Continuous transition from a Landau quasiparticle to a neutral spinon

2025-04-22 0 0 959.51KB 18 页 10玖币

侵权投诉

Jing-Yu Zhao,1Shuai A. Chen,2Rong-Yang Sun,3, 4 and Zheng-Yu Weng1

1Institute for Advanced Study, Tsinghua University, Beijing 100084, China

2Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong 999077, China

3Computational Materials Science Research Team,

RIKEN Center for Computational Science (R-CCS), Kobe, Hyogo 650-0047, Japan

4Quantum Computational Science Research Team,

RIKEN Center for Quantum Computing (RQC), Wako, Saitama 351-0198, Japan

(Dated: Thursday 9th February, 2023)

We examine a wavefunction ansatz in which a doped hole can experience a quantum transition

from a charge +eLandau quasiparticle to a neutral spinon as a function of the underlying spin-

spin correlation. As shown variationally, such a wavefunction accurately captures all the essential

features revealed by exact diagonalization and density matrix renormalization group simulations in a

two-leg t-Jladder. Hence its analytic form can provide an explicit understanding of the mechanism

for the unconventional ground state. The transition in the phase diagram is accompanied by a

change of the hole composite from a tight charge-spin binding to a loosely-bound hole-spin pair. In

the latter, the hole carries a ﬁnite spin current but with vanishing charge current in the degenerate

ground states. We show that the charge of the hole composite here is dynamically diminished due

to an internal relative hole-spin motion, which is fundamentally distinct from a simple charge-spin

separation in a one-dimensional case. We further show that the same eﬀect is also responsible for a

strong pairing between two doped holes in such a non-Landau quasiparticle regime.

CONTENTS

I. Introduction 1

A. Basic variational results 2

II. Benchmarking Wavefunction Ansatz with ED

and DMRG via VMC calculation 4

A. The two-leg anisotropic t-Jmodel 4

B. Single-hole wavefunction ansatz 4

C. Variational ground-state energy 5

D. Phase diagram 6

III. Further characterization: Novel charge

renormalization at α > αc8

A. Emergent spin-current around the hole at

α > αc: Incoherent charge component 8

B. Response to external electromagnetic ﬂux 10

C. Vanishing charge 10

IV. Discussion 11

A. QCP at αc11

B. The two-component structure at α > αc11

C. Disappearance of charge response to external

magnetic ﬂux at α > αc12

D. Binding force between two doped holes 12

V. Conclusion 13

Acknowledgments 14

A. Phase-shift operator on a two-leg ladder with

PBC 14

B. Longitudinal spin-polaron eﬀect on the

variational ansatz 15

C. Variational Monte Carlo procedure 15

1. Variational procedure for optimizing ground

state energy 16

2. Monte Carlo procedure to calculate physical

observables 16

References 17

I. INTRODUCTION

How to properly characterize a single-particle excita-

tion is one of the most essential challenges in the study

of the strongly correlated Mott insulators [1–4]. In par-

ticular, a single chargon (hole) may serve as a building

block for constructing a doped Mott insulator. The cen-

tral issue under debate is whether such a chargon mov-

ing a quantum spin background will still behave like a

conventional Landau-type quasiparticle or be fundamen-

tally renormalized into a non-Landau quasiparticle via

“twisting” the surrounding many-body spins in the back-

ground.

In analogy to the electronic “cloud” associated with

a Landau quasiparticle, it was widely believed earlier

on that such a hole in the Mott insulator may still be

the Landau-type after considering the longitudinal spin-

polaron eﬀect [5,6], which involves a distortion in the am-

plitude of the local spin magnetization around the hole.

On the other hand, it was conjectured [7] that a nontriv-

ial many-body response from the Mott insulator back-

ground may lead to an “unrenormalizable Fermi-surface

phase shift”, which can result in an “orthogonality catas-

trophe” to turn the doped hole into a non-Landau quasi-

particle.

