A microscopic model for d-wave charge carrier pairing and non-Fermi-liquid behavior in a purely repulsive 2D electron system

19 Pages·0.36 MB·

by  Mona Berciu

A microscopic model for d-wave charge carrier pairing
and non-Fermi-liquid behavior
in a purely repulsive 2D electron system
Mona Berciu and Sajeev John
0 Department of Physics, University of Toronto, 60 St. George Street, Toronto, Ontario, M5S 1A7, Canada
0 (February 6, 2008)
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2

n
a
J
We investigate a microscopic model for strongly correlated electrons with both on-site and near-
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est neighbor Coulomb repulsion on a 2D square lattice. This exhibits a state in which electrons
1
undergo a “somersault” in their internal spin-space (spin-flux) as they traverse a closed loop in

] external coordinate space. When this spin-1/2 antiferromagnetic (AFM) insulator is doped, the
n groundstateisaliquidofcharged,bosonicmeron-vortices,whichfortopologicalreasonsarecreated
o invortex-antivortexpairs. ThemagneticexchangeenergyofthedistortedAFMbackgroundleadsto
c
a logarithmic vortex-antivortex attraction which overcomes the direct Coulomb repulsion between
-
r holes localized on the vortex cores. This leads to the appearance of pre-formed charged pairs. We
p use the Configuration Interaction (CI) Method to study the quantum translational and rotational
u
motion of various charged magnetic solitons and soliton pairs. The CI method systematically de-
s
. scribes fluctuation and quantum tunneling corrections to the Hartree-Fock Approximation (HFA).
at Wefindthatthelowest energychargedmeron-antimeronpairsexhibitd-waverotationalsymmetry,
m consistent with thesymmetryof thecupratesuperconductingorderparameter. Fora single holein
the 2D AFM plane, we find a precursor to spin-charge separation in which a conventional charged
-
d spin-polaron dissociates into a singly charged meron-antimeron pair. This model provides a uni-
n fied microscopic basis for (i) non-Fermi-liquid transport properties, (ii) d-wave preformed charged
o carrier pairs, (iii) mid-infrared optical absorption, (iv) destruction of AFM long range order with
c doping and other magnetic properties, and (v) certain aspects of angled resolved photo-emission
[
spectroscopy (ARPES).

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I. INTRODUCTION been devoted to studying these charged stripes and re-
lating them to certain features of the cuprates.7
The microscopic understanding of the effect of charge Recently, a more fundamental investigation of the
carrierdopingonspin 1 antiferromagnetic(AFM)Mott many-electronproblemhassuggestedthepossibilityofan
insulators is the cent−ra2l issue of the high-temperature alternative model Hamiltonian for the cuprate physics.
superconducting cuprates.1 Many puzzling experimental This model Hamiltonian, called the spin-flux model,8
features of these systems2 suggest that a fundamental suggests that the long-range Coulomb interaction be-
law of nature remains to be recognized. Extremely low tween spin-12 electrons leads to qualitative new physics,
doping(δ 0.02 0.05chargecarrierspersite)leads to not apparent in the conventional Hubbard model (see
acomplete∼destruc−tionofthelong-rangeAFMorder,and Section 2). The results of the Hartree-Fock study9 of
a transition to an unusual non-Fermi-liquid metal. This this spin-flux model are summarized in Section 3. They
unusualmetalbecomessuperconducting,withthetransi- suggests that the undoped parent compound is also an
tiontemperatureT stronglydependentonthedopingδ. AFM Mott insulator. However, unlike the conventional
c
ThemaximumT isreachedfordopingsaroundδ =0.15. Hubbard model, the one-electron dispersion relations of
c
For higher dopings the critical temperature decreases to the AFM mean-field in the spin-flux model match those
zero, and in the overdoped region a crossover towards measured experimentally through angle-resolved photo-
a (non-superconducting) Fermi-liquid takes place. Two emission for undoped cuprates. A proper description of
central questions require resolution. The first one con- the highest occupied electronic states (as provided by
cerns the nature of the charge carriers responsible for the spin-flux model), is crucial to considerations of dop-
this non-Fermi-liquid metallic behavior. This is a fun- ing. The spin-flux model and the conventional Hubbard
damental issue, since it lies outside the framework of model differ dramatically in this regard. At the HF
Landau’s Fermi-liquid theory and it necessitates under- level, the doping holes added to the AFM background
standing the appearance of non-quasiparticle-likecharge of the spin-flux model are trapped in the core of antifer-
carriers in a system of interacting electrons. The second romagneticspinvortices. Thiscompositeobject(meron-
questionconcerns the nature ofstrong attractivepairing vortex)isabosonicchargedcollectivemodeofthemany-
between these charge carriers,given the purely repulsive electron system (the total spin of the magnetic vortex is
interactionbetweentheconstituentelectrons. Inconven- zero). The reversal of the spin-charge connection pro-
tional superconductors, the pairing attraction is due to vides a microscopic basis for non-Fermi-liquid behavior.
overscreeningofthe electron-electronCoulombrepulsion A magnetic vortex is strongly attracted to an antivor-
by the ionic lattice. In the case of the high-temperature tex. This attraction increases logarithmically with the
superconducting cuprates, it has been suggested1 that distance between the vortex cores, and is stronger than
pairing is an intrinsic property of the electron gas itself the unscreened Coulomb repulsion between the charge
mediated by AFM spin-fluctuations of the doped sys- meron-vortex cores. In effect, the increase in Coulomb
tem. Accordingly, the challenge is to identify a strong energybetweenagivenpairofholesismorethanoffsetby
attractive force based purely on repulsive Coulomb in- theloweringinexchangeenergybetweenthebackground
teractions. In this paper, we derive such a force and electrons as their vortices approach each other from far
demonstrate that it leads to d-wave pairing of charge away. As the inter-vortex distance increases, more and
carrying holes introduced by doping a quantum, spin-1, morespinsarerotatedoutoftheirAFMbackgroundori-
2
Mott-Hubbard antiferromagnet. entation and the total energy of the system increases.
