4.1 Auxiliary Dynamics

Let \(\eta _\) be the smallest number in \((2\pi /\beta )\mathbb \) such that \(\eta _ \ge \eta \). We define:

$$\begin \mathcal _(t) = \mathcal + \theta e^ t} \sum _ \in \Lambda _} \mu _(\theta x) a^_} a_}\;, \end$$

(4.1)

where \(\mu _(\theta x)\) is obtained from \(\mu (\theta x)\) after cutting off large momenta, in a smooth way:

$$\begin \mu _(\theta x) = \frac \sum _} e^ \hat_(p)\;,\qquad \hat_(p) = \hat_(p) \chi ( \theta ^ |p|_})\;, \end$$

(4.2)

with \(\alpha \in (0,1)\) and \(\chi (\cdot ) : \mathbb ^\rightarrow [0;1]\) a smooth cutoff function such that:

$$\begin \chi (t)= 1 & t\le 1\\ 0 & t\ge 2. \end\right. } \end$$

(4.3)

With respect to the original perturbation, in the definition of the function \(\mu _(\theta x)\) we are cutting off all momenta p of norm greater than \(2\theta ^\). By the smoothness of the function \(\mu (\cdot )\), this regularization will have a minor effect.

Let us denote by \(\widetilde}(t;s)\) the two-parameter unitary group generated by \(\mathcal _(t)\),

$$\begin i\partial _ \widetilde}(t;s) = \mathcal _(t) \widetilde}(t;s)\;,\qquad \widetilde}(s;s) = \mathbb \;. \end$$

(4.4)

The next result allows to control the error introduced by replacing the dynamics generated by \(\mathcal (\eta t)\) with the dynamics generated by \(\mathcal _(t)\).

Proposition 4.1

(Approximation by the auxiliary dynamics) Under the same assumptions of Theorem 3.1, it follows that, for any \(m\in \mathbb \):

$$\begin \Big \Vert \widetilde}(t;-\infty )^ \mathcal _ \widetilde}(t;-\infty ) - \mathcal (t;-\infty )^ \mathcal _ \mathcal (t;-\infty ) \Big \Vert \le C_ \frac}} + \frac\beta }\;. \end$$

(4.5)

Remark 4.2

Let us briefly comment on the reason for introducing the auxiliary dynamics. Let \(\mathcal \) be such that \([\mathcal , \mathcal ] = 0\), and let \(\tilde_(\mathcal ) = e^ \tau _(\mathcal )\), which we can view as a (non-unitary) regularization of the original dynamics, damped in the past. Then, if \(\eta \in \frac \mathbb \), we observe that \(\tilde_(\cdot )\) satisfies the KMS identity:

$$\begin \langle \tilde_(\mathcal ) \mathcal \rangle _} = \langle \mathcal \tilde_(\mathcal ) \rangle _}\;, \end$$

(4.6)

where we used the trivial but crucial identity \(e^ = e^\). The KMS identity for the regularized dynamics (4.6) plays a key role in the mapping of the real-time Duhamel series into an imaginary time expansion, which can be efficiently studied [34]. See Proposition 2.5. Due to the fact that the difference between any \(\eta > 0\) and its best approximation \(\eta _ \in \frac \mathbb \) vanishes as \(\beta \rightarrow \infty \), this approach is suitable to study zero or low temperature systems (lower than some \(\eta \)-dependent value).

Concerning the momentum-space regularization of the perturbation, this will be used to rule out the scattering of different Fermi points, at orders in the expansion that are less than some \(\theta \)-dependent (high) order.

Proof of Proposition 4.1

The proof of this proposition is a standard argument, based on Lieb–Robinson bounds. It is a slight generalization of an analogous result in [34]. We reproduce it for completeness. We start by writing:

$$\begin \begin&\Big \Vert \widetilde}(t;-\infty )^ \mathcal _ \widetilde}(t;-\infty ) - \mathcal (t;-\infty )^ \mathcal _ \mathcal (t;-\infty ) \Big \Vert \\&\qquad = \Big \Vert \mathcal _ - \mathcal _}(t;-\infty )^ \mathcal _ \mathcal _}(t;-\infty ) \Big \Vert \;, \end \end$$

(4.7)

where \(\mathcal _}(t;s) = \mathcal (t;s) \widetilde}(t;s)^\). Then, since:

$$\begin i\partial _ \mathcal _}(t;s) = -\mathcal _}(t;s) \widetilde}(t;s) \big ( \mathcal (\eta s) - \mathcal _(s) \big ) \widetilde}(t;s)^ \end$$

(4.8)

we have:

$$\begin & \mathcal _ - \mathcal _}(t;-\infty )^ \mathcal _ \mathcal _}(t;-\infty ) \nonumber \\ & \quad = \int _^ \hbox s\, \frac}s} \mathcal _}(t;s)^ \mathcal _ \mathcal _}(t;s)\nonumber \\ & \quad = i \int _^ \hbox s\, \mathcal _}(t;s)^ \Big [ \mathcal _, \widetilde}(t;s) \big ( \mathcal (\eta s) - \mathcal _(s) \big ) \widetilde}(t;s)^\Big ] \mathcal _}(t;s)\;,\nonumber \\ \end$$

