Doping a moiré Mott Insulator: A t-J model study of twisted cuprates
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Doping a moiré Mott Insulator: A t-J model study of twisted cuprates
Xue-Yang Song,1 Ya-Hui Zhang,1 and Ashvin Vishwanath1
1
Department of Physics, Harvard University,Cambridge, MA 02138, USA
(Dated: September 17, 2021)
We theoretically investigate twisted structures where each layer is composed of a strongly cor-
related material. In particular, we study a twisted t-J model of cuprate multilayers within the
slave-boson mean field theory. This treatment encompasses the Mott physics at small doping and
self consistently generates d-wave pairing. Furthermore, including the correct inter-layer tunneling
form factor consistent with the symmetry of the Cu dx2 −y2 orbital proves to be crucial for the phase
arXiv:2109.08142v1 [cond-mat.supr-con] 16 Sep 2021
diagram. We find spontaneous time reversal (T) breaking around twist angle of 45◦ , although only
in a narrow window of twist angles. Moreover, the gap obtained is small and the Chern number
vanishes, implying a non-topological superconductor. At smaller twist angles, driving an interlayer
current however can lead to a gapped topological phase. The energy-phase relation of the interlayer
Josephson junction displays notable double-Cooper-pair tunneling which dominates around 45o . The
twist angle dependence of the Josephson critical current and the Shapiro steps are consistent with
recent experiments. Utilizing the moiré structure as a probe of correlation physics, in particular of
the pair density wave state, is discussed.
Introduction- Twisted heterostructures of graphene relations. The d wave pairing ansatz is obtained self
and transition metal dichalcogenides (TMDs) have at- consistently. In contrast, in a BCS approach one in-
tracted significant attention for displaying a series of corporates attractive interaction in the d wave channel
interaction-driven phenomena such as superconductiv- by hand [13] . (II) We incorporate the Mott physics
ity [1, 2], integer and fractional Chern insulator phases in the slave boson mean field theory. Such a method
[3–7] and as simulators of the Hubbard model [8–11]. is known to be successful in capturing the d-wave su-
Although the single layers tend to be weakly correlated, perconductor in single layer[17]. In twisted bilayer, the
the twist structure reconstructs the electronic bands cre- effective inter-layer tunneling will be suppressed by a
ating a versatile platform to study strongly interacting factor x in this approach, where 0 < x < 1 is the
physics. A natural next step towards even richer phe- hole doping level. Hence the method can smoothly
nomena are twisted bilayers where each layer itself is crossover to the x = 0 limit which represents the un-
strongly correlated. A promising candidate is twisted doped Mott insulator where the inter-layer tunneling
cuprate bilayers. should be fully suppressed. This x factor makes the win-
The recent fabrication of high quality monolayer of dow of twist angle with time reversal breaking and the
Bi2 Sr2 CaCu2 O8 (Bi2212)[12] finds similar critical tem- gap significantly smaller than that within a BCS treat-
perature in single-layer and bulk cuprates, opening up ment [13]. (III) We note that the inter-layer tunneling
the experimental study of cuprates as essentially 2D between the two cuprate layers should include a form fac-
system. This also enables the experimental study of tor (cos kx,t − cos ky,t )(cos kx,b − cos ky,b )[18], where the
twisted cuprate bilayer, which in turn has spurred pi- momentum kt and kb are defined in the top and bottom
oneering theoretical proposals for two different realiza- layer separately, with the coordinate system fixed to the
tions of topological superconductor in twisted cuprate Cu-O bond of each layer.
bilayer[13, 14]. Especially, Ref. 13 proposes a topologi- Such a form is especially important to consider since it
cal superconductor with time reversal symmetry sponta- should vanish at θ = 45◦ by symmetry, or more specifi-
neously broken when the twist angle θ is around 45◦ de- cally due to the symmetry properties of the Cu dx2 −y2 or-
gree while Ref. 14 seeks to stabilize a similar state by tun- bital1 . We note that the form factor suppresses the tun-
ing near a magic angle with weak quasiparticle disper- neling around the nodal region and therefore the Dirac
sion and establishing a superflow along the ‘c-axis’. On nodes remain almost gapless even if time reversal sym-
experimental side, twisted cuprates have recently been metry is broken due to the pairing phase difference be-
fabricated and characterized [15, 16]. Ref [16] reported tween the two layers. Incorporating such momentum
a measurable second harmonic Josephson coupling at dependent tunneling terms will lead to key differences
θ ≈ 45◦ based on the Shapiro step. However, so far there from earlier studies that employ a momentum indepen-
is no experimental signature of gap opening,spontaneous dent tunneling term [13].
time-reversal symmetry breaking or topological super-
conductivity.
Here we study the twisted cuprate bilayer problem
theoretically, augmenting earlier studies in two signif- 1 Strictly speaking, the oxygen p orbital is also involved in hole
icant ways (I) we introduce a microscopic twisted t-J doped cuprate, forming the Zhang-Rice singlet. However, the
model and solve it within the self-consistent slave-boson active Zhang-Rice singlet has the same symmetry as the Cu
mean-field approximation, which accounts for strong cor- dx2 −y2 orbital and does not change our analysis.2
Our work has important implications for the ongo- (a) Ftb,1 (b) ϕ | tza0 |
ing experimental studies of twisted cuprate bilayer[16]. π
dx2−y2 s
t1 C=0
C=8
We extract the inter-layer Josephson coupling E[φ] = (I)
tbi
t2 C=4
−a cos φ + b cos 2φ for different twist angle θ, where φ is t2 (II)
the phase difference between the two layers. Then we can 0 θ
tz b1 optimal ϕ 45o
extract the critical Josephson current and demonstrate b1 (III) (c)
ϕ
similar twist angle dependence as observed in experiment optimal ϕ | tza0 |
tbi b2 Ftb,2 π
Ref. [16]. In particular we find |a| < b at θ = 45.2◦ and t2 b1 C=8
calculate the Shapiro steps at this angle. The results are tza0
consistent with the Shapiro steps measured in Ref. 16.
