Dynamical t/U Expansion of a Doped Hubbard
Model
Wenxin Ding1∗,2, Rong Yu3
1∗School of Physics and Material Science, Anhui University, Hefei, Anhui Province,
230601, China
2Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences,
Beijing, China
3Physics Department and Beijing Key Laboratory of Opto-electronic Functional
Materials and Micro-nano Devices, Renmin University, Beijing 100872, China
*wenxinding@gmail.com
July 17, 2019
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 1 / 17

Outline
1 Motivation
2 Charge SU(2) Algebra in Hubbard Model and a New Slave Spin
Representation
3 A “Dynamical Mean Field Theory” by Hand
4 The Dynamical t/U Expansion
5 Summary and Discussion
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 2 / 17

Motivation
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 3 / 17
High-TC Superconductors are generally interpretted as doped
antiferromagnetic Mott insulators
⇒ doped single band Hubbard Model (HM) with U > D
(Analytic) perturbation theories for large U HM and their difficulties:
(a) t − J model: Schrieffer-Wolff transformation.
Difficulty: operator transformation; new fermions are not local, not
physical fermions; addtional O(t2
/U) terms are ignored uncontrollably;
(b) Canonical perturbation theory in U.
Difficulty: convergence issue; divergence/nonanalyticity in vertex
functions[1];
(c) * Strong coupling perturbation theory in terms of Green’s functions[2].
Difficulty: failed for doped cases at low temperatures (T ∼ t): negative
spectral weight.
(d) * Slave-boson-type theories:
Difficulty: typically solved at mean field level; difficult to solve for
dynamical properties.
[1] O. Gunnarsson et.al PRL 119, 056402 (2017); P. Thunström et.al PRB 98, 235107 (2018).
[2] S. Pairault et.al PRL 80, 5389 (1998) & EPJB 16, 85 (2000).

Motivation
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 4 / 17
A good dynamical theory should describe (at least):
• Single-particle level: properties of local density of states (LDOS):
(a) positions of poles: LHB, Fermi energy + in-gap states (W1),
UHB (W2)
(b) spectral weight distribution over the poles
ωµ
n/2$n/2$ 1&n$
U$
Schema'c(
DOS(
(B)$Unocc$
$$$$$$LHB$
(A) Occ$$
$$$$$$LHB$
(C)$Unocc$
$$$$$$UHB$
S. Sakai et.al PRL 102, 056404(2009)

Motivation
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 4 / 17
A good dynamical theory should describe (at least):
• Single-particle level: properties of local density of states (LDOS):
(a) positions of poles: LHB, Fermi energy + in-gap states (W1),
UHB (W2)
(b) spectral weight distribution over the poles
• Fluctuation effects: evolution of spin-spin interactions with
doping
(a) recover the correct AFM interaction;
(b) its evolution with doping; the Gutzwiller approximation
for t − J model;
(c) Nagaoka’s theorem; possibility of FM phase in
thermodynamic limit, finite doping, and finite U?
(d) experiments: FM fluctuations and static FM phase in
overdoped Cuprates; FM superconductivity in magic
angle twisted bilayer graphene.

Motivation
Common and Inherent Difficulty: NON-CANONICALITY
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 5 / 17
Inherent difficulty for strong coupling perturbation:
NON-CANONICALITY
[ ˆOi, ˆO†
f ] ∝ δif for bosons, or { ˆOi, ˆO†
f } ∝ δif for fermions
(a) No Wick’s Theorem,
(b) Feynman diagram approach breaks down,

Motivation
Common and Inherent Difficulty: NON-CANONICALITY
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 5 / 17
Inherent difficulty for strong coupling perturbation:
NON-CANONICALITY
[ ˆOi, ˆO†
f ] ∝ δif for bosons, or { ˆOi, ˆO†
f } ∝ δif for fermions
(a) No Wick’s Theorem,
(b) Feynman diagram approach breaks down,
in t − J model : {X0σ1
i , Xσ20
j } = δij δσ1σ2 − (σ1σ2)X¯σ1 ¯σ2
i ,
in Hubbard model, the atomic basis involves : X0D
i , XD0
, . . .
for quantum spins:
[S+
, S−
] = 2Sz
, [Sz
, S±
] = ±S±
, {S+
, S−
} = 1, {Sz
, S±
} = 0.

