Microscopic model versus systematic low-energy
effective field theory for a doped quantum ferromagnet
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Gerber, U. et al. “Microscopic Model Versus Systematic Lowenergy Effective Field Theory for a Doped Quantum
Ferromagnet.” Physical Review B 81.6 (2010) : 064414. © 2010
The American Physical Society
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http://dx.doi.org/10.1103/PhysRevB.81.064414
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American Physical Society
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Thu May 26 04:49:02 EDT 2016
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PHYSICAL REVIEW B 81, 064414 共2010兲
Microscopic model versus systematic low-energy effective field theory
for a doped quantum ferromagnet
U. Gerber,1 C. P. Hofmann,2 F. Kämpfer,3 and U.-J. Wiese1
1
Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, Bern University,
Sidlerstrasse 5, CH-3012 Bern, Switzerland
2Facultad de Ciencias, Universidad de Colima, Bernal Díaz del Castillo 340, Colima CP 28045, Mexico
3Condensed Matter Theory Group, Department of Physics, Massachusetts Institute of Technology (MIT),
77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA
共Received 21 August 2009; revised manuscript received 29 November 2009; published 17 February 2010兲
We consider a microscopic model for a doped quantum ferromagnet as a test case for the systematic
low-energy effective field theory for magnons and holes, which is constructed in complete analogy to the case
of quantum antiferromagnets. In contrast to antiferromagnets, for which the effective field theory approach can
be tested only numerically, in the ferromagnetic case, both the microscopic and the effective theory can be
solved analytically. In this way, the low-energy parameters of the effective theory are determined exactly by
matching to the underlying microscopic model. The low-energy behavior at half-filling as well as in the singleand two-hole sectors is described exactly by the systematic low-energy effective field theory. In particular, for
weakly bound two-hole states the effective field theory even works beyond perturbation theory. This lends
strong support to the quantitative success of the systematic low-energy effective field theory method not only
in the ferromagnetic but also in the physically most interesting antiferromagnetic case.
DOI: 10.1103/PhysRevB.81.064414
PACS number共s兲: 75.30.Ds, 75.10.Pq, 75.50.Dd, 12.39.Fe
I. INTRODUCTION
Achieving a quantitative understanding of the doped antiferromagnetic precursors of high-temperature superconductors is a great challenge in condensed matter physics. In particular, away from half-filling, Monte Carlo simulations of
these strongly correlated electron systems suffer from a very
severe sign problem. Also analytic calculations in underlying
microscopic Hubbard or t-J-type models are not fully systematic but suffer from uncontrolled approximations. Particle
physicists face similar challenges in the physics of the strong
interactions between quarks and gluons. Remarkably, the
low-energy physics of pions—the pseudo-Goldstone bosons
of the spontaneously broken SU共2兲L ⫻ SU共2兲R chiral symmetry of QCD—is described quantitatively by a systematic effective field theory,1–4 known as chiral perturbation theory.
Similarly, the low-energy physics of the spin waves or
magnons—the Goldstone bosons of the spontaneously broken SU共2兲s spin symmetry in an antiferromagnet—is also
captured by a systematic effective field theory.5–11 Early attempts to include doped holes into the effective description
of antiferromagnets are described in Refs. 12–15. Motivated
by the quantitative success of baryon chiral perturbation
theory16–19 for pions and nucleons in QCD, fully systematic
low-energy effective field theories have been developed for
hole-doped antiferromagnets both on a square20,21 and on a
honeycomb lattice,22 as well as for electron-doped antiferromagnets on a square lattice.23 The resulting systematic effective field theories have been used to study magnon-mediated
two-hole21,24 and two-electron bound states23 as well as spiral phases in the staggered magnetization order
parameter.22,23,25 The quantitative correctness of the magnon
effective field theory has been demonstrated in great detail at
permille level accuracy by comparison with Monte Carlo
simulations of the quantum Heisenberg model using the very
1098-0121/2010/81共6兲/064414共16兲
efficient loop-cluster algorithm.26–28 Similarly, the singlehole sector of the t-J model has been simulated both on the
square29,30 and on the honeycomb lattice.31 Indeed, the observed location of the hole pockets in the Brillouin zone has
provided important input for the construction of the various
systematic effective field theories for doped antiferromagnets.
In general, low-energy effective field theories cannot be
derived rigorously from the underlying microscopic physics.
Instead, one performs a detailed symmetry analysis of the
underlying theory and constructs all terms in the effective
Lagrangian that are invariant, order by order in a systematic
derivative expansion. Each term is then endowed with an a
priori undetermined low-energy parameter. In particular, the
values of these parameters are not fixed by symmetry considerations, but must be determined by matching to the underlying microscopic system. This can be done by comparison with either experiment or numerical simulations. Only in
exceptional cases the underlying microscopic model can be
solved analytically and the low-energy parameters can be
determined exactly. One such case is the ferromagnetic
Heisenberg model whose low-energy physics was analytically derived by Dyson.32 The corresponding low-energy effective theory was constructed by Leutwyler33 and discussed
in great detail in Refs. 34–36. Remarkably, in contrast to the
effective theory for antiferromagnets, the effective theory for
ferromagnets contains an additional Wess-Zumino term
whose quantized prefactor is the total magnetization. The
values of the magnetization and of the spin stiffness—the
other leading order low-energy parameter of a ferromagnet—
can be easily read off from Dyson’s analytic solution of the
underlying microscopic Heisenberg model. Thanks to the
analytic solvability of the ferromagnetic Heisenberg model,
in this case the predictions of the effective theory can be
verified rigorously. Indeed, once the low-energy parameters
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PHYSICAL REVIEW B 81, 064414 共2010兲
GERBER et al.
have been fixed by matching to the underlying system, in the
low-energy domain the effective theory yields exactly the
same results as the Heisenberg model. It is interesting to note
that the calculations in the effective theory are much simpler
than those in the microscopic model.
The effective field theories for doped antiferromagnets
mentioned above have again been constructed based on symmetry considerations. However, in that case the underlying
two-dimensional Hubbard or t-J-type models cannot be
solved analytically, and one must hence rely on numerical
methods for fixing the low-energy parameters and for verifying the validity of the low-energy effective theory. In this
paper, we consider a microscopic Hubbard-type model for a
doped ferromagnet which can be solved analytically. Furthermore, the corresponding low-energy effective theory can be
constructed in exactly the same way as in the antiferromagnetic case. By showing explicitly that the microscopic and
the effective theory of the doped ferromagnet yield identical
results, we lend further support to the general construction
principle for the effective theories. For simplicity, our analytic study will be performed in one spatial dimension, but
the extension to higher dimensions is straightforward. It
should be noted that in one spatial dimension the antiferromagnetic Heisenberg model is analytically solvable by the
Bethe ansatz.37 According to Haldane’s conjecture,38 the corresponding low-energy effective theory is a two-dimensional
O共3兲 nonlinear ␴ model at vacuum angle ␪ = ␲. As first noted
by Lieb and Wu, in one dimension even the Hubbard model
can be solved analytically.39,40 In particular, these authors
have shown that this model has no Mott transition. We prefer
to consider the ferromagnetic model because it is easier to
solve analytically and because its low-energy effective
theory is similar to the one of the doped antiferromagnets. It
should be pointed out that our ferromagnetic model is not
meant to provide a realistic description of ferromagnetism in
actual materials. This would require two bands as well as
Hund rule couplings.41 A more realistic model for doped
quantum ferromagnets has been discussed and solved analytically in Ref. 42 and the related polaron problems have
been discussed in Ref. 43. For simplicity, here, we impose
ferromagnetism by including a corresponding coupling by
hand. Still, the range of applicability of the effective theory
to be constructed in this paper goes beyond our ferromagnetic model, as the effective theory applies to any system
exhibiting the same symmetries and symmetry breaking pattern as the microscopic model considered here.
While the calculations in the microscopic model are nonperturbative, some of the corresponding calculations in the
effective theory are based on perturbation theory. Indeed, it
is a big advantage of the effective theory approach to Goldstone boson physics that perturbation theory provides quantitatively correct results in a systematic low-energy expansion. While Goldstone bosons are derivatively and thus
weakly coupled at low energies, the contact interactions between two holes may very well be strong, thus requiring a
nonperturbative treatment not only of the microscopic
model, but also of the effective field theory. A similar situation arises in the effective field theory approach to the strong
interactions between nucleons and pions—the Goldstone
bosons of the spontaneously broken chiral symmetry of
QCD. Systematic low-energy effective field theories have
also been used in studying light nuclei.44–53 Then, the shortrange repulsion between two nucleons is strong, which implies that the effective theory must be treated nonperturbatively. In that case, it is still an unsettled theoretical question
how this can be achieved fully systematically. In particular,
there are various power-counting schemes, due to
Weinberg,44 as well as due to Kaplan et al.,45 which are both
not fully satisfactory. Recently, an interesting modification of
the Kaplan-Savage-Wise scheme has been proposed,53 and it
remains to be seen whether this will finally resolve this issue.
In contrast to QCD or doped antiferromagnets, the ferromagnetic model studied here has the advantage that it can be
solved analytically. Hence, one may reach a deeper understanding of the subtle nonperturbative fermion dynamics. For
this purpose, we will investigate the two-hole sector in the
effective field theory and will then compare with the analytic
results of the underlying microscopic model. Hence, the
doped ferromagnet is a system in which systematic approaches to nonperturbative problems in effective field
theory can be tested. Thus, the investigations in this or related models may also have an impact on the corresponding
issues arising in the context of the strong interactions.
