Computational condensed matter physics · Many-body theory
Alpesh SHETH, PhD
I develop rigorous, predictive computational models of quantum materials and validate them against experimental data. Most recently, my research has focused on the infinite-layer nickelates: a family of superconductors that are structural and electronic analogues to cuprates, yet exhibit divergent macroscopic behavior. I demonstrated that this discrepancy is driven by a secondary electronic band unique to the nickelate system. I implement these simulations using Python, C++, and Fortran, deployed across GPU architectures and high-performance computing (HPC) clusters.
01Research
Research
My PhD research, supervised by Sébastien Burdin at the Laboratoire Ondes et Matière d'Aquitaine (CNRS, Université de Bordeaux) and defended in November 2025, focused on the infinite-layer nickelates \(R\mathrm{NiO_2}\). Discovered to superconduct in 2019, these compounds are structural and electronic analogues to copper-oxide superconductors. However, experimental observations deviate significantly from the standard one-band cuprate model.
To address this discrepancy, I developed a two-band model: the primary nickel \(3d_{x^2-y^2}\) band hybridized with a secondary interstitial band. This second band conducts current across specific regions of the phase diagram, granting the Fermi surface a high sensitivity to applied pressure and oxygen stoichiometry—degrees of freedom fundamentally absent in one-band descriptions. Crucially, a single set of parameters within this model captures the physics of both the \(R\mathrm{NiO_3}\) and \(R\mathrm{NiO_2}\) families.
This framework yields a directly testable prediction: the characteristic charge/spin ordering wavevector of the electrons is not rigidly pinned by the crystal lattice, as previously assumed, but varies continuously with doping and pressure. This shift can be verified via scattering experiments. Simulation 2 (scroll below) allows you to adjust these physical controls and observe the peak displacement in real time. Published in J. Phys.: Condens. Matter 37, 125703 (2025); preprint arXiv:2407.16042; thesis (open access).
I have since applied this analytical framework to a broader class of computationally heavy problems governed by similar mechanics: high-dimensional state spaces driven by a tunable control parameter, leading to a critical threshold where macroscopic behavior shifts. Concretely, this includes neural networks used to represent many-body quantum states, ab initio predictions of battery aging, and mapping combinatorial optimization problems onto statistical physics models. Each simulation below isolates and visualizes the critical threshold for these respective systems.
Languages
Python · C++ · Fortran · Julia · Bash · MATLAB · LaTeX
Many-body methods
Green's functions · Dynamical mean-field theory · Exact diagonalisation · Kubo formalism
Electronic structure & dynamics
VASP · Quantum ESPRESSO · LAMMPS · CP2K · TRIQS
Data & machine learning
Predictive modelling · Neural-network methods · Data pipelines · Statistical analysis
Optimisation
Simulated annealing · Quantum-inspired solvers · Rigorous benchmarking
Infrastructure
High-performance computing (Slurm) · Git · Workflow automation
02Simulation 1 · Fermiology
Fermi-surface topology of the two-orbital model
The Fermi surface is the boundary between the electron states that are filled and those that are empty; its shape governs almost everything the material does. Two things happen as you raise the energy. First the surface tears and reconnects, changing shape entirely. About 60 meV higher, the second band drops below the cut-off and a new pocket of electrons appears. Drag the band panel to move the energy and watch both.
Simulation 1. Two-band tight-binding model with \(t = 0.40\) eV: a Ni \(d_{x^2-y^2}\) band with hoppings \(t\) and \(t'\) (in units of \(t\)), and an interlayer band of rare-earth character, coupled by hybridisation \(V\), so that \(\displaystyle E_\pm(\mathbf k) = \tfrac12\bigl(\varepsilon_1+\varepsilon_2\bigr) \pm \sqrt{\tfrac14\bigl(\varepsilon_1-\varepsilon_2\bigr)^2 + V^2}.\) Right panel: the zero level set of \(E_\pm(\mathbf k) - E_\mathrm{F}\) on a \(170^2\) grid, contoured by marching squares. Reduced from the model of J. Phys.: Condens. Matter 37, 125703 (2025).
