Hidden Order and Dynamics of SUSY VBS models...Hidden Order and Dynamics of SUSY VBS models Keisuke...
Transcript of Hidden Order and Dynamics of SUSY VBS models...Hidden Order and Dynamics of SUSY VBS models Keisuke...
![Page 1: Hidden Order and Dynamics of SUSY VBS models...Hidden Order and Dynamics of SUSY VBS models Keisuke Totsuka Yukawa Institute for Theoretical Physics, Kyoto University Kazuki Hasebe](https://reader034.fdocuments.in/reader034/viewer/2022051408/600b9f42c5f8c67fd76e0400/html5/thumbnails/1.jpg)
Hidden Order and Dynamics of SUSY VBS models
Keisuke Totsuka Yukawa Institute for Theoretical Physics, Kyoto University
Kazuki Hasebe Takuma National College of Technology
29/10/2010 “New development of numerical simulations in low-dimensional quantum systems” @ YITP
Refs: D. Arovas, KH, X-L.Qi and S-C. Zhang, Phys.Rev.B 79, 224404 (2009) KH and KT, preprint (2010) and in preparation
![Page 2: Hidden Order and Dynamics of SUSY VBS models...Hidden Order and Dynamics of SUSY VBS models Keisuke Totsuka Yukawa Institute for Theoretical Physics, Kyoto University Kazuki Hasebe](https://reader034.fdocuments.in/reader034/viewer/2022051408/600b9f42c5f8c67fd76e0400/html5/thumbnails/2.jpg)
Outline
IntroductionValence-Bond Solid (VBS) model(s)
definition, matrix product & analogiesHidden (string) order
Supersymmetric VBS stateHoles in VBS statesAnalogies (FQHE, BCS) & hidden orderLow-lying excitation (SMA)Some generalizations
Summary
![Page 3: Hidden Order and Dynamics of SUSY VBS models...Hidden Order and Dynamics of SUSY VBS models Keisuke Totsuka Yukawa Institute for Theoretical Physics, Kyoto University Kazuki Hasebe](https://reader034.fdocuments.in/reader034/viewer/2022051408/600b9f42c5f8c67fd76e0400/html5/thumbnails/3.jpg)
Introduction
VBS model ... a paradigmatic model for spin-gap ststems in 1D
generalization to other groups: SU(n), SO(n), ...
What about adding “holes” to VBS model ?✓a single spin-0 hole ... an exact eigenstate
for a t-J-like model (S=1 VBS background)
✓a single spin-1/2 hole ... a model for Y2BaNiO5
Model with many holes??Hidden order (robust against hole) ? Excitation?Possible future generalizations??
Zhang-Arovas ’89
Penc-Shiba ’95
Tu-Zhang-Xiang ’08, etc.
![Page 4: Hidden Order and Dynamics of SUSY VBS models...Hidden Order and Dynamics of SUSY VBS models Keisuke Totsuka Yukawa Institute for Theoretical Physics, Kyoto University Kazuki Hasebe](https://reader034.fdocuments.in/reader034/viewer/2022051408/600b9f42c5f8c67fd76e0400/html5/thumbnails/4.jpg)
Valence-bond solid (VBS) states
![Page 5: Hidden Order and Dynamics of SUSY VBS models...Hidden Order and Dynamics of SUSY VBS models Keisuke Totsuka Yukawa Institute for Theoretical Physics, Kyoto University Kazuki Hasebe](https://reader034.fdocuments.in/reader034/viewer/2022051408/600b9f42c5f8c67fd76e0400/html5/thumbnails/5.jpg)
Valence-bond solid modelsClass of models for which short-range valence-bondcrystals are (rigorous) ground states
Majumdar-Ghosh model (S=1/2 J1-J2 chain with J2/J1=1/2)
2D Shastry-Sutherland model, and many more...