The paper is organized as follows. In Sec. II, the underlying microscopic model is presented and its symmetry properties are investigated in detail. The model is then solved at
half-filling, as well as in the one- and two-hole sectors. In
particular, the dispersion relations of magnons and holes, as
well as the binding energy of two holes and the two-hole
scattering states are determined analytically. In Sec. III, the
corresponding low-energy effective field theory is constructed using the nonlinear realization of the spontaneously
broken SU共2兲s spin symmetry. In particular, the hole fields
are included in the same way as for a doped antiferromagnet.
In Sec. IV, magnons, single holes, as well as two-hole scattering and two-hole bound states are investigated in the effective field theory framework. The a priori undetermined
low-energy parameters are fixed by matching to the underlying microscopic system, and it is verified explicitly that the
predictions of the effective theory agree exactly with those of
the microscopic model. Finally, Sec. V contains our conclusions. Some technical details are presented in the Appendix.
II. CONSTRUCTION AND SOLUTION OF A
MICROSCOPIC MODEL FOR A
DOPED FERROMAGNET
In this section, we construct a Hubbard-type microscopic
model for a doped ferromagnet, investigate its symmetries,
and then solve it in the zero-, one-, and two-hole sectors.
A. Microscopic model for ferromagnetism
Let us construct a microscopic model describing the hopping of fermions on a one-dimensional lattice with spacing a,
with the Hamiltonian
†
cx兲 − J 兺 Sជ x · Sជ x+a +
H = − t 兺 共c†x cx+a + cx+a
x
x
U
兺 共c†cx − 1兲2 .
2 x x
共2.1兲
The creation and annihilation operators for fermions at a site
x = an, n 苸 Z, with spin s = ↑ , ↓, are given by
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MICROSCOPIC MODEL VERSUS SYSTEMATIC LOW-…
†
†
c†x = 共cx↑
,cx↓
兲,
cx =
冉 冊
cx↑
cx↓
共2.2兲
.
A displacement D by one lattice spacing is generated by
the unitary operator D, which acts as
D
They obey the standard anticommutation relations
†
,cx⬘s⬘其 = ␦xx⬘␦ss⬘,
兵cxs
†
†
兵cxs,cx⬘s⬘其 = 兵cxs
,cx⬘s⬘其 = 0. 共2.3兲
Putting ប = 1, the spin operator at the site x is given by
␴ជ
Sជ x = c†x cx ,
2
关H,Sជ 兴 = 0,
Sជ = 兺 Sជ x ,
Q+ = 兺 共−
x
Q− = 兺 共− 1兲x/acx↓cx↑ ,
Again, by relabeling the sum over the lattice points, it follows that 关H , R兴 = 0. Another important symmetry is time reversal which is implemented by an antiunitary operator T.
It is useful to introduce a matrix-valued fermion operator
CAx =
CBx =
ជ = 共Q1,Q2,Q3兲,
Q
†
cx↓
†
cx↓ − cx↑
†
cx↑ − cx↓
cx↓
†
cx↑
冊
冊
,
x 苸 A,
,
x 苸 B.
Under combined transformations
苸 SU共2兲Q, it transforms as
ជ
Q
D
g 苸 SU共2兲s
Cx⬘ = gCx⍀T .
共2.10兲
and
⍀
共2.11兲
B
CAx = Cx+a
␴ 3,
D
A
CBx = Cx+a
␴3 .
共2.12兲
The appearance of the Pauli matrix ␴3 is due to the factor
共−1兲x/a. Under the spatial reflection R, which turns x into
Rx = −x, one obtains
R
共2.6兲
H=−
The factor 共−1兲x/a distinguishes between the two sublattices
A and B of even and odd sites. Unlike for an antiferromagnet, it may seem unnatural to make such a distinction for a
ferromagnet. However, as we will see later on, the introduction of two sublattices is also important for a ferromagnet, as
it will allow us to correctly identify the transformation properties of holes and electrons in the effective theory. It is
straightforward to convince oneself that the Hamiltonian is
indeed invariant, i.e.,
ជ 兴 = 0,
关H,Q
冉
冉
cx↑
Cx = C−x .
共2.13兲
The Hamiltonian can now be expressed in a manifestly
SU共2兲s-, SU共2兲Q-, D-, and R-invariant form
x
1
Q3 = 兺 共c†x cx − 1兲.
x 2
共2.9兲
cx = R†cxR = c−x .
Under the displacement symmetry, one obtains
and is thus invariant under global SU共2兲s spin rotations. As
we will see later, the SU共2兲s symmetry is spontaneously broken down to the subgroup U共1兲s by the formation of a uniform magnetization. It should be noted that this is not in
contradiction with the Mermin-Wagner theorem. The generators of another symmetry—a non-Abelian SU共2兲Q extension
of the Abelian U共1兲Q fermion number54,55—are given by
† †
1兲x/acx↑
cx↓,
R
共2.5兲
x
共2.8兲
By relabeling the sum over the lattice points, it is easy to
show that 关H , D兴 = 0. Another discrete symmetry is the spatial reflection R, which acts as
共2.4兲
where ␴
ជ denotes the Pauli matrices. Let us discuss the various terms in the Hamiltonian above. The term proportional to
t describes hopping of fermions by one lattice spacing, i.e., it
represents the kinetic energy. The parameter J ⬎ 0 is a ferromagnetic exchange coupling constant, while the term proportional to U ⬎ 0 describes an on-site Coulomb repulsion. As
mentioned earlier, this Hamiltonian does not provide a realistic description of real ferromagnetic materials. We consider
it because it is analytically solvable at low energies and can
thus be used to test the corresponding effective theory.
The microscopic model defined by Eq. 共2.1兲 has various
symmetries, which we are going to discuss now. It is
straightforward to confirm that the Hamiltonian commutes
with the total spin
cx = D†cxD = cx+a .
Q⫾ = Q1 ⫾ iQ2 .
共2.7兲
It should be pointed out that the SU共2兲Q symmetry would be
explicitly broken down to U共1兲Q if hopping terms between
sites belonging to the same sublattice would be included in
the Hamiltonian. Furthermore, it is worth noting that the generators of SU共2兲s commute with those of SU共2兲Q.
t
†
C x兴
兺 Tr关C†x Cx+a + Cx+a
2 x
−
J
†
␴ជ Cx+a兴
兺 Tr关C†x ␴ជ Cx兴 · Tr关Cx+a
16 x
+
U
兺 Tr关C†x CxC†x Cx兴.
12 x
共2.14兲
B. Eigenstates for electrons and holes
We will now construct electron and hole states above a
half-filled ground state containing up-spin fermions at each
lattice site. The corresponding vacuum state is given by
†
兩v典 = 兿 cx↑
兩0典,
共2.15兲
x
where 兩0典 represents an empty lattice without any fermions.
Indeed, acting with the Hamiltonian, one obtains
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PHYSICAL REVIEW B 81, 064414 共2010兲
GERBER et al.
H兩v典 = Ev兩v典,
J
Ev = − N,
4
共2.16兲
i.e., 兩v典 is indeed an eigenstate, with the vacuum energy Ev
proportional to the number of lattice sites N. The total spin of
the state 兩v典 is S = N / 2. By acting with the lowering operator
S− = 兺xS−x on the vacuum state 兩v典, one can construct the other
ground states belonging to the same SU共2兲s multiplet, which
contains 2S + 1 = N + 1 degenerate states.
Let us now construct a somewhat unconventionally normalized hole state of momentum p
兩hp典 = 兺 exp共ipx兲cx↑兩v典 = c p↑兩v典.
共2.17兲
FIG. 1. Dispersion relations for electrons 共dotted curve兲 and
holes 共solid curve兲
x
Ee共p兲 =
In order to check whether this is an eigenstate, we compute
J U
J U
+ + 2t cos共pa + ␲兲 = + − 2t + ta2 p̂2 .
2 2
2 2
共2.25兲
H兩hp典 = 共关H,c p↑兴 + c p↑H兲兩v典 = 共Eh共p兲 + Ev兲兩hp典,
共2.18兲
which shows that 兩hp典 is indeed an energy eigenstate. One
obtains the energy-momentum dispersion relation of a hole
as
J U
J U
Eh共p兲 = + + 2t cos共pa兲 = + + 2t − ta2 p̂2 ,
2 2
2 2
For electrons, the minima of the dispersion relation are located at p = 2n␲ / a, with n 苸 Z. Expanding around p = 0, we
get
Ee共p兲 =
where we have introduced p̂ = a2 sin共pa / 2兲. This periodic
function has minima at p = 共2n − 1兲␲ / a, with n 苸 Z. Expanding around p = ␲ / a, we obtain
冉 冊 冉冉 冊 冊
␲
J U
+ − 2t + ta2 p −
2 2
a
2
+O
p−
␲
a
2M h⬘
+O
冉冉 冊 冊
p−
␲
a
共2.21兲
with the rest mass M h and the kinetic mass M h⬘ given by
J U
M h = + − 2t,
2 2
1
M h⬘ =
.
2ta2
As we have discussed before, in quantum ferromagnets,
the global spin rotational symmetry SU共2兲s is spontaneously
broken by the formation of a uniform magnetization. The
ground states of these systems are invariant only under spin
rotations in the subgroup U共1兲s. In this case, Goldstone’s
theorem predicts 3 − 1 = 2 massless boson fields—the
magnons—also known as ferromagnetic spin waves.
