Quantifying hole-motion-induced frustration in doped antiferromagnets by Hamiltonian reconstruction Henning Schl omer1 2 3Timon A. Hilker2 4Immanuel Bloch1 2 4

2025-05-02 0 0 2.92MB 16 页 10玖币

侵权投诉

Quantifying hole-motion-induced frustration in doped antiferromagnets by

Hamiltonian reconstruction

Henning Schl¨omer,1, 2, 3, ∗Timon A. Hilker,2, 4 Immanuel Bloch,1, 2, 4

Ulrich Schollw¨ock,1, 2 Fabian Grusdt,1, 2 and Annabelle Bohrdt3, 5, †

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

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

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

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

4Max-Planck-Institut f¨ur Quantenoptik, D-85748 Garching, Germany

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

(Dated: October 2, 2024)

Abstract. Unveiling the microscopic origins of quantum phases dominated by the interplay of

spin and motional degrees of freedom constitutes one of the central challenges in strongly correlated

many-body physics. When holes move through an antiferromagnetic spin background, they displace

the positions of spins, which induces eﬀective frustration in the magnetic environment. However, a

concrete characterization of this eﬀect in a quantum many-body system is still an unsolved prob-

lem. Here we present a Hamiltonian reconstruction scheme that allows for a precise quantiﬁcation of

hole-motion-induced frustration. We access non-local correlation functions through projective mea-

surements of the many-body state, from which eﬀective spin-Hamiltonians can be recovered after

detaching the magnetic background from dominant charge ﬂuctuations. The scheme is applied to

systems of mixed dimensionality, where holes are restricted to move in one dimension, but SU(2)su-

perexchange is two-dimensional. We demonstrate that hole motion drives the spin background into

a highly frustrated regime, which can quantitatively be described by an eﬀective J1−J2−type spin

model. We exemplify the applicability of the reconstruction scheme to ultracold atom experiments

by recovering eﬀective spin-Hamiltonians of experimentally obtained 1D Fermi-Hubbard snapshots.

Our method can be generalized to fully 2D systems, enabling promising microscopic perspectives

on the doped Hubbard model.

Introduction

Microscopically understanding the motion of mobile

charge carriers doped into Mott insulators constitutes one

of the key open problems in strongly correlated many-

body physics. When hopping through an insulating spin

environment, holes displace spins along their way, which

eﬀectively frustrates the magnetic background. The aris-

ing competition of kinetic energy gain via delocalization

and associated magnetic energy cost leads to the for-

mation of a plethora of strongly interacting many-body

phases [1,2], many of which yet seek to be explained on

a microscopic footing. The Fermi-Hubbard (FH) model,

believed to capture the essential physics of strongly cor-

related materials, has been subject to intense numer-

ical studies that can resolve the intricate competition

between various orders [3–9]. Nevertheless, despite on-

going theoretical and experimental eﬀorts over the past

decades, a precise microscopic understanding of the in-

terplay between motional and spin degrees of freedom is

still an unsolved task, whose long-sought understanding

may help to reveal the origin of high-temperature super-

conductivity and possibly lead to the discovery of novel

pairing mechanisms [10,11].

Analog quantum simulation, e.g. via ultracold atoms,

can shed new light on the microscopic mechanisms under-

lying strongly correlated quantum many-body states [12–

17] and paradigmatic Hamiltonians like the FH model

can now be experimentally explored [11,18–24]. In par-

ticular, these setups allow to perform genuine quantum

projective measurements and sample snapshots of the

many-body state in the Fock basis, which in turn allow

for insights into the wave function beyond averages and

local observables. This capability has already been used

to unveil highly non-local order parameters and hidden

correlations in many-body systems [22,25].

As we demonstrate in our work, the huge amount of

information stored in snapshots of many-body states can

further be utilized to disentangle spin and charge sectors

through non-local correlation functions, which allow us

to recover emergent eﬀective spin-Hamiltonians for parts

of the system. The problem of reconstructing a Hamil-

tonian from measured correlations via machine learning

schemes [26–30] has attracted considerable interest in re-

cent years, including certifying quantum simulation de-

vices [31].

In this article, we present a snapshot-based Hamilto-

nian reconstruction scheme for the spin channel alone,

which removes dominant charge ﬂuctuations [22,32]

from individual snapshots. This allows us to quantify

the eﬀective spin-Hamiltonian, which includes the back-

action of mobile dopants on the spin environment. We

exemplify the proposed method by considering a sys-

tem in mixed-dimensions (mixD), where hole motion

is restricted to one dimension (1D), but SU(2)spin-

superexchange is two-dimensional (2D). We ﬁnd that hole

hopping drives and stabilizes the spins in a highly frus-

arXiv:2210.02440v2 [cond-mat.quant-gas] 1 Oct 2024

trated regime, which we show to be accurately described

by a J1−J2−type spin-Hamiltonian.

