Clark Research Group

Our group connects the microscopic degrees of freedom which make up the "rules" of our universe to the emergent phenomena which arises from them. These degrees of freedom are inherently quantum-mechanical and can include the electrons in a material, spins in a lattice model, or gates in a quantum computation.

The emergent behavior is a priori unexpected from its individual aspects; a holistic effect that is both more and different from the sum of its parts. They span many of the deepest problems in science: high temperature superconductivity, heavy fermions, quantum hall, spin liquids, topological phases, and many effects in ab-initio materials. In addition to their theoretical interest, progress in emergent phenomena promises to have important practical applications ranging from designing materials with higher superconducting temperatures to making topological quantum computing possible. Our research encompasses various areas of many-body physics including condensed matter, AMO, and quantum computing.

The tools our group uses are primarily numerical including quantum Monte Carlo, tensor networks, and various methods inspired by quantum information. We not only apply but develop new algorithms to broaden the horizon of accessible many-body systems.

I'll be talking at the Stochastic Methods in Electronic Structure Theory Conference in Telluride. Please stop by and talk if you're around!

I'll be giving a summer school tutorial (as well as some lectures) on Path Integral Monte Carlo at the Telluride School on Stochastic Approaches to Electronic Structure Calculations.

I'll be giving a talk on new finite temperature methods at the Workshop on Understanding Quantum Phenomena with Path Integrals If you're around, please come chat.

'Ask the scientist' is done for the semester

Congratulations to Eli for getting a 2017 NSF Grad Fellowship Honorable Mention as well as one of the Scott Anderson Outstanding Graduate Assistant Awards for 2017 from our department. Congratulations to Dmitrii for winning the departments 2017 Jordan Asketh Award, which recognizes one of the year’s outstanding European graduate students.

I've just finished teaching graduate quantum mechanics II. *Favorite topics:* Path Integrals and quantum many-body physics.

Our paper, Finite temperature properties of strongly correlated systems via variational Monte Carlo has now come out in PRB.

I recently wrote a physics viewpoint on Hai-Jun Liao, et. al work on the KAHF spin liquid.

Our paper The mother of all states of the kagome quantum antiferromagnet has been posted. In it, we report the existence of a new analytically solvable macroscopically degenerate point on the kagome lattice (called XXZ0), mapped out the phase diagram for how XXZ0 connects to the KAHF spin liquid, and argue that the KAHF spin liquid is actually a critical point!

Come find us at the March Meeting! I'll be giving a talk on efficient algorithms for finding unitary matrix-product operators. My students/postdocs are giving talks on our work on stuffed honeycomb (Dmitrii), our work on a new macroscopically degenerate point in the kagome (Hitesh), our new finite temperature algorithm (Jahan), our new inverse QMC technique (Eli) and looking at MBL use the one-body density matrix (Benjamin).

Our paper Encoding the structure of many-body localization with matrix product operators has finally been published!

Our paper Finding Matrix Product State Representations of Highly Excited Eigenstates of Many-Body Localized Hamiltonians has been published in PRL. It's on two new algorithms for finding eigenstates of many-body localized phases.

Congratulations to David Luitz who is starting a new position as Marie Skłodowska Curie Fellow at Technische Universität München.

The semester is essentially over. I'm just finishing up teaching graduate quantum mechanics I. My favorite topics this year were quantum computing and reduced density matrices. Next semester, I'll be teaching graduate quantum mechanics II.

Our paper, Finite temperature properties of strongly correlated systems via variational Monte Carlo has been posted to the arxiv. In it, we develop an algorithm that lets us use variational wave-function to compute finite temperature properties.

An image from our bimodal paper made it to the front page of the journal website! It's an image of the cut-averaged entanglement entropy (CAEE) which takes all the disordered entanglement (the jagged curves) and turns it into a nice concave curve. It's on the PRB front page now or see it permanently here.

I've recently returned from teaching third-graders how to "think like a computational physicist." They got to simulate (in python) an ecosystem (bunnies, wolves, and flowers!), learn about positive feedback, and graph population sizes.

Our paper Bimodal entanglement entropy distribution in the many-body localization transition has come out in PRB. In it we find that the entanglement in the MBL critical regime is bimodal. (see more)

Xiongjie has passed his prelim. Congratulations!

Our paper Numerical evidence for a chiral spin liquid in the XXZ antiferromagnetic Heisenberg model on the kagome lattice at m=2/3 magnetization has come out in PRB. In it we find a new chiral spin liquid! (see more)

We've recently posted our paper, *Fixed points of Wegner-Wilson flows and many-body localization*. In this paper, we use an RG approach (Wegner-Wilson flows) to work out the flow diagram
for a MBL system. We find stable MBL and ergodic fixed points which are characterized
by power-law and narrow distributions respectively. In addition, we find a unstable
fixed point at the critical point (maybe critical phase?) which exhibits scale invariance.
We identify these things by looking at properties of l-bits. (see more)

The semester has recently started at University of Illinois and I'm teaching graduate quantum mechanics. (see more)

Statistical mechanics tells us that in a closed quantum system, the larger system should act as a heat bath for a small subsystem. The system is "ergodic." In quantum systems with interactions and sufficiently strong disorder, the system can enter a phase of matter - the many-body localized (MBL) phase - where ergodicity breaks down. This phenomena can exist out to infinite temperature and is a rare example of quantum mechanics dominating the qualitative physics at high temperatures.