03Simulation 2 · Susceptibility
Where the electrons want to order
This map asks: at which wavelength are the electrons most willing to organise themselves? A bright spot means the material is close to ordering at that spacing — magnetically, or as a static charge pattern. In this model the bright spot sits on the zone boundary and moves as the band fills, ending near the wavevector the copper oxides are known for. Move the control and watch it walk; that motion is the prediction experiments can test. The map is computed row by row when you arrive, and can be recomputed at any energy.
Simulation 2. The bare susceptibility \(\displaystyle \chi_0(\mathbf q) = \frac{1}{N}\sum_{\mathbf k} \frac{f(E_{\mathbf k}) - f(E_{\mathbf k+\mathbf q})}{E_{\mathbf k+\mathbf q} - E_{\mathbf k}},\) evaluated on a \(64^2\) \(\mathbf k\)-mesh for a \(33^2\) \(\mathbf q\)-mesh. Intraband contributions only, with matrix elements set to unity; degenerate denominators are replaced by \(-\partial f/\partial E\). The maximum arises from nesting and broadens as \(k_\mathrm{B}T\) increases. Scope of the approximation: intraband response at the bare level, without vertex corrections or self-energy.
04Simulation 3 · Optimisation
Portfolio selection as an Ising ground state
Choosing which assets to hold is the same mathematics as finding the lowest-energy arrangement of a magnet. Holding or not holding becomes spin up or spin down; expected return becomes a magnetic field on each site; the risk of holding two correlated assets becomes a coupling between them. With ten assets there are only 1024 possibilities, so every one can be checked and the true best answer is known. That matters: benchmarks of quantum-inspired solvers usually compare against another approximate method. Here the solver is scored against the proven optimum, and the gap to it is the only number worth reporting.
Simulation 3. The objective is \(\displaystyle H = -\sum_i h_i x_i + \lambda \sum_{i<j} C_{ij}\, x_i x_j + A\Bigl(\sum_i x_i - K\Bigr)^{\!2},\) with \(x_i \in \{0,1\}\), \(h_i\) the expected returns and \(C_{ij}\) the covariance matrix, minimised by single-flip Metropolis under geometric cooling (simulated annealing) from \(T = 3\) to \(T = 0.01\). Returns and covariances are fixed synthetic data and the random stream is seeded by \((\lambda, K, \text{sweeps})\), so a given setting reproduces its trace exactly. The default budget of 70 sweeps is deliberately short, so the gap to \(E^*\) stays visible and measurable. Full app opens the complete implementation in a window on this page.
05Simulation 4 · Moiré superlattices
Moiré periodicity in twisted bilayers
Stack two identical atomic lattices and rotate one by a fraction of a degree: the overlap produces a much larger repeating pattern — a moiré superlattice — in the same way two overlaid mesh screens do. The electrons feel that larger pattern as if it were the real crystal, which is how twisting a bilayer changes its electrical behaviour. Before my PhD I studied graphene on boron nitride in this geometry using density functional theory.
Simulation 4. The superlattice vectors are constructed as \(\displaystyle \mathbf L_i = \bigl(R(\theta) - I\bigr)^{-1}\mathbf a_i,\) from which \(\lambda = a / 2\sin(\theta/2)\) follows rather than being imposed; the period therefore diverges as \(\theta \to 0\). Near \(\theta \approx 1.1^\circ\) in twisted bilayer graphene the lowest folded bands become narrow, so that \(U/W\) grows large; correlated insulating and superconducting states are observed in that regime. The atom count assumes two atoms per unit cell in each of two layers.
06Simulation 5 · Criticality
The two-dimensional Ising model
The standard textbook model of a phase transition: a grid of magnets, each preferring to point the same way as its neighbours. Vermilion is up, white is down. Below the critical temperature they lock into large domains that drift slowly; above it, order breaks down and the pattern turns to noise. This is the cleanest demonstration of a threshold there is, which is why it anchors this page. It runs on your GPU. Hold a pointer over the lattice to pull the magnets one way; tap or click to push the other.
Simulation 5. The Hamiltonian is \(\displaystyle H = -J\sum_{\langle ij\rangle} s_i s_j - \sum_i h_i s_i,\) with \(J = 1\) and \(k_\mathrm{B} = 1\), for which Onsager's solution gives \(T_c = 2/\ln(1+\sqrt2) \approx 2.269\). The configuration is held in a texture and updated by a checkerboard-parallel Metropolis pass, one sublattice at a time, so that the neighbours of an updating site are fixed and detailed balance is preserved. The initial condition is seeded, so a given re-seed reproduces the same trajectory.
