Robust charge-density wave correlations in the electron-doped single-band Hubbard model Peizhi Mai1 2Nathan S. Nichols3Seher Karakuzu1 4Feng Bao5

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Robust charge-density wave correlations in the electron-doped single-band Hubbard

model

Peizhi Mai,1, 2 Nathan S. Nichols,3Seher Karakuzu,1, 4 Feng Bao,5

Adrian Del Maestro,6, 7, 8 Thomas A. Maier,1and Steven Johnston6, 7

1Computational Sciences and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee, 37831-6494, USA

2Department of Physics and Institute of Condensed Matter Theory,

University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA

3Data Science and Learning Division, Argonne National Laboratory, Argonne, Illinois 60439, USA

4Center for Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New York, New York 10010, USA

5Department of Mathematics, Florida State University, Tallahassee, Florida 32306, USA

6Department of Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee 37996, USA

7Institute of Advanced Materials and Manufacturing, The University of Tennessee, Knoxville, Tennessee 37996, USA

8Min H. Kao Department of Electrical Engineering and Computer Science,

University of Tennessee, Knoxville, Tennessee 37996, USA

(Dated: October 28, 2022)

There is growing evidence that the hole-doped single-band Hubbard and t-Jmodels do not have

a superconducting ground state reﬂective of the high-temperature cuprate superconductors but in-

stead have striped spin- and charge-ordered ground states. Nevertheless, it is proposed that these

models may still provide an eﬀective low-energy model for electron-doped materials. Here we study

the ﬁnite temperature spin and charge correlations in the electron-doped Hubbard model using

quantum Monte Carlo dynamical cluster approximation calculations and contrast their behavior

with those found on the hole-doped side of the phase diagram. We ﬁnd evidence for a charge mod-

ulation with both checkerboard and unidirectional components decoupled from any spin-density

modulations. These correlations are inconsistent with a weak-coupling description based on Fermi

surface nesting, and their doping dependence agrees qualitatively with resonant inelastic x-ray scat-

tering measurements. Our results provide evidence that the single-band Hubbard model describes

the electron-doped cuprates.

A key question in quantum materials research is

whether or not the single-band Hubbard model describes

the properties of the high-temperature (high-Tc) super-

conducting cuprates [1–3]. On the one hand, several

studies have demonstrated a direct mapping between

multi-orbital Cu-O models and eﬀective single-band de-

scriptions [4–7]. At the same time, quantum cluster

methods have found evidence for a d-wave superconduct-

ing state [6,8] in the Hubbard model, with a nonmono-

tonic Tcas a function of doping that resembles the dome

found in real materials. On the other hand, a grow-

ing number of state-of-the-art numerical studies on ex-

tended Hubbard and t-Jclusters have found evidence

for stripe-ordered ground states for model parameters

relevant to the cuprates [9–13]. While density matrix

renormalization group (DMRG) simulations of multi-leg

hole (h)-doped Hubbard ladders do obtain a supercon-

ducting ground state for nonzero values of the next-

nearest-neighbor hopping t0[14], its order parameter does

not have the correct dx2−y2symmetry [15] found in the

cuprates [16]. Conversely, DMRG calculations for six-

and eight-leg t-Jcylinders obtain the correct order pa-

rameter but only on the electron (e)-doped side of the

phase diagram [12]. These results cast signiﬁcant doubt

on the long-held belief that the Hubbard model describes

the h-doped cuprates. Nevertheless, hope remains that

it may capture the e-doped materials.

From an experimental perspective, charge-density-

wave (CDW) correlations have been established as a

ubiquitous feature of the underdoped cuprates [17,18].

Initially observed by inelastic neutron scattering in the

form of intertwined spin and charge stripes [19], short-

range CDW correlations have now been reported in

nearly all families of cuprates using scanning tunneling

microscopy [20,21] and resonant inelastic x-ray scatter-

ing (RIXS) [22–33]. Importantly, these CDW correla-

tions persist up to high temperatures, particularly on the

e-doped side of the phase diagram [25,28–30].

Given their ubiquity, these CDW correlations must be

accounted for by any proposed eﬀective model for the

cuprates. Evidence for charge modulations, both in the

form of unidirectional stripe correlations or short-range

CDW correlations, has now been found in a variety of

ﬁnite temperature quantum Monte Carlo (QMC) simu-

lations of the h-doped Hubbard model [13,34–38]. These

simulations are generally restricted to high temperatures

by the Fermion sign problem [10,34,35] (except for very

recent constrained path QMC calculations [38]) and fo-

cus on the h-doped model. The observed cuprate CDWs

exhibit a signiﬁcant electron-hole asymmetry, however.