One recent interest of our group has been using tensor networks as both a
*conceptual language* as well as a numerical tool to understand the many-body localized phase.

- In the many-body localized phase, we've recently shown that the
*entire*eigenspectrum can be represented in a beautiful compact way with only O(n) parameters (and that many of the characteristics of the MBL phase follow from the structure of these eigenstates). - The phenomenology of the fully MBL phase is mainly discussed in terms of conserved quantum numbers called l-bits. To understand properties of l-bits, one wants to get their hands on them and we've developed a numerical approach for computing l-bits as well as shown that they are quasi-local.
- We've developed an additional RG procedure to identify these l-bits and shown that there is scale invariance of the l-bits at the MBL transition.
- The "order parameter" for MBL is in terms of the highly-excited eigenstates. We've developed two new DMRG-like algorithms, ES-DMRG and SIMPSs to access very highly excited eigenvectors.
- We've used these new algorithms to verify the breakdown of ETH, to show that entanglement in the MBL phase saturates and to show the existence of a large number of local excitations.

To learn more about MBL + Tensor Networks, you might want to look at my talks from KITP (Nov. 2015), Perimeter (Oct. 2014) , or Dresden (May 2014)

In addition to working with tensor networks, we've recently discovered bimodal behavior of the entanglement entropy in the critical regime of the MBL transition both within and between disordered samples. We have also developed a new observable to examine MBL with (the cut-averaged entanglement entropy) which has the property that it is smooth even for a single disorder realization!

In another project, we explored the MBL phase analytically and numerically in infinite dimension using a wave-function approach inspired by the random energy model.

(Ferro)-magnetism was known even by the ancient Greeks. We now know that magnetism is a collective effect where a macroscopic number of spins align in the same direction. If instead, the spins are frustrated, no simple pattern of spins can can form and various exotic phenomena can arise. The prototypical way to frustrate anti-ferromagnetic spins in the Ising limit is by placing them on triangles pasted together in some way (such as in the kagome lattice on a corner sharing lattice).

Most recently, we have shown variational evidence for the existence of a chiral spin liquid in a square hopping model with pi-flux per plaquette as well as given strong numerical evidence for a chiral spin liquid in the 2/3 magnetic plateau of the kagome lattice at the XY point.

To learn more about frustrated magnetism, see my talk at the Perimeter Institute on variational methods applied to the honeycomb and kagome lattice.

In experimental physics, the design of a new apparatus has been the source of many important breakthoughs. Similarly, important results in theoretical physics are driven by new computational algorithms. Our group develops new algorithms to tackle the quantum many-body problem both improving accuracy and computational complexity. Examples of success' include

- the development of a new finite-temperature approach which is based on variational wave-functions.
- the development of two new algorithms (SIMPS and ES-DMRG) which compute matrix-product state representations of excited states high in the spectrum. We have used these algorithms for understanding aspects of the many-body localized phases.
- the combination of MPS and QMC methodology to compute many-body ground states with higher accuracy then can be computed individually.
- designing an algorithm that makes the multiSlater-Jastrow wave-function a practical one making the cost of each additional determinant above single excitations independent of particle number. We now compute on thousands of determinants and this is now a standard technique implement in all quantum Monte Carlo codes.
- understanding the sign problem and improving on FCIQMC techniques to allow unbiased calculation of fermionic systems (at a cost that, albeit exponential, signficantly improves on the state of the art)

The extended Church-Turing thesis states that anything that can be computed quickly, can also be computed quickly on a Turing machine (or your Macbook). Quantum computing is the first real challenge to this thesis. Algorithms such as factoring and finding ground states of reasonable physical systems appear to be classically difficult but possible on a quantum computer. Recently, we've been thinking about whether this additional power is only of theoretical interest or can realistically be leveraged in our lifetime (for example, in quantum chemistry or lattice models). We're also interested in questions related to topological quantum computing and the interplay between quantum information and tensor networks.

I am currently teaching: **Graduate Quantum Mechanics 2 ** [link]

Previously, I've taught various courses including
*An Algorithmic Perspective on Strongly Correlated Systems*,
*Atomic Scale Simulations *,
*Undergraduate Quantum Mechanics*,
among others (see here).

Besides courses, I've given various summer school lectures (see here). The last summer school I taught at was the Princeton Condensed Matter Physics School on MBL and Frustrated Magnetism. Thanks to all the students who came and asked great questions. Summer school lectures (especially chalk talks) don't always translate online well, but here are some of the more useful artifacts from them:

- my Boulder QMC lecture notes [VMC, DMC] or video [VMC, DMC] from the Boulder Summer school
- Cornell Summer School on Emergent Phenomena Lectures [Video1, Video2]
- my tutorials (most recently given at CECAM) on variational Monte Carlo
- my tutorials (with Ken Esler) on Path Integral Monte Carlo with relevant files.
- simple, pedagogical mathematica notebook to learn DMRG.
- Handwritten lectures notes on MBL using tensor networks and frustrated magnetism. (video)