Our method is directly applicable to experimental data

obtained from ultracold quantum gas microscopes. We

showcase this by reconstructing eﬀective Hamiltonians

from 1D measurements of the FH model [22], where spin-

charge separation governs the physics of the chains. Fur-

thermore, our insights could be used to eﬀectively simu-

late the highly frustrated J1−J2model in ultracold atom

experiments by implementing the mixD setting and post-

processing the measurements.

Our work sheds light on the long-standing question

about the interplay of spin- and motional degrees of

freedom in strongly correlated materials, and paves the

way to gain deep microscopic insights into prototypical

systems such as the 2D FH and t−Jmodel.

Results

The model. We consider the t−Jmodel in mixD [10,

33–35], described by the Hamiltonian

H=−t

⟨i,j⟩x,σ

PGW ˆc†

i,σ ˆcj,σ +h.c.ˆ

PGW +

J

⟨i,j⟩ˆ

Si⋅ˆ

Sj−ˆniˆnj

4.

(1)

Here, ˆc(†)

i, ˆniand ˆ

Siare fermionic annihilation (creation),

charge density, and spin operators on site i, respectively;

i,j(x)denotes a nearest-neighbor (NN) pair on a 2D

square lattice (with subscript xindicating a NN pair only

along the x-direction), and ˆ

PGW is the Gutzwiller oper-

ator projecting out states with double occupancy. The

mixD setting, Eq. (1), has successfully been implemented

in ultracold atom setups using strong tilted potential gra-

dients [11], which eﬀectively restrict hole motion perpen-

dicular to the gradient direction while spin-spin interac-

tions remain 2D [36,37].

Recently, we demonstrated how hidden AFM correla-

tions in the mixD t−Jmodel result in the formation of a

remarkably resilient stripe phase (i.e. a coupled charge-

and spin-density wave [38,39]), with critical tempera-

tures on the order of the magnetic coupling J[35]. Above

these critical temperatures of charge- and spin-density

wave formation, holes were found to form a deconﬁned

chargon gas, i.e., a phase without order [34,35].

In the following, we focus on the latter regime, and

study how hole motion distorts the spins in the back-

ground. The eﬀect is qualitatively depicted in Fig. 1a.

The upper panel shows an (idealized) real space snap-

shot of holes moving through an AFM N´eel background.

Bonds correspond to AFM interactions in the instan-

taneous charge conﬁguration, illustrated by gray lines.

In between holes on neighboring legs, spins are aligned,

leading to a linearly increasing magnetic energy penalty

via the formation of geometric strings [21,40–42] (de-

picted by green wiggly lines).

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[S(⇡,⇡)+S(0,⇡)]/2

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max

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J2/Jx

<latexit sha1_base64="RjgsBwrk9uini+3BCJepDWJCnIY=">AAACKHicbVDLSgMxFM34tr6qLt0Ei+CqdqRoVyK4EVcK1hamY0nSOxqax5Bk1DLMZ7jVvV/jTtz6JWZqF74uJBzOuZdz76Gp4NY1Gu/B1PTM7Nz8wmJlaXllda26vnFldWYYtJkW2nQpsSC4grbjTkA3NUAkFdChw5NS79yBsVyrSzdKIZbkRvGEM+I8FZ31w+vRXvk/9Ku1Rr0xLvwXhBNQQ5M6768H872BZpkE5Zgg1kat1MU5MY4zAUWll1lICRuSG4g8VESCjfPxygXe8cwAJ9r4pxwes98nciKtHUnqOyVxt/a3VpL/aVHmklacc5VmDhT7MkoygZ3G5f14wA0wJ0YeEGa43xWzW2IIcz6lHy7UkCE4f4eCe6alJGqQ9ygtojDO817pTBNcC4ui4qMLfwf1F1zt18ODevOiWTs+moS4gLbQNtpFITpEx+gUnaM2YkijR/SEnoOX4DV4C96/WqeCycwm+lHBxycaHKVL</latexit>

1/Jx

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AFM

<latexit sha1_base64="rccxXhHpKUaOWDZIso2j4npJC7k=">AAACJ3icbVBNSwMxEM36Va1frR69BIvgqeyWor2IBS8eFawV21WSdNaGJtklySpl2X/hVe/+Gm+iR/+Jae3Btg4MPN6b4c08mghurO9/eQuLS8srhdW14vrG5tZ2qbxzbeJUM2ixWMT6hhIDgitoWW4F3CQaiKQC2nRwNtLbj6ANj9WVHSYQSvKgeMQZsY66VXd9fIL9as2/L1X8qj8uPA+CCaigSV3cl71CtxezVIKyTBBjOo3EhhnRljMBebGbGkgIG5AH6DioiAQTZuOLc3zgmB6OYu1aWTxm/25kRBozlNRNSmL7ZlYbkf9pndRGjTDjKkktKPZrFKUC2xiP3sc9roFZMXSAMM3drZj1iSbMupCmXKgmA7DuDwVPLJaSqF7WpTTvBGGWdUfONMKVIM+LLrpgNqh5cF2rBkfV+mW90jydhLiK9tA+OkQBOkZNdI4uUAsxpNAzekGv3pv37n14n7+jC95kZxdNlff9AwGzpCM=</latexit>

