Pairing of holes by conning strings in antiferromagnets F. Grusdt1 2E. Demler3and A. Bohrdt4 5 1Department of Physics and Arnold Sommerfeld Center for Theoretical Physics ASC

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Pairing of holes by conﬁning strings in antiferromagnets

F. Grusdt,1, 2, ∗E. Demler,3and A. Bohrdt4, 5

1Department of Physics and Arnold Sommerfeld Center for Theoretical Physics (ASC),

Ludwig-Maximilians-Universit¨at M¨unchen, Theresienstr. 37, M¨unchen D-80333, Germany

2Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 M¨unchen, Germany

3Institut f¨ur Theoretische Physik, ETH Zurich, 8093 Zurich, Switzerland

4ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA

5Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

(Dated: October 6, 2022)

In strongly correlated quantum materials, the behavior of charge carriers is dominated by strong

electron-electron interactions. These can lead to insulating states with spin order, and upon doping

to competing ordered states including unconventional superconductivity. The underlying pairing

mechanism remains poorly understood however, even in strongly simpliﬁed theoretical models. Re-

cent advances in quantum simulation allow to study pairing in paradigmatic settings, e.g. in the

t−Jand t−JzHamiltonians. Even there, the most basic properties of paired states of only two

dopants, such as their dispersion relation and excitation spectra, remain poorly studied in many

cases. Here we provide new analytical insights into a possible string-based pairing mechanism of

mobile holes in an antiferromagnet. We analyze an eﬀective model of partons connected by a con-

ﬁning string and calculate the spectral properties of bound states. Our model is equally relevant for

understanding Hubbard-Mott excitons consisting of a bound doublon-hole pair or conﬁned states

of dynamical matter in lattice gauge theories, which motivates our study of diﬀerent parton statis-

tics. Although an accurate semi-analytic estimation of binding energies is challenging, our theory

provides a detailed understanding of the internal structure of pairs. For example, in a range of

settings we predict heavy states of immobile pairs with ﬂat-band dispersions – including for the

lowest-energy d-wave pair of fermions. Our ﬁndings shed new light on the long-standing question

about the origin of pairing and competing orders in high-temperature superconductors.

I. INTRODUCTION

The history of physics has repeatedly taught us that

nature tends to realize richer structures than one might

ﬁrst suggest. The most important example, by far, is

constituted by the theory of atoms which has evolved

from Thomson’s featureless plum pudding model to our

current picture of precisely quantized energy shells and

a nucleus with structures down to the level of individ-

ual quarks. The only way to reveal such structures is

to perform increasingly more precise measurements, or,

with the beneﬁt of hindsight and a microscopic Hamilto-

nian at hand, perform increasingly more precise numeri-

cal simulations. Here we address the question how much,

and which, structure paired charge carriers in correlated

quantum matter may have.

To date, important aspects of strongly correlated elec-

trons remain poorly understood. Part of the reason is

that the nature and microscopic structure of the emer-

gent charge carriers is not fully understood. Among the

most famous puzzles is the origin of pairing in high-

temperature superconductors [1,2], but similar questions

arise in heavy-fermion compounds [3], organic supercon-

ductors [4] and most recently twisted bilayer graphene [5].

However the problems are not limited to paired states of

matter: the nature of charge carriers in the exotic nor-

mal phases of these strongly correlated systems is likewise

∗Corresponding author email: fabian.grusdt@lmu.de

debated and subject of active research.

Remarkably, the prevailing picture of quantum matter

is one with more-or-less featureless charge carriers, with

little or no rigid internal structure taken into account in

calculations. Some theoretical approaches assume frac-

tionalization of quantum numbers, which leads to rich

and interesting physics, but short-range spatial ﬂuctua-

tions are rarely considered in detail. Among the reasons

for this restriction is the diﬃculty to directly experimen-

tally visualize spatial structures of quickly ﬂuctuating

charges. While ultracold atoms in optical lattices [6,7]

have taken very promising steps in this direction [8,9],

they too have not managed to fully visualize the internal

structure of doped charge carriers yet. Another reason

may be the lasting inﬂuence of Anderson’s RVB theory of

high-Tcsuperconductivity [10], which assumes point-like

charge carriers moving in a surrounding spin-liquid – in

some sense the antithesis of any theory assuming spatial

structures of charge carriers.