(4.9)

which implies:

$$\begin \begin&\Big \Vert \mathcal _ - \mathcal _}(t;-\infty )^ \mathcal _ \mathcal _}(t;-\infty ) \Big \Vert \\&\qquad \le \int _^ \hbox s\,\Big \Vert \Big [ \widetilde}(t;s)^ \mathcal _ \widetilde}(t;s), \big ( \mathcal (\eta s) - \mathcal _(s) \big )\Big ] \Big \Vert \\&\qquad \le \int _^ \hbox s\,\Big ( \Big \Vert \Big [ \widetilde}(t;s)^ \mathcal _ \widetilde}(t;s), \mathcal _(s)\Big ] \Big \Vert + \Big \Vert \Big [ \widetilde}(t;s)^ \mathcal _ \widetilde}(t;s), \mathcal _(s)\Big ] \Big \Vert \Big )\;, \end \end$$

(4.10)

where:

$$\begin \begin \mathcal _(s)&= \theta (e^ - e^ s}) \sum _ \in \Lambda _} \mu (\theta x) a^_} a_} \\ \mathcal _(s)&= \theta e^ s} \sum _ \in \Lambda _} (\mu (\theta x) -\mu _(\theta x)) a^_} a_}\;. \end \end$$

(4.11)

Consider the second term. We have:

$$\begin & \Big \Vert \Big [ \widetilde}(t;s)^ \mathcal _ \widetilde}(t;s), \mathcal _(s)\Big ] \Big \Vert \nonumber \\ & \qquad \le \theta e^ s} \sum _ \in \Lambda _} \big | \mu (\theta x) - \mu _(\theta x) \big | \Big \Vert \Big [ \widetilde}(t;s)^ \mathcal _ \widetilde}(t;s), n_} \Big ] \Big \Vert \nonumber \\ & \qquad \le \theta e^ s} \Vert \mu - \mu _ \Vert _ \sum _\in \Lambda _} \Big \Vert \Big [ \widetilde}(t;s)^ \mathcal _ \widetilde}(t;s), n_} \Big ] \Big \Vert \;. \end$$

(4.12)

By the Lieb–Robinson bound, we can further estimate:

$$\begin \Big \Vert \Big [ \widetilde}(t;s)^ \mathcal _ \widetilde}(t;s), \mathcal _(s)\Big ] \Big \Vert \le C_ \theta e^ s} (1 + |t-s|) \Vert \mu - \mu _ \Vert _\;. \end$$

(4.13)

The norm can be bounded as:

$$\begin \Vert \mu - \mu _ \Vert _ \le \frac \sum _} \big | \mu _(p)\big | \big [1 - \chi ( \theta ^ |p|_})\big ]\;. \end$$

(4.14)

Next, by the smoothness of \(\mu (x)\), we have \((1 + |q|_}^) |\hat(q)| \le C_\), which easily gives:

$$\begin \frac \sum _} \big | \mu _(p)\big | \big [1 - \chi ( \theta ^ |p|_})\big ] \le K_ \theta ^\;,\qquad \quad \forall m\in \mathbb \;. \end$$

(4.15)

Plugging this bound in (4.13), we obtain:

$$\begin \Big \Vert \Big [ \widetilde}(t;s)^ \mathcal _ \widetilde}(t;s), \mathcal _(s)\Big ] \Big \Vert \le C_ K_ \theta e^ s} (1 + |t-s|) \theta ^\;, \end$$

(4.16)

which allows to estimate the corresponding contribution to (4.10) as:

$$\begin \int _^ \hbox s\, \Big \Vert \Big [ \widetilde}(t;s)^ \mathcal _ \widetilde}(t;s), \mathcal _(s)\Big ] \Big \Vert \le C_ \frac}}\;. \end$$

(4.17)

Consider now the contribution due to \(\mathcal _(s)\) in the right-hand side of (4.10). This term can be estimated as in [34, Proposition 4.1], we omit the details. The result is:

$$\begin \int _^ \hbox s\, \Big \Vert \Big [ \widetilde}(t;s)^ \mathcal _ \widetilde}(t;s), \mathcal _(s)\Big ] \Big \Vert \le \frac\beta }\;. \end$$

(4.18)

Combining (4.7), (4.10), (4.17), (4.18), the final result (4.5) follows. \(\square \)

Therefore, by Proposition 4.1 we have:

$$\begin \,}}j_ \rho (t) = \,}}j_ \widetilde(t) + \mathcal ^_ (x,t;\eta , \theta )\;, \end$$

(4.19)

where \(\tilde(t) = \widetilde}(t;-\infty ) \rho _ \widetilde}(t;-\infty )^\) and, by Proposition 4.1:

$$\begin | \mathcal ^_ (x,t;\eta , \theta ) | \le C_ \frac}} + \frac\beta }\;. \end$$

(4.20)

We require that this error term is subleading once divided by \(\theta = a\eta \), recall (3.1). To this end, it is sufficient to consider:

$$\begin a=o\left( \eta ^-1}\right) . \end$$

(4.21)

In order to leave room for taking \(a \rightarrow \infty \) as \(\eta \rightarrow 0^\), we will choose \(\alpha \in (0,1)\) and we will pick \(m\in \mathbb \) so that \(\alpha (m-1)>2\). Later, we will have to introduce stronger constraints on a.