Soluble models which realize Haldane’s scenario:HS=1 =
�
i
�Si·Si+1 +
13
(Si·Si+1)2 +
23
�
HS=2 =�
i
�Si·Si+1 +
29
(Si·Si+1)2 +
163
(Si·Si+1)3 +
107
�
HS=3 =�
i
�Si·Si+1 +
59358
(Si·Si+1)2 +
191611
(Si·Si+1)3 +
13222
(Si·Si+1)4 +
396179
�
Affleck et al ’88, Arovas et al. ’88
Majumdar-Ghosh state: a “Kekule” structure
![Page 6: Hidden Order and Dynamics of SUSY VBS models...Hidden Order and Dynamics of SUSY VBS models Keisuke Totsuka Yukawa Institute for Theoretical Physics, Kyoto University Kazuki Hasebe](https://reader034.fdocuments.in/reader034/viewer/2022051408/600b9f42c5f8c67fd76e0400/html5/thumbnails/6.jpg)
Valence-bond solid models(Many) important features:✓short-range spin correlation:
✓ “Haldane gap” to triplon excitation
✓edge states & (exact) degeneracy in a finite open system
✓Hidden order
Arovas et al. ’88, Freitag-Müller-Hartmann ’91
�Szi Sz
j � =
(S+1)2
3
�− S
S+2
�|j−i|for i �= j
S(S+1)3 for i = j
Spin Gap (SMA) Approximation
1 2027 → 7.41× 10−1
2 15 → 2.00× 10−1
3 2646265 → 4.21× 10−2
4 71590318 → 7.92× 10−3
5 884634711 → 1.39× 10−3
6 2826251203274644 → 2.35× 10−4
1 2 3 4 5 6 7 8
10 4
0.001
0.01
0.1
1
spin
gap
KT-Suzuki ’95
![Page 7: Hidden Order and Dynamics of SUSY VBS models...Hidden Order and Dynamics of SUSY VBS models Keisuke Totsuka Yukawa Institute for Theoretical Physics, Kyoto University Kazuki Hasebe](https://reader034.fdocuments.in/reader034/viewer/2022051408/600b9f42c5f8c67fd76e0400/html5/thumbnails/7.jpg)
Supersymmetric VBS states
![Page 8: Hidden Order and Dynamics of SUSY VBS models...Hidden Order and Dynamics of SUSY VBS models Keisuke Totsuka Yukawa Institute for Theoretical Physics, Kyoto University Kazuki Hasebe](https://reader034.fdocuments.in/reader034/viewer/2022051408/600b9f42c5f8c67fd76e0400/html5/thumbnails/8.jpg)
A quick look at supersymmetry... Schwinger boson construction of SU(2)
Generators (“spin”):commutation relations:
Schwinger-boson representation:
Label for irreps. = “boson number” = spin “S”
{Sx , Sy , Sz}
[Sa , Sb] = i �abc Sc
a†a + b†b = 2S
S+ = a†b , S− = b†a , Sz = (a†a− b†b)/2 (a†a + b†b = 2S)
|S;m� =1�
(S + m)!(S −m)!(a†)S+m(b†)S−m|0�
![Page 9: Hidden Order and Dynamics of SUSY VBS models...Hidden Order and Dynamics of SUSY VBS models Keisuke Totsuka Yukawa Institute for Theoretical Physics, Kyoto University Kazuki Hasebe](https://reader034.fdocuments.in/reader034/viewer/2022051408/600b9f42c5f8c67fd76e0400/html5/thumbnails/9.jpg)
A quick look at supersymmetry ...Schwinger operator construction of UOSp(1|2)
5 Generators & 3 Schwinger op.✓ Generators-(i) (“bosonic” = spin):✓ Generators-(ii) (“fermionic”):
commutation relations:
Label of irreps.