On the other hand, it is not for a lack of ideas which

kinds of internal structures could emerge: Even before

the discovery of high-Tcsuperconductors, Bulaevksii et

al. proposed the existence of string-like states with in-

ternal vibrational excitations [11], a view taken up by

Brinkman and Rice to understand dynamical properties

of charge carriers [12]; in 1988, Trugman applied this idea

to pairs of holes [13] and in the same year Shraiman and

Siggia analyzed two-hole string states in greater detail

[14]. Their conclusions at the time were mixed: while

they found mechanisms supporting pairing, they also

identiﬁed unfavorable eﬀects such as frustration of the

arXiv:2210.02321v1 [cond-mat.str-el] 5 Oct 2022

pair-kinetic energy due to the underlying Fermi-statistics

of the holes. Today these works stand out as being among

the few and ﬁrst attempting a description starting from

the strong coupling antiferromagnetic (AFM) Mott limit

(large Hubbard-U). But it appears that the community

then focused more on approaches inspired by the weak-

coupling (small Hubbard-U) limit of the theoretical mod-

els [2], which more naturally led to the magnon-exchange

picture [15]. There, magnetic ﬂuctuations provide the

glue between point-like charge carriers, and the theoreti-

cal framework shares more similarities with the successful

BCS theory of conventional superconductors. As almost

all ideas in the ﬁeld, this picture has been debated [16].

Nevertheless, over time the idea that charge carriers

have a pronounced spatial structure was reclaimed sev-

eral times. In 1996, Laughlin and co-workers proposed a

phenomenological parton theory of doped holes, includ-

ing a conﬁning linear string tension [17,18]; diﬀerent

kinds of spatial strings, termed phase-strings, were intro-

duced in 1996 by Weng and co-workers [19,20], and their

eﬀect on pairing was recently analyzed [21]; signatures for

the more traditional Sz-string ﬂuctuations were reported

in large-scale DMRG simulations by White and Aﬄeck in

2001 [22]; in 2007 Manousakis proposed a string-based in-

terpretation of one-hole ARPES spectra and in the same

work envisioned a pairing mechanism of holes constantly

exchanging ﬂuctuating strings [23]; in 2013 exact numer-

ical simulations by Vidmar et al. in truncated bases,

closely related to the string picture, have also revealed

signatures for pairs with a rich spatial structure [24]; al-

ready in 2000 and 2001 large-scale Monte Carlo simula-

tions by Brunner et al. [25] and Mishchenko et al. [26]

have revealed long-lived vibrational excitations of indi-

vidual doped holes; and in the past few years, the present

authors have added new evidence for the existence of

long-lived rotational and vibrational string states of in-

dividual charge carriers [27–29]; Vibrational peaks have

also long been known to exist and recently further con-

ﬁrmed in linear-spin wave models of doped holes [30–33].

Finally, Hubbard-Mott excitons formed by a bound pair

of a doublon and a hole [34] have been proposed to have

a rich internal structure [35?].

In this article, we revisit the idea that mobile dopants,

the charge carriers in doped AFM Mott insulators, can

form bound states with a rich spatial structure. Speciﬁ-

cally, we derive a semi-analytical theory of pairs of holes

connected by a conﬁning string which ﬂuctuates only

through the motion of the charges at its ends. Our ap-

proach is very similar in spirit to the much earlier work

by Shraiman and Siggia mentioned above [14]; in fact we

conﬁrm several of their predictions and discuss them in

the context of three decades worth of new results includ-

ing from far advanced numerics. This includes one of

their most exciting – though often overlooked – predic-

tion that two fermionic holes can form inﬁnitely heavy

pairs with a ﬂat-band dispersion at very low energies.

In fact, we ﬁnd that that these ﬂat bands have d-wave

character. This result sheds new light on the wealth of

competing ordered states observed in cuprates and nu-

merically found in the closely related t−Jand Fermi-

Hubbard models [36–38].

Back-to-back with this article, we are publishing a sep-

arate work focusing on a numerical analysis of rotational

two-hole spectra in the t−Jzand t−Jmodels [39].

There we achieve full momentum resolution on extended

four-leg cylinders, and compare our numerical results to

the semi-analytic calculations performed in the present

article. The focus of the present article is on the semi-

analytical method itself, including its formal derivation.

