# Electrons interacting with Goldstone modes and the rotating frame

Konstantinos Vasiliou  Rudolf Peierls Centre for Theoretical Physics, Parks
Road, Oxford, OX1 3PU, UK  Yuchi He  Rudolf Peierls Centre for Theoretical
Physics, Parks Road, Oxford, OX1 3PU, UK  Nick Bultinck  Rudolf Peierls Centre
for Theoretical Physics, Parks Road, Oxford, OX1 3PU, UK  Department of
Physics, Ghent University, Krijgslaan 281, 9000 Gent, Belgium

###### Abstract

We consider electronic systems with a spontaneously broken continuous
symmetry. The scattering vertex between electrons and Goldstone modes is
calculated over the entire Brillouin zone using the random phase
approximation. This calculation reveals two things: (1) electrons always
couple to both $\phi$ and $\partial_{t}\phi$, where $\phi$ is the Goldstone
field, and (2) quasi-particles in a state with continuous symmetry breaking
have to be defined in a rotating frame, which locally follows the fluctuations
of the order parameter. The implications of these findings for electron
spectral functions in both symmetry-broken and thermally disordered systems
are discussed, and the examples of anti-ferromagnetism in the Hubbard model
and spin spiral order in the three-band model are worked out in detail.

###### Contents

  1. I Introduction
    1. I.1 Summary of results and connection to previous works
    2. I.2 Structure of the paper
  2. II Mean-field starting point
  3. III Hartree-Fock path integral
  4. IV Effective RPA interaction
  5. V Interacting electrons and Goldstone modes
    1. V.1 Properties of the Goldstone wavefunctions
    2. V.2 Electron-Goldstone scattering vertex
    3. V.3 Effective electron-boson model
    4. V.4 Transforming to the real basis
  6. VI The rotating frame
    1. VI.1 The problem: strong interband scattering in the ${\mathbf{q}}\rightarrow 0$ limit
    2. VI.2 The solution: the rotating frame
    3. VI.3 Implications for electron spectral functions
  7. VII Example I: anti-ferromagnetism in the Hubbard model
    1. VII.1 Mean-field state and Goldstone mode energies
    2. VII.2 Electron-Goldstone scattering vertices
    3. VII.3 Electron spectral function
  8. VIII Example II: spin-spiral order in the three-band model
    1. VIII.1 Mean-field state and Goldstone mode energies
    2. VIII.2 Electron spectral function
  9. IX Conclusions
  10. A Solution of the Bethe-Salpeter equation and its properties

##  I Introduction

Many strongly-correlated materials display continuous symmetry breaking in
some part of their phase diagram. Some paradigmatic examples are spin-density
wave orders in the cuprate and iron-based superconductors, and valley/spin
orders in 2D moiré materials.

As a result of the continuous symmetry-breaking, the low-energy spectrum
contains gapless Goldstone modes. In this work we set out to calculate the
scattering vertex between these Goldstone modes and the electrons.
Specifically, our goal is to obtain an expression for the scattering vertex
which is valid over the entire Brillouin zone, and not just in the long-
wavelength limit for the Goldstone modes. One reason for going beyond a long-
wavelength approximation is that just like acoustic phonons, Goldstone modes
of the electron liquid should decouple from the electrons at long wavelengths.
Hence one expects the most important scattering processes to be those where an
electron emits or absorbs a zone-boundary Goldstone mode.

The starting point of our analysis is a simple mean-field description of the
broken-symmetry state. From the optimal broken-symmetry Slater determinant, we
then construct a path-integral representation of the partition function in
order to study fluctuations on top of the mean-field state. Via the random
phase approximation (RPA)111Note that in this work RPA means summing both
bubble and ladder diagrams, such that is a conserving approximation in the
sense of Baym and Kadanoff [1]. It is also equivalent to applying the time-
dependent variational principle [2, 3], and hence is not a perturbative
calculation which is only applicable to weakly-interacting systems. we obtain
both the Bethe-Salpeter equation for the collective modes (whose solution
gives the Goldstone mode energies and wavefunctions), and an explicit
expression for the electron-Goldstone mode scattering vertex. Using these
results, we then construct an effective electron-boson model with both
fermionic and bosonic fields, which emulates the interacting electron-
Goldstone mode system. Towards the end of the paper we illustrate our
formalism with two examples: anti-ferromagnetism in the Hubbard model, and
circular spin-spiral order in the three-band model.

###  I.1 Summary of results and connection to previous works

Even though both the goal (obtaining the electron-Goldstone mode scattering
vertex) and the methods (mean-field theory and RPA) of this work are
straightforward and have previously been discussed in classic works such as
e.g. Refs. [4, 5, 6], we nevertheless find that obtaining a result which is in
line with physical expectations requires some non-trivial (and to the best of
our knowledge new) steps. In this section we briefly discuss what these non-
trivial steps are, why they are necessary, and what interesting physics is
hiding behind them.

Via the RPA analysis we obtain a scattering vertex which in the long-
wavelength limit agrees with the previous result of Ref. [7]. However, this
result is not entirely satisfactory, as it seems to imply that long-wavelength
Goldstone modes do not decouple from the electrons. Instead, they give rise to
inter-band scattering processes for the electrons with a scattering vertex of
the order of the mean-field bandgap which results from the spontaneous
symmetry breaking. Hence the long-wavelength inter-band scattering is of the
order of the interaction energy scale. Already at tree-level, exchange of
Goldstone modes therefore introduces an effective interaction with a magnitude
corresponding to the square of the bare repulsive interaction, divided by a
Goldstone propagator which vanishes in the low-energy and long-wavelength
limit. This leads to strong self-energy corrections for the mean-field
electrons, casting doubt on their validity as well-defined quasi-particles.

However, we observe that the spurious contribution to the electron-Goldstone
mode vertex is `pure gauge', meaning that it can be removed by a gauge
transformation in the path integral. This gauge transformation implements a
spatially-dependent symmetry transformation on the mean-field electrons. In
particular, the transformed electron fields are defined in a rotating frame
which follows the local fluctuations of the order parameter. Crucially,
electrons in the rotating frame completely decouple from long-wavelength
Goldstone modes. We note that a similar rotating frame has already been
introduced in previous studies of superconducting [8, 9] and magnetic [10, 11,
12] orders, but for different reasons. In particular, these previous works did
not obtain the Goldstone mode energies and wavefunctions by solving the Bethe-
Salpeter equation, and did not motivate the rotating frame by requiring a
decoupling between electrons and Goldstone modes at long wavelengths.

The mean-field electrons are related via a frequency-independent, and hence
non-dynamical, transformation to the microscopic electrons. The linear
response functions which are measured in experiment are therefore directly
related to correlation functions of the mean-field electrons. But we argued
above that the mean-field electrons are not a good approximation to the true
quasi-particles of the system – one should instead use the electrons in the
rotating frame to approximate the quasi-particles. However, the electron
propagator can be approximated as a convolution (in frequency and momentum
space) of two propagators of well-defined quasi-particles: fermions in the
rotating frame, and the Goldstone modes (we momentarily ignore the possibility
of Landau damping for the Goldstone modes). This is explained in detail in
Sec. VII.3. This result has previously also been obtained in Ref. [13], where,
similarly to Ref. [12], the rotating frame is introduced to correctly describe
fluctuations beyond mean-field theory via a Hubbard-Stratanovich decoupling of
the interaction. Again, this work did not motivate the necessity of a rotating
frame by deriving the electron-Goldstone mode scattering vertex in the RPA
formalism as we do here. We note that a similar expression for the microscopic
fermion two-point function as a convolution of two propagators is also
obtained in theories where the electron is assumed to fractionalize in a
spinon and a holon [14, 15, 16, 17, 18, 19, 20].

The RPA approach describes small fluctuations around the mean-field state. In
the effective electron-boson model, which we construct to emulate the
interacting electrons and Goldstone modes at the RPA level, the Goldstone
fields are a collection of real scalar fields. However, once the dispersion
relation of small order parameter fluctuations is known, it is straightforward
to improve the theory and incorporate the correct global structure of the
order parameter manifold by describing the Goldstone mode dynamics with a non-
linear sigma model. This is especially important in 2D, where the non-linear
sigma model is disordered at any non-zero temperature, and hence ensures that
the effective electron-boson model respects the Hohenberg-Mermin-Wagner
theorem.

As a final point, let us elaborate on the physical behaviour which can be
extracted from the expression for the microscopic fermion propagator as a
convolution of the Goldstone propagator and the propagator of the fermions in
the rotating frame. In particular, let us consider 2D systems, and focus on
temperatures $T$ which are much smaller than the mean-field bandgap. If we use
the correct non-linear sigma model action for the Goldstone mode dynamics, the
Goldstone mode propagator at finite $T$ will be invariant under global
symmetry transformations, and acquire a thermal mass. As a result, the
propagator for the microscopic fermions (which is a convolution containing the
Goldstone mode propagator) will also be symmetric (this is explained in detail
in Sec. VII.3, and illustrated via the examples in Sec. VII and VIII).
However, the spectral weight of the microscopic electrons is still
predominantly determined by the poles of the fermion propagator in the
rotating frame. As we assume $T$ to be much smaller than the mean-field
bandgap, the poles of the rotating-frame fermion propagator will be located at
the same energies of the symmetry-broken mean-field band spectrum. As a
result, we find that even though the microscopic fermion propagator is
invariant under the continuous global symmetry, it nevertheless produces a
spectral weight that is predominantly determined by the spectrum of the
broken-symmetry state – and hence is very different from the spectral weight
of the non-interacting fermions. We thus see that the introduction of a
rotating frame, which is necessary to have the electrons decouple from the
long-wavelength Goldstone modes, automatically leads to an expression for the
electron propagator which captures the physically intuitive behaviour of an
electron system with an order parameter that has well-defined magnitude, but
whose orientation is disordered by thermal fluctuations.

###  I.2 Structure of the paper

The remainder of this work is organized as follows. We start in Sec. II by
introducing the mean-field starting point of our analysis. In Sec. III we use
the optimal Slater determinant, which we obtain from the mean-field analysis,
to construct a path integral representation of the partition function, which
can be used to study fluctuations beyond mean-field. Using this path integral,
we introduce the infinite series of RPA Feynman diagrams that give rise to the
Bethe-Salpeter equation, and also the effective interaction between electrons
mediated by collective mode exchange, in Sec. IV. In Sec. V we first discuss
the general properties of the Goldstone mode wavefunctions that we obtain by
solving the Bethe-Salpeter equation. We then construct an effective electron-
boson model which mimics the interacting electron-Goldstone mode system at
tree-level. In Sec. VI we use the properties of the Goldstone mode
wavefunctions to study the electron-Goldstone mode scattering vertex obtained
from the RPA analysis in the long-wavelength limit. Here we uncover the the
problematic inter-band scattering processes which do not vanish in the long-
wavelength limit. In the same section, we then explain how this problem can be
solved by doing a gauge transformation in the path integral, and defining
fermions in a rotating frame. The implications of the rotating frame for the
electron spectral functions are discussed at the end of this section. Finally,
we illustrate the general formalism for anti-ferromagnetism in the Hubbard
model in Sec. VII, and for circular spin-spiral order in the three-band model
in Sec. VIII. The appendix reviews some properties of the Bethe-Salpeter
equation and its solutions.

##  II Mean-field starting point

In this section we set the stage for our main results presented below. In
particular, we introduce the mean-field theory which is the starting point for
our calculations. All the concepts discussed here are standard, and the main
purpose of this section is therefore to introduce the context and notation
necessary to understand the following sections.

We consider interacting electron systems described by the following general
Hamiltonian:

$H=\sum_{{\mathbf{r}},{\mathbf{r}}^{\prime}}\sum_{a,b}h({\mathbf{r}}-{\mathbf{r}}^{\prime})_{ab}c^{\dagger}_{{\mathbf{r}},a}c_{{\mathbf{r}}^{\prime},b}+\frac{1}{2}\sum_{{\mathbf{r}},{\mathbf{r}}^{\prime}}V({\mathbf{r}}-{\mathbf{r}}^{\prime}):n_{{\mathbf{r}}}n_{{\mathbf{r}}^{\prime}}:\,,$
(1)

where ${\mathbf{r}}$ denote the sites of a Bravais lattice in $d$ spatial
dimensions, $h({\mathbf{r}})$ is a general hopping Hamiltonian, and
$n_{\mathbf{r}}=\sum_{a}c^{\dagger}_{{\mathbf{r}},a}c_{{\mathbf{r}},a}$ is the
electron density at site ${\mathbf{r}}$. We also assume that $h({\mathbf{r}})$
is short range, and $V({\mathbf{r}})$ has a well-defined Fourier transform
(for example it could be a screened Coulomb potential). Our results can be
generalized to other Hamiltonians, for example with other (not density-
density) interactions, but for ease of presentation we do not consider these
generalizations here.

We will make use of the translation invariance and work in momentum space,
where the Hamiltonian in Eq. (1) takes the form

$H=\sum_{\mathbf{k}}\sum_{a,b}h({\mathbf{k}})_{ab}c^{\dagger}_{{\mathbf{k}},a}c_{{\mathbf{k}},b}+\frac{1}{2}\sum_{\mathbf{q}}V({\mathbf{q}}):n_{{\mathbf{q}}}n_{-{\mathbf{q}}}:\,,$
(2)

where

$n_{\mathbf{q}}=\frac{1}{\sqrt{N}}\sum_{\mathbf{k}}\sum_{a}c^{\dagger}_{{\mathbf{k}}+{\mathbf{q}},a}c_{{\mathbf{k}},a}\,,$
(3)

with $N$ the number of sites, is the Fourier transform of the electron
density.

We are interested in Hamiltonians with a continuous symmetry, such as spin
rotation symmetry, which gets broken spontaneously. The starting point of our
analysis is a mean-field treatment of the symmetry breaking at zero
temperature. In particular, we assume that the Slater determinants which
minimize the energy of $H$ are not invariant under the continuous symmetry of
the Hamiltonian. Let us choose one such Slater determinant and write it as

$|\psi_{0}\rangle=\prod_{{\mathbf{k}}}\prod_{i=1}^{N_{o}({\mathbf{k}})}c^{\dagger}_{{\mathbf{k}},i}|0\rangle\,,$
(4)

where $c^{\dagger}_{{\mathbf{k}},i}$ creates an electron in a mean-field
single-particle state, and $N_{o}({\mathbf{k}})$ is the number of occupied
states at momentum ${\mathbf{k}}$. Note that we have assumed that the optimal
Slater determinants preserve the translation symmetry. Also this restriction
can be relaxed, but we will not do this here. The mean-field single-particle
states (both occupied and unoccupied) will be labeled by Greek indices, and
the corresponding creation operators are defined as

$c^{\dagger}_{{\mathbf{k}},\alpha}=\sum_{a}u_{\alpha}^{a}({\mathbf{k}})c^{\dagger}_{{\mathbf{k}},a}\,,$
(5)

where $|u_{\alpha}({\mathbf{k}})\rangle$ are a set of orthonormal vectors at
every ${\mathbf{k}}$: $\langle
u_{\alpha}({\mathbf{k}})|u_{\beta}({\mathbf{k}})\rangle=\delta_{\alpha\beta}$.
We will use the convention that $i,j,k$ denote occupied mean-field states,
whereas $m,n,o$ denote the unoccupied states. The single-particle correlation
matrix of the optimal Slater determinant in Eq. (4), which by assumption is
diagonal in momentum space, is thus given by

$\displaystyle P({\mathbf{k}})_{\alpha\beta}$ $\displaystyle:=$
$\displaystyle\langle\psi_{0}|c^{\dagger}_{{\mathbf{k}},\beta}c_{{\mathbf{k}},\alpha}|\psi_{0}\rangle=\delta_{\alpha\beta}n_{\alpha}({\mathbf{k}})\,,$
(6)

with $n_{\alpha}({\mathbf{k}})\in\\{0,1\\}$, $n_{i/j/k}({\mathbf{k}})=1$ and
$n_{m/n/o}({\mathbf{k}})=0$.

Next we construct the Hartree-Fock mean-field Hamiltonian using
$P({\mathbf{k}})$. For future use we find it convenient to first rewrite the
exact Hamiltonian in the mean-field basis:

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{\mathbf{k}}h({\mathbf{k}})_{\alpha\beta}c^{\dagger}_{{\mathbf{k}},\alpha}c_{{\mathbf{k}},\beta}+\frac{1}{2}\sum_{\mathbf{q}}V({\mathbf{q}}):n_{\mathbf{q}}n_{-{\mathbf{q}}}:$
(7) $\displaystyle n_{\mathbf{q}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N_{s}}}\sum_{\mathbf{k}}\sum_{\alpha,\beta}c^{\dagger}_{{\mathbf{k}}-{\mathbf{q}},\alpha}[\Lambda_{\mathbf{q}}({\mathbf{k}})]_{\alpha\beta}c_{{\mathbf{k}},\beta}\,,$
(8)

where
$h({\mathbf{k}})_{\alpha\beta}=\sum_{a,b}u^{a*}_{\alpha}({\mathbf{k}})h({\mathbf{k}})_{ab}u^{b}_{\beta}({\mathbf{k}})$,
and

$[\Lambda_{\mathbf{q}}({\mathbf{k}})]_{\alpha\beta}:=\langle
u_{\alpha}({\mathbf{k}}-{\mathbf{q}})|u_{\beta}({\mathbf{k}})\rangle\,.$ (9)

As we are only considering translationally invariant states, the Hartree
Hamiltonian is trivial and is simply given by

$H_{H}[P]=V(0)\frac{N_{e}}{N}\sum_{\mathbf{k}}\sum_{\alpha}c^{\dagger}_{{\mathbf{k}},\alpha}c_{{\mathbf{k}},\alpha}\,,$
(10)

where $N_{e}$ is the number of electrons in the system. The Fock Hamiltonian
is given by

$H_{F}[P]=\frac{-1}{N}\sum_{{\mathbf{q}},{\mathbf{k}}}\sum_{\alpha,\beta,i}V({\mathbf{q}})[\Lambda_{-{\mathbf{q}}}({\mathbf{k}}-{\mathbf{q}})]_{\alpha
i}[\Lambda_{{\mathbf{q}}}({\mathbf{k}})]_{i\beta}c^{\dagger}_{{\mathbf{k}},\alpha}c_{{\mathbf{k}},\beta}$
(11)

The assumption that the Slater determinant in Eq. (4) is a variational energy
minimum is equivalent to the statement that the Hartree-Fock self-consistency
equation is satisfied, which in our notation takes the following form:

$\sum_{\alpha,\beta}h({\mathbf{k}})_{\alpha\beta}c^{\dagger}_{{\mathbf{k}},\alpha}c_{{\mathbf{k}},\beta}+H_{H}[P]+H_{F}[P]=\sum_{\alpha}E_{{\mathbf{k}},\alpha}c^{\dagger}_{{\mathbf{k}},\alpha}c_{{\mathbf{k}},\alpha}\,,$
(12)

where $E_{{\mathbf{k}},\alpha}$ are the mean-field single-particle energies
satisfying $\text{sgn}(E_{{\mathbf{k}},\alpha})=1-2n_{\alpha}({\mathbf{k}})$.

