Emanuel Gull, University of Michigan, Ann ArborAndrey Antipov (University of Michigan), Xi Chen (University of Michigan), Guy Cohen (Tel Aviv University), James P.F. LeBlanc (University of Michigan), Andrew J. Millis (Columbia University), David R. Reichman (Columbia University), Matthias Troyer (ETHZ), Philipp Werner (Uni Fribourg), and the Simons Collaboration on the Many-Electron Problem.

Support: DOE ER, Simons Collaboration on the Many-electron problem, Sloan Foundation

Numerical studies of Correlated Materials with Diagrammatic Monte Carlo Techniques

Speed & Accuracy

Bold Line ExpansionInteraction Expansion Hybridization Expansion

Diagrams & Monte Carlo Convergence

Maximum Fluctuations Kondo Voltage Splitting Spectral Functions

DIY DiagMCInchworm Expansion The ‘sign problem’

H = H1 +H2

Z =1X

n=0

Z

0d1 . . .

Z

n1

dnTrhe(n)H1(H2) . . . e

(21)H1(H2)e1H1

i

Hamiltonian H for finite system at non-zero temperature, split into two parts: convenient / noninteracting / exactly known part H1; ‘rest’ H2. Convergent interaction expansion of H2 with respect to H1.

Infinite sum of terms (‘diagrams’) with varying number of tau points and (potentially) internal indices. Traditionally: pick some ‘relevant’ or low-order terms, sum them up analytically, neglect the rest.

Here: perform importance sampling of diagrams: randomly generate all of them with the weight that they contribute to Z. Exact answer within Monte Carlo error, convergence guaranteed by central limit theorem.

0 25 50 75 100expansion order

0

0.01

0.02

0.03

0.04

0.05

p(or

der)

32-site cluster U/t=8, βt=2half filling

diagrams with order ≳100

exponentially suppressed

diagrams with order ≲25

exponentially suppressed

• Not necessarily ‘weak coupling’• Typical diagram orders strongly

problem dependent, maximum average orders ~3000

• No explicit truncation of diagram order

• Typical order hH2i

• Often sequence of finite systems followed by TL extrapolation.

• Finite size scattering?• Careful near phase transitions!

L ?

-0.54

-0.52

-0.5

-0.48

-0.46

0 0.05 0.1 0.15 0.2

E/t

N−2/3

DMFTDCA

extrapolation

Sequence of 3D clusters size 18 - 100

0 10 20 30 40 50 60 70βt

0

50

100

150

Mat

rix S

ize

CT-INTCT-HYBHirsch FyePrev. State of the Art(non-CTQMC)

Hybridization expansioninteraction expansion (2005) • For the same problem: problem

size reduced by ~30.• Corresponds to time speedup of

factor 303 = 27’000 or ~25 years of Moore’s law

0 0.5 1 1.5 2 2.5ω

n

-1

-0.8

-0.6

-0.4

-0.2

0

Im[Σ

(iω

n)]

HF, Δτ = 1HF, Δτ = 0.5HF, Δτ = 0.25ED, n

bath = 5, 6

CT-AUX

• Same problem solved more accurately

• Elimination of bias and systematic errors (‘Trotter discretization’ errors)

Choose H1 as the non-interacting term, H2 as the interaction. In practice limited to density-density interactions, simple band structures, best near particle hole symmetry (‘Hubbard’ models).

Ideally coupled to embedding scheme (‘DCA’, ‘CDMFT’, etc); up to 100 lattice sites at high T, on small Hubbard systems (8-site, 16-site clusters) results down to T ~ Tc/2

Energy precision close to ground state lattice methods accessible in weak to intermediate coupling regime

0 0.1 0.2 0.3 0.4 0.5T/t

-1.12

-1.08

-1.04

-1

E/t

DCADCA - TL

DiagMC [G(0)

]2Γ

(0)

DMET - TL

DiagMC G2Γ inf order

DiagMC G2Γ finite order

AFQMC - TL

FN - TLMRPHFDMRG - TLUCCSD

0 4 8iω

n

-0.5

-0.4

-0.3

-0.2

-0.1

0

Im Σ

(iω

n)

-1.07

-1.065

-1.06T=0.25α=3

α=4

α>4

n=0.8, U=4

α=5α=3

α=4

α=7

T=0.25

50

6472

‘Standard’ method for cluster dynamical mean field theory on clusters with more than four sites.

Single orbital Hubbard model, simple metallic regime: precision accessible by a range of current state of the art numerical ground state and f in i te temperature methods.

‘Standard’ method for ‘real materials’ DMFT (LDA+DMFT, GW+DMFT, etc) for quantum impurity systems.

Choose H1 as the local term, H2 as the hybridization term of the impurity with its bath. In practice limited to ~5-7 impurity orbitals but fully general local interactions, band structures.

Complementary to interaction expansion: efficient where local physics important, hybridization weak.

0 1 2 3 4 5 6 7

U/t

0

50

100

Matr

ix S

ize

CT-INTCT-HYB

Num

eric

al P

robl

em S

ize

Single site dynamical mean field calculation (Bethe lattice, t=1): Average expansion order / Matrix size as a function of interaction s t r e n g t h , i n t e r a c t i o n a n d hybridization expansion at fixed temperature. A phase transition (MIT) is visible near U~4.5. Time to solution ~ problem size3

Hybridization expansioninteraction expansion (2005)

Often analytic diagrammatic results are available (and easy to get!). Use them as a starting point for the numerics! Examples: RPA, the non-crossing approximation, the one-crossing approximation, etc.

Stochastically sample corrections to analytics.

• If analytics is good: corrections are small, Monte Carlo converges instantly.