4.2 Duhamel Series of the Auxiliary Dynamics

Consider the main term in the right-hand side of (4.19). By Proposition 2.5:

$$\begin \begin&\,}}j_ \widetilde(t) - \,}}j_ \rho _} \\&\quad = \sum _\frac e^ t}} \int _} \hbox \underline\, e^(s_ + \ldots +s_)}\langle \textbf \gamma _}(\widetilde}); \cdots ; \gamma _}(\widetilde}); j_\rangle _}\;. \end \end$$

(4.22)

It is convenient to rewrite the nth-order contribution to the expansion in Fourier space. Recalling the definition (2.38) and the form of the perturbation (2.28),

$$\begin \begin&\int _} \hbox \underline\, e^(s_ + \ldots +s_)}\langle \textbf \gamma _}(\widetilde}); \cdots ; \gamma _}(\widetilde}); j_\rangle _}\\&\quad = \int _} \hbox \underline\, e^(s_ + \ldots +s_)}\langle \textbf \gamma _}(j_(\mu _)); \cdots ; \gamma _}(j_(\mu _)); j_\rangle _}\;. \end \end$$

(4.23)

Next, using that:

$$\begin j_(\mu _) = \frac \sum _} \hat_(-p) \hat_\;,\qquad \hat_ = \sum _ \in \Lambda _} e^ \hat_}\;, \end$$

(4.24)

and writing, for \(\underline = (p_, p_)\) with \(p_ = p\) and \(p_ = \eta _\),

$$\begin \hat_} := \int _^ \hbox s\, e^ s} \gamma _(\hat_)\;, \end$$

(4.25)

we can rewrite the right-hand side of (4.23) as:

$$\begin \begin&\int _} \hbox \underline\, e^(s_ + \ldots +s_)}\langle \textbf \gamma _}(j_(\mu _)); \cdots ; \gamma _}(j_(\mu _)); j_\rangle _} \\&\qquad = \frac}} \sum _\} \in B_^} \Big [\prod _^ \hat_(-p_)\Big ] e^x} \frac\langle \textbf\, \hat__}\;; \cdots \;; \hat__}\;; \hat__} \rangle _}\;, \end \end$$

(4.26)

where \(\underline_ = -\underline_ - \ldots - \underline_\). In order to derive (4.26), we used the space-time translation invariance of the Gibbs state. Next, by Wick’s rule:

$$\begin & \frac\langle \textbf\, \hat__}\;; \cdots \;; \hat__}\;; \hat__} \rangle _\nonumber \\ & \quad = -\frac \sum _}\sum _ \in \mathbb _ \times B_} \,}}\Big [ \hat_(k, p_)\prod _^ g\Big (\underline + \sum _ \underline_\Big )\Big ] \end$$

(4.27)

where \(S_\) is the set of permutations \(\pi \) of \(\\); \(\mathbb _ = (2\pi / \beta ) (\mathbb + 1/2)\) is the set of the fermionic Matsubara frequencies, and \(g(\underline)\) is given by (2.48).

In the following, it will be convenient to decompose the fermionic propagator in a singular plus a regular part. Let \(\chi (\cdot )\) be a smooth cutoff function as in (4.3). We write:

$$\begin \begin g(\underline)&= g(\underline)\chi (|\hat(k) - \mu |/\Delta ) + g(\underline)(1-\chi (|\hat(k) - \mu |/\Delta )) \\&= g_}(\underline) + g_}(\underline)\;; \end \end$$

(4.28)

the propagator \(g_}\) satisfies the estimate:

$$\begin \Vert \text _}^} \text _}^} g_}(\underline) \Vert \le \frac, n_}}|^+1}}\qquad \text k. \end$$

(4.29)

Consider now the first term in (4.28). It is:

$$\begin g_}(\underline) = \sum _^ \frac(k) - \mu |/\Delta )}+e_(k)-\mu } P_(k)\;, \end$$

(4.30)

with \(P_(k)\) a rank-one projector,

$$\begin P_(k):= |\xi _(k) \rangle \langle \xi _(k)|\qquad \qquad \hat(k)\xi _(k)=e_(k)\xi _(k)\;. \end$$

(4.31)

We can further rewrite \(g_}(\underline)\), as follows. By assumption, the Fermi point \(k_^\) is not a critical point for \(e_\), that is \(v_ =e'_(k_^)\ne 0\). Thus, setting \(\underline := (k_, k_)\) and \(\underline_^ := (0, k^_)\),

$$\begin & \sum _^ \frac(k) - \mu |/\Delta )}+e_(k)-\mu } P_(k)\nonumber \\ & \quad =\sum _^ \frac_(\underline)}(\underline)} P_(k) + \sum _^ \Big ( \frac(k) - \mu |/\Delta )}+e_(k)-\mu } - \frac_(\underline)}(\underline)} \Big ) P_(k)\nonumber \\ \end$$

