# Walking behavior induced by $\mathcal{PT}$ symmetry breaking in a non-
Hermitian $\rm XY$ model with clock anisotropy

Eduard Naichuk  Institute for Theoretical Solid State Physics, IFW Dresden,
Helmholtzstr. 20, 01069 Dresden, Germany  Bogolyubov Institute for Theoretical
Physics, 03143 Kyiv, Ukraine  Jeroen van den Brink  Institute for Theoretical
Solid State Physics, IFW Dresden, Helmholtzstr. 20, 01069 Dresden, Germany
Institute for Theoretical Physics and Würzburg-Dresden Cluster of Excellence
ct.qmat, TU Dresden, 01069 Dresden, Germany  Flavio S. Nogueira  Institute for
Theoretical Solid State Physics, IFW Dresden, Helmholtzstr. 20, 01069 Dresden,
Germany

###### Abstract

A quantum system governed by a non-Hermitian Hamiltonian may exhibit zero
temperature phase transitions that are driven by interactions, just as its
Hermitian counterpart, raising the fundamental question how non-Hermiticity
affects quantum criticality. In this context we consider a non-Hermitian
system consisting of an $\rm XY$ model with a complex-valued four-state clock
interaction that may or may not have parity-time-reversal ($\mathcal{PT}$)
symmetry. When the $\mathcal{PT}$ symmetry is broken, and time-evolution
becomes non-unitary, a scaling behavior similar to the Berezinskii-Kosterlitz-
Thouless phase transition ensues, but in a highly unconventional way, as the
line of fixed points is absent. From the analysis of the $d$-dimensional RG
equations, we obtain that the unconventional behavior in the $\mathcal{PT}$
broken regime follows from the collision of two fixed points in the $d\to 2$
limit, leading to walking behavior or pseudocriticality. For $d=2+1$ the near
critical behavior is characterized by a correlation length exponent $\nu=3/8$,
a value smaller than the mean-field one. These results are in sharp contrast
with the $\mathcal{PT}$-symmetric case where only one fixed point arises for
$2<d<4$ and in $d=1+1$ three lines of fixed points occur with a continuously
varying critical exponent $\nu$.

The possibilities to access non-Hermitian systems [1, 2, 3] experimentally
significantly expanded in the past years, with possible incarnations ranging
from metamaterials to ultracold atoms realizations [4, 5, 6], and more
recently including electronic systems having applications as sensors [7, 8,
9]. While in any realistic physical system interactions play an important
role, from a theoretical perspective interacting non-Hermitian systems have
been little explored. Particularly pertinent is the question how non-
Hermiticity affects interaction-driven quantum phase transitions and their
universal physical properties also in absence of $\mathcal{PT}$ symmetry,
where time-evolution becomes non-unitary [10, 11, 12, 3, 13]. This motivates
considering quantum field theoretic models, where the renormalization group
(RG) provides the defining paradigm for exploration of the phase structure. In
this context it is very important to establish how the critical behavior
depends on the spacetime dimensionality $d=D+1$.

In order to address the questions above and understand how the breaking of
$\mathcal{PT}$ symmetry affects the phase structure, we investigate an
effective quantum field theory Lagrangian consisting of an $\rm XY$ model with
a non-Hermitian, $\mathcal{PT}$-symmetric clock interaction in $d=D+1$
spacetime dimensions. Its Hermitian counterpart arises in several contexts
both in high-energy physics [14, 15] and condensed matter physics [16, 17, 18,
19, 20, 21]. From an RG perspective the $N$-state clock interaction is an
irrelevant operator for $N>4$ in spacetime dimensions $2<d<4$, so one expects
a crossover behavior from the $\rm XY$ universality class to $Z_{N}$ symmetric
models. In particular, the case $N=4$ corresponds in the strong coupling clock
interaction limit to two decoupled Ising regimes [19, 20]. Since the
$\mathcal{PT}$-symmetric clock model can be mapped into its Hermitian
counterpart, the fundamental question to answer is how $\mathcal{PT}$ symmetry
breaking changes the phase structure of the system in $d=D+1$ dimensions. In
particular, we would like to establish whether a well defined phase transition
occurs. Here we show that when the $\mathcal{PT}$ symmetry is broken for
$2<d<4$ the RG flow is characterized by two fixed points rather than only one
in the $\mathcal{PT}$-symmetric case, and that near one of them complex
conjugated scaling dimensions emerge. This leads to a phase transition in
$d=2+1$ dimensions, with a near critical behavior characterized by a
correlation length exponent $\nu=3/8$. As $d\to 2$ the two fixed points
collide, leading to a so called ’walking’ behavior [22], sometimes also
referred to as pseudocriticality, a concept that has been in focus in recent
years both in high-energy and condensed matter physics [22, 23, 24, 25, 26,
27, 28, 29, 30, 31, 32, 33, 34, 35, 36]. The resulting scaling behavior for
$d\to 2$ is similar to a Berenzinskii-Kosterlitz-Thouless (BKT) one. However,
the actual phase transition is not a BKT one, since the RG flow features a
fixed point rather than a line of fixed points. We obtain that the flow lines
walk around the fixed point on RG-invariant surfaces.

Field theory — Let us consider a non-Hermitian theory consisting of a
$U(1)$-invariant Lagrangian with a $Z_{N}$ anisotropy perturbation,

$\displaystyle\mathcal{L}_{\text{NH}}$ $\displaystyle=$
$\displaystyle|\partial_{\mu}\psi|^{2}+m^{2}|\psi|^{2}+\frac{u}{2}|\psi|^{4}$
(1) $\displaystyle-$
$\displaystyle\frac{1}{2}\left[(v+w)\psi^{N}+(v-w){\psi^{*}}^{N}\right],$

where $u,v,w>0$. The $Z_{N}$ anisotropic interaction is irrelevant in the RG
sense for $N>4$ in case the spacetime dimension lies in the interval $2<d<4$.
For $w=0$ this expectation is corroborated by recent Monte Carlo simulations
[20]. Our main focus here will be on the $N=4$ case. For $d=2$ a special
treatment is required, which is one of the main topics in the subsequent
discussion. The Hermitian model for $N=4$ has been studied in different
contexts in the past decades. For example, it arises as an effective theory
emerging from breaking the $SU(N)$ symmetry in a non-Abelian Higgs model [14]
and more recently for $N=4$ and $d=3$ as theories describing the transition to
valence-bond solid phases [19].

