The Exotic Physics of Frustrated Magnets
Editor’s Note: This guest blog post is by Dr. Kingshuk Majumdar, associate professor of physics, Grand Valley State University, MI. Dr. Majumdar shares some of his research below, which was greatly facilitated via use of a supercomputing cluster. If you would like to contribute to Engineering on the Edge, please contact us.
Frustrated magnetic materials contain a wealth of interesting magnetic properties. Unlocking the mysteries of these frustrated magnets will not only deepen our understanding of the fundamental physics of these materials, but may also provide clues for potential technological applications in the near future. Therefore, these systems are presently under intense investigation by the physics community.
Besides mass and charge, the electron, an elementary particle within an atom, also has “spin.” Spin, an intrinsic property of electrons, comes in two varieties — “spin‐up” and “spin‐down.” In frustrated magnets, imbalance of these two types of spins results in magnetic frustration. With state‐of‐ the‐art 504 node supercomputing cluster “MATLAB on the TeraGrid” housed in Center for Advanced Computing at Cornell University, I am theoretically investigating the rich and exotic physics of these complex magnetic materials.
Magnetism was known to ancients. Its manifestations dazzle every school child. Its applications underlie huge industries. Yet, its understanding — even in a magnetic material like iron — is still incomplete. The phenomenon of magnetism cannot be understood without quantum physics. Besides mass and charge, the electron, an elementary particle within an atom, has spin.
Example: There are six electrons strongly bound to the nucleus in the outer shell of each iron atom. Five of these electrons have spin up and one has spin down (ignoring any quantum effects). This imbalance in spin‐up and spin‐down electrons results in a net magnetic moment of the atom, and we say the atom is magnetic.
In a magnetic material (e.g. iron, cobalt, nickel) spin‐spin interactions between magnetic atoms are responsible for their magnetic properties. There are different types of magnetism, including diamagnetism (which originates from the orbital motion of the electrons and is very weak), paramagnetism (where the atoms align in the same direction with the external magnetic field and the magnetism disappears when we remove the external field), ferromagnetism (as in iron, cobalt, nickel where the atoms align in one direction and magnetism is retained even at zero external magnetic field), and anti-ferromagnetism (as in Manganese oxide where spins align in a parallel — anti-parallel arrangement). [1, 2]
Consider the example of a ferromagnet where the majority of magnetic domains (cluster of atoms with magnetic moments pointing in the same direction) are aligned in the same direction. With an increase in temperature, these magnetic domains realign themselves randomly. Above a certain temperature, known as the Curie temperature, the material loses magnetism. In physics terminology, we say that we have a temperature-driven phase transition from an ordered ferromagnetic phase to a disordered paramagnetic phase. This is an example of a classical phase transition similar to the ice to water (solid to liquid) phase change. However, at zero temperature, it is possible to drive a magnetic material from an ordered phase to a disordered phase by changing some internal parameter(s) in the model — for example, by changing the coupling between two spins at different sites. This is known as quantum phase transition. 
Frustrated magnetic systems are of great interest in condensed matter research due to their highly unusual physical properties.  The recent discovery of superconductivity at relatively high temperatures in iron-based superconductors and the availability of new magnetic materials have refueled the interest in understanding the properties of frustrated magnets. Frustration can arise either due to the underlying geometry, or due to competing interactions. In other words, a magnetic spin system is frustrated when one cannot find a configuration of spins to fully satisfy the interaction (bond) between every pair of spins. In this case, the minimum of the total energy does not correspond to the minimum of each bond, which leads to highly degenerate ground states with a nonzero entropy at zero temperature.
The first frustrated system, which was studied in 1950, is the triangular lattice with spins interacting with each other via a nearest neighbor anti‐ferromagnetic (parallel-anti-parallel arrangements of spins) interaction. Consider a triangular lattice shown in Figure 1. Three magnetic ions reside on the corners of a triangle with anti-ferromagnetic interactions between them — the energy is minimized when each spin is aligned opposite to its neighbors. Once the first two spins align anti-parallel, the third one is frustrated because its two possible orientations, up and down, give the same energy. The third spin cannot simultaneously minimize its interactions with both of the other two. Thus the ground state is twofold degenerate.