nh=0.20

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nh=0.15

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nh=0.10

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nh=0.05

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reconstruction

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pred. ˆ⇢s⌦ˆ⇢c

FINAL FIG1

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squeezed space

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real space

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tJmodel

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e↵.Jx

1Jy

1J2model

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stripe AFM

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frustrated

FIG. 1. Hole-motion-induced spin frustration.a,

Schematic of how hole hopping induces frustration in the spin

background. Upper panel: Snapshot of holes moving through

a N´eel background. Spatial separation of holes on neighboring

legs lead to the formation of geometric strings (green wiggly

lines) costing magnetic energy. Lower panel: Upon trans-

forming the snapshot to squeezed space, originally vertical

bonds Jy

1in between two holes on neighboring legs become

eﬀective diagonal couplings J2. The resulting energy penalty

of aligned diagonal spins leads to frustration in the magnetic

background. b, Hamiltonian reconstruction results (blue) for

hole dopings nh=0.05 ...0.2 of a mixD t−Jladder with

t/J=3, Lx×Ly=20×2 and T/J=5/3∼1.67. Reconstructions

of squeezed space according to the input Jx

1−Jy

1−J2Heisen-

berg Hamiltonian, Eq. (2), are presented in a Jy

1/Jx

1, J2/Jx

diagram. Light regions in the background signal the presence

of either AFM or stripe AFM order in the purely magnetic

1−Jy

1−J2model in the ground state by plotting the sum

of the spin structure factors S(π, π)+S(0, π). Dark regions

correspond to a highly frustrated regime without apparent

order. Upon doping the system, the background spins are

driven into a strongly frustrated state. Error bars correspond

to the standard error to the mean when averaging over ten

reconstruction runs. Red connected symbols show theoretical

expectations assuming no spin-hole correlations in the mixD

t−Jmodel, i.e., ˆρ=ˆρs⊗ˆρc, evaluated via Eq. (4) and (5).

As a direct consequence of the restricted charge mo-

tion to 1D, spins can be relabeled by the new positions

they have after moving all holes to the right in each

chain – resulting in a distinct deﬁnition of squeezed

space [32,43]. More formally, consider a Fock state

yσ[1,y], σ[2,y],...,σ[Lx,y], where {0,↑,↓}∋σx,y is

the single particle basis of the t−Jmodel. These

local spin charge conﬁgurations are relabeled upon

squeezing, whereby each Fock state is now given by

y˜σ[˜

1,y], σ[˜

2,y],...,σ[˜

Lx,y]⊗ˆ

h†

[x1,y]...ˆ

h†

[xNy,y]0[34].

Here, ˜σ[˜x,y]=↑,↓(but note that ˜σ[˜x,y]≠0) denotes spins

on the squeezed lattice ˜x=1,...,Lx−Ny, where Ny

is the number of holes in rung y, and ˆ

h[x,y]creates

a hard core fermionic chargon at site i=[x, y]. By

squeezing the spins out, spins on the squeezed and real

space lattice relate as ˜σ(˜x, y)=σ(˜x+∑j<˜xnh

[j,y], y),

where nh

[x,y]refers to the number of chargons at real

space lattice site [x, y]. The lower panel of Fig. 1a

illustrates the squeezing process, where the initial

N´eel order is restored in the isolated spin background.

However, interactions on diagonal bonds emerge (ocher

lines), which cause geometric frustration of the spins in

squeezed space. From now on, we refer to lattice sites in

real and squeezed space by iand ˜

i, respectively.

Characterizing the spin state in squeezed space.

In order to quantify the arising frustration on the

squeezed lattice, we simulate the mixD t−Jmodel,

Eq. (1), at ﬁnite temperature and ﬁxed doping using

imaginary time evolution schemes (puriﬁcation) via ma-

trix product states (MPS) [44,45]. For faster, more con-

trollable numerics and to prevent post-selection of snap-

shots, we explicitly implement the system’s enhanced

U(1)symmetries in each ladder leg, i.e., we work in an en-

semble where we allow for thermal spin ﬂuctuations but

keep the number of holes in each ladder leg constant [35].

In particular, we simulate the mixD t−Jmodel at in-

termediate temperature TJ=53 (βJ =0.6), which lies

inside the chargon gas phase (i.e. no charge- and spin-

density waves form) and, furthermore, is in a tempera-

ture regime accessible for quantum gas microscopes.