two SU(2) irreps:
{Sx , Sy , Sz}
K1=12(x−1fa† + xf†b) , K2=
12(x−1fb† − xf†a)
[Sa , Sb]=i �abcSc , [Sa , Kµ] =12(σa)νµKν
{Kµ , Kν} =12(iσyσa)µνSa
S = L (nf = 0) , S = L−1/2 (nf = 1)
na + nb + nf = 2L (∈ Z) spin-1 state
1-hole (S=1/2) state
L=1
(a† , b† , f†)
![Page 10: Hidden Order and Dynamics of SUSY VBS models...Hidden Order and Dynamics of SUSY VBS models Keisuke Totsuka Yukawa Institute for Theoretical Physics, Kyoto University Kazuki Hasebe](https://reader034.fdocuments.in/reader034/viewer/2022051408/600b9f42c5f8c67fd76e0400/html5/thumbnails/10.jpg)
SUSY VBS state... construction
prepare two (super)spin-1/2s on each site:
“Physical” state @site = superspin-1Schwinger-op rep. :“spin-1” or “1-hole(S=1/2)”
Form “entangled pairs”:
SUSY-VBS state:
a†jaj + b†jbj + f†j fj = 2L = 2
(a†jb†j+1−b†ja
†j+1−r f†
j f†j+1)
Def |sVBS� =�
j
(a†jb†j+1 − b†ja
†j+1 − r f†
j f†j+1)|0� (r ≡ x2 ∈ R)
spin-↑ spin-↓ hole
☞ super-qubit... L=1/2(spin-↑, spin-↓, hole) ⇔ (a† , b† , f†)
![Page 11: Hidden Order and Dynamics of SUSY VBS models...Hidden Order and Dynamics of SUSY VBS models Keisuke Totsuka Yukawa Institute for Theoretical Physics, Kyoto University Kazuki Hasebe](https://reader034.fdocuments.in/reader034/viewer/2022051408/600b9f42c5f8c67fd76e0400/html5/thumbnails/11.jpg)
SUSY VBS state... Physical picture
Schwinger-op. rep. :
expansion w.r.t. “r”:lowest-order in “r”... S=1 VBS state
highest-order in “r”... S=1/2 Majumdar-Ghosh
r→0: VBS, r→∞: MG
a†jaj + b†jbj + f†j fj = 2L = 2
Def |sVBS� =�
j
(a†jb†j+1 − b†ja
†j+1 − r f†
j f†j+1)|0� (r ≡ x2 ∈ R)
spin-↑ spin-↓ hole
![Page 12: Hidden Order and Dynamics of SUSY VBS models...Hidden Order and Dynamics of SUSY VBS models Keisuke Totsuka Yukawa Institute for Theoretical Physics, Kyoto University Kazuki Hasebe](https://reader034.fdocuments.in/reader034/viewer/2022051408/600b9f42c5f8c67fd76e0400/html5/thumbnails/12.jpg)
SUSY VBS state ... Parent Hamiltonian
usual VBS (S=1):valence-bond = “SU(2) singlet”:addition of angular-momenta:
Projection operator:
SUSY VBS:valence-bond = “UOSp(1|2) singlet”:addition of angular momenta:
Projection operator:
(a†jb†j+1−b†ja
†j+1)
1⊗ 1 � 0⊕ 1⊕ 2
(a†jb†j+1−b†ja
†j+1−rf†
j f†j+1)
(L = 1)⊗ (L = 1) � (0)⊕ (1/2)⊕ (1)⊕ (3/2)⊕ (2)
V3/2(P†3/2P3/2) + V2(P †
2 P2) (V3/2, V2 > 0)
hlocal = V2 P2 =V2
2
�Si·Si+1 +
13
(Si·Si+1)2 +
23
�
cf) Klein model ’82
![Page 13: Hidden Order and Dynamics of SUSY VBS models...Hidden Order and Dynamics of SUSY VBS models Keisuke Totsuka Yukawa Institute for Theoretical Physics, Kyoto University Kazuki Hasebe](https://reader034.fdocuments.in/reader034/viewer/2022051408/600b9f42c5f8c67fd76e0400/html5/thumbnails/13.jpg)
Analogy to FQHE
✓coherent state w.f.
✓projection to J≧S+1 state
VBS models
HVBS =2S�
J=S+1
VJPJ(Sj ·Sj+1)
✓spin-hole coherent state w.f.✓projection to Ltot ≧ L+1/2
state
SUSY VBS models
✓Laughlin w.f. on sphere
✓two-particle int:
FQHE (Laughlin w.f.)
�
i<j
(uivj − viuj)m
H =�
i<j
�
J>2L−m
VJ PJ(i, j)
✓SUSY Laughlin w.f.
✓projection to J>2L-m state ... exact G.S.
SUSY FQHE (Laughlin w.f.)