The simplest model Hamiltonians used to investigate Thus, even at the HF level, the spin-flux model provides
the cuprate physics are the Hubbard model and the a fundamental underpinning for the origin of both non-
closely related t-J model. Unlike the 1D problem, an Fermi-liquid behavior, and strong pairing between the
exact solution for the 2D Hubbard Hamiltonian is not charge carriers.
known. As a result, various approximations are neces- While providing a good starting point, the Hartree-
sary. Although the application of the mean-field the- Fock Approximation also has serious shortcomings. For
ory has been severely criticized in this context, it pro- instance, the ground-state wavefunction in the presence
vides a valuable reference point for incorporating fluctu- of doping is non-homogeneous(the static meron-vortices
ationeffects. Moreover,evenforthe 1DHubbardmodel, of the spin-flux model, or the chargedstripes of the con-
essential features of the exact solution may be recap- ventional Hubbard model, break translational symme-
tured by judiciously incorporating fluctuation and tun- try). Physically, one expects that these charge carriers
neling effects intomean-fieldtheory.3 The moststraight- can move along the planes, resulting in a wavefunction
forward mean-field theory is the Hartree-Fock Approxi- which preserves the translational symmetry of the origi-
mation (HFA). At half-filling (δ = 0) the HFA predicts nal Hamiltonian. Quantum dynamics of the charge car-
an AFM Mott insulator ground-state. As the system riers also determines whether the doped ground-state is
is doped, HFA suggests that charge carrier holes in the really a metal. Charge carriers in the optimally doped
AFM backgroundassemble in chargedstripes, whichare cuprates are quite mobile excitations, although their
quasi-one-dimensional structures.4–6 A large effort has scattering rates are radically different from electrons in
a conventional Fermi liquid.
2
Aconsistentwayoftreating the quantumdynamicsof square lattice, n 2 c† c is the total number of
the charge carriers is provided by the Configuration In- i ≡ iσ iσ
σ=1
teraction (CI) Method,10,3 described in Section 4. Here, electrons at site i, andPVij is the Coulomb interaction
a linear combination of HF wavefunctions is used in or- between electrons at sites i and j. The dominant terms
der to restore the various broken symmetries. For in- arethenearest-neighborhoppingt =t andthe on-site
ij 0
stance, in a doped system the CI wavefunctionis chosen Coulomb repulsion V = U/2. If only these two terms
ii
to be a linearcombinationof HFwavefunctionswith the are considered, and we shift the chemical potential by
charge carrier localized at different sites. Certain types U, this reduces to the well-known Hubbard model. The
of charge carriers can lower their total energy substan- neglect of all longer range Coulombinteraction (V =0,
ij
tially by quantum mechanically hopping from one site if i = j), in the Hubbard model, is based on the Fermi-
6
to the next. We tested the accuracy of the CI method liquidtheorynotionofscreeningoftheeffectiveelectron-
againstthe exactsolution11 ofthe one-dimensionalHub- electron interaction. However, Fermi liquid theory fails
bardmodelinReference3. Inthe1DHubbardmodelthe to explain many of the crucial features of the high-T
c
CI method describes the quantum dynamics of charged cuprates. In our description, we include the nearest-
domainwallsolitonsintheAFMbackground. Byinclud- neighbor Coulomb repulsion, which we assume is on the
ingtheseeffectsasfluctuationcorrectionstotheHartree- energy scale of t. This leads to the generation of spin-
Fock mean-field theory, the CI method provides excel- flux, an entirely new type of broken symmetry in the
lent agreement with the exact Bethe Ansatz solution for many-electron system, which we show leads naturally to
theground-stateenergyofthedoped1DHubbardchain, bosonic charge carriers in the form of meron-vortices,
over the entire U/t range. The CI method also leads non-Fermi-liquidbehaviorandastrongattractivepairing
to a clear demonstration of the spin-charge separation forcebetweenholesin the AFM background. Inorderto
in 1D. Addition of one doping hole to the half-filled an- extract the relevant physics from our starting Hamilto-
tiferromagnetic chain results in the appearance of two nian,
different carriers: a charged bosonic domain-wall (which
carries the charge but no spin) and a neutral spin-1/2 = t c† c +h.c. +U n n +V n n (2)
domainwall(whichcarriesthespinbutnocharge). This H − 0 iσ jσ i↑ i↓ i j
study3 demonstrates the effectiveness of the CI method. hiX,jiσ(cid:16) (cid:17) Xi hXi,ji
InthispaperweusetheCImethodtoinvestigatedynam- we introduce bilinear combination of electron operators
ics of the chargedmeron-vortices in the spin-flux model. Λµ c† σµ c , µ = 0,1,2,3, for i = j (summation
Throughout this paper we exploit and refer to the anal- ovijer≡mulitαipαleβinjβdexes is assumed). Her6e σ0 is the 2 2
ogy between the charge excitations of the 1D Hubbard identity matrix and ~σ (σ1,σ2,σ3) are the usual Pa×uli
model and the 2D spin-flux model,12,13 apparent in the ≡
spin matrices. The notation i,j means that the sites i
CIapproach.3TheCIresultsforthespin-fluxmodel(pre- h i
and j are nearest neighbors. The quantum expectation
sented in Section 4) confirm that the meron-vortices are value ofthe Λµ operatorsareassociatedwithcharge-
very mobile, suggesting that a collection of such mobile hi ij
currents (µ = 0) and spin-currents (µ = 1,2,3). Non-
bosonic charge carriers is a non-Fermi-liquid metal. The
vanishing charge currents lead to appearance of electro-
CI method also allows us to identify the rotational sym-
magnetic fields, whichbreak the time-reversalsymmetry
metry of the meron-antimeron pair wavefunction to be
of the Hamiltonian. Experimentally, this does not occur
d-wave for the most stable pairs. An energetically more
in the cuprates. In the following, we adopt the ansatz
expensivemetastables-wavepairingisalsopossible. The
thatthereisnochargecurrentinthegroundstateΛ0 =0
possibility of spin-charge separation in 2D is elucidated. ij
butcirculatingspin-currentsmayariseandtaketheform
A summary of the results, their interpretation and con-
Λa = 2t0i∆ nˆ ,a=1,2,3,where ∆ =∆foralliand
clusions is provided in Section 5. ij V ij a | ij|
j, and nˆ is a unit vector. These spin-currents provide a
transitionstatetothespin-fluxmean-fieldthatweusein
this paper.
