Particle zoo in a doped spin chain Correlated states of mesons and magnons Petar Cubela1 2Annabelle Bohrdt3 4Markus Greiner4and Fabian Grusdt1 2 1Department of Physics and Arnold Sommerfeld Center for Theoretical Physics ASC

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Particle zoo in a doped spin chain: Correlated states of mesons and magnons

Petar ˇ

Cubela,1, 2 Annabelle Bohrdt,3, 4 Markus Greiner,4and Fabian Grusdt1, 2, ∗

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

3ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA

4Department of Physics, Harvard University, Cambridge, MA 02138, USA

(Dated: October 6, 2022)

It is a widely accepted view that the interplay of spin- and charge-degrees of freedom in doped

antiferromagnets (AFMs) gives rise to the rich physics of high-temperature superconductors. Nev-

ertheless, it remains unclear how eﬀective low-energy degrees of freedom and the corresponding

ﬁeld theories emerge from microscopic models, including the t−Jand Hubbard Hamiltonians. A

promising view comprises that the charge carriers have a rich internal parton structure on inter-

mediate scales, but the interplay of the emergent partons with collective magnon excitations of the

surrounding AFM remains unexplored. Here we study a doped one-dimensional spin chain in a stag-

gered magnetic ﬁeld and demonstrate that it supports a zoo of various long-lived excitations. These

include magnons; mesonic pairs of spinons and chargons, along with their ro-vibrational excitations;

and tetra-parton bound states of mesons and magnons. We identify these types of quasiparticles in

various spectra using DMRG simulations [1,2]. Moreover, we introduce a strong-coupling theory

describing the polaronic dressing and molecular binding of mesons to collective magnon excitations.

The eﬀective theory can be solved by standard tools developed for polaronic problems, and can be

extended to study similar physics in two-dimensional doped AFMs in the future. Experimentally,

the doped spin-chain in a staggered ﬁeld can be directly realized in quantum gas microscopes.

I. INTRODUCTION

Field theoretic approaches to quantum spin models in

lattices, such as the Heisenberg antiferromagnet (AFM),

provide very successful descriptions of these paradigmatic

quantum many-body systems [3] and have led to a thor-

ough understanding of their various quantum phase tran-

sitions in diﬀerent dimensions [4]. Key to their success is

the underlying hypothesis that the coarse-grained ﬁelds

on long length-scales feature similar behavior as the mi-

croscopic local magnetic moments underlying the spin

model. More formally, a simple renormalization-group

(RG) procedure yielding the eﬀective low-energy ﬁeld

theory does not change the particle-content of the an-

alyzed ﬁelds. However, in dimensions larger than one

and with mobile dopants included, this approach has

not been able to explain the rich phase diagram of high-

temperature superconductors so far.

In this article, we take a diﬀerent perspective and ex-

plore emergent structures, at low- to intermediate ener-

gies, in a doped quantum spin chain. The zoo of con-

stituents we ﬁnd deﬁes a naive ﬁeld-theoretic descrip-

tion: we identify emergent parton structures of spinons

and chargons, forming mesonic bound states with a rich

spectrum of ro-vibrational internal excitations. More-

over, these mesons interact with collective magnon exci-

tations in the surrounding spin system, which leads to po-

laronic dressing on the one hand and, more exotically, to

long-lived meson-magnon bound states. In a phenomeno-

∗Corresponding author email: fabian.grusdt@physik.uni-

muenchen.de

logical ﬁeld-theoretic model, each of these constituents

should be described by a separate quantum ﬁeld, with

mutual interactions between all of them. Describing how

these new ﬁelds emerge at intermediate length- or energy-

scales in a thorough RG procedure is a challenging task,

even for the simple toy model we consider. Hence we fo-

cus on a microscopic description of the individual emer-

gent bound states and analyze their characterizing prop-

erties, such as their dispersion relations, zero-point ener-

gies, and mutual interactions. To this end, we apply the

powerful theoretical tools developed for the description

of Bose polarons [5–9].

