Research Interests

General Interests:

My research field is theoretical condensed matter physics with an emphasis on correlated many-body systems. Specific topics of interest include frustrated magnetism, lattice fermion systems, and the study of computational methods applied to these areas. My work focuses on the ability of competing interactions, which arise from a competition between different energy terms in the Hamiltonian, to induce phase transitions in models of many-body systems. The low temperature phenomena of correlated materials are captured in simplified Hamiltonians and studied by numerical and analytic techniques. Investigations address questions of fundamental physics as well as specific material related issues.

General Background:

For many problems in condensed matter physics (CMP), one considers an ensemble of N particles (e.g., electrons, magnetic moments) in thermal contact with their environment, interacting through pair-wise potentials (U(r₁,r₂)), and constrained spatially to the vertices of a lattice. When interactions are turned on, exact solutions are usually only possible in reduced spatial dimensions (e.g., 1D). Hence, interacting many-body systems are studied by several powerful approximate analytic techniques (e.g., perturbation and mean-field theories) and numerical methods (e.g., Monte Carlo (MC)). In general, interacting models evolve from a collection of independent particles at high temperatures to a correlated ensemble at low temperatures. If the pair correlations extend to the infinite range limit, then the model (or material) has experienced a phase transition to a long-range ordered state. Whether a finite temperature (above zero degrees Kelvin) phase transition occurs or not depends on the spatial dimension of lattice (usually greater than 2D) and the internal symmetry of the particles (e.g., spin degree of freedom). The study of thermal phase transitions continues to be a very active field within condensed matter physics. It also has influenced other areas of physics (e.g., high energy and cosmology). Recently, the study of quantum phase transitions (occurring at zero degrees Kelvin) has generated great excitement in the field. In this case, a variable like pressure is responsible for inducing a change in phase.

Clean models in CMP are those in which the interactions take on well defined values that are not complicated by the lattice geometry, chemical dilution, or random disorder effects. A growing field of study considers what happens to the physics of these clean models, in particular their ordered states, when the interactions are disordered (U(r₁,r₂) defined by a distribution) or compete with one another. In studying these complex Hamiltonians, one grapples with some of the most perplexing problems in CMP: spin glasses, where magnetic moments freeze out of equilibrium into random orientations; spin liquids, where the magnetic moments are fluctuating down to zero degrees Kelvin; the metal-insulator transition, where the competition between electron-electron interactions and disorder in the lattice compete and favor extended or localized electronic states, respectively; and spin-charge separation in superconductors.

Research on Spin Systems:

The bulk of my research has been on magnetic spin systems in which the microscopic interactions are frustrated due to the lattice geometry. In simple terms, geometric frustration arises when the spins on a lattice can not satisfy all interactions simultaneously. Triangular or tetrahedral motifs of antiferromagnetically coupled spins (i.e., spins prefer to align antiparallel) are the canonical building blocks of many geometrically frustrated systems. Magnetic frustration can also arise from randomly distributed interactions (disorder). Research in this area has grown over the years for several reasons: most real magnets contain some degree of frustration in their interactions, frustration is often a key ingredient in the creation of novel magnetic phases, frustrated magnetism is amenable to the study of fundamental physics issues concerning interacting many-body systems.

My current research efforts, in collaboration with Michel Gingras, have focused on rare-earth pyrochlore magnets. Our emphasis has been on insulating systems, where the highly frustrated pyrochlore lattice (a network of corner sharing tetrahedra) acts in concert with the spin degrees of freedom and long-range dipole-dipole interactions to realize exotic magnetic phases like spin ice (residual entropy in the ground state) and spin liquid (dynamical fluctuations at T -> 0⁺). A battery of techniques are employed in these investigations: MC simulations, mean-field theory, random-phase approximation, high temperature series expansions, and summation methods to handle long-range dipolar interactions.

The rare-earth pyrochlores Ho₂B₂O₇ and Dy₂B₂O₇ (B=Ti,Sn) are called spin ices because from a statistical mechanics point of view they retain a similar residual (zero-point) entropy in their frozen state as hexagonal water ice (I_h). Hence, water ice and spin ice are systems frozen out of equilibrium. The pyrochlore material Tb₂Ti₂O₇, which fails to order at temperatures as low as T = 70 mK, is a candidate spin liquid. Investigations of these materials focuses on unraveling the mysteries that exist in the paramagnetic (PM), spin disordered, regime. We are also looking at the subtle physics of the Heisenberg AF pyrochlore Gd₂Ti₂O₇, which orders at T = 1 K and possibly experiences two transitions, and the complicated corner sharing triangular garnet material Gd₃Ga₅O₁₂, which has a complex magnetic field versus temperature phase diagram with spin glass, spin liquid, and antiferromagnetic phases.