##  III Hartree-Fock path integral

We are interested in fluctuations beyond mean-field theory, and in particular
in the role of the Goldstone modes associated with the spontaneous symmetry
breaking. To study these fluctuations, we will use the path integral
formalism. However, we will not use the standard path integral construction
which relies on fermionic coherent states constructed on top of the Fock
vacuum. Instead, we will construct coherent states on top of the Slater
determinant in Eq. (4). Working in the single-particle mean-field basis,
coherent states associated with unoccupied states take on the conventional
form:

$\displaystyle|\psi_{{\mathbf{k}},n}\rangle$ $\displaystyle=$
$\displaystyle(1-\psi_{{\mathbf{k}},n}c^{\dagger}_{{\mathbf{k}},n})|0\rangle$
(13) $\displaystyle\langle\bar{\psi}_{{\mathbf{k}},n}|$ $\displaystyle=$
$\displaystyle\langle 0|(1-c_{{\mathbf{k}},n}\bar{\psi}_{{\mathbf{k}},n})\,,$
(14)

where $\psi_{{\mathbf{k}},n}$ and $\bar{\psi}_{{\mathbf{k}},n}$ are Grassmann
numbers. These coherent states have the usual properties:

$\displaystyle c_{{\mathbf{k}},n}|\psi_{{\mathbf{k}},n}\rangle$
$\displaystyle=$
$\displaystyle\psi_{{\mathbf{k}},n}|\psi_{{\mathbf{k}},n}\rangle$ (15)
$\displaystyle\langle\bar{\psi}_{{\mathbf{k}},n}|\psi_{{\mathbf{k}},n}\rangle$
$\displaystyle=$ $\displaystyle
e^{\bar{\psi}_{{\mathbf{k}},n}\psi_{{\mathbf{k}},n}}\,.$ (16)

For the occupied states, we will use _particle-hole transformed_ coherent
states, which we define as

$\displaystyle|\bar{\psi}_{{\mathbf{k}},i}\rangle$ $\displaystyle=$
$\displaystyle(1-\bar{\psi}_{{\mathbf{k}},i}c_{{\mathbf{k}},i})c^{\dagger}_{{\mathbf{k}},i}|0\rangle$
(17) $\displaystyle\langle\psi_{{\mathbf{k}},i}|$ $\displaystyle=$
$\displaystyle\langle
0|c_{{\mathbf{k}},i}(1-c^{\dagger}_{{\mathbf{k}},i}\psi_{{\mathbf{k}},i})\,,$
(18)

where $\bar{\psi}_{{\mathbf{k}},i}$ and $\psi_{{\mathbf{k}},i}$ are again
Grassmann numbers. The particle-hole transformed coherent states satisfy
following properties:

$\displaystyle
c^{\dagger}_{{\mathbf{k}},i}|\bar{\psi}_{{\mathbf{k}},i}\rangle$
$\displaystyle=$
$\displaystyle\bar{\psi}_{{\mathbf{k}},i}|\bar{\psi}_{{\mathbf{k}},i}\rangle$
(19)
$\displaystyle\langle\psi_{{\mathbf{k}},i}|\bar{\psi}_{{\mathbf{k}},i}\rangle$
$\displaystyle=$ $\displaystyle
e^{\psi_{{\mathbf{k}},i}\bar{\psi}_{{\mathbf{k}},i}}\,,$ (20)

To construct a path integral representation of the partition function, we need
to insert resolutions of the identity in terms of the coherent states, which
are given by

$\displaystyle\int\mathrm{d}\psi_{{\mathbf{k}},n}\int\mathrm{d}\bar{\psi}_{{\mathbf{k}},n}e^{-\bar{\psi}_{{\mathbf{k}},n}\psi_{{\mathbf{k}},n}}|\psi_{{\mathbf{k}},n}\rangle\langle\bar{\psi}_{{\mathbf{k}},n}|$
$\displaystyle=$ $\displaystyle\mathds{1}$ (21)
$\displaystyle\int\mathrm{d}\bar{\psi}_{{\mathbf{k}},i}\int\mathrm{d}\psi_{{\mathbf{k}},i}e^{-\psi_{{\mathbf{k}},i}\bar{\psi}_{{\mathbf{k}},i}}|\bar{\psi}_{{\mathbf{k}},i}\rangle\langle\psi_{{\mathbf{k}},i}|$
$\displaystyle=$ $\displaystyle\mathds{1}$ (22)

Using the above resolutions of the identity, the partition function at
temperature $T$ can be written as

$\displaystyle Z(T)$ $\displaystyle=$
$\displaystyle\text{tr}\left(e^{-H/T}\right)$ (23) $\displaystyle=$
$\displaystyle e^{-E_{0}^{HF}/T}\int[D\psi]\int[D\bar{\psi}]\,e^{-S}\,,$ (24)

where $E_{0}^{HF}=\langle\psi_{0}|H|\psi_{0}\rangle$ is the Hartree-Fock
variational ground state energy, and the action $S$ is given by

$\begin{split}S=&\int_{0}^{1/T}\mathrm{d}\tau\,\sum_{\mathbf{k}}\sum_{\alpha}\bar{\psi}_{{\mathbf{k}},\alpha}(\partial_{\tau}+E_{{\mathbf{k}},\alpha})\psi_{{\mathbf{k}},\alpha}\\\
+\frac{1}{2N}&\sum_{{\mathbf{q}},{\mathbf{k}},{\mathbf{k}}^{\prime}}V({\mathbf{q}})\left(\bar{\psi}_{{\mathbf{k}}-{\mathbf{q}}}\Lambda_{{\mathbf{q}}}({\mathbf{k}})\psi_{{\mathbf{k}}}\right)\left(\bar{\psi}_{{\mathbf{k}}^{\prime}+{\mathbf{q}}}\Lambda_{-{\mathbf{q}}}({\mathbf{k}}^{\prime})\psi_{{\mathbf{k}}^{\prime}}\right)\,,\end{split}$
(25)

where in the last line we have suppressed the indices $\alpha,\beta,\dots$. A
few comments are in order. First, note that the kinetic term of the action
contains the mean-field single-particle energies, and not the bare band
energies (which would correspond to the eigenvalues of $h({\mathbf{k}})$). So
it will be the properties of the renormalized mean-field bands, e.g. their
nesting or density-of-states, which controls the perturbation theory in the
interaction. Secondly, we emphasize that no approximation is involved in our
derivation of the path integral – Eqs. (24), (25) constitute an exact
representation of the partition function. Thirdly, the unusual form of the
action, and the additional factor $\exp(-E_{0}^{HF}/T)$ in Eq. (24), result
from the different form of normal ordering required by the use of particle-
hole transformed coherent states: one has to normal order the Hamiltonian with
respect to the Hartree-Fock ground state, and not with respect to the Fock
vacuum state. This different choice of normal ordering produces additional
quadratic terms, which exactly correspond to the Hartree and Fock
Hamiltonians, which combined with the bare kinetic term produce the mean-field
single-particle energies via the Hartree-Fock self-consistency equation (12).

In working with the path integral in Eqs. (24), (25) the conventional
imaginary-time Feynman rules can be used, except for equal-time diagrams.
These diagrams are usually defined by inserting a factor
$e^{i\epsilon\omega_{n}}$, where $\omega_{n}$ is the fermionic Matsubara
frequency, and taking the limit $\epsilon\rightarrow 0$ at the end of the
calculation. Here, to reflect the different normal ordering, one has to add
the factor $e^{i\epsilon\omega_{n}\text{sgn}(E_{{\mathbf{k}},\alpha})}$ to the
equal-time propagators. With this regularization, the Hartree and Fock self-
energy diagrams vanish at zero temperature. This makes sense physically, as
the Hartree-Fock path integral already takes these self-energy effects into
account from the outset, and one should not double count these terms.

To conclude this section, we present the diagrammatic Feynman rules that will
be used in this work. First, the electron propagator is represented in the
usual was as a straight line with an arrow:

$=(i\omega_{n}-E_{{\mathbf{k}},\alpha})^{-1}$

The interaction will be represented as a dashed line:

${\mathbf{k}},\alpha$${\mathbf{k}}-{\mathbf{q}},\beta$${\mathbf{k}}^{\prime}-{\mathbf{q}},\sigma$${\mathbf{k}}^{\prime},\lambda$$=$$-V({\mathbf{q}})\big{[}\Lambda_{{\mathbf{q}}}({\mathbf{k}})\big{]}_{\beta\alpha}\big{[}\Lambda_{{\mathbf{q}}}^{\dagger}({\mathbf{k}}^{\prime})\big{]}_{\lambda\sigma}$

With these definitions and conventions in place, we can now turn our attention
to the RPA fluctuations on top of the mean-field result, which is the topic of
the next section.

##  IV Effective RPA interaction

In the previous section we derived the Hartree-Fock path integral, which
contains the complete bare Coulomb repulsion in the action. Here we study the
effective action which arises from summing all contributions from RPA
diagrams. In Fig. 1 we show these diagrams. They contain, among others, the
familiar bubble/polarization diagrams, and Berk-Schrieffer diagrams [21]. The
central object defined in Fig. 1 is called $G_{2}({\mathbf{q}},i\nu)$, which
is a matrix diagonal in momentum ${\mathbf{q}}$ and bosonic Matsubara
frequency $i\nu$. Writing out the indices explicitly, this matrix is
$[G_{2}({\mathbf{q}},i\nu)]^{{\mathbf{k}}\alpha\beta}_{{\mathbf{k}}^{\prime}\lambda\sigma}$.
It is defined diagramatically in Fig. 1(b) as the infinite sum of direct and
exchange diagrams concatenated with electron propagators. An explicit
expression for this infinite sum can be obtained by solving the familiar
Bethe-Salpeter equation, which is shown diagramatically in Fig. 2. Written out
explicitly, the Bethe-Salpeter equation for $G_{2}({\mathbf{q}},i\nu)$ is

$\begin{split}[G_{2}({\mathbf{q}},i\nu)]^{{\mathbf{k}}\alpha\beta}_{{\mathbf{k}}^{\prime}\lambda\sigma}&=\left[-T\sum_{i\omega_{n}}\frac{1}{i\omega_{n}-E_{{\mathbf{k}},\alpha}}\frac{1}{i(\omega_{n}-\nu)-E_{{\mathbf{k}}+{\mathbf{q}},\beta}}\right]\times\\\
\bigg{[}\delta_{\alpha\lambda}\delta_{\beta\sigma}\delta_{{\mathbf{k}},{\mathbf{k}}^{\prime}}-\frac{1}{N_{s}}\sum_{{\mathbf{k}}^{\prime\prime}\mu\nu}\bigg{(}V({\mathbf{q}})&[\Lambda_{\mathbf{q}}({\mathbf{k}})]_{\beta\alpha}[\Lambda_{-{\mathbf{q}}}({\mathbf{k}}^{\prime\prime}-{\mathbf{q}})]_{\mu\nu}-V({\mathbf{k}}^{\prime\prime}-{\mathbf{k}}^{\prime})[\Lambda_{{\mathbf{k}}-{\mathbf{k}}^{\prime\prime}}({\mathbf{k}})]_{\mu\alpha}[\Lambda_{{\mathbf{k}}^{\prime\prime}-{\mathbf{k}}}({\mathbf{k}}^{\prime\prime}-{\mathbf{q}})]_{\beta\nu}\bigg{)}[G_{2}({\mathbf{q}},i\nu)]^{{\mathbf{k}}^{\prime\prime}\mu\nu}_{{\mathbf{k}}^{\prime}\lambda\sigma}\bigg{]}\end{split}$
(26)

The details of the Bethe-Salpeter equation and its solution are presented in
Appendix A. As the Bethe-Salpeter equation is well-known, we simply present
the results here. At $T=0$, the general form of $G_{2}({\mathbf{q}},i\nu)$ is

Figure 1: Definition of the effective interaction obtained from summing RPA
diagrams. Figure 2: Diagrammatic representation of the Bethe-Salpeter equation
for $G_{2}({\mathbf{q}},i\nu)$, the solution of which represents the infinite
sum of RPA diagrams shown in Fig. 1(b).

$\big{[}G_{2}(i\nu,{\mathbf{q}})\big{]}^{{\mathbf{k}},\alpha\beta}_{{\mathbf{k}}^{\prime},\lambda\sigma}=\sum_{s}\varphi^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})\frac{1}{\omega_{{\mathbf{q}},s}-i\nu\,\eta_{{\mathbf{q}},s}}\varphi^{s*}_{{\mathbf{q}},\lambda\sigma}({\mathbf{k}}^{\prime})\,,$
(27)

where $\omega_{{\mathbf{q}},s}$ are the energies of the RPA collective modes,
which are guaranteed to be real and positive if the Slater determinant in Eq.
(4) is a local energy minimum, i.e. if it satisfies the Hartree-Fock self-
consistency equations (12). The sum in Eq. (27) runs only over those modes
which have non-zero energy. Eq. (27) shows that $G_{2}({\mathbf{q}},i\nu)$
takes the form of a propagator for the collective modes, which consist of
electron-hole pairs with a pair-wavefunction
$\varphi^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})$.

The collective mode wavefunctions
$\varphi^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})$ and energies
$\omega_{{\mathbf{q}},s}$ are obtained by solving the following generalized
eigenvalue equation:

$\begin{split}&(n_{\alpha}({\mathbf{k}})-n_{\beta}({\mathbf{k}}-{\mathbf{q}}))\varphi^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})\eta_{{\mathbf{q}},s}\omega_{{\mathbf{q}},s}=\\\
&|E_{\alpha}({\mathbf{k}})-E_{\beta}({\mathbf{k}}-{\mathbf{q}})|\varphi^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})\\\
&+V({\mathbf{q}})\frac{1}{N}\sum_{{\mathbf{k}}^{\prime}}\text{tr}\left(\varphi^{s}_{\mathbf{q}}({\mathbf{k}}^{\prime})\Lambda_{{\mathbf{q}}}({\mathbf{k}}^{\prime})\right)\left[\Lambda^{\dagger}_{{\mathbf{q}}}({\mathbf{k}})\right]_{\alpha\beta}\\\
&-\frac{1}{N}\sum_{{\mathbf{q}}^{\prime}}V({\mathbf{q}}^{\prime})\left[\Lambda^{\dagger}_{{\mathbf{q}}^{\prime}}({\mathbf{k}})\varphi^{s}_{{\mathbf{q}}}({\mathbf{k}}-{\mathbf{q}}^{\prime})\Lambda_{{\mathbf{q}}^{\prime}}({\mathbf{k}}-{\mathbf{q}})\right]_{\alpha\beta}\,.\end{split}$
(28)

Note that $\varphi^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})$ is defined to
be zero when $n_{\alpha}({\mathbf{k}})=n_{\beta}({\mathbf{k}}-{\mathbf{q}})$.
For collective modes with non-zero energy $\omega_{{\mathbf{q}},s}$, the signs
$\eta_{{\mathbf{q}},s}$ are related to the collective mode wavefunctions
through the following relation:

$\sum_{\mathbf{k}}\sum_{i,n}\left(|\varphi^{s}_{{\mathbf{q}},ni}({\mathbf{k}})|^{2}-|\varphi^{s}_{{\mathbf{q}},in}({\mathbf{k}})|^{2}\right)=\eta_{{\mathbf{q}},s}=\pm
1\,,$ (29)

where we have again used the convention that $i$ labels occupied states, and
$n$ labels empty states. The physical meaning of the $\eta_{{\mathbf{q}},s}$
sign factors and their appearance in the denominator of the collective mode
propagator in Eq. (27) can be understood as follows. The Bethe-Salpeter
equation at momentum ${\mathbf{q}}$ describes both the collective mode
creation operators, which we denote as $b_{{\mathbf{q}}}^{\dagger}$, and the
collective mode annihilation operators, denoted as $b_{-{\mathbf{q}}}$. The
collective mode propagator therefore contains contributions of the form
$\int\mathrm{d}\tau
e^{i\nu\tau}\langle\hat{T}b_{-{\mathbf{k}}}(\tau)b^{\dagger}_{-{\mathbf{k}}}(0)\rangle$
and $\int\mathrm{d}\tau
e^{i\nu\tau}\langle\hat{T}b^{\dagger}_{{\mathbf{k}}}(\tau)b_{{\mathbf{k}}}(0)\rangle$,
where $\hat{T}$ is the time-ordering operator. The latter can be rewritten as
$\int\mathrm{d}\tau
e^{-i\nu\tau}\langle\hat{T}b_{{\mathbf{k}}}(\tau)b^{\dagger}_{{\mathbf{k}}}(0)\rangle$,
hence the origin of the minus sign in front of $i\nu$ in some of the
collective mode propagators. From this we conclude that
$\eta_{{\mathbf{q}},s}=1$ means that $\varphi^{s}_{\mathbf{q}}$ corresponds to
a collective mode creation operator, whereas $\eta_{{\mathbf{q}},s}=-1$ means
that $\varphi^{s}_{\mathbf{q}}$ corresponds to a collective mode annihilation
operator.