II. THE SPIN-FLUX MODEL Using the Pauli spin-matrix identity, 1σµ (σµ )∗ =
2 αβ α′β′
δ δ , it is possible to rewrite the nearest-neighbor
αα′ ββ′
The effective 2D Hamiltonian that we use to describe
electron-electron interaction terms as n n = 2n
i j i
the strongly correlatedelectrons residing in the O(2p) 1Λµ(Λµ)+. If we neglect fluctuations in the spin−-
Cu(3dx2−y2) orbitals of the isolated CuO2 plane is th−e c2urirjentisj, we can use the mean-field factorization,
tight-binding model: Λµ(Λµ)+ Λµ (Λµ)++Λµ Λµ ∗ Λµ Λµ ∗. Thus,
ij ij →h iji ij ijh iji −h ijih iji
thequarticnearest-neighborCoulombinteractiontermis
= t c† c +h.c. + V n n (1) reducedtoaquadratictermthatisaddedtothehopping
H − ij iσ jσ ij i j
iX,j,σ(cid:16) (cid:17) Xi,j term leading to the effective Hamiltonian:
wishtehree ch†ioσppcrineagteasmapnliteuledcetrfornomatssititeejitwoitshitespiinonσ,tthiej H=−t c†iαTαijβcjβ +h.c. +U ni↑ni↓. (3)
Xhαi,βji (cid:16) (cid:17) Xi
3
Here, Tij (δ + i∆ nˆ ~σ )/√1+∆2 are spin- III. HARTREE-FOCK RESULTS FOR THE
αβ ≡ αβ ij · αβ
dependent SU(2) hopping matrix elements defined by SPIN-FLUX MODEL
the mean-field theory, and t = t √1+∆2. In deriv-
o
ing (3) we have dropped constant terms which simply The Configuration Interaction Method utilizes a lin-
change the zero of energy as well as terms proportional ear combination of judiciously chosen Hartree-Fock
to n which simply change the chemical potential. It wavefunctions.10,3 In this section, we provide a short re-
i
i viewoftherelevantHartree-Fockresultsforthespin-flux
wasPshown previously8,14 that the ground state energy model. A full comparisonbetweenthe HFA for the spin-
of the Hamiltonian of Eq.(3) depends on the SU(2) ma-
fluxmodelandtheconventionalHubbardmodelhasbeen
trices Tij only through the plaquette matrix product published elsewhere.9
T12T23T34T41 exp(inˆ ~σΦ). Here, Φ is the spin-
≡ ·
flux which passes through each plaquette and 2Φ is the
angle through which the internal coordinate system of A. The Static Hartree-Fock Approximation
the electron rotates as it encircles the plaquette. Since
the electronspinor wavefunctionis two-valued,there are
One of the most widely used approximations for the
only two possible choices for Φ. If Φ = 0 we can set
many-electron problem is the Static Hartree-Fock Ap-
Tij = δ and the Hamiltonian (3) describes conven-
αβ ij proximation (HFA). In this approximation the many-
tional ordered magnetic states of the Hubbard model.
body problem is reduced to one-electron problems in
The other possibility is that a spin-flux Φ = π pene-
which each electron moves in a self-consistent manner
trates each plaquette, leading to T12T23T34T41 = 1.
− depending on the mean-field potential of the other elec-
This means that the one-electron wavefunctions are an-
trons in the system. While this method is insufficient,
tisymmetric around each of the plaquettes, i.e. that as
by itself, to capture all of the physics of low dimensional
an electron encircles a plaquette, its wavefunctionin the
electronic systems with strong correlations,it provides a
internal spin space of Euler angles rotates by 2π in re-
valuable starting point from which essential fluctuation
sponsetostronginteractionswiththeotherelectrons. In
corrections can be included. In particular, we use the
effect, the electron performs an internal “somersault” as
Hartree-Fock method to establish the electronic struc-
it traverses a closed path in the CuO plane. This spin-
2 ture and the static energies of various magnetic soliton
fluxphaseisaccompaniedbyaAFMlocalmomentback-
structures. InthemoregeneralConfigurationInteraction
ground (with reduced magnitude relative to the AFM
(CI) variationalwavefunction,the solitons acquirequan-
phase of the conventional Hubbard model). In the spin-
tum dynamics and describe large amplitude tunneling
fluxphase,thekineticenergytermin(3)exhibitsbroken
andfluctuationeffects that gobeyondmeanfield theory.
symmetry as though a spin-orbit interaction has been
In the HF approximation, the many-body wavefunc-
added. However, it is distinct from the smaller, con-
tion Ψ is decomposed into a Slater determinant of ef-
ventional spin-orbit effects which give rise to anisotropic | i
fective one-electron orbitals. The one-electron orbitals
corrections to superexchange interactions between local-
arefoundfromthe conditionthatthe totalenergyofthe
ized spins in the AFM.15 In the presence of charge car-
system is minimized
riers this mean-field is unstable to the proliferation of
topological fluctuations (magnetic solitons) which even- Ψ Ψ
tually destroy AFM long range order. In this sense, δh |H| i =0 (4)
ΨΨ
the analysis which we present below goes beyond sim- h | i
ple mean field theory. The quantum dynamics of these In order to approximate the ground state of the spin-
magnetic solitons described by the Configuration Inter- flux Hamiltonian (3), we consider a Slater determinant
action(CI) method, correspondsto tunneling effects not trial-wavefunctionof the form
contained in the Hartree-Fock approximation. For sim-
plicity, throughoutthis paper we assume that the mean- Ne
Ψ = a† 0 , (5)
field spin-flux parameters Tij are given by the simplest | i p| i
possible choice T12 = 1,T23 = T34 = T41 = 1 (see pY=1
−
Fig. 1). In order to go beyond a mean-field description
where 0 is the vacuum state, N is the total number of
e
of the spin-flux, these matrices may also be treated as | i
electrons in the system and the one-electron states are
dynamical variables. In this paper, we go beyond mean-
given by
field theory in describing the antiferromagnetic degrees
of freedom but restrict ourselves to a mean-field model
a† = φ (i,σ)c† (6)
of the spin-flux. n n iσ
iσ
X
Here, the one-particle wave-functions φ (i,σ) form a
n
complete and orthonormalsystem.
