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Quantum simulation of the Hubbard model

CitationChiu, Christie Shinglei. 2019. Quantum simulation of the Hubbard model. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences.

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ultracold lithium-6 in an optical latticeHubbard model

new physical insight?

measurement

approximate Hamiltonian

lanthanum barium copper oxide

strange metal

AFM

doping

temperature

pseudogap Fermi liquid

Mott

cuprate superconductor phase diagram

d-wave SC

copper

lanthanum/bariumoxygen

Figure 1.1: Quantum simulationwith

the Hubbardmodel. Lathanum bar-

ium copper oxide (LBCO) was the first

high-temperature superconductor dis-

covered. It is one in a family of cuprate

materials which exhibit this phenomenon.

While thesematerials have been stud-

ied extensively and phase diagrams have

beenmapped out (here a representative

schematic is shown), many open questions

remain. The underlying physics stems from

the valence electrons of the copper atoms

which are arranged in two-dimensional

(2D) planes. Therefore, we use ultracold

neutral atoms in a 2D plane of an optical

lattice to simulate these electrons and gain

physical insights.

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H = −t∑

σ=↑,↓

∑

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(c†J,σ cK,σ + I.D.

)+ U

∑

Kc†K,↑cK,↑c

†K,↓cK,↓ ��

�

increasing particle number

increasing interaction U/t

decreasing temperature T/J

band insulatormetal

half-fillingfilledband

barely filledband

metalU≪ t U≫t

Mott insulator

Mott insulator antiferromagnet

T > J T < J

U≪Tor

at half-filling,

at half-filling and strong interactions,

Figure 2.1: Introduction to the Hubbardmodel. The

basics of the Hubbardmodel can be understood by con-

sidering its limits. In the limit of very few particles or

verymany particles (a filled band), metals and band insu-

lators can be realized, respectively. At half-filling, if the

tunneling t or temperatureT dominate, again we have

ametal; if the interactionU dominates (for repulsive

interactionsU > 0), we realize aMott insulator. At

intermediate to strong interactions, for sufficiently low

temperature an antiferromagnet emerges.

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0energy[Er]

harmonic trap frequency≈ 330 Hz

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0

trap depth ≈ 8 Er

lattice spacing a ≈ 570 nm

−10 0 10position [µm]

−18

0 with entropy redistribution patternsystem-reservoir offset ≈ 8 kHz

−17 0 17−17

0

17position[sites]

0

15

energy[kHz]

−17 0 17position [sites]

−17

0

17

0

15

0

-500

energy[kHz]

0

-500

0

-500

a

b

c

Figure 2.2: Optical lattice for

lithium-6. a,Cross section of

optical potential from retrore-

flected beams at parameters

used to study the Hubbard

model. The beam geometry

is sketched in figure 2.4 and

creates a lattice with a gaussian

envelope, which is approxi-

mately quadratic at the trap

center. b, Zoom-in of the opti-

cal lattice and corresponding

on-site energy landscape. c,

Lattice and on-site energies

with potential reshaping for

entropy redistribution. The

reshaping is donewith a digital

micromirror device and is dis-

cussed in the next chapter. The

tunneling is approximately 0.9kHz for a ground band band-

width of 7.2 kHz; the bandgapis approximately 100 kHz.

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quantum gasmicroscopy

angle-resolvedphotoemissionspectroscopy

resonantinelastic

X-rayscattering

scanning tunnelingmicroscopy

Figure 2.3: Quantum gasmicroscopy and condensed

matter approaches. Here I sketch the length- and time-

scales to which quantum gasmicroscopy and several

condensedmatter tools are sensitive, normalized to the

lattice spacing and particle tunneling times. Note that

the specifications shown here apply to the time-resolved

versions of the condensedmatter approaches discussed.

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a

b

lattice

lattice

hemispheric lens

glass cell

RamanRaman + pump

radial latticesvertical lattice

Raman

Raman+ pump

hemispheric lensglass cell

objective

imaging + DMD

atom plane(10 µm from lens)

Figure 2.4: Beam geometry of the Fermi gasmicroscope. a,

Side view. Atoms are trapped in a 2D plane of the radial lattices,

wavelength 1064 nm. They sit 10microns from the first element

of the high-resolution imaging system, a composite hemispheric

lens formed by a superpolished substrate in the glass cell, the

wall of the glass cell, and a lens outside of vacuum. Digital mi-

cromirror devices (DMDs) in the image plane project optical

potentials (650 nm, temporally incoherent) with single-site res-

olution. During imaging, a vertical lattice (1064 nm) provides

additional vertical confinement. A pair of Raman beams and

a pump beam enter the glass cell at an angle chosen to couple

to all three lattice directions. b,Bottom view. Radial lattices

reflect off of the bottom superpolished and coated surface of

the hemispheric lens before retroflecting, giving rise to an addi-

tional factor of two in trap depth which is critical for our Raman

sideband imaging.

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counts[a.u.]

0

1

singlesdensity

�6.5 0.0 6.5radius [sites]

0.00

0.15

entropy per particles [kB]

s = 0.016(3)kB

0.000.01

0.05

singles densityns

a b

Figure 3.1: Ultra-low-entropy cold-atom band insulator. a,Raw image of atom distribution with an entropy redistribu-

tion pattern designed to create a band insulator. The band insulator is in the center of the image, as doubly occupied

sites are imaged as empty, with a dilute metal at the exterior. AMott insulating ring can be seen as the chemical po-

tential increases monotonically from the edge to themiddle of the cloud. b,Average singles density in and around the

band insulator over 50 images. An azimuthal average confirms that the density is radially uniform and an average over

the entire band insulator (disk within dashed circle) yields an average entropy per particle of s = 0.016(3) k#.

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0 1 2density

0

1

8

entropy perparticle s [kB]

NLCE

0.0 0.3 0.6 0.8 1.0

0.0 0.6 1.0redistribution pattern fraction f

0.00

0.03

0.10

disk entropy perparticle s [kB]

0.18 0.28reservoir density

0.18

0.26

nearest-neighborcorrelator Cs (1)

0.00

0.01

0.02

0.03

0.04

disk singlesdensity ns

a b c

Figure 3.2: Entropy redistribution. a,NLCE numerics for the entropy per particle s in units of k# for a 2D system at

U/t = 8 andT/t = 1.3with homogenous density88. The key feature here is the sharp contrast in entropy per

particle for densities close to 0 and 2, which can be used to optimize entropy redistribution. b, Entropy per particle

within a 133-site band insulator for patterns ranging between a harmonic trap and a pattern designed for entropy

redistribution. While entropy redistribution already occurs in a harmonic trap as a result of density inhomogeneity,

under fixed global particle number and global entropy it can be improved by reshaping the potential landscape. The

final f = 1 pattern yields a slightly higher entropy per particle s than the optimum, possibly due to the loss of thermal

contact between the system and reservoir. c, Sign-corrected nearest-neighbor correlator Cs(1) for systems at half-

filling as a function of estimated reservoir density. While fluctuations in the reservoir density are large and the size

of the reservoir depends strongly on lattice alignment (here a size of 1055 sites is assumed and horizontal error bars

denote the standard deviation), there is a clear relationship between lower reservoir densities and stronger nearest-

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a b Figure 3.3: Digital micromirror device cal-

ibration. a,Density distribution of singly

occupied sites with DMD1 calibration

pattern and incident optical power of 16

mW. b, (upper) Singles density of each re-

gion highlighted in (a), and global chemical

potential, as a function of DMDpower.

(lower) Potential shift in each region

extracted from corresponding singles den-

sities, with fitted line denoting calibration.

The potential shift is expressed in units

of the tunneling t, which is approximately

0.9 kHz.

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local chemical potential

BI metal

low s high s

wall

BI metal

low s high s

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AFMmetal

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0

2

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0

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Figure 3.4: A quantum state engineering protocol. The three steps of theQSE protocol are depicted in increasing

levels of specificity. The schematic shows howwe use a coexistence of metallic and band insulating (BI) phases and

controlled separation to create a low-entropy antiferromagnet (AFM). The top view shows the physical geometry used

in our experiment, however the success of this protocol should be robust to the exact arrangement of the system and

reservoir, as long as they can be thermally connected and disconnected. The 3D cutaway shows howwe control optical

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BI

rectangular band insulator

rectangular wall

striped band insulator

+ + + +

increasing controldecreasing alignment stability

Figure 3.5: Geometries investigated for quantum state engineering.With entropy redistribution, we can create cir-

cular, rectangular, or striped band insulators. These geometries can be paired with a circular or rectangular wall to

realize configurations of varying control and sensitivity to alignment. In particular, patterns with straight edges cre-

ate well-defined boundaries and particle numbers, but must be well-aligned to the lattice sites. All of these patterns

are used for at least one of the results presented in this chapter, except for the rectangular band insulator with the

rectangular wall.

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τh

τr

Fre

eze

optical lattice

dipole trap

DMD 1

DMD 2

power

time

freeze

tl tw tr

a lattice loading with redistribution

time

freeze

optical lattice

dipole trap

DMD 1

power

tl

b quantum state engineering (QSE)

time

optical lattice

dipole trap

DMD 1

DMD 2

power

freeze

tl tw tr

Figure 3.6: Optical power ramp protocols. a,Ramp protocol for lattice loading with entropy redistribution, where

atoms are handed off from a crossed dipole trap into the combined optical lattice andDMD1 potential. The lattice is

then immediately and quickly ramped up to freeze out the tunneling in preparation for imaging. b,Ramp protocol for

quantum state engineering. After loading into the lattice andDMD1 potential, the DMD2 optical power is ramped up

to isolate the system and reservoir, after which the potential offset between doublons and holes introduced byDMD1

is ramped off. The lattice freeze follows. c, Ramp protocol for the round-trip measurement used to characterize the

quantum state engineering scheme. All quantum state engineering ramps are reversed to recover the original isolated-

system configuration. Furthermore, the extent and duration of the final ramp can be varied, as well as a hold time

before ramp reversal, to characterize the ramp adiabaticity and heating.

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0 50 250hold time [h/t]

0.000.05

0.25

0.92average singles density ns

1 site thick

1.5 sites thick

2 sites thick

3 sites thick

5 sites thick

a b

Figure 3.7: Characterization of isolation in QSE. a,Density distribution of singly occupied sites (40 realizations) and

configuration of entropy redistribution atU/t = 5.9(2) after isolation. The isolated system consists of an approx-

imately equal number of doubly occupied and unoccupied sites, whose boundary is denoted by the dashed rectangle.

Imperfections in the optical potential manifest as singly occupied sites, as seen at the upper edge of the box. Dashed

circles denote the boundaries of the isolating wall, which do not significantly modify the atom distribution from en-

tropy redistribution alone. b,Verifying wall insulation. After preparing a region containing fewer than 3 atoms, we

raise up a circular wall as in the quantum state engineering protocol. The potential used to prepare the initial state is

ramped off, leaving an atom distribution (upper left) and attractive harmonic well potential (lower left). The rate of

particle transport into the central region is measured, for varying wall thicknesses in lattice sites (right). For our QSE

protocol, we use a wall thickness of 3 lattice sites.

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0

1

singlesdensity

0

1singles density ns

−10 0 10radius [sites]

−0.21

0.00spin corr C1

0 45ramp time τr [h/t]

0.00

0.23

0.39

0.69

0.80

s [kB]

box

stripes

ln 2

0

1

singlesdensity

0

2

density

0.0 22.4 44.7time [h/t]

0.0

2.2

potentialoffset [U]

0.0

0.1

0.2

0.3ns

0.02.0final offset ∆f [U]

a b

∆f

τr

I II III

100 ms

BoxI II III

StripesI IIIII

Figure 3.8: Characterization of final state fromQSE. a,Average density map of 41 images after rampwith initial-

ization via disk pattern (upper), along with corresponding density and nearest-neighbor spin correlator profiles

versus radius after ramp (lower). The nearest-neighbor correlations are antiferromagnetic with a strength of up to

Cs(1) = −0.21(1). A simultaneous fit to both profiles (solid line) gives a temperature of k#T/t = 0.46(2).The fit is limited to radius 9, to avoid effects from the insulating wall. Error bars denote standard error of over 40

sets of correlationmaps and azimuthal averaging. HereU/t = 7.7(3). b, Examining final ramp adiabaticity at

U/t = 5.9(3). A round trip measurement beginning with an isolated box of doublons surrounded by holes demon-

strates non-adiabaticity predominantly in the second half of the offset ramp-off (upper left). Adiabaticity is not sig-

nificantly improved by initializing the holes and doublons in stripes. Horizontal lines with shading indicate reference

measurements and uncertainty, takenwith no ramp. Lower panels show schematic images for the particle density

n andmeasured average singles density maps for the box (left) and stripe (right) configurations at different times

throughout the round-trip ramp. Dashed lines indicate the wall inner edge, while dotted lines enclose band insulating

regions. Error bars are smaller than themarkers, and denote standard error of 40 (187) measurements for the box

(stripe) pattern.

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time

0.0

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0 40 80hold time τh [h /t]

0.0

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final offset ∆f [U]

0

1

2

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0.0

0.1

0.2

0.3

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0.0000

0.0010

0.0024

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0.02.0

a b c

Figure 3.9: Heating duringQSE. a,Ramp used for potential offset between the doubly occupied and unoccupied sites.

The offset is decreased to some value, at which point the sample is held for some time tomeasure a heating rate. All

measurements are taken after recovering the initial configuration for consistency, and offset ramps are set to the same

duration. b, Measured entropy per particle as a function of hold time τh for various potential offsets. Linear fits are

used to quantify heating rates. Error bars for entropymeasurements (right) denote standard error of 5measurements.

c, Extracted heating rates at various points throughout the ramp. Error bars are from the fits in (b).

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0 100.0

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initialize melt recapture

doublewell

4-sitechain

8-sitechain

14-siteladder

Figure 3.10: Adiabaticity

with smaller system sizes.

For a double well, 4-site

chain, 8-site chain, and

14-site ladder we vary the

total ramp time tomelt the

system and recapture it,

where both parts take equal

time. The resulting entropy

per particle is measured as a

function of ramp time. The

smaller two system sizes

recover adiabatic behavior

for sufficiently long times,

while the larger two do not.

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0.0

0.5

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initialize

melt recapture

isolate melt connect recapture

III

II

IIII

II

I

Figure 3.11: Investigatingmore

sophisticatedQSE protocols.We

test a protocol where pairs of sites

eachwith one doublon and one

hole are isolated and individually

melted, then connected to other

such pairs. After recapturing into

the initial configuration, the entropy

per particle is measured. Results are

similar to the original protocol but

may show slight improvement.

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2J∗

∑

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∑

σ

(eiΦ/4c†J,σ cJ+Y,σ + e−iΦ/4c†J,σ cJ+Z,σ + I.D.

)

− 1

2J∗

∑

J∈B

∑

σ

(e−iΦ/4c†J,σ cJ+Y,σ + eiΦ/4c†J,σ cJ+Z,σ + I.D.

) ��

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a b

c d

J∗2 e−iπ/4J∗

2 eiπ/4

J∗2 e−iπ/4

J∗2 eiπ/4

B

A

π −π

π−π

Néel staterestored

spinon at lower leftclockwise string =

spinon at upper rightcounter-clockwise string

indistinguishable fromfinal spin configuration

=

ℓ

n ℓ ≥ 1

multiplicity

0

1

2

3

z(z−1)ℓ−1

14

4·34·32

z = 4

C1

C1

C1C1

C1

C1

C1

C√2

C√2

C2

C1

C1

C1

C√2

C√2

C1

= J(1− 4C1)V (Σ; ℓΣ = 0)

= µhg0 +

V (Σ; ℓΣ = 1) = J(1− 7C1 )+ 2C√2

+ C2

= µhdEdℓ +

=

= J( − C1 )2 C√2V (Σ; ℓΣ = 2) V (Σ; ℓΣ = 1)−

dEdℓ

e

1 12string length ℓ[sites]

0.0

0.4

probabilityp (ℓ)

T/J = 0.00

T/J = 0.50

T/J = 0.60

T/J = 0.65

Figure 4.1: Microscopic theories of the dopedHubbardmodel. a, Staggered flux lattice forπ-flux states. The latticewith±π flux per plaquette yields states with a low variational energy, making them of interest. b, Trugman loop exam-

ple in the context of geometric string theory. Here a hole can restore the original Néel state by traversing a square one

and a half times. Equivalently, two holes at differing starting positions and the sameNéel state can create the same

end configuration, yielding two indistinguishable string states. These Trugman loops are omitted to create an approx-

imately orthonormal basis of string states. c,Mapping of string state from square to Bethe lattice. Themutiplicity of

states of length ℓ increases exponentially in ℓ. d,Derivation of string potential. By propagating a chargon and consid-ering its new and former nearest-neighbor interactions, its potential can be determined. e,Geometric string length

histogram for several temperatures, derived from geometric string theory.

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0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

0.5 69<34>

66<32>

67<33>

67<33>

0.6 2224<1122>

94<47>

527<264>

114<57>

1072<550>

511<255>

277<140>

262<131>

505<253>

210<103>

148<74>

209<104>

358<179>

149<74>

208<104>

0.7 4571<2380>

505<256>

712<357>

1610<858>

1918<1034>

1125<564>

1023<515>

450<227>

401<201>

886<446>

375<190>

357<181>

353<178>

372<190>

0.8 296<150>

78<39>

473<238>

337<169>

0.9 346<174>

191<96>

854<425>

1.0 317<160>

246<123>

273<137>

191<94>

244<122>

1.1 358<178>

76<39>

370<188>

485<243>

134<69>

1.2 430<217>

243<123>

240<120>

508<258>

463<234>

809<403>

1.3 242<123>

394<200>

71<35>

348<177>

798<399>

1.4

1.5 209<106>

147<75>

102<51>

108<54>

150<73>

1.6 839<453>

1276<704>

930<466>

460<231>

789<469>

263<131>

235<116>

1.7 289<146>

1091<648>

163<103>

100<52>

290<143>

1.8 198<100>

223<112>

784<397>

108<51>

374<187>

187<92>

91<46>

Table 4.1: Number of experimental realizations. For each doping value (columns) and temperature (rows, in units ofJ ),this table lists the number of experimental realizations with spin removal andwithout (in brackets).

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0.00 0.06 0.14 0.32doping �

0.00

0.36

spin correlatorCs (d)

dQMC T = 0.6J

dQMC T = 0.7J

0.00 0.06 0.14 0.32doping �

d = 1

d =p

2d = 2

T/J

0.5

0.7

a b Figure 4.2: Temperature de-

terminationwith the spin-spin

correlator. a, Sign-corrected

nearest-neighbor spin correlator

versus doping from experiment

and dQMC. Doping the system

does not change the temper-

ature beyond experimental

uncertainty. b, First three cor-

relation strengths for datasets

prior to grouping by doping. The

temperature is indicated by the

colorbars.

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select window with higheststaggered magnetization

images with one spin species removed

ns

1

0

no more string patterns

...

single image histogram

postselect on all data fortop 60% staggered magnetization

single image scan window across image

...+

window shape

difference fromreference checkerboard

1

0

identifylongest string pattern

count string pattern & remove

identifylongest string pattern

count string pattern & remove

experiment

simulation- geometric strings- �-flux states- sprinkled holes

or

postselection

i

reduced ROI andsample size

iteratepattern detection

ii iii

i postselection within each image and across dataset

ii comparison to reference checkerboard

single image

final histogram

1 9string-pattern length [sites]

10−1

10−2

10−3

10−4

pδ(ℓ)

0

4counts


0

4counts


0

4counts


ii identification of string patterns from difference matrix

find closer reference pattern

Figure 4.3: String detection algorithm. The string detection algorithm only uses images with one spin removed, but

these images can come from experiment or one of the simulations. After postselection, the deviation from a reference

checkerboard pattern is used to identify string patterns. String patterns consist only of sites which deviate from the

reference checkerboard andmust have an unoccupied site at one end, consistent with having a hole at that site. There-

fore, after all possible string patterns have been counted and removed, theremay be remaining sites which deviate

from the reference checkerboard but do not contribute to the string pattern observables.

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experiment geometric strings sprinkled holes ⇡-flux

0%

6%

14%

Figure 4.4: Example snapshots from experiment and simulation. Randomly selected single-spin-state images from

experiment, geometric string theory, sprinkled holes, andπ-flux states. Notice that at half-filling, images from experi-

ment, geometric string theory, and sprinkled holes are identical by construction. Snapshots from doping values of 6%and 14% are also shown.

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doping � = 0.1

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geometric strings

⇡-flux states

infinite T

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0.000

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cold

hot

1 2 3 4 5 6 7 8 9


0

1

2

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a b c

Figure 4.5: Measurement of string-pattern length histograms from site-resolved snapshots. a,Change in string-

pattern length histograms upon doping, for the “cold” dataset. The observable is sensitive to doping in this regime,

and simulated strings seem to fit the doped experimental result best. Points have been slightly offset horizontally

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sprinkled-hole pattern-length histograms, highlighting temperature-dependent sensitivity.

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��

0 6x-extent [sites]

0

6y -extent[sites]

10�3

10�2

10�1

100

0

6y -extent [sites]

�0.3

0.0

0.3

0 6x-extent [sites]

0

6y -extent [sites]

0 6x-extent [sites]

0 6x-extent [sites]

�4

0

4

doping 0.00 doping 0.06 doping 0.14 doping 0.32a

b absolute di↵erence

c relative di↵erence

Figure 4.6: String-pattern shape

distributions. a,Histogram of

the spatial extent of detected

string patterns in the two lat-

tice directions (x-extent andy-extent), for various dopingvalues. They all appear to be

isotropically distributed within

statistical fluctuation, but per-

haps with slightly more aspect

ratios close to 1. b,Absolute

difference between doped and

half-filling histograms. The (0, 0)

bins are saturated for visibility of

the remainder of the plot. c,Rel-

ative difference. Some bins are

saturated so that others are visi-

ble in the diverging colorbar. The

large values of these saturated

bins is likely due to statistical

fluctuation.

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��

1 2 3 4 5 6 7 8 9


10�1

10�2

10�3

10�4

p�(`)Heisenberg QMC � = 0.00

experimentT/J = 0.01T/J = 0.6T/J = 0.6, 8% doublon-hole pairs

1 2 3 4 5 6 7 8 9


� = 0.10

full readout, p�(`)

partial readout, p�(`) � p0(`)

a b Figure 4.7: Contributions to the half-filling

baseline and effect of partial spin readout.

a, String-pattern length distribution for

various HeisenbergQMC simulations at half-

filling, compared to experiment. b, String

pattern detection with simulated full spin

readout and partial readout (with half-filling

signal subtracted) using HeisenbergQMC

simulation with added charge fluctuations

and strings. The signals have the same shape

but are offset due to the lower detection

efficiency of partial spin readout.

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��

0.00 0.06 0.14 0.32

doping �

0.00

0.02

0.04

string-pattern countper site

experiment

sprinkled holes

geometric strings

⇡-flux states

experiment � = 0.0

0.000

0.003

absolute di↵erencep0.06(`) � p0.06

s (`)experiment

geometric strings

1 2 3 4 5 6 7 8 9


0.000

0.003

p0.14(`) � p0.14s (`)

I

II

I II

Figure 4.8: String-pattern count versus doping. a, Total number of string patterns exceeding length 2, normalized by

the system size, as a function of doping. While the stringmodel and sprinkled hole simulation both agree with experi-

ment at half filling by construction, already at low doping the stringmodel performs significantly better than sprinkled

holes. The stringmodel is quantitatively accurate across a larger doping range than forπ-flux states, but both are ingreater agreement with experiment than the sprinkled hole simulation. b,Absolute difference between experiment

and sprinkled-hole string-pattern length histograms for 6.0(5)% and 13.9(6)% doping. While the absolute differ-

ence between experiment and sprinkled-hole pattern-length histograms increases with doping, the shape remains

roughly invariant.

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��

0.00

0.02

0.04

string-pattern countper site

small window experiment

sprinkled holes

geometric stringsπ-flux states

experiment δ = 0.0 big window

0.00

0.02

0.04

fixed window moving window

0.00 0.06 0.14 0.32

doping δ

0.00

0.02

0.04

40% postselection

0.00 0.06 0.14 0.32

doping δ

80% postselection

a

b

c

z

p(z)

z

p(z)

Figure 4.9: Effect of string count

on post-selection. All param-

eters not mentioned are kept

fixed as in the procedure out-

lined in section 4.3.1. In all cases,

we see that the qualitative

features seen in figure 4.8 are

maintained. a,Using a different-

size window for the analysis

region, either 5 or 8 sites in di-

ameter. b, Fixing the window

position to the center of the

system, compared to scanning

the window position tomaximize

the staggeredmagnetization. c,

Varying the percentage of data

kept when postselecting on the

staggeredmagnetization, either

40% or 80%.

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��

0.00 0.06 0.14 0.32

doping �

0.00

0.02

0.04

string-pattern countper site experiment

T = 0.55J

T = 0.65JT = 0.60J


0.00 0.06 0.14 0.32

doping �

experiment

100%

80%50%


0.00 0.06 0.14 0.32

doping �

experiment

length 2length 3

length 4infinite length


a b c

Figure 4.10: Effect of analytic string length distribution onmeasured string-pattern count. a,Comparison of exper-

imentally measured string-pattern count and simulated geometric strings, for analytic string length distributions

corresponding to temperaturesT/J of 0.55, 0.60 (as in all other figures), and 0.65. TheT/J = 0.60 distributionmatches experiment best. b, Same, but varying the percentage of simulated holes which are then propagated to simu-

late strings. The best agreement occurs when all holes are part of strings. c, Same, but for simulated strings which are

of all of a single length, ranging from 2 sites to the infinite-site limit, instead of sampled from the analytic distribution.

0.00 0.06 0.14 0.32

doping �

0.0

0.1

0.2total number of string patternsper site

experimentsprinkled holes

geometric strings⇡-flux states


Figure 4.11: Total number of string patterns detected versus doping.

String-pattern count versus doping, as in figure 4.8, but including

all string-pattern lengths in the count. The agreement between

all simulations and experiment suggests that this quantity may be

trivially dependent on doping.

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0.000

0.008

string-pattern countdi↵erence

0.6 1.0 1.6temperature [J]

0.02

0.04

string-patterncount

experiment

sprinkled holes

Figure 4.12: String pattern sensitivity to temperature.

For a doping of 10%, the difference between the total

string-pattern count and corresponding sprinkled-hole

string-pattern count is plotted as a function of tempera-

ture. Sensitivity to strings decreases with temperature

due to decreased order in the parent antiferromagnet as

seen in the sprinkled-hole string-pattern count, inset.

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0.00 0.06 0.14 0.32doping �

1.4

2.0

2.2

averagestring-pattern length` [sites]

experimentgeometric strings

varied T to match mz + sprinkled holesvaried T to match Cs (1) + sprinkled holes

Figure 4.13: Average string-pattern length versus dop-

ing. Doped antiferromagnets exhibit longer-length string

patterns compared to heated antiferromagnets, even

when the staggeredmagnetization or nearest-neighbor

spin correlator is equal and holes are sprinkled in to

equate doping levels.

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0.0 0.3

average staggered magnetization hmzi

0.0

0.3spin correlator Cs (1)

experiment, vary dopingexperiment, vary temperatureexperiment, vary temperature + 10% sprinkled holesexperiment, vary temperature + 20% sprinkled holesexperiment, vary temperature + 30% sprinkled holes

0.00

0.05averagestring-pattern count

1.3 2.5

average string-pattern length

1.3

2.5averagestring-pattern length

0.00 0.05

average string-pattern count

Figure 4.14: Multi-parameter comparison of doped and heated antiferromagnets.Weplot pairwise the average stag-

geredmagnetization, sign-corrected nearest-neighbor spin correlator, average string pattern length, and average

string-pattern count for both doped and heated antiferromagnets, as well as density-adjusted heated antiferromag-

nets. By examining the values which can be obtained by each dataset, we identify regimes attainable by doping but not

heating, or vice-versa.

��

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10�1

10�2

10�3

10�4

probabilityp�(`)

doping � = 0.0

doping � = 0.1

sprinkled holes

geometric strings

⇡-flux states

0.00

0.04

string-patterncount per site


1 2 3 4 5 6 7 8 9


10�1

10�2

10�3

10�4

10�5

p�(`)

0.00 0.06 0.14 0.32

doping �

0.000

0.008

string-patterncount per site

a

b

Figure 4.15: Alternate string

detection schemes. a, The string-

pattern length histogram and

string-pattern count produced

with the simplified difference

pattern extraction algorithm.

Apart from an increase in values

of the string-pattern-based

observables at half-filling, the

results are similar to themain

detection algorithm. b, Same

quantities, but producedwith

the happiness method, are also

sensitive to doping but may be

too noisy to be useful.

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0.00 0.06 0.14 0.32

doping δ

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0.090.11

0.25


experiment

sprinkled holes

geometric strings

π-flux states

0.00 0.06 0.14 0.32

doping δ

0.00 0.06 0.14 0.32

doping δ

0.0 0.8staggered magnetization mz

0.0

0.2p(mz )

δ = 1.3(5)%


δ = 6.0(5)%


δ = 13.9(6)%

a

b

Figure 4.16: Spin correlations and staggeredmagnetization. a,Decay of nearest-neighbor (left), diagonal next-nearest-

neighbor (center), and straight nearest-neighbor (right) spin-spin correlation functions upon doping. While theπ-fluxtheorymost quantitatively explains Cs(1), only the stringmodel captures the sign change of Cs(

√2). In all three

cases, sprinkled holes overestimate the spin correlations. b, Full counting statistics of the staggeredmagnetization

for doping values of 1.3(5)% (left), 6.0(5)% (center), and 13.9(6)%(right). Here bothπ-flux states and geometric

strings show reasonable agreement, while sprinkled holes do not.

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doping �

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0.000

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experiment d =p2

experiment d = 2

0.00 0.06 0.14 0.32

doping �

0

1

normalized g (2)

g (2)(d)

experimentfree chargon theory

⇡-flux statesmagnetic polaron theory

1 2 4distance d [sites]

�0.015

0.005

charge correlatorCh(d)

� = 0.0(6)%


� = 6.0(5)%


� = 13.9(6)%


experiment⇡-flux states

free chargon theory

� = 19.7(6)%

a

b c

Figure 4.17: Observation of hole anti-bunching. a,Anti-moment correlation function for various doping values. The

correlation functions show a growth of hole anti-bunching with doping out to a distance of d = 2. b,Diagonal next-nearest neighbor and straight next-nearest neighbor anti-moment correlators versus doping. At both distances neg-

ative correlations growwith doping. c,Normalized anti-moment correlator at d =√2 versus doping. The experi-

mental result cannot be explained by theπ-flux or a point-likemagnetic polaron theory, but insteadmatches a free

fermionic chargon theory.

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a

b

Figure 4.18: Performance of phenomenological models. a, String length histogram, spin correlation function, and stag-

geredmagnetization distribution for experiment and a phenomenological model where spins are flipped in a random

spin distribution until the nearest-neighbor and next-nearest neighbor spin correlators match experimental values. b,

Same quantities, but comparing experiment to a phenomenological model where singlet pairs are added to a classical

Néel checkerboard and a projectivemeasurement is performed. The singlet pair density is tuned to achieve rough

agreement with the doped experimental data. For both subfigures, experimental data is shown in circles; simulated

data is shown in dashed lines.

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1 10 20 30 40

patch-pattern size `[sites]

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a b c

Figure 4.19: Expanding pattern detection to classical Néel patches. a,Histogram of classical Néel patch sizes as de-

tected after our standard postselection, shown in a linear-linear plot (top) and a linear-log plot (bottom). These his-

tograms exhibit the key features of a peak at large sizes and a plateau at more intermediate patch sizes. It also appears

that the geometric strings match the experimental data best. b, Same, but without postselection. The conclusions

remain unchanged. Not surprisingly, the peak in patch sizes is shifted to larger sizes and exhibits a reduced contrast.

c, Total number of patch patterns with size greater than two sites, with andwithout postselection. The close similar-

ities to the string-pattern count reflect a limited sensitivity of this observable to distinguishing between theories as

discussed in the text. The regions of agreement and disagreement between geometric strings and experiment are con-

sistent with that of the nearest-neighbor spin correlator and the assumption of dilute, noninteracting strings in the

geometric-string theory.

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http://dx.doi.org/10.1007/BF01303701

http://dx.doi.org/10.1007/BF01303701

http://dx.doi.org/10.1103/PhysRevLett.89.220407



http://dx.doi.org/10.1143/PTP.30.275

http://dx.doi.org/10.1098/rspa.1963.0204

http://dx.doi.org/10.1007/BF02650179



http://dx.doi.org/10.1038/374434a0

http://dx.doi.org/10.1038/374434a0

http://dx.doi.org/10.1103/PhysRevB.57.13406



http://dx.doi.org/10.1146/annurev-conmatphys-070909-104117

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http://dx.doi.org/10.1126/science.aad9041

http://dx.doi.org/10.1126/science.aag1430


http://dx.doi.org/10.1038/nature22362

http://greiner.physics.harvard.edu/assets/theses/mazurenko_thesis.pdf

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https://arxiv.org/pdf/1810.03584.pdf

http://dx.doi.org/10.1103/PhysRevA.79.033620



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http://dx.doi.org/10.1103/PhysRev.158.383



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http://dx.doi.org/10.1088/0022-3719/10/10/002

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http://dx.doi.org/10.1103/RevModPhys.82.1225



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https://www.physics.ncsu.edu/jet/theses/pdf/Gehm.pdf



https://doi.org/10.3929/ethz-a-006278918

https://d-nb.info/1018411739/34

https://d-nb.info/1018411739/34

http://dx.doi.org/10.1126/science.285.5434.1703

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http://dx.doi.org/10.1038/415039a

http://dx.doi.org/10.1038/415039a






http://dx.doi.org/10.1126/science.1165449





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http://dx.doi.org/10.1126/science.aac9812



http://dx.doi.org/10.1088/1367-2630/18/2/023016

http://dx.doi.org/10.1088/1367-2630/18/2/023016



http://dx.doi.org/10.1126/science.aam7838

http://dx.doi.org/10.1038/s41578-018-0047-2

http://dx.doi.org/10.1038/s41578-018-0047-2



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http://dx.doi.org/10.1088/0953-8984/21/16/164217

http://dx.doi.org/10.1016/S0368-2048(01)00266-3



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http://dx.doi.org/10.1098/rsta.2017.0480

http://dx.doi.org/10.1098/rsta.2017.0480

http://hoffman.physics.harvard.edu/publications/theses/2003-Jennifer-Hoffman-thesis.pdf

http://hoffman.physics.harvard.edu/publications/theses/2003-Jennifer-Hoffman-thesis.pdf

http://dx.doi.org/10.1103/PhysRevB.57.R6858

http://dx.doi.org/10.1103/PhysRevB.57.R6858

http://dx.doi.org/10.1088/0022-3727/44/46/464010

http://dx.doi.org/10.1088/0022-3727/44/46/464010

http://dx.doi.org/10.1088/0022-3727/44/46/464010



http://dx.doi.org/10.1016/j.stam.2005.06.021

http://dx.doi.org/10.1016/j.stam.2005.06.021

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http://dx.doi.org/10.1038/s41586-018-0778-7



http://dx.doi.org/10.1126/science.aat4387



http://greiner.physics.harvard.edu/assets/theses/tschuster_thesis.pdf

http://greiner.physics.harvard.edu/assets/theses/fhuber_diploma.pdf



http://dx.doi.org/10.1063/1.5066623

http://dx.doi.org/10.1063/1.5066623






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http://dx.doi.org/10.1038/ncomms1374











http://dx.doi.org/10.1088/1367-2630/9/5/134

http://dx.doi.org/10.1088/1367-2630/9/5/134



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http://dx.doi.org/10.1073/pnas.1600267113

http://dx.doi.org/10.1073/pnas.1600267113





https://doi.org/10.3929/ethz-a-010008097



http://dx.doi.org/10.1080/13642810108208566

http://dx.doi.org/10.1080/13642810108208566






http://dx.doi.org/10.1126/science.235.4793.1196


http://dx.doi.org/10.1088/0034-4885/71/1/012501

http://dx.doi.org/10.1088/0034-4885/71/1/012501

http://www.jetp.ac.ru/cgi-bin/dn/e_027_05_0836.pdf

http://www.jetp.ac.ru/cgi-bin/dn/e_027_05_0836.pdf

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http://dx.doi.org/10.1016/0550-3213(96)00196-4

http://dx.doi.org/10.1016/0550-3213(96)00196-4
















http://dx.doi.org/10.1088/0022-3719/15/1/007

http://dx.doi.org/10.1088/0022-3719/15/1/007



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http://dx.doi.org/10.1016/0003-4916(89)90077-8







http://dx.doi.org/10.1038/nphys790

http://dx.doi.org/10.1103/physrevb.39.6880


http://dx.doi.org/10.1088/0953-8984/4/23/007

http://dx.doi.org/10.1088/0953-8984/4/23/007





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http://dx.doi.org/10.1093/ptep/ptw110

http://dx.doi.org/10.1093/ptep/ptw110










http://dx.doi.org/10.1038/s41563-018-0243-x

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http://creativecommons.org/licenses/by-nc-nd/3.0/

https://github.com/suchow/Dissertate

mailto:[email protected]

mailto:[email protected]

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