(4.32)

where: \(\chi ^_(\underline):=\chi (\left\| \underline - \underline_^\right\| _/\delta )\), \(\Vert \underline\Vert _^:=q_^+v_^|q|_}^\) and \(D_(\underline_^ + \underline):=iq_+v_q\). The parameter \(\delta \) is chosen so that \(\delta \ll \inf _|k_^-k_^|_}\) and

$$\begin |e_(k)-\mu |\le \Delta \qquad \forall k\in \mathcal B^_(k_^):=\^ |_} < 2\delta / |v_|\} \end$$

(4.33)

for all \(\omega \). Therefore, we can rewrite:

$$\begin \begin g(\underline)&= g_}(\underline) + g_}(\underline)\qquad \text \\ g_}(\underline)&:= \sum _^ \frac_(\underline)}(\underline)} P_(k)\\ g_}(\underline)&:= g_}(\underline) + \sum _^ \Big ( \frac(k) - \mu |/\Delta )}+e_(k)-\mu } - \frac_(\underline)}(\underline)} \Big ) P_(k)\;. \end \end$$

(4.34)

It is not difficult to see that:

$$\begin \Vert g_}(\underline)\Vert \le \frac|}\;,\qquad \big \Vert \text _} g_}(\underline) \big \Vert \le \sum _^ \frac|} \frac - \underline^_\Vert }\;. \end$$

(4.35)

Using the above splitting we can decompose the left-hand side of (4.27) as:

$$\begin \frac\langle \textbf\, \hat__}; \cdots ; \hat__}; \hat__} \rangle _ = \widetilde^_(\underline_, \ldots , \underline_) + \widetilde^_(\underline_, \ldots , \underline_) \end$$

(4.36)

where \(\widetilde^}_(\underline_, \ldots , \underline_)\) collects the contribution due only to the singular propagators \(g_}\),

$$\begin \widetilde^}_(\underline_, \ldots , \underline_) := -\frac \sum _ \in \mathbb _ \times B_} \sum _}\,}}\left[ \hat_(k, p_)\prod _^ g_}\left( \underline + \sum _ \underline_\right) \right] , \end$$

(4.37)

while \(\widetilde^}_(\underline_, \ldots , \underline_)\) contains at least one bounded propagator \(g_}\). The next lemma allows to rewrite \(\widetilde^_\) in a more explicit way, up to subleading terms. In the following, we shall use the notation:

$$\begin \int _ \frac\underline}}\, [\cdots ] := \frac \sum _ \in \mathbb _ \times B_} [\cdots ]\;; \end$$

(4.38)

in fact, as \(\beta ,L\rightarrow \infty \), the sum converges to an integral over \(\underline \in \mathbb \times \mathbb \).

Lemma 4.3

(Structure of the singular part) Let \(v_^ = 1\) and \(v_^ = v_\). Then for any \(p_,\dots ,p_\) in the support of \(\hat_\),

$$\begin \widetilde^_(\underline_, \ldots , \underline_) = S^_(\underline_, \ldots , \underline_) + T^_(\underline_, \ldots , \underline_) \end$$

(4.39)

where, for \(g_(\underline) = \chi _^(\underline) / D_(\underline)\),

$$\begin \begin S^}_(\underline_, \ldots , \underline_)=&- \sum _^ v_^ \int _} \frac\underline}} \sum _} \prod _^ g_\Big (\underline+\sum _ \underline_\Big )\\ T^}_(\underline_, \ldots , \underline_)=&- \sum _^ \int _} \frac\underline}} \sum _}f_^(k;p_,\dots ,p_)\\&\cdot \prod _^ g_\Big ( \underline+\sum _ \underline_\Big ) \;, \end \end$$

(4.40)

where:

$$\begin |f_^(k;p_,\dots ,p_)|\le C_\Big ( \delta _|k-k_^|_}+ n\sum _^ |p_|_}\Big ). \end$$

(4.41)

Proof

We start with the following remark. Since all external momenta \(\\}\) are in the support of \(\hat_\), we have that \(|p_|_} \le \theta ^\). Furthermore, \(g_\) and \(g_\) have disjoint support for \(\omega \ne \omega '\), recall the discussion after (4.33). Thus, for \(\theta \) small enough, and for all \(i=1,\dots , n\):

$$\begin g_(\underline+\textstyle \sum _\underline_)g_(\underline+\sum _\underline_)=0\quad \mathrm . \end$$

(4.42)

Hence, \(\tilde^_\) reduces to

$$\begin \widetilde^_(\underline_, \ldots , \underline_)= & -\sum _^\sum _} \int _ \frac\underline}}\, \Big [ \prod _^ g_ \Big (\underline+\sum _ \underline_\Big )\Big ] \nonumber \\ & \cdot ~t_^(k;p_,\dots ,p_) \end$$

(4.43)

with

$$\begin t_^(k;p_,\dots ,p_):= \,}}\Big [\hat_(k, p_)\prod _^ P_\Big (k+\sum _ p_\Big )\Big ]\;. \end$$

(4.44)

Noticing that \(t_^(k;0,\dots ,0)\equiv 1\) and \(t_^(k;0,\dots ,0)=e'_(k)\), we Taylor expand \(t_^(k;\cdot )\) around \((0,\dots ,0)\), and \(e'_\) around \(k_\textrm^\), obtaining

$$\begin t_^(k;p_,\dots ,p_)= v_^ + f^_(k;p_,\dots ,p_)\;; \end$$

(4.45)

the bound (4.41) easily follows from the smoothness of \(\hat_(k,p)\) and of \(_(k)\).