Motivation
Common and Inherent Difficulty: NON-CANONICALITY
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 5 / 17
Inherent difficulty for strong coupling perturbation:
NON-CANONICALITY
[ ˆOi, ˆO†
f ] ∝ δif for bosons, or { ˆOi, ˆO†
f } ∝ δif for fermions
(a) No Wick’s Theorem,
(b) Feynman diagram approach breaks down,
A Controllable Way Around1,2:
Heisenberg Equations of Motion for Operators
⇒
Schwinger Equations of Motion (SEoM) for Green’s
functions!
1 B. S. Shastry, Phys. Rev. B, 87, 125124 (2013); 2 S. V. Tyablikov, Ukr. math. zh.
11, 287 (1959), Methods in the Quantum Theory of Magnetism, S. V. Tyablikov
(1967).

Local Charge SU(2) Algebra in Hubbard Model
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 6 / 17
Locally, the commutation relations of the fermions:
[d†
iσ, djσ ] = δijδσσ (2niσ − 1) + (nonlocal terms),
Let − (2niσ − 1)/2 → ˜Sz
iσ,
⇒ [d±
iσ, ˜Sz
iσ] = d±
iσ ⇒ d†
∼ ˜S−
, d ∼ d−
∼ ˜S+
,
up to a difference in the Casimir element:
C = S2
= (Sx
)2
+ (Sy
)2
+ (Sz
)2
: Ciσ = 1 + (niσ − 1/2)2
Spin SU(2)-rotation invariance of the Hubbard-U-term
HU =
U
2
˜Sz
iσ
˜Sz
i¯σ =
U
2
˜S z
ia
˜S z
ib,
Rotation in spin-index: ˜S z
is|s=a,b :
˜S +
is = σ f+
i,sσd−
iσ,
˜S −
is = σ f−
i,sσd†
iσ.

Local Charge SU(2) Algebra in Hubbard Model
A New Slave Spin Representation
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 7 / 17
The usual slave spin representation: d†
iσ = S+
iσf†
iσ, diσ = S−
iσfiσ,

Local Charge SU(2) Algebra in Hubbard Model
A New Slave Spin Representation
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 7 / 17
New representation: separate the spin & slave spin (charge) index
d†
iσ = (S+
ia + S+
ib )f†
iσ/
√
2, diσ = (S−
ia + S−
ib )fiσ/
√
2,
HHM−ss = HS,0 + Hf,0 + Ht,
HS,0 =
U
2
i
(
s
Sz
is)2
+ h
is
Sz
is, Hf,0 = (−h − µ)
iσ
nf
iσ,
Ht = −
ijσ
tij
2
(S+
ia + S+
ib
)(S−
ja + S−
jb
)f†
iσfjσ,
h enforces the constraint:
l=a,b
Sz
is =
σ=↑,↓
(f†
iσfiσ −
1
2
),

Local Charge SU(2) Algebra in Hubbard Model
A New Slave Spin Representation
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 7 / 17
New representation: separate the spin & slave spin (charge) index
d†
iσ = (S+
ia + S+
ib )f†
iσ/
√
2, diσ = (S−
ia + S−
ib )fiσ/
√
2,
HHM−ss = HS,0 + Hf,0 + Ht,
HS,0 =
U
2
i
(
s
Sz
is)2
+ h
is
Sz
is, Hf,0 = (−h − µ)
iσ
nf
iσ,
Ht = −
ijσ
tij
2
(S+
ia + S+
ib
)(S−
ja + S−
jb
)f†
iσfjσ,
h enforces the constraint:
l=a,b
Sz
is =
σ=↑,↓
(f†
iσfiσ −
1
2
),
Interpretation:
(a) the f-spinons: (i) auxiliary fields representing the spin-flip degrees
of freedom; (ii) carry the fermionic statistics;
(b) the slave spins: describe charge excitations - large energy pole
structures!

A “Dynamical Mean Field Theory” by Hand
Static Mean field theory of HHM−ss
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 8 / 17
HHM−ss ⇒
HS,eff = HS,0 + HS,t,
Hf,eff = Hf,0 + Hf,t.
,
HS,t = −
ij,ss
(Qf,ijS+
isS−
js
+ h.c.), Hf,t = −
ijσ
(QS,ijf†
iσfjσ + h.c.)
Qf,ij =
σ
tij
2
f†
iσfjσ , QS,ij =
ss
tij
2
S+
isS−
js
,
Mean field approximation for HS,t :
HS,tMF = −
is
(
ij s
Qf,ij S−
js
)S+
is + h.c.