Near this angle, we observe spontaneous time reversal 0
45o
θ
breaking due to φ 6= 0 as reported in earlier studies [13].
However, in contrast to the earlier studies, due to the FIG. 1: (a) A schematic plot of the twisted 4 CuO
additional ingredients in our treatment, we find no sig- layers. The sites with the same color within the top,
nificant gap nor topological chiral modes. Instead, we bottom bilayers are perfectly aligned. There is an
calculate the polar Kerr effect[19, 20] which can be used hopping across the bilayer within unit cell with
as a diagnostic of spontaneous time reversal breaking and strength tbi and interlayer hopping with strength tz .
propose future optical experiments to verify this effect. Tunneling across the twist consists of 2 processes: Ftb,1
However, following Ref. [14], we do find that topologi- is a 3 step tunneling process mediated by s orbital of
cal superconductivity (SC) can be induced by adding an Cu and Ftb,2 describes a direct tunneling between d
interlayer supercurrent by hand to give φ 6= 0. This is orbitals of Cu. (b,c) The schematic phase diagram of
despite the fact that the magic angle is suppressed due twisted cuprates as one varies twist angle θ and pairing
to the x factor in the inter-layer tunneling in slave boson phase difference φ between top and bottom layers. The
theory. This again highlights the importance of Mott two figures present cases with (b) mixed tunneling i.e.
physics in modeling twisted correlated bilayers. both cos form and uniform tunneling which vanishes at
Finally, we also point out an interesting application θ = 45o . The Chern numbers for small twist angles are
of the moire’ heterostructure as a probe of cuprates, in shown. Other Chern numbers e.g. C = 2, 6 are realized
particular the pair density wave physics, that has been with different tunneling strength; spontaneous breaking
discussed extensively [21–27]. of time reversal is shown by the red line, when it
Model for twisted double-bilayer Bi2212- We take the separates from the x-axis, revealing a nonzero optimal
t − J model to describe a single cuprate layer on square φ (c) only uniform tunneling, results are comparable to
lattice: Ref. [13]. Actual data of φ vs θ are plotted in fig 6.
X † 1 X
HtJ = −t P ci,s cj,s P +J(Si ·Sj − ni nj )+µ (ni −1),
4 i between layers inside top (bottom) bilayers (i.e. within
hiji
(1) unit cell) and n labels sites inside a Moire unit cell in a
P †
where s =↑, ↓; ni = c c i,s , S = 1 †
c σ ss 0 cs0 and monolayer. The function Ftb is the form factor for inter-
s i,s 2 s
P projects out states with double/zero occupancy on layer tunneling resulting from the orbital characteristics
any site for hole/electron doping, respectively. We take of Cu ions.
t = 2J = 0.2eV in the calculation. Note that a transfor- Interlayer tunneling between d orbitals primarily con-
mation cis → c†is relates electron to hole doping cases, sists of two processes, plotted in Fig. 1(a): The first
with the sign of t, µ changed. Here we focus on hole one is mediated by s orbitals and the second one is a di-
doping and leave electron doping for future studies. rect tunneling controlled by the dimensionless parameter
Motivated by experiment [15, 16] we consider twisted a0 [28]2 .The two processes give two factors, to wit
double bi-layers, and add an index p = t1, t2, b1, b2 de- 1 X
noting operators in the top first, top second, bottom first Ftb,1 (rij ) = ξbh ξbh0 gs (ri±bh,j±bh0 ),
2
and bottom second layers, as shown in fig 1.The twist by h0 =x0 ,y 0
h=x,y,b
b
angle θ occurs between layers t2 and b1. Each layer is ξxb = 1, ξyb = −1, gs (it2 , jb1 ) = e−(lij −ld )/ρs ,
hence defined as top (bottom) bilayers. The full Hamil-
Ftb,2 (rij ) = a0 gs (it2 , jb1 ), (3)
tonian should include interlayer tunneling with strength
tz as, where the first process has a form in momentum space
Ftb,1 (k) ∝ (cos kx,t − cos ky,t )(cos kx,b − cos ky,b ) (sub-
Ftb (rij )c†i,t2 cj,b1
X X
Htbsc = HtJ,ti + HtJ,bi − tz script t, b denotes momentum in the relative canoni-
i=1,2 ij
cal coordinates of top, bottom layers, respectively), de-
c†n,p1 cn,p2 + h.c.,(2) scending from the form (cos kx − cos ky )2 in untwisted
X
−tbi
n,p=t,b
where tz term describes interlayer tunneling which we
2 a0 = 1.6 in untwisted cuprates (appendix B)[28].
will come back to later. The second line describes tunnel3
cuprates[18]. Here lij = |it2 − jb1 | and ld ≈ 2aCu system will enter a staggered flux spin liquid for x → 0.
(aCu ≈ 0.3nm the lattice constant of CuO plane) is the Note that the eigenstate solutions are symmetric with
distance between layer, and ρ ≈ 0.5aCu is a tunneling respect to positive, negative energies due to the chiral
parameter. symmetry C (appendix A).
From a symmetry Rxy that reflects along a diagonal We use t/J = 2. For a mean-field treatment, we start
direction of one of the cuprate layer, one deduces with a d−wave pairing ansatz, i.e. ∆i,i+bx = −∆i,i+by =
δi (eiφ ). We emphasize this ansatz is not biased, i.e. it
a0 (θ) = −a0 (90o − θ), a0 (45o ) = 0. will be obtained even from random pairing ansatz. In
this way we automatically reach a mean-field minimum
Details on tunneling forms, symmetry constraints and of the free energy.
parameter choices from band structures of Bi2212 are We find that the pairing orders converge to a d− wave
listed in appendix A, B, respectively. form with a possible phase difference φ between t, b lay-
Vanishing of a0 close to 45o results in qualitative ers,
change of phase diagram for twisted cuprates plotted
in fig 1(b,c), e.g. chiral non-topological superconduc- ∆i,i+bx,t = −∆i,i+by,t = δ
tivity(SC) and smaller gap etc, compared to finite a0 ∆i,i+bx,b = −∆i,i+by,b = δeiφ , (7)
(details in app E).