�
j
(ujvj+1 − vjuj+1)S
Arovas et al ’88
Haldane ’83
![Page 14: Hidden Order and Dynamics of SUSY VBS models...Hidden Order and Dynamics of SUSY VBS models Keisuke Totsuka Yukawa Institute for Theoretical Physics, Kyoto University Kazuki Hasebe](https://reader034.fdocuments.in/reader034/viewer/2022051408/600b9f42c5f8c67fd76e0400/html5/thumbnails/14.jpg)
Matrix-product representation... SUSY VBS states
Schwinger-op. rep. :Matrix-product representation:
size = number of (L/R) edge states (up, down & 1-hole...3)KEY: “valence-bond insertion” = “matrix multiplication”
gj =
−a†jb
†j −(a†j)
2 −√
ra†jf†j
(b†j)2 a†jb
†j
√rb†jf
†j
−√
rf†j b†j −
√rf†
j a†j 0
g†j =
−ajbj (bj)2 −
√rbjfj
−(aj)2 ajbj −√
rajfj
−√
rfjaj√
rfjbj 0
|sVBS� =�
j
(a†jb†j+1−b†ja
†j+1−r f†
j f†j+1)|0�
g1 ⊗ g2 ⊗ · · ·⊗ gL⊗gL+1
|sVBS� =
�g1 ⊗ · · · ⊗ gj ⊗ · · · gL . . . for open chain
STr [g1 ⊗ · · · ⊗ gj ⊗ · · · gL] . . . for periodic chain
Fannes et al. ’92,93
hole↓ ↑
hole
↑↓
⇧Left
⇦Right
![Page 15: Hidden Order and Dynamics of SUSY VBS models...Hidden Order and Dynamics of SUSY VBS models Keisuke Totsuka Yukawa Institute for Theoretical Physics, Kyoto University Kazuki Hasebe](https://reader034.fdocuments.in/reader034/viewer/2022051408/600b9f42c5f8c67fd76e0400/html5/thumbnails/15.jpg)
Correlation functions (i) ...spin-spin correlation functions
“transfer-matrix” formalism:
spin-spin correlation function:
Correlation length:9 edge states((2L+1)2, in general)
Klümper et al ’92, KT-Suzuki ’94-95, etc..
Arovas et al. ’09
�Sαi Sα
j � =
2(r2+√
8r2+9+3)√8r2+9(
√8r2+9+3) for i = j
13r2+24+(r2+8)√
8r2+9
2√
8r2+9(√
8r2+9+3)
�− 2√
8r2+9+3
�|j−i|for i �= j
“edge states”
0 5 10 15 20
!0.5
0.0
0.5
site-j
hole
�sVBS|sVBS� = GL ,
�sVBS|OAx O
By |sVBS� = Gx−1G(OA)Gy−x−1G(OB)GL−y
Ga,a;b,b ≡ g∗(a, b)g(a, b) , G(OA)a,a;b,b ≡ g∗(a, b)OA g(a, b)
![Page 16: Hidden Order and Dynamics of SUSY VBS models...Hidden Order and Dynamics of SUSY VBS models Keisuke Totsuka Yukawa Institute for Theoretical Physics, Kyoto University Kazuki Hasebe](https://reader034.fdocuments.in/reader034/viewer/2022051408/600b9f42c5f8c67fd76e0400/html5/thumbnails/16.jpg)
Correlation functions (ii) ...”superconducting” correlation
Possible “hole pairing”...
Need to modify “transfer-matrix” formalism:
Order parameter for “hole pairing”: Arovas et al. ’09
S=1 VBS
S=1/2 MG�∆j� =
�(ajbj+1 − bjaj+1)f†
j f†j+1
�
=144r
�r2 + 3 +
√8r2 + 9
�
√8r2 + 9
�3 +
√8r2 + 9
�2 �5 +
√8r2 + 9
�
(ajbj+1 − bjaj+1)f†j f†
j+1
�sVBS|sVBS� = GL ,
�sVBS|OAx O
By |sVBS� = Gx−1G(OA) �Gy−x−1G(OB)GL−y
�Ga,a;b,b ≡ g∗(a, b)(−1)F g(a, b) , G(OA)a,a;b,b ≡ g∗(a, b)OA g(a, b)
1 2 3 j
![Page 17: Hidden Order and Dynamics of SUSY VBS models...Hidden Order and Dynamics of SUSY VBS models Keisuke Totsuka Yukawa Institute for Theoretical Physics, Kyoto University Kazuki Hasebe](https://reader034.fdocuments.in/reader034/viewer/2022051408/600b9f42c5f8c67fd76e0400/html5/thumbnails/17.jpg)
Analogy ... BCS wavefunction
BCS wavefunction:
gk=0: (trivial) vacuum0< gk <∞: superconducting gk=∞: filled Fermi sphere (normal state)