• If analytics is decent: Monte Carlo can provide the exact result.

• If analytically diagrams irrelevant: Monte Carlo will need to work as hard as bare expansion, will still always get right result.

2tF0

tF

tF2tF0

bold diagram (bottom): correction to bare series (top), with ‘bold’ lines

0 1 2 3 4 5 6 7t

-0.6

-0.4

-0.2

0

<cur

rent

>

Bold (all orders)Bold 3rd orderBold 4th orderBold 5th orderBold 6th order

0 1 2 3tB

-0.12

-0.08

-0.04

<cur

rent

>U=8, V=2, β=10

U=4, V=5, β=50

U=8, V=2, β=10

2 4 6t0

0.2

0.4

0.6

0.8

1

p(sta

te)

Bold4th order5th order6th orderHyb (bare)

0 2 4 6 8V

0

1

2

1/T 1

β=50β=1

U=8, V=1, H=0.5; | > (init | >)

U=4, V=5, H=0; | > (init | >) U=4, V=5, H=0; | > (init | >)

U=8, V=1, H=0.5; |0>

Keep diagram expansion order as low as possible (e.g. to simplify normalization to analytic results, reduce sign problem):

Simulate (exact) propagator up to some time as (exact) propagator up to a shorter time + corrections. If time difference is small: few correction diagrams.

Works well for real-time expansions of quantum impurity problems, allows access to long-time behavior at quadratic (rather than exponential) cost.

Left: A propagator of an impurity in an oscillating magnetic field on the Keldysh contour.

Top: Inchworm diagrams0.00

0.25

0.50

0.75

1.00

P |" i

P |# i

" = U/2 (Kondo) " = /2 (Mixed Valence)Order 1 (' NCA)Order 2 (' OCA)Order 3 (' 2CA)Order 4 (' 3CA)Order 5 (' 4CA)

0 1 2 3 4t

105

104

103

P |

0 i

0 1 2 3 4 5t

Right: Time evolution of density matrix elements in the Kondo and Mixed Valence regimes.

U/t

0.7 0.8 0.9 1 1.1 1.2 1.34

5

6

7

0.2

0.4

0.6

U/t

0.7 0.8 0.9 1 1.1 1.2 1.34

5

6

7

0.2

0.4

0.6

n

U/t

0.7 0.8 0.9 1 1.1 1.2 1.34

5

6

7

0.2

0.4

0.6 0

0.2βΓ=1, Ω

c/Γ=6 βΓ=3, Ω

c/Γ=6

-5.0 0.0 5.0

0

0.2

ΓA

βΓ=1, Ωc/Γ=10

-5.0 0.0 5.0

ω/Γ

βΓ=3, Ωc/Γ=10

MaxEntrt directrt auxNCA

1

3 V = 0

0.000.050.100.15

1

3

t

V = 2

0.00

0.05

0.10

10 5 0 5 10!/

1

3 V = 4

0.00

0.05

0.10

0 0.5 1 1.5 2n

-0.04

-0.02

0

P g

-0.04

-0.02

0

P g

-0.04

-0.02

0

P g

dx2-y2 dxy px (a)

(b)

(c)

Where in U/n space is the (high-T) d-wave susceptibility of the single orbital Hubbard model the largest?

8-site DCA, calculation of the vertex function, inversion of the Bethe Salpeter equation.

What about other types of superconductivity?

data obtained at T ~ 2 Tcmax, for different t’.

1. Find a convergent diagrammatic expansion (finite-T, finite system size, finite real time, etc)

2. Write the partition function in the form

3. Write the observables of interest as functions of c.4. Define a sequence of updates: given a configuration c, generate a

new configuration c’ by changing part of c (adding a new interaction vertex, changing a time, adding/removing times, etc). Make sure any diagram can be reached from any other diagram in a finite number of steps.

5. Perform a Metropolis sampling:

Z =X

c2Cw(c) C: configuration space, i.e. space of

all diagramsc: a diagramw: weight (contribution) of a diagram

Wcc0 = min

1,

w(c0)

w(c)

Diagrammatic expansion is just a Taylor series! There is no guarantee that terms are positive. In practice, this leads to the ‘sign’ problem.Most algorithms are exponential in cost (in U, 1/T, system size, real time, …) away from high symmetry points (though still numerically exact!). In practice: how far can an algorithm be pushed in the presence of a sign problem?

0 0.2 0.4 0.6 0.8doping x

0

0.2

0.4

0.6

0.8

1

<sig

n>

βt = 1βt = 2βt = 3βt = 4βt = 5βt = 6βt = 7βt = 8

average sign lattice / auxiliary fieldU/t = 4, 32-site cluster

0 0.2 0.4doping x

0

0.2

0.4

0.6

0.8

1

<sig

n>

βt = 3 impurityβt = 4 impurityβt = 5βt = 6βt = 7βt = 8

average sign impurity model / auxiliary fieldU/t = 4, 36-site cluster

Details matter. Often1. Embedding (cluster DMFT)

helps2. Changing/rotating basis may

help.3. Explicitly respecting symmetries

helps4. Adapt the algorithm to the

problem at hand.

Num

eric

al P

robl

em S

ize

Rev. Mod. Phys. 83, 349 (2011) Physical Review B 88 (15), 155108 (2013) Physical Review B 76 (23), 235123

Physical Review B 82 (7), 075109Rev. Mod. Phys. 83, 349 (2011)

Phys. Rev. Lett. 115, 266802 (2015)

Phys. Rev. Lett. 115, 116402 (2015) Phys. Rev. Lett. 112, 146802 (2014) Phys. Rev. B 89, 115139 (2014)

Phys. Rev. X 5, 041041 (2015)