Ushnish Ray

The basic idea of fundamental research in condensed matter physics (CM) is to gain insight into a range of exotic phenomena pertaining to real materials or systems and develop tools to be able to manipulate and harness them. Typically CM deals with phases of a variety of systems such as materials used to build devices. However, real materials are extremely complex and it is almost impossible to capture the full range of physics that go into describing their behavior. Instead physicists make the modest approach of trying to get an understanding of the essential ingredients that go into describing the phenomena of interest. As it turns out, this is possible by constructing models captured, for instance, via the Hamiltonian of the system.

A variety of techniques go into constructing and solving Hamiltonians. The quantum formulation makes things particularly difficult owing to the inherent probabilistic properties of the quantum world. In any case, despite such difficulties, remarkable progress has been made - opening up frontiers into a plethora of fascinating behavior such as Superfluidity, Topological superconductors and insulators, Many-Body localization, Quantum Entanglement, and the list continues to grow.

Traditionally AMO is concerned with studying the properties of atoms and molecules and its interactions with photons. These are also manipulated to study deep questions such as fundamentals of quantum mechanics, and also towards engineering marvels such as the modern atomic clock. More recently, standard techniques used to cool and trap atoms are used to construct *artificial materials - engineered Hamiltonians. *Essentially, the idea is to experimentally construct the very models that are of interest in Condensed Matter with extreme precision. Arguably, the exciting prospects with this methodology is to eliminate hard-to-control elements that feature in experiments with real systems. AMO systems can, for instance, be used to create tunable lattices, introduce controllable disorder, interactions between particles and so on - these from essential ingredients of most real materials.

Another fascinating aspect of such artificial materials is that it allows the exploration of closed quantum systems and dynamical phenomena with relative ease. These areas of physics are rife with open questions. Indeed, quantum dynamics and non-equilibrium statistical mechanics remain mysterious in many ways despite decades of inquiry.

Numerical Strategies

Exploring the effects of different ingredients that constitute the Hamiltonian and, indeed, finding or understanding the states that result from them, is the main interest of CM theorists. Towards this end there are few techniques that afford accurate and robust results that can be used to understand experiments. At the forefront of such techniques are powerful numerical algorithms - essentially theoretical calculations undertaken on powerful computers. Examples of such techniques are given below.

- Quantum Monte-Carlo (QMC) is an extremely powerful technique that utilizes the power of supercomputers and simple ideas from statistics to find solutions of the Hamiltonians. Simply put, QMC converts multi-dimensional integrals (which necessarily arise in most problems in physics) into a sampling paradigm. For example, the simple problem of solving the integral \int f(x) dx is recast into \sum_i f(x_i), where x_i is sampled from a uniform distribution (for this simple problem). This basic idea can be extended to complicated scenarios over multi-dimensions. At its core, QMC frequently employs the well known Metropolis sampling strategy to tackle problems. QMC is not without its limitations, however, for example Hamiltonians can possess certain properties (the infamous sign problem) that renders QMC inefficient. QMC algorithm development is an active area of research.

- Density Matrix Renormalization Group (DMRG) tackles the problem of solving the Hamiltonian by judiciously keeping track of the "necessary" states that contribute to the essential physics of interest. In effect, it coarse-grains out non-essential contributions and can systematically arrive at exact answers. DMRG has been remarkably successful in solving a wide class of problems in 1D and 2D where the state-space of the system possess certain properties (in particular the Area-law that encapsulates the way in which the quantum system is spatially entangled). Tensor Networks (TN) form a natural extension of the ideas of DMRG, and as it turns out, supersedes it in the sense that DMRG follows from the general ideas of TN. This numerical approach is quite recent, and is an active area of research. It has the potential to be able to access many questions in physics that have remained elusive thus far: for instance, real-time quantum dynamics.
- Density Matrix Embedding Theory (DMET) is a recent method that takes ideas from DMRG and Dynamic Mean-field Theory (DMFT) to obtain accurate measurements of static observables. The idea is to embed the full many-body Hamiltonian into an exact fragment and approximate bath formulation forming an impurity model that can then be solved to obtain a mean-field description that captures the essential features of interest (e.g. correlation functions, energy etc.)

