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Nikolay Prokofiev, Umass, Amherst

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DIAGRAMMATIC MONTE CARLO FOR CORRELATED FERMIONS. Nikolay Prokofiev, Umass, Amherst. work done in collaboration with . Lode Pollet Harvard. Matthias Troyer ETH. Evgeny Kozik ETH. Emanuel Gull Columbia. Boris Svistunov UMass. Kris van Houcke UMass Univ. Gent. Felix Werner - PowerPoint PPT Presentation

Text of Nikolay Prokofiev, Umass, Amherst

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Nikolay Prokofiev, Umass, Amherstwork done in collaboration with PITP, Dec. 4, 2009


Boris SvistunovUMass

Kris van HouckeUMassUniv. Gent Evgeny Kozik ETH

Lode PolletHarvard

Emanuel GullColumbia

Matthias TroyerETH

Felix WernerUMass1Front pageOutlineFeynman Diagrams: An acceptable solution to the sign problem? (i) proven case of fermi-polaronsMany-body implementation for (ii) the Fermi-Hubbard model in the Fermi liquid regime and (iii) the resonant Fermi gas 2Fermi-Hubbard model:

momentum representation: Elements of the diagrammatic expansion:

fifth order term: 33+The full Greens Function:

+++ ++++


Why not sample the diagrams by Monte Carlo?Configuration space = (diagram order, topology and types of lines, internal variables) 4

Diagram orderDiagram topologyMC updateMC updateMC updateThis is NOT: write diagram after diagram, compute its value, sum 5Sign-problemVariational methods

+ universal- often reliable only at T=0- systematic errors- finite-size extrapolationDeterminant MC

+ solves case - CPU expensive - not universal- finite-size extrapolation

Cluster DMFT / DCA methods+ universal- cluster size extrapolation Diagrammatic MC+ universal- diagram-order extrapolation Cluster DMFTlinear size

diagram orderDiagrammatic MC

Computational complexityIs exponential :

for irreducible diagrams6Further advantages of the diagrammatic techniqueCalculate irreducible diagrams for , , to get , , . from Dyson equations

Dyson Equation:

Make the entire scheme self-consistent, i.e. all internal lines in , , are bold = skeleton graphs


Every analytic solution or insight into the problem can be built in7Series expansion in U is often divergent, or, even worse, asymptotic. Does it makes sense to have more terms calculated?

Yes! (i) Unbiased resummation techniques (ii) there are interesting cases with convergent series (Hubbard model at finite-T, resonant fermions)

- It is an unsolved problem whether skeleton diagrams form asymptotic or convergent series Good news: BCS theory is non-analytic at U 0 , and yet this is accounted for within the lowest-order diagrams!

8Polaron problem:


E.g. Electrons in semiconducting crystals (electron-phonon polarons)ee

electronphononsel.-ph. interaction9Fermi-polaron problem:

Universal physics( independent)

10Examples: Electron-phonon polarons (e.g. Frohlich model)= particle in the bosonic environment.Too simple, no sign problem,

Fermi polarons (polarized resonant Fermi gas = particle in the fermionic environment.

Sign problem!

self-consistent and

self-consistent only


Confirmed by ENS11Exact solution:Polaron Molecule

sure, press EnterUpdates:12

2D Fermi-Hubbard model in the Fermi-liquid regimeBare series convergence:yes, after order 4

Fermi liquid regime was reached


2D Fermi-Hubbard model in the Fermi-liquid regime

Comparing DiagMC with cluster DMFT(DCA implementation)!14

2D Fermi-Hubbard model in the Fermi-liquid regime

Momentum dependence of self-energy


3D Fermi-Hubbard model in the Fermi-liquid regime

DiagMC vs high-T expansion in t/T(up to 10-th order)Unbiased high-T expansion in t/Tfails at T/t>1 before the FL regime sets in 281010816

3D Resonant Fermi gas at unitarity :

Bridging the gap between different limits17

S. Nascimb`ene, N. Navon, K. J. Jiang, F. Chevy, and C. SalomonDiagMCENS data

Seatlles Det. MC18

DiagMC fitAndre Schirotzek, Ariel Sommer, Mark Ku, and Martin Zwierlein


Universal function F0Single shot data

Andre Schirotzek, Ariel Sommer, Mark Ku, and Martin Zwierlein

20Conclusions/perspectives Bold-line Diagrammatic series can be efficiently simulated.- combine analytic and numeric tools- thermodynamic-limit results- sign-problem tolerant (small configuration space) Work in progress: bold-line implementation for the Hubbard model and the resonant Fermi-gas ( version) and the continuous electron gas or jellium model (screening version). Next step: Effects of disorder, broken symmetry phases, additional correlation functions, etc.


Configuration space = (diagram order, topology and types of lines, internal variables) 22

term orderdifferent terms ofof the same order Integration variablesContribution to the answer Monte Carlo (Metropolis-Rosenbluth-Teller) cycle:Diagram

suggest a changeAccept with probability

Collect statistics:

sign problem and potential trouble!, but 23Fermi-Hubbard model:

Self-consistency in the form of Dyson, RPA

Extrapolate to the limit.

24Large classical system Large quantum system

state described by numbers

state described by numbersPossible to simulate as is (e.g. molecular dynamics)Impossible to simulate as isapproximate Math. mapping forquantum statistical predictions Feynman Diagrams: 25

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