Melko Group

Computational quantum many-body research

Our Research

Our research tries to answer a fundamental set of questions in context of condensed-matter, quantum many-body, and quantum information theory. A non-exhaustive list of motivations might include:

  • Are there undiscovered, interesting and (potentially) useful new states of matter to be found in nature?
  • What are the microscopic mechanisms behind the variety of exotic phenomena observed in strongly-correlated matter?
  • Does undiscovered and novel universal behavior exist at certain phase transitions?
  • Can quantum systems, and all of their hosts of intriguing phenomena, be studied with classical computers?
My favorite tool for research is simulation, particularly classical and quantum Monte Carlo, Density Matrix Renormalization Group methods, and numerical series expansion methods. Below, we've provided brief blurbs for several present and future research directions, for those students considering joining my group at the University of Waterloo.

Entanglement is a quintessential feature defining quantum physics. Einstein used a two-particle system argue that entanglement leads to "spooky action at a distance". The two-particle case is by now well understood, but the situation for multi-particle systems remains unclear. In most condensed-matter systems, entanglement between two regions of a multi-particle system scales as the boundary (or area) between them. This "area law" was first understood in the context of black hole physics, and has subsequently been demonstrated for many quantum many-body systems. Whether systems obey an area law is closely related to the feasibility of the simulation of those quantum systems on a classical computer. In our group, we study numerically the existence of laws in condensed-matter systems, as well as subleading scaling that has the potential to redefine classification schemes of condensed matter through access to universal physics.

Condensed matter physics is the study of the collective behavior of infinitely-complex assemblies of electrons, magnetic moments, atoms or qubits. This complexity is reminiscent of the curse of dimensionality commonly encountered in machine learning. Despite this curse, the machine learning community has developed techniques with remarkable abilities to classify, characterize and interpret complex sets of data, such as images and natural language recordings. In my group, we are interested in developing machine learning approaches to complex quantum and classical many-body problems. Combined with conventional Monte Carlo sampling, neural networks offer a promising route towards supervised and unsupervised learning for the study of phases, phase transitions, and other phenomena in condensed matter. Many open questions regarding the classical and quantum representation power of neural networks remain, including the relationship between deep learning and the renormalization group.

Phase transitions are fascinating because they give us a window into "universal" phenomena - quantitative connections that occur between very disparate natural systems, in condensed matter and beyond. Although such critical behavior is for the large part understood in classical physical systems, quantum mechanical systems offer a variety of possibilities for new manifestations of universal behavior. Some of these quantum critical points are relatively conventional, while others involve exotic phenomena such as fractionalized particles or emergent gauge symmetries. In my group we use computer simulations of realistic quantum lattice models to discover and characterize new examples of unconventional quantum critical points. These studies give a glimpse of a strange quantum world where exotic particles such as spin-charge separated electrons can be identified through universal signatures relevant for simulation, theory, or eventually, experiment.

Frustration is the inability of a system to simultaneously satify all of its individual microscopic constraints. In classical systems, this leads to interesting collective behavior at low temperatures, such as seen in the famous examples of "spin ice" materials. However, when quantum fluctuations come in to play, these frustrated systems can realize a whole new level of exotica. Various types of valence bond solid (illustrated) or related states can arise when quantum fluctuations are present in a frustrated system. Most striking are the disordered "spin liquid" states, which are T=0 quantum collective paramagnets. These support such exotic phenomena as fractional charges, spin-charge separation, and topological order - a crutial ingredient in modern proposals to build a quantum computer from topologically protected qbits.

The possible existence of a supersolid phase of Helium, first reported in 2004 by Kim and Chan, has driven experimental and theoretical work alike. The supersolid, a state that simultaneously exhibits the properties of a solid and a superfluid, would be a new phase of matter if proven to exist experimentally. Parallel to the experimental search are theoretical efforts to find candidate systems in which the Supersolid exists. At Waterloo we use a combination of analytical and numerical approaches to study a related phase, the Superglass -- a system similar to the Supersolid except that instead of a regular repeating pattern of density (defining a solid), we have a random frozen configuration (defining a glass) with coexisting superfluidity.

Computational methods, and in particular Quantum Monte Carlo (QMC) techniques, are an important tool in statistical physics and are essential for advancing studies in almost all quantum condensed matter research. In systems with a large number of accessible states, the integrals associated with various expectation values become too complicated to calculate exactly, but Monte Carlo methods allow us to calculate these values numerically using statistical sampling. Our group works on developing algorithms that use various QMC techniques, such as Anders Sandvik's Stochastic Series Expansion (SSE) methods. We have used these types of algorithms to study quantum many-body systems, and we have also applied these Monte Carlo simulations to studies of entropy and phase transitions in classical systems.

The 'K' Computer in Kobe, Japan