Using the wavefunction (5) in equation (4), and min-
imizing with respect to the one-particle wavefunctions
4
φ (i,σ), we obtain the Hartree-Fock eigen-equations: findtheHFeigenenergiesE andwavefunctionsφ (i,α).
n n n
These are used in Eqs. (8) and (9) to calculate the new
E φ (i,α)= t Tij φ (j,β) spin and charge distribution, and the procedure is re-
n n − αβ n peated until self-consistency is reached. Numerically, we
j∈XVi,β
define self-consistency by the condition that the largest
1 variationofanyofthe chargeorspincomponentsonany
+U 2δαβQ(i)−~σαβS~(i) φn(i,β) (7) of the sites of the lattice is less that 10−9 between suc-
Xβ (cid:18) (cid:19) cessive iterations.
where (σ ,σ ,σ ) are the Pauli spin matrices and the
x y z
charge density,
B. The undoped ground state
Ne
Q(i)= Ψc† c Ψ = φ (i,α)2, (8) For the undoped system, the Hartree-Fock equations
h | iα iα| i | p |
p=1 (7) for an infinite system are easily solved. In the
X
cuprates, long-range AFM order is experimentally ob-
and the spin density, served. Accordingly,wechooseaspindistributionatthe
S~(i)= Ψc† ~σαβc Ψ = Ne φ∗(i,α)~σαβφ (i,β), (9) switheerie=~e~eixsixthae+u~enyiityaveocftotrheoffosrommSe~(ai)rb=itr(a−ry1)(dixir+eicyt)iSon~e,,
h | iα 2 iβ| i p 2 p while the charge distribution is Q(i) = 1. In the spin-
p=1
X flux phase, it is convenient to choose a square unit cell,
must be computed self-consistently. The notation j in order to simplify the description of the Tij phase-
V appearing in (7) means that the sum is performe∈d factors. We make the simplest gauge choice compatible
i
over the sites j which are nearest-neighborsof the site i. with the spin-flux condition for the T-matrices, namely
The self-consistent Hartree-Fock equations (7,8,9) must that T12 = T23 = T34 = T41 = 1 (see Fig. (1)). This
−
besatisfiedbytheoccupiedorbitalsp=1...Ne,butcan leadstothereducedsquareBrillouinzone−π/2a≤kx ≤
also be used to compute the empty (hole) orbitals. π/2a, π/2a ky π/2a. FromtheHartree-Fockequa-
− ≤ ≤
The ground-state energy of the system in the HFA is tions we find two electronic bands, characterized by the
given by dispersion relations:
E = Ψ Ψ = Ne E U 1Q(i)2 S~(i)2 Es(f±)(~k)=±Esf(~k)=± ǫ2sf(~k)+(US)2 (11)
GS p
h |H| i − 4 − q
Xp=1 Xi (cid:18) (cid:19) where each level is four-fold degenerate and ǫ (~k) =
sf
(10)
2t (cos(k a))2+(cos(k a))2 are the noninteracting
x y
−
where the single particle energies are obtained from (7). electqron dispersion relations in the presence of spin-flux.
The approximation scheme described above is called The HF ground-stateenergy of the spin-flux AFM back-
the Unrestricted Hartree-Fock Approximation, because ground is given by (see Eq. (10)):
we did not impose constraints on the wavefunction Ψ
| i 1
which would require it to be an eigenfunction of various Esf = 4 E (~k)+N2U S2+ (12)
symmetry operations which commute with the Hamilto- GS − sf 4
nian(3). Ifthesesymmetriesareenforced,themethodis X~k (cid:18) (cid:19)
called the Restricted Hartree-Fock Approximation. We where the AFM local moment amplitude is determined
use the Unrestricted HFA since it leads to lower ener- by the self-consistency condition (9)
gies. The breaking of symmetries in our case implies
that electronic correlations are more effectively taken 2 US
S = . (13)
into account.16 The restoration of these symmetries is N2 E (~k)
deferred until the CI wavefunction is introduced. X~k sf
In the undoped (half-filled) case, the self-consistent
At half-filling the valence band (E(−)(k) < 0) is com-
Hartree-Fock equations can be solved analytically for sf
the infinite system, using plane-wave one-particle wave- pletely filled, the conduction band (E(+)(k) > 0) is
sf
functions. In the unrestricted Hartree-Fock approach, completely empty, and a Mott-Hubbard gap of magni-
doping the system leads to the appearance of inhomoge- tude 2US opens between the valence and the conduc-
neoussolutions,whichbreakthetranslationalinvariance. tion bands. The ground-state of the undoped spin-flux
In this case, we solve the unrestricted self-consistent model is an AFM Mott insulator. It is interesting to
Hartree-Fock equations numerically on a finite lattice. note that the quasi-particle dispersion relation obtained
StartingwithaninitialspinandchargedistributionS~(i) in the presence of the spin-flux (Eq. (11)) closely resem-
andQ(i),we numericallysolvethe eigenproblem(7) and bles the dispersion as measured through angle-resolved
5
photo-emission spectroscopy (ARPES) in a compound 2. The meron-vortex
such as Sr CuO Cl 17 (see Fig. 2). There is a a large
2 2 2
peak centered at (π/2,π/2) with an isotropic dispersion The 2D analog of the 1D charged domain wall is the
relation around it, observed on both the (0,0) to (π,π) meron-vortex(see Fig. 4). Like the 1D domain-wall,the
and (0,π) to (π,0) lines. The spin-flux model in HFA meron-vortex is also a topological excitation, character-
exhibits another smaller peak at (0,π/2) which is not ized by a topological (winding) number 1 (the spins
resolvableinexistingexperimentaldata. This minordis- on each sublattice rotate by 2π on any c±losed contour
crepancy may be due to next nearest neighbor hopping surrounding the center of th±e meron). As such, a sin-
orotheraspectsoftheelectron-electroninteractionwhich glemeron-vortexcannotbecreatedinanextendedAFM
wehavenotyetbeenincludedinourmodel.18 Thequasi- backgroundwithcyclicboundaryconditionsbytheintro-
particle dispersion relation of the conventional Hubbard duction of a single hole (just as a single isolated charged
model (T12 =T23 =T34 =T41 =1) has a large peak at domain wall cannot be created on an AFM (even) chain
(π/2,π/2) on the (0,0) to (π,π) line (see Fig. 2), but it with cyclic boundary conditions, by the introduction of
is perfectly flat on the (0,π) to (π,0) line (which is part a single hole). From a topological point of view, this is
of the large nested Fermi surface of the conventional 2D so because the AFM background has a winding number
Hubbard model). Also, it has a large crossing from the 0, and the winding number must be conserved, unless
upper to the lower band-edge on the (0,0) to (0,π) line. topological excitations migrate over the boundary into
Thisdispersionrelationisverysimilartothatofthet J the considered region. However, excitations can be cre-
−
model (see Ref. 17). ated in pairs of total topological number 0. In the 1D
While both the conventional and spin-flux model pre- case, this means creation of pairs of domain walls, while
dict AFM insulators at half-filling (at least at the HF in 2D this means the creationof vortex-antivortexpairs.
