Introduction to Stochastic Series Expansion (SSE) Quantum ...

Introduction to Stochastic Series Expansion (SSE)Quantum Monte Carlo (QMC)

Stephan HumeniukICFO - The Institute of Photonic Sciences

UPC, 24th of January 2013

Introduction to Stochastic Series Expansion (SSE)Quantum Monte Carlo (QMC)

1 Classical Monte Carlo: Ergodicity and detailed balance

2 Quantum-to-classical mapping:

1.1 SSE representation of the partition sum2.2 Comparison between SSE and Path Integral representation

3 Update schemes

1.1 Diagonal and off-diagonal update2.2 Directed loop algorithm: detailed balance equations

4 SSE estimators for

1.1 Energy, magnetization, ...2.2 Superfluid fraction3.3 Extended ensembles and off-diagonal correlation functions

5 SSE for long-range transverse Ising systems

6 The sign problem

What is Monte Carlo ?

I An efficient method for calculating high-dim. integrals ...

Number of sampling points M,systematic error ε

I Riemann integration: ε ≈ M−k/d

I MC sampling: ε ≈ 1√M

independent of spatial

dimension (CLT)

I ... in particular expectation values in statistical physics:〈f 〉P =

∫d3N~x

∫d3N~p P(~x , ~p)f (~x , ~p).

I In quantum statistical physics there are many variants:I Path Integral MC, Determinantal MC, Stochastic Series

Expansion MC→ stochastic sampling of the partition function

I (fixed node) diffusion MC, projector MC, variational MC→ based on the wave function

I diagrammatic MC (for fermions!) → stochastic samplingof Feynman diagrams

The Metropolis Algorithm (1953)

Classical Monte Carlo

〈A〉 =

∑µ Aµe

−βEµ∑µ e−βEµ

Naive approach: Generate configurations µ randomly, compute Aµand its Boltzmann weight, and then sum.Problem: Most states will have vanishing weight.

I Importance samplingDo not pick states from a uniform distribution, but insteadperform a guided random walk in configuration space thatvisits each state as often as corresponds to its weight,i.e. pν = Z−1e−βEν .Then the expectation values are simple averages:

AM =1

M

M∑i=1

Aµi → 〈A〉 , M →∞

The random walk with transition probability P(ν → µ) must obey

1 ergodicity: Any state can be reached from any other state withnon-vanishing probability.

2 detailed balance w.r.t. the desired probability distribution {pµ}:balance of fluxes:∑

ν

pµP(µ→ ν) =∑ν

pνP(ν → µ)

detailed balance (stricter):

pµP(µ→ ν) = pνP(ν → µ)

Quantum-to-classical mappingEvery D-dimensional quantum systems corresponds to a (D+1)-dimensional effective classical system.

〈A〉 = Tr[ρA] =1

Z

∑|α〉

〈α|e−βH A|α〉

Note: The eigenenergies are not known and one needs to expand the expressionin a suitable way.

SSE representation (Taylor exp.)

〈A〉 =1

Z

∑|α〉

〈α|∞∑n=0

(−βH)n

n!A|α〉

Determinantal QMC(Hubbard-Stratonovich)

ZHS =∑i,r

auxiliaryIsing field

detM↑ · detM↓

Path integral representation (Trotter-Suzuki)

ZTS =∑

m1...m2L

〈m1|e−∆τHodd |m2L〉〈m2L|e−∆τHeven |m2L−1〉

. . . 〈m3|e−∆τHodd |m2〉〈m2|e−∆τHeven |m1〉

〈A〉 =∑C

classical weights

w(C)A(C)

SSE representation for the spin 12 XXZ model

〈A〉 =1

ZTr[e−βHA] =

1

Z

∑|α〉

〈α|∞∑n=0

(−βH)n

n!A|α〉

HXXZ = −J∑〈ij〉

{1

2(S+

i S−j + S+

j S−i ) + ∆Sz

i Szj

}− h

∑i

Szi

Decompose into diagonal (D) and off-diagonal (oD) bond operators:

H = −Nbonds∑b=1

Hb = +J

Nbonds∑b=1

(HD,b + HoD,b)

Multiplying out the nth power, we obtain

Hn =∑{Sn}

n∏i=1

Hti ,bi

where the indices ti (=operator type) and bi (=bond index) are drawn from anoperator string Sn = {[t1, b1], [t2, b2], . . . , [tn, bn]}. Then:

〈A〉 =∞∑n=0

βn

n!

