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a brief introduction of dqmc study in itinerant quantum criticalpoint
Zi Hong LiuSeptember 2018
Institute of Physics Chinese Academy of Sciences
Outlines
IntroductionModelPhysical significance
Transverse field Ising model
Mean field analysis
The free fermion Hamiltonian
Two layers constructionDeterminant Quantum Monte Carlo
Path integral of the partition function
Operator and Measurment
Sampling in DQMC
Numerical StabilizationExample
2
Introduction
∙ Strong correlated itinerant electron system;∙ Nonperturbative nature;∙ The Determinant quantum monte carlo(DQMC) method.
4
general strong correlated itinerant electron system
H = Hfermion +Hspin +Hf−s +Hf−f , (1)
Hfermion = −tij∑
ij,α c†iαcjα + h.c.− µ∑
i ni
Hspin = −J∑
ij szi s
zj − hx
∑i
sxi
Hf−s = −ξ∑
ijk,αβ c†iαszk cjβ
Hf−f = −VI
∑ijkl,αβγδ c
†iαc
†jβ ckγ clδ
, (2)
6
Spin-Fermion Model
Hfermion = −t
∑⟨ij⟩σλ c†iλσ cjλσ + h.c.− µ
∑iσλ niλσ
Hspin = −J∑⟨ij⟩
szi szj − hx
∑i s
xi
Hf−s = −ξ∑
i szi (c
†i τzσz ci) = −ξ
∑i s
zi (σ
zi1 − σz
i2)
. (3)
7
Typical phase diagram of high Tc superconductor
Keimer, B., Kivelson, S. A., Norman, M. R., Uchida, S., Zaanen, J. (2015). Nature, 518(7538), 179.
9
0 0.1 0.2 0.30
0.1
0.2
0.3
0.4
0.5
ε
1.5
1.0
2.0
YbRh2(Si
0.95Ge
0.05)
2
B ⊥c
B(T)
T(K
)
2
0 10
0.1
0.2
0.3
YbRh2Si
2
B ��c
2
B(T)
T(K
)
Nature 424, 524-527 (2003)
10
Transverse field Ising model
Hspin = −J∑⟨ij⟩
szi szj − hx
∑i
sxi (4)
Fix J = 1, at low temperature,At h → 0 situation, sysmmetry breaking phase, ⟨szi ⟩ = meiQ·ri ;At h → ∞ situation, paramagnetic phase, ⟨szi ⟩ = 0;
T
h
AFM PM
Tc
hc
11
Mean field analysis
The Ising spin and fermion coupling term make fermion behavior differently in Isingparamagentic(PM) and Ising symmetry breaking phases.
∙ At h → ∞, we can replace szi by ⟨szi ⟩ = 0.∙ At h → 0, we can replace szi by ⟨szi ⟩ = meiQ·ri .
Hf−s = −ξ∑i
szi (c†i τzσz ci) → −ξ
∑i
⟨szi ⟩ (c†i τzσz ci) (5)
12
∑⟨ij⟩
c†i1↑cj1↑ =∑i
∑l
c†i1↑ci+l,1↑ =1
N
∑i
∑l
∑kk′
c†k1↑ck′1↑e−i(k−k′)·rieik·l
=∑k
∑l
c†k1↑ck1↑eik·l =
∑k
(∑l
eik·l)nk1↑
, (8)
µ∑i
ni1↑ = µ 1N
∑i
∑kk′
c†k1↑ck′1↑e−i(k−k′)·ri
= µ∑k
nk1↑, (9)
ξ∑i
meiQ·ri ni1↑ = ξm1
N
∑i
∑kk′
c†k1↑ck′1↑e−i(k−k′−Q)·ri
= ξm∑k
c†k+Q,1↑ck1↑, (10)
14
Define ϵ(k) =
(−t∑l
eik·l + h.c.− µ
), and dk = ck+Q then
Hfermion−eff =∑
k∈BZ
(−t∑l
eik·l + h.c.− µ
)nk1↑ − ξm
∑k∈BZ
c†k+Q,1↑ck1↑
=∑
k∈BZ
ϵ(k)nk1↑ − ξm∑
k∈NBZ
d†k,1↑ck1↑ − ξm∑
k∈NBZ
c†k,1↑dk1↑
=∑
k∈NBZ
ϵ(k)c†k1↑ck1↑ +∑
k∈NBZ
ϵ(k +Q)d†k1↑dk1↑−
ξm∑
k∈NBZ
d†k,1↑ck1↑ − ξm∑
k∈NBZ
c†k,1↑dk1↑
,
(11)and
Hfermion−eff =[c†k1↑ d†k1↑
] [ ϵ(k) −ξm
−ξm ϵ(k +Q)
][ck1↑
dk1↑
]. (12)
15
Hfermion−eff =[c†k1↑ d†k1↑
] [ ϵ(k) −ξm
−ξm ϵ(k +Q)
][ck1↑
dk1↑
]. (13)
∙ The perfect nesting condition ϵ(k +Q) = −ϵ(k).∙ The eigenvalue of the matrix E±
k = ±√
ϵ2(k) + ξ2m2.∙ Fermi surface, ϵ(k) = 0, so E±
k = ±ξm the energy gap ∆ = 2ξm.