Slave boson mean field theory and self-consistent with the phase difference φ ∈ (0, π). We can also explic-
solutions- To decouple the Hamiltonian we adopt a par- itly choose the ansatz with an arbitrary φ and calculate
ton decomposition for electron operators, the energy E(φ). Details on numerics are listed in ap-
pendix C.
ci,s = b†i fi,s (4) Critical Josephson current- From the energy-phase E
vs φ relation (shown in figure 5)3 , one could extract crit-
with fermionic spinons fs and bosonic holons b. There ical Josephson current by I = 2e dE
~ maxφ ( dφ ) as shown in
P † †
is a gauge constraint s fi,s fi,s + bi bi = 1 for states fig 2(c). Fig 3(a) shows the fitting parameters a, b as
excluding double occupancy, and the electron number E = −a cos φ + b cos 2φ + const.
reads Experiments [15] on twisted Bi2212 yields a current
density Jc ≈ 100A/cm2 at θ = 45o , doping x = 0.1. An-
ni = 1 − b†i bi = 1 − xi . (5) other Josephson current measurement [16] shows strong
angular dependence of Jc , with which Fig. 2(c) quali-
Upon doping with a fraction x of holes, the condition tatively agrees, though [16] reports a smaller Jc (45o ) ≈
P †
becomes i bi bi = N x with N being the number of 40 − 120A/cm2 .Numerically we find Jc ≈ 50kA/cm2 at
sites. The holons will condense for nonzero x > 0 to tz = 0.05t, x = 0.1, 500 times larger than experiment.
√
hbi i = xi . Substituting the condensed holon operator This reduction in critical current is puzzling, and we con-
into the kinetic terms with hopping t and tunneling g in jecture that the vortex dynamics between top, bottom
the twisted bilayer t − J Hamiltonian eq(2), and further layers, rather than t − J physics, determines Jc . We also
decouple the Si · Sj by a mean-field treatment[29], one find a significant c-axis magnetization, as plotted in fig
gets (assuming spin rotation invariance) 9.
θ ∼ 45o : Chiral non-topological SC and anomalous
√ 3J ∗
χ )f † fisp + h.c.]
X X
HM F = {− [(t xip xjp + Hall effects- We found, that the system energetically fa-
8 ij,p isp vors a nonzero φ and spontaneously breaks time-reversal
p=t(b) hiji,s
3J X X symmetry close to 45o . Fig 3(a) shows the fitting pa-
− [∆∗ij,p fis,p fjs0 ,p ss0 + h.c.] + µp nip } rameters a, b as E = −a cos φ + b cos 2φ + const. At
8
hijiss0 i θ = 43.6o , 45.2o , we have 2|b| > |a| and an optimal
X √ † φ ≈ 0.16π, 0.64π, respectively. While for other angles,
− tz Ftb (rij )e−(lij −ld )/ρ xi,t2 xj,b1 fis,t2 fjs,b1
optimal φ = 0, π for θ < 41.2o , > 48.8o , respectively.
ij,x
√ The spontaneous generation of time reversal breaking
†
X
− tbi Fp (k) xn,p1 xn,p2 fkn,p1 fkn,p2 + h.c., (6) from φ 6= 0 has already been pointed out previously[13].
k,p=t,b However, the property of the chiral (T breaking) super-
conductor in our theory is qualitatively different. First,
where we mainly consider the mean field order param- due to the x factor appearing in the inter-layer tunnel-
P †
eters
P for hopping χij = s hfis fis i, pairing ∆ij = ing in the slave boson theory, the window of twist angle
ss0 ss0 hfis fjs0 i, and ignore local magnetic moments. with φ 6= 0 is significantly smaller in our calculation.
The interlayer tunneling Ftb = Ftb,1 + Ftb,2 consists of
2 processes shown in eqs (3). We use tz a0 to mark the
uniform tunneling strength hereafter and from [28] in
untwisted Bi2212 tz a0 ∼ 0.16t. 3 The energy variation upon varying φ comes mainly from inter-
The parton construction naturally incorporates the layer tunneling terms and we calculate energy only from inter-
Mott insulating state at vanishing doping x, since the layer tunneling terms hereafter.4
(a) 10
-3 (b) V(ℏω/2e)
1 ���
(a) 0.025 (b) 0.03 a
t za 0=0.2t tza0=0.8t
b*10 ���
0.02 t za 0=0.16tcos(2 ) 0.5
a0=0 ���
DOS(a.u)
a 0=0 0.02
0.015 0 ���
/t m
0.01
0.01 ���
-0.5
0.005 ���
-1 � � � � �
0 0 0 0.1 0.2 / 0.3 0.4 Id/Ic
(c) 10 20 (o) 30 40
(d) 0 0.1 E/t 0.2 0.3 (c) (d) 0.3
1 0.01 10 -3
15 6 Im[ h]
0.008 0.25
J ( )/J (0)
0.8
Re[ h]
/t
4
c
(e /h)
Chern #
0.6 10 0.006 0.2
gap
t z /t
2
2
c
0.4 0.004
h
0.15
5 0
0.2 0.002
0.1
-2
0 0 0
0 20 40 (o) 60 80 0 0.5 1 -4 0.05
/ 0.1 0.15 0.2 0.25
0 0.02 (eV) 0.04 0.06 x
FIG. 2: Data at x = 0.2, T = 0.(a) The maximal gap
∆m , obtained by varying the inter-layer phase φ at FIG. 3: Data for tz = 0.1t, x = 0.2, a0 = 0.(a) The
fixed twist angle θ, for different tunneling cases: coefficients for fitting energy phase relation
Uniform tunneling only (green). Mixed tunneling, cos E = −a cos(φ) + b cos(2φ) (E is obtained per unit cell
form factor mixed with uniform tunneling with realistic of CuO planes), with vanishing uniform tunneling
parameters (blue). Only cos form factor tunneling a0 = 0. a, b is plotted in units of t/a2Cu . At