level), the spin-flux also provides a much better agree- From Figs. 4(a) and 4(b) we can see that the meron-
ment with the dispersion relations, as measured by vortexisachargedboson. Thetotalspinofsuchaconfig-
ARPES.Asinthe1Dcase,3theeffectofdopingistheap- urationiszero,whileitcarriesthedopingchargetrapped
pearanceofdiscretelevelsdeepinsidetheMott-Hubbard in the vortex core. Moreover, from its electronic spec-
gap. These levels are drawn into the gap from the top trum (Fig. 4b) we can see that only the extended states
(bottom) of the undoped valence (conduction) bands. ofthevalencebandareoccupied. Theyaretheonlyones
Accordingly, the type of excitations created by doping contributing to the total spin. Since only one state is
depends strongly on topology of the electronic structure drawn from the valence band into the gap, to become a
near the band edges. discrete bound level, it appears that an odd (unpaired)
number of states remains in the valence band. However,
one must remember that for topologicalreasons,merons
C. Charged solitons in the doped insulator: the
must appear in vortex-antivortex pairs. Therefore, the
spin-bag and the meron-vortex
valence band has an even number of (paired) levels, and
the total spin is zero. This argument for the bosonic
1. The spin-bag characterof the meron-vortexis identical to that for the
charged domain wall in polyacetylene.13,19,20
If we introduce just one hole in the plane, the self- Unlike in the 1D case,3 we cannot directly compare
consistent HFA solution is a conventional spin-polaron the excitation energy of the spin-bag with the excita-
or “spin-bag” (see Fig. 3a). This type of excitation is tion energy of the meron-vortex. The reason is that the
the 2D analog of the 1D spin-polaron.3 The doping hole excitation energy of the latter increases logarithmically
is localized around a particular site, leading to the ap- with the size of the sample, and therefore an isolated
pearance of a small ferromagnetic core around that site. meron-vortexisalwaysenergeticallymoreexpensivethan
The spin and charge distribution at the other sites are a spin-bag. However,topology requires that merons and
only slightly affected. In fact, the localization length of antimeronsarecreatedinpairs. Theexcitationenergyof
the charge depends on U/t, and becomes very large as such a meron-antimeron pair is finite, allowing a mean-
US 0,sinceinthislimittheMott-Hubbardgapcloses. ingful comparison between excitation energies of a pair
→
ForintermediateandlargeU/t,thedopingholeisalmost of spin-bags and a meron-antimeron pair.
completelylocalizedonthefivesitesoftheferromagnetic
core.
Thespin-bagisachargedfermion,ascanbeseenbydi- 3. The meron-antimeron pair
rect inspection of its charge and spin distributions. This
is also confirmed by its electronic structure (see Fig.3b). In Figs. 5(a) and 5(b) we show the self-consistent
Thus, the 2D spin-bag is indeed the analog of the 1D spin and charge distributions for the lowest energy self-
spin-polaron.3 consistent HF configuration found when we add 2 holes
to the AFM background, in the spin-flux model, for
U/t = 5. This configuration consists of a meron and an
6
antimeroncentered on second nearest-neighborsites. As assemble in charged stripes, as opposed to the liquid of
aresultofinteractions,thecoresofthevorticesaresome- meron-antimeron pairs which is the low-energy state of
what distorted. If the vortices were uncharged, vortex- the doped spin-flux model.
antivortexpairannihilationwouldbe possible. However,
for charged vortices, the fermionic nature of the under-
lying electrons prevents two holes from being localized IV. CONFIGURATION INTERACTION
at the same site, in spite of the bosonic character of the METHOD RESULTS FOR THE 2D SYSTEM
collective excitation.
A very interesting feature of this configuration is the A. Configuration Interaction Method
strongtopologicalattractionbetweenthe vortexandthe
antivortex. The closer the two cores are to each other,
The essence of the Configuration Interaction (CI)
the smaller is the region in which the spins are rotated
method is that the ground-state wavefunction, for a sys-
outoftheirbackgroundAFMorientationbythevortices,
tem with N electrons, is not represented by just a sin-
e
and therefore the smaller is the excitation energy of the
gle N N Slater determinant (as in the HFA), but
e e
pair. Sincethe holesarelocalizedinthe coresofthevor- ×
a judiciously chosen linear combination of such Slater
tices, this topological attraction between vortices is an determinants.10 Given the fact that the set of all pos-
effective attraction between holes in the purely repulsive
sible Slater determinants (with all possible occupation
2D electron system. This effect is unique to the spin-
numbers) generated from a complete set of one-electron
flux phase. In the conventionalHubbard model, vortices
orbitals constitute a complete basis of the N -particle
e
are not stable excitations. The vortex-antivortexattrac-
Hilbert space, our aim is to pick out a subset of Slater
tion increases as the logarithm of the distance between
determinants which captures the essential physics of the
the cores. Therefore, the pair of vortices should remain
exact solution.
bound evenif full unscreened1/r Coulombrepulsionex-
Consider the CI ground-state wavefunction given by
ists between the charged cores, providing a compelling
scenario for the existence of strongly bound pre-formed N
pairs in the underdoped regime. Ψ = αi Ψi (14)
| i | i
There is another possible self-consistent state for the i=1
X
system with two holes, consisting of two spin-bags far
where each Ψ is a distinct N N Slaterdeterminant
fromeachother(suchthattheirlocalizedwavefunctions | ii e× e
andthecoefficientsα arechosentosatisfytheminimiza-
do not overlap). The excitation energy of such a pair of i
tion principle:
spin-bags is simply twice the excitation energy of a sin-
gle spin-bag. When this excitation energy is compared δ Ψ Ψ
to the excitation energy of the tightly bound meron- h |H| i =0 i=1,N (15)
δα ΨΨ
antimeron pair, we find that it is higher by 0.15t (for i (cid:18) h | i (cid:19)
U/t=5). In fact, for U/t<8 the HFA predicts that the This leads to the system of CI equations
meron-antimeron pair is the low-energy charged excita-
tion, while for U/t > 8, the spin-bag is the low-energy N N
charge carrier. This is analogous with the situation in ijαj =E ijαj i=1,N (16)
H O
1D, where the spin-bag was predicted to be the low en- j=1 j=1
X X
ergy excitation for U/t > 6.5, in the HFA.3 As in 1D,
whereE = Ψ Ψ / ΨΨ istheenergyofthesystemin
however, we expect that this conclusion will be drasti- h |H| i h | i
the Ψ state , = Ψ Ψ are the matrix elements
cally modified once the charged solitons are allowed to | i Hij h i|H| ji
of the Hamiltonian in the basis of Slater determinants
movealongthe planesandthe loweringofkinetic energy
Ψ ,i = 1,N , and = Ψ Ψ are the overlap ma-
through translations is also taken into consideration. {| ii } Oij h i| ji
trix elements of the Slater determinants (which are not
We complete this reviewof the HF results by pointing
necessarily orthogonal). The CI solution is easily found
out that the strong analogy between the 1D Hubbard
by solving the linear system of equations (16), once the
model and the 2D spin-flux model is due to the similar-
basis of Slater determinants Ψ ,i = 1,N is chosen.