Such an “unrenormalizable phase shift” has been ex-

arXiv:2210.04918v2 [cond-mat.str-el] 8 Feb 2023

plicitly identiﬁed with the phase-string eﬀect in the t-J

[8,9] and Hubbard [10] models as a singular noninte-

grable Berry phase acquired by the doped hole(s) mov-

ing in the quantum spin background of the Mott insula-

tor. Physically, it implies that the hopping of the doped

hole should generate a spin-current backﬂow. Exact di-

agonalization (ED) and density matrix renormalization

group (DMRG) simulations have recently conﬁrmed [11]

such an unconventional behavior of the doped hole in

the two-dimensional (2D) square lattice. The hidden

spin-current backﬂow overlooked by previous numerical

works has been revealed [11] to accompany and facilitate

the hopping of the hole, leading to a nontrivial quan-

tum number as a direct manifestation of a non-Landau-

like quasiparticle. Making use of the variational Monte

Carlo (VMC) method, a single-hole wavefunction ansatz

has been constructed [12], which well interprets the nu-

merical results including the nontrivial angular momenta

Lz=±1 under a C4rotational symmetry for a ﬁnite-size

2D sample up to 8 ×8 [11].

The phase-string eﬀect or the spin-current backﬂow

here leads to a transverse spin twist to renormalize the

doped hole. But previously a Landau quasiparticle be-

havior had been still inferred in a semiclassical ﬁeld-

theory approach [13,14] even with incorporating a long-

range transverse dipolar spin twist beyond the longitu-

dinal spin-polaron eﬀect. It is because a singular cou-

pling between the doped hole and the spin currents at

the short distance was omitted [15]. The latter eﬀect has

been carefully and consistently implemented in the wave-

function approach [12] to reproduce the correct behavior.

Nevertheless, in spite of the good agreement of the VMC

result [12] and exact numerics [11] at ﬁnite (small) sizes,

the 2D single-hole problem can be further complicated by

the antiferromagnetic (AF) long-range order which sets

in as the thermodynamic limit is taken, which may fur-

ther lead to a self-localization of the hole. Thus, a thor-

ough understanding of the 2D single-hole problem has

to handle both singular eﬀects in short-range and long-

range physics together, which is beyond the ﬁnite-size

exact numerical methods.

On the other hand, it would be interesting to ﬁrst fo-

cus on the singular short-range physics, which should be

generally present even though the long-range AF order

is expected to disappear at ﬁnite doping. Besides the

above VMC study on a single hole in a ﬁnite-size system

in 2D, a single-hole-doped t-Jmodel on a two-leg ladder

may be a more suitable “toy” system to examine such

an eﬀect as here the spin-spin correlation length remains

ﬁnite even at half-ﬁlling [16]. The anisotropy of a two-leg

ladder system can also provide us more tools to contin-

uously tune the correlation of the spin background. The

previous DMRG studies [16–18] have established an ex-

otic ground state for the single hole, which behaves like a

non-Landau-quasiparticle with breaking charge transla-

tional symmetry [19]. Note that the above non-Landau-

quasiparticle picture was previously contested by another

DMRG study [20], in which a ﬁnite quasiparticle spec-

tral weight Zkhas been explicitly measured. To reconcile

both DMRG results, it was pointed out [19] that the Lan-

dau’s one-to-one correspondence principle is actually bro-

ken down even though Zkstill remains ﬁnite in the non-

Landau-quasiparticle regime, where the hole object is of

a two-component structure, composed of a translation-

invariant Bloch wave component and a charge-incoherent

component. In other words, the DMRG results indicate

the existence of a new type of non-Landau quasiparticle.

Especially a quantum transition for the doped hole to be-

come a true Landau quasiparticle has been also found by

reducing the spin-spin correlation length along the lad-

der direction [17–20], with the translation symmetry and

charge coherence being eventually restored. Therefore,

a wavefunction description of such a single-hole problem

in a two-leg ladder can oﬀer a valuable proof-of-principle

on how a bare doped hole can be speciﬁcally “twisted”

into a non-Landau-quasiparticle due to the intrinsic Mott

physics even in the absence of an AF long-range order.