The connection to the $N$-state clock model is easily established after
employing the following parametrization of the scalar field, $\psi=\rho
e^{i\theta}/\sqrt{2}$, in which case the Lagrangian becomes,

$\displaystyle\mathcal{L}_{\text{NH}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\partial_{\mu}\rho)^{2}+\frac{\rho^{2}}{2}(\partial_{\mu}\theta)^{2}+\frac{m^{2}}{2}\rho^{2}+\frac{u}{8}\rho^{4}$
(2) $\displaystyle-$
$\displaystyle\frac{\rho^{N}}{2^{N/2}}\left[v\cos(N\theta)+iw\sin(N\theta)\right].$

In the representation above we see that this non-Hermitian theory is
$\mathcal{PT}$-symmetric, where $\mathcal{P}\theta\mathcal{P}^{-1}=-\theta$
and $\mathcal{T}i\mathcal{T}^{-1}=-i$. The amplitude $\rho$ is both
$\mathcal{P}$\- and $\mathcal{T}$-invariant.

Near criticality phase fluctuations are dominant, so we can consider the
amplitude field $\rho$ as being essentially frozen, and we can approximately
write $\rho^{2}=\rho_{0}^{2}=-2m^{2}/u$ for $m^{2}<0$. In this case we obtain
the effective clock model,

$\mathcal{L}_{\rm eff}=\frac{K}{2}(\partial_{\mu}\theta)^{2}-z_{r}\cos
N\theta-iz_{i}\sin N\theta,$ (3)

where $K=\rho_{0}^{2}$, $z_{r}=v\rho_{0}^{N}/2^{N/2}$, and
$z_{i}=w\rho_{0}^{N}/2^{N/2}$. For $z_{i}=0$ the model is well studied in the
literature, one seminal paper being the one by José et al. [16], where the
two-dimensional $\rm XY$ model with clock-type symmetry breaking fields was
considered.

Renormalization group analysis — A major difference with respect to the
theories discussed in Refs. [11] and [12] lies in the fact that the scalar
field $\theta$ is a periodic field. Hence, for $z_{i}=0$ the Lagrangian above
yields the effective theory for an $\rm XY$ model with a $N$-fold clock
symmetry breaking field [16].

Within a path integral formalism we can perform a shift of the field,
$\theta\to\theta+(i/N){\rm arctanh}(z_{i}/z_{r})$ [11]. This leads us to
consider the following equivalent effective Lagrangian,

$\mathcal{L}_{\rm eff}=\frac{K}{2}(\partial_{\mu}\theta)^{2}-\widetilde{z}\cos
N\theta,$ (4)

where $\widetilde{z}=\sqrt{z_{r}^{2}-z_{i}^{2}}$. When the
$\mathcal{PT}$-symmetry is broken, we simply have $\widetilde{z}\to
i\widetilde{z}$. Thus, we see that the couplings $z_{r}$ and $z_{i}$ are not
actually independent. This leads to the non-renormalization of the ratio
$z_{r}/z_{i}$, which ultimately is what will lead to the walking behavior in
the $\mathcal{PT}$ broken regime.

We will follow a procedure similar to the one by José et al. [16] in their
analysis of the $\rm XY$ model in the presence of symmetry breaking fields
(the simplest case refers to an external in-plane magnetic field). First, we
note that when $\widetilde{z}=0$ the effective Lagrangian (4) for $d=2$ is
exactly dual to a sine Gordon theory of the form [16, 37],

$\mathcal{L}_{\rm SG}=\frac{1}{8\pi^{2}K}(\partial_{\mu}\chi)^{2}-z\cos\chi,$
(5)

corresponding to the field theory of a Coulomb gas of point vortices [37].
Hence, defining the dimensionless renormalized couplings $y$ and $\kappa$
associated respectively to the bare couplings $z$ and $K$, we obtain the usual
RG equations underlying the BKT transition [16, 38].

From now on we focus on the $N=4$ and discuss the clock model symmetry
breaking perturbation for this case. Let us define the dimensionless couplings
$y_{r}$ and $y_{i}$ associated to $z_{r}$ and $z_{i}$, respectively. The RG
equations in $d$ dimensions are given by [39],

$\frac{d\kappa}{dl}=\frac{4}{\pi^{2}}(y_{r}^{2}-y_{i}^{2})-\kappa^{2}y^{2}+(d-2)\kappa,$
(6) $\frac{dy}{dl}=\left[d-f(d)\kappa\right]y.$ (7)
$\frac{dy_{r}}{dl}=\left[d-\frac{4f(d)}{\pi^{2}\kappa}\right]y_{r},$ (8)
$\frac{dy_{i}}{dl}=\left[d-\frac{4f(d)}{\pi^{2}\kappa}\right]y_{i},$ (9)

where $f(d)=(d-2)\Gamma(d/2-1)/(2\pi^{d/2-2})$.

(a) $d=3$ $\mathcal{PT}$ symmetry can be broken or not, $\widetilde{y}=0$ (b)
$d=3$ $\mathcal{PT}$ broken, $y=0$

(c) $d=3$ $\mathcal{PT}$-symmetric (d) $d=3$ $\mathcal{PT}$ broken

Figure 1: RG flows for the non-Hermitian system associated to the Lagrangian
of Eq. (3). The RG flows shown here are for $d=3$. In all panels the red dot
represents the fixed point. Panels (a) and (b) represent planes featuring
nontrivial fixed points arising in panels (c) and (d). In the
$\mathcal{PT}$-symmetric case of panel (c) the only fixed point governs the
$\rm XY$ universality class when the clock perturbation is small. In the
broken $\mathcal{PT}$ symmetry case [panel (d)], both planes of panels (a) and
(b) arise, with the fixed point of panel (b) being a highly unstable repulsive
spiral. In this case a weakly first-order phase transition occurs.