ity between the electronic structures at zero doping. As {| ii }
If we denote by φ(n)(i,σ) the p = 1,...,N one-electron
seenfromFig.2,the2Dspin-fluxmodelhasisotropicdis- p e
occupied orbitals of the Slater determinant Ψ , these
persion relations about the (π/2,π/2) point. This acts n
| i
matrix elements are given by:
as aFermi pointfor the noninteracting systemasit does
in the 1D system. The two empty discrete levels drawn
βnm ... βnm
deepinsidethe Mott-Hubbardgapinthe presenceofthe 1,1 1,Ne
. .
meron-vortexsplit fromthe (π/2,π/2)peaks ofthe elec- nm =(cid:12) .. .. (cid:12) (17)
O (cid:12) (cid:12)
tron dispersion relation. The different topology of the (cid:12) βnm ... βnm (cid:12)
large nested Fermi surface of the conventional Hubbard (cid:12)(cid:12) Ne,1 Ne,Ne (cid:12)(cid:12)
model leads to instability of the meron-antimeron pair. Thematrixelements(cid:12)ofthe Hamiltonian(cid:12) (3)canbe writ-
(cid:12) (cid:12)
In fact, in the conventionalHubbardmodel doping holes ten as:
7
pair), all possible rotations must be performed as well.
= t +U (i) (18)
nm nm nm
H − ·T V By rotation we mean changing the relative position of
i
X the meron and antimeron while keeping their center of
where the expectation values of the hopping and on-site mass fixed.
interaction terms are: Clearly,allthetranslatedHFSlaterdeterminantslead
to the same HF ground-state energy Ψ Ψ = E
βnm ... tnm ... βnm h n|H| ni GS
N 1,1 1,p 1,Ne as defined by Eq. (10). The CI method lifts the degen-
. . .
nm = (cid:12) .. .. .. (cid:12) eracybetweenstateswiththehole-inducedconfiguration
T Xp=1(cid:12)(cid:12)(cid:12)βNnme,1 ... tnNme,p ... βNnme,Ne (cid:12)(cid:12)(cid:12) lioncgatlirzaendslaattidoinffaelriennvtasriitaensc(es.eeWEeqm. a(y16i)d)e,ntthifeyretbhyerleoswtoerr--
(cid:12) (cid:12)
and (cid:12) (cid:12) ing in the total energy due to the lifting of this degen-
(cid:12) (cid:12)
eracy as quantum mechanical kinetic energy of decon-
βnm ... unm(i) ... dnm(i) ... βnm
1,1 1,p1 1,p2 1,Ne finement which the doping-induced configuration saves
. . . .
(i)= . . . . through hopping along the lattice. In addition, quan-
nm (cid:12) . . . . (cid:12)
V pX16=p2(cid:12)(cid:12)(cid:12)βNnme,1 ...unNme,p1(i)...dnNme,p2(i)...βNnme,Ne (cid:12)(cid:12)(cid:12). tsuolmitoflnucctaunatbieonisncinortphoeraitnetderbnyalisntcrluucdtiunrgetohfealomwaegsnteotric-
(cid:12) (cid:12)
Here, (cid:12)(cid:12) (cid:12)(cid:12) der excited state configurations of the static Hartree-
Fock energy spectrum. Such wavefunctions are given by
βpnhm = φh(n)∗(i,σ)φ(pm)(i,σ), aan†padh|hΨi, wNherleabpel>s tNhee lhaobleelswahnicehxcisiteldeftpabrethicinledst(asetee
e
iσ ≤
X Eq. (5)). Once again, all possible translations(and non-
trivial rotations) of this “excited” configurations must
tnm = φ(n)∗(i,α)Tij φ(m)(j,β)+h.c. , be included in the full CI wavefunction. These additions
p1,p2 p1 αβ p2 can describe changes in the “shape” of the soliton as it
Xhαi,βji(cid:16) (cid:17) undergoes quantum mechanical motion along the plane.
The CI method is described in more detail in Refer-
ence 3, where it is used to study the 1D Hubbard chain
unm (i)=φ(n)∗(i )φ(m)(i ),
p1,p2 p2 ↑ p1 ↑ in order to gauge its accuracy by comparing its results
with the exact Bethe ansatz solution. We showed that
and
the CI method recaptures the essential physical features
dnm (i)=φ(n)∗(i )φ(m)(i ). of the exact solution of the 1D Hubbard chain, such as
p1,p2 p2 ↓ p1 ↓ spin-charge separation, as well as leading to remarkable
We now consider the specific choice of the Slater de- agreement of ground state energies of doped chains for
terminant basis Ψ ,i = 1,N . Strictly speaking, one all values of U/t. The main difference between the 1D
i
{| i }
may choose an optimized basis of Slater determinants case and the 2D case is the computation time required.
from the general variational principle: Thecomputationtimeforonematrixelement nmscales
roughly like N9, where N is the number ofHsites. The
δ Ψ Ψ number of configurationsincluded in the CI set scales as
h |H| i =0 n=1,N;p=1,N (19)
δφ(pn)(i,σ)(cid:18) hΨ|Ψi (cid:19) e N!/Ns!(N−Ns)!whenNs solitonsarepresent. Forboth
an N-site chain and an N N lattice, the HF “bulk”
However, implementation of this full trial-function min- limit is reached for N 10×. In the 1D case3 we used
≥
imization scheme (also known as a multi-reference self- chains with N = 10 25, and numerical calculations
−
consistentmean-fieldapproach16)isnumericallycumber- can be easily performed. However, in 2D the smallest
some even for medium-sized systems. Instead, we select acceptable system has 100 sites, leading to an enormous
the Slater determinant basis Ψ ,i = 1,N from the increaseinthecomputationtime. Nevertheless,oursam-
i
{| i }
setofbrokensymmetry,UnrestrictedHartree-Fockwave- ple of results in 2D suggest a simple and clear physical
functions (5), their symmetry related partners and their picture which we describe below.