(a0 = 0) (red), respectively. For the latter two θ = 43.6, 45.2o (boxed) we have an energy minima at
tz = 0.1t. (b) The density of states at θ = 43.6o for nontrivial φ ≈ 0.16π, 0.64π, respectively. (b) The
(only) uniform tunneling (blue) with a gap ∼ 4%t and Shapiro step measured in resistively shunted Josephson
(only) cos form tunneling (black) with a vanishing gap junction where a DC current Id and an AC current Irf
< 0.001t. The blue curve assumes a large uniform with frequency ω. The time-averaged voltage displays
tunneling to make the gap easily discernible. (c) steps as Id is increased, in particular the half step at
Critical Josephson current density normalized by the V = 0.5, 1.5, 2.5 (in units of ~ω/2e) indicates a second
untwisted value with tz = 0.1t, a0 = 0. The smallest Jc harmonic term in energy-phase relation. Parameters
is obtained at θ = 45.2o and has a value of used are results for θ = 45.2o , b = −1.3a, with
2000kA/cm2 , while Jc (0o ) = 8 × 104 kA/cm2 with Irf = 1.25Ic , ω = 0.5Ic /e (Ic is the critical Josephson
Jc (45o )/Jc (0o ) = 0.025. (d) The Chern number (black) current). (c) Real and Imaginary parts of the Hall
and gap (orange) upon varying φ for two relatively conductivity, σh (ω) at one particular parameter set
small twist angles with mixed tunneling and φ = 3π/16, a0 = 0 (circled in (d)). The Kerr angle is
(tz = 0.1t, tz a0 = 0.16t). estimated [20] to be of order 1µrad for small frequency
ω < 0.03eV . The edge above which σh tends to 0 at
ω/t ≈ 0.2 corresponds to the maximal gap around
Fermi surface (fig 8) in untwisted cuprates. (d)The
More importantly, the chiral superconductor shows a real part of anomalous Hall conductivity Re[σh (ω = 0)]
very small gap in our theory, because of the form factor (proportional to the red disk area) at θ = 43.6o as one
in the inter-layer tunneling. The form factor suppresses varies x and tz , signaling T breaking. The blue
the inter-layer tunneling at nodal region and thus the gap diamonds denote absence of T breaking, i.e.
opening is suppressed even if there is a large interlayer Re[σh (0)] = 0.
phase φ. A significant uniform tunneling component,
added by hand, will open a larger gap as plotted in fig
2(a). The density of states(DoS) also shows qualitative
difference at small energy between cases with or without diagram in fig 7). Therefore a topological superconduc-
uniform tunneling in fig 2(b): DoS is V shaped(black) tor is absent in our treatment, in contrast to Ref. 13.
for vanishing gap with only cos form tunneling; DoS has If we ignore the form factor and just take a homoge-
a vanishing flat segment(blue) at low energy correspond- neous tunneling in our t-J calculation, we recovered re-
ing to the gap opened by uniform tunneling. We use a 4 sults in Ref.[13] qualitatively, e.g. the gap, Chern num-
band continuum model to derive the gap in appendix D ber etc. (Details in appendix E.) But unfortunately such
to explain why cos form tunneling opens a much smaller an s-wave inter-layer tunneling vanishes by symmetry in
gap than uniform tunneling at leading order. twisted cuprate bilayer at θ = 45◦ .
Besides, we find that the Chern number is zero, both The density of states (DoS) in the chiral superconduc-
for twisted double-bilayer and twisted bilayer Bi2212 tor is still V shaped (fig 2(b)) in the experimentally rel-
(i.e., retaining only t2 , b1 and interlayer tunnel, phase evant energy scale. Therefore it is not possible to detect5
the time reversal breaking using electron spectroscopy upon varying φ keeps increasing as twist angle θ is re-
such as STM or ARPES. Here we investigate the Kerr ef- duced to 9.7o , indicating flat band may exist for θ < 9.7o ,
fect to detect the chirality. We calculate the optical con- if at all. For uniform tunneling only, ∆m reaches the
ductivity tensor σij (ω), signaling time-reversal breaking, maximum at θ ≈ 13o . In ref. 14, a magic angle around
plotted in fig 3(c,d) and from this extract the Kerr an- 13.8o is predicted for Bi2212 from a BCS model analysis.
gle. The nonvanishing off-diagonal component of σij , i.e. In our parton formalism, the magic angle is reduced due
anomalous Hall effect, signals T breaking and is given by to x factor in the interlayer tunneling in HM F , as shown
the anti-symmetric part of the current-current correlator in the blue curve in fig 2(a).
πµν (q, ω), For twist angle close to 45o , surprisingly a relatively
flat portion near the top of the lowest conduction band
i is found with cos form tunnel. As plotted in fig 10, the
σh (ω) = lim (πxy (q, ω) − πyx (q, ω)). (8)
2ω q→0 energy scale is ∼ 5%t and could result in an enhanced
The current operator is given by the procedure of mini- signal in optical measurements.