Concretely, we consider doped one-dimensional spin

chains. When featuring SU(2) invariance, these systems

display spin-charge separation [10–14]: the collective ex-

citations of the spin-chain are fractionalized spinons,

which co-exist with free chargons. In this limit, non-

trivial bound states of the constituents are absent [12]

and bosonization techniques provide a powerful ﬁeld-

theoretic description of the doped system in terms of

Luttinger liquids [15]. As we demonstrate, the situa-

tion changes drastically when a staggered magnetic ﬁeld

is included, breaking the SU(2) symmetry, see Fig. 1a):

Now spinons and chargons are conﬁned [15–17], the un-

doped ground state has gapped collective magnon excita-

tions, and spin-charge separation breaks down. Despite

this conﬁnement, the situation is far from trivial: As we

will show, doped holes in this system host a zoo of exci-

tations reﬂecting their rich internal structure, and their

interaction with gapped magnons can lead to even more

complicated multi-parton bound states, see Fig. 2.

In several regards, our 1D model is motivated by the

physics of mobile holes in a SU(2)-invariant 2D Hubbard

arXiv:2210.02320v1 [cond-mat.str-el] 5 Oct 2022

FIG. 1. Particle zoo in a doped spin chain: We consider

a doped mobile hole in a spin-chain subject to a staggered

magnetic ﬁeld. a) Ignoring transverse spin ﬂuctuations, the

staggered ﬁeld leads to a conﬁning force between spinons and

chargons connected by a string of overturned spins (top row),

which leads to meson formation. A similar situation is real-

ized in a mixed-dimensional model, where a strong gradient

prevents hole motion along the direction of the gradient (bot-

tom row). b) Transverse spin couplings give rise to Holstein-

Primakoﬀ (HP) magnon ﬂuctuations in the surrounding spin

background. The latter interact with the spinon, which is sur-

rounded by the strongly ﬂuctuating but tightly bound char-

gon cloud. All constituents making up the zoo of excitations

are summarized in c), where we also indicate the background

Ising ﬁelds τz

j, aﬀected by the hole motion, around which we

expand in the generalized 1/S approximation employed here.

model. In contrast to the 1D case with SU(2) symmetry,

the ground state of the two-dimensional (2D) Heisen-

berg model has long-range magnetic order and gapless

spin-1 magnon excitations. This eﬀect is mimicked by

the external staggered magnetic ﬁeld we consider in our

model, which introduces magnetic order and leads to sim-

ilar spin-1 magnon modes, although with a non-vanishing

gap. There is strong evidence that a doped hole in the

2D AFM features a rich internal meson structure [18],

with discrete vibrational [19–21] and rotational excita-

tions [22,23]. In our 1D model, we reveal similar struc-

tures and develop an eﬀective strong-coupling descrip-

tion.

Remarkably, the coupling of mesons to collective

magnon excitations remains poorly understood in 1D and

2D, in particular around zero momentum where we show

that the competition of mesons and magnons is most pro-

nounced. In the present article we ﬁll this gap and ap-

ply a powerful theoretical framework, the so-called gen-

eralized 1/S expansion [22], to describe the coupling of

mesons to magnons in a systematic manner. As a result,

we obtain an eﬀective polaron Hamiltonian describing

the dressing, or even binding, of a spinon-chargon meson

with additional magnon excitations. Our work paves the

way for similar studies in doped 2D Mott insulators, and

may lead to a better understanding of the charge carri-

ers and their interactions with magnons in underdoped

copper oxides. In particular we expect that our formal-

ism will be useful for understanding transport measure-

ments involving magnetic polarons, such as the long-time

spreading dynamics of a hole reported in [24–27].

Experimentally, the model we consider can be realized

with ultracold atoms in optical lattices, which have re-

cently made signiﬁcant advances in studying doped quan-

tum magnets [28]. On a mean-ﬁeld level, our model more-

over maps to a doped mixed-dimensional t−Jmodel [29],

which can be realized by subjecting a Fermi-Hubbard

system to a strong tilt along one of the lattice directions

[30]. Ultracold atom realizations allow to measure spec-

tra [31–34] like the ones we calculate here to identify the

emergent zoo of excitations; moreover they can directly

visualize string patterns [13,35,36] or the dressing cloud

of magnetic polarons in conﬁguration-space [37], making

them ideal platforms to explore the emergent structures

we predict on intermediate length scales.