On the h-doped side, they can intertwine with spin-

density modulations to form stripes while they coexist

with uniform antiferromagnetic (AFM) correlations on

the e-doped side [25,28,30]. These diﬀerences have

raised questions on whether the e- and h-doped CDWs

share a common origin [29,30].

Here, we study and contrast the spin and charge

correlations of the two-dimensional Hubbard model

arXiv:2210.14930v1 [cond-mat.str-el] 26 Oct 2022

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●βt=4●6●8●10

-1.0 -0.5 0.0 0.5 1.0

1.50

1.55

1.60

1.65

1.70

(Qx,0)[π/a]

χc(Q, 0) [× 10]

0.0 0.5 1.0 1.5 2.0

1.50

1.55

1.60

1.65

1.70

(Qx,1)[π/a]

N(Q,0) [× 10]

Hole-doped, 𝑛 = 0.8 Electron-doped, 𝑛 = 1.2

Charge correlation Charge correlation

0.0 0.5 1.0 1.5 2.0

1.50

1.52

1.54

1.56

1.58

(Qx,1)[π/a]

χc(Q, 0) [× 10]

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●βt=3●4●5●6

-1.0 -0.5 0.0 0.5 1.0

1.50

1.55

1.60

1.65

1.70

(Qx,0)[π/a]

χc(Q, 0) [× 10]

0.0 0.5 1.0 1.5 2.0

0.3

0.4

0.5

0.6

0.7

0.8

(Qx,1)[π/a]

S(Q,0)

0.0 0.5 1.0 1.5 2.0

0.3

0.4

0.5

0.6

0.7

0.8

(Qx,1)[π/a]

χs(Q, 0)

Spin correlation

FIG. 1. Static spin and charge correlations in the doped single-band Hubbard model. Panels aand cshow the

static charge susceptibility χc(Q,0) along the (Qx,0) and (Qx, π/a) directions, respectively, for the hole-doped system with

t0/t =−0.2 and hni= 0.8. Panel eshows the corresponding static spin susceptibility χs(Q,0) along the Q= (Qx, π/a)

direction. The yellow dashed lines show incommensurate peaks obtained from ﬁtting multiple Lorentzian functions plus a

constant background to the βt = 6 data. (The constant contribution is not shown.) Panels b, d, f show the corresponding

results for the e-doped case with t0/t =−0.2 and hni= 1.2. These results were obtained using a 16 ×4 cluster, and the inverse

temperatures β= 1/T are reported in units of 1/t.

using the dynamical cluster approximation (DCA) [39]

and a nonperturbative QMC cluster solver [40] (see

methods). Working on large (16 ×4) rectangular clusters

that are wide enough to support large-period stripe

correlations if they are present [34], we vary the electron

density hniacross both sides of the phase diagram to

contrast the correlations in each case. Our calculations

uncover robust two-component CDW correlations on the

e-doped side, which consists of superimposed checker-

board (π/a, π/a) and unidirectional QCDW = (±δc,0)

components. These CDW correlations appear to be

decoupled from any stripe-like modulations of the spins

and instead coexist with short-range antiferromagnetic

correlations. This behavior is in direct contrast to the

h-doped case, where we ﬁnd evidence for ﬂuctuating

stripe correlations in both the charge and spin degrees

of freedom [34]. Our results agree with experimental

observations on the e-doped cuprates, including the

observed doping dependence of QCDW. This supports

the notion that the single-band Hubbard model captures

the e-doped side of the high-Tcphase diagram.

Results. Figure 1compares the static charge χc(Q, ω =

0) and spin χs(Q,0) susceptibilities for the h- (hni= 0.8)

and e-doped (hni= 1.2) Hubbard model with U/t = 6,

t0/t =−0.2, and varying temperature. In the h-doped

case (Fig. 1a,c,e), unidirectional charge and spin stripes

form as the temperature is lowered, consistent with prior

-1

-6 -4 -2 0 2 4 6 8

-0.02 -0.01 0 0.01 0.02

-1

-6 -4 -2 0 2 4 6 8

-0.002 -0.001 0 0.001 0.002

-1

-6 -4 -2 0 2 4 6 8

-0.002 -0.001 0 0.001 0.002

-1

-0.002 -0.001 0 0.001 0.002

-1

-6 -4 -2 0 2 4 6 8

-0.02 -0.01 0 0.01 0.02

-1

-0.02 -0.01 0 0.01 0.02

Hole-doped, 𝑛 = 0.8Electron-doped, 𝑛 = 1.2

a b

c d

Charge correlation Spin correlation

FIG. 2. Static spin and charge correlations in real-space. a χc(r,0) for the electron-doped system (t0=−0.2t,hni= 1.2)

at the lowest accessible inverse temperature βt = 10. bχc(r,0) for the h-doped system (t0=−0.2t,hni= 0.8) at the lowest

accessible inverse temperature βt = 6. cand dshow the staggered spin-spin correlations χs,stag(r,0) for the h- and e-doped

cases, respectively. The dashed lines indicate the approximate nodes in the spin and charge stripe modulations.