excitations. Clearly, (5) satisfies (19) by itself, provided
that the α coefficients corresponding to the other Slater
determinants in Eq.(14) are set to zero (see Eq. (4)). B. Spin-bag Dissociation:
Since this Unrestricted HF wavefunction is not trans- Spin-Charge Separation in 2D
lationally invariant (the doping hole is always localized
somewhere on the lattice), we can restore the transla- The charged spin-bag carries a spin of 1/2. Let
tionalinvariance ofthe CI ground-statewavefunctionby Ψ , Ψ be the HFdeterminantsforthe spin-bagcen-
+ −
also including in the basis of Slater determinants all the |teredi a|taniytwonearestneighborsites,respectively,and
possiblelatticetranslationsofthisUnrestrictedHFwave- let Sˆ = Sˆ (i)= 1 σc† c be the total spin op-
z i z 2 i,σ iσ iσ
function. Furthermore, if the self-consistent configura-
erator in the z-direction. Then, Sˆ Ψ = 1 Ψ while
tionisnotrotationally-invariant(e.g. ameron-antimeron P P z| +i 2| +i
8
Sˆ Ψ = 1 Ψ (or viceversa), since moving the cen- From Fig. 6 we also see that the dispersion relations
z| −i −2| −i
ter of the spin-bag by one site leads to a flip of its total for the spin-bag in the two different models are very dif-
spin (see Fig. 3(a)). Consequently, Ψ Ψ = 0. Since ferent. The dispersion relations over the full 2D Bril-
− +
h | i
the Hubbard Hamiltonian commutes with Sˆ , it follows louin zone are shown in Fig. 7, and they are seen to
z
that Ψ Ψ = 0. From the CI equation (16) we mimic the electronic dispersion relation of the underly-
− +
concluhdet|Hha|tthiereisnomixingbetweenstateswithdif- ing undoped AFM background, shown in Fig. 2. This is
ferent total spin. As a result, it is enough to include in consistent with the quasi-particle nature of this charged
the CI set only those configurations with the spin bag spin-1/2 spin-bag. In the conventional Hubbard model,
localizedonthesamemagneticsublattice. Letusdenote the undoped AFM background has a large nested Fermi
by Ψ the initial static Hartree-Fock configuration, surface along the (0,π) to (π,0) line, and it is exactly
(0,0)
| i
and by Ψ the configuration obtained through its along this line that the spin-bag dispersion band has a
(n,m)
translati|on by ni sites in the x-direction and m sites in minimum. Similarly,thelowestenergyofthespin-bagof
they-direction(cyclicboundaryconditionsareimposed). thespin-fluxmodelisat(π/2,π/2),correspondingtothe
The condition that only configurations on the same sub- Fermi points of the underlying undoped spin-flux AFM
lattice are included means that n+m must be an even background.
number, and the cyclic boundary conditions mean that The extra kinetic energy Esb(π2,π2)−EsHbF saved by
0 n N 1,0 m N 1, for a N N lat- the spin-bag through quantum hopping is 0.37t in the
tic≤e. As≤expla−ined in≤detai≤l in th−e 1D analysis,×3 mixing conventionalmodel and 0.56t in the spin-flux model (for
configurationswiththechargedspin-baglocalizedatdif- U/t = 5). Since the spin-bag is confined to one mag-
ferent sites and then subtracting out the contribution of netic sublattice, it must tunnel two lattice constants to
the undopedAFMbackgroundallowsus tocalculatethe the next allowed site. Consequently, the energy gained
dispersion band of the spin-bag itself: through hopping (of order t2/U) is small. This is dis-
played, for the spin-bag of the spin-flux model, in Fig.
Esb(~k)=E(~k,N) N2eGS. (20) 8, where we plot the lowering in kinetic energy of the
−
deconfined spin-bag E (π,π) EHF as a function of
Here, the total energy of the lattice with the spin-bag sb 2 2 − sb
t/U. A similar dependence for the spin-bag of the con-
Ψ Ψ ventionalHubbardmodelispresentedelsewhere.10 As in
E(~k,N)= h ~k|H| ~ki the 1D case, we conclude that the spin-bag in 2D is a
Ψ Ψ
h ~k| ~ki rather immobile quasiparticle-like excitation.
and the CI wave-function In the 1D model it is energetically favorable for the
immobile spin-bag to decay into a charged bosonic do-
|Ψ~ki= exp(i(kxn+ky|m)a)Ψ(n,m)i mainwallandaneutralfermionic domainwall,resulting
(Xn,m) inspin-chargeseparation.3 Theanalogofthe1Dcharged
bosonic domain wall is the 2D charged bosonic meron-
arethe solutionsoftheCIequations(16). Thefinite size
vortexofthespin-fluxmodel. Ifthespin-bagdecaysinto
ofthelatticeandcyclicboundaryconditionsrestrictsthe
achargedmeron-vortex,amagneticantivortexmustalso
calculation to ~k-points of the form ~k = 2π (α~e +β~e ),
Na x y be created for topological reasons. Unlike the pair of
where (α,β) is any pair of integer numbers. As usual,
domain walls in the 1D case, the vortex-antivortex pair
only ~k-points inside the first Brillouin zone need to be is tightly bound by a topological binding potential that
considered. increases as the logarithm of the vortex-antivortex sep-
An analysis of the dependence of the spin-bag disper- aration. Therefore, we expect that the doping charge
sion relation E (~k) on the size N N of the lattice is is shared between the two magnetic vortices. One tech-
sb
×
shown in Fig. 6, for the conventional Hubbard model nical problem for testing this hypothesis is that such a
(upper panel) and spin-flux model (lower panel), and configuration(a vortex-antivortexpair sharingone dop-
U/t = 5. We used 6x6, 8x8, 10x10 and (only for the ing hole) is not self-consistent at the static Hartree-Fock
spin-fluxmodel)12x12lattices. Thedispersionrelationis level. In the static approximationwe require two doping
plotted alonglines ofhighsymmetryof the full Brillouin holes to stabilize two vortex cores and create a meron-
zone. Forcomparison,wealsoshowtheexcitationenergy antimeronpair. Wecan,however,constructatrialwave-
EHF obtained in the static HFA as a full line. For both functiontodescribethe singly-chargedvortex-antivortex
sb
models, we see that the spin-bag dispersion band is al- pair,byaddingoneelectroninthefirstemptystateofthe
mostconverged,eventhoughweusedquitesmalllattices. self-consistentdoubly-chargedmeron-antimeronconfigu-
Theconvergenceissomewhatslowerinthespin-fluxcase, ration. The first empty levels of the meron-antimeron
asseenmostclearlyatthe(0,0)point. Althoughtheval- pair are the localized levels bound in the vortex cores,
ues obtained from the four lattices all differ at (0,0), the two for each vortex (see Fig. 4(b)). Because of degener-
