October c.-
SLAC-PUB-5027 1989 CT)
FERMIONS SIGMA
AND MODEL
SOLITONS
IN THE
O(3) NONLINEAR DIMENSIONS*
IN 2 + 1 SPACE-TIME
PURUSHOTHAM VORUGANTI~
Stanford Linear Accelerator Center Stanford University, Stanford, CA 94309
and
._.
Stanford University, Department of Physics Stanford, CA 94305
SEBASTIAN DONIACH
Stanford University, Department of Applied Physics Stanford, CA 94305 ABSTRACT
The field theory O(3) nonlinear term that solitons. lim it of antiferromagnetism with holes is described using the to the spin-wave gives a Pauli charge density potential for
sigma m o d e l. The m inimal
coupling
couples the hole charge density
to the topological
This term leads to drastic consequences; an attractive spin-charge coupling
for holes The
and solitons,
for the holes and zero-momentum solutions
modes.
zero-momentum equations connections
modes are exact nonperturbative
to the full
coupled term, problem
of motion
for spin-waves and holes. The effect of a Chern-Simons approaches and the quantum boundstate
to other mean-field
are discussed.
If the zero-modes are the states of lowest energy, then the holes in are attached to skyrmions.
this field theory lim it
Submitted
to Physical Review B.
* Work supported by Department of Energy contract DE-AC03-76SF00515. o Now at University of Toronto, 60 St. George St., Toronto, Ontario, Canada
M55 lA7.
-
.-
I.
MOTIVATION
f
--I
Many
of the new high-Tc After
superconductors the addition
exhibit
a N&eel antiferromagnetic
state at zero doping. this ordering ducting. importance
of roughly
one hole per ten Cu atoms, the system becomes superconproperties2 hint at the
becomes short-ranged,' magnetic
and moreover,
The anisotropic
and superconducting
of dimensionality.
If the feature of antiferromagnetism description
is at all crucial state beonce is the to If
to the superconducting comes necessary. the connection -
state, an adequate
of the magnetic
The unique feature of two-dimensional
antiferromagnetism, limit,
to the nonlinear excitations.
sigma model is made in the continuum
_.finite energy soliton the superconducting
3 If these are to play any role in the transition with holes must be fully addressed. explanation
state, their interaction
the holes and the solitons form bound range magnetic
pairs,* a natural
for the short-
order emerges, since each soliton presents a small area of disorder. state could be equivalently viewed as the soliton condensed
Thus the disordered
-
state.5 If the soliton could bound two holes and no more, then the other half of the picture, namely Cooper pairs, provide an explanation for the superconductivity. written as solutions to
-` --
Interestingly
enough, many of these notions can be explicitly Although the solutions are difficult
various equations. quantum
to extract,
the classical and are
regimes will be systematically energetics
discussed.
Even though our conclusions the possibility The attractive
that soliton-hole that new methods this description magnetic
do not favor the above picture, the above scenario.
remains feature of to the
may demonstrate
is its uniqueness
to two dimensions
and its deep connection
properties. 2
xr.
-.
FIELD
THEORY
OF THE
2D HEISENBERG
MODEL
numerous to
f --I
With evidence
the exception 6 of the 30K exists that
Bismuth-oxide
superconductors, undergo
the most of the high-Tc
compounds
a transition
the Neel state at low doping. - 2D Heisenberg by Dyson,
The Neel state is a variational
state for the S = l/2
model and is the exact ground state for spin S 2 1, due to a theorem simulations of the nonlinear sigma model Starting around the
Lieb and Simon. 7 Numerical model show that
and the Heisenberg with
it is N&e1 ordered'
for S = l/2.
the 2D Heisenberg
model
on the square lattice
and expanding operators
N&e1 State, Haldane the long wavelength -model.