16
The band folding and gap opening due to magnetic order with wave vector in onedimensional system, ϵ(k) = − cos(k) and Q = π, ϵ(k +Q) = cos(k).
- 0
Ef
- 0
Ef
In two dimension, in hall filling square lattice, ϵ(k) = −cos(kx)− cos(ky), Q = (π, π),ϵ(k +Q) = −ϵ(k) satisfy in the whole Fermi-surface.
17
Two layers construction
fermion site
Ising site
coupling
Ising spin
fermion
superposition of spin up and spin down
λ=1
λ=2
-t
Phys. Rev. B, 98:045116
19
Determinant Quantum Monte Carlo
∙ R. Blankenbecler, D. J. Scalapino, and R. L. Sugar. i. Phys. Rev. D, 24:2278–2286, Oct1981.
∙ Hirsch, Phys. Rev. B 28, 4059(R) (1983)∙ Hirsch, Phys. Rev. B 31, 4403 (1985)
∙ F.F. Assaad and H.G. Evertz. Springer Berlin Heidelberg, 2008.∙ Xiao Yan Xu. PhD thesis, Institute of Physics (IOP), Chinese Academy of Sciences,2017.
21
Free fermion partition function
Free fermion HamiltonianH0 =
∑ij
c†iTij cj = c†T c (15)
Z = Tr{e−βH} = Tr{e−βc†T c} = Tr{e−β∑
ij c†iTij cj}
= Tr{e−β(Uc)†U†TUc} = Tr{e−β∑
k ϵ(k)nk}=
∏k
∑nk=0,1
e−βϵ(k)nk = det(1+ e−βT ). (16)
22
Hubbard-Stratonovich(HS) transformation
H = H0 +HI . (17)The general four fermion interaction is
Hf−f = −VI
∑ijkl,αβγδ
c†iαc†jβ ckγ clδ, (18)
The HS transformation
∙ Continuous version
eK2
2 =1√2π
∞∫−∞
dϕe−ϕ2
2−ϕK . (19)
∙ Four components
e∆τWK2
=1
4
∑l=−2,−1,1,2
γ(l) exp(√∆τWη(l)K) +O(∆τ4). (20)
F.F. Assaad, arXiv:cond-mat/9806307
23
Trotter decomposition
Z = Tr{e−βH} = Tr{(e−∆τHI e−∆τH0)M}+O(∆τ2)
=∑C
WSC Tr{
1∏τ=M
ec†V (C)ce−∆τc†T c}+O(∆τ2)
. (21)
Define
U(τ2, τ1) =
n2∏n=n1+1
ec†V (C)ce−∆τc†T c, (22)
B(τ2, τ1) =
n2∏n=n1+1
eV (C)e−∆τT . (23)
We can write down the partition function to a much concise form,
Z =∑C
WSC Tr{U(β, 0)} (24)
After tracing out fermion degrees of freedom, we obtain
Z =∑C
WSC det[1+B(β, 0)]. (25)
24
Path integral to transverse field Ising model
To the bare transverse field Ising model Hspin.
Hspin = −J∑ij
szi szj − hx
∑i
sxi (26)
Z = Tr{e−βHspin}
=
∏τ
∏⟨ij⟩
e∆τJ
∑⟨ij⟩s
zi,τs
zj,τ
∏i
∏⟨τ,τ ′⟩
Λeγ∑
⟨ij⟩szi,τs
zj,τ ′
+O(∆τ2).