A “Dynamical Mean Field Theory” by Hand
Static Mean field theory of HHM−ss
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 8 / 17
HHM−ss ⇒
HS,eff = HS,0 + HS,t,
Hf,eff = Hf,0 + Hf,t.
,
HS,t = −
ij,ss
(Qf,ijS+
isS−
js
+ h.c.), Hf,t = −
ijσ
(QS,ijf†
iσfjσ + h.c.)
Qf,ij =
σ
tij
2
f†
iσfjσ , QS,ij =
ss
tij
2
S+
isS−
js
,
Mean field approximation for HS,t :
HS,tMF = −
is
(
ij s
Qf,ij S−
js
)S+
is + h.c.
HS,MF = HS,0 + HS,tMF =
i,s
U
2
Sz
isSz
i¯s + hSz
is − hxSx
is ,
hx = DMxQf .
(a) for a 2D square lattice with only nn hopping;
(b) assume uniform values for Qf,ij = Qf , D = 4 is the number of nn bond;
(c) Mx = Sx
a + Sx
b /2 is to be determined self-consistently from solving HS,MF

A “Dynamical Mean Field Theory” by Hand
Static Mean field theory of HHM−ss
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 9 / 17
U-1
0.05 0.08 0.1 0.12
0 0.005 0.01 0.015 0.02
2
3
4
5
6
7
8
δ
h
a)
U-1
0.05
0.08
0.1
0.12
0 0.005 0.01 0.015 0.02
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
δ
Z=Mx
2
/2
b)
δ
0.001
0.005
0.008
0.01
0.00 0.02 0.04 0.06 0.08 0.10
1.0
1.1
1.2
1.3
1.4
1.5
1.6
U-1
Z/δ
c)

A “Dynamical Mean Field Theory” by Hand
Expression for Physical Green’s functions:
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 10 / 17
iGd,σσ [i, f] (
ss
iG−+
B,Sss [i, f])(iGf,σσ [i, f])/2 →
ρd[ω, x] = dω (
ss
ρ−+
B,Sss [ω + ω , x])ρf [ω ]/2,
• Consider ρf [ω ] δ(ω ) since QS Qf , ⇒
ρMF
d [ω, x]
ss
ρ−+
B,Sss [ω, x]/2.
G−+
B,Sss [ω, x] for HS,tMF
with good analytic properties!

A “Dynamical Mean Field Theory” by Hand
Solution for Gαα
B,ss [ω]: Schwinger’s Equation of Motion
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 11 / 17
Atomic (Ising) limit:
−i∂tS−
is = [HS
S,0, S−
is] = −USz
i¯sS−
is − hS−
is ⇒
−i∂ti
G−+
Sss [i, f] = −2 Sz
is δ[i, f]δss − hG−+
B,S,ss [i, f] − U Sz
i¯sS−
isS+
fs ,
in the Ising limit: Sz
i¯sS−
isS+
fs = U Sz
i¯s G−+
Sss [i, f]
⇒ G−+
B,S[ω] =
−(Mσ0 + mσz)
ω + (h + U(Mσ0 − mσz)/2)
,
here Sz
a + Sz
b = M = −δ, Sz
a − Sz
b = m.
At half filling, h = 0, M = 0, m = 1:
G−+
B,S,aa =
−1
ω − U/2
, G−+
B,S,bb =
1
ω + U/2
,
GX ⇔
s
G−+
B,S,ss agrees with slave rotor theory

A “Dynamical Mean Field Theory” by Hand
Solution for Gαα
B,ss [ω]: Schwinger’s Equation of Motion
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 11 / 17
With the hopping (hx, transverse field) term:
(1) renormalizing |ψ :
(i) as hx → 0+, |ψ → |M = 0, m = 0, ∆m2 = 1 ; lifting the
degeneracy for hx = 0 case;
(ii) in general, Mx ∝ hx, M ∝ h2
x, can be solved numerically,
or perturbatively from the Green’s function;
(2) correction to the SEoM and the Green’s functions:
Sz
i¯sS−
isS+
fs = Sz
i¯s G−+
ss [i, f];

A “Dynamical Mean Field Theory” by Hand
Solution for Gαα
B,ss [ω]: Schwinger’s Equation of Motion
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 11 / 17
Expressions for an arbitrary slave spin state of a Hubbard atom:
|ψM,m,∆m = a1| ↑↓ + a2| ↓↑ + a3| ↑↑ + a4| ↓↓ ,
M = a2
3 − a2
4, m = a2
1 − a2
2, ∆m = ˆm2
= a2
1 + a2
2,
Construct G−+
B,S,|ψ
as:
G−+
B,S,0,|ψ
= a2
1GB,S,|ψ1,0
+ a2
2GB,S,|ψ−1,0
+ a2
3GB,S,|ψ0,1
+ a2
4GB,S,|ψ0,−1
,
G−+
B,S,|ψ
[ω] =
−((1 − ∆m2
)σ0 + mσz)(ω − h) + (M − ∆m2
)Uσ0/2
(ω − h)2 − U2/4

A “Dynamical Mean Field Theory” by Hand
Solution for Gαα
B,ss [ω]: Schwinger’s Equation of Motion
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 11 / 17
Correction to SEoM:
− i∂tSα
s = α(USz
¯s Sα
s + hSα
s + hxSz
s ),
− i∂tSz
s = ihxSy
s =
hx
2
(S+
s − S−
s ),
− i∂ti
G−+
B,S,ss [i, f] = − 2 Sz
s δss δ[i, f] − hG−+
B,S,ss [i, f]
+ U Γz−;+
B,S,¯ss;s [i, f] + hxGz+
B,S,¯ss [i, f] ,
− i∂tGz ¯α
B [ω] =
1
2
(α Ix + hx
α
αGα ¯α
B [i, f]), Ix = Mxσ0,
Sz
i¯sS−
isS+
fs = Γz−;+
B,S,¯ss;s [i, f] Sz
i¯s G−+
ss [i, f] + S−
s Gz+
¯ss [i, f].

A “Dynamical Mean Field Theory” by Hand
Solution for Gαα
B,ss [ω]: Schwinger’s Equation of Motion
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 11 / 17
Perturbative iteration in hx:
G−+
B,S[ω] G−+
B,S,0,|ψ
[ω] + G−+
B,S,1[ω],
G−+
B,S,1,ss [ω] =
−1
ω − h
(U Sx
s + hx)Ix
2
+
Uhx
2ω
Sx
σ
α
α Gα +
B,S,0,¯ss [ω]
+
h2
x
2ω
α
α Gα +
B,S,0,ss [ω] ,
RPA: G−+
B,S,ss[ω] G−+
B,S,0,ss[ω]−1
− G−+
B,S,1,ss[ω]
−1
,
this is allowed as long as h2
xU 1,
i.e. G−+
B,S,1 is much smaller than G−+
B,S,0.

A “Dynamical Mean Field Theory” by Hand
Local electron spectral functions
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 12 / 17
Left panel: Local spectral function of the electrons;
Right panel: δ-dependence of spectral weight of each pole with
linear-fit (dashed lines) and total spectral weight (dotted line) of
1st order RPA-Gd.
δ 0.008 0.02 0.05 0.1
U=10W1
W2
W3
W4
-4 -2 0 2 4 6 8 10
0.0
0.5
1.0
1.5
2.0
2.5
ω
ρd
MF
[ω]
U=10
1/4
(1-δ)/2
δ
2δ
(1-5 δ)/2
ΣiWi/4
0.00 0.02 0.04 0.06 0.08 0.10
0.0
0.1
0.2
0.3
0.4
0.5
δ
W
W1 W2 W3 W4

A “Dynamical Mean Field Theory” by Hand
Local electron spectral functions
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 12 / 17
Left panel: Local spectral function of the electrons;
Right panel: δ-dependence of spectral weight of each pole with
linear-fit (dashed lines) and total spectral weight (dotted line) of
1st order RPA-Gd.

A “Dynamical Mean Field Theory” by Hand
Local electron spectral functions
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 12 / 17
Analytic properties:
(a) poles: LHB ∼ E1 = −h, UHB ∼ E4 = h + U, Fermi energy
E2 = 0, in-gap states: ∼ E3 = h!
(b) W1 (1 − δ)/2 , W2 δ, W3 2δ, W4 (1 − 5δ)/2;
⇒ doublon density ndoublon 2δ for δ 1, in CDMFT ,
ndoublon 1.x δ
(c) Gz+
B,S introduces a dynamical term ∝ 1/ω for time-ordered
(causal) G−+,
Im[1/ω] ∝ δ (ω) ⇒ ∂ωρf (ω) in ρd(ω)
• the origin of dynamical paritcle-hole asymmetry near the
Fermi energy;
• generic behavior, NOT mean field artifact.

The Dynamical t/U Expansion
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 13 / 17
Integrate out f-fermions, or the slave spin S±
respectively from LHM :
LHM ⇒



Leff,f
= Lf,0 − ln det
(G0
S)−1
tij f†
iαfjα
tij f†
jαfiα (G0
S)−1
i,j
,
Leff,S
= LS,0 − ln det
(G0
f )−1
tijS+
i S−
j
tijS+
j S−
i (G0
f )−1
i,j
,

The Dynamical t/U Expansion
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 13 / 17
Integrate out f-fermions, or the slave spin S±
respectively from LHM :
LHM ⇒



Leff,f
= Lf,0 − ln det
(G0
S)−1
tij f†
iαfjα
tij f†
jαfiα (G0
S)−1
i,j
,
Leff,S
= LS,0 − ln det
(G0
f )−1
tijS+
i S−
j
tijS+
j S−
i (G0
f )−1
i,j
,
Let G0
f → Gf , and G0
S → GS where
Gf = (Leff,f
)−1
, GS = (Leff,S
)−1
,
and require self-consistency between Leff,f
⇔ Leff,S
.