was able to show that the relevant continuum limit
that appear in sigma
are just given by the O(3) nonlinear ground
The assumption which,
of this particular after making
state then leads to a relativistic
2+1 field theory,
the standard
CP1 rnap,l' is given by
7
JkP with
-
=
j(D,Z)+(D' Z)
- q(Z+.Z' - 1)
two-spinor
(1)
(a, /3). The theory
p = 0,1,2;
D, = a,, + iA,;
coupling
and 2 a complex constant
has one dimensionful
j and constraint ground
fields ~(2) and A,(z). and
The Neel state corresponds -` -p2 = 1 -02. spinwave
to the field theory
state of CI = constant the same number version,
The field theory description version,
has precisely
of massless has a space-
modes as the lattice symmetry.lr
but unlike the lattice
time Lorentz The thing
Hamiltonian
in the presence Hamiltonian limit
of holes on the lattice with onsite repulsion Model
is probably
some-
like the Hubbard
and nearest-neighbor to
hopping. I2 The field theory derive although
of the Hubbard models exist.13 3
is much more difficult
many mean-field
Writhe limit i I. with hopping
of large onsite repulsion, parameter
one can instead spin-stiffness
start with the t-J
model
t and a Heisenberg
term J. At zero doping, which in the large
the t term U limit
is irrelevant
since it will lead to double suppressed. The Heisenberg the two-dimensional four mean-field
occupancy interaction lattice variables
is energetically
is then mapped
onto the sigma model by dividing plaquettes
into nonintersecting for each plaquette. then
of four sites each giving
The massless modes in the large S limit describe the sigma model. The number
are just the $ and n,#, fields which of massless modes in the continuum the Neel state. interaction
is two, even though in agreement The expansion with the
four sites lead to four degrees of freedom Goldstone
-
theorem
upon expanding
around
around
_.the N&e1 state thus describes For the problem extremely small with
the spin-spin nonzero doping,
portion
of the t - J mode. the case of is still descripa
it is simplest observed
to discuss
doping
where
the experimentally
Ndel state
good description
of the ground
state and its symmetries. with spin-waves
The mean-field has attracted
tion of the small number attention.` 4t25 identify
of vacancies
considerable states will as well as zone
Like the two spin-wave number
modes,
the single hole.ground Lattice calculations
the relevant
of degrees of freedom.
other mean-field locatedI theory.
descriptions f7r/2).
show that there are four minima
in the Brillouin
at z = (&n/2, Let us introduce
Th ese four states should appear in any mean-field operators Cl , Cl which live on each plaquette. a Dirac Spinor, $J = (CT , CJ). to two such Dirac fermions but
two fermion
From these two degrees of freedom, The number of low-energy
we can construct
hole states corresponds physics
none of our results
on soliton-hole
will be affected
if we just concentrate degrees of
on one species of holes. Interactions
of the spin-waves 4
with the fermion
free&-m
should
flip down spins to up spins in the plaquette of a hole from the up sublattice theory
and this corresponds In order we
to the hopping f -1
to the down sublattice. coupled
to write the mean-field follow
for the Dirac fermion of the spin-wave theory
to the spin-waves,
closely the symmetries
which like the Hamiltonian of an extremely small
for the vacancies number
comes from the t - J model. the mean-field
In the limit
of holes, we expect
Lagrangian
to have the same symmesymmetry with of the N&e1 state be-
tries as the system without is of course the Lorentz ing the spin-wave ground -
any holes. The surprising a pseudorelativity
symmetry,
the speed of light
velocity.
At large doping, arguments
the NCel state is no longer
the valid the The to
state and no symmetry regime
are valid in this regime, but luckily of the spin-disordered symmetry our attention state. related
low-density other
may determine
the nature
symmetry where
of the nonlinear G is an element
sigma model is the internal of SU(2). By restricting
2 -+ GZ
to just
the one Dirac fermion, symmetry. -.