(27)
25
Partition function of spin-fermion model
Hfermion = −t
∑⟨ij⟩σλ c†iλσ cjλσ + h.c.− µ
∑iσλ niλσ
Hspin = −J∑⟨ij⟩
szi szj − hx
∑i s
xi
Hf−s = −ξ∑
i szi (c
†i τzσz ci) = −ξ
∑i s
zi (σ
zi1 − σz
i2)
. (28)
Z =∑{Sz}
WT IC WF
C +O(∆τ2) . (29)
WT IC =
∑{Sz}
(∏τ
∏⟨ij⟩e
∆τJ∑
⟨ij⟩szi,τs
zj,τ
)(∏i
∏⟨τ,τ ′⟩Λe
γ∑
⟨ij⟩szi,τs
zj,τ ′)
(30)
WFC = det[1+B(β, 0)] (31)
26
Operator and Measurment
In statistical mechanics, the ensemble average of physical observable can be expressas ⟨
O⟩=
Tr{e−βHO}Tr{e−βH}
=∑C
PC
⟨O⟩C+O(∆τ2) (32)
PC =WS
C det[1+B(β, 0)]∑C WS
C det[1+B(β, 0)], (33)
⟨O⟩C=
Tr{U(β, τ)OU(τ, 0)}Tr{U(β, 0)}
. (34)
Equal time Green’s function
(Gij)C =⟨cic
†j
⟩C= (1+B(τ, 0)B(β, τ))−1
ij . (35)
27
The time dependent Green’s function
The time dependent Green’s function is
(Gij(τ1, τ2))C =⟨Tτ ci(τ1)c
†j(τ2)
⟩C, (36)
where Tτ is time-ordering operator. When τ1 > τ2, we can obtain
(Gij(τ1, τ2))C =⟨ci(τ1)c
†j(τ2)
⟩C
=Tr{U(β,τ1)ciU(τ1,τ2)c
†j U(τ2,0)}
Tr{U(β,0)}
=Tr{U(β,τ2)U
−1(τ1,τ2)ciU(τ1,τ2)c†j U(τ2,0)}
Tr{U(β,0)}
. (37)
(Gij(τ1, τ2))C = [B(τ1, τ2)GC(τ2, τ2)]ij . (38)
28
Sampling in DQMC
If we can make the configurations generated according to the distribution define byPC , the ensemble average of observable will be⟨
O⟩=
1
Nsample
sample∑i=1
⟨O⟩Ci
, (39)
1. Detail Balance Condition:
WCT (C → C′) = WC’T (C′ → C). (40)
2. Ergodic Condition: All states are aperiodic and positive recurrent
29
Sign problem
Map probability to PC , when PC = PC
⟨O⟩=
Tr{e−βHO}Tr{e−βH}
=∑C
PC
⟨O⟩C+O(∆τ2) (41)
Reweight scheme in measurement, Map probability to PC ,
⟨O⟩=
⟨sign e−S
R{e−S} ⟨O⟩⟩
P⟨sign⟩
P
, (42)
30
Hfermion = −t∑
⟨ij⟩σλ c†iλσ cjλσ + h.c.− µ∑
iσλ niλσ
Hf−s = −ξ∑
i szi (c
†i τzσz ci) = −ξ
∑i s
zi (σ
zi1 − σz
i2). (43)
The Hamiltonian is block diagonal as four orbitals, which is
(τz, σz) = [↑ 1, ↓ 1, ↑ 2, ↓ 2]. (44)
We can see H↑1 = H↓2, H↑2 = H↓1. Regroup four orbitals into two superpositionorbitals
(α1, α2) = [(↑ 1, ↓ 2), (↑ 2, ↓ 1)]. (45)
In the two regroup orbitals, Hα1 = Hα2 , so
det(1+B(β, 0)) =∏2
i=1 det(1+Bαi(β, 0)) = |det(1+Bα1(β, 0))|2 . (46)
So our designer model is free of sign problem.
31
Fast Update
Local update, Metropolis-Hastings algorithm.
A(C → C′) = min
[1,
WC′
WC
]. (47)
For the onsite coupling feature, V (C) is a diagonal matrix with
exp(V 1↑i (C)) = exp(V 1↑(si)) = exp(∆τξsi). (48)
In single auxillary field update scheme,
eV1↑i (C′) = exp(∆τξs′i) = (1 + (e∆τξs′ie−∆τξsi − 1))e∆τξsi = (1 +∆ii)e
V1↑i (C), (49)
where ∆ii = e∆τξs′ie−∆τξsi − 1.
32
In the onsite coupling situation, ∆ is a diagonal matrix in real space basis. Then theratio of the weight is
R =WS
C′ det(1+BC′ (β,0))
WSC det(1+BC(β,0))
=WS
C′WS
CRf , (50)
Rf =det(1+BC′ (β,0))det(1+BC(β,0))
= det(1+BC(β,τ)(1+∆)BC(τ,0))det(1+BC(β,0))
= det[1+∆(1− (1+BC(τ, 0)BC(β, τ))
−1)]
= det [1+∆(1−GC(τ, τ))]
, (51)
33
If update is accepted, we also need update Green’s function
GC′(τ, τ) = [1+ (1+∆)BC(τ, 0)BC(β, τ)]−1 (52)
= [1+BC(τ, 0)BC(β, τ)]−1 × (53)[
(1+ (1+∆)BC(τ, 0)BC(β, τ))((1+BC(τ, 0)BC(β, τ))
−1)]−1
as GC(τ, τ) = [1+BC(τ, 0)BC(β, τ)]−1, we denote AC ≡ BC(τ, 0)BC(β, τ) ≡ G−1
C − 1,
GC′(τ, τ) = GC [(1+ (1+∆)AC)GC ]−1
= GC[(1+ (1+∆)
(G−1
C − 1))
GC]−1 (54)
= GC [1+∆(1−GC)]−1
34
Rf = 1 +∆ii(1−GCii) (55)
GC′(τ, τ) = GC(τ, τ) + αiGC(:, i) (GC(i, :)− ei) (56)
, αi = ∆ii/Rf .Computational complexity: O(N2) for each auxillary field update, O(βN3) for onesweep.