07Summary
Models, control parameters, and checks
Five of these seven rows are the simulations above; two are still in progress. The last column says what each model was checked against, which is what makes the rest verifiable.
| System | Reduced to | Control parameter | Threshold | Validated against |
|---|---|---|---|---|
| Infinite-layer nickelates | Two-orbital model: Ni \(d_{x^2-y^2}\) and interlayer \(d_{z^2}\) band | EF, hybridisation V, pressure | second Fermi pocket opens | DFT band structures, ARPES, published spectra |
| Electronic instabilities | Lindhard susceptibility χ0(q) | Doping, temperature | boundary nesting, X → M with filling | Cuprate comparison, JPCM 37, 125703 |
| Portfolio selection | Ising Hamiltonian, binary holdings | Risk aversion λ, cardinality K | exact optimum reached | Enumeration over 2N |
| Twisted bilayers | Moiré superlattice, folded mini-zone | Twist angle θ | bandwidth collapse at 1.1° | DFT on graphene / hBN |
| Spin lattices | 2D Ising model, nearest-neighbour exchange | Temperature T, applied field h | order lost at Tc | Onsager's exact solution |
| Many-body ground states | Neural ansatz |ψθ⟩, variational Monte Carlo | Network capacity, sampler quality | variational energy converged | Exact diagonalisation |
| Battery ageing | DFN cell model with ab initio priors | Migration barriers, site capacities | residual systematic, not random | Laboratory cycling data |
→ table scrolls sideways
08Method
Method
Every model above was built in the same four steps, in that order. The order matters more than any single technique: most of the work is in the first step, and most of the mistakes are caught in the last.
Reduce
Identify the minimal set of moving parts that still produces the behaviour — two bands for a nickelate, holdings and correlations for a portfolio. This step sets the ceiling on everything after it: the reduction decides what the computation is able to tell you.
Formulate
Write it in a form that established mathematics already knows how to attack — an energy function, a landscape to minimise. Then what you can say about the answer is a result, not just a number the code returned.
Compute
Vectorise the inner loops, move the expensive parts onto GPUs or the cluster, and check against cases with known exact answers before running anything large.
Validate
Compare against experiment, independent data, or an exact solution — specifically against data the model was not built to reproduce. Where prediction and measurement diverge, the discrepancy identifies precisely which assumption to revisit.
09Record
Publications and training
Download PDFOne peer-reviewed paper, an open-access thesis, a preprint, and the positions behind them. The thesis and both preprints are free to read.
10What is next
Where the work is going
Four active threads. Each has the same shape as the work above — a large space of possibilities, one dial, a threshold worth finding. For each I have said what remains to be established.
Neural-network quantum states
Writing a quantum state down explicitly costs twice as much memory for every particle added, which puts the practical reach at a few tens of particles. A neural network can represent the state compactly instead, trained by Monte Carlo and checked against exact answers wherever those are still computable. Open directions: finite temperature, real-time dynamics, and neutral-atom hardware.
Ab initio battery modelling
Quantities computed from first-principles chemistry — how easily lithium moves, how much a site holds, what voltage results — fed as priors into the Doyle–Fuller–Newman cell model, the standard engineering description of a battery. Where that model and real cycling data disagree systematically, the disagreement is the signal, and it becomes the target for predicting cell lifetime.
Benchmarking quantum-inspired optimisation
Classical and quantum-inspired solvers run on identical problems, under one metric, against exact baselines — so that a reported speed-up reflects the solver and not a quietly changed convention. Simulation 3 is the public version.
Statistical physics of market microstructure
An order book is a large number of agents each making a buy-or-sell decision, pushed far from equilibrium — formally close to a spin system. Whether the sudden shifts in market behaviour can be described in the same language as phase transitions is an open question, and establishing the answer either way is the useful result.
11Contact
Contact
If you are wrestling with a complex system that stubbornly resists being reduced to its mathematical essentials, or a critical threshold currently hiding in the noise, I would like to hear about it. Email is the most reliable way to reach me.
- Emailalpeshsheth.phy@gmail.com
- LinkedInalpesh-sheth
- ThesisOpen access · HAL
- ScholarPublications
- CVDownload PDF