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)