We note that $\ln(y_{r}/y_{i})$ is an RG invariant, so we can write
$y_{i}=ky_{r}$, where $k$ is a constant. We see that for $k=1$ we obviously
recover when $d=2$ the RG flow equations corresponding to a BKT behavior. In
the $\mathcal{PT}$-symmetric case ($y_{r}>y_{i}$ or, equivalently, $k<1$) and
$d=2$ these equations are easily shown to be equivalent to the ones obtained
in Ref. [16]. This can be seen by deriving the RG equation for
$\widetilde{y}=\sqrt{y_{r}^{2}-y_{i}^{2}}$, which is obtained directly from
Eqs. (8) and (9) as,

$\frac{d\widetilde{y}}{dl}=\left[d-\frac{4f(d)}{\pi^{2}\kappa}\right]\widetilde{y},$
(10)

along with the corresponding $d=2$ result,
$d\widetilde{y}/dl=[2-4/(\pi\kappa)]\widetilde{y}$.

If the $\mathcal{PT}$ symmetry is not broken we obtain for $d=3$ and small
$\widetilde{y}$ the usual three-dimensional $\rm XY$ universality class. This
critical behavior is governed by the nontrivial fixed point for
$\widetilde{y}=0$ shown in panel (a) of Fig. 1. Our RG equations yield in this
case the $\epsilon$-expansion ($\epsilon=d-2$) result
$\nu=1/(2\sqrt{\epsilon})+1/8+\mathcal{O}(\sqrt{\epsilon})$, which reproduces
for $\epsilon=1$ one-loop value, $\nu\approx 0.625$ [40]. The RG flow for
nonzero and large $\widetilde{y}$ exhibits a critical behavior where the four-
state clock interaction becomes very large, as illustrated in panel (c) of
Fig. 1. This situation is well known to lead to two decoupled Ising models
[20, 16].

When the $\mathcal{PT}$-symmetry is broken the situation changes considerably.
Since in this case $y_{r}<y_{i}$, Eq. (6) becomes,

$\frac{d\kappa}{dl}=-\frac{4}{\pi^{2}}\widetilde{y}^{2}-\kappa^{2}y^{2}+(d-2)\kappa,$
(11)

while Eqs. (7) and (10) remain unchanged. The first thing to note is that now
we have two nontrivial fixed points, namely,
$P_{1}=(\kappa_{1},y_{1},\widetilde{y}_{1})=(d/f(d),\sqrt{(d-2)/\kappa_{1}},0)$
and
$P_{2}=(\kappa_{2},y_{2},\widetilde{y}_{2})=(4f(d)/(\pi^{2}d),0,(\pi/2)\sqrt{(d-2)\kappa_{2}})$.
The fixed point $P_{1}$ is the same as shown in panel (a) of Fig. 1. The fixed
point $P_{2}$, on the other hand, is highly unstable, corresponding to the
repulsive spiral shown in panel (b) of Fig. 1, with the full flow diagram
featuring both fixed points shown in panel (d). Although the fixed point
$P_{1}$ is the same as the one for $\widetilde{y}=0$ of panel (c) in Fig. 1,
$P_{1}$ does not govern the three-dimensional $\rm XY$ universality class as
it is the case in panel (c). Indeed, after linearization of the RG equations
in the $\mathcal{PT}$ broken regime, we obtain the eigenvalues of the linear
system, $\lambda_{0}=d-f(d)\kappa_{2}$ and $\lambda_{\pm}=[d-2\pm
i\sqrt{(d-2)(7d+2)}]/2$. The complex conjugated eigenvalues $\lambda_{\pm}$
are associated to the repulsive spiral fixed point $P_{2}$. The critical
exponent $\nu$ is determined by the eigenvalue $\lambda_{0}$ as
$\nu=1/\lambda_{0}$. For $d=3$ it yields $\nu=3/8$ (the $\epsilon$-expansion
practically yields the same result in this case). We note that this is smaller
than the mean-field value 1/2, thus indicating that a near critical behavior,
or pseudocriticality, when the $\mathcal{PT}$ symmetry is broken.

When $d\to 2$ the fixed points $P_{1}$ and $P_{2}$ collide, representing a
typical characteristic of walking behavior [22]. In Fig. 2 we compare the
scaling behaviors of the $\mathcal{PT}$-symmetric with the $\mathcal{PT}$
broken case. In the $\mathcal{PT}$-symmetric case both $dy/dl$ and
$d\widetilde{y}/dl$ vanish for $\kappa_{*}=2/\pi$, leading to three lines of
fixed points, as $\widetilde{y}^{2}_{*}=y^{2}_{*}$ provides one more fixed
line in addition to the two BKT ones determined by the vanishing of both $y$
and $\widetilde{y}$. In panels (a) and (b) of Fig. 2 we illustrate the dual RG
BKT flows for the situations where $\widetilde{y}=0$ and $y=0$, respectively.
Overall, however, the phase transition is not a BKT one in this case: the
correlation length exhibits a power-law with a continuously varying critical
exponent $\nu(y_{*})$ [16]. Panel (c) of Fig. 2 shows a view of the full RG
flow diagram. In the $\mathcal{PT}$ broken regime a BKT flow is still present
for $\widetilde{y}=0$, but the BKT flow for $y=0$ disappears, as it is
illustrated in panel (d) of Fig. 2. The resulting RG flow is better understood
by introducing the variables $X=2-\pi\kappa$, $Y=2y/\sqrt{\pi}$, and
$\widetilde{Y}=2\widetilde{y}/\sqrt{\pi}$, in which case we can rewrite the RG
equations in the form, $dX/dl\approx\widetilde{Y}^{2}+Y^{2}$, $dY/dl=XY$, and
$d\widetilde{Y}/dl\approx-X\widetilde{Y}$. We obtain that the family of
hyperboloids, $X^{2}-Y^{2}+\widetilde{Y}^{2}=c^{2}={\rm const}$, are RG-
invariants, see Fig. 2-(e). At the critical point the constant $c$ vanishes,
thus defining the cone with apex at the fixed point shown in Fig. 2-(f) as a
separatrix. Using the hyperboloid RG invariant one can proceed to
approximately solve the RG flow equations [39] to obtain the correlation
length. From the solution, $X(l)=X(0)\cos(\sqrt{2}cl)$, the correlation length
$\xi$ is obtained by demanding that $X(l_{*})\sim\mathcal{O}(1)$ for some
scale $l_{*}$ such that $\ln(\xi/a)=l_{*}$. This leads to a BKT scaling,
$\xi^{-1}\sim\exp(-{\rm const}/\sqrt{K-K_{c}})$, where $K_{c}=\kappa_{*}$. The
critical behavior is thus essentially governed by the BKT flow diagram of Fig.
2-(b). Strictly speaking the phase transition is not exactly a BKT one though.
The flow lines do not stop at the fixed line that exists for $\widetilde{Y}=0$
when the latter is nonzero, no matter how small. They follow parabolic
trajectory around the conical separatrix shown in Fig. 2-(f). In other words,
it is a BKT-like scaling occurring by means of a running around the fixed
point, with no fixed lines present. In view of these results the
$\mathcal{PT}$ broken regime evidences a scaling behavior known as walking or
pseudocritical, which was studied in detail by Gorbenko et al. [22] (see also
Refs. [41, 25, 28, 31], a concept closely related to ’conformality loss’ [42,
23, 24, 26, 29]).