35
Global Update
∙ Critical slowing down;∙ Self learning Monte Carlo;
MachineLearning
(ii)Trial simulation by local update(i)
Ηeff
Learning
Ηeff
Detailedbalance
Η
(iii) (iv)
Proposetrial Conf.
Simulating
Phys. Rev. B, 95:041101
36
Numerical Stabilization
∙ The condition number of fermion determinant;∙ The singular value decompositions (SVD);∙ O(N3) computational complexity;
B(nτw, 0) = Un
X
XX
X
︸ ︷︷ ︸
Dn
Vn, (57)
τw = Nst∆τ
37
B((n+ 1)τw, 0) = B((n+ 1)τw, nτw)B(nτw, 0)
= B((n+ 1)τw, nτw)Un
X
XX
X
︸ ︷︷ ︸
Dn
Vn
=
X X X X
X X X X
X X X X
X X X X
︸ ︷︷ ︸B((n+1)τw,nτw)UnDn
Vn = Un+1
X
XX
X
︸ ︷︷ ︸
Dn+1
V ′Vn
= Un+1Dn+1Vn+1
, (58)
38
G(τ, τ)
= [1 +B(τ, 0)B(β, τ)]−1
= [1 + URDRVRVLDLUL]−1
= U−1L
[(ULUR)
−1 +DR (VRVL)DL
]−1U−1
R
= U−1L
[(ULUR)
−1 +DmaxR Dmin
R (VRVL)DminL Dmax
L
]−1
U−1R
= U−1L (Dmax
L )−1[(Dmax
R )−1 (ULUR)−1 (Dmax
L )−1 +DminR VRVLD
minL
]−1
(DmaxR )−1 U−1
R
= UL (DmaxL )−1
[(Dmax
R )−1(U†
LUR
)−1
(DmaxL )−1 +Dmin
R VRV†LD
minL
]−1
(DmaxR )−1 U−1
R
where DR = DmaxR Dmin
R , DL = DminL Dmax
L
39
Example
Hfermion = −t1∑
⟨ij⟩σλ
c†iλσ cjλσ − t2∑
⟨⟨ij⟩⟩σλ
c†iλσ cjλσ−
t3∑
⟨⟨⟨ij⟩⟩⟩σλ
c†iλσ cjλσ + h.c.− µ∑iσλ
niλσ
Hspin = −J∑⟨ij⟩
szi szj − hx
∑i
sxi
Hf−s = −ξ∑i
szi (c†i τzσz ci) = −ξ
∑i
szi (σzi1 − σz
i2)
, (59)
arXiv:1808.08878
41
0
0.2
0.4
0.6
0.8
1
2.6 2.8 3 3.2 3.4 3.6 3.8
T
h
Pure Boson hc=3.044(3)DQMC hc=3.32(2)EQMC hc=3.355(5)
b cafermion site
Ising site
couplingIsing spin
fermion
λ=1
λ=2
-t22
-t1
-t3
hJ
‘
(-π,π)
(-π,−π)
(π,π)
(π,−π)
K1
K1
K2 K3
K4
‘
K3
‘
K2
‘
K4
Q1 Q
3
Q2
Q4
k
k
x
y
Q1
Q3
4 2
K1
K
’
K
4K
2
K
3
K
1 K
2
K4
’
’’
3
arXiv:1808.08878
42
m =
∫dτ∑i
szi (τ)eiQ·ri , (60)
m(h, L) = L−β/νf(L1/ν(h− hc)), (61)
Lβ/νm(h, L) = f(L1/ν(h− hc)). (62)
2.9 3 3.1 3.2 3.3 3.4 3.5
h
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
m(h,L)
β = 4
L=8
L=12
L=16
L=20
L=24
2.9 3 3.1 3.2 3.3 3.4 3.5
h
0
0.2
0.4
0.6
0.8
1
Lβ/νm(h,L
)
β = 4
L=8
L=12
L=16
L=20
L=24
arXiv:1808.08878
43
G(k, τ > 0) =
∫ ∞
−∞dω
e−ω(τ−β/2)
2 cosh(βω/2)A(k, ω). (63)
limβ→∞
G(k, τ =β
2) = βA(k, ω = 0). (64)
High
Lowhkjjkj
High (a) (b)
h<hc h=hcarXiv:1808.08878
44