II. MODEL AND MAIN RESULTS

In this article, we study a simple but rich one-

dimensional model of a doped AFM. Our starting point is

an SU(2)-invariant Heisenberg spin chain. An additional

staggered magnetic ﬁeld of strength ±hon alternating

sites along the z-direction breaks the SU(2) symmetry,

introduces long-range magnetic correlations, and leads

to collective magnon excitations with a tunable gap con-

trolled by |h|. To describe mobile holes doped into this

model, we use a t−JHamiltonian:

H=−tX

j,σ

Pˆc†

j+1,σ ˆcj,σ + h.c.ˆ

+JX

Sj+1 ·ˆ

Sj−hX

(−1)jˆ

j.(1)

Since we will only consider a single doped hole in

this article, we dropped the nearest-neighbor interaction

−J/4 ˆnj+1 ˆnjtypically included in the t−Jmodel [38].

A similar model, including phonons, has been studied in

Ref. [39].

A. Lattice gauge Hamiltonian

For later purposes, we ﬁnd it convenient to write the

Hamiltonian as a sum of two separate parts: (i) a t−Jz

part which conserves each individual spin in the so-called

squeezed space [12,13,40] obtained by removing holes

from the chain:

Ht−Jz=−tX

j,σ

Pˆc†

j+1,σ ˆcj,σ + h.c.ˆ

+JzX

j+1 ˆ

j−hX

(−1)jˆ

j.(2)

To keep our analytical formalism later on general, we

introduced the coupling Jz, which is simply Jz=Jfor

the original model in Eq. (1).

Remarkably, the Hamiltonian in Eq. (2) is exactly

equivalent to a Z2lattice gauge theory (LGT), as shown

in Refs. [17,41]. In this mapping, chargons (i.e. spin-

less holes) and spinons (i.e. Ising domain walls) carry

Z2gauge charges and are connected by a Z2electric

string τx

hi,ji. The staggered ﬁeld ±hleads to a term

hPhi,jiˆτx

hi,jiin the Z2gauge invariant Hamiltonian. The

latter has been shown to cause spinon-chargon conﬁne-

ment for any inﬁnitesimal h6= 0 [16,17]. This Z2LGT

formalism forms the basis for our mesonic description of

a doped hole.

In addition, the full Hamiltonian in Eq. (1) includes

(ii) transverse spin ﬂuctuations,

H=ˆ

Ht−Jz+ˆ

HJ⊥,(3)

where we ﬁnd it most convenient to write

HJ⊥=J⊥

jˆ

S+

j+1 ˆ

S−

j+ h.c..(4)

Again we introduced the more general coupling strength

J⊥in this term, although for our original model in Eq. (1)

J⊥=J. Later on, we will include such transverse

spin ﬂuctuations on top of a N´eel ordered ground state

distorted by the hole motion by introducing Holstein-

Primakoﬀ bosons (magnons), see Fig. 1b).

Finally, we note that in the limit h/J⊥→ ∞, the trans-

verse ﬂuctuations ˆ

HJ⊥can always be treated perturba-

tively, independent of the ratios Jz/J⊥or t/J⊥. To low-

est order, only the t−Jzpart of the Hamiltonian, Eq. (2),

remains and it follows that the model has an emergent

Z2gauge structure for large values of h.

B. Main results: particle zoo in the spin chain

The separation of the Hamiltonian in two components

lends a natural understanding of our results. Our main

goal is to understand the ground and excited states of a

mobile dopant in the spin chain. As described in detail

below, we ﬁnd that the Z2gauge structure of the t−Jz

part of the Hamiltonian, or equivalently (in our model)

the string-picture of magnetic polarons [22,42,43], intro-

duces parton constituents, namely spinons and chargons,

which are conﬁned by the linear string tension generated

by the staggered ﬁeld h. The resulting mesonic spinon-

chargon bound state has a rich internal structure con-

stituted by inversion-even and inversion-odd vibrational

modes of the Z2electric string, or equivalently the string

of overturned Ising spins, connecting the spinon and the

chargon. We probe these states directly in spectra cal-

culated by time-dependent matrix product states (td-

MPS), see Sec. III, and compare to an eﬀective strong-

coupling description that we develop here, see Sec. IV.