From the thermal MPS at inverse temperature β,

we sample uncorrelated snapshots of the corresponding

Gibbs state [46,47]. After post-processing the individual

measurements by squeezing out the holes, spin-spin cor-

relations ˆ

iˆ

jcan directly be evaluated in squeezed

space. In Fig. 2a, nearest-neighbor as well as diag-

onal spin-spin correlations are shown on the squeezed

lattice. In the bulk of the squeezed ladder, both near-

est neighbor as well as diagonal correlators are negative

and comparable in magnitude, signaling strong frustra-

tion in the spin background induced by the motion of

the holes. In contrast, at both edges of the ladder, low

average hole concentrations lead to only marginal per-

turbations of the spin background – resulting in AFM

type correlations that are negative (positive) along near-

est (diagonal) neighbors.

Fig. 2bshows nearest-neighbor rung (blue) and di-

agonal (red) correlators. Dashed-dotted lines correspond

to rung and diagonal correlations for a Heisenberg lad-

der with solely nearest-neighbor couplings (where βJx

βJy

1=0.44), which describe the physics at the edges qual-

itatively well, but fail to reproduce the measured cor-

relations in the bulk. Due to the frustrating eﬀect of

the hopping holes, diagonal couplings need to be taken

into account to accurately capture the physics of the

squeezed background. To this end, we introduce an ef-

fective Jx

1−Jy

1−J2Heisenberg model [48], given by the

Hamiltonian

H{JH}=

µ=x,y

Jµ

1

⟨˜

i,˜

j⟩µ

S˜

i⋅ˆ

S˜

j+J2

⟪˜

i,˜

j⟫diag

S˜

i⋅ˆ

S˜

j.(2)

Here, ˆ

S˜

iis the spin-1/2 operator at site ˜

i, and {JH}=

{Jx

1, Jy

1, J2}are the coupling strengths of neighbouring

spins in x, y and diagonal direction on the squeezed lat-

tice, respectively.

To quantitatively pin down the strength of the aris-

ing frustration, we perform a Hamiltonian reconstruction

with input Hamiltonian ˆ

H{JH}, Eq. (2), together with the

measured spin-spin correlations in squeezed space. Cou-

plings in ˆ

H{JH}are chosen to be homogeneous through-

out the bulk of squeezed space, justiﬁed by the approx-

imately constant behaviour of correlations in the bulk

region of Fig. 2b– solid lines are averages of the cor-

relations over the marked box. Due to the underlying

SU(2)symmetry of the mixD t−Jmodel, many-body

snapshots along a single spin axis – here chosen along z

– are suﬃcient to reconstruct the full eﬀective Hamilto-

nian, Eq. (2). Results of the reconstruction correspond

to the parameter conﬁguration {JH}that best describe

the measured correlations.

On a more formal footing, we follow the procedure in-

troduced in [30] and minimize the objective function G

over all possible coupling parameters {JH},

G=ln Z(β, {JH})+3β



µ=x,y

Jµ

1Mµ

1+J2M2

,(3)

with Z=Tr[e−βˆ

H{JH}]the partition function and Mµ

∑′

⟨˜

i,˜

j⟩µˆ

iˆ

j,M2=∑′

⟪˜

i,˜

j⟫diag ˆ

iˆ

jthe summed cor-

relations along nearest- and diagonal neighbors within

the considered window in the bulk of squeezed space.

Fig. 2cshows how approximations for Mx,y

1,M2quickly

saturate with the number of used snapshots, suggesting

a qualitatively satisfactory proxy for the spin-spin cor-

relators after a few thousand projective measurements.

For the rest of the analysis, we use sample sizes of 7,000

snapshots for each approximation of the correlations.

The minimization process is done via standard gradi-

ent descent (GD) methods, where in each iteration the

parameters are updated according to the gradient ∇′G

within the considered bulk window of squeezed space.

The temperature β−1of the Jx

1−Jy

1−J2Heisenberg Hamil-

tonian is chosen identically to the underlying simulations

of the mixD t−Jsystem during the GD. Note that this

choice might not reﬂect the actual eﬀective temperature

of the spin background. However, the relevant ratios

1Jx

1,J2Jx

1that quantify the frustration in the system

AFM

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J1J2

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Hamiltonian reconstruction

<latexit sha1_base64="bUGVD0z1iga0L9NNWhAWWdjNESY=">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</latexit>

# snapshots

<latexit sha1_base64="lMyJ5v2C7ymkfwNb9OR6+hFJp8g=">AAACJnicbVDLSgNBEJz1FY3P6NHLYBA8hawEzVHw4lHB+CC7yMxsrw6ZmV1metWw7Fd41btf403Em5/iJObgq6GhqOqmuovnSjpst9+DqemZ2bna/EJ9cWl5ZXWtsX7mssIK6IlMZfaCMwdKGuihRAUXuQWmuYJzPjgc6ee3YJ3MzCkOc4g1uzYylYKhpy4jlCqB8r66Wmu2W+1x0b8gnIAmmdTxVSOoRUkmCg0GhWLO9bs5xiWzKIWCqh4VDnImBuwa+h4apsHF5fjgim57JqFpZn0bpGP2+0bJtHNDzf2kZnjjfmsj8j+tX2DajUtp8gLBiC+jtFAUMzr6nibSgkA19IAJK/2tVNwwywT6jH64cMsGgP4PA3ci05qZpIw4r/phXJbRyJmntBlWVd1HF/4O6i84222Fe63OSad50JmEOE82yRbZISHZJwfkiByTHhFEkwfySJ6C5+AleA3evkangsnOBvlRwccnUSClew==</latexit>

˜x

<latexit sha1_base64="UNtJ27tk3nebpL0RcR1tK6m+J7E=">AAACLnicbVBNSwMxFMz6bf1o1aOXYBE8la4U9Sh48SIoWCu0a3lJ37ahSXZJskpZ9pd41bu/RvAgXv0ZprUHqw4Ehpn3eJNhqRTW1etvwdz8wuLS8spqaW19Y7Nc2dq+sUlmODZ5IhNzy8CiFBqbTjiJt6lBUExiiw3Pxn7rHo0Vib52oxQjBX0tYsHBealbKXcUuAEHmV8U3fBu1K1U67X6BPQvCaekSqa47G4Fy51ewjOF2nEJ1rZPUhflYJzgEotSJ7OYAh9CH9uealBoo3wSvKD7XunRODH+aUcn6s+NHJS1I8X85Dim/e2Nxf+8dubikygXOs0cav59KM4kdQkdt0B7wiB3cuQJcCN8VsoHYIA739XMFWZgiM7/Q+MDT5QC3cs7jBXtMMrzSXksptWwKEq+uvB3UX/JzWEtPKo1rhrV08a0xBWyS/bIAQnJMTkl5+SSNAknGXkkT+Q5eAleg/fg43t0Lpju7JAZBJ9fRLWn7A==</latexit>

<latexit sha1_base64="buAslP9JI0NbPjLb4tmSQa86B6k=">AAACLnicbVBNSwMxFMz6bf2qevQSLIKn0pWiHgUvXgQFq4V2LS/pWw1NskuSVcuyv8Sr3v01ggfx6s8wrT3Y1oHAMPMebzIslcK6Wu0jmJmdm19YXFourayurW+UN7eubZIZjg2eyMQ0GViUQmPDCSexmRoExSTesN7pwL95QGNFoq9cP8VIwZ0WseDgvNQpb7QVuHsOMj8vOuHtU6dcqVVrQ9BpEo5IhYxw0dkMFtvdhGcKteMSrG0dpy7KwTjBJRaldmYxBd6DO2x5qkGhjfJh8ILueaVL48T4px0dqn83clDW9hXzk4OYdtIbiP95rczFx1EudJo51Pz3UJxJ6hI6aIF2hUHuZN8T4Eb4rJTfgwHufFdjV5iBHjr/D42PPFEKdDdvM1a0wijPh+WxmFbCoij56sLJoqbJ9UE1PKzWL+uVk/qoxCWyQ3bJPgnJETkhZ+SCNAgnGXkmL+Q1eAveg8/g63d0JhjtbJMxBN8/Qvun6w==</latexit>

<latexit sha1_base64="aDKiTiwRktEoIk8/68vcFWW6IpQ=">AAACKnicbVBNSwMxFMz6Weu3Hr0Ei+CpdKVoj4IXL0IF2wrtUl7StzU0ya5JVinL/g6vevfXeBOv/hDT2oNWBwLDzHu8ybBUCutqtfdgYXFpeWW1tFZe39jc2t7Z3WvbJDMcWzyRibllYFEKjS0nnMTb1CAoJrHDRhcTv/OAxopE37hxipGCoRax4OC8FPUUuDsOMr8q+if9nUqtWpuC/iXhjFTIDM3+brDaGyQ8U6gdl2Btt5G6KAfjBJdYlHuZxRT4CIbY9VSDQhvl09AFPfLKgMaJ8U87OlV/buSgrB0r5icnIe28NxH/87qZixtRLnSaOdT8+1CcSeoSOmmADoRB7uTYE+BG+KyU34EB7nxPv64wAyN0/h8aH3miFOhB3mOs6IZRnk+rYzGthEVR9tWF80X9Je2TanharV/XK+f1WYklckAOyTEJyRk5J5ekSVqEk3vyRJ7JS/AavAXvwcf36EIw29knvxB8fgEAAqbR</latexit>