A general ansatz for an electron-hole state is given by
兩ehp典 = 兺 exp共ipy兲f共x兲c†y↓cy+x↑兩v典.
共2.22兲
Since the theory is nonrelativistic, the rest mass M h and the
kinetic mass M h⬘ need not to be the same.
Similarly, we construct electron states
†
兩ep典 = 兺 exp共− ipx兲cx↓
兩v典 = c†p↓兩v典,
冏 冉 冊冔
Q−兩ep典 = − h − p +
␲
a
Here, x is the distance between the electron and the hole and
f共x兲 is the corresponding wave function of their relative motion. Indeed, magnons are massless bound states of an electron and a hole. We now consider
H兩ehp典 = H 兺 exp共ipy兲f共x兲c†y↓cy+x↑兩v典
共2.23兲
x,y
=
.
冉冋
H, 兺 exp共ipy兲f共x兲c†y↓cy+x↑
x,y
冊
册
+ 兺 exp共ipy兲f共x兲c†y↓cy+x↑H 兩v典.
共2.24兲
x,y
The SU共2兲Q symmetry then implies that the energy of an
electron is given by 共Fig. 1兲
共2.28兲
x,y
x
and we compute
共2.27兲
C. Gap equation for magnon states
4
,
p2
.
2M e⬘
Due to the SU共2兲Q symmetry, the rest and kinetic masses
M h,e and M h,e
⬘ of holes and electrons are identical.
.
The holes are massive objects and their dispersion relation is
given by
共p − ␲/a兲2
Ee共p兲 = M e +
4
共2.20兲
Eh共p兲 = M h +
共2.26兲
Again, for small momenta
共2.19兲
Eh共p兲 =
J U
+ − 2t + ta2 p2 + O共p4兲.
2 2
共2.29兲
The last term on the right-hand side represents the vacuum
energy. The electron-hole energy Eeh共p兲 is given by
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MICROSCOPIC MODEL VERSUS SYSTEMATIC LOW-…
冋
册
冢冣 冢
H, 兺 exp共ipy兲f共x兲c†y↓cy+x↑ 兩v典 = Eeh共p兲兩ehp典.
x,y
A
I4 I6 I2
J
I6 I5 I3
I B =−
2
zI2 zI3 zI1
C
共2.30兲
A somewhat tedious evaluation of Eq. 共2.30兲 implies that
兩ehp典 is an eigenstate only if
Eeh共p兲f共x兲 = − t关f共x − a兲共e
+
冉
z=2+2
共2.31兲
in the generic case x ⫽ 0 , ⫾ a, as well as
ipa
冊
− 1兲 + f共x + a兲共e
−ipa
3
J + U f共x兲,
4
− 1兲兴
i1 = − sign共␥兲
in the special case x = ⫾ a, and
Eeh共p兲f共x兲 = − t关f共x − a兲共eipa − 1兲 + f共x + a兲共e−ipa − 1兲兴
冉
冊
i2 = − sign共␥兲
共2.33兲
A cos共qa兲 + B sin共qa兲 + C
,
Eeh共p兲 − 2t关cos共qa兲 − cos共qa − pa兲兴 − J − U
共2.34兲
with
A=−
B=−
冉
C= J
J 1
2 2␲
冕
J 1
2 2␲
冕
i3 = − sign共␥兲
␲/a
dqf共q兲sin共qa兲,
−␲/a
dqf共q兲.
−␲/a
共2.35兲
These are three coupled gap equations which must be solved
self-consistently for A , B , C, and Eeh共p兲. We are going to do
this in the next section. The denominator in Eq. 共2.34兲 can be
rewritten as
Eeh共p兲 − 2t关cos共qa兲 − cos共qa − pa兲兴 − J − U
= ␣ cos共qa兲 + ␤ sin共qa兲 + ␥ ,
共2.36兲
共2兲
共p兲 = −
Eeh
共4J + 16U兲t2 2 2 3
p a + J + U + O共p4兲.
4
J共3J + 4U兲
As before, this is indeed the energy of an electron-hole state,
but this state is not a magnon either. Finally, 共again for ␥
⬍ 0兲 the condition i2 = 1 implies
共3兲
Eeh
共p兲 =
共2.37兲
D. Solution of the gap equation
Let us now solve the gap Eq. 共2.35兲. Inserting this equation into Eq. 共2.35兲, we obtain an eigenvalue problem with
eigenvalue 1,
共2.41兲
共2.42兲
␤ = 2t sin共pa兲,
␥ = Eeh共p兲 − J − U.
共2.40兲
Although this is the energy of an electron-hole state with
total momentum p, the corresponding eigenstate does not
共1兲
共p兲 does not vanish for zero
represent a magnon because Eeh
momentum. Similarly, the condition i3 = 1 can be fulfilled
only for ␥ ⬍ 0, which then implies
where
␣ = 2t关cos共pa兲 − 1兴,
1
J
关兩␥兩 + z共兩␥兩 + s兲
4 s共兩␥兩 + s兲
3
4t2 p̂2a2
共1兲
Eeh
.
共p兲 = J + U −
4
J
−␲/a
␲/a
1
J
关兩␥兩 + z共兩␥兩 + s兲
4 s共兩␥兩 + s兲
The solutions of the gap Eq. 共2.35兲 correspond to eigenvalues 1 in Eq. 共2.38兲. The condition i1 = 1 can be fulfilled only
for ␥ ⬍ 0 and then implies
dqf共q兲cos共qa兲,
冊 冕
J 1
,
2 兩␥兩 + s
− 冑− 4s共兩␥兩 + s兲z + 关兩␥兩 + z共兩␥兩 + s兲兴2兴.
␲/a
1
p̂2a2
−J−U
2
2␲
共2.39兲
+ 冑− 4s共兩␥兩 + s兲z + 关兩␥兩 + z共兩␥兩 + s兲兴2兴,
in the special case x = 0. These three equations represent the
lattice Schrödinger equation for an electron-hole pair with
wave function f共x兲. In order to solve these equations, we
transform to momentum space and obtain
f共q兲 =
U
− p̂2a2 .
J
The quantities I1 , I2 , . . . , I6 are integrals, which are evaluated
in the Appendix. The eigenvalues of the matrix I of Eq.
共2.38兲 are given by
共2.32兲
a2 p̂2
+ J
+ U f共x兲,
2
共2.38兲
where
Eeh共p兲f共x兲 = − t关f共x − a兲共eipa − 1兲 + f共x + a兲共e−ipa − 1兲兴
+ 共J + U兲f共x兲,
冣冢 冣 冢 冣
A
A
B = B ,
C
C
J共3J + 4U兲 − 16t2 2 2
p a + O共p4兲.
2共3J + 4U兲
共2.43兲
This energy vanishes at zero momentum. Hence, the corresponding electron-hole eigenstate can be identified as a magnon state. Indeed, the nonrelativistic dispersion relation
共3兲
共p兲 ⬀ p2 is characteristic for ferromagnetic spin waves. In
Eeh
momentum space the wave function for the relative motion
of the electron and hole forming the massless magnon takes
the form
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GERBER et al.
f共q兲 =
A cos共qa兲 + B sin共qa兲 + C
共3兲
Eeh
共p兲
− 2t关cos共qa兲 − cos共qa − pa兲兴 − J − U
冉
,
冊冉 冊 冉 冊
冉 冊 冉 冊
− cos共pa兲 − sin共pa兲
− sin共pa兲
共2.44兲
which turns into
f共x兲 = N
冉 冊
␣ + i␤
兩␥兩 + s
x/a
,
共2.45兲
We now introduce C, such that
冉 冊
pa
,
2
C=
J 1
2 2␲
冕
冉
␲/a
共2.53兲
冊
共2.54兲
dqg共q兲sin qa −
−␲/a
pa
.
2
pa
,
2
with
g共q兲 =
兩hhp典 = 兺 exp共ipy兲g共x兲cy↑cy+x↑兩v典,
− C sin共qa − pa/2兲
.
Ehh共p兲 − 2t关cos共pa − qa兲 + cos共qa兲兴 − J − U
共2.55兲
x,y
g共− x兲 = − g共x兲exp共− ipx兲.
共2.46兲
The antisymmetry condition g共−x兲 = −g共x兲exp共−ipx兲 follows
from the Pauli principle. In complete analogy to the particlehole states, one derives the Schrödinger equation
Let us now solve the gap equation. Inserting Eq. 共2.55兲
into Eq. 共2.54兲 yields
1=−
J 1
2 2␲
冕
␲/a
dq
−␲/a
sin2共qa兲
.
Ehh共p兲 − 4t cos共qa兲cos共pa/2兲 − J − U
共2.56兲
Ehh共p兲g共x兲 = t关g共x + a兲共1 + e−ipa兲 + g共x − a兲共1 + eipa兲兴
+ 共J + U兲g共x兲,
共2.47兲
Using
1
2␲
for a generic situation with x ⫽ 0 , ⫾ a. In the special case x
= ⫾ a, one obtains
冕
␲/a
−␲/a
dq
sin2共qa兲
1
= sign共␥兲
,
␣ cos共qa兲 + ␥
兩␥兩 + 冑␥2 − ␣2
共2.57兲
Ehh共p兲g共x兲 = t关g共x + a兲共1 + e−ipa兲 + g共x − a兲共1 + eipa兲兴
+
冉
冊
3
J + U g共x兲,
4
共2.48兲
with
␥ = Ehh共p兲 − J − U,
冉 冊
␣ = − 4t cos
while for x = 0, the Schrödinger equation takes the form
pa
,
2
共2.49兲
J
= J + U − Ehh共p兲 + 冑关Ehh共p兲 − J − U兴2 − 16t2 cos2共pa/2兲.
2
Going to momentum space, one obtains the gap equation
B=−
J 1
2 2␲
冕
J 1
2 2␲
冕
共2.59兲
␲/a
Squaring this equation, we find the two-hole energy
dqg共q兲cos共qa兲,
3
16t2 4t2 p2a2
+
+ O共p4兲.
Ehh共p兲 = J + U −
4
J
J
−␲/a
␲/a
−␲/a
dqg共q兲sin共qa兲,
共2.58兲
for ␥ ⬍ 0, one thus obtains
Ehh共p兲g共x兲 = t关g共x + a兲共1 + e−ipa兲 + g共x − a兲共1 + eipa兲兴.
A=−
共2.52兲
冉 冊
B = − C cos
The gap equation can thus be written as
E. Two-hole states
Similar to the particle-hole spin-wave states, we now derive a Schrödinger equation for two-hole bound states. We
make the ansatz
A
A
=
B
B
pa
pa
A = − sin
B.
2
2
⇒ cos
A = C sin
for x ⱖ 0, where N is a normalization factor. For x ⱕ 0, one
finds f共x兲 = f共−x兲ⴱ.
cos共pa兲
共2.50兲
From this expression, we read off the total rest mass of the
two-hole bound state as
3
16t2
,
M hh = J + U −
4
J
with
A cos共qa兲 + B sin共qa兲
.
g共q兲 =
Ehh共p兲 − 2t关cos共qa兲 − cos共qa − pa兲兴 − J − U
共2.60兲
共2.61兲
while the corresponding kinetic mass is given by
共2.51兲
The antisymmetry condition g共−x兲 = −g共x兲exp共−ipx兲 implies
g共−q兲 = −g共p + q兲. Imposing this condition on the gap equation leads to
⬘ =
M hh
J
.
8t2a2
共2.62兲
The binding energy of the two-hole state hence takes the
form
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EB = 2M h − M hh = J
冉 冊
1 4t
−
2 J
2
共2.63兲
.
As we will see below, the two-hole bound-state wave function is normalizable only for 0 ⬍ t ⬍ J / 8, while for t ⬎ J / 8
the two-hole bound state disappears.
Let us now consider the wave function in coordinate
space. For ␥ ⬍ 0, we obtain
冉 冊冋
g共x兲 = C0 exp
ipx
2
冉 冊册
pa
t
− 8 cos
J
2
x/a
,
共2.64兲
where C0 is a normalization constant. The expression Eq.
共2.64兲 indeed satisfies the antisymmetry condition g共−x兲
= −g共x兲exp共−ipx兲.
Let us also consider scattering states of two holes described by the ansatz
g共x兲 = Ã exp共iqx兲 + B̃ exp共− iqx兲.
共2.65兲
For x ⫽ 0 , ⫾ a and p = 0, the Schrödinger equation is given
by
2t关g共x + a兲 + g共x − a兲兴 + 共J + U兲g共x兲 = Eg共x兲. 共2.66兲
Inserting the ansatz of Eq. 共2.65兲 into Eq. 共2.66兲 leads to
E = 4t cos共qa兲 + J + U,
metries of the underlying microscopic system. Therefore, we
now construct magnon fields and discuss how they transform
under those symmetries.
As we have mentioned in Sec. II, in a quantum ferromagnet the global spin rotation symmetry G = SU共2兲s is spontaneously broken by the formation of a uniform magnetization.
The ground state of these systems is invariant only under
spin rotations in the unbroken subgroup H = U共1兲s. As a consequence of the spontaneous symmetry breaking, there are
two massless Goldstone boson fields, leading to the ferromagnetic spin wave or magnon. We already discussed magnons in the microscopic model, where we calculated the dispersion relation in Eq. 共2.43兲. In the effective field theory, the
direction of the magnetization is described by a unit-vector
field
eជ 共x兲 = 共e1共x兲,e2共x兲,e3共x兲兲 苸 S2,
in the coset space G / H = SU共2兲s / U共1兲s = S , where x = 共x1 , t兲
is a point in 共1 + 1兲-dimensional Euclidean space-time.
Beyond the O共3兲 vector representation eជ 共x兲, it is useful to
introduce an alternative CP共1兲 representation of the magnon
field using 2 ⫻ 2 Hermitean projection matrices P共x兲 that
obey
=
共2.68兲
This implies that the ansatz of Eq. 共2.65兲 is purely real. Finally, we obtain
B̃
=
sin共qa兲 + i关cos共qa兲 + 8t/J兴
.
sin共qa兲 − i关cos共qa兲 + 8t/J兴
P共x兲2 = P共x兲,
1
P共x兲 = 共1 + eជ 共x兲 · ␴
ជ兲
2
J cos共qa兲 + 8t
J
,
+i
à =
16t sin共qa兲
16t
Ã
Tr P共x兲 = 1,
共2.69兲
Later, we will compare this result with the corresponding one
obtained in the effective field theory.
共3.2兲
and are given by
while the amplitudes are given by
J cos共qa兲 + 8t
J
= Ãⴱ .
−i
B̃ =
16t sin共qa兲
16t
共3.1兲
2
P共x兲† = P共x兲,
共2.67兲
eជ 共x兲2 = 1,
冉
冊
1 + e3共x兲
e1共x兲 − ie2共x兲
1
.
2 e1共x兲 + ie2共x兲
1 − e3共x兲
共3.3兲
The first symmetry we encountered in Sec. II was the
global spin rotation symmetry SU共2兲s under which the magnon field transforms as
P共x兲⬘ = gP共x兲g† .
共3.4兲
Note that the magnon field P共x兲 is invariant under the Abelian and non-Abelian fermion number symmetries U共1兲Q and
SU共2兲Q, i.e.,
ជ
Q
P共x兲 = P共x兲.
共3.5兲
Unlike in an antiferromagnet, in a ferromagnet the order parameter eជ 共x兲 is invariant under the displacement symmetry
D, i.e.,
III. CONSTRUCTION OF THE EFFECTIVE FIELD
THEORY FOR THE DOPED FERROMAGNET
The effective field theory we are going to discuss now
captures the low-energy physics of the underlying microscopic system, order by order in a systematic low-energy
expansion. The effective field theory is constructed in complete analogy to the corresponding cases of hole- or electrondoped antiferromagnets.20,21,23
A. Symmetry properties of magnon fields
In this section, we are going to investigate the symmetries
of magnon fields. At the beginning of Sec. II, we have studied the symmetries of the microscopic model describing ferromagnetism. The effective field theory must share the sym-
D
eជ 共x兲 = eជ 共x兲 ⇒ D P共x兲 = P共x兲.
共3.6兲
Under the spatial reflection R, which turns the point x
= 共x1 , t兲 into the reflected point Rx = 共−x1 , t兲, the magnon field
transforms as
R
P共x兲 = P共Rx兲.
共3.7兲
Another important symmetry is time reversal T, which turns
x into Tx = 共x1 , −t兲. The spin transforms like the orbital angular momentum Lជ of a particle. The momentum pជ changes
sign under time reversal and so does Lជ , i.e., TLជ = −Lជ . Consequently, under T the magnetization vector 共which is a sum of
microscopic spins兲 transforms as
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GERBER et al.
T
eជ 共x兲 = − eជ 共Tx兲 ⇒ T P共x兲 = 1 − P共Tx兲.
共3.8兲
SWZ关eជ 共1兲兴 − SWZ关eជ 共2兲兴
= − im
B. Effective action for magnons
Since the low-energy physics is dominated by terms with
the smallest possible number of derivatives, we construct an
effective Lagrangian according to a systematic derivative expansion. All terms in the Lagrangian must be invariant under
the symmetry transformations considered in the previous
section. In contrast to an antiferromagnet, a ferromagnet has
a conserved order parameter—the total spin. In the effective
theory, this manifests itself by the presence of a WessZumino term, which gives rise to a nonrelativistic magnon
dispersion relation.33 Indeed, as we have seen from the calculations in the microscopic model, the ferromagnet has a
nonrelativistic spectrum, i.e., E ⬃ p2. The leading order Euclidean effective action for an undoped ferromagnet derived
in Ref. 33 takes the form
S关eជ 兴 =
冕 冕
␤
dx1
dt
0
␳s
⳵1eជ · ⳵1eជ + SWZ关eជ 兴,
2
共3.9兲
with ␳s being the spin stiffness. The Wess-Zumino term is
given by
SWZ关eជ 兴 = − im
冕 冕 冕
␤
dx1
0
= − im
dx1
1
4␲
H2
S2
dtd␶eជ 共2兲 · 共⳵teជ 共2兲 ⫻ ⳵␶eជ 共2兲兲
dtd␶eជ · 共⳵teជ ⫻ ⳵␶eជ 兲.
e2共x, ␶兲 = ␶e2共x兲,
共3.13兲
冕
S2
dtd␶eជ · 共⳵teជ ⫻ ⳵␶eជ 兲 = n 苸 Z
共3.14兲
is the integer winding number of the field eជ which maps S2
共parameterized by t and ␶兲 into the order parameter sphere
S2. Indeed, the corresponding second homotopy group is
given by ⌸2关S2兴 = Z. Hence, the extrapolation ambiguity is
given by
SWZ关eជ 共1兲兴 − SWZ关eជ 共2兲兴 = − im
冕
dx14␲n.