(a) $d=2$ $\mathcal{PT}$ symmetry can be broken or not, $\widetilde{y}=0$ (b)
$d=2$ $\mathcal{PT}$-symmetric, $y=0$

(c) $d=2$ $\mathcal{PT}$-symmetric (d) $d=2$ $\mathcal{PT}$ broken (e) RG-
invariant surfaces $\mathcal{PT}$ broken (f) Conical separatrix $\mathcal{PT}$
broken

Figure 2: RG flows for the non-Hermitian system associated to the Lagrangian
of Eq. (3). The RG flows shown here are for $d=2$. In all panels the red dot
represents the fixed point $\kappa_{*}=2/\pi$ and $\widetilde{y}=y=0$. Panels
(a) and (b) show separately the BKT planes appearing in panel (c) for the
$\mathcal{PT}$-symmetric case. Note that these planes are dual to each other
in the sense that the fixed line is for $\kappa\geq\kappa_{*}$ in panel (a)
and for $\kappa\leq\kappa_{*}$ in panel (b). When the $\mathcal{PT}$ symmetry
is broken only the BKT plane of panel (a) survives, causing the BKT transition
seems to dominate the critical behavior, as shown in panel (d). Panel (e)
shows the RG-invariant surfaces. In panel (f) the conical separatrix with apex
at the fixed point is shown. In both panels (e) and (f) we have defined the
variables, $X=2-\pi\kappa$, $Y=2y/\sqrt{\pi}$, and
$\widetilde{Y}=2\widetilde{y}/\sqrt{\pi}$.

Conclusions and outlook — We have determined the phase structure and critical
behavior of a non-Hermitian $\rm XY$ model with a four-state clock
perturbation both in 1+1 and 2+1 dimensions. Our RG analysis gives information
about both $\mathcal{PT}$-symmetric and $\mathcal{PT}$-broken regimes, where
the former phases are associated with real energy eigenvalues giving rise to a
unitary time-evolution and the latter not. In view of the exact equivalence of
the $\mathcal{PT}$-symmetric Lagrangian to a Hermitian clock model, our RG
analysis in this case essentially corroborates existing numerical works for
the Hermitian clock model [17, 20, 21]. On the other hand, our RG analysis
also gives access to the $\mathcal{PT}$-broken regime and we show that its
phase structure and critical behavior are very different from its
$\mathcal{PT}$-symmetric counterpart, as discussed in the main text. The main
point we make here is that the $\mathcal{PT}$ broken regime exhibits walking
(or pseudocritical) behavior in the sense defined in Ref. [22]. This analysis
paves the way to consider the role of dangerously irrelevant operators [43] in
non-Hermitian systems, which are pertinent for non-Hermitian deconfined
quantum critical points, where in a number of cases can be qualified as
pseudocritical [22, 24, 25, 28]. From an experimental point of view, we expect
that the effective theories investigated here may for instance be realized
using Josephson junction arrays with an asymmetric current bias [44]. Other
interesting experimental possibilities arise when certain non-Hermitian
conductance matrices are interpreted as non-Hermitian Hamiltonians, as
recently discussed in Refs. [7, 8].

###### Acknowledgements.

We thank Slava Rychkov for useful comments and discussions on the concept of
walking. We acknowledge financial support by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation), through SFB 1143
project A5 and the Würzburg-Dresden Cluster of Excellence on Complexity and
Topology in Quantum Matter-ct.qmat (EXC 2147, Project Id No. 390858490).

##  Appendix A Supplemental Material

###  A.1 Derivation of the RG equations

Given the renormalized counterparts $\kappa$ and $y$ of $K$ and $z$,
respectively, we obtain the usual RG equations associated to the BKT phase
transition [38, 16],

$\frac{d\kappa^{-1}}{dl}=y^{2},$ (12) $\frac{dy}{dl}=(2-\pi\kappa)y.$ (13)

Switching on the $n=4$ clock model perturbation modifies these RG equations
considerably [16]. In order to obtain the result, we simply disregard the
periodicity of $\phi$ in Eq. (4), which makes it equivalent to an ordinary
sine-Gordon theory, which is known to be exactly equivalent to a Coulomb gas
in $d$ dimensions [37]. Note that in this case we are not specializing to
$d=2$. The reason for this will become clear below. Importantly, we have to
adapt the derivation as given in Ref. [45] such that it allows for a non-
Hermitian sine-Gordon interaction. Crucial in this regard is the role played
by the spatially dependent dielectric constant,