<latexit sha1_base64="HycxHRgZ42BHbH8otmgvOElrUk8=">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</latexit>

Mx/y

1,M2

FINAL FIG2

AFM

<latexit sha1_base64="4+WoQDmCJDv5rmRMIwVqY7t2rFg=">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</latexit>

1(˜x),C

2(˜x)

<latexit sha1_base64="q2pkWTi+Yo5j8dO0qEr7Cnz9x2U=">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</latexit>

1(˜x)

<latexit sha1_base64="n3t+l/2nIjqhMZXjyX8yqBm7+D0=">AAACK3icbVDLSgMxFM34tr516SZYBN2UjtTHUnDjUsGq0BlLkrmjoUlmSO6oZZj/cKt7v8aV4tb/MK1d+DoQOJxzL+fm8FxJh83mazA2PjE5NT0zW5ubX1hcWl5ZPXdZYQW0RaYye8mZAyUNtFGigsvcAtNcwQXvHQ38i1uwTmbmDPs5xJpdG5lKwdBLV0fdna0IpUqgvK+2u8v1ZqM5BP1LwhGpkxFOuivBdJRkotBgUCjmXOcgx7hkFqVQUNWiwkHORI9dQ8dTwzS4uBxeXdFNryQ0zax/BulQ/b5RMu1cX3M/qRneuN/eQPzP6xSYHsSlNHmBYMRXUFooihkdVEATaUGg6nvChJX+VipumGUCfVE/UrhlPUD/DwN3ItOamaSMOK86YVyW0SCZp7QeVlXNVxf+LuovOd9phHuN1mmrfrg7KnGGrJMNskVCsk8OyTE5IW0iiCUP5JE8Bc/BS/AWvH+NjgWjnTXyA8HHJxTRptM=</latexit>

C2(˜x)

<latexit sha1_base64="z7Z1NN2dDdXLdJ6q04w8veNk9sw=">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</latexit>

hC2(˜x)i

<latexit sha1_base64="qsp3U4+q3hlKyotPC/uvbg40kj8=">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</latexit>

hCy

1(˜x)i

<latexit sha1_base64="sGvZaKGAmEPPozpPiPGLX45pYcI=">AAACHnicbVBNS8NAEN34bf1q9ehlMQieSiKiHgtePFqwWkhD2d1MdOnuJuxulBLyC7zq3V/jTbzqv3Fbc9DWgYHHezO8mUdzwY0Ngi9vYXFpeWV1bb2xsbm1vdNs7d6YrNAMeiwTme5TYkBwBT3LrYB+roFIKuCWji4m+u0DaMMzdW3HOcSS3Cmeckaso7rBsOkH7WBaeB6ENfBRXVfDlrc6SDJWSFCWCWJMdJ7buCTaciagagwKAzlhI3IHkYOKSDBxOT20woeOSXCaadfK4in7e6Mk0pixpG5SEntvZrUJ+Z8WFTY9j0uu8sKCYj9GaSGwzfDka5xwDcyKsQOEae5uxeyeaMKsy+aPC9VkBNb9oeCRZVISlZQDSqsojMtyMHGmKfbDqmq46MLZoObBzXE7PG2fdE/8TliHuIb20QE6QiE6Qx10ia5QDzEE6Ak9oxfv1Xvz3r2Pn9EFr97ZQ3/K+/wGvYKheg==</latexit>

<latexit sha1_base64="XTGNxyJnDCDMRSBEk2WQvAIPUVw=">AAACInicbVBNSwMxEM3Wj2r9avXoJVgET2VXi/ZY8OJRwdVCu0iSzrahSXZJskpZ9jd41bu/xpt4EvwxZrUHbR0Iebw3w5t5NBXcWN//8CpLyyur1bX12sbm1vZOvbF7Y5JMMwhZIhLdo8SA4ApCy62AXqqBSCrglk7OS/32HrThibq20xQiSUaKx5wR66jQb/n+yV29Wf5l4UUQzEATzeryruFVB8OEZRKUZYIY0++kNsqJtpwJKGqDzEBK2ISMoO+gIhJMlH8vW+BDxwxxnGj3lMXf7O+JnEhjppK6Tkns2MxrJfmf1s9s3IlyrtLMgmI/RnEmsE1weTkecg3MiqkDhGnudsVsTDRh1uXzx4VqMgHr7lDwwBIpiRrmA0qLfhDl+aB0pjFuBkVRc9EF80EtgpvjVnDaal+1m91gFuIa2kcH6AgF6Ax10QW6RCFiiKNH9ISevRfv1Xvz3n9aK95sZg/9Ke/zC6EoomM=</latexit>

