# 2017 Many Electron Collaboration Summer School

Date & Time

Location

June 16-23, 2017

Coding Workshop: June 16-18
Summer School: June 19-23

REGISTER HERE
Note: Attendance is by invitation only.

Organizers:
Coding Workshop: Emanuel Gull, University of Michigan
Summer School: Boris Svistunov, University of Massachusetts, Amherst

• Agenda

### Coding Workshop Agenda:

Friday, June 16
9:00 AM E. Gull: Programming Best Practices I
9:45 AM E. Gull: Programming Best Practices II
10:30 AM Break
11:00 AM M. Stoudenmire: Design
12:30 PM Lunch
1:30 PM M. Stoudenmire: iTensor Tutorial
3:30 PM Break
3:40 PM Student Presentations
Saturday, June 17
9:00 AM M. Stoudenmire: Julia
9:45 AM O. Parcollet: Zero cost abstraction for HPC by examples, gcc explorer
10:30 AM Break
11:00 AM E. Gull: Introduction to Hardware
12:30 PM Lunch
1:30 PM A. Gaenko: ALPS Tutorial
3:30 PM Break
3:40 PM Student Presentations
Sunday, June 18
9:00 AM O. Parcollet: Modern C++
9:45 AM O. Parcollet: Debugging
10:30 AM Break
11:00 AM S. Iskakov: Revision control and build tool tutorials
12:30 PM Lunch
1:30 PM O. Parcollet: Triqs tutorial

### Summer School Agenda:

Monday, June 19
9:00 AM A. Millis: Welcome
9:15 AM N. Prokof’ev: Introduction to Diagrammatic Monte Carlo
10:30 AM Break
11:00 AM E. Gull: DCA and Dual Fermions
12:30 PM Lunch
2:00 PM O. Parcollet: Non-equilibrium Correlated Electron Physics and the Numerical Challenges
3:30 PM Break
4:00 PM B. Svistunov: Basics of Feynman Diagrammatics
6:00 PM Break
6:30 PM Opening Banquet (at SCGP)
Tuesday, June 20
9:00 AM G. Kotliar: The Road to First Principles Electronic Structure Studies of Correlated Solids
10:30 AM Break
11:00 AM O. Parcollet: Out-of-equilibrium Quantum Impurity Models
12:30 PM Lunch
2:00 PM N. Prokof’ev
3:30 PM Break
4:00 PM Student Presentations
6:30 PM Free time and dinner on your own
Wednesday, June 21
9:00 AM M. Dean: Resonant Inelastic X-Ray Scattering as a Probe of Quantum Materials
10:00 AM Coffee Break
10:30 AM P. Johnson: Photoemission: A Key Probe of Condensed Matter
12:00 PM Group Outing – Smith Point Beach (Boxed lunch will be provided)
7:00 PM Group Dinner at the Fifth Season (34 E. Broadway, Port Jefferson)
Thursday, June 22
9:00 AM A. Alavi: Full Configuration Interaction Quantum Monte Carlo and Beyond
10:30 AM Break
11:00 AM S. Zhang: Real Materials
12:30 PM Lunch
2:00 PM R. Rossi: Shifted-action and Determinantal Diagrammatic Monte Carlo Techniques
3:30 PM Break
4:00 PM L. Pollet: Solving high-dimensional integral equations by a homotopy-transform: field-theoretical applications
6:00 PM Free time and dinner on your own
Friday, June 23
9:00 AM S. Zhang: Auxiliary-field Quantum Monte Carlo
10:30 AM Break
11:00 AM R. Rossi: Advanced Resummation Techniques
12:30 PM Lunch and Departure
• Student Presentations

Tuesday, June 20, 2017 (4:00-6:30 PM)

1. Hao Shi,College of William and Mary
Computations of fermion pairing by stochastic sampling in Hartree-Fock-Bogoliubov space

We describe the computational ingredients for an approach to treat interacting fermion systems in the presence of pairing fields, based on path-integrals in the space of Hartree-Fock-Bogoliubov (HFB) wave functions. The path-integrals can be evaluated by Monte Carlo, via random walks of HFB wave functions whose orbitals evolve stochastically. The approach combines the advantage of HFB theory in paired fermion systems and many-body quantum Monte Carlo (QMC) techniques. A constrained-path or phaseless approximation can be applied to the random walks of the HFB states if a sign problem or phase problem is present. With these techniques, we study the nature of the superconducting order in the two-dimensional Hubbard model by applying an external pairing finning field.

2. Mario Motta, College of William & Mary
Towards the solution of the many-electron problem in real materials: equation of state of the hydrogen chain with state-of-the-art many-body methods

We present numerical results for the equation of state of an infinite chain of hydrogen atoms. A variety of modern many-body methods are employed, with exhaustive cross-checks and validation. Approaches for reaching the continuous space limit and the thermodynamic limit are investigated, proposed, and tested. The detailed comparisons provide a benchmark for assessing the current state of the art in many-body computation, and for the development of new methods. The ground-state energy per atom in the linear chain is accurately determined versus bondlength, with a confidence bound given on all uncertainties.

Authors: Mario Motta, David M. Ceperley, Garnet Kin-Lic Chan, John A. Gomez, Emanuel Gull, Sheng Guo, Carlos Jimenez-Hoyos, Tran Nguyen Lan, Jia Li, Fengjie Ma, Andrew J. Millis, Nikolay V. Prokof’ev, Ushnish Ray, Gustavo E. Scuseria, Sandro Sorella, Edwin M. Stoudenmire, Qiming Sun, Igor S. Tupitsyn, Steven R. White, Dominika Zgid, Shiwei Zhang

3. Natalia Chepiga, UC Irvine
Excitation spectrum and Density Matrix Renormalization Group iterations

We show that, in certain circumstances, exact excitation energies appear as locally site-independent (or flat) modes if one records the excitation spectrum of the effective Hamiltonian while sweeping through the lattice in the variational Matrix Product State formulation of the Density Matrix Renormalization Group (DMRG), a remarkable property since the effective Hamiltonian is only constructed to target the ground state. Conversely, modes that are very flat over several consecutive iterations are systematically found to correspond to faithful excitations. We suggest to use this property to extract accurate information about excited states using the standard ground state algorithm. The results are spectacular for critical systems, for which the low-energy conformal tower of states can be obtained very accurately at essentially no additional cost, as demonstrated by confirming the predictions of boundary conformal field theory for two simple minimal models -the transverse-field Ising model and the critical three-state Potts model. This approach is also very efficient to detect the quasi-degenerate low-energy excitations in topological phases, and to identify localized excitations in systems with impurities.

4. Kai Guther, Max Planck Institute for Solid State Research
Computation of spectral functions using real-time evolution with FCIQMC

Kai Guther, Olle Gunnarsson, Werner Dobrautz and Ali Alavi

We report on a new method to perform quantum evolution of a fermionic system utilizing the full conﬁguration interaction quantum Monte Carlo method [1]. We employ this technique to compute Green’s functions and therefore also spectral weight functions. We demonstrate the applicability of the algorithm using amongst others the examples of the 2D-Hubbard model and the carbon dimer, showing that the algorithm can in principle be used as an impurity solver for cluster approaches and is as well capable of obtaining photoemission spectra of ab-initio systems.

[1] G.H. Booth, A.J.W. Thom and A. Alavi, J. Chem. Phys. 131, 054106 (2009)

5. Werner Dobrautz, Max Planck Institute for Solid State Research
Implementation of the SU(2) Symmetry in FCIQMC using the Graphical Unitary Group Approach

The Full Configuration Interaction Quantum Monte Carlo (FCIQMC) algorithm is a projector QMC method, previously formulated in the total anti-symmetric space of Slater Determinants, based on the imaginary-time Schrödinger equation to obtain the ground state of a system in the long-time limit.