共3.15兲
Since m is the magnetization density,
M=m
冕
dx1
共3.16兲
is the total spin of the entire magnet and hence an integer or
a half-integer. Since
exp共− SWZ关eជ 共1兲兴 + SWZ关eជ 共2兲兴兲 = exp共4␲iMn兲 = 1,
共3.17兲
the factor exp共−S关eជ 兴兲 that enters the path integral of Eq.
共3.12兲 is thus unambiguously defined, irrespective of the arbitrarily chosen extrapolation eជ 共x , ␶兲.
In the CP共1兲 representation, the leading order low-energy
Euclidean action takes the form
共3.11兲
S关P兴 =
The action of Eq. 共3.9兲 enters the Euclidean path integral
冕
dx1
dtd␶eជ 共1兲 · 共⳵teជ 共1兲 ⫻ ⳵␶eជ 共1兲兲
0
e3共x, ␶兲 = 冑1 − e1共x, ␶兲2 − e2共x, ␶兲2 .
Z=
H2
The two extrapolations eជ 共1兲 and eជ 共2兲 over the two hemispheres
H2 共which are differently oriented due to the minus sign
between the Wess-Zumino terms兲 are combined to an extrapolation eជ over an entire compact sphere S2. We now use
the fact that
d␶eជ · 共⳵teជ ⫻ ⳵␶eជ 兲. 共3.10兲
Here, m is the magnetization density. This term contains only
one temporal derivative and hence leads to a nonrelativistic
dispersion relation. The coordinates t and ␶ parameterize a
disk or two-dimensional hemisphere H2, which is bounded
by the compactified Euclidean time interval S1. The magnon
field eជ 共x兲 at physical space-time points x 苸 R ⫻ S1 is extended to a field eជ 共x1 , t , ␶兲 in the three-dimensional domain
共x1 , t , ␶兲 苸 R ⫻ H2. The integrand of the Wess-Zumino term is
a total derivative and hence only receives contributions from
the boundary, which coincides with the physical space-time
where eជ 共x , ␶ = 1兲 = eជ 共x兲. A possible extrapolation of the physical magnon field into the additional dimension with
eជ 共x , ␶ = 0兲 = 共0 , 0 , 1兲 is given by
e1共x, ␶兲 = ␶e1共x兲,
dx1
+ im
1
dt
冕 冕
冕 冕
冕 冕
冕 冕
冕
␤
dx1
dt Tr关␳s⳵1 P⳵1 P
0
1
Deជ exp共− S关eជ 兴兲,
+ 2m
共3.12兲
which should depend only on the physical magnon field and
not on a particular extrapolation into the additional dimension. In order to show that this is indeed the case, we compare two arbitrary extrapolations eជ 共1兲共x , ␶兲 and eជ 共2兲共x , ␶兲 and
we consider the difference between the two corresponding
Wess-Zumino terms
d␶ P共⳵t P⳵␶ P − ⳵␶ P⳵t P兲兴.
共3.18兲
0
From Eq. 共3.9兲, one can derive the Landau-Lifshitz equation
for spin waves in a ferromagnet56
␳seជ ⫻ ⳵21eជ = m⳵teជ .
共3.19兲
We assume a magnetization in the three-direction with small
perturbations in the one- and two-directions, i.e.,
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MICROSCOPIC MODEL VERSUS SYSTEMATIC LOW-…
eជ 共x兲 =
冉冑
冊
m1共x兲 m2共x兲
,
,1 + O共m2兲.
␳ s 冑␳ s
Expanding up to linear powers in the magnon fluctuations m1
and m2, one obtains the equation
␳s
i⳵t共m1 + im2兲 = − ⳵21共m1 + im2兲,
m
eជ 共x兲 = 共sin ␪共x兲cos ␸共x兲,sin ␪共x兲sin ␸共x兲,cos ␪共x兲兲,
共3.20兲
共3.21兲
共3.27兲
one obtains
u共x兲 =
=
which implies the nonrelativistic magnon dispersion relation
Em共p兲 =
␳s p2
.
m
1
冑2共1 + e3共x兲兲
冉
1
.
2a
sin关␪共x兲/2兴exp关− i␸共x兲兴
− sin关␪共x兲/2兴exp关i␸共x兲兴
cos关␪共x兲/2兴
冊
.
u11共x兲⬘ ⱖ 0,
共3.29兲
which implicitly defines the nonlinear symmetry transformation
h共x兲 = exp关i␣共x兲␴3兴 =
冉
exp关i␣共x兲兴
0
0
exp关− i␣共x兲兴
冊
苸 U共1兲s .
共3.30兲
The transformation h共x兲 is uniquely defined since we demand that u11共x兲⬘ is again real and non-negative.
Since in a ferromagnet the order parameter is invariant
under the displacement symmetry D, we have
D
共3.31兲
u共x兲 = u共x兲.
In order to couple electrons and holes to the magnons, it is
necessary to introduce the anti-Hermitean traceless field
v␮共x兲 = u共x兲⳵␮u共x兲† ,
共3.25兲
for the spin stiffness. For t = 0, the microscopic model reduces to the Heisenberg model and the spin stiffness takes
the familiar value ␳s = Ja / 4. It should be noted that the ferromagnetic vacuum becomes unstable when 16t2 ⬎ J共3J
+ 4U兲.
v␮共x兲⬘ = h共x兲u共x兲g†⳵␮关gu共x兲†h共x兲†兴 = h共x兲关v␮共x兲 + ⳵␮兴h共x兲† .
共3.33兲
Since the field v␮共x兲 is traceless, it can be written as a linear
combination of the Pauli matrices ␴a
v␮共x兲 = iv␮a 共x兲␴a,
In order to couple electron or hole fields to the order
parameter, a nonlinear realization of the SU共2兲s symmetry
has been constructed in Refs. 20, 21, and 23. A local transformation h共x兲 苸 U共1兲s is then constructed from the global
transformation g 苸 SU共2兲s as well as from the local magnon
field P共x兲 as follows. First, one diagonalizes the magnon
field by a unitary transformation u共x兲 苸 SU共2兲s, i.e.,
a 苸 兵1,2,3其,
v␮a 共x兲 苸 R.
共3.34兲
The factor i is needed to make v␮共x兲 anti-Hermitean. Introducing
v␮⫾共x兲 = v␮1 共x兲 ⫿ iv␮2 共x兲,
we write
u11共x兲 ⱖ 0.
v␮共x兲 = i
共3.26兲
共3.32兲
which under SU共2兲s transforms as
D. Nonlinear realization of the SU(2)s spin symmetry
冉 冊
cos关␪共x兲/2兴
共3.28兲
共3.24兲
共3兲
Identifying Eeh
共p兲 with Em共p兲, we read off the value
1 0
1
,
u共x兲P共x兲u共x兲† = 共1 + ␴3兲 =
0 0
2
冊
共3.23兲
J共3J + 4U兲 − 16t2 2 2
p a + O共p4兲.
2共3J + 4U兲
J共3J + 4U兲 − 16t2
a,
4共3J + 4U兲
1 + e3共x兲
u共x兲⬘ = h共x兲u共x兲g†,
The magnon dispersion relation obtained in the microscopic
model was given by
␳s =
e1共x兲 − ie2共x兲
Under a global SU共2兲s transformation g, the diagonalizing
field u共x兲 transforms as
At this point, we can match the low-energy parameters m
and ␳s of the effective field theory to the coupling constants
t, J, and U of the underlying microscopic system. First of all,
in the ferromagnetic ground state all spins are up, and hence
the magnetization density is given by
共3兲
共p兲 =
Eeh
1 + e3共x兲
− e1共x兲 − ie2共x兲
共3.22兲
C. Determination of the low-energy parameters
m=
冉
冉
v␮3 共x兲
v␮+ 共x兲
v␮− 共x兲 − v␮3 共x兲
冊
共3.35兲
.
This leads to the transformation laws for v␮3 共x兲
Note that, due to its projector properties, P共x兲 has eigenvalues 0 and 1. In order to make u共x兲 uniquely defined, we
demand that the element u11共x兲 is real and non-negative.
Otherwise, the diagonalizing matrix u共x兲 would be defined
only up to a U共1兲s phase. Using Eq. 共3.3兲 and spherical coordinates for eជ 共x兲, i.e.,
064414-9
SU共2兲s:v␮3 共x兲⬘ = v␮3 共x兲 − ⳵␮␣共x兲,
U共1兲Q: Qv␮3 共x兲 = v␮3 共x兲,
D: Dv␮3 共x兲 = v␮3 共x兲,
共3.36兲
PHYSICAL REVIEW B 81, 064414 共2010兲
GERBER et al.
R: Rv31共x兲 = − v31共Rx兲,
R 3
vt 共x兲
= v3t 共Rx兲,
T: Tv31共x兲 = − v31共Tx兲,
T 3
vt 共x兲
= v3t 共Tx兲,
ជ
Q
共3.37兲
as well as for v␮⫾共x兲
SU共2兲s:v␮⫾共x兲⬘ = exp关⫾2i␣共x兲兴v␮⫾共x兲,
U共1兲Q: Qv␮⫾共x兲
=
Q A,B
␺x⫾
v␮⫾共x兲,
共3.42兲
A,B
= exp共i␻兲␺x⫾
.