The collective mode propagator is guaranteed to have a particle-hole symmetry,
which implies that if $\varphi^{s}_{{\mathbf{q}}}$ is a solution to the
generalized eigenvalue equation (28) with energy $\omega_{{\mathbf{q}},s}$,
then so is its particle-hole conjugate partner
$\varphi^{s^{\prime}}_{-{\mathbf{q}}}$ defined as

$\varphi^{s^{\prime}}_{-{\mathbf{q}},\alpha\beta}({\mathbf{k}}):=\mathcal{P}\left[\varphi^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})\right]:=\varphi^{s*}_{{\mathbf{q}},\beta\alpha}({\mathbf{k}}+{\mathbf{q}})\,,$
(30)

where $\mathcal{P}$ is the (anti-unitary) particle-hole conjugation operator.
Note that the particle-hole transformation inverts the momentum
${\mathbf{q}}$, but preserves the energy $\omega_{{\mathbf{q}},s}$. From Eq.
(29) we also see that the particle-hole transformation flips the signs
$\eta_{{\mathbf{q}},s}$.

When the self-consistent Hartree-Fock state has a time-reversal symmetry
$\mathcal{T}$, the combined action $\mathcal{PT}$ is a unitary symmetry which
preserves momentum but flips the signs $\eta_{{\mathbf{q}},s}$ defined in Eq.
(29). The $\mathcal{PT}$ symmetry guarantees that non-zero energies appear in
degenerate pairs $\omega_{{\mathbf{q}},s}=\omega_{{\mathbf{q}},s^{\prime}}$.
Furthermore, the corresponding wavefunctions are related by

$\varphi^{s^{\prime}}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}}):=\mathcal{PT}[\varphi^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})]:=\varphi^{s}_{{\mathbf{q}},\beta\alpha}(-{\mathbf{k}}+{\mathbf{q}})\,,$
(31)

where we have without loss of generality chosen to work in a gauge where
$\mathcal{T}|u_{\alpha}({\mathbf{k}})\rangle=|u_{\alpha}(-{\mathbf{k}})\rangle$.
Below, we will focus on time-reversal symmetric self-consistent Hartree-Fock
states and make use of this property.

At this point we have introduced all the properties of the collective mode
propagator in Eq. (27) that we will need for our further analysis. It is now
straightforward to obtain the RPA effective interaction by plugging in the
collective mode propagator in the diagrams in Fig. 1 (a). The last four
diagrams in this figure can then be interpreted as the effective interaction
between electrons which results from the exchange of collective modes. In the
following section, we will return to the main topic of this work, which is
interacting electron systems with a spontaneously broken continuous symmetry.
The collective modes of interest will be the Goldstone modes, and we will use
the general formalism discussed in this section to obtain an expression for
the scattering vertex between electrons and the Goldstone modes.

##  V Interacting electrons and Goldstone modes

When the self-consistent Hartree-Fock state spontaneously breaks a continuous
symmetry the collective mode spectrum is guaranteed to have gapless branches
corresponding to the Goldstone modes. From now on we will exclusively focus on
these Goldstone modes, and ignore the other collective modes. In this section
we (1) elaborate on the properties of the Goldstone mode wavefunctions, (2)
construct the scattering vertex between electrons and Goldstone modes, and (3)
synthesize our findings in an effective electron-boson model.

###  V.1 Properties of the Goldstone wavefunctions

Assuming that the mean-field state preserves time-reversal symmetry, for every
non-zero Goldstone mode energy $\omega_{{\mathbf{q}},s}$, there are two
collective mode wavefunctions $\varphi^{s}_{{\mathbf{q}}}$ which are related
by the action of $\mathcal{PT}$, and which have opposite signs
$\eta_{{\mathbf{q}},s}$ as defined in Eq. (29). We will use the convention
that for every $s>0$, the Goldstone mode wavefunctions corresponding to
$\omega_{{\mathbf{q}},s}$ are denoted as $\varphi^{\pm s}_{\mathbf{q}}$, where
the sign is determined by the corresponding value of $\eta_{{\mathbf{q}},s}$.
In this work, we are mainly interested in linearly dispersing Goldstone modes
(quadratically dispersing Goldstone modes are simpler, and do not require the
full power of the formalism that we develop here). So we assume that at small
momenta
$\omega_{{\mathbf{q}},s}=c_{s}|{\mathbf{q}}|+\mathcal{O}({\mathbf{q}}^{2})$,
where $c_{s}$ are the velocities of the Goldstone modes.

For a Goldstone branch corresponding to broken symmetry generator $Q_{s}$, we
can write down the exact analytic expression for the zero mode at
${\mathbf{q}}=0$ 222Note that as we are only considering linearly dispersing
Goldstone modes, the broken symmetry generators $Q_{s}$ are in one-to-one
correspondence with the Goldstone modes [22].:

$\tilde{Q}_{s,\alpha\beta}({\mathbf{k}})=i\langle
u_{\alpha}({\mathbf{k}})|Q_{s}|u_{\beta}({\mathbf{k}})\rangle\left[n_{\alpha}({\mathbf{k}})-n_{\beta}({\mathbf{k}})\right]\,,$
(32)

where the factor $i$ is added to ensure that
$\tilde{\varphi}^{s}_{{\mathbf{q}}=0}$ is an eigenstate of the particle-hole
conjugation operator $\mathcal{P}$ defined in Eq. (30) with eigenvalue $1$.
Given that $Q_{s}$ commutes with $h({\mathbf{k}})$, one can check that
$\tilde{Q}_{s}$ is indeed a zero mode of the generalized eigenvalue equation
in Eq. (28). Furthermore, due to the factor
$\left[n_{\alpha}({\mathbf{k}})-n_{\beta}({\mathbf{k}})\right]$ the
wavefunction $\tilde{Q}_{s,\alpha\beta}({\mathbf{k}})$ is non-zero only when
the symmetry generator $Q_{s}$ is broken (if it is not broken, then $Q_{s}$
commutes with the mean-field Hamiltonian and hence is diagonal in the mean-
field basis $|u_{\alpha}({\mathbf{k}})\rangle$). The zero mode is also an
eigenstate of the $\mathcal{PT}$ symmetry in Eq. (31) with eigenvalue
$(-1)^{\kappa+1}$, where $\kappa$ encodes whether $Q_{s}$ is time-reversal
even or odd:

$\mathcal{T}Q_{s}\mathcal{T}^{-1}=(-1)^{\kappa}Q_{s}\,.$ (33)

At non-zero ${\mathbf{q}}$, for every broken generator $Q_{s}$ there are two
collective mode wavefunctions $\tilde{\varphi}^{\pm s}_{\mathbf{q}}$ which are
mapped to each other by the action of $\mathcal{PT}$. In order to have a non-
singular and smooth ${\mathbf{q}}\rightarrow 0$ limit, we define the following
rescaled collective mode wavefunctions at non-zero ${\mathbf{q}}$:

$\tilde{\varphi}^{\pm
s}_{{\mathbf{q}}}:=\sqrt{\frac{2\omega_{{\mathbf{q}},s}aw_{s}N}{c_{s}}}\varphi_{{\mathbf{q}}}^{\pm
s}\,,$ (34)

where $a$ is the lattice constant, and $w_{s}$ a dimensionless number that
will be determined below. The components $\tilde{\varphi}^{\pm
s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})$ are generically order one numbers
which for fixed non-zero ${\mathbf{q}}$ do not go to zero in the thermodynamic
limit. Moreover, in the appendices we show that in the thermodynamic limit
$\tilde{\varphi}^{\pm s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})$ also
remains finite as ${\mathbf{q}}\rightarrow 0$.

The two wavefunctions $\tilde{\varphi}^{\pm
s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})$ at non-zero ${\mathbf{q}}$ become
identical (possibly up to a phase factor) as ${\mathbf{q}}\rightarrow 0$. This
is possible because the Goldstone mode wavefunctions are _not_ orthogonal
eigenvectors obtained from a Hermitian eigenvalue problem, but instead are
obtained from the _generalized_ eigenvalue equation in Eq. (28). Exactly at
${\mathbf{q}}=0$, there is only one Goldstone wavefunction – which corresponds
to the exact zero mode in Eq. (32). This happens because in the presence of
zero modes, solutions to the generalized eigenvalue equation (28) do not form
a complete basis [23]. As a next step, we fix the phase and norm of the
rescaled wavefunctions such that

$\lim_{{\mathbf{q}}\rightarrow 0}\tilde{\varphi}^{\pm
s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})=\tilde{Q}_{s,\alpha\beta}({\mathbf{k}})\,,$
(35)

where $\tilde{Q}_{s}$ is defined in Eq. (32). As a first consistency check,
let us note that it follows from Eq. (29) that the rescaled wavefunctions
satisfy

$\frac{1}{N}\sum_{\mathbf{k}}\sum_{i,n}\left(|\tilde{\varphi}^{\pm
s}_{{\mathbf{q}},ni}({\mathbf{k}})|^{2}-|\tilde{\varphi}^{\pm
s}_{{\mathbf{q}},in}({\mathbf{k}})|^{2}\right)=\pm
2\omega_{{\mathbf{q}},s}\frac{aw_{s}}{c_{s}}\,.$ (36)

The exact zero mode $\tilde{Q}_{s}$ defined in Eq. (32) also satisfies this
equation with $\omega_{{\mathbf{q}}=0,s}=0$.

To satisfy Eq. (35) we have to adjust the phase and norm of the wavefunctions
$\tilde{\varphi}^{\pm s}_{{\mathbf{q}}}$. The norm we fix via the
dimensionless parameter $w_{s}$ defined in Eq. (34). The phase can be
partially fixed using the symmetries $\mathcal{P}$ and $\mathcal{PT}$. In
particular, as the zero mode $\tilde{Q}_{s}$ is even under $\mathcal{P}$ and
has eigenvalue $(-1)^{\kappa+1}$ under $\mathcal{PT}$, we require that

$\displaystyle\mathcal{PT}[\tilde{\varphi}^{\pm s}_{\mathbf{q}}]$
$\displaystyle=$ $\displaystyle(-1)^{\kappa+1}\tilde{\varphi}^{\mp
s}_{\mathbf{q}}$ (37) $\displaystyle\mathcal{P}[\tilde{\varphi}^{\pm
s}_{\mathbf{q}}]$ $\displaystyle=$ $\displaystyle\tilde{\varphi}^{\mp
s}_{-{\mathbf{q}}}\,.$ (38)

After this partial gauge fixing, there is a remaining phase freedom over half
of the Brillouin zone for one of the two wavefunctions, say
$\tilde{\varphi}_{\mathbf{q}}^{+s}$. We fix this phase by requiring
$\tilde{\varphi}_{\mathbf{q}}^{+s}$ to be a continuous function of
${\mathbf{q}}$, and Eq. (35) to hold. Note that if there is an additional
inversion symmetry $\mathcal{I}$ under which
${\mathbf{q}}\rightarrow-{\mathbf{q}}$, then we can fix the remaining phase of
$\tilde{\varphi}_{\mathbf{q}}^{+s}$ over half of the Brillouin zone up to a
minus sign by requiring that $\tilde{\varphi}_{\mathbf{q}}^{+s}$ is either
even or odd under $\mathcal{IT}$ (depending on whether $\tilde{Q}_{s}$ is even
or odd under $\mathcal{IT}$). The requirements of continuity and a smooth
${\mathbf{q}}\rightarrow 0$ limit as in Eq. (35) then uniquely fix the phase.

With the gauge choice in Eq. (37), it follows that
$(\tilde{\varphi}^{s}_{\mathbf{q}}\pm\tilde{\varphi}^{-s}_{{\mathbf{q}}})/2$
is an eigenstate of $\mathcal{PT}$ with eigenvalue $\pm(-1)^{\kappa+1}$. In
the limit ${\mathbf{q}}\rightarrow 0$, the symmetric combination therefore
becomes the unique zero mode $\tilde{Q}_{s}$, which has $\mathcal{PT}$
eigenvalue $(-1)^{\kappa+1}$. The anti-symmetric combination becomes a zero
mode with $\mathcal{PT}$ eigenvalue $-(-1)^{\kappa+1}$, which must be the zero
vector. From this it follows that with the gauge choice in Eq. (37), equation
(35) indeed holds.

This concludes our discussion of the Goldstone wavefunctions. Below we will
make use of the properties of these wavefunctions when we go to the rotating
frame in Sec. VI.

###  V.2 Electron-Goldstone scattering vertex

In Sec. IV we defined the effective RPA interaction, which contains the
collective mode propagator $G_{2}({\mathbf{q}},i\nu)$ that we obtained by
solving the Bethe-Salpeter equation. We also discussed the properties of the
solutions to the Bether-Salpeter equation (the details of which are contained
in the appendices), and in the previous subsection we focused in particular on
solutions that correspond to Goldstone modes. In this section, we come back to
the effective RPA interaction. In particular, now that we understand better
the properties of $G_{2}({\mathbf{q}},i\nu)$ and the associated collective
mode wavefunctions, we will plug $G_{2}({\mathbf{q}},i\nu)$ into the diagrams
in Fig. 1 (a) and study the interaction between electrons which results from
the exchange of Goldstone modes.

Our starting point is Eq. (27), which is the general form for
$G_{2}({\mathbf{q}},i\nu)$ as a solution to the Bethe-Salpeter equation.
Taking this expression, and plugging it into the diagrams in Fig. 1 (a), we
see that the effective RPA interaction is given by the bare interaction (the
left most diagram in Fig. 1 (a)), plus an interaction where the electrons emit
and absorb a Goldstone mode, which can be denoted by the following single
diagram:

(39)

In this diagram, the wavy line denotes the boson progagator
$(i\nu\eta_{{\mathbf{q}},s}-\omega_{{\mathbf{q}},s})^{-1}$, and the scattering
vertex is given by

$\includegraphics[scale={0.9}]{vertex.pdf}\,,$ (40)

where the box represents the Goldstone wavefunction
$\varphi^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})$. Written out in
equations, the vertex is given by

$\begin{split}\hat{g}^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})=V({\mathbf{q}})\frac{1}{N}\sum_{{\mathbf{k}}^{\prime}}&\text{tr}\left(\varphi^{s}_{\mathbf{q}}({\mathbf{k}}^{\prime})\Lambda_{{\mathbf{q}}}({\mathbf{k}}^{\prime})\right)\left[\Lambda^{\dagger}_{{\mathbf{q}}}({\mathbf{k}})\right]_{\alpha\beta}\\\
-\frac{1}{N}\sum_{{\mathbf{q}}^{\prime}}V({\mathbf{q}}^{\prime})&\left[\Lambda^{\dagger}_{{\mathbf{q}}^{\prime}}({\mathbf{k}})\varphi^{s}_{{\mathbf{q}}}({\mathbf{k}}-{\mathbf{q}}^{\prime})\Lambda_{{\mathbf{q}}^{\prime}}({\mathbf{k}}-{\mathbf{q}})\right]_{\alpha\beta}\end{split}$
(41)

Note that $\hat{g}^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})$ is
generically non-zero for all $\alpha$ and $\beta$, whereas
$\varphi^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})$ is only non-zero if
$n_{\alpha}({\mathbf{k}})\neq n_{\beta}({\mathbf{k}}-{\mathbf{q}})$. The terms
in Eq. (41) which define $\hat{g}_{\mathbf{q}}^{s}$ have appeared previously
as part of the generalized eigenvalue equation in Eq. (28). Since the
Goldstone wavefunctions $\varphi_{{\mathbf{q}}}^{s}$ are obtained as solutions
to that equation, we immediately obtain from Eqs. (41) and (28) that

$\begin{split}\hat{g}^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})&=\bigg{(}[n_{\alpha}({\mathbf{k}})-n_{\beta}({\mathbf{k}}-{\mathbf{q}})]\eta_{{\mathbf{q}},s}\omega_{{\mathbf{q}},s}\\\
&-|E_{\alpha}({\mathbf{k}})-E_{\beta}({\mathbf{k}}-{\mathbf{q}})|\bigg{)}\varphi^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})\\\
\text{ if }&n_{\alpha}({\mathbf{k}})\neq
n_{\beta}({\mathbf{k}}-{\mathbf{q}})\,.\end{split}$ (42)

For the components $\hat{g}^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})$ with
$n_{\alpha}({\mathbf{k}})=n_{\beta}({\mathbf{k}}-{\mathbf{q}})$ we have no
general analytic expression. Using the $\mathcal{P}$ symmetry in the gauge of
Eq. (38), Eq. (42) also makes it explicit that the electron-Goldstone vertex
satisfies

$\hat{g}^{s*}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})=\hat{g}^{-s}_{-{\mathbf{q}},\beta\alpha}({\mathbf{k}}-{\mathbf{q}})\,,$
(43)

where we have used that $\eta_{{\mathbf{q}},s}=-\eta_{-{\mathbf{q}},-s}$ and
$\omega_{{\mathbf{q}},s}=\omega_{-{\mathbf{q}},-s}$. This property ensures
that the Goldstone-mediated interaction between electrons is Hermitian.

###  V.3 Effective electron-boson model

In the previous section we have derived an explicit expression for the
interaction between electrons mediated by the exchange of Goldstone modes. The
Goldstone modes are collective excitations of the electron fluid, and hence
are not described by independent microscopic degrees of freedom. In this
section, however, we will write down an effective theory which does contain
both independent electronic and bosonic degrees of freedom, and which mimics
the behaviour of the coupled electron-Goldstone-boson system. The action for
the effective electron-boson model is given by

$S=S_{el}+S_{V}+S_{B}+S_{el-B}+S_{C}\,,$ (44)

where $S_{el}$ and $S_{V}$ respectively denote the quadratic and interacting
part of the electronic action that appeared previously in the Hartree-Fock
path integral in Eq. (25).