By formulating the method in eigenfunctions of the S^2 total spin operator via the Graphical Unitary Group Approach we can make use of the block-diagonal form of spin-preserving, non-relativistic Hamiltonians for different values of the total spin. This allows us to lift possible near degeneracies of low-lying excitations of different spin sectors, calculate spin-gaps more easily and obtain the physical correct ground-state, without spin-contamination, and easily identify its total spin quantum number.

6. Ettore Vitali, College of William & Mary
Ground state properties of the three-band Hubbard model

The three-band Hubbard model, or Emery model, is a minimal model for the Copper-Oxygen planes of the Cuprates. I will present Generalized Hartree Fock and Auxiliary-Field Quantum Monte Carlo results for the Ground State of the model. In particular I will address the dependence of the physical properties on the charge transfer energy, which controls the density of holes surrounding the Copper and Oxygen atoms. Experiments have shown that such density is correlated with the critical temperature of the materials, making thus very interesting to explore its role within microscopic calculations.

7. Edwin Huang, Stanford University
Evidence for fluctuating stripes in cuprates from finite temperature quantum Monte Carlo

A microscopic understanding of the strongly correlated physics of the cuprates must account for the translational and rotational symmetry breaking present across all cuprate families. In this talk I will present our determinant quantum Monte Carlo results showing fluctuating stripes in both the single-band and the three-band Hubbard models, which represent minimal models of the strongly correlated electrons in copper-oxide planes. I will discuss the relevance of our findings for interpreting experimental results and comment on the implications of our work for other numerical studies on the Hubbard model.

References:
E. W. Huang, C. B. Mendl, S. Liu, S. Johnston, H.-C. Jiang, B. Moritz, and T. P. Devereaux, arXiv:1612.05211 (2016).

8. Hanna Terletska, University of Michigan
Phase transitions in 2D extended Hubbard model

In this talk, I will present our recent results for the half-filled extended Hubbard model in 2D. Using the cluster dynamical mean field theory, we find that the model exhibits metallic, Mott insulating, and charge ordered phases. Using the broken symmetry solution, we determine the location of the charge ordering phase transition line and the properties of the charge ordered and charge disordered phases as a function of temperature, local interaction, and nearest neighbor interaction. Our results show strong non-local correlations in the uniform phase and a pronounced screening effect in the vicinity of the phase transition to the charge ordered phase, where nonlocal interactions push the system towards metallic behavior. In contrast, correlations in the ordered phase are mostly local to the unit cell. Finally, I will also discuss how doping affects the charge ordered phase.

9. Markus Wallerberger, University of Michigan
Hypothesis testing of quantum Monte Carlo simulations

The large implementation complexity of modern quantum Monte Carlo solvers makes careful testing of the algorithm as well as verification of the results an imperative.  While deterministic unit tests are unsuitable and visual inspection is error-prone, statistical hypothesis testing provides a non-deterministic alternative: we choose an exact result (which exists for certain limits) as the null hypothesis and compute the statistical significance score for the Monte Carlo data.  Rejection of the null hypothesis then amounts to a failed test, thus providing a test criterion for both the Monte Carlo estimate and its error bars.  We develop a testing framework and illustrate the procedure for Continuous-time quantum Monte Carlo data for the single impurity Anderson model by verifying them against exact diagonalization results.

10. Peter Rosenberg, College of William & Mary
Attractive fermions in a 2D optical lattice with spin-orbit coupling: Charge order, superfluidity, and topological signatures

Exotic states of matter, including high-Tc superconductors, and topological phases, have long been a focus of condensed matter physics. With the recent advent of artificial spin-orbit coupling in ultracold gases, and the remarkable experimental control and enhanced interactions provided by optical lattices, a broad range of novel strongly correlated systems are quickly becoming experimentally accessible. One system of particular interest, given its potential impact on spintronics and quantum computation, is a 2D optical lattice of fermionic atoms with attractive interaction. Here we examine the combined effects of Rashba spin-orbit coupling and interaction in this system, with a focus on the pairing, charge, and spin properties of the ground state, which are computed using the numerically exact auxiliary-field quantum Monte Carlo technique. In addition to illuminating the behavior of this exotic charge-ordered superfluid state, our results serve as high-accuracy benchmarks for the coming generation of precision experiments with ultracold gases.