SUSY VBS state:
“vacuum” = spin-1 VBS stater=0: pure VBS state0<r<∞: “hole-pair condensate”r=∞: Majumdar-Ghosh state
electron pair
hole pair|sVBS� =
�
j
(a†jb†j+1 − b†ja
†j+1 − r f†
j f†j+1)|0� (r ≡ x2 ∈ R)
|ΨBCS� =�
k
(1 + gkc†k,↑c†−k,↓)|0�
![Page 18: Hidden Order and Dynamics of SUSY VBS models...Hidden Order and Dynamics of SUSY VBS models Keisuke Totsuka Yukawa Institute for Theoretical Physics, Kyoto University Kazuki Hasebe](https://reader034.fdocuments.in/reader034/viewer/2022051408/600b9f42c5f8c67fd76e0400/html5/thumbnails/18.jpg)
Hidden Order... (usual) VBS state
Order in VBS state:No ordinary (local) magnetic LRO, no dimerization...But there is “hidden” order (“string order”)
How to detect ?Néel order:VBS:
den Nijs-Rommelse ’89, Tasaki ’91Pérez-García et al ’08
Oshikawa ’92, KT-Suzuki ’95
S=2
Szj (−1)nSz
j+n ⇒ ferro
Girvin-Arovas ’90
cf) Gu-Wen ’09
Ozstring = lim
n�∞�sVBS|Sj exp
iπj+n−1�
k=j
Szk
Szj+n|sVBS�
![Page 19: Hidden Order and Dynamics of SUSY VBS models...Hidden Order and Dynamics of SUSY VBS models Keisuke Totsuka Yukawa Institute for Theoretical Physics, Kyoto University Kazuki Hasebe](https://reader034.fdocuments.in/reader034/viewer/2022051408/600b9f42c5f8c67fd76e0400/html5/thumbnails/19.jpg)
Hidden Order... SUSY VBS state
“string order” in SUSY VBS ?matrix-product rep. ⇒ ”almost” flat surface
string order parameter captures hidden order...
gj =
−a†jb
†j −(a†j)
2 −√
ra†jf†j
(b†j)2 a†jb
†j
√rb†jf
†j
−√
rf†j b†j −
√rf†
j a†j 0
k�
j=1
gj =
|Sz
tot=0� |Sztot= + 1� |Sz
tot= + 1/2�|Sz
tot=− 1� |Sztot=0� |Sz
tot=− 1/2�|Sz
tot=− 1/2� |Sztot= + 1/2� |Sz
tot=0�
Sztot ≡
k�
j=1
Szj
-1 0
Ozstring = �sVBS|Sj exp
iπj+n−1�
k=j
Szk
Szj+n|sVBS�
S=1 VBS
“S”=1 sVBS
![Page 20: Hidden Order and Dynamics of SUSY VBS models...Hidden Order and Dynamics of SUSY VBS models Keisuke Totsuka Yukawa Institute for Theoretical Physics, Kyoto University Kazuki Hasebe](https://reader034.fdocuments.in/reader034/viewer/2022051408/600b9f42c5f8c67fd76e0400/html5/thumbnails/20.jpg)
Low-lying excitations ...“spin”
Failure of “Lieb-Schultz-Mattis” like excitation = want for alternative way to physical excitations
triplon = “crackion”
single-mode approximation:
Fáth-Sólyom ’93, KT-Suzuki ’94-95, Rommer-Östlund ‘95,’97
Szj |sVBS� =
12
�−|ψ(0)
j−1�+ |ψ(0)j �
�
ωzSMA(k) = −
12�sVBS| [[H, Sz(k)], Sz(−k)] |sVBS�
�sVBS|Sz(k)Sz(−k)|sVBS�
=�ψ(0)(k)|H|ψ(0)(k)��ψ(0)(k)|ψ(0)(k)�
=12
(1 − cos k)Szz(k)
�ψ(0)x |hx,x+1|ψ
(0)x �
SUSY valence bond
triplet bond
![Page 21: Hidden Order and Dynamics of SUSY VBS models...Hidden Order and Dynamics of SUSY VBS models Keisuke Totsuka Yukawa Institute for Theoretical Physics, Kyoto University Kazuki Hasebe](https://reader034.fdocuments.in/reader034/viewer/2022051408/600b9f42c5f8c67fd76e0400/html5/thumbnails/21.jpg)
Low-lying excitations ...“hole”
In SUSY, we have two more (fermionic) generators
“charege-crackion” = “spinon-hole pair”
can be treated by MPS, toolack of unitary operationconnecting “spin” and “charge” ➡ different spectra ...