\(\square \)

Remark 4.4

This decomposition allows to isolate, at order n, the most singular contribution from the nth-order term in the Duhamel expansion. Indeed, for \(\theta \propto \eta \), a rescaling shows that each summand appearing in \(\theta ^ S^}_\) is O(1), whereas the summands in \(\theta ^ T^}_\) and \(\theta ^ \widetilde^}_\) are \(O(\theta )\). We will see that the smallness of \(\theta ^ S^}_\) will be guaranteed by a crucial cancellation.

Let \(R^}_ := \widetilde^}_ + T^}_\). We obtained the following representation for the Duhamel expansion of the response functions, recall (4.22)–(4.26),

$$\begin \chi ^}_(x;\eta ,\theta )= & -\sum _^\infty \frac} \frac} \sum _\} \in B_^} \Big [\prod _^ \hat_(-p_)\Big ] e^x}\nonumber \\ & \cdot [S^}_(\underline_,\dots , \underline_)+R^}_(\underline_,\dots , \underline_)] + E^}_(x;\eta ,\theta )\;,\nonumber \\ \end$$

(4.46)

where: \(\underline_ := (\eta _, p_)\) and \(\underline_ := -\sum _^ \underline_\); the term \(S^_\) is given by the first of (4.40); the term \(R^}_(\underline_,\dots , \underline_)\) will be proven to be subleading as \(\theta \rightarrow 0\); and, recall (4.20):

$$\begin | E^}_(x;\eta ,\theta )|\le \frac}} + \frac\beta }\;. \end$$

(4.47)

Our next task will be to evaluate explicitly the linear response, and to bound the higher-order terms, starting from the identity (4.46).

4.3 Evaluation of the Linear Response

In this subsection we will evaluate the main term in the expression for the response function, Eq. (3.2), as \(\beta ,L\rightarrow \infty \). The analysis is based on lattice conservation laws, regularity of correlations, and explicit computation of the scaling limit contribution. It has been already used to determine the linear response of 1d and quasi-1d systems, [2, 10, 13, 45, 46], in a setting that also allow to include many-body interactions. For completeness, we shall reproduce all the steps here, for general 1d non-interacting systems.

Let \(S_ = \lim _\rightarrow \infty }S^}_\) and \(R_ = \lim _\rightarrow \infty }R^}_\). In this limit, the \(n=1\) term in (4.46) reads, setting \(\underline=\theta \underline\), with \(\underline=(a^,q)\),

$$\begin \chi _^}(x;\eta ,\theta ) = - \int __}} \fracq}\, \hat_\alpha (-q)e^ \big [S_(\theta \underline) + R_(\theta \underline)\big ]\;, \end$$

(4.48)

with \(\hat_(q) = \hat(q) \chi ( \theta ^ |q|)\), and \(\mathbb _}\) is the torus \(\mathbb \) rescaled by a factor \(1/\theta \). Let us start by discussing the \(R_\) term.

Lemma 4.5

(Continuity of the remainder). For \(\alpha \in (0,1)\), there exists \(C_ > 0\) such that:

$$\begin |R_(\underline)| \le C,\qquad |R_(\underline)-R_(\underline)|\le C_ \Vert \underline\Vert ^\;. \end$$

(4.49)

Proof

We have:

$$\begin \widetilde_(\underline)= & \int _\times \mathbb } \frac\underline} \,}}\Big (\hat_(k, -p)(g_}(\underline) g_}(\underline+\underline)\nonumber \\ & +g_}(\underline)g_}(\underline+\underline)+g_}(\underline)g_}(\underline+\underline))\Big )\;. \end$$

(4.50)

By the estimate (4.35), \(\widetilde_(\underline)\) is finite. Furthermore, by the smoothness of \(\hat_(k, p)\) and using the bound, with \(0<\alpha <1\),

$$\begin \big \Vert g_}(\underline + \underline) - g_}(\underline) \big \Vert \le C\sum _ \frac|^}\frac\Vert ^} - \underline_^\Vert ^}\, \end$$

(4.51)

which follows from the second of (4.35), we easily get:

$$\begin \Big | \widetilde_(\underline) - \widetilde_(\underline) \Big | \le C_ \Vert \underline\Vert ^\;. \end$$

(4.52)

Consider now the term \(T_\). From the estimate (4.41), and recalling that \(g_(\underline) = \chi _^(\underline) / D_(\underline)\), we have:

$$\begin \Big | T_(\underline) \Big |\le & C\sum _^ \int \hbox \underline\, \big | f_^(k;p)\big | | g_(\underline + \underline) | | g_(\underline)| \nonumber \\\le & K \sum _^ \int \hbox \underline\, (|k - k_^|_} + |p|_}) | g_(\underline + \underline)| | g_(\underline)|\nonumber \\\le & \widetilde(1 + \Vert \underline\Vert \big |\log \Vert \underline\Vert \big |)\;. \end$$

