# Yang-Le Wu

ph.d. in physics ⇢ quantitative trading

ph.d. in physics ⇢ quantitative trading

I am a Senior Vice President and quantitative analyst at the D. E. Shaw group in New York. I apply statistical inference techniques to study global markets and develop machine learning algorithms to uncover and correct market inefficiencies.

In a previous life, I was a theoretical physicist. I worked on strongly correlated electrons and topological phases of condensed matter, and in particular, I had expertise in certain mathematical and numerical aspects of the fractional quantum Hall effect. This page showcases relics and memorabilia from those years in academia.

I graduated with a PhD in Physics from Princeton University in 2014. My advisor was Professor B. Andrei Bernevig, and my informal co-advisor was Professor Nicolas Regnault at École Normale Supérieure. From 2014 to 2017 I worked at the Condensed Matter Theory Center of the University of Maryland as a JQI Postdoctoral Fellow, mentored by Professor Sankar Das Sarma.

Links to my research papers can be found at the bottom of this page, on arXiv, and also on my Google Scholar profile.

You can reach me at (PGP key).

Non-Abelian anyons
are among the most striking manifestations of topological order.
These exotic excitations have been at the center of a huge research effort in recent years,
partly driven by the possibility
of constructing topologically fault-tolerant quantum computers.
In this context, of particular importance are the Fibonacci anyons.
They provide a representation of the
braid group
rich
enough to carry out *universal* quantum computations,
which cannot be achieved with simpler non-Abelian anyons such as the Majorana fermions.

It had long been conjectured that the quasiholes in the ℤ₃ Read-Rezayi state, which may describe the filling ν = 12/5 fractional quantum Hall plateau, are Fibonacci anyons when sufficiently separated. However, this conjecture had not been supported by any microscopic evidence, due to the sheer complexity of the ℤ₃ Read-Rezayi quasihole wave functions. In fact, beyond the conjectured universal topological properties, very little was known about these elusive excitations.

In Phys. Rev. Lett. 113, 116801, we settled this long-standing problem and explicitly demonstrated the Fibonacci nature of the ℤ₃ Read-Rezayi quasiholes. Through a numerical study of the model wave functions, we established the exponential convergence of the braiding matrices with increasing quasihole separations, and we extracted the associated length scales as well as the quasihole radii. This puts an upper bound on the desirable quasihole density in interferometer devices — at a higher density, the ℤ₃ Read-Rezayi quasiholes exhibit clear non-universal deviations from the Fibonacci anyons.

In addition, we also provided
a microscopic diagnosis for the pathology of the
Gaffnian wave function,
which was conjectured *not* to give rise to sensible braiding statistics due to its root in non-unitary conformal field theories.
We explicitly demonstrated that the non-universal, path-dependent contributions to the braiding matrices
follow a power-law rather than exponential decay as quasihole separations increase.
This signals the failure of plasma screening
and highlights the gapless nature of the Gaffnian.
Our results largely ruled out the possibility of salvaging this pathological wave function as the description of a gapped phase with topological order.

The above progress was enabled by the recent breakthrough of the exact matrix product states for fractional quantum Hall effect. Essentially, the matrix product state is a factorization of the many-body wave function. This factorization makes it possible to exploit the entanglement area law and to store quantum information compactly, and it also greatly facilitates the calculation of physical observables. Its appearance in the quantum Hall context is deeply rooted in the bulk-edge correspondence. Namely, a large class of quantum Hall trial states are described by many-point correlation functions in chiral conformal field theories (CFT). Each electron / quasihole in the quantum Hall wave function is represented by a primary field operator in the conformal correlator. Since a field operator is essentially an infinite-dimensional matrix over the CFT Hilbert space, the conformal correlator, and thus the quantum Hall trial state, can be cast in the form of a matrix product state (MPS). After a truncation of descendants in the CFT Hilbert space, a field operator (or more precisely, the corresponding 3-point function in the CFT) can be well approximated by a finite-size (albeit large) matrix. Such matrices can be constructed numerically from the Virasoro algebra, even for interacting CFTs.

This provides an extremely powerful numerical tool to study the strongly-correlated physics in fractional quantum Hall phases. For our purposes, the MPS technique enables a brute-force evaluation of the conformal-block wave functions with an arbitrary number of quasiholes, without relying on bosonization tricks. It grants access to much larger system sizes than previously attainable, and opens an avenue to the direct characterization of physical properties without confronting the exponentially large many-electron Hilbert space. In Phys. Rev. B 92, 045109, we provided a pedagogical description of the MPS construction. In particular, we discussed in detail how to build MPS for non-Abelian quasiholes with conformal-block normalization and monodromy.

Stabilizing non-Abelian fractional quantum Hall states often requires careful softening of the Coulomb repulsion between electrons. In realistic setups, this task can be extremely challenging. One possible avenue for highly tunable interactions comes from multi-component quantum Hall systems, such as bilayers or wide quantum wells. These systems typically feature a rich variety of topological phases and interaction-driven topological phase transitions. Unfortunately, the internal degrees of freedom that provide tunability also impose a heavy burden on microscopic numerical calculations. As a result, many such systems have not been thoroughly explored.

In Phys. Rev. B 92, 035103, we initiated a systematic numerical study of quantum Hall bilayers at filling ν = 2/3. Using a combination of exact diagonalization and variational Monte Carlo, we mapped out the phase diagram in both the lowest and the second Landau levels as a function of interlayer separation and tunneling strength. We found that the ℤ₄ Read-Rezayi state is highly competitive in the second Landau level.

Interactions
can
stabilize
strongly-correlated phases in
topological
flat bands with nonzero Chern number *C*.
These so-called fractional Chern insulators (FCI) exhibit fractional quantum Hall effect at *zero* magnetic field.
In Phys. Rev. B 85, 075116,
using exact diagonalization and particle entanglement
spectrum,
we demonstrated numerically the existence of the FCI phase
in
an
array
of
lattice models with a *C = 1* flat band.
This includes both Abelian and non-Abelian quantum Hall states.
We found a correlation between the stability of the strongly-correlated phase
and the uniformity of the Berry curvature in the band structure.

The nature of the FCI ground state at *C = 1* can be
understood by the Wannier mapping
between a Chern band and the lowest Landau level (LLL).
In Phys. Rev. B 86, 085129,
after a proper gauge fixing, we transcribed the continuum Laughlin state to the lattice,
and achieved high overlaps with the FCI ground state.

For
FCI
with
*C*
> 1, however,
numerical studies revealed anomalous features
distinct from the usual multicomponent quantum Hall states.
In Phys. Rev. Lett. 110, 106802, we found
that the correct one-body mapping for *C > 1* involves a specially crafted set of boundary conditions for the multicomponent LLL.
This new boundary condition sews together the *C* components into a single manifold with Chern number *C*.
Using the modified one-body mapping,
we constructed pseudopotential Hamiltonians and model wave functions
for FCI with an arbitrary Chern number.
Our model wave functions correctly capture the subtle differences between the lattice FCI states
and the usual multicomponent quantum Hall states.
In Phys. Rev. B 89, 155113,
we analyzed the FCI pseudopotential Hamiltonian in
the
thin-torus
limit.
This revealed a generalized Pauli principle for the degeneracy of the FCI ground states in each Bloch momentum sector.
A reference implementation of the corresponding counting rule is available here.

In the past, I have also studied the pairing mechanism of iron-selenide superconductor using functional renormalization group.

Before coming to the US for graduate school, I worked with Professor Qi Ouyang as an undergraduate Chun-Tsung Scholar at Peking University, on the origin of the dynamical robustness of regulatory networks in living cells, and on the non-linear dynamics of reaction-diffusion systems.