The transverse couplings introduced by ˆ

HJ⊥lead to

vacuum ﬂuctuations of magnons in the absence of a

doped hole. This eﬀect can be captured by a simple lin-

ear spin-wave expansion around the classical N´eel state,

012345

hin units of t

2

1

G.S. Energy in units of t

Bare Mes

LLP + Gauss

DMRG

h in units of t

Ground state energy in units of t

bare meson

LLP + Gauss

DMRG

energy in units of Jz

momentum k/π

FIG. 2. Polaronic bands in the presence of meson-magnon

interactions at low energies: The overall ground state at

momentum k=π/2 is realized by a weakly dressed meson

(solid blue line). Before the broad meson-magnon continuum

is reached at higher energies (wide red band), we predict a

weakly dispersing meson-magnon bound state (dark red line),

corresponding to a tetra-parton conﬁguration. The black lines

indicate the bare meson dispersion (solid) and edges of the

meson-magnon continuum (dashed) in the absence of meson-

magnon interactions, respectively. At higher energies (not

shown) we ﬁnd ro-vibrational internal meson excitations. Cal-

culations were performed using the strong-coupling general-

ized 1/S approximation introduced in the text; we chose pa-

rameters h= 0.6Jand t= 5J.

which we achieve by a Holstein-Primakoﬀ approximation.

In the vicinity of the meson, the distortion of the N´eel

background caused by the spinon-chargon pair introduces

additional couplings to magnons which give rise to addi-

tional rich physics: On one hand, they lead to polaronic

dressing and weak mass renormalization of the meson

around the dispersion minimum at momentum k=π/2.

This is shown for the lowest-energy mesonic state (solid

blue line) in Fig. 2.

More dramatically, the interactions with magnons can

give rise to meson-magnon bound states. Since the

magnon itself can be viewed as a bound state of two con-

ﬁned spinons, this state constitutes an emergent tetra-

parton composite. As demonstrated in Fig. 2, for suﬃ-

ciently large values of hour eﬀective model of the meson-

magnon coupling predicts a low-lying meson-magnon

bound state at relatively low energies below the meson-

magnon scattering continuum. This should be contrasted

with the higher excitation energies of ro-vibrational inter-

nal meson modes. We conﬁrm our prediction of meson-

magnon bound states in td-MPS calculations of one-hole

spectra in a sector with total spin Sz= 3/2, see Sec. III.

Finally, meson-magnon interactions can have a pro-

nounced eﬀect on the quasiparticle dispersion of the

dressed meson around momentum k= 0. In this re-

gion of momentum space, the bare meson dispersion ap-

proaches the meson-magnon scattering continuum most

closely, as indicated by the dashed and solid black lines

in Fig. 2. Without meson-magnon interactions and for

suﬃciently weak ﬁelds ht, J, we ﬁnd that they can

even cross, leading to a decaying bare meson state inside

the meson-magnon scattering continuum. However, in

Sec. V B we analyze our eﬀective meson-magnon Hamil-

tonian and ﬁnd indications that meson-magnon interac-

tions in the 1D chain are strong enough to avoid such

quasiparticle decay [44]. Namely, the meson and magnon

bands repel and an isolated quasiparticle band of the

mesonic magnetic polaron survives even around k= 0.

This prediction is further supported by td-MPS simula-

tions at small ﬁelds where the eﬀect is most pronounced.

Methodologically, we deviate from the standard ap-

proach typically used to describe magnetic polaron for-

mation in an AFM [45–47]. As mentioned above, we ﬁrst

take into account how the mobile hole distorts the N´eel

background with pure Ising interactions. This allows us

to make a direct connection to the Z2LGT and identify

the parton content of the meson. Moreover, we can rela-

tively easily capture the competition between the tunnel-

ing term tand the linear string tension ∝h, to all orders

in t/h. This is achieved within a strong-coupling theory.

Next we introduce generalized Holstein-Primakoﬀ bosons

(loosely speaking, magnons) by expanding around the al-

ready distorted N´eel state (we refer to this approach as

the generalized 1/S approximation [22]). This yields ad-

ditional couplings of the meson to the magnons; impor-

tantly, the strength of these couplings is only of order

J, and a fraction of tfor some further corrections we

identify. Hence, perturbative or simple variational ap-

proaches are suﬃcient to capture the additional meson-

magnon interactions. This should be contrasted with

the traditional 1/S approximation [45–47] where the hole

hopping titself leads to magnon creation: as a result, the

eﬀective Hamiltonian is strongly coupled when t>hand

direct analytical insights are harder to obtain.

C. Possible experimental realizations

Experimentally, the model in Eq. (1) we study can be

realized in diﬀerent ultracold atom setups. We propose

to use ultracold fermionic Lithium or Potassium atoms

which have very successfully explored the SU(2)-invariant

2D Fermi-Hubbard model [28]. The main obstacle in

these systems is to implement the staggered magnetic

ﬁeld, which requires local addressability on the scale of

an optical wavelength, see e.g. [48], and sizable magnetic

moments in order to distinguish diﬀerent spin states, in a

regime close to an atomic Feshbach resonance to realize

super-exchange couplings.

A ﬁrst option is to use Potassium atoms in a quantum

gas microscope [49] which have a sizable magnetic mo-

ment [50], allowing for a local modulation of the magnetic

ﬁeld. A second option is to work in a mixed-dimensional

setting where tunneling is strongly suppressed by strong

gradients along all but one lattice direction [29]. More-

over, we assume that nearest-neighbor AFM Ising cou-

plings between all spins are present, which dominate over

the weak super-exchange couplings along the gradient di-

rections. This can be realized in an optical lattice by

adding Rydberg dressing [51]. When doping only the

central chain with one hole and keeping all neighboring

chains at half ﬁlling, the surrounding spin chains can gen-

erate an eﬀective staggered ﬁeld term ±hif they are suﬃ-

ciently cold. Here we assumed, in a mean-ﬁeld spirit, that

the wavefunctions of the diﬀerent chains approximately

factorize. Similarly, in mixed-dimensional settings with

SU (2) invariant spin-exchange interactions [29,30] we

expect a ground state with broken SU (2) symmetry in

qualitatively very similar physics.

Finally, we note that the 1D model in Eq. (1) can be

equally realized with bosons as long as one ensures to

have AFM Heisenberg couplings between the spins [52,

53]. The statistics of the dopants is irrelevant, as can be

shown by a Jordan-Wigner transformation. Hence the

model in Eq. (1) can also be simulated in qubit arrays

or digital quantum computers [54], without the need to

incorporate fermionic statistics.

III. NUMERICAL DMRG SPECTRA

In this section we present our numerical results, largely

based on td-MPS simulations [55], which support our

main ﬁndings about the structure and interactions of

doped holes in the 1D spin chain with a staggered ﬁeld.

We already compare our numerical results to predictions

by the semi-analytical strong-coupling meson-magnon

theory introduced in the subsequent sections. This the-

ory provides a uniﬁed understanding of all our key nu-

merical observations.

Detailed descriptions of the numerical td-MPS simu-

lations we performed can be found in Refs. [21,23]; our

algorithm builds upon the earlier works [56–58]. To en-

sure proper convergence of the MPS calculations, we per-

formed the same convergence checks, in time and bond-

dimension, as described in [21,23].

A. Ground state: Dressed hole

In Fig. 3we start by showing the standard one-hole

angle-resolved photoemission spectrum (ARPES), de-

ﬁned by

S(k, ω) = −1

πImZ∞

dtei(ω−E0)t

√LX

eikj Gj(t),(5)

with the Green’s function

Gj(t) = X

σhΨ0|ˆc†

j,σe−iˆ

Htˆc0,σ|Ψ0i(6)

0.0 0.2 0.4 0.6 0.8 1.0

k/π

−8

−7

−6

−5

−4

−3

−2

−1

ω/J

S(kx, ky= 0.0π, ω)

0.2

0.4

0.6

0.8

1.0

1.2

FIG. 3. The standard one-hole ARPES spectrum reveals

a pronounced quasiparticle peak at the lowest energy. The

dispersion minimum is located at k=π/2, as predicted by

our semi-analytical theory (solid blue line). Here we consider

h= 1.0Jand t= 5J; the color scale is in a.u..

where |Ψ0iis the ground state with energy E0of ˆ

Hwith

zero holes.