In rare-earth pyrochlores, the rare-earth
ions (e.g., Ho³⁺,Dy³⁺,Tb³⁺,Gd³⁺) sit on
the vertices of the tetrahedra. Exchange
and dipolar interactions are strong in these
materials. Single-ion anisotropy is significant
in some of these systems (e.g., the spin-ices). pyrochlore lattice

ggg lattice The lattice of Gd₃Ga₅O₁₂ (or GGG)
consists of two interpenetrating
sublattices of corner sharing triangles.
The rare-earth ion Gd³⁺ sits on the
vertices of this lattice. The single-ion
anisotropy is negligible, but exchange and
and dipolar interactions are strong.

Earlier work on spin systems focused on bilayer magnets with frustration caused by lattice mismatch between layers and was done in collaboration with Richard Scalettar and Susan Kauzlarich. Using MC methods and mean-field techniques, we found that lattice off-set could induce collinear Ne^'el states, canted phases, or orthogonal order between layers. This work was motivated by experimental evidence of orthogonal ordering in several mixed layer pnictide oxide materials (Sr₂Mn₃Pn₂O₂, Pn=As,Sb).

Magnetic unit cell of pnictide oxide. The magnetic unit cell of Sr₂Mn₃Sb₂O₂.
The manganese ions Mn²⁺ (S=5/2) (red and blue circles)
reside on two inequivalent square lattices. Neutron powder
diffraction suggests a magnetic structure in which the moments
in alternating layers order, antiferromagnetically, along
orthogonal directions.

Research on Fermi Systems:

My endeavors into this field are rooted in a joint collaboration with Richard Scalettar , George Batrouni, Shiwei Zhang and Frederic Hebert to study interacting fermions on a lattice in the presence of disorder. The aim of the project was to study the phase diagram of the half-filled 2D Hubbard model (a simple model for electrons on a lattice with kinetic energy and on-site repulsion terms) with random hopping (bond disorder) and chemical potential (site disorder), with the goal to develop a better understanding of the interplay between interactions and disorder in electronic systems (e.g., doped semiconductors and thin superconducting films). Our particular focus was on the magnetic and charge correlations in this model.

To model disorder in a real material, calculations have to consider the average effect of disorder on the physics of your Hamiltonian. This means that calculations (numerical here) are very CPU intensive because an average over disorder realizations must be performed in addition to a thermal average (configuration average at T=0). We employed several techniques in our study: determinant quantum Monte Carlo (DQMC), constrained path quantum Monte Carlo (CPQMC), numerical Hartree-Fock, and exact diagonalization for small lattices. Among these, the CPQMC algorithm was crucial because it avoids the sign problem (see below) present in MC simulations of the site disordered model. We were successful in our efforts to study the effects of disorder on the magnetic correlations in the Hubbard model, finding that antiferromagnetic order is destroyed at a critical value of the disorder strength for each case considered. We also extended the limits of validity of the CPQMC algorithm to disordered lattice models. The effects of disorder on the charge correlations in the 2D Hubbard model are planned for the future.

A schematic of bond disorder for
fermions on a lattice. Singlets form
along the strong bonds, magnetism
is destroyed, but the local moments
are preserved. Bond disorder in the Hubbard model.

Site disorder in the Hubbard model A schematic of site disorder for fermions on a lattice.
Electrons pair on sites with deep potential wells. The
loss of magnetic order is accompanied by the destruction
of local moments. The electron wave function is localizes,
an insulator.

Research on Numerical Methods:

My efforts in numerical methods development have centered on investigations to extend the range of applicability of the CPQMC algorithm and to possible solutions to the quantum MC (QMC) sign problem. In short, the sign problem arises when the signal (average of an observable, < f > ) to noise (variance of that observable Var{f}) in a QMC simulation vanishes (i.e., < f >²/Var{f} -> 0). The sign problem occurs in MC simulations of fermions, bosons, and quantum spin models. Solving the sign problem remains a challenge in QMC research. My work was done in collaboration with Malvin Kalos, Shiwei Zhang, and Richard Scalettar and entailed the study of the symmetry properties in simple continuum quantum models using diffusion Monte Carlo with correlated signed walkers. To learn more about the QCM sign problem, take a look at some notes by David Ceperley.

In addition, I am interested studying and applying sophisticated classical MC methods (e.g., cluster or loop algorithms, tempering and multi-canonical methods) to problems in frustrated magnetism, spin glasses, water ice, and possibly biological systems.

In rare-earth pyrochlores, the rare-earth ions (e.g., Ho³⁺,Dy³⁺,Tb³⁺,Gd³⁺) sit on the vertices of the tetrahedra. Exchange and dipolar interactions are strong in these materials. Single-ion anisotropy is significant in some of these systems (e.g., the spin-ices).
	The lattice of Gd₃Ga₅O₁₂ (or GGG) consists of two interpenetrating sublattices of corner sharing triangles. The rare-earth ion Gd³⁺ sits on the vertices of this lattice. The single-ion anisotropy is negligible, but exchange and and dipolar interactions are strong.

A schematic of bond disorder for fermions on a lattice. Singlets form along the strong bonds, magnetism is destroyed, but the local moments are preserved.
	A schematic of site disorder for fermions on a lattice. Electrons pair on sites with deep potential wells. The loss of magnetic order is accompanied by the destruction of local moments. The electron wave function is localizes, an insulator.