mal coupling k → k + eA/~ and taking derivatives with Moiré as a Probe of Correlation Physics- We also pro-
respect to EM field A, ji = Ω1 ∂Htbsc,A /∂Ai , where Ω is pose to detect the pair-density-wave (PDW)[21] super-
the volume. Note pairing function ∆k does not appear in conductor in twisted cuprate bilayer using the inter-layer
the current operator since it does not involve the center pair-tunneling. We can apply a bias to give different
of mass motion of Cooper pairs, details in appendix F. doping in the two layers. This makes it possible to be in
Nonzero anomalous Hall conductivity σh (q → 0) hap- the regime T > Tct , T < Tcb , where Tct , Tcb are the criti-
pens only if T is broken and occurs in a system that cal temperature for the uniform d wave superconductor
breaks Galilean invariance[30–32]4 . of the two layers. Therefore the bottom layer is in the
Due to the parton approach with the x factor in in- d-wave superconductor (SC) phase, while the top layer
terlayer tunneling, the peak in σh occurs at smaller fre- is in the pseudogap (PG) phase. Hence we have a PG-
quency ∼ 0.02eV than in optical measurements. The SC junction instead of the SC-SC junction. It can be
Kerr angle when the sample thickness h λ (λ the shown[34, 35] that the inter-layer tunneling I − V curve
wavelength of incident light) reads[20] encodes the information of superconductor fluctuation in
the top layer:
σxy
θK = Re arctan[ 2 + σ2 )
]. (9)
σxx + 4π(σxx xy X 2eV
I(V ) ∝ |a(Q)|2 |∆b |2 ImχR (q̃ − Q, ω = ) (10)
At the typical frequency in fig 3(c), the behavior for σxx Q
~
is complicated [33]. Though σh is of same order as the
Hall conductivity in [20], albeit in lower frequency range, where a(Q) is the Fourier transform of the Josephson
we expect the order of magnitude of θK is the same, i.e. coefficient a(r). q̃ = 2e~ At , where At is the gauge field
∼ µrad hence, measurable in experiments with Terahertz in the top layer, which can be changed by either in-
radiation. plane magnetic field or a solenoid[35]. χR (q, ω) is the
Small twist angles: Topological superconductivity and retarded pair susceptibility in the top layer at momen-
Flat band- At small angles, we take tz a0 = 0.16t to ac- tum q and frequency ω, which is a Fourier transforma-
count for some homogeneous tunneling and find T is pre-
Rr
tion of χR (r, t) = h[∆t (r, t), ∆†t (0, 0)]iθ(t)e−i ~ 0 At ·dl .
2e
served. If one adds by hand an interlayer supercurrent Here ∆t (r, t) is the Cooper pair field in the top layer.
(to model a uniform current state along the c-axis) by For untwisted bilayer, a(Q) 6= 0 only for Q = 0. In
fixing φ 6= 0, π, topological superconductivity emerges this case the PDW fluctuation in the top layer can not
an in Ref. 14 and indicated by Chern number plotted couple to the uniform d-wave superconductor in the bot-
in fig 2 (d). Chern number for θ = 19o , 28o reaches the tom layer unless we provide an unreasonably large At to
maximal value of 7.7, 4.5, the non-quantization is due to compensate the momentum QPDW ≈ ( 2π 8a , 0). In our
numerical precision and is assigned to have C = 8, 4 re- twisted cuprate bilayer, we expect that the Josephson
spectively. The gap for the topological superconductor coefficient a(r) to vary within the moiré unit cell and
is also shown fig 2 (d). we have a(GM ) 6= 0, where GM is a reciprocal vec-
For locating the flat band, since the x factor suppresses tor of the moiré lattice. At twist angle θ ≈ 7.18◦ , we
tunneling, we expect the magic angle is suppressed from notice that GM = 2π 2π 1
a (sin θ, cos θ − 1) ≈ a ( 8 , −0.007)
θM A ≈ 2vTFzQ , where Tz is effective tunneling strength in the coordinate of the top layer and is close to the
and Q the node momentum. Indeed in fig 2(a), the max- expected QPDW . The small deviation may be compen-
imal possible gap ∆m (blue curve for mixed tunneling) sated by q̃ = 2π a (0, −0.007) given by an in-plane mag-
|q̃|
netic field Bx . From 2π/a = eBda
h we estimate that
we need Bx = 84 T, assuming the inter-layer distance
4 We explicitly check that for zero twist angle (single-band d− d = 0.95 nm and a = 0.36 nm. This seems to be unreal-
wave pairing), even with nontrivial φ 6= 0, π and tunnel, σh (q → istic, but we expect to see PDW fluctuation with a small
0) vanishes. deviation because ImχR (q, ω) should have a Gaussian6
form around QPDW . Thus a small momentum deviation reference frame t̃. Given the d orbitals are odd under re-
δq = 0.007 2πa does not kill the signal. Alternatively we flection Rxy for the bottom layers, while d orbital wave-
can follow Ref. 35 to generate a larger At with a solenoid. function transforms to still d orbitals for the top layers
Conclusions- Through a self-consistent mean field with the rotated reference frame, a minus sign is added
study of t-J model for twisted double-bilayer cuprates, for cb transforms in the first line. This reflection relates
we find a time reversal breaking but non-topological su- the twist angle θ → π/2 − θ and the interlayer tunneling
perconductor at twist angle around 45◦ . Our conclu- Ftb (rij ) → −F̃tb (Rxy (rij )).
sions rely on incorporating Mott physics and form fac- In particular, for twist angle θ = 45o , the system stays
tor in the inter-layer tunneling, highlighting their impor- invariant under Rxy , imposing the constraint
tance in the modeling of twisted cuprate bilayer. With
microscopic calculations, we also provide twist angle de- Ftb,45o (rij ) = −Ftb,45o (Rxy (rij )). (A2)
pendence of the critical Josephson current, in qualitative We next derive the interlayer tunneling on microscopic
agreement with the recent experiment[16]. Finite tem- grounds and use Rxy to restrict the parameters. There
perature extensions of this theory and the interplay of are 2 major processes that cause the d orbital electrons
correlation phenomena with the moireśtructure will be to hop between top (t2) and bottom (b1) layers: first
interesting to explore in the future. one is mediated by s orbitals and second one is a di-
Acknowledgements- We thank Philip Kim, Xiaomeng rect tunneling from d orbitals active in the t − J model.