In the spectrum, we observe a pronounced quasiparti-

cle peak at low energy which corresponds to the magnetic

polaron. The comparison with our semi-analytical the-

ory shows that it is located around the expected energy,

and shows the same dispersion relation with a minimum

at k=π/2. At higher energies the spectrum is relatively

featureless for the considered value of h/J = 1.0 in Fig. 3.

As we show next, additional features becomes visible for

larger values of h.

B. Ro-vibrational excitations: Mesonic states

Now we calculate a rotational variant of the ARPES

spectrum, where spinon-chargon excitations with odd

(ξ=−1) and even (ξ= +1) inversion symmetry can

be detected. It is deﬁned as in Eq. (5) but using the

rotational Green’s function [23]

Grot

j,ξ (t) = X

σhΨ0|ˆc†

j,σ ˆ

X†

j,ξe−iˆ

Htˆ

X0,ξ ˆc0,σ|Ψ0i(7)

where ˆ

Xj,ξ =Pσˆc†

j,σ(ˆcj+1,σ +ξˆcj−1,σ ) creates an addi-

tional excitation of the spinon-chargon pair.

In Fig. 4we show our results for h= 4J. In both parity

sectors ξ=±1 we observe pronounced vibrational peaks,

which correspond to vibrational modes of the spinon-

chargon string. The absence of even (odd) peaks in the

odd (even) spectrum indicates that the parity ξis a good

emergent quantum number at all momenta, not only at

k= 0, π/2 where the system is strictly inversion sym-

metric. This is a direct indication for the existence of an

internal meson structure [23].

0.0 0.2 0.4 0.6 0.8 1.0

k/π

−10

−5

ω/J

S(kx, ky= 0.0π, ω)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.0 0.2 0.4 0.6 0.8 1.0

k/π

−10

−5

ω/J

S(kx, ky= 0.0π, ω)

0.0

0.2

0.4

0.6

0.8

FIG. 4. The rotational one-hole ARPES spectrum reveals a

series of long-lived vibrational excitations with even (ξ= +1,

top) and odd (ξ=−1, bottom) parity. We compare the td-

MPS spectra with bare meson resonances calculated from our

strong-coupling theory (gray solid lines: ξ= +1 even; gray

dashed lines: ξ=−1 odd). Here we consider h= 4.0Jand

t= 5J; the color scale is in a.u..

In Fig. 4we also compare the peak positions observed

in td-MPS with predictions by our strong-coupling the-

ory. The observed peaks in our full numerical spectra

are in excellent agreement with our semi-analytical pre-

dictions. In the latter, for simplicity, we neglected cor-

rections from magnon-dressing which are weak at large

values of h. Nevertheless, note that signiﬁcant charge

ﬂuctuations are present since we consider t>hin the

ﬁgure.

C. Meson-magnon bound states

Next we show that even more complex excitations can

arise when the mesonic hole interacts with its spin envi-

ronment. Speciﬁcally, the meson can form a stable bound

state with a magnon excitation. To demonstrate the ex-

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Particlezooinadopedspinchain:CorrelatedstatesofmesonsandmagnonsPetarCubela,1,2AnnabelleBohrdt,3,4MarkusGreiner,4andFabianGrusdt1,2,1DepartmentofPhysicsandArnoldSommerfeldCenterforTheoreticalPhysics(ASC),Ludwig-Maximilians-UniversitatMunchen,Theresienstr.37,MunchenD-80333,Germany2MunichCenterfor...

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Particle zoo in a doped spin chain Correlated states of mesons and magnons Petar Cubela1 2Annabelle Bohrdt3 4Markus Greiner4and Fabian Grusdt1 2 1Department of Physics and Arnold Sommerfeld Center for Theoretical Physics ASC

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