# Path-integral Bosonization of $d=2$ ${\cal P}{\cal T}$ symmetric models

C.M. Naón and F.A. Schaposnik
Departamento de Física, Facultad de Ciencias Exactas,
Universidad Nacional de La Plata
Instituto de Física La Plata-CONICET, UNLP
C.C. 67, 1900 La Plata, Argentina

###### Abstract

We discuss bosonization of non-Hermitian ${\cal P}{\cal T}$ invariant fermion
models in $d=2$ space-time dimensions within the path-integral approach in
which the generating functionals associated to the fermion and boson models
can be related. We first discuss the ${\cal P}{\cal T}$ symmetric Thirring-
sine-Gordon connection and then extend the treatment to bosonize the Gross-
Neveu model.

The possibility that non-Hermitian (NH) quantum Hamiltonians could have real
energy eigenvalues was a subject of study since the 1940 decade. Seminal
results were obtained more recently by Bender and collaborators, starting from
the work [1] and followed by the study of a series of NH Hamiltonian invariant
under both parity (${\cal P}$) and time reversal (${\cal T}$) transformations
leading to real energy eigenvalues and conserved probabilities (see [2] and
[3] for a more complete list of references). Among the many applications to
quantum field theories, one should mention the case of NH quantum
electrodynamics [4]-[5], dual quantum field theories [6], and many other
models.

More recently it was observed that ${\cal P}{\cal T}$-symmetry in topological
insulators guarantees unitarity of states in the bulk but lost at the boundary
thus violating the bulk-boundary correspondence of topological phases; in
connection with this, the role of chiral symmetry was also investigated (see
[7] and references therein). There were also studies of ${\cal P}{\cal T}$
symmetric models in condensed matter physics concerning quantum critical
points [8], tight binding models [9], quantum entanglement [10]-[11] and
quantum dynamics without dissipation [12].

In view of the recent developments mentioned above, it seems relevant to study
boson and fermion quantum field theories within the path-integral framework
which, as recently observed, is particularly useful to study ${\cal P}{\cal
T}$ models for general space-time dimensions [13]. It is the purpose of this
work to analyze bosonization of $d=2$ space-time models in which Lagrangians
are modified in order to have, ${\cal P}{\cal T}$ symmetry in the absence of
hermiticity. In the path-integral approach this implies to show the connection
between generating functionals of the boson and fermion models and this will
be achieved for different $d=2$ models.

The plan of this work is the following: we first discuss, inspired in ref.[6]
the connection between the $d=2$, ${\cal P}{\cal T}$ sine-Gordon and Thirring
models [14]-[15], extending, within the path-integral framework, the
bosonization results presented in [16] for the Hermitian models and then
extend the treatment to the case of the ${\cal P}{\cal T}$ Gross-Neveu model.

## The ${\cal P}{\cal T}$ sine-Gordon model

The generating functional for the ${\cal P}{\cal T}$ extension of the original
Hermitian sine-Gordon model proposed in ref.[6] reads

$Z_{sG}^{{\cal P}{\cal T}}={\cal N}\int D\phi\exp\left(i\int
d^{2}xL_{sG}^{{\cal P}{\cal T}}\right)$ (1)

with $L_{sG}^{{\cal P}{\cal T}}$ given by

$L_{sG}^{{\cal P}{\cal
T}}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{\alpha_{0}}{\beta^{2}}\left(\cos\beta\phi+i\epsilon\sin\beta\phi\right)$
(2)

Note that we have not included the external source coupled to the scalar field
since it does not play any role in the bosonization process.

The addition of the sine term in (2) breaks the original sine-Gordon model
hermiticity but its imaginary coefficient guarantees that ${\cal P}{\cal T}$
symmetry holds.