∑|α〉

∑{Sn}

〈α|n∏

i=1

Hti ,biA|α〉 =∞∑n=0

∑|α〉

∑{Sn}

w(α,Sn)A(α,Sn),

which is a sum over classical weights.

SSE simulation cell for the spin 12 XXZ model

Note that there is no branching:

Hti ,bi |α(p)〉 = |α(p + 1)〉,

i.e. all |α(p)〉 are basis states and no superpositions are created.

I One SSE configuration is spefiedby an initial state and an operatorstring (|α〉, Sn).

I Periodic boundary conditions inimaginary time due to the tracestructure of the partition sum.

I MC update consists in exchangingoperators: diagonal andoff-diagonal update.

I For convenience we truncate theexpansion order to nmax = M andfill smaller expansion orders upwith identity operators.

Path integral formulation of Z

I is an integral over differentclosed propagation paths inimaginary time.

I The quantum operatordriving the propagation isalways the same, e−βH , sothe integration runs overinitial and intermediatestates.

I “Schrodinger picture” ofQM

I Trotter error

SSE formulation of Z

I is an integral over differentclosed propagation routesuniquely specified by anoperator string. Theintegral runs overinitial/final states of thepropagation, and over theoperator string driving thepropagation.

I “Heisenberg picture” of QM

I No intrinsic approximationerror

|α〉 = |α(p = 0)〉 → |α(1)〉 →. . .→ |α(p = L) = α(0)〉

Comparison between SSE and PI

I The distribution of expansion orders shows that there is nointrinsic approximation involved in SSE.

I There is a statistical correspondence between worldlineconfigurations within SSE formulation and worldlineconfigurations in PI. Imaginary-time intervals and propagationintervals tend to coincide for β →∞.

Diagonal update: id ↔ D

Exchange identity and diagonal operators with Metropolisacceptance probabilities

Padd = P([I , b]p → [D, b]p) = min

(1,

NbondsW (. . . [D, b]p . . . ;α)

W (. . . [I , b]p . . . ;α)

)= min

(1,βNbonds

(M − n)· 〈α(p)|HD,b|α(p)〉

),

Premove = P([D, b]p → [I , b]p) = min

(1,

W (. . . [I , b]p . . . ;α)

NbondsW (. . . [D, b]p . . . ;α)

)= min

(1,

M − (n − 1)

βNbonds· 1

〈α(p)|HD,b|α(p)〉

)This changes the expansion order, which is related to the energy

〈E 〉 = − 〈n〉β . The magnetization is not changed so that the diagonal

update needs to be complemented by the off-diagonal update to satisfy

ergodicity.

Off-diagonal update (“worm” or loop update): D ↔ oDI Energy remains fixed, grand-canonical moves

I The worm travels on the linked list flipping spins as it goes and therebyconverting diagonal into off-diagonal operators and vice versa (D ↔ oD).

I It has to close on itself. This ensures that the replacements D ↔ oDoccur an even number of times which implies that the periodicboundary conditions in imaginary time are preserved, i.e. a newconfiguration with non-vanishing weight is generated during the update.

Off-diagonal update (“worm” or loop update)

Directed loop equationsI Transition probabilities for the worm must sum to unity

p(1→ 6) + p(1→ 2) + p(1, b)︸ ︷︷ ︸bounce probability

= 1

etc. for all independent transition processes

I Multiply with the weight of the initial vertex and introduce the notation

w(i → j) = w(i)p(i → j)

The detailed balance conditions take the simple form

w(i → j) = w(j → i)

and allow to indentify different coefficients with each other.

I ⇒ (under-determined) set of equations which have to be solved

I minimizing the bounce probabilities whileI keeping all transition rates w(i → j) positive.

... in detail... for the spin 12 XXZ model

hb = hzJ

: magnetic field∆ : spin space anisotropy parameterregion I: bounce free solution

SSE estimators

I Energy

E = −〈n〉β, 〈n〉 : average expansion order

I Specific heatC = [〈n2〉 − 〈n〉 − 〈n〉2]

This shows that the fluctuations of a quantity are not the same asthe fluctuations of its estimator.