The terms $S_{B}$, $S_{el-B}$, and $S_{C}$ are not present in the Hartree-Fock
action. Of these, the first two describe the Goldstone dynamics and the
electron-Goldstone-mode scattering. The Goldstone dynamics is described by

$S_{B}=\int_{0}^{1/T}\mathrm{d}\tau\,\sum_{s>0}\sum_{\mathbf{q}}\bar{b}_{{\mathbf{q}},s}(\partial_{\tau}+\omega_{{\mathbf{q}},s})b_{{\mathbf{q}},s}\,,$
(45)

where $\bar{b}_{{\mathbf{q}},s},b_{{\mathbf{q}},s}$ are bosonic fields
corresponding to the Goldstone creation and annihilation operators. The
electron-Goldstone coupling is given by

$\begin{split}S_{el-B}&=\int_{0}^{1/T}\mathrm{d}\tau\,\frac{1}{\sqrt{N}}\sum_{{\mathbf{q}},{\mathbf{k}}}\bar{\psi}_{{\mathbf{k}},\alpha}\psi_{{\mathbf{k}}-{\mathbf{q}},\beta}\times\\\
\sum_{s>0}&\sqrt{\frac{c_{s}}{2\omega_{\mathbf{q}}aw_{s}}}\left(\tilde{g}^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})\bar{b}_{-{\mathbf{q}},s}+\tilde{g}^{-s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})b_{{\mathbf{q}},s}\right)\,,\end{split}$
(46)

where for future use we have defined the vertex functions
$\tilde{g}^{s}_{{\mathbf{q}}}$ using the same rescaling as in Eq. (34), i.e.
$\tilde{g}^{s}_{{\mathbf{q}}}=\hat{g}^{s}_{{\mathbf{q}}}\sqrt{2\omega_{\mathbf{q}}aw_{s}N/c_{s}}$.
It follows from Eq. (43) that tree-level boson exchange in the effective
electron-boson model indeed generates the interaction in Eq. (39).

So far we have taken the Hartree-Fock action, and supplemented it with the
bosonic degrees of freedom. The theories with and without the bosonic fields
are obviously not the same. In particular, the bosonic degrees of freedom are
designed to capture the RPA fluctuations which result from the electronic
interaction. As a result, working at tree level with the effective action in
Eq. (44) (without $S_{C}$) is the same as doing the infinite sum of RPA
diagrams in the original theory. Once loops are taken into account in the
effective theory, however, one has to be careful that the loop diagrams do not
correspond to a double counting of RPA diagrams in the original theory. For
example, $\omega_{{\mathbf{q}},s}$ is already the complete boson dispersion
which is generated by the electron dynamics, and it should therefore not be
further renormalized by coupling to the electrons. To ensure this, we have to
add a counter term which is quadratic in the boson fields, and which removes
all self-energy diagrams for the bosons. Another counter term quartic in the
fermion fields has to be added to remove the RPA renormalization of the
electron repulsion, as this effect is already contained in the interaction
mediated by boson exchange. All these counter terms are contained in $S_{C}$.
It is not necessary to construct these terms explicitly – all one needs to
know is that they are present, and that they eliminate certain diagrams such
that working with the effective theory beyond tree level is the same as
working with the original theory.

###  V.4 Transforming to the real basis

In this final section we bring the effective electron-boson model introduced
in the previous section in a more standard form. First, we rescale the boson
fields as follows

$\bar{b}_{{\mathbf{q}},s},b_{{\mathbf{q}},s}\rightarrow\sqrt{\frac{aw_{s}}{c_{s}}}\bar{b}_{{\mathbf{q}},s},\sqrt{\frac{aw_{s}}{c_{s}}}b_{{\mathbf{q}},s}\,,$
(47)

and then define the usual (Fourier transformed) canonically conjugate real
fields as

$\displaystyle\phi_{{\mathbf{q}},s}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\omega_{\mathbf{q}}}}(\bar{b}_{{\mathbf{q}},s}+b_{-{\mathbf{q}},s})$
(48) $\displaystyle\pi_{{\mathbf{q}},s}$ $\displaystyle=$ $\displaystyle
i\sqrt{\frac{\omega_{\mathbf{q}}}{2}}(\bar{b}_{{\mathbf{q}},s}-b_{-{\mathbf{q}},s})\,,$
(49)

where we adopted the standard notation. In terms of the new fields, the
quadratic boson action becomes

$\begin{split}S_{B}=\int_{0}^{1/T}\mathrm{d}\tau&\sum_{s>0}\frac{aw_{s}}{c_{s}}\sum_{\mathbf{q}}\bigg{(}-i\pi_{{\mathbf{q}},s}\partial_{\tau}\phi_{-{\mathbf{q}},s}\\\
&+\frac{1}{2}\pi_{{\mathbf{q}},s}\pi_{-{\mathbf{q}},s}+\frac{1}{2}\omega_{{\mathbf{q}},s}^{2}\phi_{{\mathbf{q}},s}\phi_{-{\mathbf{q}},s}\bigg{)}\,,\end{split}$
(50)

and the electron-boson vertex is given by

$\begin{split}S_{el-B}=&\int_{0}^{1/T}\mathrm{d}\tau\,\frac{1}{\sqrt{N}}\sum_{{\mathbf{q}},{\mathbf{k}}}\bar{\psi}_{{\mathbf{k}},\alpha}\psi_{{\mathbf{k}}-{\mathbf{q}},\beta}\times\\\
\sum_{s>0}&\left(g^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})\phi_{-{\mathbf{q}},s}+f^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})\pi_{-{\mathbf{q}},s}\right)\,,\end{split}$
(51)

with

$\displaystyle g^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(\tilde{g}^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})+\tilde{g}^{-s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})\right)$
(52) $\displaystyle f^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})$
$\displaystyle=$
$\displaystyle\frac{-i}{2\omega_{{\mathbf{q}},s}}\left(\tilde{g}^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})-\tilde{g}^{-s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})\right)\,.$
(53)

As a final step, we integrate out the $\pi_{{\mathbf{q}},s}$ fields. This
produces the following quadratic boson action:

$S_{B}=\frac{1}{2}\int\mathrm{d}\tau\,\sum_{s>0}\frac{aw_{s}}{c_{s}}\sum_{\mathbf{q}}\left(-\phi_{-{\mathbf{q}}}\partial^{2}_{\tau}\phi_{\mathbf{q}}+\omega_{{\mathbf{q}},s}^{2}\phi_{{\mathbf{q}},s}\phi_{-{\mathbf{q}},s}\right)\,.$
(54)

In the long-wavelength continuum limit, using $\omega_{{\mathbf{q}},s}\sim
c_{s}^{2}{\mathbf{q}}^{2}$, we can write this as the standard action for
relativistic real scalar fields:

$S_{B}=\frac{1}{2}\int\mathrm{d}\tau\int\mathrm{d}^{2}{\mathbf{r}}\,\sum_{s>0}\chi_{s}(\partial_{\tau}\phi_{s})^{2}+\rho_{s}(\nabla\phi_{s})^{2}\,,$
(55)

with $\chi_{s}=w_{s}/ac_{s}$, and $\rho_{s}=w_{s}c_{s}/a$ is the stiffness.
After integrating out $\pi_{{\mathbf{q}},s}$, the electron-Goldstone vertex
becomes

$\begin{split}S_{el-B}=&\int_{0}^{1/T}\mathrm{d}\tau\,\frac{1}{\sqrt{N}}\sum_{{\mathbf{q}},{\mathbf{k}}}\bar{\psi}_{{\mathbf{k}},\alpha}\psi_{{\mathbf{k}}-{\mathbf{q}},\beta}\times\\\
\sum_{s>0}&\left(g^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})\phi_{-{\mathbf{q}},s}+f^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})i\partial_{\tau}\phi_{-{\mathbf{q}},s}\right)\,,\end{split}$
(56)

where in the second term $\pi_{{\mathbf{q}},s}$ is now replaced with
$i\partial_{\tau}\phi_{{\mathbf{q}},s}$. Finally, we also obtain following
two-body interaction from integrating out $\pi_{{\mathbf{q}},s}$:

$\begin{split}S_{V\pi}=\int\mathrm{d}\tau\sum_{s>0}\frac{-c_{s}}{2aw_{s}N}&\sum_{{\mathbf{k}},{\mathbf{k}}^{\prime},{\mathbf{q}}}\big{(}\bar{\psi}_{\mathbf{k}}f^{s}_{\mathbf{q}}({\mathbf{k}})\psi_{{\mathbf{k}}-{\mathbf{q}}}\big{)}\times\\\
&\left(\bar{\psi}_{{\mathbf{k}}^{\prime}}f^{s}_{-{\mathbf{q}}}({\mathbf{k}}^{\prime})\psi_{{\mathbf{k}}^{\prime}+{\mathbf{q}}}\right)\,,\end{split}$
(57)

where we have suppressed the Greek indices, and instead adopted a matrix
notation. From Eqs. (53) and (43) it follows that we can equivalently write
Eq. (57) as

$\begin{split}S_{V\pi}=\int\mathrm{d}\tau\sum_{s>0}\frac{-c_{s}}{2aw_{s}N}&\sum_{{\mathbf{k}},{\mathbf{k}}^{\prime},{\mathbf{q}}}\big{(}\bar{\psi}_{\mathbf{k}}f^{s}_{\mathbf{q}}({\mathbf{k}})\psi_{{\mathbf{k}}-{\mathbf{q}}}\big{)}\times\\\
&\left(\bar{\psi}_{{\mathbf{k}}^{\prime}-{\mathbf{q}}}f_{{\mathbf{q}}}^{s\dagger}({\mathbf{k}}^{\prime})\psi_{{\mathbf{k}}^{\prime}}\right)\,,\end{split}$
(58)

which makes it explicit that this is a negative definite interaction.

##  VI The rotating frame

We continue to work with the effective electron-boson model derived in the
previous section. Even though this model was shown to exactly reproduce the
RPA result, we will argue below that it has an (apparent) unphysical property,
which can be removed by working in the so-called `rotating frame'.

###  VI.1 The problem: strong interband scattering in the
${\mathbf{q}}\rightarrow 0$ limit

Using Eqs. (52), (53) and (42) we see that the vertices $g^{s}_{\mathbf{q}}$
and $f^{s}_{{\mathbf{q}}}$ which appear in the effective electron-boson model
for $n_{\alpha}({\mathbf{k}})\neq n_{\beta}({\mathbf{k}}-{\mathbf{q}})$ are
given by

$\displaystyle g^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\bigg{(}[n_{\alpha}({\mathbf{k}})-n_{\beta}({\mathbf{k}}-{\mathbf{q}})]\omega_{{\mathbf{q}},s}\left(\tilde{\varphi}^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})-\tilde{\varphi}^{-s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})\right)-|E_{\alpha}({\mathbf{k}})-E_{\beta}({\mathbf{k}}-{\mathbf{q}})|\left(\tilde{\varphi}^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})+\tilde{\varphi}^{-s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})\right)\bigg{)}$
(59) $\displaystyle f^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})$
$\displaystyle=$
$\displaystyle\frac{-i}{2\omega_{{\mathbf{q}},s}}\bigg{(}[n_{\alpha}({\mathbf{k}})-n_{\beta}({\mathbf{k}}-{\mathbf{q}})]\omega_{{\mathbf{q}},s}\left(\tilde{\varphi}^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})+\tilde{\varphi}^{-s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})\right)-|E_{\alpha}({\mathbf{k}})-E_{\beta}({\mathbf{k}}-{\mathbf{q}})|\left(\tilde{\varphi}^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})-\tilde{\varphi}^{-s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})\right)\bigg{)}\,,$

where we have used that $\eta_{{\mathbf{q}},s}=-\eta_{{\mathbf{q}},-s}$. From
these results, and Eqs. (35) and (32), we obtain an explicit expression for
the ${\mathbf{q}}\rightarrow 0$ limit of $g^{s}_{\mathbf{q}}$:

$\begin{split}\lim_{{\mathbf{q}}\rightarrow
0}&g^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})=i[E_{\alpha}({\mathbf{k}})-E_{\beta}({\mathbf{k}})]\langle
u_{\alpha}({\mathbf{k}})|Q_{s}|u_{\beta}({\mathbf{k}})\rangle\\\ &\text{ if
}n_{\alpha}({\mathbf{k}})\neq n_{\beta}({\mathbf{k}})\,,\end{split}$ (60)

where we used that

$\begin{split}[n_{\alpha}({\mathbf{k}})-n_{\beta}({\mathbf{k}})]\times|E_{\alpha}({\mathbf{k}})-E_{\beta}({\mathbf{k}})|=\\\
-|n_{\alpha}({\mathbf{k}})-n_{\beta}({\mathbf{k}})|\times[E_{\alpha}({\mathbf{k}})-E_{\beta}({\mathbf{k}})]\,.\end{split}$
(61)

Equation (60) shows that ${\mathbf{q}}=0$ Goldstone modes induce scattering
between electrons in different mean-field bands. This is because the mean-
field bands have gaps which result from the presence of a non-zero symmetry-
breaking order parameter. The way the original, non-interacting bands are
split to generate the mean-field bands depends on the orientation of the order
parameter. So a ${\mathbf{q}}=0$ Goldstone mode, which corresponds to a global
rotation of the order parameter, will induce a mixing of the mean-field bands
which enter the Hartree-Fock path integral, as they are defined using an order
parameter with a chosen fixed orientation. We note Eq. (60) has previously
also been obtained in Ref. [7], although the authors of that work did not use
the RPA approach adopted here.

The exchange of Goldstone modes thus generates an interaction between the
mean-field electrons which contains a contribution of the form
$\sim\Delta^{2}/(\chi_{s}(i\nu)^{2}-\rho_{s}{\mathbf{q}}^{2})$, where $\Delta$
is the gap in the mean-field bands induced by the symmetry breaking. This
interaction is singular in the $\nu,{\mathbf{q}}\rightarrow 0$ limit.
Moreover, $\Delta$ is of the interaction energy scale. So for strongly
interacting electron systems, one finds that the energy scale of the
interaction mediated by Goldstone exchange is much larger than the energy
scale of the bare density-density interaction. This results in a situation
where the mean-field electrons acquire large self-energy corrections, which
casts doubt on their existence as well-defined quasi-particles. In the next
section we will show that electrons which are defined in a rotating frame are
not affected by ${\mathbf{q}}=0$ Goldstone modes, and do not experience any
singular interactions.

Finally, from Eq. (59) it follows that also $\lim_{{\mathbf{q}}\rightarrow
0}f^{s}_{\mathbf{q}}\neq 0$. However, as $f^{s}_{\mathbf{q}}$ is the vertex
which couples the mean-field electrons to $\partial_{\tau}\phi$, this vertex
does not induce a singular interaction.

###  VI.2 The solution: the rotating frame

To remove the singular interaction between the mean-field electrons discussed
in the previous section we perform a change of integration variables in the
path integral of the electron-boson model. In particular, in the local orbital
basis we change the Grassmann fields as

$\psi_{a}({\mathbf{r}})\rightarrow\sum_{b}R_{ab}({\mathbf{r}})\psi_{b}({\mathbf{r}})\,,$
(62)

with

$R({\mathbf{r}})=\exp\left(-i\sum_{s>0}\phi_{s}({\mathbf{r}})Q_{s}\right)\,.$
(63)

As this is a unitary transformation, the corresponding change of integration
variables has a trivial Jacobian. In momentum space, and working in the mean-
field basis, the transformation in Eq. (62) can be written as

$\psi_{\alpha}({\mathbf{k}})\rightarrow\psi_{\alpha}({\mathbf{k}})-\frac{i}{\sqrt{N}}\sum_{s,{\mathbf{q}},\beta}\phi_{{\mathbf{q}},s}Q^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})\psi_{\beta}({\mathbf{k}}-{\mathbf{q}})+\mathcal{O}(\phi^{2})\,,$
(64)

where we have defined

$Q^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})=\langle
u_{\alpha}({\mathbf{k}})|Q_{s}|u_{\beta}({\mathbf{k}}-{\mathbf{q}})\rangle\,.$
(65)

Performing this change of variables in the kinetic term of the mean-field
fermions,
$\sum_{{\mathbf{k}},\alpha}\bar{\psi}_{{\mathbf{k}},\alpha}(\partial_{\tau}+E_{\alpha}({\mathbf{k}}))\psi_{{\mathbf{k}},\alpha}$,
produces terms which can be absorbed in $S_{el-B}$. In particular, using Eq.
(64) we find that the change of Grassmann variables induces following shift in
the vertex functions:

$\displaystyle g^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})$
$\displaystyle\rightarrow$ $\displaystyle
g^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})-iQ^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})[E_{\alpha}({\mathbf{k}})-E_{\beta}({\mathbf{k}}-{\mathbf{q}})]$
(66) $\displaystyle=:g^{Rs}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})$
$\displaystyle f^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})$
$\displaystyle\rightarrow$ $\displaystyle
f^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})-Q^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})$
(67) $\displaystyle=:f^{Rs}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})$

The change of variables also introduces higher-order interaction terms of the
form $\phi^{n}\bar{\psi}\psi$ with $n>1$, but we ignore these terms here.

From Eq. (60) we see that

$\lim_{{\mathbf{q}}\rightarrow
0}g^{Rs}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})=0\,,$ (68)

i.e. the $\nu,{\mathbf{q}}=0$ Goldstone modes decouple from the electrons.
This shows that the electrons defined in Eq. (62), which live in a rotating
frame that locally follows the order parameter fluctuations, will have much
smaller self-energy corrections than the mean-field electrons, and hence will
be much closer to the true quasi-particles of the symmetry-broken system.

Before concluding this section let us mention the effect of going to the
rotating frame on the other terms in the action of the electron-boson model.
First, the change of Grassmann variables in Eq. (62) leaves the bare
interaction term $S_{V}$ invariant. However, in general it will change the
interaction contained in $S_{V\pi}$ (58). For some applications it might be
important to keep this in mind, but in the remainder of this work we will not
use the interactions contained in $S_{V}$ and $S_{V\pi}$ anymore.

