Dichotomy of heavy and light pairs of holes in the tJmodel A. Bohrdt1 2 3 4 E. Demler5and F. Grusdt6 7 1ITAMP Harvard-Smithsonian Center for Astrophysics Cambridge MA 02138 USA

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Dichotomy of heavy and light pairs of holes in the t−Jmodel

A. Bohrdt,1, 2, 3, 4, ∗E. Demler,5and F. Grusdt6, 7

1ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA

2Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

3Institut f¨ur Theoretische Physik, Universit¨at Regensburg, D-93035 Regensburg, Germany

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

5Institut f¨ur Theoretische Physik, ETH Zurich, 8093 Zurich, Switzerland

6Department of Physics and Arnold Sommerfeld Center for Theoretical Physics (ASC),

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

7Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 M¨unchen, Germany

(Dated: July 24, 2023)

A key step in unraveling the mysteries of materials exhibiting unconventional superconductivity

is to understand the underlying pairing mechanism. While it is widely agreed upon that the pairing

glue in many of these systems originates from antiferromagnetic spin correlations [1–3], a microscopic

description of pairs of charge carriers remains lacking. Here we use state-of-the art numerical

methods to probe the internal structure and dynamical properties of pairs of charge carriers in

quantum antiferromagnets in four-legged cylinders. Exploiting the full momentum resolution in our

simulations, we are able to distinguish two qualitatively diﬀerent types of bound states: a highly

mobile, meta-stable pair, which has a dispersion proportional to the hole hopping t, and a heavy

pair, which can only move due to spin exchange processes and turns into a ﬂat band in the Ising

limit of the model. Understanding the pairing mechanism can on the one hand pave the way to

boosting binding energies in related models [4], and on the other hand enable insights into the

intricate competition of various phases of matter in strongly correlated electron systems [5,6].

Introduction.– Following the discovery of high tem-

perature superconductivity in the cuprates, understand-

ing the mechanism by which pairs of charge carriers can

form in a system with repulsive interactions has been a

key question in the ﬁeld. Motivated by experimental re-

sults on the cuprate materials, a lot of theoretical and

numerical work has focused on identifying the potential

pairing symmetry [7,8] as well as the binding energies

in these microscopic models [9,10]. Despite a vast re-

search eﬀort over several decades, the existence of a su-

perconducting phase in the simplest model describing in-

teracting electrons, the Fermi-Hubbard model, remains

debated [6]. Competing orders, such as charge density

waves and stripes, contribute to the diﬃculty in realizing

as well as understanding superconductivity [5]. In order

to unravel the competition between diﬀerent orders, and

thus the conditions for the existence of a superconduct-

ing phase, it is essential to gain a deeper understanding

of the nature of individual pairs of charge carriers. The

existence of pairs close to half-ﬁlling does not imply that

for a ﬁnite density of holes, the system necessarily realizes

ad-wave paired state. Instead, a ﬁnite number of charge

carriers can for example self-organize into a charge or pair

density wave state [11]. However, understanding whether

and how pairs form in the two-hole problem is crucial

to the subsequent understanding of self-organization of

many holes.

Here we approach the question of the underlying bind-

ing mechanism from a new perspective: through novel

∗Corresponding author email: annabelle.bohrdt@physik.uni-

regensburg.de

spectroscopic tools, we search for bound states of charge

carriers in a quantum antiferromagnet and directly probe

their internal structure. In particular, we numerically

simulate rotational two-hole spectra, where diﬀerent an-

gular momenta can be imparted on the system, using

time-dependent matrix product states. Crucially, these

rotational spectra go beyond the standard pairing corre-

lations through the momentum resolution they provide.

The momentum dependence of the peaks in the spectral

function enables direct insights into the eﬀective mass

of the pairs, which is an essential property for under-

standing their ability to condense at ﬁnite doping and

temperature.

