Paul C. Bressloff- Stochastic Neural Field Theory and the System-Size Expansion
Introduction to Stochastic Series Expansion (SSE) Quantum ...
Transcript of 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)