共3.43兲
Under the displacement symmetry, we obtain
D
B,A
B,A
D
D A,B
⌿A,B
x = u共x兲 Cx = u共x + a兲Cx+a␴3 = ⌿x+a␴3 .
共3.44兲
⫾
R: Rv⫾
1 共x兲 = − v1 共Rx兲,
R ⫾
vt 共x兲
= v⫾
t 共Rx兲,
T: Tv⫾
1 共x兲
T ⫾
vt 共x兲
v⫾
t 共Tx兲.
=−
ជ
Here, we have used the fact that u共x兲 is invariant under the
ជ
fermion number symmetries U共1兲Q and SU共2兲Q, i.e., Qu共x兲
= u共x兲. In particular, under the U共1兲Q subgroup of SU共2兲Q,
the components transform as
D: Dv␮⫾共x兲 = v␮⫾共x兲,
v⫾
1 共Tx兲,
ជ
A,B T
Q
Q
T
⌿A,B
x = u共x兲 Cx = u共x兲Cx⍀ = ⌿x ⍀ .
=
Expressed in terms of components, this implies
D A,B
␺x⫾
共3.38兲
B,A
= ␺x+a⫾
.
共3.45兲
F. Effective fields for charge carriers
E. Microscopic operators in a magnon background field
In the context of the effective field theory, until now we
have only discussed magnons, which correspond to states at
half-filling in the microscopic model. In this section, we will
begin to include doped electrons and holes. For this purpose,
we must establish a connection between the microscopic degrees of freedom and the low-energy effective fields describing electrons or holes. Following Refs. 20, 21, and 23, we
now discuss how this connection is established. It is a virtue
of the completely analytically controlled ferromagnetic case
that this connection can be tested rigorously.
As discussed in detail in Ref. 20, 21, and 23 in order to
define new operators ⌿Ax and ⌿Bx , it is useful to introduce the
matrix-valued fermion operator Cx. We have already used the
operator Cx in Eq. 共2.14兲 to rewrite the Hamiltonian in a
manifestly SU共2兲s-, SU共2兲Q-, D-, and R-invariant form. Now,
we write
⌿Ax = u共x兲Cx = u共x兲
⌿Bx = u共x兲Cx = u共x兲
冉
冉
cx↑
†
cx↓
cx↓ −
†
cx↑
†
cx↑ − cx↓
cx↓
†
cx↑
冊冉
冊冉
=
=
A
␺x+
A
␺x−
A†
␺x−
−
A†
␺x+
B†
B
␺x+
− ␺x−
B
␺x−
B†
␺x+
冊
冊
In the low-energy effective field theory, we will use a
Euclidean path integral description instead of the Hamiltonian description used in the microscopic model. The lattice
A,B
A,B†
and ␺x⫾
are then replaced by Grassmann
operators ␺x⫾
A,B
A,B†
共x兲, which are completely indenumbers ␺⫾ 共x兲 and ␺⫾
pendent of each other. Therefore, in the effective field theory
the electron and hole fields are represented by eight indepenA,B
A,B†
共x兲 and ␺⫾
共x兲, which can be
dent Grassmann numbers ␺⫾
combined to
⌿A共x兲 =
⌿B共x兲 =
x 苸 A,
⌿A†共x兲 =
,
x 苸 B.
⌿B†共x兲 =
Note that ⌿ denotes a matrix while ␺ denotes a matrix element. The new lattice operators inherit their transformation
properties from the operators of the microscopic model, i.e.,
we use the transformation properties of Cx discussed in Sec.
II. It should be noted that here the continuum field u共x兲 is
evaluated only at discrete lattice points x. According to Eqs.
共2.11兲 and 共3.29兲, under the SU共2兲s symmetry, one obtains
⬘ = u共x兲⬘C⬘ = h共x兲u共x兲g†gC = h共x兲⌿A,B . 共3.40兲
⌿A,B
x
x
x
x
In components, this relation takes the form
A,B⬘
A,B
␺x⫾
= exp关⫾i␣共x兲兴␺x⫾
.
Similarly, under the SU共2兲Q symmetry, one obtains
共3.41兲
␺+A共x兲
␺−A†共x兲
␺−A共x兲 − ␺+A†共x兲
␺+B共x兲 − ␺−B†共x兲
␺−B共x兲
␺+B†共x兲
冊
冊
,
.
共3.46兲
For notational convenience, we also introduce the fields
,
共3.39兲
冉
冉
冉
冉
␺+A†共x兲 ␺−A†共x兲
␺−A共x兲 − ␺+A共x兲
␺+B†共x兲 ␺−B†共x兲
− ␺−B共x兲 ␺+B共x兲
冊
冊
,
.
共3.47兲
We should note that ⌿A,B†共x兲 is not independent of ⌿A,B共x兲,
A,B
共x兲 and
since both contain the same Grassmann fields ␺⫾
A,B†
␺⫾ 共x兲. It should also be pointed out that the continuum
fields of the low-energy effective theory cannot be derived
explicitly from the lattice operators of the microscopic
model. Still, the Grassmann fields ⌿A,B共x兲 describing electrons and holes in the low-energy effective theory transform
discussed before. In conjust like the lattice operators ⌿A,B
x
trast to the lattice operators, the fields ⌿A,B共x兲 are defined in
the continuum. Hence, under the displacement symmetry D
one no longer distinguishes between the points x and x + a.
We now list the transformation properties of the effective
fields under the various symmetries, which can be derived
using the transformation properties discussed above
SU共2兲s:⌿A,B共x兲⬘ = h共x兲⌿A,B共x兲,
064414-10
⌿A,B†共x兲⬘ = ⌿A,B†共x兲h共x兲† ,
PHYSICAL REVIEW B 81, 064414 共2010兲
MICROSCOPIC MODEL VERSUS SYSTEMATIC LOW-…
ជ
ជ
Q
SU共2兲Q: Q⌿A,B共x兲 = ⌿A,B共x兲⍀T,
D: D⌿A,B共x兲 = ⌿B,A共x兲␴3,
D
R: R⌿A,B共x兲 = ⌿A,B共Rx兲,
R
␺−0†共x兲⬘ = exp关i␣共x兲兴␺−0†共x兲,
†
⌿A,B†共x兲 = ⍀T ⌿A,B†共x兲,
␺+␲共x兲⬘ = exp关i␣共x兲兴␺+␲共x兲,
⌿A,B†共x兲 = ␴3⌿B,A†共x兲,
␺+␲†共x兲⬘ = exp关− i␣共x兲兴␺+␲†共x兲,
⌿A,B†共x兲 = ⌿A,B†共Rx兲,
U共1兲Q: Q␺−0共x兲 = exp共i␻兲␺−0共x兲,
T: T⌿A,B共x兲 = − 关⌿A,B†共Tx兲T兴␴3,
T
⌿A,B†共x兲 = ␴3关⌿A,B共Tx兲T兴.
Q 0†
␺− 共x兲
共3.48兲
Q ␲
␺+ 共x兲
Note, that an upper index T on the right denotes transpose,
while on the left it denotes time reversal. In components, the
symmetry transformations read
Q ␲†
␺+ 共x兲
A,B
A,B
共x兲⬘ = exp关⫾i␣共x兲兴␺⫾
共x兲,
SU共2兲s:␺⫾
= exp共i␻兲␺+␲共x兲,
= exp共− i␻兲␺+␲†共x兲,
D: D␺−0共x兲 = ␺−0共x兲,
A,B†
A,B†
␺⫾
共x兲⬘ = exp关⫿i␣共x兲兴␺⫾
共x兲,
D 0†
␺− 共x兲
A,B
A,B
U共1兲Q: Q␺⫾
共x兲 = exp共i␻兲␺⫾
共x兲,
Q A,B†
␺⫾ 共x兲
= exp共− i␻兲␺−0†共x兲,
A,B†
= exp共− i␻兲␺⫾
共x兲,
= ␺−0†共x兲,
D ␲
␺+ 共x兲
= − ␺+␲共x兲,
D ␲†
␺+ 共x兲
= − ␺+␲†共x兲,
A,B
B,A
D: D␺⫾
共x兲 = ␺⫾
共x兲,
D A,B†
␺⫾ 共x兲
B,A†
= ␺⫾
共x兲,
R: R␺−0共x兲 = ␺−0共Rx兲,
A,B
A,B
R: R␺⫾
共x兲 = ␺⫾
共Rx兲,
R A,B†
␺⫾ 共x兲
A,B†
= ␺⫾
共Rx兲,
R 0†
␺− 共x兲
= ␺−0†共Rx兲,
R ␲
␺+ 共x兲
= ␺+␲共Rx兲,
R ␲†
␺+ 共x兲
= ␺+␲†共Rx兲,
A,B
A,B†
T: T␺⫾
共x兲 = − ␺⫾
共Tx兲,
T A,B†
␺⫾ 共x兲
A,B
= ␺⫾
共Tx兲.
共3.49兲
G. Fermion fields in momentum space pockets
As we know from the microscopic model, the electrons
live in a momentum space pocket around p = 0 and have a
spin opposite to the total magnetization, while the holes live
in a pocket around p = ␲ / a and have a spin parallel to the
magnetization. In order to describe these low-energy fermion
degrees of freedom, we perform a discrete Fourier transform
from the sublattice indices A and B to the momentum space
pocket indices 0 and ␲. Again, this is in complete analogy to
the antiferromagnetic case discussed in Refs. 21 and 23,
␺−0共x兲 =
␺−0†共x兲 =
␺+␲共x兲 =
␺+␲†共x兲 =
1
A
1
A†
冑2 关␺− 共x兲 + ␺− 共x兲兴,
冑2 关 ␺ −
B
T 0†
␺− 共x兲
T ␲
␺+ 共x兲
A
1
A†
B
共3.51兲
The effective Lagrangian to be constructed in the next section must be invariant under all these symmetry transformations as well as under the SU共2兲Q transformations. The latter
do not have a simple form in terms of the momentum space
pocket fields 共and have thus not been listed here兲, but they
follow from Eq. 共3.48兲.
共3.50兲
We now construct the leading terms of the effective action
for charge carriers, which must be invariant under the symmetries SU共2兲s, SU共2兲Q, D, R, and T. We use the indices nt,
nx, and n␺ in a contribution to the Lagrangian Lnt,nx,n␺ to
denote the number of temporal derivatives nt, the number of
spatial derivatives nx, and the number of fermion fields n␺.
The effective Lagrangian then takes the form
L=
We obtain the transformation rules
SU共2兲s:␺−0共x兲⬘ = exp关− i␣共x兲兴␺−0共x兲,
= ␺+␲共Tx兲.
H. Effective action for charge carriers
冑2 关␺+ 共x兲 − ␺+ 共x兲兴,
共x兲 − ␺+B†共x兲兴.
= ␺−0共Tx兲,
= − ␺+␲†共Tx兲,
T ␲†
␺+ 共x兲
共x兲 + ␺−B†共x兲兴,
1
冑2 关 ␺ +
T: T␺−0共x兲 = − ␺−0†共Tx兲,
兺
nt,nx,n␺
The mass term is given by
064414-11
Lnt,nx,n␺ .
共3.52兲
PHYSICAL REVIEW B 81, 064414 共2010兲
GERBER et al.
L0,0,2 = M共␺+␲†␺+␲ − ␺−0†␺−0兲.
共3.53兲
In order to express the terms with spatial or temporal derivatives, we introduce the covariant derivative D␮, which acts as
D␮␺−0共x兲 = 关⳵␮ − iv␮3 共x兲兴␺−0共x兲,
Eh共p兲 = M +
Eh共p兲 =
共3.54兲
Using the transformation laws of v␮3 共x兲 listed in Eq. 共3.37兲,
one arrives at the terms
L1,0,2 =
␺−0†Dt␺−0 ,
One should keep in mind that the holes live in momentum
space pockets centered at p = ␲ / a, which must be taken into
account in the matching of the parameters. Indeed, we had
already identified the rest and kinetic masses as
M=
共3.55兲
as well as
L0,2,2 =
共3.56兲
In contrast to an antiferromagnet, there is no fermion-singlemagnon vertex. Instead, all vertices contain at least two magnons. This implies that the fermion-magnon interactions in a
doped ferromagnet are of higher order than in an antiferromagnet.
Using the algebraic manipulation program FORM, we have
also constructed all terms involving four fermion fields and
up to one temporal or two spatial derivatives. They are not
very illuminating and we thus do not list them here. Instead
we just concentrate on the fermionic Lagrangian in the twohole sector, which will be used later and which takes the
form
L = M ␺+␲†␺+␲ + ␺+␲†Dt␺+␲ +
1
D1␺+␲†D1␺+␲ + N␺+␲†v+1 v−1 ␺+␲
2M ⬘
+ G␺+␲†␺+␲D1␺+␲†D1␺+␲ ,
J U
+ − 2t,
2 2
M⬘ =
1
.
2ta2
共3.60兲
IV. NONPERTURBATIVE SOLUTION OF THE
EFFECTIVE THEORY IN THE TWO-HOLE SECTOR
1
共D1␺+␲†D1␺+␲ − D1␺−0†D1␺−0兲
2M ⬘
+ N共␺−0†v−1 v+1 ␺−0 + ␺+␲†v+1 v−1 ␺+␲兲.
J U
+ − 2t + ta2共p − ␲/a兲2 + O共共共p − ␲/a兲4兲兲.
2 2
共3.59兲
D␮␺+␲共x兲 = 关⳵␮ + iv␮3 共x兲兴␺+␲共x兲,
␺+␲†Dt␺+␲ +
共3.58兲
with the rest mass M and the kinetic mass M ⬘. In the microscopic model in Eq. 共2.20兲, we calculated the dispersion relations of holes
D␮␺−0†共x兲 = 关⳵␮ + iv␮3 共x兲兴␺−0†共x兲,
D␮␺+␲†共x兲 = 关⳵␮ − iv␮3 共x兲兴␺+␲†共x兲.
p2
,
2M ⬘
共3.57兲
where G is a four-fermion coupling constant. It should be
noted that the effective coupling constants M, M ⬘, N, and G
are real. Since in the above Lagrangian, we have omitted the
electron degrees of freedom, it is no longer
SU共2兲Q-invariant. Interestingly, the low-energy effective Lagrangian has an emergent Galilean boost symmetry, despite
the fact that the underlying microscopic model does not possess this invariance.
I. Determination of the fermion mass parameters
We now like to match the fermion mass parameters to the
parameters of the underlying microscopic system. Due to the
SU共2兲Q symmetry the masses of electrons and holes are identical. Here, we concentrate on the holes whose dispersion
relation is given by
In this section, we will investigate the two-hole sector in
the effective field theory and will then again compare with
the analytic results of the underlying microscopic model.
Solution of the two-hole Schrödinger equation
In order to calculate the bound- and scattering-states of
two holes in the effective theory, one can derive a two-hole
potential from the effective Lagrangian of Eq. 共3.57兲. The
four-fermion contact term of strength G gives rise to a potential that is proportional to the second derivative of a ␦
function. Such potentials are ultraviolet divergent and require
renormalization even in quantum mechanics. In order to
avoid the corresponding subtleties, it is more efficient to apply the technique of self-adjoint extensions. In particular, it is
then not even necessary to explicitly construct the potential.
It should be mentioned that the method of self-adjoint extensions would be applicable also in higher dimensions.
Since the effective theory has an emergent Galilean boost
symmetry, we may consider the two-hole system in its rest
frame. Introducing the relative coordinate x between the two
holes, the Schrödinger equation reduces to a single particle
equation with the reduced mass M ⬘ / 2. For kinematical reasons, the two holes cannot exchange magnons. Instead, they
just experience their four-fermion contact interaction. Away
from the contact point x = 0, the two-hole Schrödinger equation thus describes free particles and is simply given by
−
1 2
⳵ ␺共x兲 = E␺共x兲.
M⬘ x
共4.1兲
In the theory of self-adjoint extensions, contact interactions
in one-dimensional quantum mechanics are treated by removing the contact point x = 0 from the physical space. The
effect of the four-fermion interaction is then represented by a
boundary condition on the wave function. The most general
self-adjoint extension has four independent parameters and is
characterized by the boundary condition
064414-12
PHYSICAL REVIEW B 81, 064414 共2010兲
MICROSCOPIC MODEL VERSUS SYSTEMATIC LOW-…
冉 冊
冉 冊冉
␺共⑀兲
a b
= exp共i␪兲
⳵ x␺ 共 ⑀ 兲
c d
冊
␺共− ⑀兲
.
⳵x␺共− ⑀兲
共4.2兲
Here, a, b, c, and d are real numbers with the constraint
ad − bc = 1, and ⑀ is an infinitesimal displacement from the
point x = 0. Since our system is parity-invariant 共against the
reflection R兲, the situation simplifies further and one obtains
冉 冊 冉 冊冉
␺共⑀兲
a b
=
⳵ x␺ 共 ⑀ 兲
c a
冊
␺共− ⑀兲
,
⳵x␺共− ⑀兲
共4.3兲
i.e., ␪ = 0 and d = a. Since we are dealing with fermions, the
Pauli principle implies a parity-odd wave function obeying
␺共−x兲 = −␺共x兲. The boundary condition on the wave function
then implies
␬␺共⑀兲 + ⳵x␺共⑀兲 = 0,
␬=
a+1
.
b
共4.4兲
It should be pointed out that the wave function will in general not be continuous at x = 0. Alternatively to the fourfermion coupling G, the strength of the two-hole contact interaction can be characterized by the parameter ␬. Relating ␬
to G would require the ultraviolet regularization of the second derivative of a ␦-function potential. We avoid this unnecessary step by matching the value of ␬ directly to the
parameters of the underlying microscopic model.
Let us first search for two-hole bound states 共with E ⬍ 0兲.
The wave function then takes the form
␺共x兲 = A exp共− ␬x兲,
␺共− x兲 = − ␺共x兲, 共4.5兲
x ⬎ 0,
which is indeed discontinuous at x = 0. The corresponding
wave function in the microscopic model was calculated in
Eq. 共2.64兲 and 共in the rest frame, i.e., for p = 0兲 is given by
g共x兲 = C0共− 1兲x/a
冉冊
8t
J
x/a
共4.6兲
.