We study pairing between two individual holes doped

into the two-dimensional t−Jmodel, which corresponds

to the enigmatic Fermi-Hubbard model to second order in

t/U (up to next-nearest neighbor hopping terms, where

Uis the on-site interaction) and describes electrons in

cuprates [12]:

Ht−J=−tˆ

⟨i,j⟩X

σˆc†

i,σ ˆcj,σ + h.c.ˆ

P+

+JX

⟨i,j⟩

Si·ˆ

Sj−J

⟨i,j⟩

ˆniˆnj,(1)

where ˆ

Pprojects to the subspace with maximum sin-

gle occupancy per site; ˆ

Sjand ˆnjdenote the on-site

spin and density operators, respectively. In our numer-

ical simulations, we consider a 40 site long, four-legged

cylinder, which is suﬃciently long to ensure that the two-

hole wavefront in the time-evolution we consider below

does not reach the edges of the system. This also means

that the thermodynamic limit is essentially reached in

arXiv:2210.02322v2 [cond-mat.str-el] 21 Jul 2023

the long direction, and our resulting spectra correspond

to predictions at zero doping.

In order to probe a possible bound state of two charges,

we consider an extension of conventional angle-resolved

photoemission spectroscopy (ARPES). In particular, we

excite the initially undoped antiferromagnet by creating

not one, but two charges while simultaneously imparting

angular momentum on the system. The resulting spec-

tra thus directly contain information about the existence

of possible bound states, their ground state energy, as

well as their dispersion relation. In our numerical matrix

product state calculations, we ﬁnd well-deﬁned peaks in

the rotational spectral function for all angular momenta,

for spin singlet as well as triplet pairs, and throughout

an extended frequency range.

In order to gain a deeper understanding of the rota-

tional two-hole spectra, we also consider the conceptually

simpler t−Jzmodel, where the SU (2) invariant spin in-

teractions are replaced by Ising type interactions. With-

out additional spin dynamics, a direct comparison of our

numerical results to an eﬀective theory describing pairs

of charge carriers bound by strings is possible, yielding

excellent agreement in terms of the existence as well as

the dispersion of the various bound states we observe. In

particular, we discover a strongly dispersive low-energy

peak, with a dispersion scaling with the hole hopping t,

as well as completely ﬂat bands at competitive energies.

We attribute the ﬂat bands to destructive interference of

pairs with d-wave symmetry [13].

Upon introducing spin dynamics, the ﬂat bands de-

velop into weakly dispersive bands, whereas the t-

dependent feature remains largely unchanged. We thus

discover two qualitatively diﬀerent kinds of bound states:

highly dispersive peaks, including a high energy feature

with strong spectral weight in the s-wave spectra; and

a weakly dispersive band, which has a high amount of

spectral weight in the d-wave spectra. The dispersion

of the latter is determined by the spin coupling J. The

emergence of a slow time-scale set by Jis intuitive and

well-known in the case of a single hole [14], which forms

a spinon-chargon bound state and can thus only move as

fast as the spin excitation [15]. In contrast, it is surpris-

ing to ﬁnd a coherent bound state peak of two holes in

the spectrum with a dispersion ∝textending over a wide

range of energies without decaying into incoherent pairs

of individual holes.

The remainder of this paper is organized as follows.

We start by introducing the rotational two-hole spectra.

We then discuss results for the t−Jzmodel, where the

SU(2) invariant spin interactions are replaced by Ising-

type interactions. We discuss the features found in the

numerically obtained spectra in detail and compare to a

semi-analytical theoretical description of pairs of charge

carriers [1]. Finally, we consider the full t−Jmodel.

Rotational Spectra.– In order to probe the internal

structure of pairs of charge carriers, we study rotational

spectra. We deﬁne an operator ˆ

∆m4(j, σ, σ′) that cre-

ates a pair of holes on the bonds adjacent to site jwith

°10

!/J

°10

!/J

(0,k

y)(º/2,k

y)(º,k

momentum k

°10

!/J

0.00

0.05

0.10

0.15

0.20

0.25

0.30

°10

°5

!/J

°10

°5

!/J

(0,k

y)(º/2,k

y)(º,k

momentum k

°10

°5

!/J

0.00

0.05

0.10

0.15

0.20

0.25

ky= 0

ky=π/2

ky=π

(a)

(b)

°10

!/J

°10

!/J

(0,k

y)(º/2,k

y)(º,k

momentum k

°10

!/J

0.00

0.05

0.10

0.15

0.20

0.25

0.30

°10

!/J

°10

!/J

(0,k

y)(º/2,k

y)(º,k

momentum k

°10

!/J

0.00

0.05

0.10

0.15

0.20

0.25

0.30

(a)