$\varepsilon(r)=1+S_{d}\chi(r),$ (14)

where $S_{d}=2\pi^{d/2}/\Gamma(d/2)$ is the surface area of the unit sphere in
$d$ dimensions and the electric susceptibility of the Coulomb gas is given by,

$\chi(r)=S_{d}\int_{a}^{r}dss^{d-1}\alpha(s)n(s).$ (15)

In the above expression $\alpha(r)$ represents the polarizability and $n(r)$
is the local average number of dipoles. It is in this quantity that the
couplings $z_{r}$ and $z_{i}$ appearing in Eq. (3) of the main text will
enter. Approximately (i.e., for small dipoles) we have,

$n(r)\approx(z_{r}^{2}-z_{i}^{2})e^{-U(r)},$ (16)

where $U(r)$ is the renormalized electric potential. From the self-consistent
analysis of $U(r)$ the RG equations (8) and (9) in the main text are obtained
for the dimensionless couplings $y_{r}$ and $y_{i}$ associated to $z_{r}$ and
$z_{i}$. Note that for broken $\mathcal{PT}$ symmetry where $z_{r}<z_{i}$ the
density (16) becomes negative (in this sense $n(r)$ is best interpreted as a
density imbalance), and the dielectric constant $\varepsilon(r)<1$, leading to
an anti-screening in the Coulomb gas.

Defining $\kappa=4/(\pi^{2}\widetilde{\kappa})$ and $\widetilde{y}$ as the
dimensionless renormalized counterpart of $\widetilde{z}$, we obtain using the
results of Refs. [46, 47, 45] the leading order RG equations in $d$ dimensions
in the form,

$\frac{d\widetilde{\kappa}^{-1}}{dl}=\widetilde{y}^{2}+(d-2)\widetilde{\kappa}^{-1},$
(17)
$\frac{d\widetilde{y}}{dl}=\left[d-f(d)\widetilde{\kappa}\right]\widetilde{y},$
(18)

where $f(d)=(d-2)\Gamma(d/2-1)/(2\pi^{d/2-2})$ and $l=\ln(r/a)$ is the RG
logarithmic scale.

In order to obtain the full set of RG equations for $d>2$ we have to surmount
the technical difficulty imposed by the fact that in this case we are not
dealing with an ensemble of point vortices any longer. In fact, for $d=3$ the
duality leads to an Abelian Higgs theory describing an ensemble of vortex
loops rather than a sine-Gordon theory [48, 49, 50]. However, it has been
argued in the past [51, 52] that near $d=2$ the RG equations for $d>2$ simply
amount to a modification of Eq. (12) in which a term $-(d-2)\kappa^{-1}$ is
added to its right-hand side. Crucially, we note that this term has a sign
opposite to the similar term appearing in Eq. (17) for the Coulomb gas case.
The latter leads to the well known result that the Coulomb gas for $d>2$ does
not undergo a phase transition, reflecting the absence of a fixed point and
the runaway flow [46, 45].

Doing an $\epsilon$-expansion ($\epsilon=d-2$) in the way just mentioned,
which follows the prescription in Refs. [51, 52] has the drawback that it
yields a correlation length critical exponent
$\nu=1/(2\sqrt{\epsilon})[1+\mathcal{O}(\sqrt{\epsilon})]$, which does not
deviate from the mean-field theory value $\nu=1/2$ when $\epsilon=1$. However,
this can be improved in the following way. We can argue that the RG equation
in $d$ dimensions for $y$ has the same form as the one for $\widetilde{y}$
given in Eq. (18), i.e., we do not perform the $\epsilon$-expansion in the RG
equation. This would lead to the following RG equations in absence of the
symmetry breaking field,

$\frac{d\kappa^{-1}}{dl}=y^{2}-(d-2)\kappa^{-1},$ (19)
$\frac{dy}{dl}=\left[d-f(d)\kappa\right]y.$ (20)

Linearizing the above equations around the fixed point with coordinates
$\kappa_{*}=d/f(d)$ and $y_{*}\sqrt{(d-2)/\kappa_{*}}$, we find the
eigenvalues $a_{\pm}=[2-d\pm\sqrt{(d-2)(9d-2)}]/2$, leading to $1/\nu=a_{+}$.
While setting $d=3$ directly in this expression yields exactly $\nu=1/2$, we
have to remember that the procedure assumes that near $d=2$ we can still use
point vortices as a leading order approximation to vortex loops. Hence, our
expression for $\nu$ should also be considered within the framework of an
$\epsilon$-expansion. As a matter of fact, we obtain,
$\nu=1/(2\sqrt{\epsilon})+1/8+\mathcal{O}(\sqrt{\epsilon})$, which in turns
yields $\nu\approx 0.625$ when we set $\epsilon=1$ at the end. This result
precisely agrees with the one-loop RG result obtained from a Landau-Ginzburg-
Wilson analysis of a $|\psi|^{4}$ theory. There are other approaches to
improve the strategy followed by Nelson and Fisher [51], like for example
dealing more directly with the scaling behavior of vortex loops [53], which
ultimately leads to the more accurate result, $\nu\approx 2/3$ [40]. However,
our approach has the virtue of being technically simpler, as it relies more on
the original idea of Refs. [51, 52].

After the above preliminary discussion, we are now finally in position to
write down the RG equations for the Lagrangian of Eq. (4), which simply
amounts to combining the results of Eqs. (17) and (18) with those of Eqs. (19)
and (20) to obtain,

$\frac{d\kappa}{dl}=\frac{4}{\pi^{2}}\widetilde{y}^{2}-\kappa^{2}y^{2}+(d-2)\kappa,$
(21)
$\frac{d\widetilde{y}}{dl}=\left[d-\frac{4f(d)}{\pi^{2}\kappa}\right]\widetilde{y},$
(22)

while the flow equation for $y$ is given by Eq. (20).

