the critical point these spin ½ chains condense and form the AFM order.