(b)

, -wave

ky=π

, -wave

ky= 0

FIG. 1. Rotational spectroscopy of two holes in a

singlet state in the t−Jmodel with t/J = 3, on a 40 ×4

cylinder, based on a time evolution up to Tmax/J = 3 and

bond dimension χ= 1200. Energies are measured relative

to the undoped parent antiferromagnet. (a) Sketch of the

response probed by the rotational spectrum. (b) The upper

(lower) plot corresponds to ky= 0(π) and a d-wave (s-wave)

excitation. Data is shown as a function of momentum kxand

frequency ω/J. Gray dashed lines correspond to a cosine dis-

persion −2Jα cos(kx) + bJ, black line corresponds to a cosine

dispersion −2tα cos(kx) + bt, where α= 0.33 in both cases,

bJ=−11J, and bt=−9J.

discrete C4angular momentum m4= 0,1,2,3 as

∆m4(j, σ, σ′) = X

i:⟨i,j⟩

eim4φi−jˆci,σ′ˆcj,σ ,(2)

with φr= arg(r) the polar angle of r; see Fig. 1(a) for

an illustration. In order to annihilate a spin-singlet, we

deﬁne the singlet pair operator (and similar for triplets)

∆(s)

m4(j) = ˆ

∆m4(j,↑,↓)−ˆ

∆m4(j,↓,↑).(3)

The simplest term creating a spin-singlet excitation

with discrete angular momentum m4, charge two, and

total momentum kis directly given by the spatial Fourier

transform of the singlet pair operator as

∆(s)

m4(k) = X

e−ik·j

√V

∆(s)

m4(j) (4)

with volume V. The discrete angular momentum m4

is a good quantum number at C4invariant momenta

k= (0,0),(π, π) only. Based on this operator, we now

consider the rotational Green’s function

G(m4)

rot (k, t) = θ(t)⟨Ψ0|ˆ

∆(s)†

m4(k, t)ˆ

∆(s)

m4(k,0)|Ψ0⟩,(5)

which we calculate using time-dependent matrix product

states [11,17,19]. The corresponding two-hole rotational

spectrum, −π−1ImG(m4)

rot (k, ω), in Lehmann representa-

tion is

A(m4)

rot (k, ω) = X

δω−En+E0

0|⟨Ψn|ˆ

∆(s)

m4(k)|Ψ0

0⟩|2,

(6)

where |Ψ0

0⟩(E0

0) is the ground state (energy) of the un-

doped system and |Ψn⟩(En) are the eigenstates (eigenen-

ergies) with two holes.

The two-hole rotational spectral function deﬁned

above is closely related to the dynamical pairing correla-

tions frequently considered in the literature [6,20,21],

P(ω) = Zdt eiωt Dˆ

∆(s)†

m4(t)ˆ

∆(s)

m4(0)E,(7)

where ˆ

∆(s)

m4=Pjˆ

∆(s)

m4(j). Here, however, we consider

the full momentum dependence of the pairing correla-

tions, which enables direct insights into the center-of-

mass dispersion of pairs of charge carriers.

The resulting rotational spectra thus directly probe the

existence of bound states and their internal structure:

If a bound state of two holes with long-lived rotational

excitations exists, the rotational spectra should exhibit

well-deﬁned coherent peaks. If on the other hand such

bound states do not exist, the excitation with the rota-

tional operator ˆ

∆m4(k) will lead to a broad continuum

in the corresponding spectral function.

In Fig. 1(b), we show the two-hole spectral func-

tion with angular momentum, i.e. m4= 0 (s-wave)

and m4= 2 (d-wave) for the t−Jmodel for momenta

0≤kx≤πand ky=πand ky= 0, respectively. We

ﬁnd a well-deﬁned coherent peak at low energies for all

momenta, indicating the existence of a bound state. The

spectrum furthermore reveals a plethora of diﬀerent fea-

tures, including a highly dispersive band (black line, s-

wave excitation) as well as bands with a dispersion pro-

portional to the spin-exchange J(gray dashed lines, d-

wave excitation). At momentum k= (π, π), the spectral

weight vanishes for all energies for the s-wave excitation

since ˆ

∆(s)

0(k= (π, π)) = 0.