- Yang-Le Wu and S. Das Sarma, "Decoherence of two coupled singlet-triplet spin qubits", Physical Review B, 96(16):165301, 2017. [arXiv:1708.07539]
- Yang-Le Wu, Dong-Ling Deng, Xiaopeng Li, and S. Das Sarma, "Intrinsic decoherence in isolated quantum systems", Physical Review B, 95(1):014202, 2017. [arXiv:1610.03058]
- Xiaopeng Li, Dong-Ling Deng, Yang-Le Wu, and S. Das Sarma, "Statistical Bubble Localization with Random Interactions", Physical Review B, 95(2):020201(R), 2017. [arXiv:1609.01288]
- Dong-Ling Deng, Xiaopeng Li, J. H. Pixley, Yang-Le Wu, S. Das Sarma, "Logarithmic Entanglement Lightcone in Many-Body Localized Systems", Physical Review B, 95(2):024202, 2017. [arXiv:1607.08611]
- Kiryl Pakrouski, Matthias Troyer, Yang-Le Wu, Sankar Das Sarma, and Michael R. Peterson, "The enigmatic 12/5 fractional quantum Hall effect", Physical Review B, 94(7):075108, 2016. [arXiv:1604.04610]
- S. Das Sarma, Robert E. Throckmorton, and Yang-Le Wu, "Dynamics of two coupled semiconductor spin qubits in a noisy environment", Physical Review B, 94(4):045435, 2016. [arXiv:1604.06110]
- Yang-Le Wu and S. Das Sarma, "Understanding analog quantum simulation dynamics in coupled ion-trap qubits", Physical Review A, 93(2):022332, 2016. [arXiv:1512.00848]
- Edwin Barnes, Dong-Ling Deng, Robert E. Throckmorton, Yang-Le Wu, and S. Das Sarma, "Noise-induced collective quantum state preservation in spin qubit arrays", Physical Review B, 93(8):085420, 2016. [arXiv:1510.03862]
- Yang-Le Wu, B. Estienne, N. Regnault, and B. Andrei Bernevig, "Matrix product state representation of non-Abelian quasiholes", Physical Review B, 92(4):045109, 2015. [arXiv:1504.06620]
- Michael R. Peterson, Yang-Le Wu, Meng Cheng, Maissam Barkeshli, Zhenghan Wang, and Sankar Das Sarma, "Abelian and non-Abelian states in ν=2/3 bilayer fractional quantum Hall systems", Physical Review B, 92(3):035103, 2015. [arXiv:1502.02671]
- Yang-Le Wu, B. Estienne, N. Regnault, and B. Andrei Bernevig, "Braiding Non-Abelian Quasiholes in Fractional Quantum Hall States", Physical Review Letters, 113(6):116801, 2014. [arXiv:1405.1720]
- Yang-Le Wu, N. Regnault, and B. Andrei Bernevig, "Haldane statistics for fractional Chern insulators with an arbitrary Chern number", Physical Review B, 89(15):155113, 2014. [arXiv:1310.6354]
- Xiaomeng Zhang, Bin Shao, Yangle Wu, Qi Ouyang, "A Reverse Engineering Approach to Optimize Experiments for the Construction of Biological Regulatory Networks", PLoS ONE, 8(9):e75931, 2013.
- Yang-Le Wu, N. Regnault, and B. Andrei Bernevig, "Bloch Model Wave Functions and Pseudopotentials for All Fractional Chern Insulators", Physical Review Letters, 110(10):106802, 2013. [arXiv:1210.6356]
- Yang-Le Wu, N. Regnault, and B. Andrei Bernevig, "Gauge-fixed Wannier wave functions for fractional topological insulators", Physical Review B, 86(8):085129 [Editor's Suggestion], 2012. [arXiv:1206.5773]
- Yang-Le Wu, B. Andrei Bernevig, and N. Regnault, "Zoology of fractional Chern insulators", Physical Review B, 85(7):075116 [Editor's Suggestion], 2012. [arXiv:1111.1172]
- Chen Fang, Yang-Le Wu, Ronny Thomale, B. Andrei Bernevig, and Jiangping Hu,
"Robustness of
*s*-Wave Pairing in Electron-Overdoped*A*_{1-y}Fe_{2-x}Se_{2}(*A*=K,Cs)", Physical Review X, 1(1):011009, 2011. [arXiv:1105.1135] - Yangle Wu, Xiaomeng Zhang, Jianglei Yu, and Qi Ouyang, "Identification of a Topological Characteristic Responsible for the Biological Robustness of Regulatory Networks", PLoS Computational Biology, 5:1000442, 2009.
- Huimin Liao, Yangle Wu, Jianglei Yu, and Qi Ouyang, "Local wave grouping in a parameter-gradient system and its formation mechanism", Physical Review E, 77(7):016206, 2008.