(4.53)

Similarly,

$$\begin \Big | T_(\underline) - T_(\underline) \Big |\le & K \sum _^ \int \hbox \underline\, |k - k_^|_} | \big ( g_(\underline + \underline) - g_(\underline)\big ) | | g_(\underline)|\nonumber \\ & + K \sum _^ \int \hbox \underline\, |p|_} | g_(\underline + \underline) | |g_(\underline)|\;; \end$$

(4.54)

the second term is bounded proportionally to \(\Vert \underline\Vert \left| \log \Vert \underline\Vert \right| \). Consider the first term. Using that, for \(0<\alpha <1\):

$$\begin | g_(\underline + \underline) - g_(\underline) |\le & C\left( \frac(\underline)} - \underline_^\Vert ^} + \frac(\underline + \underline)} - \underline_^ + \underline\Vert ^}\right) \nonumber \\ & \cdot ~\frac\Vert ^} - \underline_^ + \underline\Vert ^ \Vert \underline - \underline_^ \Vert ^} \end$$

(4.55)

we obtain:

$$\begin \begin&\int \hbox \underline\, |k - k_^| | \big ( g_(\underline + \underline) - g_(\underline)\big ) | |g_(\underline)| \\&\quad \le C\int d\underline\,\left( \frac(\underline)} - \underline_^\Vert ^} + \frac(\underline + \underline)} - \underline_^ + \underline\Vert ^}\right) \frac\Vert ^} - \underline_^ + \underline\Vert ^ \Vert \underline - \underline_^\Vert ^} \\&\quad \le \widetilde_ \Vert p\Vert ^. \end \end$$

(4.56)

Thus, we found:

$$\begin \Big | T_(\underline) - T_(\underline) \Big | \le C_ \Vert \underline\Vert ^\;. \end$$

(4.57)

This concludes the proof of (4.49). \(\square \)

Next, we shall consider the singular term \(S_\) in (4.48), whose explicit form is

$$\begin \begin S_(\theta \underline)&= -\sum _^ v_^ \int _\times \mathbb } g_(\underline)g_(\underline+\theta \underline) \frac\underline} \\&\equiv \sum _^ v_^ \mathfrak B_^(\theta \underline)\;, \end \end$$

(4.58)

where \( \mathfrak B_^\) is the relativistic bubble diagram:

$$\begin \mathfrak B_^(\theta \underline):= -\int _^} \frac^(\underline)\chi _^(\underline+\theta \underline)}(\underline)D_(\underline+\theta \underline)} \frac\underline}\;. \end$$

(4.59)

The next proposition allows to compute it. The result is well known, and we reproduce it here for completeness.

Proposition 4.6

(The relativistic bubble diagram). Let \(\alpha \in (0,1)\). For any \(\underline=(q_,q)\ne \underline\) such that \(\Vert \underline\Vert \le \theta ^\), we have:

$$\begin \mathfrak B_^(\theta \underline) = \frac|}\fracq}q} + O(\theta ^)\;. \end$$

(4.60)

Proof

First of all, by performing the change of coordinates \(\underline\rightarrow \theta \underline\) we obtain

$$\begin \mathfrak B_^(\theta \underline) = \mathfrak B_^(\underline)\;. \end$$

(4.61)

Then, we rewrite:

$$\begin \begin \mathfrak B_^(\underline)&= -\int _^} \frac^(\underline)\chi _^(\underline+\underline)}(\underline)D_(\underline+\underline)} \frac\underline} \\&= - \frac(\underline)} \int _^} \chi _^(\underline)\chi _^(\underline+\underline)\left( \frac(\underline)} - \frac(\underline + \underline)} \right) \frac\underline} \\&= - \frac(\underline)} \int _^} \frac^(\underline)\chi _^(\underline+\underline) - \chi _^(\underline - \underline)\chi _^(\underline)}(\underline)}\frac\underline}\;. \end \end$$

(4.62)

Observing that:

$$\begin \chi _^(\underline+\underline) - \chi _^(\underline - \underline) = 2\underline \cdot \nabla _} \chi _^(\underline) + r_(\underline, \underline) \end$$

(4.63)

with \(| r_(\underline, \underline) | \le C_ \theta ^\Vert \underline\Vert ^2 \), we see that, for \(\Vert \underline\Vert \le \theta ^\), using that \(r_(\underline, \underline)\) is supported for \(\Vert \underline\Vert \sim (\delta / \theta )\):

$$\begin \left| \frac(\underline)} \int _^} \frac^(\underline)r_(\underline, \underline)}(\underline)} \frac\underline} \right| \le C_\theta ^ \end$$

(4.64)

which vanishes for \(\theta \rightarrow 0\). Consider now the main term,

$$\begin \begin&- \frac(\underline)} \int _^} \frac^(\underline) \underline\cdot \nabla _} \chi _^(\underline)}(\underline)} \frac\underline} \\&\qquad = - \frac}(\underline)} \int _^} \frac^(\underline) \partial _ \chi _^(\underline)}(\underline)} \frac\underline} \\&\qquad \qquad - \frac}(\underline)} \int _^} \frac^(\underline) \partial _ \chi _^(\underline)}(\underline)} \frac\underline}\;. \end \end$$