In order to gain a deeper understanding of these in-

triguing results, we take a step back and analyze the

conceptually simpler t−Jzmodel in the following sec-

tion.

The t−XXZ model.– We now consider a modiﬁca-

tion of the t−Jmodel, where the SU(2) invariant spin

interactions are replaced by in-plane and Ising-type spin

°10

!/Jz

°10

!/Jz

(0,k

y)(º/2,k

y)(º,k

momentum k

°10

!/Jz

0.00

0.05

0.10

0.15

0.20

0.25

0.30

°10

!/Jz

°10

!/Jz

(0,k

y)(º/2,k

y)(º,k

momentum k

°10

!/Jz

0.00

0.05

0.10

0.15

0.20

0.25

0.30

°10

°5

!/Jz

°10

°5

!/Jz

(0,k

y)(º/2,k

y)(º,k

momentum k

°10

°5

!/Jz

0.0

0.1

0.2

0.3

0.4

0.5

ky= 0

ky=π/2

ky=π

, -wave

ky=π

, -wave

ky= 0

°10

!/Jz

°10

!/Jz

(0,k

y)(º/2,k

y)(º,k

momentum k

°10

!/Jz

0.00

0.05

0.10

0.15

0.20

0.25

0.30

FIG. 2. Two hole rotational spectra in the t−XXZ

model for t/Jz= 3 and J⊥/Jz= 0.1 on a 40 ×4 cylinder,

based on time evolution up to Tmax/Jz= 10 and bond dimen-

sion χ= 600. The colormap corresponds to numerical matrix

product state simulations of the singlet two-hole rotational

spectrum, blue lines are geometric string theory predictions

for the position of states (all shifted by −0.35Jz), and the

black line is a cosine ﬁt. The upper (lower) plot corresponds

to ky= 0 (ky=π) at m4= 2, d-wave (m4= 0, s-wave) and

data is shown as a function of momentum kxand frequency

ω/Jz. In the top panel the overall ground state energy for two

holes is marked by orange circles for ky= 0, and the green

dashed line corresponds to twice the energy of a single hole

(indicating a small pairing gap on the order of Jz).

interactions with coupling constants J⊥and Jz:

Ht−XXZ =X

⟨i,j⟩J⊥ˆ

iˆ

j+ˆ

iˆ

j+Jzˆ

iˆ

j−

−tˆ

⟨i,j⟩X

σˆc†

i,σ ˆcj,σ + h.c.ˆ

P − Jz

⟨i,j⟩

ˆniˆnj.(8)

In the limit of J⊥≪Jz, also called the t−Jzmodel, the

lack of spin dynamics facilitates our theoretical under-

standing. Experimentally, the anisotropic interactions

can for example be realized by employing Rydberg inter-

actions [22,23] or using ultracold molecules in tweezer

arrays [23].

Remarkably, the two-hole spectral function, Fig. 2, ex-

hibits a highly dispersive peak with a mass proportional

to 1/t, best identiﬁed at ky=π(bottom panel); I.e., we

ﬁnd a long-lived, tightly bound state of two holes, which

can move as fast as the hole hopping t. This is in stark

contrast to the case of a single hole in the same model,

which has a very high eﬀective mass ≫1/t and thus an

almost ﬂat dispersion [24], since it can only move due to

Trugman loops [25], which are higher order processes.

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Dichotomyofheavyandlightpairsofholesinthet−JmodelA.Bohrdt,1,2,3,4,∗E.Demler,5andF.Grusdt6,71ITAMP,Harvard-SmithsonianCenterforAstrophysics,Cambridge,MA02138,USA2DepartmentofPhysics,HarvardUniversity,Cambridge,Massachusetts02138,USA3Institutf¨urTheoretischePhysik,Universit¨atRegensburg,D-93035Regensb...

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Dichotomy of heavy and light pairs of holes in the tJmodel A. Bohrdt1 2 3 4 E. Demler5and F. Grusdt6 7 1ITAMP Harvard-Smithsonian Center for Astrophysics Cambridge MA 02138 USA

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