SUSY valence bond
spinon-hole pair
0.0
0.5
1.0
0.0
0.5
1.0
0
1
2
3
K1(j) =12
�1√ra†jfj +
√rf†
j bj
�, K2(j) =
12
�1√rb†jfj −
√rf†
j aj
�
K1,j |sVBS� =12√
r�
|ψ(1/2)j−1 � − |ψ(1/2)
j ��
![Page 22: Hidden Order and Dynamics of SUSY VBS models...Hidden Order and Dynamics of SUSY VBS models Keisuke Totsuka Yukawa Institute for Theoretical Physics, Kyoto University Kazuki Hasebe](https://reader034.fdocuments.in/reader034/viewer/2022051408/600b9f42c5f8c67fd76e0400/html5/thumbnails/22.jpg)
Generalizations
![Page 23: Hidden Order and Dynamics of SUSY VBS models...Hidden Order and Dynamics of SUSY VBS models Keisuke Totsuka Yukawa Institute for Theoretical Physics, Kyoto University Kazuki Hasebe](https://reader034.fdocuments.in/reader034/viewer/2022051408/600b9f42c5f8c67fd76e0400/html5/thumbnails/23.jpg)
On Square Lattice...
model-I: 1-species of fermion
model-II: 3-species of fermion
=
|Ψ2D-(I)sVBS � =
�
�i,j�
(a†i b†j−b†ia
†j−r f†
i f†j )|0�
|Ψ2D-(II)sVBS � =
�
�i,j�
(a†i b†j−b†ia
†j−r1 f†
i f†j−r3 g†i g
†j−r3 h†
ih†j)|0�
r1,r2,r3→∞−−−−−−−−→�
all possible dimer coverings
Rokhsar-Kivelson RVB liquid Rokhsar-Kivelson ’88
+ ... +
![Page 24: Hidden Order and Dynamics of SUSY VBS models...Hidden Order and Dynamics of SUSY VBS models Keisuke Totsuka Yukawa Institute for Theoretical Physics, Kyoto University Kazuki Hasebe](https://reader034.fdocuments.in/reader034/viewer/2022051408/600b9f42c5f8c67fd76e0400/html5/thumbnails/24.jpg)
More fermions...
Case with two species of holes (1D)...
Matrix-product rep.:
Matrix-product formalism:finite string ordersymmetric behavior w.r.t. “r”
|sVBS� =�
j
�a†jb
†j+1−b†ja
†j+1−r(f†
j f†j+1 + h†
jh†j+1)
�|0�
gj =
−a†jb†j −(a†j)
2 −√
ra†jf†j −
√ra†jh
†j
(b†j)2 a†jb
†j
√rb†jf
†j
√rb†jh
†j
−√
rh†jb
†j −
√rh†
ja†j −rh†
jf†j 0
−√
rf†j b†j −
√rf†
j a†j 0 −rf†j h†
j
|0�j
![Page 25: Hidden Order and Dynamics of SUSY VBS models...Hidden Order and Dynamics of SUSY VBS models Keisuke Totsuka Yukawa Institute for Theoretical Physics, Kyoto University Kazuki Hasebe](https://reader034.fdocuments.in/reader034/viewer/2022051408/600b9f42c5f8c67fd76e0400/html5/thumbnails/25.jpg)
Summary
SU(2) -> UOSp(1|2) generalization of VBS model(s)✓ “spin-L” or “spin-(L-1/2)+one (spin-1/2) hole”
Matrix-product representation in terms of (2L+1)×(2L+1) matrix:
✓ short-range spin-spin cor., finite hole-pairing correlation✓ hidden string order (symmetry-protected phase ?)✓ single-mode approx. ⇒ gapped magnetic excitations
Generalizations:
✓ 2D case ⇒ relation to Rokhsar-Kivelson point...
✓ physics of QDM, topological order ??
(a†jb†j+1−b†ja
†j+1) ⇒ (a†jb
†j+1−b†ja
†j+1−rf†
j f†j+1)