I Magnetization

mz =1

NL

L−1∑p=0

〈Sz~r [p]〉

Due to the cyclic structure of the partition function one can averageover propagation steps p to obtain more statistics.

I Helicity modulus, superfluid density

Γα =kBT

2JxyLd−2〈w2

α〉

where wα =∑

b‖α

(N+

b −N−b

L

)is the winding number for α = x , y , z .

SSE estimators and extended ensemble techniques

I (Propagation-time) off-diagonal correlator: Ratio of amodified partition function and of the original partitionfunction.

〈S+i [m]Si+r [0]〉 =

Z ′

Z

Z ′: modified partition function with two discontinuities of theworm ends at (i ,m) and (i + r , 0).

Transverse field Ising model with arbitrary interactions

I HamiltonianH =

∑ij

Jijσzi σ

zj − hx

∑i

σxi

I Jij arbitrary (long-range, frustrated, random)

I Define the bond operators in such a way that the stateevolution is deterministic and all weights are positive:

H0,0 = 1

Hi ,0 = h(σ+i + σ−i ), i > 0

Hi ,i = h, i > 0

Hi ,j = |Jij | − Jijσzi σ

zj , i , j > 0, i 6= j

I Observations: No loop update possible as there are nooff-diagonal pair interactions.Constants added in a clever way.

TFI model

Observations:

I The arbitrary-rangeinteractions in space havebeen transformed intocompletely local constraintsin imaginary time.

I Summing over allinteractions requires ≈ N2

operations. Here thediagonal update at allpositions in SL requires≈ L ln(N) ≈ βN ln(N)IN(J)operations.

The sign problemQMC cannot simulate

I fermions

I frustrated spin systems (i.e. AFM on non-bipartite lattices)

as the weights cannot be chosen to be positive definite, e.g.

HXXZ = −1

2

∑〈ij〉

(JzSzi S

zj + 2hSz

i +C)

︸ ︷︷ ︸≥0

+1

2

∑〈ij〉

Jxy (S+i S−j + S−i S+

j )︸ ︷︷ ︸sign problem

The sign problem affects only the off-diagonal part since here we cannot add aconstant to make the matrix elements positive definite.

In a bipartite lattice, we need an off-diagonal bond operator to act an evennumber of times to restore the original configurationt on |α〉.If the lattice is non-bipartite (e.g. triangular or J1 − J2 chain), there can be aproduct of an odd number of off-diagonal operators. Similarly for fermions.The sign problem is NP-hard. [Troyer and Wiese, PRL 2005]

The sign problem

How does a negative sign in some configuration weights affect theQMC simulation ?

w(α,SL) → |w(α,SL)|Any average of an observable takes the form

〈O〉 =

∑α,SL

O(α,SL)sign(α,SL)|w(α,SL)|∑α,SL

sign(α,SL)|w(α,SL)|

shift the sign from the weight onto the observable

〈O〉 =〈sign(α,SL)O(α,SL)〉|w|〈sign(α,SL)〉|w|

where 〈. . .〉|w| denotes the average∑α,SL

(...)|w(α,SL)|∑α,SL

|w(α,SL)| .

The sign problem

In particular:

〈sign(α,SL)〉|w| =

∑α,SL

w(α,SL)∑α,SL|w(α,SL)| =

Zw

Z|w|

=e−βVfw

e−βVf|w|= exp[−βV (fw − f|w|)],

where fw and f|w| are the freee energy densities of the systems with weightw(α,SL) and |w(α,SL)|, respectively.Now we have that

Zw ≤ Z|w|

because∑α,SL

w(α,SL) ≤∑α,SL|w(α,SL)|, and since f = − 1

βVlnZ ,

fw ≥ f|w|, ∆f = fw − f|w| ≥ 0

so that 〈sign(α,SL)〉|w| = exp(−βV∆f ) is an exponentially decreasingquantity when β,V →∞.The miserable signal-to-noise ratio of 〈sign〉 propagates on all the otherestimates.

References

I [1] Phys. Rev. B 59, R14157 (1999)

I [2] Phys.Rev.E 68, 056701 (2003)