In this paper, we shall present a simple variational

wavefunction ansatz by incorporating the aforementioned

phase-string or transverse spin-current eﬀect. It can ac-

curately capture all the essential ground-state features of

a single-hole-doped two-leg t-Jladder under a periodic

boundary condition (PBC) via the VMC method. In par-

ticular, it will describe a doped hole as a Landau quasi-

particle with charge q= +ein the strong anisotropic

limit of α1 (αis the ratio of chain direction coupling

and rung direction coupling, cf. Fig. 2). In the isotropic

regime of α'1, the doped hole will acquire nonzero total

momenta with ﬁnite spin currents, but its charge current

will vanish in the large sample limit. The results clearly

indicate that the doped hole is a charge-neutral “spinon”

in the regime of α > αcwith αcdenoting a quantum criti-

cal point (QCP), as schematically illustrated in Fig. 1(a),

which are in excellent agreement with the DRMG calcula-

tion [cf. Fig. 1(b)]. In other words, we shall demonstrate

by an analytic wavefunction that an exotic transition re-

sembling a metal-insulator transition beyond a Landau

paradigm [21] can happen in a doped Mott insulator.

The speciﬁc form of such a variational wavefunction and

its VMC results are brieﬂy outlined as follows.

A. Basic variational results

Such a single-hole wavefunction ansatz is given by

|ΨGi1h ∝X

ei(k0xi−ˆ

Ωi)ci↓|φ0i,(1)

where |φ0idenotes an undoped spin-singlet background

of the two-leg ladder, and the electron annihilation op-

erator ci↓removes an electron of spin ↓(without loss of

generality) to create a bare hole at site i. The ground

state momentum k0is along the quasi-1D ladder direc-

tion, with each rung labeled by xi. Note that if one

turns oﬀ the phase-shift operator ˆ

Ωiin Eq. (1), the wave-

function is simply reduced to a conventional Bloch-wave

0.1 1 10

0.2

0.4

0.6

0.8

Q0=:

(b)

VMC

DMRG

q= +e q = 0

Landau qp Spinon

ck<~ck<

(a)

0:2:

0.3

0.6

,= 0:4

0:2:

0.1

0.2

,= 1:0

FIG. 1. The phase diagram of the single-hole variational

ground state as a function of the underlying spin-spin cor-

relation controlled by an anisotropic parameter αin a two-leg

t-Jladder (see text). (a) A schematic illustration of a quan-

tum transition point at α=αcfor a Landau quasiparticle of

charge q= +estate to become a “twisted” quasiparticle with

diminished charge q= 0 to be shown in this paper; (b) Q0[cf.

Eq. (3)] as a function of α, which measures the momentum

splitting in the ground state (the insets) as calculated by both

DMRG and VMC methods on a 48 ×2 ladder under PBC.

The insets show the quasiparticle weight Zkat two typical

α= 0.4< αcand α= 1.0> αc, respectively, determined by

VMC with αc≈0.68 at t/J = 3.

state as the leading term of a spin polaron wavefunction

[13,22],

|ΨB(k0)i1h ∝X

eik0xici↓|φ0i,(2)

which is explicitly translational invariant (with the trans-

lationally invariant spin background |φ0i) to describe a

Landau-like quasiparticle of momentum k0. The phase-

shift ﬁeld ˆ

Ωias a nonlocal spin operator is thus the only

unconventional quantity in the ground state of Eq. (1),

which is to incorporate the phase-string eﬀect as stated

above. An excellent agreement of the VMC calculation

based on Eq. (1) and its variant (i.e., a further incorpo-

ration of the longitudinal spin-polaron eﬀect) with the

DMRG and ED results will be demonstrated in this pa-

per.