extremum values correspond to the 6x6 and the 8x8 lat- acy between the two lower discrete levels of the pair, we
tices, while the values for the 10x10 and 12x12 lattices haveinfacttwodistincttrialwave-functions,obtainedby
are indistinguishable. We conclude that the fit (20) is adding one electronin either ofthese twolowerlocalized
legitimate. gapelectronicstatesofaself-consistentmeron-antimeron
9
pair. These wavefunctions are not invariant to rotations the system evolves with doping. If each hole is dressed
( see Fig. 5). Therefore we must include in the CI set of into a singly-charged vortex-antivortex pair, when two
Slater determinants the configurations obtained through suchpairsoverlapitispossiblethatbothdopingcharges
π/2rotationsofthevortex-antivortexpairaboutitsfixed move to the same pair, creating a meron-antimeron pair
center of mass in addition to translated configurations. of charged bosons. Such pre-formed charge pairs may
Asaresult,wehaveatotalof8N2configurationsdescrib- condense into a superconducting state at low tempera-
ingthe singlychargedvortex-antivortexpairlocalizedat tures. The other uncharged vortex-antivortex pair may
all possible sites with all possible orientations about the either collapse and disappear (this is likely to happen
center of mass. at low-temperatures)or remain as a magnetic excitation
WeperformedthisCIanalysisfora10 10latticeand of the system (at higher temperatures), mediating the
×
U/t = 5. The HF energy of a simple static spin-bag is destruction of the long-range AFM order, the renormal-
0.82t (measured with respect to the HF energy of the izationofthespin-wavespectrumandthe openingofthe
−
undoped AFM background, equal to 76.76t). The en- spin pseudogap.
−
ergy of the static singly-charged vortex-antivortex pair
is 0.23t. Thus, we see that because this singly-charged
−
pairtrialwavefunctionisnotself-consistent,inthestatic C. D-wave pairing of charge carriers
case this configurationis energetically much more costly
thantheself-consistentspin-bagconfiguration. However, From the static HF analysis we found that the most
if we allow for quantum motion of these configurations, stable static self-consistent configuration with two dop-
the situation changes dramatically. Performing the CI ing holes added to the AFM backgroundof the spin-flux
analysis for the set of all possible translated spin-bag model is the meron-antimeron pair, for 3<U/t<8. At
configurations, we find that the energy of the spin-bag larger U/t, two charged spin-bags become more stable,
is lowered to 1.24t. Performing the CI analysis for the in the static HF approximation. This is in close anal-
−
set of all translated and rotated singly-charged vortex- ogy to the prediction that the spin-bag is energetically
antivortexpairswefindthatthisconfiguration’senergyis more favorable than the static charged domain-wall for
loweredto 2.18t. Thisshowsthatthevortex-antivortex U/t>6.5, in the HFA of the 1D Hubbard model.3 How-
−
pair has lowered its translational and rotational kinetic ever, in the 1D case the charged domain-wall is consid-
energy by almost 2t, thereby becoming the low-energy erably more mobile than the chargedspin bag, gaining a
chargecarrier. This largenumberis notsurprising,since kineticenergyoftheorderoftasopposedtot2/U energy
unlikethespin-bag,thevortex-antivortexpairisnotcon- gained by the spin-bag. As a result, when this kinetic
strained to motion on one magnetic sublattice. As a re- energyofdeconfinementis takenintoaccountwithin the
sult, suchconfigurationslowertheir kinetic energyby an CI method, the charged domain-wall is found to be the
amount on the scale of t, as opposed to t2/U for the relevant charged excitation for all values of U/t. A sim-
spin-bag configuration. For larger U/t values this effect ilar picture emerges in the 2D case, because the meron-
is even more pronounced. vortices are much more mobile than the spin-bags.
Weconcludethattheseresultsstronglysupportthehy- For the 2D system, we have shown that the charged
pothesisofspin-bagdissociationintoamuchmoremobile spin-bag has very similar behavior to the 1D charged
singly-charged vortex-antivortex pair, analogous to the spin-bag. Theanalogofthe1Dchargedbosonicdomain-
1Dspin-bagdissociationinto a pairofa chargedbosonic wall is the 2D charged bosonic meron-vortex. We now
domain-wall and a neutral fermionic domain-wall.3 Un- consider the properties of the doubly-charged meron-
like in the 1D case, however, we do not have distinct antimeron pair. All the numerical results quoted in the
chargeandspincarriersforthecompositeexcitation. In- rest of this section refer to a meron-antimeron pair on a
steadthespinandthechargearesharedequallybetween 10x10 lattice, in the spin-flux model with U/t=5.
the vortex and the antivortex. If, on the other hand, As already discussed, the meron-antimeronpair is not
there was a mechanism whereby the vortices became rotationally invariant. We can find the rotational ki-
unbound, complete spin-charge separation could occur, netic energy saved by the pair as it rotates about its
in which one vortex traps the hole (and is therefore a centerofmass. Inthe presentcase,only4configurations
chargedmeron)andtheothervortexcarriesthespin,ina need to be included, corresponding to the four possible
lotus-flower12,13(orundopedmagneticmeron)configura- self-consistentarrangementsofthemeronandantimeron
tions. Inthe absenceofthe correspondingself-consistent abouttheirfixedcenterofmass(seeFig. 9). Simplerota-
staticHFconfigurationwearenotabletosettlethisques- tionbyπ/2oftheone-particleorbitalsφ (i,σ)aboutthe
p
tion. At very low doping, the strong vortex-antivortex center of mass is not, however, sufficient to generate the
topologicalattractionbindsthespinandchargetogether. rotatedconfigurations. First ofall, the π/2 rotationalso
This is different from the 1D case, where the absence of changesthespin-fluxparameterization. Ifthespin-fluxof
long-range interactions between the domain-walls allow the initialconfigurationisT12 = 1,T23=T34 =T41 =
for a complete spin-charge separationat any doping and 1, a π/2 rotation leads to a stat−e corresponding to the
even at zero temperature. rotated configurationT12 =1,T23 = 1,T34 =T41 =1.
Thisscenarioopensanewavenueforresearchintohow −
10