0.003

<latexit sha1_base64="lv2i0HJQ+MfYPxqjJHaHzNotCHw=">AAACInicbVBNSwMxEM3Wr1q/Wj16CS6CF0tXinosePGo4KrQLiXJzrahSXZJskpZ9jd41bu/xpt4EvwxprUHWx0Iebw3w5t5NBPc2Fbr06ssLa+srlXXaxubW9s79cburUlzzSBkqUj1PSUGBFcQWm4F3GcaiKQC7ujoYqLfPYA2PFU3dpxBJMlA8YQzYh0VHrearaBf9903LfwXBDPgo1ld9RveWi9OWS5BWSaIMd3zzEYF0ZYzAWWtlxvICBuRAXQdVESCiYrpsiU+dEyMk1S7pyyesr8nCiKNGUvqOiWxQ7OoTcj/tG5uk/Oo4CrLLSj2Y5TkAtsUTy7HMdfArBg7QJjmblfMhkQTZl0+cy5UkxFYd4eCR5ZKSVRc9Cgtu0FUFL2JM02wH5RlzUUXLAb1F9yeNIPTZvu67XeCWYhVtI8O0BEK0BnqoEt0hULEEEdP6Bm9eK/em/fuffy0VrzZzB6aK+/rG5h8ol4=</latexit>

0.01

<latexit sha1_base64="KaeBAe3y9wg3V4nf1R95mdXMv0Q=">AAACHXicZVDLSgMxFE18W9+6dBMsghvLjPhaieDGjaBoq9AOkqR3NDaPIckoZZh/cKsbv8aduBX/xox2YfVAuIebnHtuDsukcD6KPvHI6Nj4xOTUdG1mdm5+YXFpueVMbjk0uZHGXjHqQAoNTS+8hKvMAlVMwiXrHVX3l/dgnTD6wvczSBS90SIVnPrQam1GjSjauV6sV7UC+U/iAamjAU6vlzDudA3PFWjPJXWuvZ/5pKDWCy6hrHVyBxnlPXoD7UA1VeCS4nvbkqyHTpekxoajPfnu/lYUJ/q8r5iRQ3MKqpyi/jboq+KGPXy6nxRCZ7kHzX8s0lwSb0j1adIVFriX/UAotyJsSfgttZT7EM2QC7O0Bz44a3jgRimqu0WHsbIdJ0XRqZxZSupxWdZCaPHfiP6T1lYj3m1sn23XDw8G8U2hVbSGNlCM9tAhOkanqIk4ukOP6Ak94xf8it/w+8/TETzQrKAh4I8vsXWgWA==</latexit>

−0.005

<latexit sha1_base64="RPZ1SItMpBS1tEInJQKSIbKZajk=">AAACHXicZVDLSgMxFE18W5/VpZtgEdxYOlIfKym4cSMo2iq0gyTpHY3NY0gyShnmH9zqxq9xJ27FvzGtXVg9EO7hJueem8NSKZyv1b7wxOTU9Mzs3HxpYXFpeWW1vNZyJrMcmtxIY68ZdSCFhqYXXsJ1aoEqJuGK9Y4H91cPYJ0w+tL3U4gVvdUiEZz60Grt1Kq1aO9mtRLqEOQ/iUakgkY4uylj3OkaninQnkvqXPsw9XFOrRdcQlHqZA5Synv0FtqBaqrAxflw24JshU6XJMaGoz0Zdn8r8lN90VfMyLE5OVVOUX8X9IPixj18chjnQqeZB81/LJJMEm/I4NOkKyxwL/uBUG5F2JLwO2op9yGaMRdmaQ98cNbwyI1SVHfzDmNFO4rzvDNwZgmpREVRCqFFfyP6T1q71Wi/Wj+vVxpHo/jm0AbaRNsoQgeogU7QGWoiju7RE3pGL/gVv+F3/PHzdAKPNOtoDPjzG7MqoFk=</latexit>

−0.015

<latexit sha1_base64="59yz13Vp5LHMscZdXBw/TcBDbyQ=">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</latexit>

˜x

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˜y

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ˆ

iˆ

j

FIG. 2. Correlations in squeezed space & Hamiltonian reconstruction. a, Spin-spin correlations ⟨ˆ

iˆ

j⟩on the

squeezed lattice of an original 20 ×2 mixD t−Jladder, with nh=0.2, T/J=5/3∼1.67, and using 20,000 snapshots. In the

bulk of squeezed space, hole motion distorts the spin background, leading to negative correlations across diagonals. In this

region, eﬀective J1−J2physics is expected, as captured by the Hamiltonian Eq. (2). As holes are rarely located at the open

boundaries of the system, correlations are left almost undisturbed and are of AFM type. Correlations along nearest neighbors

in xgo beyond the cutoﬀ of the colorbar. b, Rung Cy

1(˜x)=⟨ˆ

[˜x,˜

0]ˆ

[˜x,˜

1]⟩and diagonal C2(˜x)=⟨ˆ

[˜x,˜

0]ˆ

[˜x+1,˜

1]⟩+⟨ˆ

[˜x+1,˜

0]ˆ

[˜x,˜

1]⟩

correlations. In the central bulk region of the ladder, correlations are approximately constant, the average being used as the

input for Jx

1−Jy

1−J2Hamiltonian reconstructions. In particular, we discard the two outer sites in squeezed space, as illustrated

by the yellow box. Dashed-dotted lines correspond to rung and diagonal correlations for a nearest-neighbor Heisenberg model

with βJx

1=βJy

1∼0.44, which captures the physics at the edges, but fails to describe the correlations in the bulk of squeezed

space. The introduction of diagonal (frustrating) bonds is hence an essential step to describe the spin system on the squeezed

lattice. c, Summed correlations in the boxed bulk in balong rungs, legs and diagonals for varying snapshot set sizes. Light

regions correspond to the standard error to the mean. After a few thousand measurements, convergence of the correlator proxies

is reached. d, Results show perfect agreement between the correlations emerging from a reconstructed eﬀective Jx

1−Jy

1−J2

Heisenberg model (black crosses) and bulk averaged correlations of the doped mixD model in squeezed space (solid lines).

are independent of the true temperature of the squeezed

magnetic environment.

Intermediate temperature regimes TJ≳1 – as also

chosen in our simulations – have been shown to work

best for reconstructions of the underlying coupling pa-

rameters, as both in the low and high temperature limit

the energy landscape deﬁned by Gis entirely ﬂat [30].

Given a size Lx×Lyof the mixD system, the dimensions

of the reconstructed Jx

1−Jy

1−J2Heisenberg ladder on

the squeezed lattice is given by ˜

Lx×˜

Ly=(1−nh)Lx×Ly.

Hamiltonian reconstruction results for a single run are

presented in Fig. 2d. Evaluated correlations of the best

ﬁtting Jx

1−Jy

1−J2model are seen to perfectly match

the measured mean correlations in the bulk of squeezed

space, hence strongly supporting that the physics of the

magnetic background in the mixD t−Jmodel is well cap-

tured by Jx

1−Jy

1−J2Heisenberg interactions on a square

lattice. We have explicitly checked that independent of

the initially chosen parameter values for the GD, {JH}

always converge to identical points in parameter space,

underlining the robustness of the GD scheme – see Sup-

plementary Note 1.

To characterize and classify the reconstructed spin

states in squeezed space as a function of doping, we per-

form ground state calculations of the Jx

1−Jy

1−J2Heisen-

berg model and evaluate the static spin structure fac-

tor (SSSF) given by S(qx, qy)=1

xL2

y∑i,jeiq⋅(i−j)ˆ

Si⋅ˆ

Sj.

For Jx

1=Jy

1=J1, it has been demonstrated that a highly

frustrated magnetic regime exists for 0.4≲J2J1≲0.6

that is sandwiched by a N´eel and stripe AFM phase [49–

54]. Though the exact nature of the non-magnetic ground

state in the frustrated regime is still controversial, it re-

mains a promising candidate for the realization of a quan-

tum spin liquid phase possibly described by Anderson’s

resonating valence bond (RVB) paradigm [55–62]. We

evaluate the hybrid order parameter S(π, π)+S(0, π)in

the Jy

1Jx

1−J2Jx

1parameter space, signaling whether

AFM or stripe AFM order exists in the system. Dark

regions in the background of Fig. 1bcorrespond to no

apparent spin ordering, and hence signal the existence of

a strongly frustrated spin state akin to the observations

in the homogeneous J1−J2model.

The reconstruction process, consisting of (i) approx-

imating correlations in squeezed space using snapshots

and (ii) performing the GD, is repeated a total number

of ten times. Averaging over the converged results of all

runs leads to the main result of this paper, presented in

Fig. 1bby blue connected symbols. We observe how

the spin state in squeezed space rapidly approaches the

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摘要：

Quantifyinghole-motion-inducedfrustrationindopedantiferromagnetsbyHamiltonianreconstructionHenningSchl¨omer,1,2,3,∗TimonA.Hilker,2,4ImmanuelBloch,1,2,4UlrichSchollw¨ock,1,2FabianGrusdt,1,2andAnnabelleBohrdt3,5,†1DepartmentofPhysicsandArnoldSommerfeldCenterforTheoreticalPhysics(ASC),Ludwig-Maximilian...

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Quantifying hole-motion-induced frustration in doped antiferromagnets by Hamiltonian reconstruction Henning Schl omer1 2 3Timon A. Hilker2 4Immanuel Bloch1 2 4.pdf

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Quantifying hole-motion-induced frustration in doped antiferromagnets by Hamiltonian reconstruction Henning Schl omer1 2 3Timon A. Hilker2 4Immanuel Bloch1 2 4

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