As a matter of fact, a QCP is found in the ground state

as a function of the anisotropic parameter α, by which

the AF correlation in |φ0ican be continuously tuned. At

strong rung limit (α1), k0is found at πmod 2π(tak-

ing the lattice constant as the unit), which corresponds to

a non-degenerate state essentially the same as the Bloch-

wave state in Eq. (2). A double-degeneracy in the ground

state arises on the other side of the QCP at larger α’s,

which is characterized by nontrivial momentum splitting

k±

0=π±κmod 2π. The split as characterized by the

wavevector:

Q0≡2κ(3)

is shown in Fig. 1(b) as a function of αcalculated by

VMC and DMRG, respectively, which both indicate a

QCP at αc'0.68 where Q0vanishes as determined by

DMRG [17–20].

In the insets of Fig. 1(b), the sharp peak(s) of the

quasiparticle spectral weight Zkspeciﬁes k0in the ground

state. Here Zkis deﬁned as the absolute-value squared

of the overlap between the wavefunctions in Eqs. (1) and

(2) (after normalization) as follows

Zk≡1

2|hΨB(k)|ΨGi1h|2=



hφ0|c†

k↓|ΨGi1h



2,(4)

where c†

k↓= 1/N Pic†

i↓eikxiis the kspace electron

with ky= 0. On both sides of the QCP in Fig. 1(b),

Zkis always ﬁnite, indeed consistent with the DMRG

result ﬁrst shown in Ref. 20. In particular, the non-

degenerate ground state at α < αcis a Landau quasipar-

ticle, which can be smoothly connected to the Bloch-wave

state in Eq. (2). However, we shall show that the charge

of the doped hole will actually disappear at α > αc,

whereas its spin-1/2 remains unrenormalized. In other

words, the QCP represents a fundamental transition of

the doped hole from a Landau-like quasiparticle to a pure

charge-neutral spinon, which is schematically illustrated

in Fig. 1(a). Such a non-Landau-like quasiparticle with

a ﬁnite Zk±

0indicates a two-component structure in the

wavefunction where the Landau’s one-to-one correspon-

dence hypothesis fails at α > αc. Indeed, besides a ﬁnite

amplitude of the Bloch-wave component (with Zk±

06= 0),

another many-body component is also explicitly identi-

ﬁed in the ground state, in which a spin current pattern

associated with the doped charge is always present. The

latter is found to be charge incoherent as the total mo-

mentum k±

0is now continuously shared between the hole

and spin degrees of freedom.

Finally, it is brieﬂy discussed that the pairing between

two doped holes also becomes substantially enhanced at

α > αcas previously revealed by the DMRG calculation

[17,23]. An explicit pairing-mediated spin current pat-

tern is shown based on the present wavefunction ansatz,

which illustrates how a strong binding can be indeed re-

alized by eliminating the phase-string eﬀect through the

pairing of two holes.

The rest of the paper is organized as follows. In Sec. II,

we introduce the two-leg anisotropic t-Jmodel and con-

struct a single-hole-doped wavefunction ansatz under the

PBC. A systematical comparison between the DMRG

and VMC methods are shown on both sides of the QCP

at αc. In Sec. III, the properties of the wavefunction

at α > αcare further analyzed to show that, diﬀerent

3i(l1)3i(l2)

l1l2

FIG. 2. Illustration of a two-leg ladder with anisotropic cou-

pling parameters tij and Jij of the t-Jmodel under the

PBC (see text). Here the total number of the ladder sites

is N=Nx×2 with Nxdenoting the total number along each

of the two legs, which are embedded in 2D with a spatial

ring conﬁguration. Note that the two legs of the ladder as

rings are with diﬀerent radii: rin = 2 −λand rout = 2 + λ,

respectively, in which λis a variational parameter to spec-

ify the phase-string operator ˆ

Ωiof Eq. (9). Here the angle

ﬁeld θi(l), satisfying Eq. (10), is deﬁned accordingly in the

2D conﬁguration.

from a Landau quasiparticle, here the “twisted” quasi-

particle carries a ﬁnite spin current in the degenerate

ground state but vanishing charge current in the ther-

modynamic limit. In Sec. IV, a further discussion of the

underlying physics of the non-Landau quasiparticle be-

havior is made. In particular, how the incoherent charge

component is crucial to the pairing between doped holes

is pointed out. Finally, the conclusion and perspectives

are given in Sec. V.