###  VI.3 Implications for electron spectral functions

In this section we investigate the consequences of the fact that quasi-
particles are defined in a rotating frame for the correlation functions of
microscopic fermions. In particular, we are interested in the two-point
functions

$\displaystyle G_{ab}(\tau,{\mathbf{r}}-{\mathbf{r}}^{\prime})$
$\displaystyle=$
$\displaystyle-\langle\hat{T}c_{{\mathbf{r}},a}(\tau)c^{\dagger}_{{\mathbf{r}}^{\prime},b}(0)\rangle_{\beta}$
(69) $\displaystyle=$
$\displaystyle\frac{-1}{Z}\int_{\bar{\psi},\psi,\phi}\psi_{{\mathbf{r}},a}\bar{\psi}_{{\mathbf{r}}^{\prime},b}\,e^{-S[\bar{\psi},\psi,\phi]}\,,$
(70)

where $\hat{T}$ is again the time-ordering operator,
$\langle\cdot\rangle_{\beta}$ is the thermal average, and
$\int_{\bar{\psi},\psi,\phi}$ a path integral for the fields $\bar{\psi},\psi$
and $\phi$. Note that the fermion fields in Eq. (70) are in the original,
unrotated frame. Performing the change of integration variables as in Eq.
(62), we obtain

$G_{ab}(\tau,{\mathbf{r}})=-\sum_{c,d}\langle
R^{*}_{ca}(\tau,{\mathbf{r}})\psi_{c,{\mathbf{r}}}(\tau)R_{db}(0,0)\bar{\psi}_{d,0}(0)\rangle_{\beta}\,,$
(71)

where $\bar{\psi},\psi$ now live in the rotated frame, and hence describe the
quasi-particles of the broken-symmetry state. As a first approximation, we
ignore the interaction between the electrons and the Goldstone modes, such
that the correlation function of the microscopic electrons factorizes as

$G_{ab}(\tau,{\mathbf{r}})=-\sum_{c,d}\langle
R^{*}_{ca}(\tau,{\mathbf{r}})R_{db}(0,0)\rangle\,\langle\psi_{c,{\mathbf{r}}}(\tau)\bar{\psi}_{d}(0,0)\rangle_{\beta}\,.$
(72)

By working with this approximation we make a physical assumption that
corrections resulting from electron-Goldstone mode interactions will be
subleading compared to the above `zeroth order' contribution. This assumption
is partly justified by the fact that the fermions in the rotating frame
decouple from the Goldstone modes at long wavelengths. Going to frequency and
momentum space we then obtain

$\begin{split}G_{ab}(i\omega,{\mathbf{k}})=\frac{-T}{N}\sum_{i\nu,{\mathbf{q}}}&\sum_{cd\alpha}D^{R}_{ab,cd}(i\nu,{\mathbf{q}})\\\
&\times\frac{u^{c}_{\alpha}({\mathbf{k}}-{\mathbf{q}})u^{d*}_{\alpha}({\mathbf{k}}-{\mathbf{q}})}{i(\omega-\nu)-E_{{\mathbf{k}}-{\mathbf{q}},\alpha}}\,,\end{split}$
(73)

where $D^{R}_{ab,cd}(i\nu,{\mathbf{q}})=-\langle
R^{*}_{ca}(i\nu,{\mathbf{q}})R_{db}(i\nu,{\mathbf{q}})\rangle$. We thus find
that the fermion correlation function in frequency-momentum space is a
convolution of the fermionic quasi-particle propagator with the propagator of
the Goldstone modes. Note that an expression very similar to Eq. (73) is
obtained for the electron Green's function in theories where the electron is
assumed to fractionalize in a spinon and a holon [14, 15, 16, 17, 18, 19, 20].

Up to now our effective electron-boson model describes a collection of
independent generalized `spin-waves', and does not take topological order
parameter configurations, such as e.g. vortices or skyrmions, or interactions
between Goldstone modes into account. To remedy this, one can improve the
model to correctly capture the compact nature of the order parameter manifold
by rewriting the Goldstone action as a non-linear sigma model. We will do this
explicitly in the next section for the anti-ferromagnetic ground state of the
Hubbard model. With the non-linear sigma model description of the Goldstone
modes, it is possible that thermal fluctuations destroy the long-range order
at a much lower temperature than the mean-field transition temperature at
which the symmetry-breaking gap disappears from the Hartree-Fock band
spectrum. This is especially relevant in 2D, where the Hohenberg-Mermin-Wagner
theorem states that long-range order from continuous symmetry breaking
disappears at any non-zero temperature. In the resulting thermally disordered
phase, the propagator $D^{R}_{ab,cd}(i\nu,{\mathbf{q}})$ is symmetric, i.e.
$D^{R}_{ab,cd}(i\nu,{\mathbf{q}})\propto\delta_{ab}$, and acquires a mass.
From Eq. (73), we see that in that case the electron Green's function
$G_{ab}(i\omega,{\mathbf{k}})$ also becomes symmetric in the thermally
disordered phase, even though the order-parameter-induced gap can still be
present in the mean-field spectrum of the fermions in the rotating frame. This
effect therefore naturally leads to `pseudo-gap' physics in thermally
disordered broken-symmetry states. Below we illustrate this explicitly in the
two example sections.

##  VII Example I: anti-ferromagnetism in the Hubbard model

In this first example section we apply the general formalism introduced above
to the anti-ferromagnetic ground state of the square-lattice Hubbard model.

###  VII.1 Mean-field state and Goldstone mode energies

We are interested in the Hubbard model on the square lattice, which is defined
by the following Hamiltonian:

$H=-t\sum_{\langle
ij\rangle}\sum_{s}c^{\dagger}_{i,s}c_{j,s}-t^{\prime}\sum_{\langle\langle
ij\rangle\rangle}c^{\dagger}_{i,s}c_{j,s}+h.c.+U\sum_{i}n_{i,\uparrow}n_{i,\downarrow}\,,$
(74)

where the first sum is over nearest neighbors, and the second sum over next
nearest neighbors. In the interaction term, $n_{i,s}=c^{\dagger}_{i,s}c_{i,s}$
is the density of electrons with spin $s$ at site $i$. We choose units of
energy such that $t\equiv 1$, and take $t^{\prime}=-0.35$.

For our purposes, it suffices to only consider half filling, where it is well-
known that the ground state is an insulating anti-ferromagnet for sufficiently
large $U$. An anti-ferromagnet (AFM) breaks translation symmetry, so at first
sight the general formalism introduced above, which explicitly assumes
translation invariance, does not seem to apply. However, an AFM is invariant
under the combined action of translating by one lattice site and flipping all
the spins. Let's call this modified translation symmetry $T_{x/y}^{\prime}$.
If we consider the AFM state with periodic boundary conditions, i.e. on a
torus of size $L_{x}\times L_{y}$ (note that both $L_{x}$ and $L_{y}$ have to
be even), then we see that
$T_{x}^{{}^{\prime}L_{x}}=T_{y}^{{}^{\prime}L_{y}}=\mathds{1}$. This shows
that the eigenvalues of $T^{\prime}_{x}$ ($T^{\prime}_{y}$) are phases
$e^{i\tilde{k}_{x}}$ ($e^{i\tilde{k}_{y}})$, with $\tilde{k}_{x}=\frac{2\pi
n}{L_{x}}$ ($\tilde{k}_{y}=\frac{2\pi n}{L_{y}}$), just as for the
conventional translation operators. Because the AFM is invariant under
$T^{\prime}_{x/y}$, the correlation functions satisfy $\langle
c^{\dagger}_{\tilde{{\mathbf{k}}}}c_{\tilde{{\mathbf{k}}}^{\prime}}\rangle\propto\delta_{\tilde{{\mathbf{k}}},\tilde{{\mathbf{k}}}^{\prime}}$,
i.e. the single-particle density matrix is diagonal in $\tilde{{\mathbf{k}}}$.
So by working in the basis where $T^{\prime}_{x/y}$ is diagonal, we can treat
the AFM in the same way as conventional translationally invariant states, by
making use of the conserved pseudo-momentum $\tilde{{\mathbf{k}}}$.

Let us assume without loss of generality that the AFM order is in the XY
plane. $T^{\prime}_{x/y}$ can then be defined as a translation followed by a
$\pi$ rotation along the $z$ axis:

$T^{\prime}_{{\mathbf{r}}}=e^{i{\mathbf{r}}\cdot{\mathbf{Q}}\,s^{z}/2}T_{{\mathbf{r}}}\,,$
(75)

where ${\mathbf{r}}=(n,m)$ with $n,m\in\mathbb{Z}$ describes a general lattice
vector, ${\mathbf{Q}}=(\pi,\pi)$, and $s^{i}$ are the Pauli matrices acting on
the spin indices. In this case, the connection between the pseudo-momentum
$\tilde{{\mathbf{k}}}$ and the crystal momentum ${\mathbf{k}}$ is given by:

$\displaystyle c^{\dagger}_{\tilde{{\mathbf{k}}},\uparrow}$ $\displaystyle=$
$\displaystyle c^{\dagger}_{{\mathbf{k}}-{\mathbf{Q}}/2,\uparrow}$ (76)
$\displaystyle c^{\dagger}_{\tilde{{\mathbf{k}}},\downarrow}$ $\displaystyle=$
$\displaystyle c^{\dagger}_{{\mathbf{k}}+{\mathbf{Q}}/2,\downarrow}$ (77)

The self-consistent Hartree-Fock mean-field Hamiltonian is diagonal in the
pseudo-momentum $\tilde{{\mathbf{k}}}$, and, assuming AFM order along the
$x$-direction, can be written as

$H_{HF}=\sum_{\tilde{{\mathbf{k}}}}c^{\dagger}_{\tilde{{\mathbf{k}}}}\left(\begin{matrix}\varepsilon_{\tilde{{\mathbf{k}}},\uparrow}&\Delta_{\tilde{{\mathbf{k}}}}\\\
\Delta_{\tilde{{\mathbf{k}}}}&\varepsilon_{\tilde{{\mathbf{k}}},\downarrow}\end{matrix}\right)c_{\tilde{{\mathbf{k}}}}\,,$
(78)

where
$c^{\dagger}_{\tilde{{\mathbf{k}}}}=(c^{\dagger}_{\tilde{{\mathbf{k}}},\uparrow},c^{\dagger}_{\tilde{{\mathbf{k}}},\downarrow})$.
We have numerically solved the Hartree-Fock self-consistency equations on a
$34\times 34$ pseudo-momentum grid using $U=5$, and we find an AFM order
parameter $\Delta_{\tilde{{\mathbf{k}}}}=\Delta\approx 1.93$.

Figure 3: Goldstone energy $\omega_{\tilde{{\mathbf{q}}}}$, obtained by
solving the Bethe-Salpeter equation (28) for the square-lattice Hubbard model
with $U=5$ on a $34\times 34$ system, as a function of the pseudo-momentum.

We next obtain the Goldstone mode energies and wavefunctions by numerically
solving the generalized eigenvalue equation in Eq. (28), where each momentum
label ${\mathbf{k}}$ is to be replaced with a pseudo-momentum label
$\tilde{{\mathbf{k}}}$. The numerically obtained energy for the lowest-energy
collective mode (i.e. the Goldstone mode) is shown in Fig. 3 over the entire
pseudo-momentum Brillouin zone, and along two cuts $\tilde{q}_{y}=0$ and
$\tilde{q}_{y}=\pi$. As expected, we find two linearly dispersing Goldstone
modes, one at pseudo-momentum $\tilde{{\mathbf{q}}}=(0,0)$, and one at
$\tilde{{\mathbf{q}}}=(\pi,\pi)$. As the AFM order is along the $x$-direction,
we can understand this by noting that the two broken-symmetry generators
$s^{z}$ and $s^{y}$ at crystal momentum ${\mathbf{q}}=(0,0)$ respectively have
pseudo-momentum $\tilde{{\mathbf{q}}}=(0,0)$ and
$\tilde{{\mathbf{q}}}=(\pi,\pi)$. From a fit to the linear part of the
dispersion relation $\omega_{\tilde{{\mathbf{q}}}}$, we find a Goldstone
velocity $c\approx 0.988$.

###  VII.2 Electron-Goldstone scattering vertices

By solving the generalized eigenvalue equation (28), we also obtain the
collective mode wavefunctions
$\varphi^{\pm}_{\tilde{{\mathbf{q}}},\alpha\beta}(\tilde{{\mathbf{k}}})$,
which can be used to construct the electron-Golstone scattering vertices
$g_{\tilde{{\mathbf{q}}},\alpha\beta}(\tilde{{\mathbf{k}}})$ and
$f_{\tilde{{\mathbf{q}}},\alpha\beta}(\tilde{{\mathbf{k}}})$ as explained in
Sec. V. Recall that $\alpha$ and $\beta$ label the mean-field bands, i.e. the
eigenstates of $H_{HF}$ defined in Eq. (78), and not the spin indices.

To go to the rotating frame, we use

$R({\mathbf{r}})=\exp\left(-i\left[\phi_{z}({\mathbf{r}})s^{z}+\phi_{y}({\mathbf{r}})(-1)^{{\mathbf{Q}}\cdot{\mathbf{r}}}s^{y}\right]\right)\,,$
(79)

where $\phi_{z}({\mathbf{r}})$ and $\phi_{y}({\mathbf{r}})$ contain pseudo-
momenta $\tilde{{\mathbf{k}}}$ which lie in the magnetic Brillouin zone, i.e.
the Brillouin zone with reciprocal vectors $(\pi,\pm\pi)$. It thus follows
that $Q_{\tilde{{\mathbf{q}}},\alpha\beta}(\tilde{{\mathbf{k}}})$, as defined
in Eq. (64), is given by

$Q_{\tilde{{\mathbf{q}}},\alpha\beta}(\tilde{{\mathbf{k}}})=\langle
u_{\alpha}(\tilde{{\mathbf{k}}})|s^{z}|u_{\beta}(\tilde{{\mathbf{k}}}-\tilde{{\mathbf{q}}})\rangle\,,$
(80)

if $\tilde{{\mathbf{q}}}$ lies in the first magnetic Brillouin zone, and

$Q_{\tilde{{\mathbf{q}}},\alpha\beta}(\tilde{{\mathbf{k}}})=\langle
u_{\alpha}(\tilde{{\mathbf{k}}})|s^{y}|u_{\beta}(\tilde{{\mathbf{k}}}-\tilde{{\mathbf{q}}})\rangle\,,$
(81)

if $\tilde{{\mathbf{q}}}$ lies in the complement of the first magnetic
Brillouin zone in the full pseudo-momentum Brillouin zone. From Eq. (78), it
is clear that the mean-field states satisfy
$|u_{\alpha}(\tilde{{\mathbf{k}}}+{\mathbf{Q}})\rangle=\pm
s^{x}|u_{\alpha}(\tilde{{\mathbf{k}}})\rangle$ (assuming that we work in a
gauge where the mean-field states are real). From this it follows that
$Q_{\tilde{{\mathbf{q}}}}$ is periodic under shifts by ${\mathbf{Q}}$ and
multiplication by $i$, up to gauge-dependent factors $\pm 1$.

The rotating frame is introduced to ensure that the scattering vertex
$g^{R}_{\tilde{{\mathbf{q}}},\alpha\beta}(\tilde{{\mathbf{k}}})$, defined in
Eq. (66), is zero as $\tilde{{\mathbf{q}}}\rightarrow 0,{\mathbf{Q}}$. As
explained in Sec. V.1, this requires both a rescaling of the collective mode
wavefunctions by a factor $w^{1/2}$, and a gauge fixing procedure. The gauge
fixing procedure is simplified by choosing AFM order along the $x$-direction,
such that the collective mode wavefunctions $\varphi_{\tilde{{\mathbf{q}}}}$
can initially be taken to be real. We then fix the gauge of the
$\varphi_{\tilde{{\mathbf{q}}}}$ by multiplication with $\pm i$ when
$\tilde{{\mathbf{q}}}$ lies in the first magnetic Brillouin zone, and $\pm 1$
otherwise, such that the phase of an arbitrary off-diagonal element in
$\varphi_{\tilde{{\mathbf{q}}},\alpha\beta}(0)$ agrees with the phase of the
corresponding off-diagonal element in
$iQ_{\tilde{{\mathbf{q}}},\alpha\beta}(0)[n_{\alpha}(0)-n_{\beta}(\tilde{{\mathbf{q}}})]$.
After the gauge fixing, we find that $\lim_{\tilde{{\mathbf{q}}}\rightarrow
0,{\mathbf{Q}}}g^{R}_{\tilde{{\mathbf{q}}}}=0$ if we take $w\approx 0.922$.

Figure 4: Averaged electron-Goldstone scattering vertices $\langle
g_{\tilde{{\mathbf{q}}}}\rangle$. (a) Intra-band vertex in the mean-field
basis [Eq. (82)]. (b) Inter-band vertex in the mean-field basis [Eq. (83)].
(c) Intra-band vertex in the rotating frame [Eq. (84)]. (d) Inter-band vertex
in the rotating frame [Eq. (85)]. Figure 5: Averaged electron-Goldstone
scattering vertices $\langle f_{\tilde{{\mathbf{q}}}}\rangle$. (a) Intra-band
vertex in the mean-field basis. (b) Inter-band vertex in the mean-field basis.
(c) Intra-band vertex in the rotating frame. (d) Inter-band vertex in the
rotating frame.