The Dynamical t/U Expansion
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 13 / 17
Integrate out f-fermions, or the slave spin S±
respectively from LHM :
LHM ⇒



Leff,f
= Lf,0 − ln det
(G0
S)−1
tij f†
iαfjα
tij f†
jαfiα (G0
S)−1
i,j
,
Leff,S
= LS,0 − ln det
(G0
f )−1
tijS+
i S−
j
tijS+
j S−
i (G0
f )−1
i,j
,
Let G0
f → Gf , and G0
S → GS where
Gf = (Leff,f
)−1
, GS = (Leff,S
)−1
,
and require self-consistency between Leff,f
⇔ Leff,S
.
Mean field Theory is recovered at the lowest order.
Flucations at higher order
by further expanding the ln det[. . . ]

The Dynamical t/U Expansion
Effective spin-spin interactions
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 14 / 17
Expand Leff,f
to O(t2
):
HHeisenberg = J/2 f†
i,αfi,βf†
j,βfj,α,
J = J(0)
+ J(1)
+ . . . ,
J(0)
= −t2 dω
2π
ss
G+−
B,S,0,ss G−+
B,S,0,ss ,
J(1)
= −t2 dω
2π
ss
(G+−
B,S,1,ss G−+
B,S,0,ss + G+−
B,S,0,ss G−+
B,S,1,ss ).
fjα
f†
iα
f†
jβ
fiβ S+
is S−
is′
S−
js S+
js′
At half-filling with h = 0,
J0 = J(0)
(h = 0) = 2t2
/U,

The Dynamical t/U Expansion
Effective spin-spin interactions
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 14 / 17
Expand Leff,f
to O(t2
):
HHeisenberg = J/2 f†
i,αfi,βf†
j,βfj,α,
J = J(0)
+ J(1)
+ . . . ,
J(0)
= −t2 dω
2π
ss
G+−
B,S,0,ss G−+
B,S,0,ss ,
J(1)
= −t2 dω
2π
ss
(G+−
B,S,1,ss G−+
B,S,0,ss + G+−
B,S,0,ss G−+
B,S,1,ss ).
fjα
f†
iα
f†
jβ
fiβ S+
is S−
is′
S−
js S+
js′
At finite doping, asymptotically, we have
J(0)
J∗
0 (1 − 2δ), J(1)
−
(1 + δ)M2
xt2
1 + 4a
,
where a = h/U, J∗
0 is J(0)
(h = hc) which is smaller than J0 = 4t2
/U.
M2
x ∝ Z, Z|U→∞ = δ! J(1)
survives in the U → ∞ limit !

The Dynamical t/U Expansion
Effective spin-spin interactions
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 14 / 17
U-1
0.05 0.08 0.1 0.12
J
J(0)
0.25+J(1)
0 0.005 0.01 0.015 0.02
0.15
0.20
0.25
0.30
δ
J

The Dynamical t/U Expansion
Effective Hamiltonian at O(t2
/U):
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 15 / 17
HHM ⇒
HS,eff = HS,0 + HS,t,
Hf,eff = Hf,0 + Hf,t + HJ0 + HJ1 .
(1) Hf,t + HJ0 forms a t − J-type model for the spinons with
renormalization factors in agreement with the Gutzwiller
approximation;
(2) HJ1 is a FM term which survives the U → ∞ limit;
(3) HS,eff is formed by two copies of XXZ model with external
field and an onsite Ising interaction (U);
(4) HS,eff dominates the electron dynamical properties.

Summary and Discussion
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 16 / 17
Summary:
(1) new slave spin representation by algebraic consideration;
(2) local pole-structure and the corresponding spectral weights in
agreement with DMFT;
(3) recovered t − J model with Gutwiller approximation from
fluctuation effects;
(4) obtained FM spin-spin interaction in the doped regime;
Discussion:
(1) relevance to FM fluctuations (in hole-overdoped) or phase (in
electron-overdoped) of cuprates
(2) relation to Nagaoka theorem?
(3) this new representation and the solution can be used in DMFT as
an efficient impurity solver away from half-filling;
(4) solutions beyond mean field, magnetism, superconductivity, etc.

The End
Wenxin Ding (AHU) 9th
QMBC@RUC July 17, 2019 17 / 17
THANK YOU!