Cui and Mohit Randeria for discussions. This research The first one consists further of 3 steps: d electrons hop
was funded by a Simons Investigator award (AV), the to s orbitals in the same CuO plane, i.e. within the
Simons Collaboration on Ultra-Quantum Matter, which t2, b1 layers, with a form factor cos kx − cos ky and s or-
is a grant from the Simons Foundation (651440, AV), bitals hop between t2, b1 layers, with exponential decay
and a 2019 grant from the Harvard Quantum Initiative gs (i, j) = e−(lij −ld )/ρs (the s orbital distance lij and ld
Seed Funding program. the distance between top t2 and bottom b1 layers). The
Note added- During the finalization of the manuscript, second one assumes a form of
we are aware of two preprints [36, 37] which also study
the twisted cuprate bilayer and discusses the critical Ftb,2 (rij ) = a0 e−(lij −ld )/ρd c†i,t2 cj,b1 . (A3)
Josephson current. ρs,d is the spatial extent for s, d orbitals, respectively.
At θ = 45o , the first process obeys the reflection Rxy
automatically, since the form factor cos kx − cos ky takes
Appendix A: Interlayer Tunneling and Symmetries into account the d orbital symmetries. The second pro-
cess is even for lij → Rxy (lij ) while Rxy requires it be-
1. Interlayer Tunneling and Reflection ing odd in eq (A2). Hence we conclude the a0 = 0 for
θ = 45o . This is intuitive from the d orbital symmetries:
We note a reflection transform Rxy that relates twist at θ = 45o , d orbitals in bottom layer is odd under Rxy ,
angel θ → π/2−θ and imposes certain conditions on such while d orbitals in top layer is even under Rxy , so they
interlayer tunneling form. In particular for twist angle cannot mix directly.
θ = 45o , Rxy leaves the system invariant, i.e. becomes a For the first process, the tunneling form reads in real
symmetry. The reflection with respect to x = y diagonal space as
axis of the bottom layer reads 1 X
Ftb,1 (rij ) = ξbh ξbh0 gs (ri±bh,j±bh0 ),
2
(cb↑,ib , cb↓,ib )T → −Sxy (cb↑,Rxy(ib ) , cb↓,Rxy(ib ) )T , h0 =x0 ,y 0
h=x,y,b
b
(ct↑,it , ct↓,it )T → Sxy (ct↑,Rx(it̃ ) , ct↓,Rx(it̃ ) )T , ξxb = 1, ξyb = −1, gs (it2 , jb1 ) = e−(lij −ld )/ρs , (A4)
it = (it,x , it,y ), it̃ = (it̃,x , −it̃,y ), (A1) where we write the cos kx − cos ky form as neighboring
hopping with direction-dependent phase ξbh . So d elec-
√
where Sxy = i(σ x + σ y )/ 2. Rxy(x) denotes the reflec- trons hop to s orbitals in neighboring ions, and s orbitals
tion for spatial coordinates that sends Rxy : ix ↔ iy , tunneling between layers and this effectively realizes tun-
Rx : (ix → ix , iy → −iy ), respectively and the electron neling of d electrons between t2, b1 layers.
operators coordinates are written in the Cartesian co-
ordinates of the layer they are in (see fig 4), hence the
subscript it,b . The reflection is along diagonal direction 2. Relation between (θ, φ) and (π/2 − θ, π − φ)
of the bottom layer and transforms the top layer to a
different twist angle θ̃ = π/2 − θ with respect to the We note the transformation Rxy that sends the system
bottom layer, denoted by b̃. The electron coordinates in with twist angle θ to π/2−θ. For the parton Hamiltonian
top layer is written in the transformed reference frame HM F , this reflection sends
t̃ (note this is not the reflected frame, but rather a ro- ∆i,i+bx(by),b → ∆i,i+by(bx),b
tated frame, hence Rx (it̃ )), i.e. an electron (it,x , it,y )
in the original top layer is reflected to (ix̃ , −it̃,y ) in the ∆i,i+bx(by),t → ∆i,i+x̃(ỹ),t , (A5)7
bilayers tz = 0.1t. The ‘z’ dispersion is stated to be:
(a) (b) ym
yb Rxy yb
yt xt̃ yt Ez = Tz (kk , cos kz c/2)
1
[cos(kx a) − cos(ky a)]2 + ã0
yt̃ 4
xt xt C2x q
90o − θ xm Tz = ± t2bi + A2 + 2tbi A cos kz c/2
θ
xb xb
A = 4t̃z cos kx a/2 cos ky a/2 (B1)
(xt, yt) → (xt̃ , − yt̃ )
(xb, yb) → (yb, xb) (xb(t), yb(t)) → (xt(b), − yt(b))
and ã0 = 0.4 which gives some tunneling along the nodal
direction. In the main text, we have a0 = 4ã0 to account
FIG. 4: (a)Illustration of Rxy that sends θ → 90o − θ. for the factor 4 in expression for A and hence a0 = 1.6.
The straightforward way is to write coordinates in the Note there is some statement that the tunneling tz
local reference frame of each layer, i.e.xt,b , yt,b , etc. (b) which connects different bilayers has some structure cor-
C2x that exchanges t, b layers. responding to displacements (corresponding to A above).
We take the geometry as no displacement between 2 lay-
ers and find a displacement of site close to (a/2, a/2) for
and leaves homogeneous mean-field χ’s invariant. The θ ∼ 45o .
phase difference φ therefore becomes φ − π given the d- For the twisted setting, we modify the factor
wave pairing ansatz. For the special case of θ = 45o , the cos(kx a)−cos(ky a)]2 in Ez to (cxt −cyt )(cxb −cyb ) where
reflection leaves the two layers unchanged, with pairing cx(y) a) and we write the two factors in reference frame of
phase difference φ → φ − π, further related to π − φ by top, bottom layers, hence the subscript t, b. This gives
T. Hence the reflection symmetry Rxy means the mean- rise to the form factor in eq (A4) in real space. The uni-
field energy vs φ should be symmetric for φ, π − φ, which form a0 factor corresponds to the second process Ftb,2
our results for θ = 45.2o approximately obeys in fig 5. mentioned in eq (A3).