###  A.2 Aproximate analysis of the BKT-like scaling in the $\mathcal{PT}$
broken regime

RG equations for $\mathcal{PT}$ broken region in $1+1$ dimensions are given
by,

$\frac{d\kappa}{dl}=-\frac{4}{\pi^{2}}\widetilde{y}^{2}-\kappa^{2}y^{2},\qquad\frac{dy}{dl}=\left[2-\pi\kappa\right]y,\qquad\frac{d\widetilde{y}}{dl}=\left[2-\frac{4}{\pi\kappa}\right]\widetilde{y}.$
(23)

Let’s introduce the new variables,

$X=2-\pi\kappa,\qquad
Y=\frac{2}{\sqrt{\pi}}y,\qquad\widetilde{Y}=\frac{2}{\sqrt{\pi}}\widetilde{y}.$
(24)

The RG equations become

$\frac{dX}{dl}=\widetilde{Y}^{2}+\left(1-\frac{X}{2}\right)^{2}Y^{2}\approx\widetilde{Y}^{2}+Y^{2},\qquad\frac{dY}{dl}=XY,\qquad\frac{d\widetilde{Y}}{dl}=2\left(1-\frac{1}{1-\frac{X}{2}}\right)\widetilde{Y}\approx-X\widetilde{Y}.$
(25)

Thus, the family of hyperboloid

$X^{2}-Y^{2}+\widetilde{Y}^{2}=c^{2}=\text{const}$ (26)

are RG invariants. For $X^{2}+\widetilde{Y}^{2}<Y^{2}$, we parametrize the
hyperboloid of two sheets by

$X=c\sinh{u}\cos{v},\qquad
Y=c\cosh{u},\qquad\widetilde{Y}=c\sinh{u}\sin{v},\qquad
u\in\mathbb{R}^{2},\quad v\in\left[0,\pi\right).$ (27)

It is convenient to define new variable

$X(l)\approx X_{0}+\widetilde{X}(l).$ (28)

and rewrite the Eqs. (25) in terms of $X_{0}$ and $\widetilde{X}$

$\frac{d\widetilde{X}}{dl}\approx\widetilde{Y}^{2}+Y^{2},\qquad\frac{dY}{dl}\approx
X_{0}Y,\qquad\frac{d\widetilde{Y}}{dl}\approx-X_{0}\widetilde{Y}.$ (29)

In this approximation, we can obtain a solution in closed form

$u(l)=\text{arccosh}\left(e^{X_{0}l}\right),\qquad
v(l)=\arcsin\left(\frac{e^{-2X_{0}l}}{\sqrt{e^{-2X_{0}l}-1}}\right).$ (30)

In the large-distance limit, $l$ goes to infinity. Therefore, if $X_{0}$ is
positive, then $u(l)$ increases and $v(l)$ goes to $0$. As a result $X(l)$ and
$Y(l)$ also increas , until the RG trajectory leaves the perturbative regime
and $\widetilde{Y}(l)$ goes to $0$. By contrast, if $X_{0}<0$ then $u(l)$ goes
to $0$ and $v(l)$ increases. In this case, our approximation does not work,
since $v$ is limited by the interval $\left[0,\pi\right)$.

For $X^{2}+\widetilde{Y}^{2}>Y^{2}$, we parametrize the hyperboloid of one
sheet by

$X=c\cosh{u}\cos{v},\qquad
Y=c\sinh{u},\qquad\widetilde{Y}=c\cosh{u}\sin{v},\qquad
u\in\mathbb{R}^{2},\quad v\in\left[0,\pi\right).$ (31)

The solution of Eqs. (29) is then

$u(l)=\text{arcsinh}\left(e^{X_{0}l}\right),\qquad
v(l)=\arcsin\left(\frac{e^{-2X_{0}l}}{\sqrt{X_{0}(e^{-2X_{0}l}+1)}}\right).$
(32)

Therefore, if $X_{0}>0$, then we have exactly the same behavior as in the
previous case.

Since the critical point is characterized by $X=0$, $Y=0$ and
$\widetilde{Y}=0$, the constant $c^{2}$ has to vanish there. Thus, $c^{2}$
controls the distance to the critical point, which we can capture in terms of
the parameter $K$, according to

$c^{2}=b\left(K-K_{c}\right),$ (33)

where $b$ is an unimportant prefactor. Using Eqs. (25), we can write

$\frac{d^{2}X}{dl^{2}}=2\widetilde{Y}\frac{d\widetilde{Y}}{dl}+2Y\frac{dY}{dl}=-2X(-Y^{2}+\widetilde{Y}^{2})=-2X(c^{2}-X^{2})\approx-2c^{2}X,$
(34)

which can be satisfied by the next ansatz

$X(l)=X(0)\cos(\sqrt{2}cl).$ (35)

We evaluate the left-hand side when $X(l)$ is of order 1. This happens at a
scale $l=l_{*}$ which is related to the correlation length $\xi=ae^{l_{*}}$.
Equation

$X(0)\cos(\sqrt{2}cl_{*})=1$ (36)

has a solution

$l_{*}=\frac{1}{\sqrt{2b(K-K_{c})}}\arccos\left[1/X(0)\right],\qquad|X(0)|\geqslant
1$ (37)

and finally

$\xi^{-1}=a^{-1}\exp\left(-\frac{1}{\sqrt{2b(K-K_{c})}}\arccos\left[1/X(0)\right]\right).$
(38)

## References

  * Bender and Boettcher [1998] C. M. Bender and S. Boettcher, Real spectra in non-hermitian hamiltonians having $\mathcal{PT}$ symmetry, Phys. Rev. Lett. 80, 5243 (1998).
  * Bender [2005] C. M. Bender, Introduction to $\mathcal{PT}$-symmetric quantum theory, Contemporary Physics 46, 277 (2005), https://doi.org/10.1080/00107500072632 .
  * Ashida  _et al._ [2020] Y. Ashida, Z. Gong, and M. Ueda, Non-hermitian physics, Advances in Physics 69, 249 (2020), https://doi.org/10.1080/00018732.2021.1876991 .
  * Stegmaier  _et al._ [2021] A. Stegmaier, S. Imhof, T. Helbig, T. Hofmann, C. H. Lee, M. Kremer, A. Fritzsche, T. Feichtner, S. Klembt, S. Höfling, I. Boettcher, I. C. Fulga, L. Ma, O. G. Schmidt, M. Greiter, T. Kiessling, A. Szameit, and R. Thomale, Topological defect engineering and $\mathcal{P}\mathcal{T}$ symmetry in non-hermitian electrical circuits, Phys. Rev. Lett. 126, 215302 (2021).
  * Wang  _et al._ [2023] A. Wang, Z. Meng, and C. Q. Chen, Non-hermitian topology in static mechanical metamaterials, Science Advances 9, eadf7299 (2023), https://www.science.org/doi/pdf/10.1126/sciadv.adf7299 .
  * Liang  _et al._ [2022] Q. Liang, D. Xie, Z. Dong, H. Li, H. Li, B. Gadway, W. Yi, and B. Yan, Dynamic signatures of non-hermitian skin effect and topology in ultracold atoms, Phys. Rev. Lett. 129, 070401 (2022).
  * Ochkan  _et al._ [2024] K. Ochkan, R. Chaturvedi, V. Könye, L. Veyrat, R. Giraud, D. Mailly, A. Cavanna, U. Gennser, E. M. Hankiewicz, B. Büchner, J. van den Brink, J. Dufouleur, and I. C. Fulga, Non-hermitian topology in a multi-terminal quantum hall device, Nature Physics 10.1038/s41567-023-02337-4 (2024).
  * Könye  _et al._ [2023] V. Könye, K. Ochkan, A. Chyzhykova, J. C. Budich, J. van den Brink, I. C. Fulga, and J. Dufouleur, Non-hermitian topological ohmmeter (2023), arXiv:2308.11367 [cond-mat.mes-hall] .
  * Li  _et al._ [2023] J. Li, M. Tang, J. Duan, X. Xu, K. Xu, L. Ma, and J. Wang, Exceptional points in a spiral ring cavity for enhanced biosensing, Journal of Lightwave Technology 41, 2870 (2023).
  * Hatano and Nelson [1996] N. Hatano and D. R. Nelson, Localization transitions in non-hermitian quantum mechanics, Phys. Rev. Lett. 77, 570 (1996).
  * Bender  _et al._ [2005] C. M. Bender, H. Jones, and R. Rivers, Dual $\mathcal{PT}$-symmetric quantum field theories, Physics Letters B 625, 333 (2005).
  * Ashida  _et al._ [2016] Y. Ashida, S. Furukawa, and M. Ueda, Parity-time symmetric quantum critical phenomena, Nature Communications 8 (2016).
  * Zhang  _et al._ [2022] S.-B. Zhang, M. M. Denner, T. c. v. Bzdušek, M. A. Sentef, and T. Neupert, Symmetry breaking and spectral structure of the interacting hatano-nelson model, Phys. Rev. B 106, L121102 (2022).
  * ’t Hooft [1978] G. ’t Hooft, On the phase transition towards permanent quark confinement, Nuclear Physics B 138, 1 (1978).
  * Elitzur  _et al._ [1979] S. Elitzur, R. B. Pearson, and J. Shigemitsu, Phase structure of discrete abelian spin and gauge systems, Phys. Rev. D 19, 3698 (1979).
  * José  _et al._ [1977] J. V. José, L. P. Kadanoff, S. Kirkpatrick, and D. R. Nelson, Renormalization, vortices, and symmetry-breaking perturbations in the two-dimensional planar model, Phys. Rev. B 16, 1217 (1977).
  * Hove and Sudbø [2003] J. Hove and A. Sudbø, Criticality versus q in the $(2+1)$-dimensional ${Z}_{q}$ clock model, Phys. Rev. E 68, 046107 (2003).
  * Oshikawa [2000] M. Oshikawa, Ordered phase and scaling in ${Z}_{n}$ models and the three-state antiferromagnetic potts model in three dimensions, Phys. Rev. B 61, 3430 (2000).
  * Pujari  _et al._ [2015] S. Pujari, F. Alet, and K. Damle, Transitions to valence-bond solid order in a honeycomb lattice antiferromagnet, Phys. Rev. B 91, 104411 (2015).
  * Shao  _et al._ [2020] H. Shao, W. Guo, and A. W. Sandvik, Monte carlo renormalization flows in the space of relevant and irrelevant operators: Application to three-dimensional clock models, Phys. Rev. Lett. 124, 080602 (2020).
  * Patil  _et al._ [2021] P. Patil, H. Shao, and A. W. Sandvik, Unconventional u(1) to ${Z}_{q}$ crossover in quantum and classical $q$-state clock models, Phys. Rev. B 103, 054418 (2021).
  * Gorbenko  _et al._ [2018] V. Gorbenko, S. Rychkov, and B. Zan, Walking, weak first-order transitions, and complex cfts, Journal of High Energy Physics 2018, 108 (2018).
  * Kaplan  _et al._ [2009] D. B. Kaplan, J.-W. Lee, D. T. Son, and M. A. Stephanov, Conformality lost, Phys. Rev. D 80, 125005 (2009).
  * Nogueira and Sudbø [2013] F. S. Nogueira and A. Sudbø, Deconfined quantum criticality and conformal phase transition in two-dimensional antiferromagnets, Europhysics Letters 104, 56004 (2013).
  * Nahum  _et al._ [2015] A. Nahum, J. T. Chalker, P. Serna, M. Ortuño, and A. M. Somoza, Deconfined quantum criticality, scaling violations, and classical loop models, Phys. Rev. X 5, 041048 (2015).
  * Nogueira  _et al._ [2019] F. S. Nogueira, J. van den Brink, and A. Sudbø, Conformality loss and quantum criticality in topological higgs electrodynamics in $2+1$ dimensions, Phys. Rev. D 100, 085005 (2019).