The oscillating factor 共−1兲x/a is not present in the effective
field theory because of the momentum shift in the effective
hole fields which are located near p = ␲ / a in the Brillouin
zone. Matching the exponential decays, we identify
冉冊
1
8t
␬ = − log
.
a
J
␬2
.
M⬘
共4.8兲
The corresponding expression in the microscopic model was
calculated in Eq. 共2.63兲 and is given by
EB =
冉 冊
8t
J
1−
4
J
2
=
冉 冊
8t
J
1−
2
J
8ta M h⬘
2
.
共4.9兲
Here, we have used the value M h⬘ = 1 / 2ta2 for the kinetic hole
mass. Hence, from this expression, one would conclude that
1
a
冑冉 冊
8t
J
1−
.
8t
J
共4.10兲
This is consistent with Eq. 共4.7兲 only when J ⬇ 8t, i.e., when
the binding energy of Eq. 共4.9兲 is small. In fact, the two
expressions even coincide up to second order in the perturbation ␦ = 1 − 8t / J. The effective field theory thus provides a
correct description of the bound state only when the binding
is weak. This is not surprising. If two holes form a bound
state with a large binding energy, this bound state must be
introduced in the effective theory as an independent degree
of freedom. Only when the bound state resembles a weakly
coupled “molecule” in which the constituent holes can be
identified as relevant low-energy degrees of freedom, the effective field theory 共without explicit bound-state fields兲 is
appropriate. In this context, it is interesting to note that the
kinetic mass of two holes calculated in Eq. 共2.62兲 was given
by
⬘ =
M hh
J
.
8t2a2
共4.11兲
Only for J = 8t, this corresponds to the sum of the kinetic
masses of two holes 2M h⬘ = 1 / ta2. This is consistent, because
the emergent Galilean boost invariance of the effective
theory indeed implies this relation.
Finally, let us consider the scattering states of two holes
共with E ⬎ 0兲. We make the ansatz
␺共x兲 = A exp共ikx兲 + B exp共− ikx兲,
共4.12兲
insert it into Eq. 共4.4兲, and find
A k + i␬
=
.
B k − i␬
共4.13兲
Let us now compare this result with the one obtained in the
microscopic model given in Eq. 共2.69兲. For low energies, i.e.,
for q → ␲ / a, one obtains
Ã
B̃
=
k + i共1 – 8t/J兲
+ O共k2兲.
k − i共1 – 8t/J兲
共4.14兲
Indeed, using the value of ␬ given in Eq. 共4.7兲, we finally get
A Ã
= .
B B̃
共4.7兲
In the effective theory, the bound-state energy is given by
EB = − E =
␬=
共4.15兲
We conclude that also here, the effective field theory makes
correct predictions, provided that the energies of both the
bound state and the scattering states are small.
V. CONCLUSIONS
We have investigated a Hubbard-type model for a doped
ferromagnet. While this model does not provide a realistic
description of actual ferromagnetic systems, since it can be
solved completely analytically, it provides a stringent test of
the corresponding low-energy effective field theory for magnons and doped electrons or holes. Similar effective theories
have been constructed for magnons and charge carriers in the
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GERBER et al.
antiferromagnetic precursors of high-temperature superconductors. Since, in that case, the underlying microscopic models cannot be solved analytically, the correctness of the effective field theory can only be tested in Monte Carlo
simulations. Indeed, such tests provide excellent numerical
evidence for the validity of the effective field theory approach. In the ferromagnetic case discussed here, the exact
agreement between the analytic results of the microscopic
and the effective theory lends further support to the validity
of the systematic low-energy effective field theory technique.
In particular, we like to stress once more that the basic principles behind the construction of the effective theory are the
same for ferro- and for antiferromagnets.
While in this work we have investigated bound and scattering states of two holes, another case of interest concerns
the interaction between a spin wave and a hole. Indeed, in
the microscopic theory one is then lead to a Faddeev-type
equation and it would be instructive to confront the microscopic result with the effective theory prediction also for this
case.
Effective field theories are also being used in the description of light nuclei. In that case, a low-energy effective field
theory of pions and nucleons must be solved nonperturbatively, and it is currently not completely clear how to do this
in a fully systematic manner, i.e., based on a consistent
power-counting scheme. In this context, it is interesting that
the two-hole sector of the ferromagnetic model discussed
here can be solved nonperturbatively both in the microscopic
and in the effective field theory treatment. Both approaches
agree as long as the two-hole binding energy is small. On the
other hand, when the binding becomes strong, the bound
state should be described by an independent effective field.
The analytically solvable test case of the ferromagnet may
also provide valuable insights into the subtle power-counting
issues that arise in the context of the strong interactions.
1
=sign共␥兲 ,
s
I2 =
=−
I3 =
I4 =
I5 =
In order to solve the gap Eqs. 共2.35兲, we needed the integrals
−␲/a
dq
1
␣ cos共qa兲 + ␤ sin共qa兲 + ␥
−␲/a
冕
␲/a
dq
−␲/a
I6 =
cos共qa兲
␣ cos共qa兲 + ␤ sin共qa兲 + ␥
sin共qa兲
␣ cos共qa兲 + ␤ sin共qa兲 + ␥
␤
,
s共兩␥兩 + s兲
1
2␲
冕
␲/a
dq
−␲/a
1
2␲
冕
1
2␲
dq
−␲/a
冕
cos2共qa兲
␣ cos共qa兲 + ␤ sin共qa兲 + ␥
␣2 + s共兩␥兩 + s兲
,
s共兩␥兩 + s兲2
␲/a
=sign共␥兲
sin2共qa兲
␣ cos共qa兲 + ␤ sin共qa兲 + ␥
␤2 + s共兩␥兩 + s兲
,
s共兩␥兩 + s兲
␲/a
dq
−␲/a
cos共qa兲sin共qa兲
␣ cos共qa兲 + ␤ sin共qa兲 + ␥
␣␤
,
s共兩␥兩 + s兲2
s = 冑␥2 − ␣2 − ␤2 .
共A1兲
The aim of this appendix is to show how these integrals can
be evaluated. Since
␣I2 + ␤I3 + ␥I1 = 1,
I4 + I5 = I1 ,
共A2兲
one only needs to do four integrals, for example, I1, I2, I4,
and I6. These four integrals can be evaluated by using the
residue theorem. We now explicitly present the calculation
for
APPENDIX: INTEGRALS FOR THE GAP EQUATION
␲/a
dq
=sign共␥兲
I1 =
冕
␲/a
= sign共␥兲
The authors would like to thank C. Brügger and M. Pepe
for contributions at an early stage of this work. C.P.H. would
like to thank the members of the Institute for Theoretical
Physics at Bern University for their hospitality during a visit
at which this project was completed. F.K. is supported by the
Schweizerischer Nationalfonds. The work of C.P.H. is supported by CONACYT under Grant No. 50744-F. The work
of U.G. is supported in part by funds provided by the Schweizerischer Nationalfonds. The “Albert Einstein Center for
Fundamental Physics” at Bern University is supported by the
“Innovations- und Kooperationsprojekt C-13” of the Schweizerischer Nationalfonds.
1
2␲
冕
␣
,
s共兩␥兩 + s兲
1
2␲
=−
ACKNOWLEDGMENTS
I1 =
1
2␲
1
2␲
冕
␲/a
−␲/a
dq关␣ cos共qa兲 + ␤ sin共qa兲 + ␥兴−1 .
共A3兲
We integrate around the unit circle C, and thus make the
substitution
z = exp共iqa兲.
Then, we can write
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MICROSCOPIC MODEL VERSUS SYSTEMATIC LOW-…
I1 =
1
2␲i
冖 冋
dz
C
␤
␣ 2
共z + 1兲 + 共z2 − 1兲 + ␥z
2
2i
册
−1
.
共A4兲
zA =
− ␥ + 冑␥ − ␣ − ␤
,
␣ − ␤i
zB =
− ␥ − 冑␥2 − ␣2 − ␤2
.
␣ − ␤i
2
兩zB兩2 ⬎ 1 ⇔ ␥ ⬎ 0,
共A7兲
兩zA兩2 ⬎ 1 ⇔ ␥ ⬍ 0,
兩zB兩2 ⬍ 1 ⇔ ␥ ⬍ 0.
共A8兲
as well as
The integrand has two singularities at
2
兩zA兩2 ⬍ 1 ⇔ ␥ ⬎ 0,
Hence, for ␥ ⬎ 0 only zA lies within the unit circle C and for
␥ ⬍ 0 only zB lies within C. The residues of the poles at zA
and zB are given by
2
共A5兲
RA =
We get a contribution to the integral only if the singularities lie within the unit circle C. Therefore, we consider the
absolute values of zA and zB,
RB =
兩zA兩2 =
共冑␥2 − ␣2 − ␤2 − ␥兲2
,
␣2 + ␤2
共冑␥2 − ␣2 − ␤2 + ␥兲2
.
兩zB兩 =
␣2 + ␤2
2
2␣冑␥2 − ␣2 − ␤2 − 2i␤冑␥2 − ␣2 − ␤2
2␥ + 2冑␥2 − ␣2 − ␤2
2␣冑␥2 − ␣2 − ␤2 − 2i␤冑␥2 − ␣2 − ␤2
,
.
共A9兲
Collecting the results, we obtain
I1 = sign共␥兲
共A6兲
1
冑␥2 − ␣2 − ␤2
.
共A10兲
The integrals I2, I4, and I6 can be obtained completely analogously.
We find
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