3. Other symmetries Appendix C: Details on numerical iteration
The system has an anti-unitary chiral symmetry C: We iterate for a self-consistent solution for χ, ∆, x’s,
z † i.e. substitute the order parameters by the expectation
C : cis,p → Kσss cis,p value of the corresponding operators,
C † Htbsc C = −Htbsc , (A6) X †
χij = nf (n )hψk,n |fis fis |ψk,n i,
z
where the spin-dependent factor takes care of the σss k,n
singlet pairing terms to become negative upon the trans- XX
formation. ∆ij = nf (n )ss0 hψk,n |fis fjs0 |ψk,n i,
We focus on commensurate twist, i.e. when the en- k,n ss0
†
X X
larged, exact, square unit cell contain a finite number xi = nf (n )hψk,n |(1 − fis fis )|ψk,n i (C1)
of sites, characterized by a twist vector (n, m) with the k,n s
angle θ = 2arctan(n/m) as the twist angle. In this case,
a four-fold rotation C4 and two-fold rotation while inter- where n labels the eigenstate at a certain momentum k,
changing t, b layers C2x are present, acting as and nf (n ) is the Fermi function for the eigenenergy n .
z In numerics, we switch to an equivalent model setting
C4 : cis,p → σss cC4 (i)s,p where the pairing is enforced to be real numbers, and
C2x : (cb↑,ib , cb↓,ib )T → −(ct↑,Rx(ib ) , ct↓,Rx(ib ) )T , phase difference φ enters through a phase for the inter-
φ
(ct↑,it , ct↓,it )T → −(cb↑,Rx(it ) , cb↓,Rx(it ) )T , layer tunneling Ftb → Ftb ei 2 . By a phase rotation of
φ
the spinon in bottom layer fis,b → fis,b e− 2 , one comes
Rx ((ix , iy )) = (ix , −iy ),(A7)
back to real Ftb ’s with pairing difference φ. This trans-
where C4 (i) is the C4 rotated site of the original site i. formation allows us to obtain self-consistent solutions
C2x (illustrated in fig 4(b))looks like a reflection when with a fixed φ, likely with a nonvanishing interlayer su-
restricting to the 2D cuprate planes, where Rx (i) = percurrent. We also find that if one takes a simplified
(ix , −iy ) is the transformed coordinates of C2x , written model of twisted bilayer cuprates,with only t, b layers,
in the transformed layer t,b = b, t reference frame. the results stay qualitatively invariant compared with
twisted double-bilayer (4 layers in total) settings, i.e. re-
garding time-reversal breaking. The gap obtained in 2
Appendix B: Some Band Structure Details layers is larger (roughly by a factor of 2) than that for
twisted double-bilayers. The Chern number results are
Taken from Ref [28], in Bi2212 there are two kinds different for 2, 4 layers calculation in some cases, e.g.
of hoppings, within each bilayer tbi = 0.3t and between at small twist angles with nontrivial φ, 2, 4 layers give8
are obtained in a twisted bilayer (2 layers) setting.
We focus on commensurate twist, i.e. when the en-
Twisted 4 layers Bi2212,x=0.2,t z=0.1t, Energy-phase relation
2
10 -3
larged, exact, square unit cell contain a finite number
of sites, characterized by a twist vector (n, m) with the
=0 o
1.5 angle θ = 2arctan(n/m) as the twisted angle.
1
Appendix D: Continuum model and gap
0.5
E/t
0
=45.2 o We use a linearized model near the Dirac cone of
d−wave SC and add interlayer tunnel[14] to see how
-0.5
interlayer tunneling affects the size of gap. To further
-1 simplify, we consider 2− band models in the top and
=73.8 o bottom cuprates, respectively. The Dirac Hamiltonian
-1.5
-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 for one of the Dirac fermions, for the 2 layers read (σ
/
acts in Nambu spinor space, τ acts in layer space) in the
† †
basis (ψk↑,t , ψ−k↓,t , ψk↑,b , ψ−k↓,b )T ,
HDirac (k) = (vF k⊥ + µ)σ 3 + v∆ kk σ 1 + v∆ |Q|σ 1 τ(D1)
3
,
FIG. 5: The energy phase relations for various twisting
angles θ (increasing from top to bottom), at where kk,⊥ represents momenta parallel, perpendicu-
tz = 0.1t, a0 = 0. At θ = 43.6o , 45.2o , T is broken, lar to the Fermi surface, vF,∆ are velocities for Dirac
otherwise φ = 0(π) for θ < (>)45o , respectively. fermions in the normal state, from gap function, respec-
tively. 2Q is the twist vector, 2Q ≈ θb z × K, where
K(= (±π/2, ±π/2)) is the momentum of Dirac point.
The last term accounts for the mismatch of the Dirac
points momenta in two layers due to the twist, i.e. we
C = 4, 8, respectively. The results of hall conductivity are expanding around K + Q.
The tunneling with a phase φ/2, equivalent to a pairing phase difference φ, reads in the low-energy space,
φ φ
Htunnel (k) = Tz (k)(cos σ 3 τ 1 + sin τ 2 ), (D2)
2 2
where Tz (k) = tz xFtb (k) is the effective tunneling matrix. The eigen energy is
E 2 = (vF k⊥ + µ)2 + (v∆ kk )2 + (v∆ Q)2 + (Tz (k))2
q
± 2 (vF k⊥ + µ)2 (Tz (k))2 + (v∆ Q)2 ((Tz (k) cos φ/2)2 + (v∆ kk )2 ) + (v∆ kk Tz (k) sin φ/2)2 .