Moreover, the calculations we perform here are applica-

ble in a larger class of models: as explained below, our

main assumption is that two partons on a square lattice

are connected by a rigid string Σ which creates a mem-

ory of the parton’s motion in the wavefunction. In the

case of mobile holes doped into an AFM Mott insula-

tor, the string directly encodes the prevalent spin-charge

correlations. But similar situations can be found in lat-

tice gauge theories in a strongly conﬁning regime where

the dynamics of the corresponding electric ﬁeld strings is

dominated by fast charge ﬂuctuations.

The goal of our present work is primarily to understand

the properties of pairs of mobile dopants, i.e. their spatial

structure and energy spectrum, including their rotational

quantum numbers and eﬀective mass. Previously much

emphasis has been on the question whether the dopants

pair up; i.e. about the magnitude and sign of the binding

energy

Ebdg = 2(E1−E0)−(E2−E0) (1)

where Enis the ground state energy in the presence of

ndopants. While we also view this as an important is-

sue, we argue that it is often not a well-suited question

for a semi-analytical approach, since it strongly depends

on details; e.g. addressing it requires precise knowledge

of the one-hole energy. In this article we take the view

that the structure of the paired state can be very diﬀer-

ent from the structure of one-dopant states. Our goal

is to understand the former, and we leave the question

of how and when excited (or ground) states of pairs can

decay into individual single-dopant states to future anal-

ysis (only a brief discussion within our model will be pro-

vided). Moreover, we note that the question of pairing is

not identical to a question about the existence supercon-

ductivity: instead of condensing into a superconductor,

pairs of holes may also crystalize and form a pair-density

wave at ﬁnite doping [40].

Nevertheless, we will address the origins of pairing

within our semi-analytical framework. To this end we

identify competing eﬀects which tend to increase or de-

crease the binding energy. Our results shed new light

on earlier predictions [13,14]: (i) fermionic statistics of

the dopants frustrates the pair’s kinetic energy, which

is unfavorable for pairing; (ii) the most detrimental ef-

fect for pairing comes from the hard-core property of

two dopants, which cannot occupy the same site; More-

over, we reveal (iii) a geometric spinon-chargon repul-

sion in dimensions d≥2 which enhances the one-dopant

energy E1[41] and thus favors pairing. In addition to

all of these, further contributions stemming from low-

energy spin-ﬂuctuations in the background are expected,

whose quantitative eﬀects are more challenging to predict

and will thus be left to future work to explore. Finally,

we note that in speciﬁcally tailored settings our simpli-

fying assumptions become quantitatively accurate: In

Ref. [42] we proposed a mixed-dimensional bilayer model

with strong rung singlets where we demonstrated strong

string-based pairing of doped holes with a binding en-

ergy scaling as Ebdg 't1/3J2/3, when the hole tunneling

tJexceeds the rung super-exchange J. Recently, in a

closely related mixed-dimensional two-leg ladder, ultra-

cold fermions directly observed strong hole binding [43],

realizing a decades old toy model of pairing [44,45]. The

internal structure of hole pairs in these models can also

be described by the theoretical model we develop here.

This article is organized as follows. In Sec. II we brieﬂy

discuss the microscopic models motivating our analysis.

Sec. III constitutes the main body of our article: there

we develop the eﬀective string model to describe bound

states of two mobile partons connected by a strongly con-

ﬁning string. Our focus is on two holes, but as we discuss

in detail in Appendix A, the formalism we develop is also

applicable to pairs of more general partons, in particular

spinon-chargon pairs [29]. In the second main part of the

article, Sec. IV, we present results from our analytical

formalism and discuss possible implications for general

pairing mechanisms in doped AFMs. We close with a

summary and outlook in Sec. V.

II. MICROSCOPIC MODELS

In this article we introduce and solve an eﬀective the-

ory describing bound states of holes. As described in de-

tail in Sec. III, we will make approximations on the level

of both the Hilbert space and the Hamiltonian. Nev-

ertheless, our starting point are microscopic models of

doped AFM Mott insulators to which, we argue, our re-

sults apply within some approximations. Critical minds

should simply view these models as motivating our eﬀec-

tive theory, although our numerical analysis in Ref. [39]

indicates remarkable similarities with the semi-analytical

predictions derived here.