(4.65)

After a rescaling and a change of variable in the \(k_\) variable, we get;

$$\begin \begin&- \frac(\underline)} \int _^} \frac^(\underline) \underline\cdot \nabla _} \chi _^(\underline)}(\underline)} \frac\underline} \\&\quad = - \frac}(\underline)} \frac|}\int _^} \frac) \partial _ \chi (\underline)} + k_} \frac\underline} - \frac}(\underline)} \frac}|} \int _^} \frac) \partial _ \chi (\underline)} + k_} \frac\underline}\;, \end \end$$

(4.66)

where \(\chi (\underline) \equiv \chi (\Vert \underline\Vert )\), see Eq. (4.3). Then, one observes that this expression can be further rewritten as:

$$\begin & - \frac(\underline)} \int _^} \frac^(\underline) \underline\cdot \nabla _} \chi _^(\underline)}(\underline)} \frac\underline}\nonumber \\ & \qquad = - \frac + i v_q_)}(\underline) |v_|} \int _^} \frac) \partial _ \chi (\underline)} + k_} \frac\underline} \nonumber \\ & \qquad = - \frac + i v_q_)}(\underline) |v_|} \int _^} \frac}\Vert }\frac\Vert ) \chi '(\Vert \underline\Vert )} + k_} \frac\underline} \nonumber \\ & \qquad = - \frac + i v_q_}(\underline) |v_|} \int _^} \frac}\Vert }\frac(\Vert \underline\Vert ))'} + k_} \frac\underline}\;. \end$$

(4.67)

The last integral can be computed switching to polar coordinates, and one finds:

$$\begin - \frac(\underline)} \int _^} \frac^(\underline) \underline\cdot \nabla _} \chi _^(\underline)}(\underline)} \frac\underline} = \frac + i v_q_}(\underline) |v_|} \frac \end$$

(4.68)

which reproduces the main term in (4.60). \(\square \)

The evaluation of the relativistic bubble diagram, combined with lattice conservation laws, allow to compute (4.48).

Proposition 4.7

(Evaluation of the linear response). We have:

$$\begin \chi _^}(x;\eta ,\theta ) =-\sum _^\chi _\int _\mathbb \hat_(q)e^\frac\,\fracq} + O(\theta ^)\;, \end$$

(4.69)

with \(\chi _=v_^/2\pi |v_\omega |\). Let \(\theta = a \eta \). We can distinguish three regimes.

(i)

Suppose that \(a\rightarrow 0\) as \(\eta \rightarrow 0^\). Then \(\chi _^}(x;\eta ,\theta ) = O(a)\).

(ii)

Suppose that a is constant in \(\eta \). Then:

$$\begin \chi _^}(x;\eta ,\theta )= & -\sum _^\chi _\int _\mathbb \mu _(\theta x-y)\nonumber \\ & \cdot \left[ \delta (y)-\frac|} e^}\Theta (y/v_)\right] \,\textrmy+ O(\eta ^)\;. \end$$

(4.70)

(iii)

Suppose that \(a\rightarrow \infty \) as \(\eta \rightarrow 0^\), so that \(a\eta \rightarrow 0\). Then:

$$\begin \chi _^}(x;\eta ,\theta )= - \mu _(\theta x)\sum _^\chi _ + O(\theta ^)\;. \end$$

(4.71)

Proof

By (4.58), we rewrite the linear response (4.48) as:

$$\begin \begin \chi _^}(x,\eta ,\theta )&= - \int __}} \fracq}\, \hat_\alpha (-q)e^ \big [S_(\theta \underline) + R_(\theta \underline)\big ] \\&= -\int _} \fracq}\, \hat_(-q)e^\left[ \sum _^ v_^ \mathfrak B_^(\theta \underline) + R_(\underline)\right] \\&\qquad + E_(x;\eta ,\theta )\;, \end \end$$

(4.72)

where, by (4.49), and by the regularity properties of the function \(\mu _(x)\):

$$\begin \Big |E_(x;\eta ,\theta ) \Big | \le K \theta \;. \end$$

(4.73)

Next, by (4.60), (4.61), \(\mathfrak B_^(\theta \underline) = \mathfrak B_^(\underline) + O(\theta ^)\), with:

$$\begin \mathfrak B_^(\underline)=\frac|}\frac+v_q}+ v_q}\;; \end$$

(4.74)

thus, recalling that \(\underline = (a^,q)\) and performing the change of variables \(q\rightarrow -q\),

$$\begin \begin \chi _^}(x;\eta ,\theta )&= -\int _} \fracq}\, \hat_(q)e^\\&\qquad \cdot \left[ \sum _^ v_^ \frac|}\fracq}q} + R_(\underline)\right] + \sum _ E_(x;\eta ,\theta ) \end \end$$

(4.75)

where:

$$\begin | E_(x;\eta ,\theta ) | \le C\theta ^\;. \end$$

(4.76)