ﬁnite-temperature studies [34,35,38]. These correla-

tions manifest as incommensurate peaks in the static

susceptibility at wave vectors Qc= (±δc,0) and Qs=

(π/a ±δs, π/a) for the charge and spin channels, respec-

tively. For the spin channel in Fig. 1e, the dashed line

shows a ﬁt of the lowest temperature data using two

Lorentzian functions. Fig. 1cplots the charge correla-

tions for the h-doped case along the (Qx, π/a) direction,

where we observe a weak (π/a, π/a) peak emerging at

the lowest accessible temperature. This modulation is

weaker than the Qcstructure, such that the charge cor-

relations are predominantly stripe-like.

We observe qualitatively diﬀerent correlations in the

e-doped case shown in Figs. 1b,d, and f. At high tem-

perature (β≤6/t), χc(Q,0) has a single broad peak cen-

tered at q= (0,0), which can again be decomposed into

two incommensurate Lorentzian peaks centered at ±δc,

indicative of a unidirectional charge stripe. As the tem-

perature is lowered, these peaks sharpen and become dis-

cernible without ﬁts while the q-independent background

remains constant. The charge correlations also have a rel-

atively temperature-independent (π/a, π/a) component

(Fig. 1e) of similar strength as the stripe-like charge cor-

relations. In contrast to the h-doped case, we ﬁnd no

indication of a spin-stripe at this doping; the spin sus-

ceptibility has a single peak centered at (π/a, π/a) for all

accessible temperatures.

Comparing panels aand b, we see that the charge

stripe correlations in the h- and e-doped systems de-

velop diﬀerently as the temperature decreases. In the

h-doped case (Fig. 1a), the incommensurate peaks grow

while δcshifts to smaller values as the temperature de-

creases [34]. In the e-doped case, the incommensurate

peaks grow (Fig. 1b) while the value of χc(Q,0) near

zone center is suppressed, resulting in well-deﬁned peaks

centered at ≈(±π/2a, 0). In addition, the (π/a, π/a)

component is signiﬁcantly stronger in the e-doped case

(Fig. 1d). However, since the height of the incommen-

surate peaks in Fig. 1bdoes not appear to level oﬀ, the

stripe correlations could dominate at lower temperatures.

The corresponding correlation functions in real-space,

obtained at the lowest accessible temperatures (βt = 6

and 10 for the h- and e-doped cases, respectively) are

plotted in Fig. 2, with the data for the h-doped case

reproduced from Ref. 34. The e-doped case (Fig. 2a)

has a clear short-range (π/a, π/a) checkerboard-like pat-

tern near r= 0 , superimposed over a stripe-like compo-

nent. In contrast, the h-doped case (Fig. 2b) only shows

a stripe pattern. Figs. 2cand dshow the staggered spin

correlation function. Here, the blue region in the mid-

dle represents one AFM domain, while the red region on

both sides indicates neighboring AFM domains with the

opposite phase (note that the staggered spin correlation

function that is plotted contains an additional negative

sign on the B-sublattice as explained in the Methods sec-

tion). Consistent with Fig. 1and as observed before [34],

the h-doped case has a clear stripe pattern whose pe-

riod is roughly twice that of the charge modulations. In

contrast, the e-doped cuprates are dominated by short-

range AFM correlations with only a single phase inver-

sion appearing at longer distances. To summarize, the

h-doped system has intertwined spin and charge stripe

correlations, while the e-doped system manifests CDW

correlation with stripe-like Q= (δc,0) and checkerboard

(π/a, π/a) components and nearly uniform AFM spin

correlations.

Figure 3examines how the stripe component of the

charge correlations develops for the e-doped case with

density and t0for a ﬁxed inverse temperature β= 8/t.

Fig. 3ashows χc(Q,0) along the (Qx,0) direction for

various densities, while holding t0/t =−0.2 ﬁxed. At

hni= 1.125, the charge susceptibility has a single peak

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Robustcharge-densitywavecorrelationsintheelectron-dopedsingle-bandHubbardmodelPeizhiMai,1,2NathanS.Nichols,3SeherKarakuzu,1,4FengBao,5AdrianDelMaestro,6,7,8ThomasA.Maier,1andStevenJohnston6,71ComputationalSciencesandEngineeringDivision,OakRidgeNationalLaboratory,OakRidge,Tennessee,37831-6494,USA2Dep...

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Robust charge-density wave correlations in the electron-doped single-band Hubbard model Peizhi Mai1 2Nathan S. Nichols3Seher Karakuzu1 4Feng Bao5

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