  * Ma and Wang [2020] R. Ma and C. Wang, Theory of deconfined pseudocriticality, Phys. Rev. B 102, 020407 (2020).
  * Ihrig  _et al._ [2019] B. Ihrig, N. Zerf, P. Marquard, I. F. Herbut, and M. M. Scherer, Abelian higgs model at four loops, fixed-point collision, and deconfined criticality, Phys. Rev. B 100, 134507 (2019).
  * Ma and He [2019] H. Ma and Y.-C. He, Shadow of complex fixed point: Approximate conformality of $q>4$ potts model, Phys. Rev. B 99, 195130 (2019).
  * Burgelman  _et al._ [2023] L. Burgelman, L. Devos, B. Vanhecke, F. Verstraete, and L. Vanderstraeten, Contrasting pseudocriticality in the classical two-dimensional heisenberg and ${\mathrm{rp}}^{2}$ models: Zero-temperature phase transition versus finite-temperature crossover, Phys. Rev. E 107, 014117 (2023).
  * Weber and Vojta [2023] M. Weber and M. Vojta, Su(2)-symmetric spin-boson model: Quantum criticality, fixed-point annihilation, and duality, Phys. Rev. Lett. 130, 186701 (2023).
  * Hawashin  _et al._ [2023] B. Hawashin, A. Eichhorn, L. Janssen, M. M. Scherer, and S. Ray, The nordic-walking mechanism and its explanation of deconfined pseudocriticality from wess-zumino-witten theory (2023), arXiv:2312.11614 [cond-mat.str-el] .
  * Weber [2024] M. Weber, Tunable quantum criticality and pseudocriticality across the fixed-point annihilation in the anisotropic spin-boson model (2024), arXiv:2403.02400 [cond-mat.str-el] .
  * Zhou  _et al._ [2024] Z. Zhou, L. Hu, W. Zhu, and Y.-C. He, The $\mathrm{SO}(5)$ deconfined phase transition under the fuzzy sphere microscope: Approximate conformal symmetry, pseudo-criticality, and operator spectrum (2024), arXiv:2306.16435 [cond-mat.str-el] .
  * Jacobsen and Wiese [2024] J. L. Jacobsen and K. J. Wiese, Lattice realization of complex cfts: Two-dimensional potts model with $q>4$ states (2024), arXiv:2402.10732 [hep-th] .
  * Tang  _et al._ [2024] Y. Tang, H. Ma, Q. Tang, Y.-C. He, and W. Zhu, Reclaiming the lost conformality in a non-hermitian quantum 5-state potts model (2024), arXiv:2403.00852 [cond-mat.stat-mech] .
  * Fröhlich and Spencer [1981] J. Fröhlich and T. Spencer, On the statistical mechanics of classical coulomb and dipole gases, Journal of Statistical Physics 24, 617 (1981).
  * Nelson and Kosterlitz [1977] D. R. Nelson and J. M. Kosterlitz, Universal jump in the superfluid density of two-dimensional superfluids, Phys. Rev. Lett. 39, 1201 (1977).
  * [39] See Supplemental Material at [URL will be inserted by publisher] for a derivation of this result, and where Refs. [46, 47, 45, 48, 49, 51, 52, 53] are cited.
  * Zinn-Justin [2021] J. Zinn-Justin,  _Quantum Field Theory and Critical Phenomena: Fifth Edition_, International series of monographs on physics (Oxford University Press, 2021).
  * Herbut [2016] I. F. Herbut, Chiral symmetry breaking in three-dimensional quantum electrodynamics as fixed point annihilation, Phys. Rev. D 94, 025036 (2016).
  * Miransky and Yamawaki [1997] V. A. Miransky and K. Yamawaki, Conformal phase transition in gauge theories, Phys. Rev. D 55, 5051 (1997).
  * Amit and Peliti [1982] D. J. Amit and L. Peliti, On dangerous irrelevant operators, Annals of Physics 140, 207 (1982).
  * Bøttcher  _et al._ [2018] C. G. L. Bøttcher, F. Nichele, M. Kjaergaard, H. J. Suominen, J. Shabani, C. J. Palmstrøm, and C. M. Marcus, Superconducting, insulating and anomalous metallic regimes in a gated two-dimensional semiconductor–superconductor array, Nature Physics 14, 1138 (2018).
  * Nogueira and Kleinert [2008] F. S. Nogueira and H. Kleinert, Compact quantum electrodynamics in $2+1$ dimensions and spinon deconfinement: A renormalization group analysis, Phys. Rev. B 77, 045107 (2008).
  * Kosterlitz [1977] J. M. Kosterlitz, The d-dimensional coulomb gas and the roughening transition, Journal of Physics C: Solid State Physics 10, 3753 (1977).
  * Kleinert  _et al._ [2003] H. Kleinert, F. S. Nogueira, and A. Sudbø, Kosterlitz–thouless-like deconfinement mechanism in the (2+1)-dimensional abelian higgs model, Nuclear Physics B 666, 361 (2003).
  * Peskin [1978] M. E. Peskin, Mandelstam-’t Hooft duality in abelian lattice models, Ann. Phys. (N. Y). 113, 122 (1978).
  * Dasgupta and Halperin [1981] C. Dasgupta and B. I. Halperin, Phase transition in a lattice model of superconductivity, Phys. Rev. Lett. 47, 1556 (1981).
  * Kleinert [1989] H. Kleinert,  _Gauge Fields in Condensed Matter: Vol. 1: Superflow and Vortex Lines (Disorder Fields, Phase Transitions) Vol. 2: Stresses and Defects (Differential Geometry, Crystal Melting)_ (World Scientific, 1989).
  * Nelson and Fisher [1977] D. R. Nelson and D. S. Fisher, Dynamics of classical $\mathrm{XY}$ spins in one and two dimensions, Phys. Rev. B 16, 4945 (1977).
  * Cardy and Hamber [1980] J. L. Cardy and H. W. Hamber, $o(n)$ heisenberg model close to $n=d=2$, Phys. Rev. Lett. 45, 499 (1980).
  * Shenoy [1989] S. R. Shenoy, Vortex-loop scaling in the three-dimensional xy ferromagnet, Phys. Rev. B 40, 5056 (1989).