II. BENCHMARKING WAVEFUNCTION

ANSATZ WITH ED AND DMRG VIA VMC

CALCULATION

A. The two-leg anisotropic t-Jmodel

In this paper, we shall study the single-hole-doped

ground state of the t-Jmodel on an anisotropic two-

leg ladder with system size N=Nx×2. Here the t-J

Hamiltonian is given by H=Ps(Ht+HJ)Ps, where

Ht=−X

hiji,σ

tij (c†

iσcjσ + H.c.),(5)

HJ=X

hiji

Jij Si·Sj−1

4ninj,(6)

with hijidenoting a nearest-neighbor (NN) bond. Here

Siand niare spin and electron number operators on

site i, respectively. The strong correlation nature of the

t-Jmodel originates from the no double occupancy con-

straint Pσc†

iσciσ ≤1 on each site, which is imposed via

the projection operator Ps. Generally, a two-leg ladder

is anisotropic along the chain direction (denoted as xdi-

rection) and the rung direction (denoted as ydirection)

as illustrated in Fig. 2under PBC, where we choose the

rung-direction couplings as tij =tand Jij =J, and the

chain-direction couplings as tij =αt and Jij =αJ, with

α > 0 as the anisotropic parameter. The superexchange

coupling constant Jis taken as the unit and the hopping

term t/J = 3 is used throughout the paper. The DMRG

calculation of this paper is done with 2500 saved states

to ﬁt a truncation error up to 10−10 with 200 sweeps for

convergence. Most of the VMC calculations in this paper

are done on a 48×2 lattice, but no obvious change of the

results is seen as the system size changes up to 64 ×2.

B. Single-hole wavefunction ansatz

At half-ﬁlling, where the t-Jmodel is reduced to the

Heisenberg spin model on a bipartite square lattice, the

ground state is a spin singlet state, with a ﬁnite spin-

gap opened up for the two-leg ladder case [16]. In the

following, we shall denote it as |φ0i.

Then, based on a bare hole state created at site iby

removing an electron of spin ↓from the spin-singlet back-

ground, i.e., ci↓|φ0i, a Bloch-wave-like single-hole state

may be constructed as

|ΨBi1h =X

ϕB

h(i)ci↓|φ0i,(7)

where the variational wavefunction ϕB

h(i)∝eikxiis a

Bloch-wave with a momentum kalong the quasi-1D lad-

der direction under the translation symmetry. In general,

the doped hole will induce a many-body response from

the spin background, known as the phase-string eﬀect

[8,9], such that the single-hole state can be signiﬁcantly

renormalized beyond the Bloch-wave-like one in Eq. (7).

How to treat such an eﬀect is therefore the central issue

in the study of the doped Mott physics.

An ansatz ground state has been previously proposed

for the t-Jmodel, which is generally given in the one-hole

case as follows [12,24,25]

|ΨGi1h =X

ϕh(i)e−iˆ

Ωici↓|φ0i,(8)

where a new phase factor e−iˆ

Ωiis explicitly introduced to

represent the many-body phase shift or the phase-string

eﬀect from the spin background when a hole is created

at site i. In other words, the corresponding spin back-

ground is modiﬁed from |φ0ito e−iˆ

Ωi|φ0i. Here ϕh(i)

is a variational wavefunction to be optimized, and ˆ

Ωiis

explicitly given by [12,24,25]

Ωi=X

l(6=i)

θi(l)nl↓,(9)

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摘要：

ContinuoustransitionfromaLandauquasiparticletoaneutralspinonJing-YuZhao,1ShuaiA.Chen,2Rong-YangSun,3,4andZheng-YuWeng11InstituteforAdvancedStudy,TsinghuaUniversity,Beijing100084,China2DepartmentofPhysics,HongKongUniversityofScienceandTechnology,ClearWaterBay,HongKong999077,China3ComputationalMateria...

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