• Meeting Location

Simons Center for Geometry and Physics
John S. Toll Drive
Stony Brook University
Stony Brook, NY 11794

• Travel, Local Transport and Parking

### Getting to Long Island

#### From NYC Penn Station:

Board a Long Island Railroad train bound for Port Jefferson. Exit at Port Jefferson Station.

#### From John F. Kennedy Airport:

1. Take the AirTrain JFK towards Jamaica.
2. Transfer at Jamaica Station to a Long Island Railroad train to Port Jefferson.
3. Travel to the Port Jefferson Station.
4. Once you have arrived you can walk, take a taxi or call Danfords (631.928.5200) and a free shuttle will be sent for you. Please note that the free shuttle can take a while to pick you up, so taking a taxi or walking may be faster. Taking a taxi will cost $7. #### From La Guardia Airport: For those flying into LGA you will have to catch the Long Island Railroad at Jamaica Station in Queens. You can take public transportation to Jamaica via the Q33 bus and then the E train; however, taking a taxi may be more convenient. Once you arrive at Jamaica Station follow the directions above starting at #2. #### From Newark Airport: Take the Air Train to the New Jersey Transit and then take the New Jersey Transit to NYC Penn Station. Once at Penn Station, board a Port Jefferson bound Long Island Rail Road train. Exit at Port Jefferson Station. ### Transportation to and from Simons Center for Geometry & Physics Hired coaches will transport participants from the hotel to SCGP and back daily. ### Parking SCGP has a gated parking lot. If you intend to use your car each day, please contact Emily Klein who will provide you with an entrance code for the gate. • Hotel Danfords Hotel & Marina 25 E Broadway, Port Jefferson, NY 11777 (631) 928-5200 www.danfords.comAccommodations will be made for you at the Danfords hotel at the expense of the Simons Foundation. Checkout will be on June 23 before noon. If you have any questions about hotel arrangements, please contact Emily Klein to confirm. • Meals While Traveling On days when participants are traveling to or from summer school, the foundation will reimburse up to US$115 per day for meals.Breakfast will be served daily at Danfords. Breakfast is included with your stay and begins at 7:00 AM daily.

Lunch will be provided for all participants at the SCGP Café.

Tea will take place on weekday afternoons from 3:30 – 4:00 PM at the SCGP Café.

Dinner will be on your own at any number of restaurants located in downtown Port Jefferson. Except for dinner on Monday, June 19 and Wednesday, June 21, participants will be reimbursed for their dinner expenses.

Monday, June 19th: Dinner
A welcome dinner will be held at the SCGP Café at 6:30 PM on Monday, June 19. All participants are expected to attend.

Wednesday, June 21st: Trip
Following morning sessions, transportation to the beach (weather permitting) will be arranged.

Wednesday, June 21st: Dinner
A dinner event will take place at the Fifth Season restaurant on Wednesday, June 21. The restaurant is located across the street from the hotel.

• Contacts

### Travel Assistance

Elise Volpe, Protravel International
elise.volpe@protravelinc.com
(516) 465-1006

### Registration, Hotel and General Meeting Assistance

Emily Klein
Program Associate, Simons Foundation
eklein@simonsfoundation.org
(646) 751-1262
• Reimbursement and Travel Policy

Any expenses not directly paid for by the Simons Foundation are subject to reimbursement based on the travel policy. An email will be sent within a week following the conclusion of the symposium with further instructions on submitting your expenses via the foundation’s web-based expense reimbursement platform.Receipts are required for any expenses over \$50 USD and are due within THIRTY DAYS (30) after the conclusion of the symposium. Should you have any questions, please contact Emily Klein.

### Guest Policy

It is the foundation’s policy to discourage spouses, families or others from joining participants during the meeting. Participants who choose to invite guests to come before or stay on after the meeting do so at their own expense. The Simons Foundation will not cover expenses associated with such extensions.
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