Let us now determine \(R_(\underline)\). We shall compute it by exploiting lattice conservation laws. Thanks to the continuity equation (2.33), the following equality holds:

$$\begin i\partial _}\langle \textbf n_};j_}\rangle _ + \textrm_ \langle \textbf j_};j_}\rangle _ = i\delta (x_-y_) \langle [n_},j_}]\rangle _\;, \end$$

(4.77)

where \(\mathcal O_}:=\gamma _}(\mathcal O_})\). The contact term on the right-hand side arises due to the definition of time-ordering. Taking the Fourier transform of left-hand side and right-hand side, we obtain:

$$\begin p_ \frac\langle \textbf \hat_};\hat_}\rangle _ + (1-e^) \frac\langle \textbf \hat_};\hat_}\rangle _ = i\sum _} e^\langle [n_,j_]\rangle _\;. \end$$

(4.78)

Let \(\underline = (p_, 0)\). Using that \([\sum _} n_,j_]=0\), we obtain \(\langle \textbf \hat_};\hat_}\rangle _ = 0\). Hence, recalling that \((1/\beta L)\langle \textbf \hat_};\hat_}\rangle _ = S^_(\underline)+R^_(\underline)\), we obtain, in the \(\beta , L \rightarrow \infty \) limit, and by the continuity of \(R_(\underline)\) at \(\underline = \underline\):

$$\begin & S_((p_,0)) + R_((p_,0))=0\qquad \Longrightarrow \nonumber \\ & R_(\underline)=-\lim _\rightarrow 0} S_((p_,0))=\sum _^ \frac^}|}. \end$$

(4.79)

Plugging this identity in (4.75), we obtain:

$$\begin \chi _^}(x;\eta ,\theta ) =-\sum _^\frac^} \int _\mathbb \fracq}\, \hat_(q)e^\frac + \sum _ E_(x;\eta ,\theta ) \end$$

(4.80)

which proves (4.69). Let us further analyse the integral. From (4.80), it is clear that if \(a \rightarrow 0\), the integral is vanishing. More generally, we can rewrite (4.80) as:

$$\begin \chi _^}(x;\eta ,\theta ) =-\sum _^\chi _\Big [\mu _(\theta x)+\frac \int _\mathbb \fracq}\, \frac_(q)e^}\Big ] + \sum _ E_(x;\eta ,\theta )\;. \end$$

(4.81)

To compute the integral, we employ the good decay properties of \(\hat_(q)\) to rewrite it as:

$$\begin \int _\mathbb \frac_(q)e^}\,\fracq}= & \lim _ \int _ \frac_(q)e^}\,\fracq} \nonumber \\= & \lim _ \int dy\, \mu _(y) \int _ \frace^}\,\fracq} \nonumber \\= & \lim _ \int dz\, \mu _(z + \theta x) \int _ \frac}\,\fracq}\;.\nonumber \\ \end$$

(4.82)

Let \(z>0\). By Cauchy theorem for holomorphic functions:

$$\begin \begin \int _ \frac}\,\fracq}&= \frac} \int _ \frac}) + q}\, \textrmq \\&= -\frac} (2\pi i) e^)} \mathbb (v_ < 0) + g^_(z) \end \end$$

(4.83)

where \(g^_(z)\) is bounded uniformly in z and \(\lim _g^_(z) = 0\). Similarly, if \(z<0\):

$$\begin \int _ \frac}\,\fracq} = \frac} (2\pi i) e^)} \mathbb (v_ > 0) + g^_(z) \end$$

(4.84)

with \(g^_(z)\) is bounded uniformly in z and \(\lim _g^_(z) = 0\). All in all, for \(z\ne 0\):

$$\begin \lim _ \int _ \frac}\,\fracq} = \frac|} e^)} \mathbb ((z/v_)<0). \end$$

(4.85)

Plugging this in (4.82), we obtain, performing a change of variable \(z\rightarrow -z\):

$$\begin \int _\mathbb \frac_(q)e^}\,\fracq} = \frac|} \int \hbox z\, \mu _(\theta x - z) e^)} \Theta (z/v_\omega ) \end$$

(4.86)

with \(\Theta (\cdot )\) the Heaviside step function, equal to 1 for positive argument and zero otherwise. Inserting this formula in (4.81), we get:

$$\begin \chi _^}(x;\eta ,\theta )= & -\sum _^\chi _\Big [\mu _(\theta x) - \frac|} \int \hbox z\, \mu _(\theta x - z) e^)} \Theta (z/v_)\Big ]\nonumber \\ & + \sum _ E_(x;\eta ,\theta ) \end$$

(4.87)

which proves (4.70). Finally, the last claim (4.71) follows after taking the \(a\rightarrow \infty \) limit, and using that \(\mu \) is integrable. This concludes the proof of Proposition 4.7. \(\square \)

4.4 Estimates for the Higher-Order Corrections

Let:

$$\begin \mathfrak B_^(\underline_,\dots ,\underline_) := -\int _^} \prod _^ g_\Big (\underline+\sum _ \underline_\Big ) \frac\underline}\;, \end$$

(4.88)

with the understanding that \(\underline_ = -\sum _^ \underline_\). Thus, from (4.40), in the \(\beta , L\rightarrow \infty \) limit: