# Products of Daniell integrals

Gerhard Rompf, Götz Kersting Institut für Mathematik, Goethe Universität,
Frankfurt<EMAIL_ADDRESS>
###### Abstract

We show that for any two Daniell integrals $J$ and $K$, given on some Riesz
spaces $S$ and $T$, there exists a product integral $I$ on the space $R$,
which is the smallest Riesz space containing the tensor product of $S$ and
$T$. The integral $I$ is uniquely characterized by the property $I(f\otimes
g)=J(f)K(g)$ for all $f\in S$, $g\in T$. Also a Fubini-type theorem is
presented.

Keywords and phrases. Daniell integral, product integral, Fubini-Stone theorem

MSC 2020 subject classification. Primary 28C05, Secondary 28A35.

##  1 Introduction and main results

In this paper we close a structural gap in the theory of Daniell integration.
Let $J$ and $K$ be two Daniell integrals. Their domains are Riesz spaces
(vector lattices) $S$ and $T$ of real-valued functions. Then one would like to
dispose of a product integral $I$ of these two integrals on a suitable Riesz
space $R$. By now, this task has been resolved only for integrals $J$ and $K$
stemming from Lebesgue measures. Here we show that a product integral $I$ can
be uniquely obtained for any two Daniell integrals, regardless whether they
are elementary or completed. Its domain is the smallest Riesz space $R$
containing the tensor product of $S$ and $T$, which certainly is the natural
candidate. A main obstacle to our construction is that in general there is no
lucid way available in order to build up this space $R$ from $S$ and $T$.
Therefore, we first tackle the problem for more amenable Riesz spaces closed
under bounded pointwise convergence, which we like to call Lebesgue lattices,
and then reduce the general case to this special configuration. At the same
time, a Fubini-type theorem arises. – This paper builds on a manuscript [5] of
one of the authors (G. R.), which already contains both theorems below.

Let us recall the major notions. Let $F(Z)$ be the set of all function from
some non-empty set $Z$ into the real numbers. Consider a set $R\subset F(Z)$,
$R\neq\emptyset$, and a mapping $I:R\to\mathbb{R}$. If the conditions

  1. (i)

(Additivity) If $f,g\in R$, then $f+g\in R$ and $I(f+g)=I(f)+I(g)$.

  2. (ii)

(Homogeneity) If $f\in R$ and $r\in\mathbb{R}$, then $rf\in R$ and
$I(rf)=rI(f)$.

  3. (iii)

(Monotonicity) If $f\in R$, then $|f|\in R$ and $I(f)\leq I(|f|)$.

  4. (iv)

(Continuity) If $f_{1},f_{2},\ldots\in R$ with $0\leq f_{n+1}\leq f_{n}$ for
all $n\in\mathbb{N}$ and if $\inf_{n}f_{n}=0$, then $\inf_{n}I(f_{n})=0$.

are fulfilled, then we call $R$ a Riesz space over $Z$ and $I$ an (elementary)
Daniell integral, or briefly an integral on $R$. As usual, all operations or
relations of functions are pointwisely defined. From the celebrated
integration theory of Daniell, as presented e.g. in [1, 2, 6, 8], we shall
rest upon the extention of such integrals and upon their convergence
properties in the spirit of Lebesgue’s integration theory.

Now let $S\subset F(X)$ and $T\subset F(Y)$ be two Riesz spaces. Their tensor
product is the vector space

$S\otimes T=\Big{\\{}\sum_{i=1}^{k}f_{i}\otimes g_{i}:k\in\mathbb{N},f_{i}\in
S,g_{i}\in T\text{ for all }1\leq i\leq k\Big{\\}},$

where $f\otimes g\in F(X\times Y)$ with $f\in S$, $g\in T$ is given by
$f\otimes g(x,y):=f(x)g(y)$, $x\in X,y\in Y$. In general, this tensor product
may (severely) fail to be a Riesz space. The smallest Riesz space within
$F(Z)$ containing some set $M\subset F(Z)$ is denoted by $R(M)$.

Here comes our first main result.

### Theorem 1.

Let $J:S\to\mathbb{R}$ and $K:T\to\mathbb{R}$ be two integrals. Then there
exists an integral $I:R\to\mathbb{R}$ on the Riesz space $R=R(S\otimes T)$
such that for all $f\in S$, $g\in T$

$I(f\otimes g)=J(f)K(g).$

By this property $I$ is uniquely determined.

We name $I$ the product integral of $J$ and $K$ and write $I=J\otimes K$.

Next, the question arises whether as well a Fubini-type theorem can be
obtained, reducing the product integral to iterated integrals. In the
literature Fubini theorems for the Daniell integral have been obtained under
the consistent assumption that a product integral exists, see [3, 4, 7]. This
is now guaranteed by our theorem. Thus, we may link to Stone’s [7] Fubini
theorem. To this end we introduce some notations. For any $f\in F(X\times Y)$
define $f^{x}\in F(Y)$, $x\in X$, by $f^{x}(y):=f(x,y)$ for all $y\in Y$. If
$f^{x}\in T$ for all $x\in X$, let $Kf\in F(X)$ be given by $Kf(x):=K(f^{x})$.
Guided by Stone’s way of writing we set

$S\ast K:=\\{f\in F(X\times Y):f^{x}\in T\text{ for all }x\in X\text{ and
}Kf\in S\\}$

and analoguously (with self-explaining notation)

$J\ast T:=\\{f\in F(X\times Y):f_{y}\in S\text{ for all }y\in Y\text{ and
}Jf\in T\\}.$

Now, Stone’s Fubini theorem applies to the product integral $I$ provided
$R\subset S\ast K$ or $R\subset J\ast T$. However, in general these
requirements might be too much to ask for. Instead, we proceed from $J$ to its
unique extension $\bar{J}$ on the Riesz space $\bar{S}$ of $J$-integrable
real-valued functions on $X$. (In order to make $\bar{S}$ a Riesz space, we
leave aside those integrable functions, taking the values $\pm\infty$ or not
being defined everywhere.) Similarly, we obtain $\bar{T}$ and $\bar{K}$ from
$K$.

### Theorem 2.

For any two integrals $J:S\to\mathbb{R}$, $K:T\to\mathbb{R}$ and their product
integral $I:R\to\mathbb{R}$ we have $R\subset\bar{S}\ast\bar{K}$,
$R\subset\bar{J}\ast\bar{T}$ and for all $f\in R$

$I(f)=\bar{J}(\bar{K}f)=\bar{K}(\bar{J}f).$

Within this constellation we may now apply the Fubini-Stone theorem to the
triple $I$, $\bar{J}$, $\bar{K}$. The theorem tells that, by taking into
account certain null-sets, also the extended integral $\bar{I}$ may be
obtained by iterated integration of $\bar{J}$ and $\bar{K}$. For details we
refer to [7, 8].

##  2 Lebesgue lattices

### Definition.

Let $M\subset F(Z)$, $M\neq\emptyset$. Then $P(M)$ denotes the set of all
functions $f\in F(Z)$, which possess a sequence $f_{0},f_{1},f_{2},\ldots\in
M$ such that $|f_{n}|\leq f_{0}$ for all $n\in\mathbb{N}$ and
$f(z)=\lim_{n\to\infty}f_{n}(z)$ for all $z\in Z$.

If $R$ is a Riesz space over $Z$ and $P(R)\subset R$, then we call $R$ a
Lebesgue lattice. The smallest Lebesgue lattice within $F(Z)$ containing some
set $M\subset F(Z)$ is denoted by $L(M)$.

We shall use the following facts on Lebesgue lattices.

### Lemma 1.

Let $f$ and $g$ be elements of a Lebesgue lattice $L$ over $Z$. Denote by
$U(g)$ the set $\\{z\in Z:g(z)\neq 0\\}$ and by $1_{U(g)}$ its characteristic
function. Then $f\cdot 1_{U(g)}\in L$.

###### Proof.

If $f\geq 0$, set $f_{n}:=\inf(f,n|g|)\in L$. Then $0\leq f_{n}\leq
f_{n+1}\leq f$, thus $f\cdot 1_{U(g)}=\lim_{n}f_{n}\in L$. In general, $f\cdot
1_{U(g)}=f^{+}\cdot 1_{U(g)}-f^{-}\cdot 1_{U(g)}\in L$. ∎

### Lemma 2.

Let $R$ be a Riesz space over $Z$ and let the set $M\subset F(Z)$ satisfy
$R\subset M$ and $P(M)\subset M$. Then $L(R)\subset M$.

###### Proof.

Let $L\subset F(Z)$ be the smallest set containig $R$ and obeying $P(L)\subset
L$. Thus $R\subset L\subset M$. We show that $L$ is a Lebesgue lattice.

First, let $L_{R}:=\\{f\in L:|f|\leq h$ for some $h\in R\\}$. Obviously,
$R\subset L_{R}$ and $P(L_{R})\subset L_{R}$. Thus $L_{R}=L$, saying that for
each $f\in L$ there is a $h\in R$ such that

$|f|\leq h.$

Next, for $f\in L$ set $L_{f}:=\\{g\in L:g+f\in L\\}$. Then $P(L_{f})\subset
L_{f}$. Indeed, consider a convergent sequence $g_{0},g_{1}\ldots$ in $L_{f}$
with limit $g$ and with $|g_{n}|\leq g_{0}$ for all $n\in\mathbb{N}$. Then
there are $h,k\in R$ such that $|f|\leq h$, $|g_{0}|\leq k$. It follows that
$g\in L$, $g_{n}+f\in L$ and $|g_{n}+f|\leq h+k$ with $h+k\in R\subset L$.
Also $g+f=\lim_{n}(g_{n}+f)$. Since $P(L)\subset L$, we obtain $g+f\in L$ and
thus $g\in L_{f}$.

Moreover, $R\subset L_{f}$ for $f\in R$. Then $L_{f}=L$, meaning that $f+g\in
L$ for $f\in R$ and $g\in L$. This says that $R\subset L_{g}$ for any $g\in
L$, which in turn implies $L_{g}=L$ for any $g\in L$. Consequently $f+g\in L$
for any $f,g\in L$.

Finally, for $c\in\mathbb{R}$ let $L_{c}:=\\{f\in L:|f|\in L$ and $cf\in
L\\}$. Then, again using $|f|\leq h\in R$ and $ch\in R$, we obtain $R\subset
L_{c}$ and $P(L_{c})\subset L_{c}$, yielding $L_{c}=L$. Thus $|f|\in L$ and
$cf\in L$ for any $f\in L$.

Altogether, $L$ is a Riesz space with $P(L)\subset L$, and consequenty
$L=L(R)$. This concludes the proof. ∎

Next we consider integrals $I$ on a Lebesgue lattice $L$. We point out the
well-known fact that for such integrals the theorem of dominated convergence,
the analog of Lebesgue’s convergence theorem holds: If $f_{0},f_{1},\ldots\in
L$ with $|f_{n}|\leq f_{0}$ and $f=\lim_{n}f_{n}$, then

$I(f)=\lim_{n}I(f_{n}).$ (1)

### Lemma 3.

Let $I:R\to\mathbb{R}$ be an integral on the Riesz space $R$. Then there
exists a unique integral $\bar{I}:L(R)\to\mathbb{R}$ such that
$\bar{I}(f)=I(f)$ for all $f\in R$.

###### Proof.

It is an immediate consequence of the theory of Daniell that the space of all
$I$-integrable real-valued functions forms a Lebesgue lattice and that the
common extension of $I$ makes an integral on this Lebesgue lattice.
Restricting this object to $L(R)$ gives the desired integral $\bar{I}$.
Moreover, if $I^{\prime}$ is another extension of $I$ to $L(R)$, then
$M:=\\{f\in L(R):I^{\prime}(f)=\bar{I}(f)\\}$ obeys $R\subset M$ and because
of (1) $P(M)\subset M$. Lemma 2 implies $L(R)\subset M$, hence
$I^{\prime}=\bar{I}$. ∎

##  3 Proof of the theorems

Basically we are faced with the task to extent a functional $I$ from a vector
space $U\subset F(Z)$ (here a tensor product) to a Riesz space $R\supset U$.
Denoting

$|U|:=\\{|f|:f\in U\\},$

we approach this task by constructing an infinite sequence $U=U_{1}\subset
U_{2}\subset\cdots$ of vector spaces which satisfy $|U_{m}|\subset U_{m+1}$
for all $m\geq 1$. Then $R:=\bigcup_{m=1}^{\infty}U_{m}$ is apparently a Riesz
space.

### Lemma 4.

Let $U$ be a vector space. Then the set $P(U)$ is a vector space, too. If
additionally $|U|\subset P(U)$, then we have $U\subset P(U)$ and
$|P(U)|\subset P(P(U))$.

###### Proof.

It is straightforward to show that $P(U)$ is a vector space. Thus let
$|U|\subset P(U)$. This means that for any $f\in U$ there is a sequence
$f_{0},f_{1},\ldots$ in $U$ converging to $|f|$ and obeying $|f_{n}|\leq
f_{0}$ for all $n\geq 1$, which implies $|f|\leq f_{0}$. Hence, by considering
the sequence $f_{0},f,f,\ldots$ in $U$ we realize that $f\in P(U)$, too. Thus
$U\subset P(U)$.

Now, let $g\in|P(U)|$. Then $g=|f|$, where $f$ is the limit of some sequence
$f_{0},f_{1},\ldots$ in $U$ satisfying $|f_{n}|\leq f_{0}$. From $|U|\subset
P(U)$ we obtain that $|f_{n}|\in P(U)$ and $f_{0}=|f_{0}|\in P(U)$. Hence,
considering the sequence $f_{0},|f_{1}|,|f_{2}|,\cdots$ yields $|f|\in
P(P(U))$. Thus $|P(U)|\subset P(P(U))$. ∎

### Lemma 5.

Let $U\subset F(Z)$ be a vector space fulfilling $|U|\subset P(U)$. Define
recursively $U_{1}:=U$, $U_{m+1}:=P(U_{m})$ for $m\geq 1$ and set
$R:=\bigcup_{m=1}^{\infty}U_{m}$. Then we have:

  * (i)

$R$ is a Riesz space containing $U$.

  * (ii)

If $U\subset M$ and $P(M)\subset M$ for some $M\subset F(Z)$, then $R\subset
M$.

  * (iii)

If integrals $I$ and $I^{\prime}$ on $R$ coincide on $U$, then $I=I^{\prime}$.

###### Proof.

By assumption $|U_{1}|\subset U_{2}$ and by means of Lemma 4 $U_{1}\subset
U_{2}$ and $|U_{2}|\subset U_{3}$. Induction yields $U_{1}\subset
U_{2}\subset\cdots$ and $|U_{m}|\subset U_{m+1}$ for all $m\geq 1$.
Consequently, as stated above, (i) holds true.

Moreover, from $U\subset M$ we obtain $U_{2}=P(U)\subset P(M)\subset M$ and
via induction $U_{m}\subset M$ for all $m\geq 1$. Similar, from
$I|U=I^{\prime}|U$ it follows by means of (1) that
$I|U_{2}=I|P(U)=I^{\prime}|P(U)=I^{\prime}|U_{2}$ and by induction
$I|U_{m}=I^{\prime}|U_{m}$. These statements imply (ii) and (iii). ∎

### Example.

If $U\subset F(Z)$ is an algebra of bounded functions and if there is a
$f_{0}\in U$ with $\inf f_{0}>0$, then $U$ meets the assumptions of Lemma 5.
For the proof recall (from proofs of the Stone-Weierstrass theorem) that the
sequence of polynomials $p_{n}(t)$, $0\leq t\leq 1$, given recursively by
$p_{1}(t)=0$, $p_{n+1}(t)=(t-p_{n}^{2}(t))/2+p_{n}(t)$, satisfies $0\leq
p_{n}(t)\leq p_{n+1}(t)\leq\sqrt{t}$ and $\lim_{n}p_{n}(t)=\sqrt{t}$. For
$f\in U$ also $f_{n}:=p_{n}(f^{2})\in U$. Without loss we assume that
$\sup|f|\leq 1$. Then $\lim_{n}f_{n}=\sqrt{f^{2}}=|f|$. Also $0\leq
f_{n}\leq|f|\leq\lambda f_{0}$ for a suitable $\lambda>0$, hence $|f|\in
P(U)$.

This framework applies to product spaces as follows.

### Lemma 6.

Let $V$ and $W$ be Lebesgue lattices over $X$ and $Y$ and let $U=V\otimes W$.
Then $|U|\subset P(U)$. Also, for any $f\in U$ there are $g\in V^{+}$, $h\in
W^{+}$ such that $|f|\leq g\otimes h$.

###### Proof.

Let $f=\sum_{i=1}^{k}g_{i}\otimes h_{i}$ with $g_{1},\ldots,g_{k}\in V$ and
$h_{1},\ldots,h_{k}\in W$. We set

$g=\sum_{i=1}^{k}|g_{i}|,\quad h:=\sum_{i=1}^{k}|h_{i}|,$

thus $g\in V^{+}$, $h\in W^{+}$. Obviously, $|f|\leq g\otimes h$. Further, for
any $n\in\mathbb{N}$ we put

$A_{a_{1}\ldots a_{k}n}:=\Big{\\{}\frac{a_{1}}{n}\cdot g\leq
g_{1}<\frac{a_{1}+1}{n}\cdot
g\Big{\\}}\cap\cdots\cap\Big{\\{}\frac{a_{k}}{n}\cdot g\leq
g_{k}<\frac{a_{k}+1}{n}\cdot g\Big{\\}}$

with integers $a_{1},\ldots,a_{k}\in\\{-n,\ldots,n\\}$. Apparently, these sets
yield a partition of the domain $X$. Similar the sets

$B_{b_{1}\ldots b_{k}n}:=\Big{\\{}\frac{b_{1}}{n}\cdot h\leq
h_{1}<\frac{b_{1}+1}{n}\cdot
h\Big{\\}}\cap\cdots\cap\Big{\\{}\frac{b_{k}}{n}\cdot h\leq
h_{k}<\frac{b_{k}+1}{n}\cdot h\Big{\\}}$

with $b_{1},\ldots,b_{k}\in\\{-n,\ldots,n\\}$ make a partition of $Y$. Note
that

$1_{A_{a_{1}\ldots a_{k}n}}=\prod_{i=1}^{k}(1_{U(((a_{i}+1)\cdot g-n\cdot
g_{i})^{+})}-1_{U((a_{i}\cdot g-n\cdot g_{i})^{+})}).$

Thus, expanding the right-hand product and applying Lemma 1 iteratively, we
obtain that $g\cdot 1_{A_{a_{1}\ldots a_{k}n}}$ belongs for all
$a_{1},\ldots,a_{k}$ to the Lebesgue lattice $V$, and $h\cdot
1_{B_{b_{1}\cdots b_{k}n}}$ to $W$. Therefore

$f_{n}:=\sum_{a_{1},b_{1},\ldots
a_{k},b_{k}=-n}^{n}\Big{(}\frac{a_{1}b_{1}}{n^{2}}+\cdots+\frac{a_{k}b_{k}}{n^{2}}\Big{)}\Big{(}(g\cdot
1_{A_{a_{1}\ldots a_{k}n}})\otimes(h\cdot 1_{B_{b_{1}\ldots b_{k}n}})\Big{)}$

is an element of $V\otimes W$. Since the sets $A_{a_{1}\ldots a_{k}n}\times
B_{b_{1}\ldots b_{k}n}$ make a partition of $X\times Y$, there will be for
each $f_{n}(x,y)$ with $x\in X$ and $y\in Y$ at most one non-vanishing summand
in the previous sum. Hence

$|f_{n}|=\sum_{a_{1},b_{1},\ldots
a_{k},b_{k}=-n}^{n}\Big{|}\frac{a_{1}b_{1}}{n^{2}}+\cdots+\frac{a_{k}b_{k}}{n^{2}}\Big{|}\Big{(}(g\cdot
1_{A_{a_{1}\ldots a_{k}n}})\otimes(h\cdot 1_{B_{b_{1}\ldots b_{k}n}})\Big{)},$

implying that $|f_{n}|$ belongs to $V\otimes W$, too. Moreover, $|f_{n}|\leq
k\cdot g\otimes h$.

Finally, for $x\in A_{a_{1}\ldots a_{k}n}$ and $y\in B_{b_{1}\ldots b_{k}n}$,
since $|g_{i}|\leq g$,

$\displaystyle\Big{|}g_{i}(x)h_{i}(y)$
$\displaystyle-\frac{a_{i}b_{i}}{n^{2}}g(x)h(y)\Big{|}$
$\displaystyle\leq\Big{|}g_{i}(x)\Big{(}h_{i}(y)-\frac{b_{i}}{n}h(y)\Big{)}\Big{|}+\Big{|}\frac{b_{i}}{n}h(y)\Big{(}g_{i}(x)-\frac{a_{i}}{n}g(x)\Big{)}\Big{|}$
$\displaystyle\leq\frac{2}{n}g(x)h(y).$

Therefore, in view of

$\displaystyle f-f_{n}=\sum_{a_{1},b_{1},\ldots
a_{k},b_{k}=-n}^{n}\sum_{i=1}^{k}\Big{(}g_{i}\otimes
h_{i}-\frac{a_{i}b_{i}}{n^{2}}g\otimes h\Big{)}1_{A_{a_{1}\ldots a_{k}n}\times
B_{b_{1}\ldots b_{k}n}},$

we obtain

$\big{|}|f|-|f_{n}|\big{|}\leq|f-f_{n}|\leq\frac{2k}{n}g\otimes h.$

Consequently, $\lim_{n}|f_{n}|=|f|$. Recalling $|f_{n}|\in V\otimes W$ and
$|f_{n}|\leq k\cdot g\otimes h$, we end up with $|f|\in P(V\otimes W)$. ∎

### Lemma 7.

Let $V$ and $W$ be Lebesgue lattices with domains $X$ and $Y$, and let
$J:V\to\mathbb{R}$ and $K:W\to\mathbb{R}$ be integrals. Then $L(V\otimes
W)\subset S\ast K$, and by

$I(f):=J(Kf),$

$f\in L(V\otimes W)$, an integral $I:L(V\otimes W)\to\mathbb{R}$ is given. It
is uniquely determined by the condition

$I(g\otimes h)=J(g)K(h)$

for all $g\in V$, $h\in W$.

###### Proof.

Let $M:=\\{f\in S\ast K:|f|\leq g\otimes h\text{ for some }g\in V^{+},h\in
W^{+}\\}$. We first show $P(M)\subset M$. Indeed: Let $f_{0},f_{1},\ldots\in
M$ with $|f_{n}|\leq f_{0}$ for $n\geq 1$, and let $f=\lim_{n}f_{n}$. Then
$|f_{n}|\leq g\otimes h$ for some $g\in V^{+}$, $h\in W^{+}$. Thus
$|f_{n}^{x}|\leq g(x)\cdot h$, therefore $f^{x}=\lim_{n}f_{n}^{x}\in W$ and by
(1) $K(f^{x})=\lim_{n}K(f_{n}^{x})$ for all $x\in X$. Moreover, $Kf_{n}\in V$
and $|Kf_{n}|\leq K(h)\cdot g$. Therefore, $Kf=\lim_{n}Kf_{n}\in V$ implying
$f\in S\ast K$. Since $|f|\leq g\otimes h$, also $f\in M$. Thus $P(M)\subset
M$. Note also that because of (1)

$\displaystyle J(Kf)=\lim_{n}J(Kf_{n}).$ (2)

Now, set $U=S\otimes T$. Because of Lemma 6 we may resort to Lemma 5. Define
$R\supset U$ as in this lemma. Since $P(L(U))\subset L(U)$, we have $R\subset
L(U)$ in view of Lemma 5 (ii). Therefore, $L(U)=L(R)$. Also, $U\subset M$,
hence $R\subset M$ because of Lemma 5 (ii), and $L(R)\subset M$ by means of
Lemma 5 (i) and Lemma 2. This implies, $L(U)\subset M\subset S\ast K$, thus
the functional $I$ is well-defined and in view of (2) an integral.

As to uniqueness, let $I^{\prime}$ be another integral on $L(U)$ with
$I^{\prime}|U=I|U$. Then by means of Lemma 5 (iii) $I^{\prime}|R=I|R$ and by
Lemma 3 $I^{\prime}|L(R)=I|L(R)$, thus $I^{\prime}=I$. ∎

###### Proof of Theorem 1.

Let $J:S\to\mathbb{R}$ and $K:T\to\mathbb{R}$ be integrals on the Riesz spaces
$S$ and $T$. Define the induced Lebesgue spaces $V=L(S)$ and $W=L(T)$. By
means of Lemma 3 we have unique extensions $\bar{J}:V\to\mathbb{R}$ and
$\bar{K}:W\to\mathbb{R}$ of $J$ and $K$. Lemma 7 gives an integral
$\bar{I}:L(V\otimes W)\to\mathbb{R}$, which is obtained by double integration.
In particular, $\bar{I}(g\otimes h)=\bar{J}(g)\cdot\bar{K}(h)$ for $g\in V$,
$h\in W$. Restricting this integral to $R(S\otimes T)$ gives the existence of
the desired integral $I$.

It remains to prove uniqueness. So, let $I:R(S\otimes T)\to\mathbb{R}$ be any
integral obeying the condition of Theorem 1, and let $\bar{I}$ its extension
to $L(R(S\otimes T))=L(S\otimes T)$ as given by Lemma 3. For $g\in S$ let

$L_{g}:=\big{\\{}h\in L(T):g\otimes h\in L(S\otimes T)\text{ and
}\bar{I}(g\otimes h)=J(g)\bar{K}(h)\big{\\}}.$

Then $T\subset L_{g}$, also $P(L_{g})\subset L_{g}$ because of (1). Thus by
Lemma 2 $L(T)\subset L_{g}$, indeed $L_{g}=L(T)$.

Thus, if $g\in S$ and $h\in L(T)$, then

$\bar{I}(g\otimes h)=J(g)\bar{K}(h).$

So, if we define for $h\in L(T)$

$L^{h}:=\big{\\{}g\in L(S):g\otimes h\in L(S\otimes T)\text{ and
}\bar{I}(g\otimes h)=\bar{J}(g)\bar{K}(h)\\},$

then by the same reasoning we get $L^{h}=L(S)$.

Thus, if $g\in L(S)$ and $h\in L(T)$, then

$\bar{I}(g\otimes h)=\bar{J}(g)\bar{K}(h).$

Employing Lemma 7 we obtain

$\bar{I}(f)=\bar{J}(\bar{K}f)$

for any $f\in L(L(S)\otimes L(T))=L(S\otimes T)$. Thus $\bar{I}$ is completely
determined by $\bar{J}$ and $\bar{K}$, respectively by $J$ and $K$. By
restricting $\bar{I}$ to $I$, uniqueness follows. ∎

###### Proof of Theorem 2.

The formula $I(f)=\bar{J}(\bar{K}f)$ has already been obtained in the
preceding proof. By interchanging the roles of $J$ and $K$, and because of
uniqueness, we also have $I(f)=\bar{K}(\bar{J}f)$. ∎

## References

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  * [3] J. Mikusiński, The Bochner Integral. Acad. Press, 1978.
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  * [8] A. E. Taylor, General Theory of Functions and Integration. Blaisdell Publishing Co., 1965.


aainstitutetext: Departamento de Física, Universidad Técnica Federico Santa
María, Casilla 110, Valparaíso, Chile.bbinstitutetext: Nordita, KTH Royal
Institute of Technology and Stockholm University, Hannes Alfvéns väg 12,
SE-106 91 Stockholm, Sweden

# Tilted Dirac superconductor at quantum criticality: Restoration of Lorentz
symmetry

Pablo Reiser a,b Vladimir Juričić<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Lorentz symmetry appears as a quite robust feature of the strongly interacting
Dirac materials even though the lattice interactions break such a symmetry. We
here demonstrate that the Lorentz symmetry is restored at the quantum-critical
point (QCP) separating the tilted Dirac semimetal, breaking this symmetry
already at the noninteracting level, from a gapped $s-$wave superconducting
instability. To this end, we employ a one-loop $\epsilon=(3-D)-$expansion
close to the $D=3$ upper critical dimension of the corresponding Gross-Neveu-
Yukawa field theory. In particular, we show that the tilt parameter is
irrelevant and ultimately vanishes at the QCP separating the two phases. In
fact, as we argue here, such a Lorentz symmetry restoration may be generic for
the strongly interacting tilted Dirac semimetals, irrespective of whether they
feature mirror-symmetric or mirror-asymmetric tilting, and is also insensitive
to whether the instability represents an insulator or a gapped superconductor.
The proposed scenario can be tested in the quantum Monte Carlo simulations of
the interacting tilted Dirac fermion lattice models.

††arxiv: 2311.12797

##  1 Introduction

The Dirac crystals featuring Dirac quasiparticles as their low-energy
excitations exhibiting a linear dispersion opened up a new paradigm in the
modern condensed matter physics CastroNeto_09 ; Balatsky2014 ; armitage2018 .
They also bridge the gap between the seemingly unrelated phenomena in the
realms of high-energy and condensed-matter physics through the emergence of
the Dirac and Weyl quasiparticles, being the prime examples in this respect.
Such Dirac materials feature an emergent relativistic-like Lorentz symmetry
because of the Dirac quasiparticles’ linear energy-momentum dispersion,
$E({\bf k})=v_{F}k$, with the Fermi velocity, $v_{F}$, typically a few hundred
times smaller than the speed of light, playing the role of an effective
velocity of light in their crystalline universe. Such a relativistic-like
Lorentz symmetry turns out to be quite a robust feature of the Dirac materials
even when the electron-electron interactions that explicitly break it are
taken into account. For instance, the Lorentz symmetry emerges in the deep
infrared regime for the two-dimensional (2D) Dirac fermions coupled via the
long-range Coulomb interaction, where the speed of light takes over as the
common velocity for both the Dirac fermions and the photons Gonzalez1994 .
Furthermore, for Yukawa (short-range) interacting Dirac fermions, the quantum-
critical point (QCP) separating a semimetallic and a strongly-coupled gapped
(insulating or superconducting) phase is Lorentz symmetric SSLee2007 ;
Isobe2012 ; Roy2016 ; Roy2018 ; Roy2020 . This issue has been also recently
addressed beyond the realm of the usual Hermitian Dirac materials, by taking
into account the effects of the non-Hermiticity, that may arise, e.g. from the
coupling with an environment, resulting in the Lorentz symmetry emerging at
low energies juricic2023yukawalorentz ; murshed2023quantum . These results
point towards a ubiquitous emergence of the Lorentz symmetry at low energy in
the Dirac materials, even though such a symmetry is absent at the lattice (UV)
energy scale.

To obtain further insights into the ubiquity of such an emergent Lorentz
symmetry, we here consider tilted Dirac materials, featuring low-energy
quasiparticles with tilted Dirac dispersion, explicitly breaking this symmetry
at the lattice scale already at the noninteracting level. Such an explicit
symmetry breaking is realized by the direction of the tilt term in the
dispersion of the tilted Dirac fermions (TDFs), which is dictated by the
underlying crystalline lattice, and is represented by a single term
$\sim\alpha$, with $\alpha$ as the tilt parameter, in a Dirac Hamiltonian, as
shown in Eqs. (3) and (4). Introduction of such a tilt term may be considered
as possibly the simplest Lorentz-symmetry-breaking deformation of an otherwise
Lorentz-symmetric noninteracting Dirac Hamiltonian. Its effects have been
studied in various contexts, for instance, it was argued that TDFs may
represent a platform for the simulation of the curved space-time in condensed-
matter systems Volovik2016 ; Nissinen2017 ; Jafari-PRB2019 ; Ojanen-PRR2019 ;
Jafari-PRR2020 ; Schmidt-SciPost2021 ; Volovik2021 ; Meng-PRR2022 ; vanWezel-
PRR2022 ; vanderBrink-PRB2023 , while the quantum transport Bergholtz-PRB2015
; Carbotte-PRB2016 ; Stoof-PRB2017 ; Rodriguez-Lopez2020 ; Maytorena-PRB2021 ;
Hao-Ran-PRB2022 ; Portnoi-PRB2022 , the effects of the long- and short-range
components of the Coulomb interaction Fritz-PRB2019 ; Rostami-PRR2020 ;
RudneiRamos-PRB2021 ; RudneiRamos-PRB2023 , charged impurities Bergholtz-
PRB2017 ; Fritz-PRB2017 ; GZLiu-PRB2018 ; YWLee-PRB2019 , and the magnetic
field Goerbig-PRB2008 ; Carbotte-PRB-2018 ; Nag_2021  have also been
addressed. We emphasize that the tilted Dirac dispersion has been
theoretically investigated in the materials using first-principle techniques
Soluyanov2015 ; Schnyder2017 ; Geilhufe-2018 . In fact, the TDFs have been
realized in a plethora of quantum materials Koji-2006 ; Akito-2007 ; Koji-2014
; Deng2016 ; Yun-Wu2016 ; ChenluWang-2016 ; Kenan-Zhang2017 ; Bihlmayer-
PRB2017 , both with the subcritical tilt (type-1), featuring the point-like
Fermi surfaces, and in overtilted materials (type-2), with the electron- and
hole-like Fermi surfaces, putting the tilted Dirac crystals within the
landscape of the quantum materials.

In this article, we explore the effects of the $s-$wave superconducting
instability in the type-1 tilted Dirac materials. Our main result is that at
the strong-coupling QCP separating the tilted Dirac semimetal and the $s-$wave
SC phase, the tilt operator becomes _irrelevant_ and therefore the Lorentz
symmetry gets restored. To address this problem, we employ the Gross-Neveu-
Yukawa (GNY) theory Zinn_book ; Rosenstein1993 ; sslee-2007 ; RJH-PRB2013 ;
RY2013 ; Ponte_2014 ; Jian2015 ; Zerf2016 , since this instability necessarily
occurs at strong coupling, as can be seen already at the mean-field level [Eq.
(9)], due to a vanishing density of states (DOS) at zero energy [Eq. (10)]. To
account for the effects of both fermionic and order-parameter (OP)
fluctuations in the vicinity of such a QCP, since both the Yukawa and
$\Phi^{4}-$couplings [Eq. (19)] are marginal in three spatial dimensions
($D=3$), we use the renormalization group (RG) analysis within the
$\epsilon=(3-D)$-expansion. We demonstrate that the tilt parameter in the
mirror-symmetric [Eq. (3)] and mirror-symmetry-breaking [Eq. (4)] tilted Dirac
semimetals are irrelevant at the superconducting QCP, with the corresponding
RG $\beta-$function given by Eq. (25) [see also Fig. 1], and the flow shown in
Fig. 2, therefore implying the restoration of the Lorentz symmetry at such a
QCP. To further corroborate the ubiquity of this scenario, we show that the
transition into a CDW phase also features the emergent Lorentz symmetry at low
energy, albeit with a slower rate of the restoration, as shown in Eq. (34),
see also Fig. 3 and Fig. 1. We argue that such an emergence of the Lorentz
symmetry is generic for the TDFs interacting via a short-range Coulomb
interaction, with the rate that depends on whether the OP commutes or
anticommutes with the tilt term. Finally, the anticommutation makes the rate
of the restoration faster, as can be seen explicitly in the case of the SC and
CDW instabilities.

The paper is organized as follows. In Sec. 2, we introduce the low-energy
theory for the 2D TDFs, while in Sec. 3, we analyze the short-range
interactions at the mean-field level. We then proceed with the setup of the
quantum-critical GNY field theory in Sec. 4, and in Sec. 5 we perform its RG
analysis. Our conclusions are presented in Sec. 6, and additional technical
details are relegated to the Appendix A.

##  2 Tilted Dirac superconductor: The low-energy theory

We consider a low-energy effective theory for the Dirac fermion quasiparticles
around the two inequivalent valleys, as for instance, two Dirac points at the
two inequivalent corners of the Brillouin zone in graphene. To construct the
Nambu-doubled basis for the Dirac superconductor, we first write an eight-
component Dirac spinor given by Szabo-RoyPRB2021

$\Psi_{\omega,\mathbf{k}}=\begin{pmatrix}c^{(+)}_{\uparrow}&c^{(-)}_{\uparrow}&c^{(+)}_{\downarrow}&c^{(-)}_{\downarrow}\end{pmatrix}^{T},\quad
c^{v}_{s}=\begin{pmatrix}a^{v}_{s},&b^{v}_{s}\end{pmatrix},$ (1)

where $a^{v}_{s}$ and $b^{v}_{s}$ are the fermionic fields corresponding to
two different ($A$ and $B$) sublattices with valley index $v=(\pm)$, and spin
projection $s=\uparrow,\downarrow$. Then, we introduce a particle-hole degree
of freedom and write a doubled, sixteen-component Dirac-Nambu spinor as

$\Psi_{\omega,\mathbf{k}}\rightarrow\Psi_{\text{Nam}}=\begin{pmatrix}\Psi_{\omega,\mathbf{k}}\\\
\Gamma_{210}\Psi^{*}_{-\omega,-\mathbf{k}}\end{pmatrix}.$ (2)

We conveniently write matrices acting on the sixteen-dimensional Dirac-Nambu
spinors as
$\Gamma_{\mu\nu\lambda\rho}=\eta_{\mu}\otimes\sigma_{\nu}\otimes\tau_{\lambda}\otimes\alpha_{\rho}$.
Here, $\\{\alpha\\},\\{\tau\\},\\{\sigma\\}$ and $\\{\eta\\}$ are sets of
Pauli matrices operating on the sublattice, valley, spin and particle-hole
degrees of freedom, respectively, with $\mu,\nu,\lambda,\rho=0,1,2,3$, and the
index $0$ corresponding to the $2\times 2$ unity matrix. Notice that the lower
component of the Dirac-Nambu spinor in Eq. (2) is a time-reversal partner of
the upper one, since the matrix
$\Gamma_{210}=\sigma_{2}\otimes\tau_{1}\otimes\alpha_{0}$ is the unitary part
of the time reversal operator in the undoubled basis. Furthermore, the form of
the $\Gamma-$matrices dictates that any two of them either commute or
anticommute. To simplify notation, we suppress subscript “Nam” hereafter,
$\Psi\equiv\Psi_{\text{Nam}}$ and
$\Psi^{\dagger}\equiv\Psi_{\text{Nam}}^{\dagger}$.

Given this Dirac-Nambu representation, we can write the noninteracting low
energy effective Hamiltonian for the tilted Dirac fermions in the form
$\hat{H}=\sum_{\mathbf{k}}\Psi^{\dagger}(\mathcal{\hat{H}}(\mathbf{k})-\mu_{\rm
tot}\Gamma_{3000})\Psi$, with $\mu_{\rm tot}$ as the chemical potential, and

$\mathcal{\hat{H}}(\mathbf{k})=v\Gamma_{3031}k_{x}+v\Gamma_{3002}k_{y}+\alpha
v\Gamma_{3030}k_{x},$ (3)

where $v$ is the Fermi velocity and $\alpha$ represents the (dimensionless)
tilt parameter. We here consider only case with $|\alpha|<1$ (type-1)
featuring the point-like Fermi surfaces with the TDF quasiparticle
excitations, while the overtilted (type-2) system with $|\alpha|>1$ hosts a
particle- and a hole-like Fermi surfaces. Notice that the tilt term can be
thought of as a momentum-dependent chiral chemical potential, and it commutes
with the (total) chemical potential represented by the matrix $\Gamma_{3000}$.
We hereafter set $v=1$ and consider the system at half filling, $\mu_{\rm
tot}=0$, unless otherwise stated.

The form of the single-particle Hamiltonian implies its invariance under time
reversal, generated by $\mathcal{T}=\Gamma_{0210}{\rm K}$, with ${\rm K}$ as
the complex conjugation, SU(2) spin rotations generated by $\Gamma_{0s00}$,
with $s=\\{1,2,3\\}$, as well as the valley (sublattice) exchange, represented
by $V=\Gamma_{0010}$, $k_{x}\rightarrow-k_{x}$ ($S=\Gamma_{0001}$,
$k_{y}\rightarrow-k_{y}$) HJR-PRB2009 . Finally, the unitary particle-hole
symmetry is represented by $\mathcal{C}=\Gamma_{1000}$, so that
$\\{\mathcal{C},\mathcal{\hat{H}}({\bf k})\\}=0$, implying that its anti-
unitary counterpart is represented by the operator $\mathcal{C}\mathcal{T}$.
Notice that $\tilde{\mathcal{C}}=\Gamma_{2000}$ can also be taken as a unitary
particle-hole operator, which is a consequence of the U(1) symmetry of the
noninteracting Hamiltonian generated by the matrix $\Gamma_{3000}$
representing the electric charge operator in the Dirac-Nambu representation
employed here. Finally, we emphasize that the choice of the tilt term
($\sim\alpha$) in Eq. (3) ensures its invariance under all the mentioned
symmetries.

We point out that the mirror-symmetry breaking tilted Dirac Hamiltonian, with
the sign of the tilt equal for all the valleys (or other fermion flavors) is
obtained by breaking the valley exchange or mirror symmetry discussed above,
and we refer to it as _asymmetric_ tilt. The corresponding Hamiltonian then
takes the form

$\mathcal{\hat{H}_{\rm
AS}}(\mathbf{k})=v\Gamma_{3031}k_{x}+v\Gamma_{3002}k_{y}+\alpha
v\Gamma_{3000}k_{x},$ (4)

with the tilt term proportional to the matrix representing the chemical
potential. Therefore, the asymmetric tilt term effectively represents a
momentum-dependent chemical potential. As we show in the following, our
conclusions regarding the restoration of the Lorentz symmetry in the case of
the interacting fully symmetric tilted Dirac Hamiltonian [Eq. (3)] equally
apply to the asymmetrically tilted Hamiltonian in Eq. (4), as follows from the
form of the corresponding propagator, Eq. (18).

##  3 Mean-field analysis

To set the stage for the analysis of the strong-coupling superconducting
instability of the tilted Dirac semimetal, we first discuss mean-field (MF)
theory for a transition from the semimetallic to a fully gapped ordered state,
which is independent of the nature of the state, i.e. whether the state is an
insulator or a superconductor. For concreteness, we here subscribe to the
$s-$wave superconducting order parameter, which reads

$\phi(\mathbf{r},\tau)=\left\langle{\Psi^{\dagger}(\mathbf{r},\tau)(\Gamma_{1000}\cos{\varphi}+i\Gamma_{2000}\sin{\varphi})\Psi(\mathbf{r},\tau)}\right\rangle,$
(5)

where $\varphi$ is the U(1) phase of the superconducting OP. This OP
anticommutes with the particle number operator $N=\Gamma_{3000}$, and commutes
with all the three generators of spin rotations, $\Gamma_{0s00}$, as dictated
by its s-wave nature. Notice that both matrices in the $s$-wave OP
_anticommute_ with the tilted Dirac Hamiltonian in Eq. (3) which ensures that
the underlying ordered state is fully gapped. In the MF analysis, we set the
OP to be a constant, $\phi(\mathbf{r},\tau)\equiv m$, and fix the SC phase
$\varphi=0$. We assume that such a condensation is propelled by an effective
local density-density interaction of the form

$\mathcal{H}_{\text{int}}=\tilde{g}\int
d^{2}\mathbf{r}\left[\Psi^{\dagger}(\mathbf{r},\tau)\,\Gamma_{1000}\,\Psi(\mathbf{r},\tau)\right]^{2}.$
(6)

Then, we formulate the MF theory for the transition into such a state by
taking the Hamiltonian

${H}_{\rm MF}=\int d^{2}{\bf
r}\,\Psi^{\dagger}(\mathbf{r})\,\Big{\\{}\mathcal{\hat{H}}(\mathbf{k}\rightarrow-i\nabla_{{\bf
r}})+\mathcal{H}_{\text{int}}\Big{\\}}\Psi(\mathbf{r}),$ (7)

with the first term representing the free tilted Dirac Hamiltonian in Eq. (3),
together with the short-range interaction part, given by Eq. (6). We then
perform a Hubbard-Stratonovich transformation in the $s-$wave channel to
obtain the corresponding energy of the ground state at half filling ($\mu=0$)
in the form,

$\displaystyle\mathcal{E}[m]$
$\displaystyle=-\frac{1}{2\tilde{g}}m^{2}+2\sum_{w=\pm}\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}\left\\{w\alpha
k_{x}+\sqrt{k^{2}+m^{2}}\right\\}$
$\displaystyle=-\frac{1}{2\tilde{g}}m^{2}+4\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}\sqrt{k^{2}+m^{2}}.$
(8)

Therefore, the MF ground state energy is tilt-independent, yielding the same
form of the gap equation as in the untilted case, after minimization with
respect to $m$, and implies that

$m\sim(\tilde{g}-\tilde{g}_{c}),$ (9)

with the tilt-indenpendent critical coupling $\tilde{g}_{c}\sim\Lambda$
scaling with the ultraviolet cutoff, defining the energy scale up to which the
Dirac dispersion holds. Therefore, the transition into the $s-$wave (and, more
generally, any gapped) phase takes place at a strong coupling, which is a
consequence of the vanishing density of states (DOS) for the tilted Dirac
fermions

$~{}\rho(E,\alpha)=-\frac{1}{\pi}\lim_{\delta\to
0}\Im{\text{Tr}\,G_{F}(i\omega\rightarrow\omega+i\delta,\mathbf{k})}=\frac{4}{\pi}\frac{|E|}{(1-\alpha^{2})^{3/2}},$
(10)

for a single 16-component Dirac flavor. It is therefore expected that at any
finite-tilt fixed point of the RG, the critical value of the interaction
decreases with respect to the untilted ($\alpha=0$) value. Indeed this is the
case for unstable fixed points at finite tilt corresponding to the $s-$wave
and the CDW OPs, which can be obtained from Eq. (26) and (35), respectively,
which are, however, rendered unstable, as the tilt is irrelevant for both the
$s-$wave and the CDW OPs, see Eqs. (25) and (34).

##  4 Gross-Neveu-Yukawa quantum-critical field theory

We now address the behavior of the $(2+1)-$dimensional tilted $s-$wave Dirac
superconductor at the quantum-critical point within the framework of the
Gross-Neveu-Yukawa (GNY) field theory.

###  4.1  $S-$wave superconducting instability

First, since both noninteracting Hamiltonian and the OP are spin singlets, we
discard the spin subspace hereafter. We therefore consider eight-component
massless Dirac fermions endowed with $N_{F}$ copies, which interact with a
complex bosonic order parameter field via Yukawa interaction ($g$). The
$s-$wave superconducting OP is in the form given by Eq. (9), which after
discarding the spin subspace, reads

$\phi(\mathbf{r},\tau)=\left\langle{\Psi^{\dagger}(\mathbf{r},\tau)(\Gamma_{100}\cos{\varphi}+i\Gamma_{200}\sin{\varphi})\Psi(\mathbf{r},\tau)}\right\rangle,$
(11)

Hereafter, $\Psi\equiv\Psi(\mathbf{r},\tau)$,
$\Psi^{\dagger}\equiv\Psi^{\dagger}(\mathbf{r},\tau)$ and
$\phi\equiv\phi(\mathbf{r},\tau)$. The dynamics of the system is then
described by an imaginary time (Euclidean) action

$S=S_{F}+S_{Y}+S_{B},$ (12)

with the noninteracting fermionic part of the form

$S_{F}=\int d\tau
d^{D}\mathbf{r}\,\,\Psi^{\dagger}\left[\partial_{\tau}+\mathcal{\hat{H}}(\mathbf{k}\rightarrow-i\nabla_{\mathbf{r}})\right]\Psi,$
(13)

where $\tau$ is the imaginary time and $\mathcal{\hat{H}}(\mathbf{k})$
represents the Hamiltonian for noninteracting massless TDFs in Eq. (3). The
coupling of the massless Dirac fermions to the OP fluctuations takes the
characteristic Yukawa form,

$S_{Y}=g\int d\tau
d^{D}\mathbf{r}[(\text{Re}\phi)\Psi^{\dagger}\Gamma_{100}\Psi+(\text{Im}\phi)\Psi^{\dagger}i\Gamma_{200}\Psi].$
(14)

Lastly, the dynamics of the OP is described by the standard Ginzburg-Landau
action

$S_{B}=\int d\tau
d^{D}\mathbf{r}\left[\phi(-\partial^{2}_{\tau}-\nabla^{2}+m_{B}^{2})\phi+\frac{\lambda}{2}\phi^{4}\right],$
(15)

where $m_{B}^{2}$ is the tuning parameter for the continuous phase transition
with $m_{B}^{2}>0$ ($m_{B}^{2}<0$) in the $U(1)$ symmetric (symmetry broken)
phase. We furthermore set the bosonic and fermionic velocities to be equal to
unity in the critical region Roy2016 , see also Sec. A.1.3.

The fermionic and the bosonic propagators read, respectively,

$\displaystyle G_{F}(i\omega,\mathbf{k})=$ $\displaystyle\big{\\{}i\omega
A_{-}+B(k_{x}\Gamma_{001}+k_{y}\Gamma_{032})+A_{-}(\alpha
k_{x}\Gamma_{330}+\Gamma_{331}k_{x}+\Gamma_{302}k_{y})$ (16)
$\displaystyle+2\alpha k_{x}[\Gamma_{330}k^{2}+\alpha
k_{x}(k_{x}\Gamma_{331}+k_{y}\Gamma_{302})]\big{\\}}\frac{1}{A_{+}^{2}-B^{2}},$

and

$G_{B}(i\omega,\mathbf{k})=\frac{1}{\omega^{2}+k^{2}+m_{B}^{2}}.$ (17)

where $A_{\pm}\equiv A_{\pm}(i\omega,{\bf
k})=-\omega^{2}\pm\alpha^{2}k_{x}^{2}-k^{2}$ and $B\equiv B(i\omega,{\bf
k})=2\alpha i\omega k_{x}$.

We point out that the form of the propagator corresponding to the
asymmetrically tilted system, described by the Hamiltonian in Eq (4),
preserves the form in Eq. (16), since the asymmetric tilt commutes with the
untilted piece of the Dirac Hamiltonian. Explicitly, the propagator of the
asymmetric TDFs reads

$\displaystyle G_{F,AS}(i\omega,\mathbf{k})=$ $\displaystyle\big{\\{}i\omega
A_{-}+B(k_{x}\Gamma_{031}+k_{y}\Gamma_{002})+A_{-}(\alpha
k_{x}\Gamma_{300}+\Gamma_{331}k_{x}+\Gamma_{302}k_{y})$ (18)
$\displaystyle+2\alpha k_{x}[\Gamma_{300}k^{2}+\alpha
k_{x}(k_{x}\Gamma_{331}+k_{y}\Gamma_{302})]\big{\\}}\frac{1}{A_{+}^{2}-B^{2}},$

and such a structure of the propagator, as we show in the following, implies
that all the conclusions derived for the symmetric tilt equally apply to the
asymmetric case. We therefore consider only the fully symmetric TDFs.

##  5 Renormalization group analysis

Next, we proceed with the analysis of the U(1) GNY field theory. The
engineering (bare) dimensions of the couplings are
$\text{dim}[g^{2}]=\text{dim}[\lambda]=3-D$ and for the tilt parameter
$\text{dim}[\alpha]=0$, implying that $D=3$ is the upper critical dimension of
the theory. We therefore employ an $\epsilon$-expansion about the upper
critical dimension, $\epsilon=3-D$, to access the quantum-critical behavior in
$D=2$. To this end, we define the action $S_{R}=\int d\tau d\mathbf{r}L_{R}$,
where the renormalized Lagrangian is

$\displaystyle L_{R}=$
$\displaystyle\Psi^{\dagger}\left[Z_{\Psi}\left(\partial_{\tau}-iZ_{v}v\Gamma_{331}\partial_{x}-iZ_{v}v\Gamma_{302}\partial_{y}-iZ_{v}Z_{\alpha}\alpha
v\Gamma_{330}\partial_{x}\right)\right]\Psi+Z_{g}L_{Y}$ (19)
$\displaystyle+Z_{\phi}\phi(-\partial^{2}_{\tau}-\nabla^{2})\phi+Z_{m_{B}}m_{B}^{2}\phi^{2}+Z_{\lambda}\frac{\lambda}{2}\phi^{4},$

where $Z_{j}$’s are renormalization constants, and we restore the Fermi
velocity for completeness.

###  5.1 $\beta-$functions for a tilted Dirac $s-$wave superconductor

To obtain RG flow of the couplings, we integrate out the fermionic and bosonic
modes with frequency $-\infty<\omega<\infty$ and use dimensional
regularization in $D=3-\epsilon$ spatial dimensions within the minimal
subtraction scheme to handle the ensuing divergences. The one-loop computation
of the self-energy diagrams for the fermions (App. A.1) and for the bosonic
order parameter (App. A.2) then yields the fermion field renormalization (App.
A.1.1), the boson field renormalization (A.2), and the renormalization
constant for the tilt parameter (App. A.1.2), which, respectively, read

$Z_{\Psi}=1-\frac{g^{2}}{(4\pi)^{2}}G(\alpha)\frac{1}{\epsilon},$ (20)

$Z_{\Phi}=1-\frac{g^{2}}{4\pi^{2}}N_{F}F(\alpha)\frac{1}{\epsilon}$ (21)

$Z_{\alpha}=1+\frac{g^{2}}{(4\pi)^{2}}\left[G(\alpha)-I(\alpha)\right]\frac{1}{\epsilon},$
(22)

while the renormalization constant for the velocity $Z_{v}=1$, as shown in
App. A.1.3. Here, the functions are defined as
$F(\alpha)=(1-\alpha^{2})^{-2}$, $G(\alpha)=(1-(\alpha/2)^{2})^{-1}$, while
$I(\alpha)$ is given by Eq. (50).

The computation of the vertex diagrams to the same order gives the
renormalization conditions for the Yukawa coupling constant (App. A.3)

$Z_{\phi}^{1/2}\,Z_{\Psi}\,g_{0}\,\mu^{-\epsilon/2}=g,$ (23)

and for for the $\Phi^{4}$ coupling (App. A.4)

$Z_{\phi}^{2}\lambda_{0}\mu^{-\epsilon}-\left((N_{B}+4)\frac{\lambda^{2}}{2\pi^{2}}-\frac{g^{4}}{2\pi^{2}}N_{F}F(\alpha)\right)\frac{1}{\epsilon}=\lambda.$
(24)

Here, the couplings with (without) the subscript are the bare (renormalized)
ones. Furthermore, $\mu$ is the RG (momentum) scale, and we consider $N_{B}$
complex components for the bosonic OP.

Using the above renormalization conditions, we obtain the RG $\beta$ functions
to the leading order in the $\epsilon$-expansion, in the critical hyperplane
$m_{B}^{2}=0$, see Eq. (19). We define the $\beta$ function for a coupling
constant $X$ as $\beta_{X}\equiv-dX/d\ln{(\mu)}|_{\\{X_{0}\\}}$, with all the
bare couplings ($\\{X_{0}\\}$) fixed. Since the renormalization constant for
the velocity $Z_{v}=1$, the renormalization condition $Z_{v}v=v_{0}$ implies
that the Fermi velocity is marginal, $\beta_{v}=0$. Furthermore, the
renormalization condition for the tilt term, $Z_{\alpha}\alpha=\alpha_{0}$
implies that

$\beta_{\alpha}=-\alpha\frac{g^{2}}{2}[G(\alpha)-I(\alpha)]\equiv-\alpha
g^{2}K(\alpha),$ (25)

Figure 1: The form of the functions $K(\alpha)$ and ${\tilde{K}}(\alpha)$
governing the renormalization-group flow of the tilt parameter at the
transition from the tilted Dirac semimetal to the $s-$wave superconducting
(SC) and the charge-density wave (CDW) phases, respectively. The corresponding
$\beta$ function is given by Eq. (25) [Eq. (34)] for the $s-$wave SC [CDW]
instability.

with the function $K(\alpha)$ as strictly positive for $|\alpha|<1$, as shown
in Fig. 1. Therefore, the tilt parameter is _irrelevant_ , and flows to zero
as the system approaches the critical point governing the transition into the
$s$-wave superconducting state. Ultimately, the Lorentz symmetry is restored
at the QCP and the quantum-criticality is controlled by the GNY QCP for
$\alpha=0$. To confirm this scenario, using the renormalization condition in
Eq. (23), we furthermore find the $\beta-$function for the Yukawa coupling

$\beta_{g^{2}}=\epsilon g^{2}-[2N_{F}F(\alpha)+G(\alpha)]g^{4},$ (26)

where we rescaled $g^{2}/8\pi^{2}\rightarrow g^{2}$. The RG flow in the
$(g^{2},\alpha)$ plane is shown in Fig. 2, which indeed corroborates the
irrelevance of the tilt parameter, and the ensuing restoration of the Lorentz
symmetry at the nontrivial QCP with the value given by Eq. (26),

$g^{2}_{\star}=\frac{\epsilon}{[2N_{F}F(0)+G(0)]}.$ (27)

For completeness, we also show the RG flow equation for the
$\Phi^{4}-$coupling for the tilted Dirac superconductor,

$\beta_{\lambda}=\epsilon\lambda-(4+N_{B})\lambda^{2}+4N_{F}g^{2}F(\alpha)(g^{2}-\lambda),$
(28)

which is obtained from the renormalization condition in Eq. (24), after
rescaling $X/8\pi^{2}\rightarrow X$, with $X=\\{g^{2},\lambda\\}$. Notice that
the $\beta$ functions in Eqs. (26) and (25) for the untilted Dirac fermion
($\alpha=0$) coincide with the ones obtained in Ref. RJH-PRB2013 . The
corresponding fixed points of the RG flow are obtained from the
$\beta-$functions for the Yukawa and the $\lambda$ couplings after taking
$\alpha=0$, as implied by the irrlevance of the tilt term [Eq. (25)]. Only one
of them is stable in the critical plane ($m_{B}^{2}=0$), and therefore
represents the critical point, which is located at RJH-PRB2013

$~{}(g_{\star}^{2},\lambda_{\star})=\frac{\epsilon}{X+4N_{F}}\left(1,X+\sqrt{X^{2}+8(1-X)(4+N_{B})}\right),$
(29)

where $X=1-2N_{F}$. Notice that $\lambda_{\star}$ is positive, and therefore
corresponds to a second-order phase transition, only for $N_{F}>1/2$, or
equivalently, in the Dirac system featuring at least one eight-component Dirac
fermion flavor.

We emphasize here that identical renormalization factors and vertex
corrections are obtained for the asymmetric TDFs, since the tilt matrix in
this case anticommutes with the SC OP and commutes with the untilted part of
the Dirac Hamiltonian, which is the same as for the symmetric TDFs, while all
the form factors in the propagators for the asymmetric [Eq. (18)] and
symmetric [Eq. (16)] TDFs are identical.

(a)

(b)

(c)

Figure 2: One-loop RG flow of the tilt parameter $\alpha$ and the Yukawa
coupling $g^{2}$ at fixed $\epsilon=1$ in the case of the $s-$wave
instability. The red dot corresponds to the Gross-Neveu-Yukawa superconducting
quantum-critical point (QCP), for different number of eight-component Dirac
flavors ($N_{F}$). RG flow is shown for (a) $N_{F}=1$; (b) $N_{F}=2$; (c)
$N_{F}=3$. The analytical form of the Lorentz-symmetric QCP ($\alpha=0$) is
given by Eq. (29).

###  5.2 Comparison with charge-density-wave instability

We here use the opportunity to revise the problem of a charge-density wave
(CDW) instability breaking the sublattice symmetry of a tilted Dirac semimetal
Rostami-PRR2020 , in light of the restoration of the Lorentz symmetry at the
superconduting QCP. The CDW OP takes the form,

$\Phi(\mathbf{r},\tau)=\left\langle{\Psi^{\dagger}(\mathbf{r},\tau)\Gamma_{303}\Psi(\mathbf{r},\tau)}\right\rangle,$
(30)

implying that the Yukawa action reads

$S_{Y}=g\int d\tau d^{D}\mathbf{r}\,\,\Phi\,\,\Psi^{\dagger}\Gamma_{303}\Psi,$
(31)

with the OP matrix anticommuting with the untilted part of the Dirac
Hamiltonian in Eq. (3), and therefore gaps it out. We are interested here only
in the RG flow of the tilt parameter and the Yukawa coupling, and therefore
consider the fermionic self-energy, which reads

$\displaystyle\Sigma_{F}(i\Omega,{\bf k})=$ $\displaystyle
g^{2}\int[d\omega]\int[d^{D}\mathbf{q}]\,\Gamma_{303}\,G_{F}(i\omega,\mathbf{q})\,\Gamma_{303}\,G_{B}(i\Omega-i\omega,\mathbf{q})$
(32) $\displaystyle=$ $\displaystyle
g^{2}\int[d\omega]\int[d^{D}\mathbf{q}]\,\left[i\omega
A_{-}-B(q_{x}\Gamma_{001}+q_{y}\Gamma_{032})+\alpha
q_{x}(A_{-}+2q^{2})\Gamma_{330}\right.$
$\displaystyle\left.-(A_{-}+2\alpha^{2}q_{x}^{2})(q_{x}\Gamma_{331}+q_{y}\Gamma_{302})\right]\frac{1}{[A_{+}^{2}+B^{2}][(\omega-\Omega)^{2}-({\bf
q}-{\bf k})^{2}]},$

which is obtained directly from the self-energy for the SC OP, Eq. (A.1),
taking into account that the CDW OP commutes with the tilt matrix,
$\Gamma_{330}$. The functions $A_{\pm}$ and $B$ are defined in Eq. (3). This
form of the fermionic self-energy, in turn, yields the renormalization
constant for the field ($Z_{\Psi}$) in the form given by Eq. (20), while the
factor $Z_{\alpha}$ reads

$Z_{\alpha}=1+\frac{g^{2}}{(4\pi)^{2}}\left[G(\alpha)+I(\alpha)\right]\frac{1}{\epsilon},$
(33)

with the function $G(\alpha)$ [$I(\alpha)$] defined in Eq. (22) [Eq. (50)].
The corresponding RG flow equation for the tilt is then straightforwardly
obtained from the renormalization condition $Z_{\alpha}\alpha=\alpha_{0}$,
yielding

$\beta_{\alpha}=-\alpha\frac{g^{2}}{2}[G(\alpha)+I(\alpha)]\equiv-\alpha
g^{2}{\tilde{K}}(\alpha),$ (34)

with the strictly positive function ${\tilde{K}}(\alpha)$ shown in Fig. 1.
Therefore, the Lorentz symmetry gets restored for the CDW instability, but
much slower than in the SC case, since ${\tilde{K}}(\alpha)<K(\alpha)$, for
$|\alpha|<1$, as can be seen in Fig. 1. Furthermore, the flow of the Yukawa
coupling in this case is given by

$~{}\beta_{g^{2}}=\epsilon g^{2}-(N_{F}+3)G(\alpha)g^{4},$ (35)

with the stable QCP realized at

$~{}g_{\star}^{2}=\frac{\epsilon}{(3+N_{F})G(0)}.$ (36)

The corresponding RG flow in the $(g^{2},\alpha)$ plane is displayed in Fig.
3, which corroborates the restoration of the Lorentz symmetry at the QCP
governing the transition into the CDW phase, and making the line of critical
points found in Ref. Rostami-PRR2020  eventually unstable.

(a)

(b)

(c)

Figure 3: One-loop RG flow of the tilt parameter $\alpha$ and the Yukawa
coupling $g^{2}$ at fixed $\epsilon=1$ in the case of the charge-density-wave
(CDW) instability. The red dot corresponds to the Gross-Neveu-Yukawa
superconducting quantum-critical point (QCP), for different number of eight-
component Dirac flavors ($N_{F}$). RG flow is shown for (a) $N_{F}=1$; (b)
$N_{F}=2$; (c) $N_{F}=3$. The analytical form of the Lorentz-symmetric QCP
($\alpha=0$) is given by Eq. (36).

We point out that the sign in front of the strictly negative function
$I(\alpha)$ (for $|\alpha|<1$) in Eq. (34), is a direct consequence of the
_commutation_ of the tilt matrix ($\Gamma_{330}$) and the CDW OP
$\Gamma_{303}$, which is diagonal in the Nambu space. This _positive_ sign is
responsible for slowing down the RG flow of the tilt with respect to the SC
case, implying that the restoration of the Lorentz symmetry takes longer RG
time.

In general, we can observe that the rate of the RG flow for the tilt
parameter, and therefore the rate of the Lorentz symmetry restoration is
directly related to the (anti)commutation relation of the OP with the tilt
matrix. Namely, since all the $\Gamma-$matrices square to unity and either
commute or anticommute with each other, an OP that commutes with the tilt
matrix is expected to make the tilt parameter _less irrelevant_ than an
anticommuting OP. In particular, an insulating (particle-hole) instability
diagonal in other flavor subspaces (valley, sublattice, etc.) should render
the tilt less irrelevant than a superconducting (particle-particle)
instability, as the examples of the CDW and the $s-$wave instabilities
explicitly demonstrate.

Finally, the form of the propagator of the asymmetric TDFs implies that the
$\beta-$functions at CDW QCP for the asymmetric TDFs are of the same form as
for the symmetric TDFs.

##  6 Conclusions and outlook

In this paper, we have shown that the Lorentz symmetry is restored at the QCP
separating the tilt Dirac semimetal from a gapped $s-$wave SC instability
through the irrelevance of the tilt operator and its ultimate vanishing at the
QCP. As we further argued such a symmetry restoration is a quite generic
feature for the strongly interacting TDFs, irrespective of whether they break
a discrete mirror lattice symmetry at the noninteracting level, i.e. whether
they feature symmetric [Eq. (3)] or asymmetric [Eq. (4)] tilt.

Our results can be routinely tested in the lattice quantum Monte Carlo
simulations Sorella1992  of the Hubbard interacting TDFs. In light of this, we
emphasize that that even though the tilt parameter is irrelevant, and
ultimately vanishes at the QCP, it may take long RG time to reach this value.
Therefore in any finite-size system the effects of a finite tilt may still be
appreciable, as given by the solution of the flow equations (25), (26) and
(28). See also Fig. 2.

Directly pertinent to the SC instability of the TDFs is whether the coupling
to a dynamical $U(1)$ gauge field can lead to the breaking of the Lorentz
symmetry, which is an issue that we plan to investigate in future. In
particular, it would be interesting to see whether such a coupling could help
stabilization of a charged QCP at a finite coupling.

Our work should motivate further studies of the interaction effects in the
non-Hermitian tilted Dirac materials with the coupling to the environment
explicitly taken into account through non-Hermitian terms in the Hamiltonian-
like operator. In particular, it will be important to address whether the
restoration of the Lorentz symmetry is also featured in the non-Hermitian
Dirac semimetals, similarly to the case of untilted Dirac fermions
juricic2023yukawalorentz .

Finally, we would like to point out that these results may be relevant for
future studies of the strong coupling instabilities of type-2 Dirac semimetals
with electron- and hole Fermi surfaces, and should spur further research
efforts in this direction.

###### Acknowledgements.

We thank Bitan Roy for fruitful discussions. This work was supported by the
Swedish Research Council Grant No. VR 2019-04735 (V.J.), Fondecyt (Chile)
Grant No. 1230933 (V.J.) and ANID/ACT210100 (V.J.).

##  Appendix A Renormalization group analysis the Gross-Neveu-Yukawa theory

In this appendix we present the details of the renormalization group
calculation to the one-loop order within the $\epsilon=3-D$ expansion about
$D=3$ upper critical dimension of the quantum-critical theory describing
interacting TDFs. The short-range interaction is encoded through the Yukawa
coupling between the Dirac fermion quasiparticles and the bosonic degrees of
freedom representing the fluctuations of the $s$-wave superconducting phase

$S_{Y}=g\int d\tau
d^{D}\mathbf{r}[(\text{Re}\phi)\Psi^{\dagger}\Gamma_{100}\Psi+(\text{Im}\phi)\Psi^{\dagger}i\Gamma_{200}\Psi]\equiv\int
d\tau d^{D}\mathbf{r}\,L_{Y}.$ (37)

It is convenient to rewrite the coupling of the bosonic order parameter to the
fermionic field explicitly in terms of the real and imaginary parts of the OP,

$L_{Y}=g\left[(\text{Re}\phi)\Psi^{\dagger}\Gamma_{100}\Psi+(\text{Im}\phi)\Psi^{\dagger}i\Gamma_{200}\Psi\right]=g\left[\phi\Psi^{\dagger}P_{+}\Psi+\phi^{*}\Psi^{\dagger}P_{-}\Psi\right],$
(38)

where the matrices $P_{\pm}$ are defined as

$P_{\pm}=\frac{1}{2}\left(\Gamma_{100}\mp i\Gamma_{200}\right),$ (39)

and we employ eight-dimensional representation of the $\Gamma-$matrices.

To address the quantum-critical behavior in $D=2$, we employ $\epsilon=3-D$,
the deviation from the upper critical $D=3$ spatial dimensions, as an
expansion parameter.

(a)

(b)

(c)

(d)

(e)

Figure 4: Feynman diagrams to the leading order in the $\epsilon$ expansion:
(a) fermionic self-energy, (b) bosonic self-energy, (c) correction to the
Yukawa vertex arising from fermion excitations, (d) correction to the
$\lambda$ vertex from the boson self-interaction, and (e) correction to the
$\lambda$ vertex from the Yukawa interaction. The solid line represents the
fermionic propagator, while the dashed line stands for the bosonic propagator.

###  A.1 Fermionic self energy

We compute to the leading-order the zero-temperature ($T=0$) self-energy for
tilted Dirac fermions (Fig. 4(a)) in the critical hyperplane (all masses set
to zero)

$\Sigma_{F}(i\Omega,\mathbf{k})=g^{2}\int[d\omega]\int[d^{D}\mathbf{q}]\\{P_{+}G_{F}(i\omega,\mathbf{q})P_{-}+P_{-}G_{F}(i\omega,\mathbf{q})P_{+}\\}G_{b}(i\Omega-i\omega,\mathbf{q}-\mathbf{k})$
(40)

where $[d\omega]=d\omega/2\pi$ and
$[d^{D}\mathbf{q}]=d^{D}\mathbf{q}/(2\pi)^{D}$. The bosonic and fermionic
propagators are given in the main text, Eqs. (16) and (17). The explicit form
of the fermionic self-energy is

$\displaystyle\Sigma_{F}(i\Omega,{\bf k})=$ $\displaystyle
g^{2}\int[d\omega]\int[d^{D}\mathbf{q}]\\{P_{+}G_{F}(i\omega,\mathbf{q})P_{-}+P_{-}G_{F}(i\omega,\mathbf{q})P_{+}\\}G_{B}(i\Omega-i\omega,\mathbf{q})$
(41) $\displaystyle=$ $\displaystyle
g^{2}\int[d\omega]\int[d^{D}\mathbf{q}]\\{P_{+},P_{-}\\}\left[i\omega
A_{-}+B(q_{x}\Gamma_{001}+q_{y}\Gamma_{032})-\alpha
q_{x}(A_{-}+2q^{2})\Gamma_{330}\right.$
$\displaystyle\left.-(A_{-}+2\alpha^{2}q_{x}^{2})(q_{x}\Gamma_{331}+q_{y}\Gamma_{302})\right]\frac{1}{(\omega^{2}-\alpha^{2}q_{x}^{2}+q^{2})^{2}+(2\alpha\omega
q_{x})^{2}}$ $\displaystyle\times\frac{1}{(\omega-\Omega)^{2}-({\bf q}-{\bf
k})^{2}},$

with the anticommutator $\\{P_{+},P_{-}\\}=1$, as can be directly obtained
from Eq. (39); $A_{\pm}\equiv A_{\pm}(i\omega,{\bf
k})=-\omega^{2}\pm\alpha^{2}k_{x}^{2}-k^{2}$ and $B\equiv B(i\omega,{\bf
k})=2\alpha i\omega k_{x}$. We here also set Fermi velocity $v=1$, since it
does not get renormalized due to the relativistic invariance of the Yukawa
interaction [Eq. (14)], as also explicitly shown in App. A.1.3. The algebraic
structure of the integrand allows us to systematically calculate the
corresponding renormalization constants ($Z^{\prime}s$), which we explicitly
show in the following. Notice that the negative signs multiplying the last two
terms in the above self-energy result from the anticommutation of the OP
matrices ($P_{\pm}$) with the tilt matrix $\Gamma_{330}$.

####  A.1.1 Fermionic field renormalization ($Z_{\Psi}$)

For the renormalization constant $Z_{\Psi}$, only the terms proportional to
the unity matrix contribute. Thus after setting the external momentum to zero,
${\bf k}=0$, and performing the integral over $\omega$, we obtain

$\Sigma_{F,000}(i\Omega,0)=-g^{2}\int[d^{D}\mathbf{q}]\frac{i\Omega\left(4q^{2}+\alpha^{2}q_{x}^{2}+\Omega^{2}\right)}{2q\left(16q^{4}+8q^{2}\left(\Omega^{2}-\alpha^{2}q_{x}^{2}\right)+\left(\alpha^{2}q_{x}^{2}+\Omega^{2}\right)^{2}\right)}.$
(42)

Now, to find the fermion field renormalization $Z_{\Psi}$, we take

$\lim_{i\Omega\to
0}\frac{\partial\Sigma_{F,000}(i\Omega,0)}{\partial(i\Omega)}=-\frac{g^{2}}{(4\pi)^{2}}\,G(\alpha)\int_{\lambda}^{\Lambda}dq\frac{1}{q},$
(43)

with $\lambda=\Lambda/b$, where the Wilsonian RG parameter $b>1$, and $b-1\ll
1$. Then, we use the correspondence between the hard cutoff and dimensional
regularization

$\int_{\lambda}^{\Lambda}\frac{dq}{q}\rightarrow\frac{1}{\epsilon},$ (44)

and the function $G(\alpha)$ reads

$G(\alpha)=\int_{0}^{2\pi}\frac{d\phi}{\pi}\int_{0}^{\pi}d\theta\sin(\theta)\frac{\left(4+\alpha^{2}\sin^{2}(\theta)\cos^{2}(\phi)\right)}{\left(4-\alpha^{2}\sin^{2}(\theta)\cos^{2}(\phi)\right)^{2}}\\\
=\frac{4}{4-\alpha^{2}},$ (45)

Notice that the function $G(\alpha)$ is strictly positive for $|\alpha|<1$. We
then obtain the renormalization constant for the fermion field through the
following renormalization condition

$Z_{\Psi}=1+\lim_{i\Omega\to
0}\frac{\partial\Sigma_{f,000}(i\Omega,0)}{\partial
i\Omega}=1-\frac{g^{2}}{(4\pi)^{2}}G(\alpha)\frac{1}{\epsilon}.$ (46)

####  A.1.2 Renormalization factor for the tilt parameter ($Z_{\alpha}$)

The renormalization constant for the tilt parameter, $Z_{\alpha}$, is obtained
from the term in the fermionic self-energy proportional to the matrix
$\Gamma_{330}$, as dictated by the form of the tilt term in the bare theory
[see Eq. (3)]. This, together with the form of the self-energy, in turn,
dictates the following renormalization condition for the tilt,

$\left.-Z_{\Psi}Z_{\alpha}\alpha
k_{x}-k_{x}\frac{\partial{\Sigma}_{F,330}(i\Omega,\mathbf{k})}{\partial
k_{x}}\right|_{i\Omega\to 0,\mathbf{k}\to 0}=-\alpha k_{x},$ (47)

where ${\Sigma}_{F,330}(i\Omega,\mathbf{k})$ is the term proportional to the
matrix $\Gamma_{330}$ in the fermionic self-energy in Eq. (41),

${\Sigma}_{F,330}(i\Omega,\mathbf{k})=g^{2}\int[d\omega]\int[d^{D}\mathbf{q}]\frac{\alpha
q_{x}(\omega^{2}+\alpha^{2}q_{x}^{2}-q^{2})}{[(\omega^{2}-\alpha^{2}q_{x}^{2}+q^{2})^{2}+(2\alpha\omega
q_{x})^{2}][(\omega-\Omega)^{2}-({\bf q}-{\bf k})^{2}]}.$ (48)

We notice that we here explicitly used marginality of the Fermi velocity $v$
in Eq. (3), due to the relativistic invariance of the Yukawa term, see Eq.
(14).

We then take a finite external momentum $\mathbf{k}=k_{x}{\bf e}_{x}$ in Eq.
(48), and keep only the terms proportional to $\Gamma_{330}$, which, by virtue
of the renormalization condition in Eq. (47), yields,

$Z_{\alpha}Z_{\Psi}=1-\frac{g^{2}}{(4\pi)^{2}\epsilon}I(\alpha),$ (49)

where

$I(\alpha)=\int_{0}^{2\pi}\frac{d\phi}{\pi}\int_{0}^{\pi}d\theta\,\,\frac{\cos^{2}\phi\sin^{3}\theta[-8+\alpha^{2}\cos^{2}\phi\sin^{2}\theta]}{(4-\alpha^{2}\cos^{2}\phi\sin^{2}\theta)^{2}}.$
(50)

Finally, using the form of the fermion-field renormalization $Z_{\Psi}$ in Eq.
(46), we find

$Z_{\alpha}=1+\frac{g^{2}}{(4\pi)^{2}\epsilon}\left[G(\alpha)-I(\alpha)\right],$
(51)

with the function $G(\alpha)$ given by Eq. (45), which is identical to Eq.
(22) in the main text.

####  A.1.3 Renormalization factor for the Fermi velocity ($Z_{v}$)

We here show that the renormalization factor of the Fermi velocity, $Z_{v}$,
is equal to unity, as a consequence of the Lorentz invariance of the Yukawa
vertex in Eq. (14). Therefore, the Fermi velocity remains marginal, as is the
case in the tilt-free case. To show this feature, we consider only one
component of the velocity, taken to be along the $x-$axis for concreteness.
The corresponding renormalization condition reads

$\left.Z_{\Psi}Z_{v}+\frac{\partial\Sigma_{F,331}(i\Omega,\mathbf{k})}{\partial
k_{x}}\right|_{i\Omega\to 0,\mathbf{k}\to 0}=1,$ (52)

where the term $\Sigma_{f,331}$ is the part of the self-energy in Eq. (41)
multiplied by the matrix $\Gamma_{331}$, which explicitly reads

$\Sigma_{F,331}(i\Omega,\mathbf{k})=g^{2}\int[d\omega]\int[d^{D}\mathbf{k}]\,\,\frac{q_{x}(\omega^{2}+q^{2}-\alpha^{2}q_{x}^{2})}{[(\omega^{2}+q^{2}-\alpha^{2}q_{x}^{2})^{2}+(2\alpha\omega
q_{x})^{2}][(\omega-\Omega)^{2}+(\mathbf{q}-\mathbf{k})^{2}]}.$ (53)

An explicit calculation then yields

$\left.\frac{\partial\Sigma_{F,331}(i\Omega,\mathbf{k})}{\partial
k_{x}}\right|_{i\Omega\to 0,\mathbf{k}\to
0}=\frac{g^{2}}{2(2\pi)^{3}\epsilon}\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta\,\,\frac{\cos^{2}\phi\sin^{3}\theta(12-\alpha^{2}\cos^{2}\phi\sin^{2}\theta)}{(4-\alpha^{2}\cos^{2}\phi\sin^{2}\theta)^{2}},$
(54)

which, after performing the angular integrals, yields

$\left.\frac{\partial\Sigma_{F,331}(i\Omega,\mathbf{k})}{\partial
k_{x}}\right|_{i\Omega\to 0,\mathbf{k}\to
0}=\frac{g^{2}}{(4\pi)^{2}}G(\alpha)\frac{1}{\epsilon},$ (55)

with the function $G(\alpha)$ given by Eq. (45). Using Eq. (46) and the
renormalization condition (54) , we obtain that $Z_{v}=1$, implying that the
Fermi velocity remains marginal (unrenormalized) for the TDFs to the leading
order in the $\epsilon-$expansion, once the velocity isotropy is reached.
Notice that we here considered only the $x-$component of the velocity, but the
same conclusion holds for other components after invoking the irrelevance of
the tilt parameter.

###  A.2 Bosonic self energy

In order to obtain the RG equations for the Yukawa coupling, we compute the
self energy for the bosonic field and the correction to the Yukawa vertex.
First, we compute the bosonic self energy (Fig. 4(b)), which reads

$\Sigma_{B}(i\Omega,\mathbf{k})=-{g^{2}}\int[d\omega]\int[d^{D}\mathbf{q}]\,\text{Tr}\left[P_{+}G_{F}(i\omega+i\Omega,\mathbf{q+k})P_{-}G_{F}(i\omega,\mathbf{q})\right],$
(56)

with the negative sign arising from the trace over the fermionic fields, and
we set external momentum to zero ($\mathbf{k}=0$). After taking the trace, the
integral is of the form

$\displaystyle\Sigma_{B}(i\Omega,0)=$
$\displaystyle-4{g^{2}N_{F}}\int[d\omega]\int[d^{D}\mathbf{q}]\,\,\Big{(}4k_{x}^{4}k^{2}\alpha^{4}-B(\omega,\mathbf{q})B(\omega+\Omega,\mathbf{q})q^{2}$
(57)
$\displaystyle+4k_{x}^{2}k^{2}\alpha^{2}(A_{-}(\omega+\Omega,\mathbf{q})+k^{2})+4k_{x}^{2}k^{2}\alpha^{2}A_{-}(\omega,\mathbf{q})$
$\displaystyle+A_{-}(\omega,\mathbf{q})A_{-}(\omega+\Omega,\mathbf{q})(k^{2}+\alpha^{2}k_{x}^{2}+\omega(\omega+\Omega))\Big{)}$
$\displaystyle\times\Bigg{(}\frac{1}{[A_{+}(\omega,\mathbf{q})]^{2}-B^{2}(\omega,\mathbf{q})}\Bigg{)}\Bigg{(}\frac{1}{[A_{+}(\omega+\Omega,\mathbf{q})]^{2}-B^{2}(\omega+\Omega,\mathbf{q})}\Bigg{)},$

with $N_{F}$ as the number of eight-component Dirac fermion flavors. Taking
the derivative, after performing the integral over the frequency and the
momentum, we obtain

$\lim_{\Omega\to
0}\frac{\partial\Sigma_{B}(i\Omega,0)}{\partial\Omega^{2}}=-\frac{g^{2}}{4\pi^{2}}N_{F}F(\alpha)\frac{1}{\epsilon},$
(58)

where function $F(\alpha)$ is defined as

$F(\alpha)=\frac{1}{(1-\alpha^{2})^{2}}.$ (59)

With the above definition of the bosonic self-energy, the bosonic wave-
function renormalization is determined by the renormalization condition

$Z_{\Phi}-\left.\frac{\partial\Sigma_{B}(i\Omega,0)}{\partial\Omega^{2}}\right|_{\Omega\to
0}=1,$ (60)

implying that the renormalization constant for the bosonic fields is

$~{}Z_{\Phi}=1+\left.\frac{\partial\Sigma_{B}(i\Omega,0)}{\partial\Omega^{2}}\right|_{\Omega\to
0}=1-\frac{g^{2}}{4\pi^{2}}N_{F}F(\alpha)\frac{1}{\epsilon}.$ (61)

###  A.3 Correction to Yukawa vertex

The correction to the Yukawa vertex ((Fig. 4(c)) reads

$\displaystyle\delta g=$ $\displaystyle
g^{2}\int[d\omega]\int[d^{D}\mathbf{q}]\left\\{\left(P_{+}G_{F}(i\Omega_{1}+i\Omega_{2}-i\omega,\mathbf{k+p-q})P_{+}G_{F}(i\Omega_{1}-i\omega,\mathbf{k-q})P_{-}\right)\right.$
(62)
$\displaystyle\left.+\left(P_{-}G_{F}(i\omega+i\Omega_{1}+i\omega_{2},\mathbf{q+k+q})P_{+}G_{F}(i\omega+i\Omega_{1},\mathbf{q+k})P_{+}\right)\right.$
$\displaystyle+\left(P_{+}G_{F}(i\Omega_{1}-\Omega_{2}-i\omega,\mathbf{k-p-q})P_{-}G_{F}(i\Omega_{1}-i\omega,\mathbf{k-q})P_{-}\right)$
$\displaystyle\left.+\left(P_{-}G_{F}(i\Omega_{1}-i\Omega_{2}+i\omega,\mathbf{k-p-q})P_{-}G_{F}(i\omega+i\Omega_{1},\mathbf{q+k})P_{+}\right)\right\\}\frac{1}{\omega^{2}+q^{2}}.$

Using the relation $P_{\pm}P_{\pm}=0$, the form of the fermionic propagator in
Eq. (16), and that all the matrices either commute or anticommute with the
matrices $P_{\pm}$, it can be readily shown that the correction of the Yukawa
vertex identically vanishes, $\delta g=0$, after taking all the external
momenta and frequencies to be equal to zero. Therefore, the renormalization
for the Yukawa coupling has the form

$Z_{\phi}^{1/2}Z_{\Psi}g_{0}\mu^{-\epsilon/2}=g,$ (63)

which, together with the form of the renormalization constants $Z_{\Psi}$ and
$Z_{\phi}$, in Eqs. (46) and (61), respectively, implies the form of the RG
flow equation of the Yukawa coupling in Eq. (26) of the main text.

###  A.4 Correction to the $\phi^{4}$ vertex

The first contribution to the $\lambda$ coupling comes from the already know
result form the $\phi^{4}$ vertex (Fig. 4(d)), which reads

$\delta\lambda_{\lambda}=(N_{B}+4)\frac{\lambda^{2}}{8\pi^{2}}\frac{1}{\epsilon},$
(64)

where, $N_{B}$ is the number of complex components of the bosonic order
parameter field.

The correction to the $\lambda$ coupling due to the Yukawa interaction (Fig.
4(e)), with all external momenta and frequencies set to zero, reads

$\delta\lambda_{Y}=-g^{4}\int[d\omega]\int[d^{D}\mathbf{q}]\,\text{Tr}\left[P_{+}G_{F}(i\omega,\mathbf{q})P_{-}G_{F}(i\omega,\mathbf{q})P_{+}G_{F}(i\omega,\mathbf{q})P_{-}G_{F}(i\omega,\mathbf{q})\right].$
(65)

After taking the trace and integrate over the frequency and momentum, we
obtain the divergent part of this correction

$\delta\lambda_{Y}=-\frac{g^{4}}{2\pi^{2}}N_{F}F(\alpha)\frac{1}{\epsilon},$
(66)

where the function $F(\alpha)$ is defined in Eq. (59). Finally, the
renormalization condition of the $\lambda$ reads

$Z_{\phi}^{2}\lambda_{0}\mu^{-\epsilon}-(\delta\lambda_{Y}+\delta\lambda_{\lambda})=\lambda,$
(67)

which ultimately yields the $\beta$ function for the $\lambda$ coupling in Eq.
(28).

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