we can consistently
treat it as a singlet under this internal of fermions various on each plaquette, other
Had we used the two "flavors" exist. l3 After considering preserves
rich possibilities Lagrangian that
II, - 2 interactions,
the simplest of
all the symmetries
and contains
the lowest number
-` --
derivatives
and includes
the spin-flips
is given by
L = j(D,Z)+(DV)
+ r&Z+2 - 1) + &"(ia,
-id,
+ m)ti
9
(2)
where we consider holes of any general coupling is a complex The coupling two-spinor
g and mass m (ti = 1, c = 1). The 1c, that will be defined below. This Lagrangian spin
and the yp is a 2 x 2 spin matrix
g begins as S but receives the usual renormalization. has a rigorous formulation 5
for holes at low density
for the one-dimensional
chair?* c .--I fermion coupling fermions with
Neither
the sign of the mass nor charge nor the relativistic is crucial to the behavior around solitons. fermions
version We consider
of the this
Lagrangian
for its generality,
and the cases of same-sign Having
and nonrelativistic of holes
are subcases of what follows. we next examine arbitrary
discussed
the interactions
spin-waves,
the soliton
sector of the theory.
The solutions, of the energy
however, functional. solitons
come with This unstable
size and thus are not true minima physics
property, against
in the absence of additional quantum or l/S corrections is examined
like holes, makes a-model.
to the nonlinear in Sec. III.
This problem
in the presence of fermions
III.
SOLITONS
AND
SIZE
INSTABILITY
can be brought
Solitons of the O(3) nonlinear in cylindrical coordinates
sigma model in the CP' language
to the formI*
z sol
where r ' = (r/X)lNl, excitations with
=
(l+J&,2
ezPGw )
(
,
(3)
number. The
(1+&/2
X being
the size and N the winding solutions
lowest-energy is independent paper. I4 Two
are the N = fl
with energy 27r j. that was addressed
The energy in an earlier First, as sugin
of X and this is the size instability solutions have been suggested Polyakov
to cure this problem. topological
gested by Dzyaloshinskii, the long-distance either limit,
and Wiegmann,*
terms, relevant
like the Chern-Simons
term or the Hopf term,
can arise
from fermions limit
being integrated Model
out of the path integral in two dimensions. the self-consistent 6
or from a complete another pos-
field theory sibility
of the Hubbard stability
However, trapping
arises for soliton
through
of holes in
solitis.
The first scenario
will be shown easily not to hold true. here, provides a consistent
Our idea on the mechanism.
second possibility, f
as we demonstrate
As we saw above, -1 interacting _ freedom,r6
the effect of holes is to introduce
two species of fermions out the fermion degrees of to lowest-
with the CP1 gauge field. l5 If we integrate we obtain the following effective action
for the spin-waves
order momentum,
L eff
= Lcp + k&,,xA"d"AX
,
(4
mass. The same-sign Chern-Simons stability. Our
with k = g2/4.rr since we have two species with the same-sign mass has been suggested term17 results Dilation and our purpose on soliton-hole invariance as a way of realizing is to study pairing
the parity-violating
only its effect on the soliton upon
do not depend
the signs of the mass terms. term
of the energy functional paper,r*
is broken when the Chern-Simons a parameterization
is present.
In an earlier
we suggested solutions. to infinite
of the 2 fields in the presence of with the
for determining -
the new soliton
Here, we show that
this new term, solitons Chern-Simons
are pushed
size. The energy functional
term present
becomes
-`
--
f
= f
J
k2 d d2+(" x x)2+
(D$)+(DiZ)]
,
(5)
Law gives A0 = in the presence scale
where
we consider
static
configurations
Z(z)
only
and Gauss' s of motion
@/f)(~x~~*
SUPPose an exact solution
term is known,
to the equations
of the Chern-Simons transformation,
called Z(Z). can be brought z(z), 7
By doing an appropriate
the energy functional Forth e solution
to the form & = j J &z[( a x If we then
Z)2+(DiZ)+(DiZ)]e
this becomes & = j(Er +E2).