The system most closely related to our eﬀective theory

is constituted by the 2D t−Jzmodel on a square lattice,

with Hamiltonian

Ht−Jz=−tˆ

hi,jiX

σˆc†

i,σ ˆcj,σ + h.c.ˆ

P+

+JzX

hi,ji

iˆ

j−Jz

hi,ji

ˆniˆnj.(2)

Here ˆcj,σ deﬁnes the underlying particles, with spin-index

σ=↑,↓and density ˆnj=Pσˆc†

j,σ ˆcj,σ; the tunneling

FIG. 1. We work in an eﬀective Hilbertspace consisting of

pairs of dopants (red and blue) connected by a string Σ on a

square lattice. Every state |x1,Σiavoiding double occupan-

cies of any site with two dopants is associated with a unique

state |Ψ(x1,Σ)iin the microscopic model. (a) A typical ex-

ample with string length `Σ= 3. (b) Rare loop conﬁgurations

leading to double-occupancies of dopants have no correspon-

dence in the microscopic model.

amplitude tdescribes hopping of these particles, and ˆ

projects on a sector with a given total number of dou-

blons, holes and singly-occupied sites. Jz>0 is an AFM

Ising coupling between the spins ˆ

j=Pσ(−1)σˆc†

j,σ ˆcj,σ

which we assume to be antiferromagnetic throughout.

In principle the Hamiltonian in Eq. (2) can be deﬁned

with particles ˆcj,σ of any exchange statistics, bosonic or

fermionic. While this makes no diﬀerence for zero and

one mobile dopant, the statistics plays an important role

if two dopants of the same type are considered. Since the

fermionic case is more closely related to the celebrated

Fermi-Hubbard model with its AFM ground state at half

ﬁlling, it usually takes center stage. However, we ﬁnd it

instructive to consider the bosonic version – without any

direct connection to a Hubbard Hamiltonian – as well.

In fact, quantum simulators using ultracold atoms have

been proposed which allow the realization of both cases

in experiments [46–49].

Likewise, we can consider the AFM t−Jmodel in a

2D square lattice with arbitrary underlying statistics. Its

Hamiltonian is given by

Ht−J=−tˆ

hi,jiX

σˆc†

i,σ ˆcj,σ + h.c.ˆ

P+

+JX

hi,ji

Si·ˆ

Sj−J

hi,ji

ˆniˆnj.(3)

Now J > 0 is the strength of antiferromagnetic SU(2)-

invariant Heisenberg interactions between spins ˆ

Sj=

Pα,β ˆc†

j,α 1

2σαβ ˆcj,β.

In both the t−Jand t−Jzmodels, diﬀerent types

of dopants can be considered. The most often studied

case, which is also our primary focus, constitutes pairs

of two indistinguishable holes; owing to the particle-hole

symmetry of the models, one can interchangeably con-

sider two indistinguishable doublons however. A second,

closely related case corresponds to Hubbard-Mott exci-

tons [35], where pairs of doublons and holes can form. In

this case, exchange statistics again plays no role on the

level of the t−J(z)model, since the dopants deﬁne distin-

guishable conserved particles. Experimentally, all these

situations can be addressed by state-of-the-art ultracold

atom experiments [6,7].

III. EFFECTIVE STRING MODEL

In this section, we introduce an eﬀective string model

to describe tightly bound pairs of two holes. Our ap-

proach is motivated by considering two dopants moving

in a N´eel state, modeled by the 2D t−Jzor t−Jmodel.

To make analytical progress, we perform approximations

on the eﬀective Hilbert space (see Fig. 1) as well as the

eﬀective Hamiltonian. While we expect our approximate

description to be most accurate for the t−Jzmodel, it

should also capture the essential physics of related mod-

els, such as the t−Jmodel, as long as charge ﬂuctuations

dominate, tJ, and they feature strong local AFM

correlations at zero doping. In these cases, the geometric

string approach, developed originally for single dopants,

can be applied [8,11,12,14,27].

In the subsequent sections, we will always talk about

paired holes and consider diﬀerent kinds of statistics.

However, as discussed in Sec. II, these can interchange-

ably be considered as general types of dopants, or even

more generally as two partons.

A. String Hilbert space and eﬀective Hamiltonian

In a perfect N´eel state the motion of a hole leaves be-

hind a string Σ of displaced spins, which allows to as-

sociate distinct hole trajectories with orthogonal states

|Ψ(Σ)iin the quantum many-body system (Fig. 1); here

strings denote hole trajectories with all self-retracing

components removed. Exceptions, where two diﬀerent

strings Σ16= Σ2correspond to identical many-body

states |Ψ(Σ1)i=|Ψ(Σ2)i, are associated with so-called

Trugman loops [13]. Since the number of Trugman loops

is small compared to the exponentially growing number

of string states (Tab. I), their eﬀect is generally found

to be small [13,27]. Although in exceptional cases the

small eﬀect of loops can still dominate, e.g. for the very

narrow center-of-mass dispersion of a single hole in the

2D t−Jzmodel [13,50], we neglect such loops in the

following and study only the dominant eﬀects of string

formation. Loop eﬀects can be re-introduced perturba-

tively in the end [27].

1. Hilbert space

While the set of string states {|Ψ(Σ)i} deﬁnes an over-

complete basis, we approximate our Hilbertspace and for-

mally deﬁne a set of two-hole string states:

|x1,Σi,x1∈Z2,Σ∈BL.(4)

Here x1denotes the location of the ﬁrst hole in the 2D

square lattice, and Σ is the string which connects x1to

`Σno. of states d(`Σ) double-occupancies Trugman loops

1 4 0 0

2 12 0 0

3 36 0 0

4 108 7.4% 0

5 324 0 0

6 972 2.5% 0

7 2,916 0 0.55%

8 8,748 0.91% 1.3%

TABLE I. Imperfections of the string model. The string

basis with two distinguishable holes includes states |x1,Σi

corresponding to unphysical double-occupancies of holes in

the associated microscopic states |Ψ(x1,Σ)i. Their relative

fraction of all string states N(`Σ) of a given length `Σis indi-

cated. The relative number of Trugman loop conﬁgurations

[13] is also shown.

x2=x1+RΣat its opposite end (Fig. 1). The strings Σ

can be represented by the sites of a Bethe lattice (BL),

or Cayley tree, with coordination number z= 4, see

Fig. 2(a). Similar to the construction of the celebrated

Rokhsar-Kivelson quantum dimer model [51], we postu-

late that the new basis states are orthonormal,

hx0

1,Σ0|x1,Σi=δΣ,Σ0δx1,x0

1.(5)

Every new basis state is associated with a unique mi-

croscopic two-hole state |Ψ(x1,Σ)iin the original t−Jz

or t−Jmodel. Some states in the new model describe un-

physical double occupancies with holes: in this case the

associated state becomes |Ψ(x1,Σ)i= 0 (Fig. 1). The

fraction of such strings is relatively small, however, and

decreases with increasing string lengths (Tab. I).

So far we work in ﬁrst quantization and assign separate

labels to the two holes. Later (see III D) we will gener-

alize our approach to situations with indistinguishable

holes, with bosonic or fermionic statistics.

2. Eﬀective Hamiltonian

Next we deﬁne the eﬀective Hamiltonian ˆ

Heﬀ in the

approximated Hilbert space of the string model. To this

end we require that the matrix elements satisfy:

hx0

1,Σ0|ˆ

Heﬀ |x1,Σi=hΨ(x0

1,Σ0)|ˆ

H|Ψ(x1,Σ)i,(6)

where ˆ

Hcorresponds to the respective microscopic model

Hamiltonian (t−Jz,t−J,...).

As a result of the microscopic nearest-neighbor (NN)

hopping t2of hole 2 at site x2on the lattice, we obtain

NN hopping on the Bethe lattice (Fig. 2(b)):

Heﬀ

t,2=t2X

x1|x1ihx1| ⊗ X

hΣ0,Σi|Σ0ihΣ|+ h.c. (7)

The NN hopping t1of hole 1 gives rise to a correlated

NN tunneling of x1and a simultaneous change of the

FIG. 2. The hopping part of the eﬀective Hamiltonian ˆ

Heﬀ

describes NN tunneling of holes 1 (t1) and 2 (t2) on the square

lattice. The string Σ from hole 1 to 2 changes accordingly. We

illustrate a typical initial state (a), which is coupled to neigh-

boring string states on the Bethe lattice by t2(b). The cou-

pling t1creates a string state Σ(1) which is a further neighbor

of Σ on the Bethe lattice, either by re-tracing (b) or extending

(d) the ﬁrst string segment.

ﬁrst string segment of Σ →Σ(1)(x0

1,x1,Σ):

Heﬀ

t,1=t1X

hx0

1,x1iX

Σ|x0

1ihx1|⊗|Σ(1)(x0

1,x1,Σ)ihΣ|+ h.c..

(8)

If the ﬁrst string segment in Σ points along x0

1−x1,

it is removed to obtain Σ(1) (Fig. 2(c)); otherwise, the

string is extended by adding a new string segment point-

ing along x0

1−x1at the beginning of Σ to obtain Σ(1)

(Fig. 2(d)).

Similarly, we obtain the potential terms ˆ

Heﬀ

Jin the ef-

fective Hamiltonian. They do not change the positions

x1,2of the holes, and we neglect oﬀ-diagonal matrix el-

ements (6) for which Σ06= Σ. Hence, formally we can

write:

Heﬀ

J=X

x1X

VΣ|x1,Σihx1,Σ|,(9)

where VΣis a function of Σ on the Bethe lattice only.

Note that Trugman loops correspond to local minima in

the Bethe lattice potential VΣ, which allows for a system-

atic tight-binding treatment of Trugman loops within our

model so far, even when t1,2Jor Jz[27].

The complete eﬀective Hamiltonian we consider is

Heﬀ =ˆ

Heﬀ

t,1+ˆ

Heﬀ

t,2+ˆ

Heﬀ

J.(10)

3. Linear string approximation

Since the dimension of the string Hilbert space grows

exponentially with the maximum string length `max,

dtot(`max) =

`max

`Σ=1

4×3`Σ−1

| {z }

=d(`Σ)

,(11)

further approximations are required to make analytical

progress. For a general string potential VΣ, all states

in the string Hilbert space are coupled to each other by

the tunneling terms. Next, by simplifying the poten-

tial VΣ, many symmetry sectors emerge which are only

weakly coupled and can be described by a simpler eﬀec-

tive Hamiltonian that will be derived.

Since we are mostly interested in the regime tJ,

we expect that inhomogeneities of VΣon the scale of J

play a sub-dominant role. Such ﬂuctuations of VΣfrom

string to string (or site to site on the Bethe lattice) re-

sult from string-string interactions [52], and appear like

a weak disorder potential. We can include their eﬀect

on a mean-ﬁeld level by averaging the potential over all

strings of a given length:

V(`) = 1

d(`)X

Σ:`Σ=`

VΣ.(12)

The resulting problem is highly symmetric since all

branches on the Bethe lattice corresponding to the same

string length are equivalent.

As a further approximation, we can estimate the

string-length potential V(`)≈VLST(`) by considering

only straight strings in Eq. (12). Since string-string in-

teractions are always attractive, this linear string theory

(LST) estimate also deﬁnes an upper bound for the av-

eraged potential:

V(`)≤VLST(`) = dE

d` ×(`Σ−1) + g(0)

cc δ`Σ,1+µcc.(13)

Here dE/d` denotes the linear string tension, g(0)

cc is a

nearest-neighbor hole-hole interaction, and µcc an over-

all energy shift. The overall energy of the two holes is

measured relative to the undoped parent antiferromag-

netic state.

In the case of a microscopic t−Jzmodel, we obtain:

d` =Jz, g(0)

cc =−Jz

2, µcc = 4Jz.(14)

More generally, we can derive the LST potential by ap-

plying the frozen spin approximation and expressing the

potential in terms of local spin-spin correlations of the

undoped parent antiferromagnet, see Refs. [27,41]. For

a doped J1−J2spin model on a square lattice, with NN

hopping of the holes, this yields:

d` = 2J1(C2−C1)+2J2(C1+C4−2C2),(15)

g(0)

cc =J1C1−J

4,(16)

µcc =−8J1C1+J2C2−J

4.(17)

Here J1and J2denote the NN and diagonal NNN Heisen-

berg couplings on the square lattice; Jdenotes the

strength of the local attraction −J/4ˆniˆnjin Eq. (3) and

is treated as an independent parameter here. The cor-

relators are C1=hˆ

Siˆ

Si+exi(NN), C2=hˆ

Siˆ

Si+ex+eyi

(NNN), C3=hˆ

Siˆ

Si+2exiand C4=hˆ

Siˆ

Si+2ex+eyi; they

depend on the ratio J1/J2[53].

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PairingofholesbyconningstringsinantiferromagnetsF.Grusdt,1,2,E.Demler,3andA.Bohrdt4,51DepartmentofPhysicsandArnoldSommerfeldCenterforTheoreticalPhysics(ASC),Ludwig-Maximilians-UniversitatMunchen,Theresienstr.37,MunchenD-80333,Germany2MunichCenterforQuantumScienceandTechnology(MCQST),Schellingst...

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Pairing of holes by conning strings in antiferromagnets F. Grusdt1 2E. Demler3and A. Bohrdt4 5 1Department of Physics and Arnold Sommerfeld Center for Theoretical Physics ASC

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