Similarly in three dimensions, four spins arranged in a tetrahedron (Figure 2) may experience geometric frustration. If there is an anti-ferromagnetic interaction between spins, then it is not possible to arrange the spins so that all interactions between spins are anti‐parallel. There are six nearest‐neighbor interactions, four of which are anti-parallel and thus favorable, but two of which (between 1 and 2, and between 3 and 4) are unfavorable. It is impossible to have all interactions favorable, and the system is frustrated.
There are many examples of frustrated magnetic systems. At low temperatures, many of these systems exhibit exotic quantum phases such as spin liquids, spin‐ice states, and clusters or loops of a finite number of spins. [1, 2] Many of these highly complex systems can be well described by a simple model known in the literature as the Heisenberg spin model, with nearest neighbor anti-ferromagnetic coupling (J1) and next‐nearest‐neighbor anti-ferromagnetic coupling (J2). Experimentally, by applying high pressures, the low-temperature phases of these frustrated systems can be explored from low frustration (low J2/J1 ratio) to large frustration (large J2/J1 ratio).
For example, Li2VOSiO4 is an insulating vanadium oxide, with spin 1⁄2 V4+ ions arranged in a square lattice planes at the centers of VO4 pyramids. These are linked by SiO4 tetrahedra with lithium ions occupying the space between the V‐O planes. X‐ray diffraction, nuclear magnetic resonance, and muon spin‐rotation measurements on this compound reveal that it undergoes a phase transition at a low temperature T=2.8 K from an anti-ferromagnetic ordered phase to a collinear stripe phase. In the collinear stripe phase, the nearest neighbor spins have a parallel orientation in the vertical direction and anti-parallel orientation in the horizontal direction or vice versa. 
Unveiling the physics of these complex magnetic compounds is an exciting and challenging endeavor, which may provide clues for potential technological applications in the near future. That is what draws me to theoretically investigate the exotic physics of these magnetic materials. There are many analytical and numerical approaches to study the physics of quantum phases in these magnetic systems. One such technique is the spin‐wave expansion, in which the quantum fluctuations are treated in a systematic way in powers of 1/S (S=spin). Higher‐order corrections to the thermodynamics properties such as energy and magnetization are notoriously difficult to calculate, as the calculations involve six- (for two‐dimensional systems) or nine‐dimensional integrals (for three‐dimensional systems). For each value of energy or magnetization, the integrals are done by summing over billions of points. Such calculations are extremely time consuming and are almost impossible to do in a personal computer or in a workstation.
Presently I am using the state‐of‐the‐art 504 node supercomputing cluster “MATLAB on the TeraGrid” housed in the Center for Advanced Computing (CAC) at Cornell University. This computing resource is funded by the National Science Foundation (NSF) in partnership with Purdue University, Dell, MathWorks, and Microsoft. MATLAB’s seamlessly accessible parallel computing interactive environment enables me to write and run my code without being a parallel programming expert. In “MATLAB on the TeraGrid,” it now takes 2‐3 days to obtain a set of magnetization or energy data. This computing resource has proven to be invaluable to my research. I have been able to complete and publish my work in peer‐reviewed journals [3‐7] in a timely manner which would not have been possible without the use of the TeraGrid at Cornell.
1. H. T. Diep, “Frustrated Spin Systems,” World Scientific, Singapore, 2004, 1st ed.
2. S. Sachdev, “Quantum Phase Transitions,” Cambridge University Press, Cambridge, UK, 2001, 1st ed.
3. Kingshuk Majumdar, “Spin‐wave energy dispersion of a frustrated spin‐1/2 Heisenberg antiferromagnet on a stacked square lattice,” J. Phys.: Condens. Matter 23, 116004 (2011).
4. Kingshuk Majumdar, “Magnetic phase diagram of spatially anisotropic, frustrated spin‐1/2 Heisenberg anti-ferromagnet on a stacked square lattice,” J. Phys.: Condens. Matter 23, 046001 (2011).
5. Kingshuk Majumdar, “Second‐order quantum corrections for the frustrated, spatially anisotropic, spin‐1/2 Heisenberg antiferromagnet on a square lattice,” Phys. Rev. B 82, 144407 (2010).
6. Kingshuk Majumdar and T. Datta, “Zero temperature phases of the frustrated J1‐J2 antiferromagnetic model on a simple cubic lattice,” J. Stat. Phys. 139, 714‐726 (2010).
7. Kingshuk Majumdar and T. Datta, “Non‐linear spin wave theory results for the frustrated S=1/2 Heisenberg antiferromagnet on a bcc lattice,” J. Phys: Condens. Matt. 21, 406004 (2009).