cons+&& a solution and we see that
f
z(ar)
for some scale factor is reduced
a, the energy
is & = j(u2Er the addition Lagrangian
+ E2), of the gives
the energy term,
for a -+ 0. Therefore, to the sigma model
Chern-Simons -1 minimum
whatever
the reason,
energy to the solitons for small a or equivalently in the presence of the Chern-Simons invariance
infinite
size. The energy
_ of the minimum
term is still 27r j. The Chernbut since it the
Simons term breaks the dilatation is a higher-derivative Chern-Simons correction,
of the energy functional, to infinite
pushes the solitons
size. Although
term does not appear in the covariant
energy-momentum
tensor due through the
to the absence of the metric
s gPV, it affects the Gauss' law condition
A0 equation
-
of motion l8 and changes the form of the canonical corrections to the soliton elsewhere; mass"-
momenta.
Leading
_.quantum
effects-zero-point
in the presence of that large to be
the Chern-Simons size solitons
term were considered Quantum finite
again, it was found
were preferred.` s
effects to the first order in k remain size solitons will arise. Lattice
done and it is not clear whether also stabilize -. solutions motion -` -the soliton
effects may
when the Chern-Simons
term is present, Instead,
but the solitons we consider the
on the lattice
even for k = 0 are not known. of soliton
of a single hole in the background stabilization.
and see if the eigenspectrum
reveals any nontrivial
To begin the analysis, fermions. through theory
we start with the original of motion couple
Lagrangian
for spinwaves
and
The full equations
the fermions
and the spinwaves
the gauge field A,. We will examine and treat the A,, that
the one soliton
sector of the spinwave a single fermion theory (although for
arises as the background
in which
is to move. quantum
Since there is no Chern-Simons
term in the original
effects could generate
it at order h), the classical equation
8
of motion *
A0 b&comes A0 = 0 for static solitons.
urations fermions, ,$ = 0, B f 0. If the background then the requirement that
Therefore, magnetic
the soliton
presents field configfor mass
field has bound eigenstates energy E plus the soliton
the binding
27rj be less than
zero could place a condition
on the size.
The size of the solibecomes number N
ton effects boundstate narrower
energies, since the potential Using the soliton
well for the fermions solutions of winding
as the size decreases.
we wrote earlier, we are automatically written as B = -V2$, winding
in the gauge e -2 = 0. B(z)
can thus be
and A; becomes Ai = ciiajd number
for some scalar field $(cc). For 4(z) is
general soliton
N, the scalar potential
- _.
The fermion solitoa equation
d(x) =
of motion
sgn~N)In(l
+ r21Ni)
.
(6)
of motion in the above
x
is the usual dirac equation
background
&lC, field Ai or ir'
+ +/` (i& - gAi)t,b + mlC, = 0. Using the 2
2
Pauli matrices
-.
with y" = Q,, y' = ia,) r2 = igY, we get the Schroedinger
equation
-` --
where
or
=
-fYy,cY2
= u2. For
fermions
$J =
components the equation
are coupled.
In order to analyze form,
the eigenstate
0
2.4 when
7
V
m f
0, the two we square
spectrum,
to get the usual decoupled
[a2 + 2igi.
9 + ((E2 - m2) - g2A2 - ga,B)]$
= 0
.
(8)
would
Had we started
with nonrelativistic
fermions,
essentially Pauli
the same equation term
have arisen if we had added
the two-dimensional
s,B to the usual
9
(p <A)2/2772
nonrelativistic
term. picture,
Although
such a term
has to be put
in my hand
in the
the internal moving
spin degree of freedom external
of the holes requires it field, just as in the
in the case of electrons Quantum symmetries Z(z)
in a constant
magnetic
Hall phenomena.
The addition
of this Pauli term preserves all the gauge and written purely in terms of the
of the nonrelativistic
field theory,
fields, appears as
where in the nonrelativistic This interaction -. logical represents
case, 1c, is a single component a coupling of the fermion
anticommuting
field.
charge density conserved current
to the topois given by poten-
charge density
where the general topological 2o This additional
Jp
= c,,,P' Z+PZ.
term is the origin of the attractive cases.
tial for both
the relativistic
and the nonrelativistic
This higher-derivative
interaction
between spin-waves and holes has been missed in all other mean-field
Having now a wave equation for the
approaches, but here it emerges naturally.
holes in a soliton To generalize background, for a moment,
we look for eigenstates. it is known 21 that for A0 = 0, static fermion to it is continuous. zero These
modes exist for any B as long as the c$(x ) corresponding zero-momentum modes are, in the gauge 9 . x,
$0 = 9
for o,lc) = II+, the integrability
(rd(x,
Y>> exp We) 2
,
k =
O,l,...
,
(10)
where the restriction of the wavefunctions.
on k to the nonnegative Therefore, 10
integers
arises from modes in
the zero-momentum
the s&ton
background
have E = fm
and are given by
f
w$% $0 = (1 + .2lNl)F T-'exp (&ikO)
.
(11)
lkj > 1.
exist
_ Normalizability soliton sgn(s,g) with
of the zero-modes number N
therefore
requires
IgNl -
Given only
a for
of winding > 0 where
= +IN I, normalizable of a,.
zero-modes
sZ is the eigenvalue
Hence,
for the fermion
species
g > 0, spin up electrons the opposite
for normalizable
zero-modes
and vice versa for with
g < 0 and exactly N = -INI. -
holds in the background soliton number,
of an antisoliton
Th is coupling momentum
between
charge and spin also extends N, there will be IgNl zeroall these modes number of zero-
_. to the angular modes
of the fermion. - 1).
For general
for k = O,l,..
. (IgNl only.
Wh en g and N are fixed, flux for a soliton
will be of one chirality N is Jd2xB = -27rN.
The integrated
of winding
If it were not for the dependence relating number
of the number
modes on g, there would be an index theorem opposite of solitons chirality zero-modes numbers to the winding
the difference
of .normalizable In the case for
of the gauge field.
of winding
one and two, the potential
and the zero-modes
the s-states is plotted form for the holes,
in Fig. 1. Had we instead started
with the full nonrelativistic
&$-ia+,~)2
+ gs,B]$
= ,Q/,
,
(12)
then the final zero-mode tions on sgn(gNs,)
equation
would be exactly
as before, and similar momentum
restricstates,
would arise for s, = fl. 11
For higher angular
we tite
c.-
+(r,S)
= #` x(r).
Th e S c h roe"d'mger equation
for x(r)
becomes
x"
+
3
+
[(E2
-
m2)
-
K&)]x
=
0
,
(13)
-1
where _ giving
the effective
potential
is determined
by summing
all the relevant
terms,
2&kr21NI-2
1 + r2lNI Previously, we analyzed number the restrictions
rWI-2
+ (1 + GINI)
((~N)~r~l~l
- 2glNlNsJ
.
(14)
on g, sZ and N arising from normalizability. corresponds of r$a2(r) precisely to the angular re-
The k quantum momentum _. quires
for the zero-modes and integrability
quantum
number,
for the zero-modes
1gNI - lkl > 1 and k > -l/2. only when sgn(gNs,)
The effective
potential
shows attraction behavior is like (see > 0.
in the s-channel +l/r21NI.
< 0 and the long-distance but attractive
Th e p o t en t' 1 is thus short-ranged la arises whenever simultaneously potential
at small distances < 0 and sgn(gNs,)
Fig. 1). For k > 0, attraction Requiring
-.
sgn(gkN)
both
conditions the effective
thus couples
k to s,. Had we dropped
as
the Pauli term,
could be written
v,ff
which is clearly positive
=
r41NI(gN + k)2 + k2
?-2(1 + ANI) The Pauli term ' therefore
(15)
gives an attractive and is crucial the potential and the now to
for all r.
piece to the effective for the existence is attractive existence
potential
in all angular
momentum determines
channels whether
of the zero-modes. Having potential
Sgn(gks,)
or repulsive. of an attractive
discussed the hole-soliton and unexpected
wave equation we turn
zero-modes,
the size stability
of the solitons. quantum 12
ClZssically, f
the hole would motion
drop
into the minimum
of the potential to the energy.
and the The zero-
zero-point
would
be the first correction
point motion around simplicity, soliton
can be estimated
by making
a saddle point expansion the relevant oscillator
of the potential frequency w. For the
the minimum we consider
and thus determine
the case k = 0 and N = 1, for which, potential has the form
upon
restoring
size X, the effective
V eff
There is an attractive
=
gr
2
(iI2
w*
- x2(1
1
+ r2)2 *
sz = +1
( x > (1 +r2)2
(16)
and
minimum
for any sgn(gs,)
> 0, and we pi&
g > 0. The classical binding
-
energy becomes
_
E class
Solitons would be spontaneously to offset the cost of creating restriction constrains X to
=
-- 2l!Pz x2
I '
produced the soliton,
only if the binding which
energy gain is enough to be 2rf. This
we computed
A<$. In physical units X < g/3 limit A, which, to soliton
(17)
` f 1 we use values of g in the range of one to size of the order of one lattice fluctuations, spacing. Since
ten, gives an upper
the sigma model represents one-lattice
the long wavelength to accept.
details at the level of of holes invariance
spacing are more difficult binding, an upper
In any case, the interaction
and subsequent and establish
at least at the classical level, break the dilation bound on soliton size. Although
the classical result favors size to a finite
X + 0, the exact quantum range, as we will see below.
problem,
in principle,
fixes the soliton
13
-
.-
Iv.
equation
QUANTUM
EFFECTS
after resealing
f
The Schroedinger
for k = 0, N = 1, s, = 1 becomes,
-.
by A,
x"+ $+[-c+ (g2r2 (1 +;2;:)lx = 0 7
where -6 G (E2 - m 2 )X 2. For boundstates, the lowest eigenvalue, Requiring the eigenvalues satisfy
(18)
E > 0. If EO is
then the energy of the hole becomes E = m(1 -c~/rn~X~)r/~. X > A/m. The binding energy in the to in
E2 > 0 gives the lower bound
is given by -(co/2n2X2) rest energy, an upper ere f ore, positivity of soliton-hole
large m limit
+ 0(mF3). bound
Since this must be enough regime results
offset the soliton _. Xdm.Th and the energetics
like the classical
of the energy-squared
for stationary
states size A,
binding
gives a finite range for the soliton
@
m
Solutions
-.
<A<
J-
60
47rjm
'
constructed
(19)
above. E
for eigenvalues
E = 0 were the zero-modes we explicitly binding to demonstrate
It is necessary for soliton-hole also exist.
that solutions
for positive
We turn now to the search for boundstates.
-`
` -
Saddle-Point
Approximation.
expansion
Corrections around
to the classical
trajectory
can be In the the new is at
done by a saddle-point harmonic approximation
the minimum
of the potential.
to the potential
V(r)
= V(0) + (1/2)V"(0)r2,
energy is E = Eclass + (1/2)tiw.
We expand
about
zero, since the minimum
r = 0 for k = 0, N = 1, sZ = 1, g > 0. In the case of a minimum is necessary problem to analytically coordinates extend
at zero, it since the problem.
V(r) to th e unphysical
region of -r,
in radial
can also be thought 14
as a one-dimensional
Usingthe
effective
potential
we wrote earlier,
V"(0)
= 2g2 + 8g - 20~. Therefore, becomes
in the harmonic f
approximation,
the X(T) equation
-1
- where c' = -c - V(0)
x" + 6 + (6'-a2r2)x
and V(0) = -29. This
= 0 ,
equation has a spectrum
(20)
c' =
4a(n + l/2).
g Th e er` envalue equation
then becomes
-e = -2g+4&TT&i n+i . ( >
Even for n = 0, we find no positive, for k f -` find that 0, and separately the quantum N f i.e., boundstate, solutions to c
(21)
The case
1, cannot
be done so easily;
but even here, we into the
fluctuations
push the hole out of the minimum The saddle-point approximation
continuum, anharmonic
i.e., E is always negative. r3 and higher-order
is sensitive to goes
terms; especially
so here, since our potential
to zero quickly differential
for T > 1. Since these higher-order harder to solve, perhaps
effects make the approximate variational methods
equation
Rayleigh-Ritz
are more efficient.
Variational
the leading x(r),
Methods.
The asymptotic
behavior r t
of x(r)
is obtained
by keeping
powers in V(r)
for the two limits
0 and 1" +
00. This gives, for
the Bessel functions
x(r)
-+ Jo(r&Z) -+ K,(r&)
as r -b 0 (22) as r + 00 .
width
This is the asymptotic
behavior
for wavefunctions
in a box of finite
a with
V = -2g for r 5 a and V = 0 for r > a. The eigenvalues
15
in this case are obtained
by matching parameter c .--I
the wavefunctions a, an entire
and their
derivatives
at r = a. By varying which, in turn, function
the can r$, we
set of wavefunctions functions
can be obtained
be used as variational compute the variational
for our potential.
For a variational
energy by inverting
the Schroedinger
equation,
.
E vat =
s d2xKg)2 + ew sd2xp
from the box,
*
Jo(Jzs-Er) matching
(23)
+ con-
In the case of the wavefunctions cKo( &), ditions. where both
we have 4 = by the boundary
E and c are determined
In the entire
range of a and the accompanying for Coulomb functions like rPeBar,
wavefunctions, Gaussian
we find like
E 2)ar > 0. Similarly,
_.rpemffr2 , hyperbolic we always find E,,, but apparently
functions
functions
sech(or),
and rational
functions
like P/(1
+ (p~)y)~, modes
> 0. It is peculiar
that the system admits
zero-energy
no negative-energy
quantum
states. The above variational state through
functions choice
were able to get arbitrarily of the parameters.
-.
close to the zero-mode methods
a suitable
Variational
have the drawback
that a small change in of a
the parameter complete
effects the function
significantly
for all r. A linear combination to the coefficients
set is desirable Alternatively,
and minimization exact differential
with respect integrators
is often
-` ` -
tractable. linear
in numerical
packages handle
differential method,
equations
like this one well. Using an ODE solver based on the the eigenvalue of the solutions. Although was found. methods, 16 we are unable to find any boundstates in increments of a thousandth, eigenvalue we makes
Gear-stiff examined the function correctly,
by varying
the large r behavior diverge at large r. 6 solution
An unphysical identified
the program
the zero-mode
no positive
Therefore,
using three different
for thG soliton-hole form f -1 soliton for a general is simply
problem. coupling
Starting
from either the relativistic boundstate equation
or nonrelativistic is the same. A
g, the essential
a small region of nonzero magnetic
flux B(r, t9). Spinless particles, In the presence of spinto the coupling g. It should gives
i.e., dropping coupling, would
the Pauli term, feel no attractive portion
potential.
an attractive
arises which is very sensitive limit of constant problem start magnetic
seem that
the extreme
flux everywhere field problem
also have boundstates the well-known Landau
if the soliton
does. The constant at zero energy,
levels which
yet have an attractive is overly simplistic fields. We can
piece for V&f in the presence of the Pauli term. and still does not rule out boundstates _. write the s-wave equation for x(r) as
This argument
for fermions
in magnetic
for s, = 1 in the Coulomb
gauge as before,
using the scalar field d(x)
x"+ 6 + (-e - g2(t7$)2 gv2qqx = cl . +
The Coulomb states. energy -` ` states in the s-channel netic field cannot mogenous form is called [VI, the statement boundstates differential implies equation that that the two-dimensional for any v, the nonlinear, potential in two dimensions the system is an instructive number example
(24)
for boundwith
For V, = -o/r,
has an infinite
of boundstates
E = -o/(1
+ 2n) (Appendix).
If the class of potentials
that have boundmaginho-
second-order
g2(&) *(c$) -&TV24= v ,
has no real solutions. 17
(25)
--
F` the symmetric or first order equation, f y2 = -V
case v p' + p/r
= V(r),
we can write Defining equation
p(r)
= gd$/dr
giving
the
- p2 = -V.
p = y + 1/2r, and making
we get y' -
+ 1/4r2 G p(r).
Th'is is Riccati' s
the substitution
u = e-J' Y(~)&) we obtain
the linear second-order
d% p+pu
equation
=
0
.
(26)
quantum
The solution problem
of this equation
is the zero-mode
of the one-dimensional potential
u" + (E - v)u
= 0, w h ere the attractive
ug, the corresponding
f7 = -p = V - l/4r2. problem
If we call the zero-mode then becomes _
solution
4 for the original
c$(r) =
This is readily integrable
$jd7' (
-$;+$)
= (l/g)
.
ln( fi/uo(r)).
(27)
It is not unrea-
and gives 4(r)
sonable to expect
zero-modes
for appropriate potentials
choices of V, since by assumption with boundstates. The solutions boundstates to
V belongs to the class of attractive
this equation for fermions are found for V, a member moving
of [VI, is very interesting
and implies
in static magnetic
fields in two dimensions. would
Even if solutions addressed.
for some V, our soliton methods
problem
still not be directly
The conventional existence
do not favor the existence of boundstates, the differential equation field theory
but a proof of
may lie in considering
for 4. The above analysis description of holes and and soliton
made simple assumptions solitons. quantum soliton-hole
of the continuum
It leads to a rich coupling numbers; we explicitly
of spin, charge, angular momentum, demonstrated zero-energy beyond
states for the original the above treatment
interaction.
The prospects
for binding
are summarized
in Sec. V. 18
-
.-
V.
description nondynamical
CONCLUSIONS
of spin-waves and fermions is the 2 + 1 QED-
f
A long wavelength -1 like Lagrangian whether with
A,.
Aside from
the signs of masses, and regime or how many species
one works in the relativistic
or nonrelativistic
of fermions
there are, one always finds attraction
of holes to solitons in all channels of the soliton as a sta-
when the Pauli term is present and in the approximation tionary fermion Curiously, background. The attraction
places precise constraints momentum and the soliton
on the signs of the winding number.
charge, its spin, its angular it is straightforward
to prove that zero-modes Had boundstates existed,
exist in any background
where A0 = 0, Ai = &ijdj$.
two opposite charge holes of
provided the bindother of one
-. spin up and spin down would tumble to the lowest eigenstate,
ing energy thus gained is sufficient rich possibilities soliton exist. to offset the soliton problem
rest energy.
Many
The boundstate
was analyzed
in the limit
and one hole. Since both the hole and the soliton carry A, charge, the two of motion the limit become coupled through
equations considered
A,, = (-i/2)(Zt13PZ)
+ g$y,$.
We
where A, is entirely could not be found. the potential
given by the soliton Perhaps self-consistent
background. corrections
To this of A,
order, boundstates through
1c,will modify
sufficient
to find boundstates.
This path was equations
not pursued, of motion,
since the same consistency
requires we resolve the soliton
etc. We showed that the very existence of boundstates invariance spacings. of the soliton With
in the relativistic
regime breaks the dilation the range of a few lattice the soliton 4(r). background,
very nicely and gives sizes in lack of boundstates field parameterized for by so-
the apparent
we considered
a general magnetic
The existence
of s-wave boundstates 19
in this case depends on zero-mode
lutic&
of a new equation
with a purely
attractive
potential
energy
V(r).
Deriving is curone
the boundstates f rently under
after finding
the zero-modes
for interesting
choices of V(r)
progress.
The relation
of these new configurations
to the soliton
is not clear but it may shed light on what the self-consistent _ The importance by Ya. Kogan22 with though of the Pauli term for binding in a system described
potential
may be like. context
was discovered
in another
by charged fermions
of mass m interacting term. Even
a gauge field that the two fermions
obtains
a mass M through
a Chern-Simons
have equal sign charge, th...