To illustrate our numerical results for the electron-Goldstone scattering
vertices, we define

$\displaystyle\langle g_{{\mathbf{q}},D}\rangle^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{L_{x}L_{y}}\sum_{\tilde{{\mathbf{k}}}}|g_{\tilde{{\mathbf{q}}},00}(\tilde{{\mathbf{k}}})|^{2}+|g_{\tilde{{\mathbf{q}}},11}(\tilde{{\mathbf{k}}})|^{2}\,,$
(82) $\displaystyle\langle g_{{\mathbf{q}},O}\rangle^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{L_{x}L_{y}}\sum_{\tilde{{\mathbf{k}}}}|g_{\tilde{{\mathbf{q}}},01}(\tilde{{\mathbf{k}}})|^{2}+|g_{\tilde{{\mathbf{q}}},10}(\tilde{{\mathbf{k}}})|^{2}\,,$
(83)

as a pseudo-momentum averaged version of respectively the intra-band
(diagonal) and inter-band (off-diagonal) scattering vertices in the mean-field
basis. Similarly, we define the averaged scattering vertices in the rotating
frame as

$\displaystyle\langle g^{R}_{{\mathbf{q}},D}\rangle^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{L_{x}L_{y}}\sum_{\tilde{{\mathbf{k}}}}|g^{R}_{\tilde{{\mathbf{q}}},00}(\tilde{{\mathbf{k}}})|^{2}+|g^{R}_{\tilde{{\mathbf{q}}},11}(\tilde{{\mathbf{k}}})|^{2}\,,$
(84) $\displaystyle\langle g^{R}_{{\mathbf{q}},O}\rangle^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{L_{x}L_{y}}\sum_{\tilde{{\mathbf{k}}}}|g^{R}_{\tilde{{\mathbf{q}}},01}(\tilde{{\mathbf{k}}})|^{2}+|g^{R}_{\tilde{{\mathbf{q}}},10}(\tilde{{\mathbf{k}}})|^{2}\,.$
(85)

In Fig. 4 (a) and (b) we show the averaged scattering vertices in the mean-
field basis. As expected, the intra-band scattering vertex $\langle
g_{\tilde{{\mathbf{q}}},D}\rangle$ goes to zero as
$\tilde{{\mathbf{q}}}\rightarrow 0,{\mathbf{Q}}$, whereas the inter-band
vertex $\langle g_{\tilde{{\mathbf{q}}},O}\rangle$ remains non-zero. We see
that $\langle g_{\tilde{{\mathbf{q}}}=0,O}\rangle\approx U=5$, which
illustrates the strong inter-band scattering for mean-field electrons induced
by $\tilde{{\mathbf{q}}}=0,{\mathbf{Q}}$ Goldstone modes. Fig. 4 (c-d) shows
the averaged scattering vertices in the rotating frame. From Fig. 4 (d) we see
that the inter-band scattering in the rotating frame indeed goes to zero as
$\tilde{{\mathbf{q}}}\rightarrow 0,{\mathbf{Q}}$, and is now maximal at the
magnetic Brillouin zone boundary. We also see that the intra-band scattering
vertex $\langle g^{R}_{\tilde{{\mathbf{q}}}}\rangle$ is slightly enhanced in
the rotating frame, and is almost identical (up to an overall scale factor) to
the Goldstone dispersion $\omega_{\tilde{{\mathbf{q}}}}$ shown in Fig. 3.

We can also similarly define the averaged intra/inter-band scattering vertices
$\langle f_{\tilde{{\mathbf{q}}},D/O}\rangle$ and $\langle
f^{R}_{\tilde{{\mathbf{q}}},D/O}\rangle$ in the mean-field and rotating frame
respectively. These quantities are shown in Fig. 5. Both the intra- and inter-
band scattering vertices become smaller in the rotating frame, are maximal
near the magnetic Brillouin zone boundary, but remain non-zero and order one
as $\tilde{{\mathbf{q}}}\rightarrow 0,{\mathbf{Q}}$.

###  VII.3 Electron spectral function

In this last section, we go back to the original crystal momentum basis, and
again assume that the AFM order is along the $x$-direction. The free fermion
part of the effective action can then be written as

$\begin{split}S_{\psi}=\int_{0}^{\beta}\mathrm{d}\tau&\sum_{\mathbf{k}}\sum_{s}\bar{\psi}_{{\mathbf{k}},s}(\partial_{\tau}+\varepsilon_{\mathbf{k}})\psi_{{\mathbf{k}},s}\\\
&+\Delta(\bar{\psi}_{{\mathbf{k}},\uparrow}\psi_{{\mathbf{k}}+{\mathbf{Q}},\downarrow}+\bar{\psi}_{{\mathbf{k}}+{\mathbf{Q}},\downarrow}\psi_{{\mathbf{k}},\uparrow})\,.\end{split}$
(86)

We will assume that these fermion fields are already in the rotated frame,
such that their scattering with the Goldstone modes vanishes as
${\mathbf{q}}\rightarrow 0$. The components of the fermion Green's function in
the rotated frame which are diagonal in the spin-$z$ indices are given by

$\left[G^{R}_{\uparrow\uparrow}(i\omega,{\mathbf{k}})\right]^{-1}=\left[G^{R}_{\downarrow\downarrow}(i\omega,{\mathbf{k}})\right]^{-1}=i\omega-\varepsilon_{\mathbf{k}}-\frac{\Delta^{2}}{i\omega-\varepsilon_{{\mathbf{k}}+{\mathbf{Q}}}}\,.$
(87)

In experiments one obviously does not measure the electrons in the rotated
frame. One measures the original physical fermions in the unrotated frame,
which are given by $R^{\dagger}({\mathbf{r}})\bar{\psi}_{{\mathbf{r}}}$, with

$R({\mathbf{r}})=\exp\left(i[\varphi_{y}({\mathbf{r}})s^{y}+\varphi_{z}({\mathbf{r}})s^{z}]/2\right)\,,$
(88)

where $\varphi_{y}$ and $\varphi_{z}$ are the Goldstone boson fields. To
encode the compactness of the Goldstone modes we need to write the quadratic
Goldstone mode action in terms of a general SU$(2)$ matrix $R({\mathbf{r}})$
(and not in terms of $\varphi_{y}$ and $\varphi_{z}$). For this, it is easiest
to start with the continuum version of the Goldstone mode action

$S_{G}=\frac{1}{2}\int\mathrm{d}\tau\int\mathrm{d}^{2}{\mathbf{r}}\sum_{n=y,z}\chi(\partial_{\tau}\varphi_{n})^{2}+\rho(\nabla\varphi_{n})^{2}\,.$
(89)

As a next step, we note that

$R^{\dagger}(-i\partial_{\mu})R=\frac{1}{2}\sum_{n=y,z}\partial_{\mu}\varphi_{n}s^{n}+\mathcal{O}(\varphi^{2})$
(90)

As a result, the following action agrees with Eq. (89) to lowest order in
$\varphi$:

$S_{G}=\frac{1}{2}\int_{\tau,{\mathbf{r}}}\sum_{n=y,z}\chi\,\left[\text{tr}(R^{\dagger}i\partial_{\tau}Rs^{n})\right]^{2}+\rho\,\left[\text{tr}(R^{\dagger}i\nabla
Rs^{n})\right]^{2}$ (91)

This is the improved action which correctly incorporates the compactness of
the order parameter manifold. Note that this action is invariant under

$R({\mathbf{r}})\rightarrow R({\mathbf{r}})e^{i\theta({\mathbf{r}})s^{z}}\,,$
(92)

with $\theta({\mathbf{r}})$ an arbitrary function. This is the U$(1)$ gauge
invariance which is familiar from the CP1 formulation of the AFM Goldstone
action [24].

The Green's function of the physical fermions is given by

$G_{s_{1}s^{\prime}_{1}}(\tau,{\mathbf{r}})=-\langle
R^{*}_{s_{2}s_{1}}(\tau,{\mathbf{r}})\psi_{s_{2},{\mathbf{r}}}(\tau)R_{s^{\prime}_{2}s^{\prime}_{1}}(0,0)\bar{\psi}_{s^{\prime}_{2}}(0,0)\rangle_{\beta}\,,$
(93)

with repeated indices summed over. As a first approximation, we ignore the
interactions between the electrons and the Goldstone modes, in which case the
Green's function factorizes:

$G_{s_{1}s^{\prime}_{1}}(\tau,{\mathbf{r}})\approx-\langle
R^{*}_{s_{2}s_{1}}(\tau,{\mathbf{r}})R_{s^{\prime}_{2}s^{\prime}_{1}}(0,0)\rangle\langle\psi_{s_{2},{\mathbf{r}}}(\tau)\bar{\psi}_{s^{\prime}_{2}}(0,0)\rangle_{\beta}\,.$
(94)

This is expected to be a reasonable approximation as the fermions in the
rotating frame decouple from the low-energy Goldstone modes.

An approximate expression for the Goldstone propagator can be obtained from a
large-N analysis of the CP1 model [25], which leads to the following result
[13]:

$\langle
R^{*}_{s_{2},s_{1}}(\tau,{\mathbf{r}})R_{s^{\prime}_{2},s^{\prime}_{1}}(0,0)\rangle=-D(\tau,{\mathbf{r}})\delta_{s_{1}s^{\prime}_{1}}\delta_{s_{2},s^{\prime}_{2}}\,,$
(95)

where $D(\tau,{\mathbf{r}})$ is the Fourier transform of

$D(i\nu,{\mathbf{q}})=\frac{\chi^{-1}}{(i\nu)^{2}-c^{2}{\mathbf{q}}^{2}-m^{2}}\,.$
(96)

Here $m^{2}$ is a small mass for the Goldstone modes generated by thermal
fluctuations, which restore the symmetry in $2$ spatial dimensions due to the
Hohenberg-Mermin-Wagner theorem.

Putting everything together, we find that the fermion Green's function is
given by

$\begin{split}&G_{ss^{\prime}}(i\omega,{\mathbf{k}})=-\delta_{ss^{\prime}}T\sum_{i\nu}\frac{1}{L_{x}L_{y}}\times\\\
&\sum_{\mathbf{q}}\frac{\chi^{-1}}{(i\nu)^{2}-\omega_{\mathbf{q}}^{2}-m^{2}}2G^{R}_{\uparrow\uparrow}(i\omega-i\nu,{\mathbf{k}}-{\mathbf{q}})\end{split}$
(97)

This result has previously also been obtained by Borejsza and Dupuis [13]
starting from the rotating-frame mean-field approach motivated by Schultz
[12]. To simplify Eq. (97), we first rewrite $G^{R}_{\uparrow\uparrow}$ as

$\displaystyle G^{R}_{\uparrow\uparrow}(i\omega,{\mathbf{k}})$
$\displaystyle=$
$\displaystyle\frac{i\omega-\varepsilon_{{\mathbf{k}}+{\mathbf{Q}}}}{(i\omega-
E_{\mathbf{k}}^{+})(i\omega-E_{\mathbf{k}}^{-})}$ (98) $\displaystyle=$
$\displaystyle\frac{|u_{+}({\mathbf{k}})|^{2}}{i\omega-
E_{{\mathbf{k}}}^{+}}+\frac{|u_{-}({\mathbf{k}})|^{2}}{i\omega-
E_{{\mathbf{k}}}^{-}}\,,$ (99)

where

$\displaystyle E_{\mathbf{k}}^{\pm}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\varepsilon_{\mathbf{k}}+\varepsilon_{{\mathbf{k}}+{\mathbf{Q}}}\pm\sqrt{(\varepsilon_{\mathbf{k}}-\varepsilon_{{\mathbf{k}}+{\mathbf{Q}}})^{2}+4\Delta^{2}}\right)\,,$
$\displaystyle|u_{\pm}({\mathbf{k}})|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(1\pm\frac{\varepsilon_{{\mathbf{k}}}-\varepsilon_{{\mathbf{k}}+{\mathbf{Q}}}}{\sqrt{(\varepsilon_{\mathbf{k}}-\varepsilon_{{\mathbf{k}}+{\mathbf{Q}}})^{2}+4\Delta^{2}}}\right)\,.$
(100)

It is now straightforward to perform the summation over $i\nu$ in Eq. (97),
which gives

$\begin{split}G_{ss^{\prime}}(i\omega,{\mathbf{k}})=&\frac{\delta_{ss^{\prime}}}{L_{x}L_{y}}\sum_{\mathbf{q}}\sum_{\sigma=\pm}\frac{|u_{\sigma}({\mathbf{k}}-{\mathbf{q}})|^{2}}{\chi\tilde{\omega}_{\mathbf{q}}}\times\\\
&\left(\frac{n(\tilde{\omega}_{\mathbf{q}})+f(-E^{\sigma}_{{\mathbf{k}}-{\mathbf{q}}})}{i\omega-E^{\sigma}_{{\mathbf{k}}-{\mathbf{q}}}-\tilde{\omega}_{\mathbf{q}}}+\frac{n(\tilde{\omega}_{\mathbf{q}})+f(E^{\sigma}_{{\mathbf{k}}-{\mathbf{q}}})}{i\omega-E^{\sigma}_{{\mathbf{k}}-{\mathbf{q}}}+\tilde{\omega}_{\mathbf{q}}}\right)\,,\end{split}$
(101)

where $n(\omega)$ and $f(E)$ are respectively the Bose-Einstein and Fermi-
Dirac distributions, and
$\tilde{\omega}_{\mathbf{q}}=\sqrt{c^{2}{\mathbf{q}}^{2}+m^{2}}$.

The spectral function, defined for spin-rotation invariant systems as,

$\mathcal{A}(\omega,{\mathbf{k}})=\frac{2}{\pi}\text{Im}\,G_{\uparrow\uparrow}(\omega-i\epsilon,{\mathbf{k}})$
(102)

is now easily obtained [13]:

$\begin{split}\mathcal{A}&(\omega,{\mathbf{k}})=\frac{2}{L_{x}L_{y}}\sum_{\mathbf{q}}\sum_{\sigma=\pm}\frac{|u_{\sigma}({\mathbf{k}}-{\mathbf{q}})|^{2}}{\chi\tilde{\omega}_{\mathbf{q}}}\times\\\
&\frac{\epsilon}{\pi}\left(\frac{n(\tilde{\omega}_{\mathbf{q}})+f(-E^{\sigma}_{{\mathbf{k}}-{\mathbf{q}}})}{(\omega-E^{\sigma}_{{\mathbf{k}}-{\mathbf{q}}}-\tilde{\omega}_{\mathbf{q}})^{2}+\epsilon^{2}}+\frac{n(\tilde{\omega}_{\mathbf{q}})+f(E^{\sigma}_{{\mathbf{k}}-{\mathbf{q}}})}{(\omega-E^{\sigma}_{{\mathbf{k}}-{\mathbf{q}}}+\tilde{\omega}_{\mathbf{q}})^{2}+\epsilon^{2}}\right)\,.\end{split}$
(103)

In Fig. 6 (a) we show the numerically obtained spectral weight
$\mathcal{A}(\omega,{\mathbf{k}})$ at frequency $\omega=-1.73$. To obtain this
result, we have kept a small non-zero $\epsilon=T=0.05$ to smear out the
numerical results obtained on a finite-size discrete momentum grid. We see
that at this negative frequency, the spectral weight is located along contours
demarcating the boundaries of `hole pockets' centered at
$(\pm\pi/2,\pm\pi/2)$. The part of the contour oriented towards the center of
the Brillouin zone is brighter than the backside facing $(\pi,\pi)$ as a
result of the coherence factors $|u_{\sigma}({\mathbf{k}})|^{2}$ in Eq. (103).

Fluctuations will cause the AFM order to decrease. To illustrate the effect of
a reduced order in the rotating frame on the spectral function, we show the
spectral weight at $\omega=-0.53$ obtained using a smaller $\Delta=0.5$ in
Fig. 6 (b). Note that both in Fig. 6 (a) and (b), the area inside the `hole
pockets' is $\sim 10\%$ of the total Brillouin zone area. From Fig. 6 (b) we
see that reducing $\Delta$ elongates the contours of high spectral weight in
the directions toward $(0,\pm\pi)$ and $(\pm\pi,0)$. Also the spectral weight
on the backside of the contours is further reduced.

Figure 6: Spectral weight $\mathcal{A}(\omega,{\mathbf{k}})$ defined in Eq.
(103) with $T=\epsilon=0.05$ and $m=0.002$, normalized such that the maximal
value is one. (a) Spectral weight at frequency $\omega=-1.73$, using the mean-
field value $\Delta\approx 1.93$. (b) Spectral weight at frequency
$\omega=-0.53$, using a renormalized value $\Delta=0.5$. In both (a) and (b),
the total area contained in the pockets centered at $(\pm\pi/2,\pm\pi/2)$ is
$\sim 10\%$ of the total Brillouin zone area. The results were obtained by
numerically interpolating the mean-field results obtained on a $34\times 34$
momentum grid to a $170\times 170$ grid, using a bivariate spline
interpolation.

Before concluding we want to reiterate that the spectral weights shown in
Figs. 6 (a-b) are those of a thermally disordered state, with no long-range
AFM order. Nevertheless, the spectral weight retains important features of the
$T=0$ AFM state, and is strikingly different from the spectral weight of the
conventional Fermi liquid at small $U$.

##  VIII Example II: spin-spiral order in the three-band model

Having discussed the anti-ferromagnetic Mott insulator in the Hubbard model in
the previous section, we now turn to the slightly more involved example of
spin spiral order in the hole-doped three-band Hubbard model [26].

###  VIII.1 Mean-field state and Goldstone mode energies

The three-band model is described by the following Hamiltonian:

$\begin{split}&H=\sum_{i}\epsilon_{d}c^{\dagger}_{1i}c_{1i}+\epsilon_{p}\left(c^{\dagger}_{2i}c_{2i}+c^{\dagger}_{3i}c_{3i}\right)\\\
&+\sum_{\langle
ij\rangle}t^{ij}_{pd}\left(c^{\dagger}_{1i}c_{2j}+c^{\dagger}_{1i}c_{3j}+h.c.\right)+\sum_{\langle
ij\rangle}t^{ij}_{pp}\left(c^{\dagger}_{2i}c_{3j}+h.c.\right)\\\
&+\sum_{i}U_{d}n_{1i\uparrow}n_{1i\downarrow}+U_{p}\left(n_{2i\uparrow}n_{2i\downarrow}+n_{3i\uparrow}n_{3i\downarrow}\right)\,,\end{split}$
(104)

where $a=1,2,3$ respectively refers to the Copper $d_{x^{2}-y^{2}}$, Oxygen
$2p_{x}$ and $2p_{y}$ orbitals, $s$ denotes the spin, and $i,j$ label the unit
cells. Note that following most of the literature on this model, we have
formulated the Hamiltonian in terms of the hole degrees of freedom. The first
two lines in Eq. (104) represent a potential energy difference for the Copper
and Oxygen orbitals, and nearest and next-nearest neighbour hopping (in these
terms, the spin summation is implicit). Note that the signs of the hopping
parameters $t^{ij}_{pd}$ and $t^{ij}_{pp}$ depend on the relative orientation
of the sites $i$ and $j$. See Fig. 7 for a graphical representation of the
different hopping processes in the three-band model, and how the corresponding
signs depend on the orientation. The last line in Eq. (104) contains the on-
site Hubbard interactions for electrons in the Copper and Oxygen orbitals.

$-t_{\text{pd}}$$+t_{\text{pd}}$$+t_{\text{pd}}$$-t_{\text{pd}}$$+t_{\text{pp}}$$-t_{\text{pp}}$$-t_{\text{pp}}$$+t_{\text{pp}}$
Figure 7: Hopping processes in the three-band model. The copper
$d_{x^{2}-y^{2}}$ orbitals are denoted by the black dots while the Oxygen
$2p_{x}$ and $2p_{y}$ orbitals are denoted by the red and blue crosses
respectively. The notation for the hopping amplitudes is the same as in Eq.
(104).

An important parameter in the three-band model is the charge-transfer
parameter $\delta=\epsilon_{p}-\epsilon_{d}>0$. Previous works [27, 28] have
investigated the groundstate properties of the three-band model for a range of
values of $\delta$ using different numerical methods, including Hartree Fock,
and identified a parameter regime with spin-spiral order. To make contact with
the results of [27], we choose the following values for the parameters of the
three-band model:
$U_{d}=8.4,U_{p}=2.0,\epsilon_{d}=-7.6,\epsilon_{p}=-6.1,|t^{ij}_{pd}|=1.2,|t^{ij}_{pp}|=0.7$.
Note that for these parameter values the charge-transfer parameter is
$\delta=1.5$. At half filling, these parameters lead to an insulating AFM
ground state, with orbital hole densities $n_{1}\approx
0.46$,$n_{2}=n_{3}\approx 0.27$. Upon hole doping, the anti-ferromagnet
becomes a circular spin spiral with wave vector
${\mathbf{Q}}=(\pi,\pi-2\pi\eta)$ or ${\mathbf{Q}}=(\pi-2\pi\eta,\pi)$, where
$\eta$ is the incommensurability of the spiral. In this work we investigate
the ground state at a hole doping of $1/8$.

Let us assume without loss of generality that the spiral order is in the XY
plane. In this case, the spin expectation value is given by

$\langle\mathbf{S}_{i}\rangle=\Delta\left(\cos({\mathbf{Q}}_{\text{sp}}\cdot\mathbf{r}_{i})\mathbf{\hat{x}}+\sin({\mathbf{Q}}\cdot\mathbf{r}_{i})\mathbf{\hat{y}}\right)\,,$
(105)

where
$\mathbf{S}_{i}=\frac{1}{2}\sum_{a,s,s^{\prime}}c^{\dagger}_{j,s,a}\mathbf{\sigma}_{ss^{\prime}}c_{j,s^{\prime},a}$.
Similar to the AFM, the circular spiral order breaks both SU$(2)$ spin
symmetry and translation symmetry, but is invariant under the modified
translation symmetry

$T^{\prime}_{{\mathbf{r}}}=e^{i{\mathbf{r}}\cdot{\mathbf{Q}}s^{z}/2}T_{{\mathbf{r}}}\,,$
(106)

which allows us to define a conserved pseudo-momentum $\tilde{\mathbf{k}}$ as
in Eq.(77).

The self-consistent Hartree-Fock mean-field Hamiltonian can then be written as

$H_{HF}=\sum_{\tilde{{\mathbf{k}}}}c^{\dagger}_{\tilde{{\mathbf{k}}}}\left(\begin{matrix}\varepsilon_{\tilde{{\mathbf{k}}},\uparrow}&\Delta_{\tilde{{\mathbf{k}}}}\\\
\Delta_{\tilde{{\mathbf{k}}}}&\varepsilon_{\tilde{{\mathbf{k}}},\downarrow}\end{matrix}\right)c_{\tilde{{\mathbf{k}}}}\,,$
(107)

where
$c^{\dagger}_{\tilde{{\mathbf{k}}}}=(c^{\dagger}_{1,\tilde{{\mathbf{k}}},\uparrow},c^{\dagger}_{2,\tilde{{\mathbf{k}}},\uparrow},c^{\dagger}_{3,\tilde{{\mathbf{k}}},\uparrow},c^{\dagger}_{1\tilde{{\mathbf{k}}},\downarrow},c^{\dagger}_{2\tilde{{\mathbf{k}}},\downarrow},c^{\dagger}_{3\tilde{{\mathbf{k}}},\downarrow})$
and
$\Delta_{\tilde{{\mathbf{k}}}}=\text{diag(}\Delta_{1,{\tilde{\mathbf{k}}}},\Delta_{2,{\tilde{\mathbf{k}}}},\Delta_{3,{\tilde{\mathbf{k}}}}\text{)}$.
Here, $\varepsilon_{\tilde{{\mathbf{k}}}}$ are not the bare band energies
since the contribution from the Hartree term is not just a simple overall
energy shift, but rather, each orbital gets shifted by a different amount,
corresponding to the orbital's hole density.

To solve the HF self-consistency equations, we have applied the following
procedure. First, we used restricted HF with conserved pseudo-momentum, and
minimized the energy for different values of ${\mathbf{Q}}$. Even though the
optimal ${\mathbf{Q}}$ varies slightly with system size, it was consistently
found to be ${\mathbf{Q}}\approx(\pi,0.80\pi)$ (assuming spiral order along
the $y$-direction). As a next step we used the optimal solution of the
restricted HF as an initial seed for completely unrestricted HF, which did not
assume any translational symmetry. The converged unrestricted HF yielded
ground states with the same spiral order as the restricted HF, but with a
small $\delta n/n\sim 10^{-5}-10^{-3}$ charge density modulation, which may be
a result of the incommensurability of the exact optimal ${\mathbf{Q}}$ with
the finite system size. The difference in energy between the spiral states
with and without charge density modulation was found to decrease with system
size. Our results are thus consistent with a circular spin spiral state with
${\mathbf{Q}}=(\pi,\pi-2\pi\eta)$ and $\eta\approx 0.10$, in agreement with
Refs. [27, 29]. The spin hybridization parameters in the mean-field
Hamiltonian were found to be
$\left(\Delta_{1,{\tilde{\mathbf{k}}}},\Delta_{2,{\tilde{\mathbf{k}}}},\Delta_{3,{\tilde{\mathbf{k}}}}\right)=\left(\Delta_{1},\Delta_{2},\Delta_{3}\right)\approx\left(1.31,0.00,0.013\right)$.

The spiral order has three broken symmetry generators, and three Goldstone
modes [30, 31]. Two of the modes are associated with rotating the plane of the
spiral order (out-of-plane modes), and one mode corresponds to in-plane
rotations. For a $XY$ spiral, the out-of-plane modes are related to the broken
$s^{x},s^{y}$ generators while the in-plane mode is related to the broken
$s^{z}$ generator. We can understand the location of the Goldstone modes in
the pseudo-momentum frame by relating them to the original frame, where they
are all at ${\mathbf{q}}=(0,0)$. The in-plane mode remains at zero,
$\tilde{{\mathbf{q}}}=(0,0)$, while the two out-of-plane modes go to
$\tilde{{\mathbf{q}}}=\pm{\mathbf{Q}}$. We obtain the collective mode spectrum
by numerically solving Eq.(28) in the pseudo-momentum basis. This was first
done over the entire pseudo-momentum Brillouin zone on a $24\times 24$ grid,
and subsequently on a larger $40\times 40$ grid near the locations of the
Goldstone modes. The energies for the low-energy collective modes are shown in
Fig.8, near $\tilde{{\mathbf{q}}}=(0,0)$ and
$\tilde{{\mathbf{q}}}=\pm{\mathbf{Q}}$. We see that the Goldstone modes near
$\pm{\mathbf{Q}}$, which correspond to the out-of-plane fluctuations, lie
outside the particle-hole continuum and hence will not be damped. The
Goldstone mode near $\tilde{{\mathbf{q}}}$, on the other hand, does lie in the
particle-hole continuum, and hence will be Landau damped. These observations
agree with Refs. [30, 31, 32]. The velocities of the out-of-plane Goldstone
modes are anisotropic, and from a linear fit we find $c_{x}\approx 0.32$ and
$c_{y}\approx 0.27$.

Figure 8: Plots of the collective modes of the spin spiral state on a
$40\times 40$ grid, across different cuts of the Brillouin Zone. The red
points show the Goldstone modes where they can be distinguished from the
particle-hole continuum . The linear low-energy dispersion of the Goldstone
modes near $\tilde{{\mathbf{q}}}=\pm{\mathbf{Q}}=(\pi,\pm 0.8\pi)$ is clear in
(a), (b) and (c). In (d), the Goldstone mode at $\tilde{{\mathbf{q}}}=(0,0)$
is dissolved in the particle-hole continuum.

###  VIII.2 Electron spectral function

Working in the original crystal momentum basis, and taking the spiral order to
be in the XY plane, the fermion action is given by form

$\begin{split}&S_{\psi}=\int_{0}^{\beta}\mathrm{d}\tau\sum_{\mathbf{k}}\sum_{s,a,b}\bar{\psi}_{{\mathbf{k}},s,a}(\delta_{ab}\partial_{\tau}+\varepsilon_{{\mathbf{k}},ab})\psi_{{\mathbf{k}},s,b}\\\
&+\sum_{a}\Delta_{a}(\bar{\psi}_{{\mathbf{k}},\uparrow,a}\psi_{{\mathbf{k}}+{\mathbf{Q}},\downarrow,b}+\bar{\psi}_{{\mathbf{k}}+{\mathbf{Q}},\downarrow,b}\psi_{{\mathbf{k}},\uparrow,a})\,.\end{split}$
(108)

As before, we assume that these fermion fields are already in the rotated
frame. In a basis
$(\psi_{{\mathbf{k}},\uparrow},\psi_{{\mathbf{k}}+{\mathbf{Q}},\downarrow})$,
suppressing the orbital indices, the rotated-frame Green's function is given
by

$[G^{R}(i\omega,{\mathbf{k}})]^{-1}=\begin{pmatrix}i\omega\mathbb{1}-\varepsilon_{{\mathbf{k}}}&-\Delta\\\
-\Delta&i\omega\mathbb{1}-\varepsilon_{{\mathbf{k}}+{\mathbf{Q}}}\\\
\end{pmatrix}\,,$ (109)

where $\Delta,\varepsilon_{{\mathbf{k}}}$ are $3\times 3$ matrices with
orbital indices, with values as described in the previous section.

The spin-diagonal part of the Green's function can be written as

$G^{R\hskip 1.42271ptab}_{\hskip
7.11317ptss}(i\omega,{\mathbf{k}})=\sum_{\alpha}\frac{u^{as}_{\alpha}({\mathbf{k}})u^{bs\hskip
0.28436pt*}_{\alpha}({\mathbf{k}})}{i\omega-E_{\alpha}({\mathbf{k}})}\,,$
(110)

which is the equivalent of Eq. (98) in the one-band Hubbard model case. Note
that for the spiral order, it no longer holds that
$G^{R}_{\uparrow\uparrow}(i\omega,{\mathbf{k}})=G^{R}_{\downarrow\downarrow}(i\omega,{\mathbf{k}})$.

The microscopic fermions $R^{\dagger}({\mathbf{r}})\bar{\psi}_{{\mathbf{r}}}$
are obtained by acting with

$\begin{split}R({\mathbf{r}})&=\exp\left(i[\varphi_{y}({\mathbf{r}})\mathbb{1}\otimes
s^{y}+\varphi_{z}({\mathbf{r}})\mathbb{1}\otimes s^{z}]/2\right)\\\
&=\exp\left(i[\varphi_{x}({\mathbf{r}})\bar{s}^{x}+\varphi_{y}({\mathbf{r}})\bar{s}^{y}+\varphi_{z}({\mathbf{r}})\bar{s}^{z}]/2\right)\,,\end{split}$
(111)

on the fermions in the rotating frame. Here we have defined
$\bar{s}^{i}=\mathbb{1}\otimes s^{i}$ are the correct generators which acts
trivially on the orbital indices. The quadratic Goldstone action is then given
by

$\begin{split}S_{G}=\frac{1}{2}\int\mathrm{d}\tau&\int\mathrm{d}^{2}{\mathbf{r}}\sum_{n=x,y}\left(\chi^{\perp}(\partial_{\tau}\varphi_{n})^{2}+\sum_{i=x,y}\rho^{\perp}_{i}(\partial_{i}\varphi_{n})^{2}\right)\\\
&+\chi^{\Box}(\partial_{\tau}\varphi_{z})^{2}+\sum_{i=x,y}\rho^{\Box}_{i}(\partial_{i}\varphi_{z})^{2}\,,\end{split}$
(112)

where the out-of-plane and in-plane parameters are denoted by $\perp$ and
$\Box$. Note that terms involving $\partial_{i}\varphi\partial_{j}\varphi$
with $i\neq j$ are forbidden by the reflection symmetry of the spiral state.

By again using Eq. (90), we obtain following non-linear sigma action for the
Goldstone modes:

$\begin{split}S_{G}=&\frac{1}{2}\int_{\tau,{\mathbf{r}}}\sum_{n=x,y}\chi^{\perp}\left[\text{tr}(R^{\dagger}i\partial_{\tau}Rs^{n}\right]^{2}+\rho_{j}^{\perp}\left[\text{tr}(R^{\dagger}i\partial_{j}Rs^{n}\right]^{2}\\\
&+\chi^{\Box}\left[\text{tr}(R^{\dagger}i\partial_{\tau}Rs^{z}\right]^{2}+\rho_{j}^{\Box}\left[\text{tr}(R^{\dagger}i\partial_{j}Rs^{z}\right]^{2}\,,\end{split}$
(113)

where the sum over $j=x,y$ is implicit. A large-N analysis of the non-linear
sigma model [20], leads to the following result for the Goldstone propagator:

$\langle R^{c,a\hskip
2.84544pt*}_{s_{2},s_{1}}(\tau,{\mathbf{r}})R^{d,b}_{s^{\prime}_{2},s^{\prime}_{1}}(0,0)\rangle=-D(\tau,{\mathbf{r}})\delta_{s_{1}s^{\prime}_{1}}\delta_{s_{2}s^{\prime}_{2}}\delta_{ad}\delta_{cb}\,,$
(114)

where $D(\tau,{\mathbf{r}})$ is the Fourier transform of

$D(i\nu,{\mathbf{q}})=\frac{\hskip
8.5359pt(\chi^{\perp})^{-1}}{(i\nu)^{2}-\tilde{\omega}_{{\mathbf{q}}}^{2}}\,.$
(115)

Here,
$\tilde{\omega}_{{\mathbf{q}}}=\sqrt{\rho^{\perp}_{\alpha}q_{\alpha}^{2}/\chi^{\perp}+m^{2}}$
is the Goldstone mode dispersion, including the exponentially small (in $T$)
spin-gap mass generated by the thermal fluctuations at non-zero temperature,
as determined via the saddle point equations. The velocities
$c_{\alpha}=\sqrt{\rho^{\perp}_{\alpha}/\chi^{\perp}}$ are obtained from the
collective mode spectrum.

Finally, we express the fermion Green's function as

$\begin{split}G^{ab}_{s_{1}s^{\prime}_{1}}(\tau,{\mathbf{r}})&=-\langle
R^{ca\hskip
2.84544pt*}_{s_{2}s_{1}}(\tau,{\mathbf{r}})\psi^{c}_{s_{2}}(\tau,{\mathbf{r}})\bar{\psi}^{d}_{s_{2}^{\prime}}(0,0)R^{db}_{s^{\prime}_{2}s^{\prime}_{1}}(0,0)\rangle_{\beta}\\\
&\approx-\langle R^{ca\hskip
2.84544pt*}_{s_{2}s_{1}}(\tau,{\mathbf{r}})R^{db}_{s^{\prime}_{2}s^{\prime}_{1}}(0,0)\rangle\,G^{\text{R}\hskip
1.42271ptcd}_{\hskip
6.544pts_{2}s_{2}^{\prime}}(\tau,{\mathbf{r}})\end{split}$

where we have again ignored the interaction between the Goldstone modes and
the electrons in the rotating frame, so that the Green's function factorizes.
Going to momentum space, and using Eqs. (114) and (115), we find that the
fermion Green's function is given by

$\begin{split}G^{ab}_{ss^{\prime}}(i\omega,{\mathbf{k}})&=-\delta_{ss^{\prime}}T\sum_{i\nu}\frac{1}{N_{x}N_{y}}\times\\\
&\sum_{{\mathbf{q}}}\frac{\hskip
8.5359pt(\chi^{\perp})^{-1}}{(i\nu)^{2}-\tilde{\omega}_{{\mathbf{q}}}^{2}}\sum_{\sigma,\alpha}\frac{u^{a,\sigma}_{\alpha}({\mathbf{k}}-{\mathbf{q}})u^{b,\sigma\hskip
1.42271pt*}_{\alpha}({\mathbf{k}}-{\mathbf{q}})}{i(\omega-\nu)-E_{{\mathbf{k}}-{\mathbf{q}},\alpha}}\,,\end{split}$
(116)

from which we obtain the spectral weight.

In Fig 9(a) we show the Fermi surfaces of the spiral state mean-field band
structure in the pseudo-momentum Brillouin zone. The band spectrum clearly
breaks $C_{4}$ symmetry, in contrast to the AFM case, but retains a reflection
symmetry about $x$. The spectral weight of the physical electrons involves a
spin sum, as seen in Eq. (116), which adds two copies of the Fermi levels of
Fig 9(a), with a relative shift of ${\mathbf{Q}}$ and the associated coherence
factors. This restores the reflection symmetry about $y$, but not the $C_{4}$
symmetry. To illustrate this, we have calculated the spectral weight of the
following fermions:

$f^{\dagger}_{{\mathbf{r}}}(\theta)=\cos\theta\,c^{\dagger}_{1,{\mathbf{r}}}+\frac{\sin\theta}{2}\left(c^{\dagger}_{2,{\mathbf{r}}}-c^{\dagger}_{3,{\mathbf{r}}}-c^{\dagger}_{2,{\mathbf{r}}-x}+c^{\dagger}_{3,{\mathbf{r}}-y}\right)\,,$
(117)

where the combination of orbitals in this operator is chosen to obey all the
point group symmetries of the three-band model. We have optimized the spectral
weight at the Fermi energy as a function of $\theta$, and found that the
maximal spectral weight occurs at $\theta=\theta^{*}\approx 0.27\pi$. In Fig.
9(b) we plot the spectral weight of $f^{\dagger}_{\mathbf{r}}(\theta^{*})$,
which is indeed reflection symmetric, but breaks $C_{4}$. Similar to the AFM
case discussed above, the highest spectral weight is along four contours with
a suppressed backside due to the coherence factors
$|u^{a}_{\alpha}({\mathbf{k}})|^{2}$. However, compared to the AFM, the
centers of the high spectral-weight contours are shifted along $k_{y}$. Also,
the fraction of the Brillouin zone area contained in the contours is _twice_
the number of doped holes per unit cell relative to half filling, i.e. 1/4 for
the doping that we consider here. The implications of this were previously
discussed in Ref. [33].

Figure 9: (a) Plot of the mean-field Fermi surfaces. (b) Normalized spectral
weight $\mathcal{A}(\omega=0,{\mathbf{k}})$ at the Fermi energy of the
fermions $f^{\dagger}_{\mathbf{r}}(\theta^{*}=0.27\pi)$ as defined in (117),
using $T=\epsilon=0.05$. The results were obtained by interpolating the mean-
field results on a $40\times 40$ momentum grid to a $160\times 160$ grid.

##  IX Conclusions

We have shown that a naive application of mean-field theory + RPA produces
scattering vertices between electrons and Goldstone modes which do not vanish
in the long-wavelength limit. However, fermions which live in a frame that
locally follows the order parameter fluctuations do decouple from
${\mathbf{q}}=0$ Goldstone modes. This has important consequences for electron
correlation functions, which by making use of the rotating frame naturally
reflect the properties of thermally disordered systems which nevertheless
retain features associated with a non-zero magnitude for the order parameter.

Although we have illustrated our formalism only for square-lattice Hubbard and
three-band models, its applicability is far more general. In particular, it
can also be used to study the broken-symmetry states in moiré materials, such
as e.g. the Incommensurate Kekulé Spiral (IKS) state [34, 35, 36] which is
observed experimentally in both magic-angle twisted bilayer [37] and trilayer
[38] graphene. The IKS order is in fact very similar to the circular spin-
spiral order that we have studied in the three-band model in Sec. VIII. We
leave these applications for near-future work.

_Acknowledgements –_ N.B. would like to thank Patrick Ledwidth for helpful
discussions about the manuscript, and Steve Kivelson for bringing Ref. [8] to
our attention. This work was supported by a Leverhulme Trust International
Professorship grant [number LIP-202-014]. For the purpose of Open Access, the
authors have applied a CC BY public copyright licence to any Author Accepted
Manuscript version arising from this submission. The research was also in part
supported by the National Science Foundation under Grant No. NSF PHY-1748958.
N.B. was supported by a University Research Fellowship of the Royal Society,
and thanks École Normale Supérieure Paris, where part of this work was
completed, for hospitality. YH acknowledges support from the European Research
Council (ERC) under the European Union Horizon 2020 Research and Innovation
Programme (Grant Agreement Nos. 804213-TMCS)

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— Supplementary Material —

##  Appendix A Solution of the Bethe-Salpeter equation and its properties

In this appendix we elaborate on the Bethe-Salpeter equation, its solutions,
and their properties. We will closely follow the seminal works [39, 23]. To
ease the presentation, we will consider a general Hartree-Fock action of the
form

$\int_{0}^{1/T}\mathrm{d}\tau\,\sum_{\alpha}\bar{\psi}(\partial_{\tau}+E_{\alpha})\psi_{\alpha}+\frac{1}{2}\sum_{\alpha\beta\lambda\sigma}V_{\alpha\lambda,\beta\sigma}\bar{\psi}_{\alpha}\bar{\psi}_{\lambda}\psi_{\sigma}\psi_{\beta}\,,$
(118)

where the Greek indices run over the Hartree-Fock single-particle states. Note
that compared to the main text we have dropped the momentum labels, which one
can think of as being absorbed into the Greek indices.

As explained in the main text, the RPA collective mode propagator is obtained
by summing the infinite set of diagrams in Eq.(140). Summing this infinite
series is equivalent to solving the Bethe-Salpeter equation, whose graphical
representation we repeat here:

(119)

Written out explicitly, the Bethe-Salpeter (BS) equation is

$\displaystyle G_{2}(\alpha,\beta;\lambda,\sigma;i\nu)$ $\displaystyle=$
$\displaystyle
T\sum_{\omega_{n}}\left(-\delta_{\beta\sigma}\delta_{\lambda\alpha}\frac{1}{i(\omega_{n}-\nu)-E_{\beta}}\frac{1}{i\omega_{n}-E_{\alpha}}\right)$
$\displaystyle+\sum_{\mu\nu}V_{\alpha\mu,\beta\nu}G_{2}(\nu,\mu;\lambda,\sigma;i\nu)T\sum_{\omega_{n}}\frac{1}{i(\omega_{n}-\nu)-E_{\beta}}\frac{1}{i\omega_{n}-E_{\alpha}}$
$\displaystyle-\sum_{\mu\nu}V_{\alpha\mu,\nu\beta}G_{2}(\nu,\mu;\lambda,\sigma;i\nu)T\sum_{\omega_{n}}\frac{1}{i(\omega_{n}-\nu)-E_{\beta}}\frac{1}{i\omega_{n}-E_{\alpha}}$
$\displaystyle=$ $\displaystyle-\frac{f(E_{\beta})-f(E_{\alpha})}{i\nu-
E_{\alpha}+E_{\beta}}\left(\delta_{\alpha\lambda}\delta_{\beta\sigma}-\sum_{\mu\nu}\left(V_{\alpha\mu,\beta\nu}-V_{\alpha\mu,\nu\beta}\right)G_{2}(\nu,\mu;\lambda,\sigma;i\nu)\,\right),$

where $f(E)$ is the Fermi-Dirac distribution. By going to zero temperature we
find that $G_{2}$ satisfies the following equation:

$\displaystyle\sum_{\mu,\nu}\left((n_{\beta}-n_{\alpha})(E_{\alpha}-E_{\beta}-i\nu)\delta_{\alpha\nu}\delta_{\beta\mu}+V_{\alpha\mu,\beta\nu}-V_{\alpha\mu,\nu\beta}\right)G_{2}(\nu,\mu;\lambda,\sigma;i\nu)=\delta_{\alpha\lambda}\delta_{\beta\sigma}\,,$
(121)

where $n_{\alpha}\in\\{0,1\\}$ are the fermion occupation numbers. Again
adopting the notation where $i,j,k,l$ represent occupied states, and $m,n,o,p$
unoccupied states, we can rewrite the BS equation as

$\left(\begin{matrix}A-i\nu\mathds{1}&B\\\
B^{*}&A^{*}+i\nu\mathds{1}\end{matrix}\right)\left(\begin{matrix}G^{(1)}(i\nu)&G^{(2)}(i\nu)\\\
G^{(3)}(i\nu)&G^{(4)}(i\nu)\end{matrix}\right)=\left(\begin{matrix}\mathds{1}&0\\\
0&\mathds{1}\end{matrix}\right)\,,$ (122)

where we have defined

$\displaystyle G^{(1)}_{mi,nj}(i\nu)$ $\displaystyle=$ $\displaystyle
G_{2}(m,i;n,j;i\nu)$ (123) $\displaystyle G^{(2)}_{mi,nj}(i\nu)$
$\displaystyle=$ $\displaystyle G_{2}(m,i;j,n;i\nu)$ (124) $\displaystyle
G^{(3)}_{mi,nj}(i\nu)$ $\displaystyle=$ $\displaystyle G_{2}(i,m;n,j;i\nu)$
(125) $\displaystyle G^{(4)}_{mi,nj}(i\nu)$ $\displaystyle=$ $\displaystyle
G_{2}(i,m;j,n;i\nu)\,,$ (126)

and the matrices $A$ and $B$ are given by

$\displaystyle A_{mi,nj}$ $\displaystyle=$
$\displaystyle(E_{m}-E_{i})\delta_{ij}\delta_{mn}+V_{mj,in}-V_{mj,ni}$ (127)
$\displaystyle B_{mi,nj}$ $\displaystyle=$ $\displaystyle V_{mn,ij}-V_{mn,ji}$
(128)

The complex conjugate in the lower rows in Eq. (122) comes from the following
property of the interaction potential:

$V^{*}_{\alpha\lambda,\beta\sigma}=V_{\beta\sigma,\alpha\lambda}\,,$ (129)

which is a simple consequence of hermiticity of the Hamiltonian. The anti-
symmetry of the fermion creation and annihilation operators implies that the
interaction potential also satisfies

$V_{\alpha\lambda,\beta\sigma}=V_{\lambda\alpha,\sigma\beta}\,,$ (130)

from which it follows that $A$ is hermitian, and $B$ is symmetric. As a
result, the following matrix

$H_{B}=\left(\begin{matrix}A&B\\\ B^{*}&A^{*}\end{matrix}\right)$ (131)

is hermitian. It also has a particle-hole symmetry:

$H_{B}^{*}=XH_{B}X\,,\text{ with }X=\left(\begin{matrix}0&\mathds{1}\\\
\mathds{1}&0\end{matrix}\right)\,.$ (132)

Note that this is a true particle-hole transformation for the excitations on
top of the HF state, as $X$ maps from $(i,m)$ space to $(m,i)$ space, so it
maps electron excitations to hole excitations and vice versa.

The solution to Eq. (122) is given by

$G_{2}(i\nu)=\left(ZH_{B}-i\nu\mathds{1}\right)^{-1}Z\,,$ (133)

where

$Z=\left(\begin{matrix}\mathds{1}&0\\\ 0&-\mathds{1}\end{matrix}\right)\,.$
(134)

The matrix $ZH_{B}$ satisfies $X(ZH_{B})^{*}X=-ZH_{B}$, so all its non-zero
eigenvalues come in pairs $\pm\omega$. Here we used that $\omega$ is real
whenever $H_{B}$ is a positive matrix, which is the condition for the Hartree-
Fock state to be a variational energy minimum [39, 23]). Let us now assume
that $ZH_{B}$ has no zero modes, which will generically be true if we add for
example a very small symmetry breaking field to the Hamiltonian which gaps the
Goldstone modes. In the absence of zero modes, the eigenvectors of $ZH_{B}$
span a complete basis [23], which means that the matrix $S$ whose columns
consist of these eigenvectors is invertible. We can therefore write $ZH_{B}$
as

$ZH_{B}=S(Z_{\eta}\Omega)S^{-1}\,,$ (135)

where $\Omega$ is a positive diagonal matrix, and $Z_{\eta}\Omega$ is a
diagonal matrix containing the eigenvalue pairs $\pm\omega$ ($Z_{\eta}$ is
thus a diagonal matrix with $\pm 1$ on the diagonal). The particle-hole
symmetry in Eq. (132) implies that if $S_{(\alpha\beta),s}$ is an eigenvector
with eigenvalue $\omega_{s}$, then $[XS^{*}]_{(\alpha\beta),s}$ is an
eigenvector with eigenvalue $-\omega_{s}$.

Using the hermiticity of $H_{B}$ and Eq. (135), we can write

$\displaystyle[S^{\dagger}H_{B}S]_{ss^{\prime}}$ $\displaystyle=$
$\displaystyle[S^{\dagger}ZS]_{ss^{\prime}}[Z_{\eta}\Omega]_{s^{\prime}s^{\prime}}$
(136) $\displaystyle=$
$\displaystyle[Z_{\eta}\Omega]_{ss}[S^{\dagger}ZS]_{ss^{\prime}}$ (137)

Comparing Eqs. (136) and (137), we conclude that if
$[Z_{\eta}\Omega]_{s^{\prime}s^{\prime}}\neq[Z_{\eta}\Omega]_{ss}$, then
$[S^{\dagger}ZS]_{ss^{\prime}}=0$. Moreover, as $H_{B}$ is positive if the
Hartree-Fock state is a variational energy minimum, which we assume to be the
case here, it follows that

$\text{sgn}([S^{\dagger}ZS]_{ss})=[Z_{\eta}]_{ss}\text{ if
}\Omega_{ss}=\omega_{s}\neq 0.$ (138)

As we assume the Goldstone modes to be gapped, and hence that there are no
zero modes, we can normalize the columns of $S$ (i.e. the eigenvectors of
$ZH_{B}$) such that the following equation holds:

$S^{\dagger}ZS=Z_{\eta}\,,$ (139)

Equation (139) is Eq. (29) in the main text. To see this, note that if we
restore the momentum labels, we have that
$[S({\mathbf{q}})]_{({\mathbf{k}}\alpha\beta),s}=\varphi^{s}_{{\mathbf{q}},\alpha\beta}({\mathbf{k}})$,
i.e. the columns of $S$ (which is block-diagonal in momentum ${\mathbf{q}}$)
are labeled by $s$ and correspond to the collective mode wavefunctions. The
diagonal elements of $Z_{\eta}$ are the signs $\eta_{{\mathbf{q}},s}$ defined
in the main text.

Using the eigenbasis of $ZH_{B}$, normalized such that Eq. (139) holds, we can
write the collective mode propagator as

$\displaystyle G_{2}(i\nu)$ $\displaystyle=$ $\displaystyle
S(Z_{\eta}\Omega-i\nu\mathds{1})^{-1}S^{-1}Z$ (140) $\displaystyle=$
$\displaystyle S(\Omega-i\nu Z_{\eta})^{-1}Z_{\eta}S^{-1}Z$ (141)
$\displaystyle=$ $\displaystyle S(\Omega-i\nu Z_{\eta})^{-1}S^{\dagger}\,.$
(142)

The last line is Eq. (27) in the main text.

Let us now consider translationally invariant systems, for which $H_{B}$ is
block-diagonal in the momentum ${\mathbf{q}}$. We will write the momentum-
dependent blocks as $H_{B}({\mathbf{q}})$. We are interested in the case where
$H_{B}({\mathbf{q}})$ has a zero mode at ${\mathbf{q}}=0$, and we want to
understand the behaviour of the eigenvectors contained in $S({\mathbf{q}})$ as
a function of ${\mathbf{q}}$. Much can be learned from studying the following
simple example:

$H_{B}({\mathbf{q}})=\left(\begin{matrix}1+\epsilon_{\mathbf{q}}^{2}&1\\\
1&1+\epsilon_{\mathbf{q}}^{2}\end{matrix}\right)\,,$ (143)

where $\epsilon_{{\mathbf{q}}=0}=0$, and $\epsilon_{\mathbf{q}}\ll 1$ for all
${\mathbf{q}}$. For this example we have $Z=\sigma^{z}$ and $X=\sigma^{x}$,
where $\sigma^{i}$ are the Pauli matrices. Working to second order in
$\epsilon_{\mathbf{q}}$, we find that the eigenvalues of
$ZH_{B}({\mathbf{q}})$ are given by
$\omega_{\pm,{\mathbf{q}}}=\pm\sqrt{2}\epsilon_{\mathbf{q}}$. The
corresponding eigenvectors, normalized to have unit Euclidean norm, are given
by

$\displaystyle S({\mathbf{q}})_{:,+}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}-\epsilon_{\mathbf{q}}+3\sqrt{2}\epsilon_{\mathbf{q}}^{2}/4}\left(\begin{matrix}1\\\
-1+\sqrt{2}\epsilon_{\mathbf{q}}-\epsilon_{\mathbf{q}}^{2}\end{matrix}\right)$
(144) $\displaystyle S({\mathbf{q}})_{:,-}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}-\epsilon_{\mathbf{q}}+3\sqrt{2}\epsilon_{\mathbf{q}}^{2}/4}\left(\begin{matrix}1-\sqrt{2}\epsilon_{\mathbf{q}}+\epsilon_{\mathbf{q}}^{2}\\\
-1\end{matrix}\right)\,,$ (145)

where we have again only kept terms up to second order in
$\epsilon_{\mathbf{q}}$. Note that as ${\mathbf{q}}\rightarrow 0$, and thus
$\epsilon_{\mathbf{q}}\rightarrow 0$, the two eigenvectors
$S({\mathbf{q}})_{:,\pm}$ become identical, reflecting the fact that the
generalized eigenvalue equation $H_{B}(0)|v\rangle=\lambda Z|v\rangle$ has
only one solution in the presence of a zero mode, and the matrix
$S({\mathbf{q}})$ becomes degenerate. The eigenvectors also satisfy

$\displaystyle[S^{\dagger}({\mathbf{q}})ZS({\mathbf{q}})]_{++}$
$\displaystyle=$ $\displaystyle\sqrt{2}\epsilon_{\mathbf{q}}$ (146)
$\displaystyle\left[S^{\dagger}({\mathbf{q}})ZS({\mathbf{q}})\right]_{--}$
$\displaystyle=$ $\displaystyle-\sqrt{2}\epsilon_{\mathbf{q}}\,.$ (147)

These equations show that if we want to adopt the normalization of the
eigenvectors such that Eq. (139) holds, then the Euclidean norm of the
eigenvectors will diverge in the ${\mathbf{q}}\rightarrow 0$ limit as
$\epsilon_{\mathbf{q}}^{-1/2}$. This is a generic feature of the generalized
eigenvalue problem considered here, and is not an artefact of the simple
example. To see this, let us write $ZH_{B}=ZUDU^{\dagger}$, where $U$ is the
unitary matrix which diagonalizes $H_{B}$. If $H_{B}$ has a zero mode with
eigenvector corresponding to the first column of $U$, denoted as $U_{:,0}$,
then $U_{:,0}$ is also a zero mode of $ZH_{B}$, i.e. we can take
$S_{:,0}=U_{:,0}$. Furthermore, as $H_{B}$ is particle-hole symmetric [Eq.
(132)], it follows that $[XU^{*}]_{:,0}=e^{i\alpha}U_{:,0}$, where
$e^{i\alpha}$ is an unimportant phase. As $X$ anti-commutes with $Z$, the
particle-hole symmetry implies that

$[S^{\dagger}ZS]_{ss}=0\text{ if }\Omega_{ss}=\omega_{s}=0\,.$ (148)

This shows that the zero mode eigenvectors cannot be normalized such that Eq.
(139) holds. Previously, we assumed a small non-zero symmetry breaking field
to gap the Goldstone modes at ${\mathbf{q}}=0$. In this case, the Euclidean
norm of the Goldstone mode eigenvectors at ${\mathbf{q}}=0$, normalized such
that Eq. (139) holds, will diverge as $\omega_{\mathbf{q}}^{-1/2}$ when we
take this symmetry-breaking field to zero at the end of the calculation. To
avoid this divergence, we defined the rescaled wavefunctions in Eq. (34) of
the main text.