This page gives a basic overview of the different types of projects that I have been working on. It will help in putting my published papers in context. At the very outset it is probably worth mentioning that this is a work in progress and it will probably take me some time to flesh out the details and to polish the contents, so apologies in advance!

Figure 1: (A) Example of the cuprates, (B) the 2D layered structure of the Cu and O electron orbitals and (C) Actual crystal of YBCO.

e)

**A tunable lattice can be constructed by a combination of lasers and magnetic fields. The collection cell dispenses the requisite atoms with necessary symmetry properties (e.g. Rubidium 85, an alkali atom is Bosonic whereas Potassium 40 is Fermionic). These atoms are trapped via a combination of lasers and magnetic fields that are used to construct a Magneto-Optical Trap (MOT). Atoms in the MOT can then be transferred into a 3D-lattice constructed from counter-propagating lasers. Experiments generally are undertaken in ultra-low pressure to minimize losses from collisions with other atoms. In this manner a closed system is constructed. **

**The figure to the right shows an example of such a construction. a) Vacuum system that is used to create a ultra-low pressure environment needed for these systems. b) The science cell into which atoms are transported from the Magneto-Optical trap (MOT) used to collect atoms in the collection cell. c) Is an example of atoms trapped in the MOT. In this case Dysprosium atoms are visible as a bright blue spot in the image. d) Example of the laser system constructed from 421 nm light used to trap and cool Dysprosium atoms.** [Images a)-b) is courtesy of DeMarco Lab at the University of Illinois at Urbana-Champaign. c)-d) courtesy of Lev Lab at Stanford University.]

**The science cell into which atoms are transported harbors an periodic potential that is similar to the potential created by ions in crystalline lattices. The exact form can be manipulated by manipulating the laser profile. e) Shows an example of a cubic potential in 2D into which atoms are loaded. Notice that in this manner the continuum equivalent of the lattice potential discussed earlier in the context of the Fermi-Hubbard model can be constructed.**

Figure 2: Phase diagram of high Tc superconductors.

[Source: Nature Materials 10, 259–261 (2011)]

[Sources: PNAS 2013 110 (30) 12235-12240 (2013) and Wikipedia]

C

**For example, consider the cuprates that constitute a class of materials associated with high temperature superconductivity. The Fig. (1A) on the right shows multiple types of cuprates. The electron orbitals of copper and oxygen atoms arranged on a plane (shown in Fig. 1B) separated by the interstitial atoms (Ba and Y in the case of YBCO). There are many open questions and this material is still an active areas of research. The full Hamiltonian of the system is given by: **

******The terms of the Hamiltonian represent the energetics due to the interactions between the particles (electrons and ions) and their motion or kinetics. Frequently, solving this full Hamiltonian is an impossible task and we must simplify it using reasonable approximations. For instance, in the case of cuprates a reasonable direction might be to ignore the interstitial atoms and the possible interactions between the Cu-O layers. Each layer then constitutes a 2D problem. Additionally, since we are interested in the low temperature properties, the ground state and low lying excitations of the system should be a good starting point. As such, we can coarse-grain out the contribution due to high energy components of the system. The Fermi-Hubbard model is a lattice system that is supposed to capture the essential physics. It is given by:**

**Although this model is considerably simpler than the full Hamiltonian, it is still a very challenging problem. If this simplification is indeed correct, then the properties of Hamiltonian should be able to reproduce the phase diagram shown in Fig. 2. The remarkable consequence is that if this works, then (barring pathologies) any high Tc-superconducting material could be characterized via the simple parameters of effective interactions, doping and the temperature. **