The minimum of E requires dE 2 /dk⊥ = 0, dE 2 /dkk = 0. The first derivative is satisfied at vF k⊥ + µ = 0. We have
minimum of |E| at the minimum of the following expression,
q
(v∆ kk )2 + (v∆ Q)2 + (Tz (k))2 − 2 (v∆ Q)2 ((Tz (k) cos φ/2)2 + (v∆ kk )2 ) + (v∆ kk Tz (k) sin φ/2)2 + (v∆
2 Qk )2 ,
k
which is always positive for φ 6= 0, π. The minimum is reached when
(v∆ kk,m )2 ((v∆ Q)2 + (Tz (k) sin φ)2 ) = ((v∆ Q)2 + (Tz (k) sin φ)2 )2 − (Tz (k) cos φ)2 (v∆ Q)2 , (D3)
leading to kk,m close to Q given v∆ Q Tz (kmin ), with the minimal energy
(Tz (kmin ))2 sin φ
Emin ≈ p . (D4)
2 (v∆ Q)2 + [(Tz (k) sin φ/2]2
Crucially for not so large µ, Tz , kmin lies close to the original Dirac point K where Ftb,1 vanishes. Hence the gap
should be vanishingly small if the tunneling is purely Ftb,1 (k) = (cos kx,t − cos ky,t )(cos kx,b − cos ky,b ).
Another saddle point solution with nonzero vF k⊥ + µ gives a saddle point energy E = 2|v∆ Q sin φ2 |, and at not so
small twist angle, is larger than the above Emin given Tz is suppressed by a factor of x ∼ 0.1.
From this crude analysis we see how the cos form factor leads to vanishing gap. In reality, higher-order processes
that scatter between momenta k, k + Kmoire will also contribute.
Appendix E: Comparison with BCS calculation differences from the BCS calculation [13]:
results
The x factor from parton approach and cos form fac-
tor in tunneling in t-J calculation result in several key9
1
larger Jc = 7000kA/cm2 (Jc = 800kA/cm2 for the same
0.9
parameters except setting a0 = 0, tz = 0.1t instead).
0.8 tz 0.2t
0,t za 0 =0.4
0.7
optimal / 0.6
0.5
0.4
0.3
0.2
0.1
0
20 30 40 50 60 70
FIG. 6: The optimal phase difference φ as a function of
twisting angle with only uniform tunneling
tz → 0, tz a0 = 0.2t.The results agree with those in [13]
qualitatively.
Twisted bilayer, tz=0.1t
ϕ
optimal ϕ | tza0 |
π
C=4 C=2 C=0 C=2 C=4
0.16t
FIG. 8: The band dispersion for untwisted d-wave
0 45o 90o θ models with maximal gap ≈ 0.2t along the Fermi
surface for reference.
FIG. 7: A schematic phase diagram for twisted bilayer
Bi2212 calculation. The results agree qualitatively with
those of twisted double-bilayer Bi2212, except Chern The T breaking twist angle range is significantly larger
numbers are halved. as shown in fig6. This resembles [13] results with expo-
nential decaying tunneling for ∆ = 0.022t, φ = 0.5π at
θ = 43.6o .
1. The d− wave solution with φ 6= 0 survives to tz as
large as 0.3t(fig 3(d)), while in BCS calculation d + is or Appendix F: Details on Hall conductivity
other mean-field solution set in at tz ≈ 0.15t. calculations
2.The θ with spontaneous T breaking falls in a nar-
rower window than in BCS calculation,i.e. 43o < θ < Expand in the eigenbasis of the mean-field Hamilto-
47o ,since here a commensurate twist θ = 41o already nian HM F , the hall conductivity reads (η → 0)
yields φ = 0.
3.The minimal gap resulting from interlayer tunneling ∂H
hn| ∂k ∂H
|mihm| ∂k |ni
e2 X
orders of magnitudes smaller ∆ ≈ 0.0004t for θ ≈ 45o , σh (ω) = 2
(F (Em ) − F (E n )) x y
while BCS calculation gives a gap ∆ ≈ 0.02t. i~ ωΩ n,m ~ω + iη + Em − En
4. Despite the T breaking, the chern number is 0 −[kx ↔ ky ],
identically.For small twist angle θ < 40o , a uniform tun- (F1)
neling component is allowed by symmetry and we take
tz a0 = 0.16t from [28]. We found that T is preserved. If where F (E) is the Fermi function.
we add by hand a nonzero pairing phase difference φ, the A simplification gives,
system enters topological superconductivity phase wit-
nessed by chern number C = 8, 4 plotted in fig 3(d),as e2 X
predicted by [14]. σh (ω) = (F (Em ) − F (En ))
i~Ω
En >Em
We have also calculated cases with only an exponen-
∂H ∂H
tially decaying and homogeneous tunneling t-J calcula- Imhn| ∂k x
|mihm| ∂ky
|ni
tion,i.e. setting the process with form factor cos kx − , (F2)
(~ω + iη)2 − (Em − En )2
cos ky, Ftb,1 = 0 identically. Operationally it is achieved
in HM F by taking tz → 0, tz a0 = 0.2t. Then we recov- in practice to regulate the singularity when En −Em = ω,
ered results in ref[13]: Gap ∆ ≈ 0.01t for tz a0 = 0.2t, θ = we take the small parameter η = 0.004t as a damping
43.6o (gap as θ varies is plotted in fig 2(a)), with a much factor.10
|M z,k | at =45 o ,x=0.2,tz =0.1t
10 -3
7
4
6
3
2 5
1
|M z,k |(a.u)
4
ky,ma
0
3
-1
-2 2
-3
1
-4
-4 -2 0 2 4
kx,m a
FIG. 9: Magnetization density Mz (along c axis) from
interlayer currents at θ = 45.2o . Dashed lines are guide
for the eyes of the two cuprate layers Brillouin zones.
The magnetization respects C2x and (approximate)
Rxy . The scattering between the bright yellow spots
located around e.g. (2.4, 2.4), (3.1, 0) etc will give peaks
along reciprocal vector direction of the two cuprate
layers,i.e. parallel to one of the dashed lines. This is
illustrated by the white lines connection the yellow
spots. In-plane Magnetization is an order smaller than
|Mz |.
FIG. 10: Lowest conduction band at θ = 41o , 43.6o for the left and right panel, respectively, with tz = 0.1t, a0 = 0.
Although no flat band in the lowest energy, we find a relatively flat band top at E ≈ 0.05, 0.06t, respectively, which
is signaled by a peak in optical conductivity measurement around ω = 400, 500cm−1 .
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