The interaction term in $L_{sG}^{{\cal P}{\cal T}}$ can be written as

$\frac{\alpha_{0}}{\beta^{2}}\left(\cos\beta\phi+i\epsilon\sin\beta\phi\right)=\frac{\alpha_{0}}{\beta^{2}}\frac{1}{2}\left(\vphantom{\frac{1}{2}}(1+i\epsilon)\exp(i\beta\phi)+(1-i\epsilon)\exp(-i\beta\phi)\right)$
(3)

In order to perform the path-integral defining $Z_{sG}^{{\cal P}{\cal T}}$ by
means of a power expansion in $\alpha_{0}$ we shall then have to consider
vacuum expectations of the form

$\left\langle\prod_{i}\exp\left(i\beta_{i}\phi(x_{i})\right)\right\rangle_{0},$
(4)

where $\langle~{}\rangle_{0}$ indicates v.e.v.’s for the case of free boson
fields $\phi$, and

$\beta_{i}=\pm\beta$ (5)

The result of such v.e.v.’s involves scalar propagators which have serious
infrared divergences. Indeed, the $d=2$ scalar propagator for a free scalar of
mass $\mu$ takes, at short distances the form

$\Delta_{\mu}(x;\mu)=-\frac{1}{2\pi}\log(c\mu|x|)$ (6)

where $c=(1/2)\exp\gamma$ with $\gamma$ the Euler constant. One has also to
take care of ultraviolet divergencies introducing a cutoff $\Lambda$. Using
the Wick product and adopting an appropriate normal ordering (which in the
path-integral consists in the addition of appropriate counterterms to kill
tadpoles) one can easily see that the v.e.v. in (4) takes the form

$\left\langle\prod_{i}\exp\left(i\beta_{i}\phi(x_{i})\right)\right\rangle_{0}=\exp\left(-\frac{1}{2}\sum_{i,j}\beta_{i}\beta_{j}\Delta_{\rho}(x_{i}-x_{j};\mu)\right)$
(7)

where the regularized propagator $\Delta_{\rho}(x;\mu)$ takes the form

$\Delta_{\rho}(x;\mu)=\Delta(\mu|x|)-\Delta_{\rho}(\rho|x|)$ (8)

Here $\rho$ is a mass that takes care of divergencies arising when $x\to 0$.
Indeed,

$2\pi\lim_{\mu|x|\to 0}\Delta_{\rho}(|x|)=-\log(\mu c)+\log(\rho
c)=-\log(\mu/\rho)$ (9)

Now, one has also to take care of ultraviolet divergencies and this is
achieved by introducing a mass $\Lambda$ which will act as a cutoff by
introducing a propagator $\tilde{\Delta}_{\rho}(x;\mu)$ in the form

$\tilde{\Delta}_{\rho}(x;\mu)=\Delta_{\rho}(x;\mu)-\Delta_{\rho}(x;\Lambda)$
(10)

so that finally eq.(7) takes the form

$\displaystyle\left\langle\prod_{i}\exp\left(i\beta_{i}\phi(x_{i})\right)\right\rangle_{\\!\\!0}=\exp\left(-\frac{1}{2}\sum_{i,j}\beta_{i}\beta_{j}\tilde{\Delta}_{\rho}(x_{i}-x_{j};\mu)\right)$
$\displaystyle=$
$\displaystyle\exp\left(-\frac{1}{2}\sum_{i=j}\beta_{i}\beta_{j}\tilde{\Delta}_{\rho}(x_{i}-x_{j};\mu)\right)\exp\left(-\frac{1}{2}\sum_{i\neq
j}\beta_{i}\beta_{j}\tilde{\Delta}_{\rho}(x_{i}-x_{j};\mu)\right)$

where we have separated the $x_{j}=x_{i}$ and the $x_{j}\neq x_{i}$ factors.
Using eq.(9) the former one takes the form

$\displaystyle\lim_{x_{i}\to
x_{j}}\exp\left(-\frac{1}{2}\sum_{i}\frac{\beta_{i}^{2}}{8\pi}\tilde{\Delta}(x_{i}-x_{j})\right)$
$\displaystyle=$
$\displaystyle\left(\frac{\mu}{\rho}\right)^{\sum_{i}\beta_{i}^{2}/(8\pi)}\left(\frac{\rho}{\Lambda}\right)^{-\sum_{i}\beta_{i}^{2}/(8\pi)}$
(12) $\displaystyle=$
$\displaystyle\left(\frac{\mu}{\Lambda}\right)^{\left({(\sum_{i}\beta_{i})^{2}/(4\pi)}\right)}$

If we now include the $i\neq j$ terms one can prove that the complete
$\mu$-dependent factor becomes

$\langle\exp
i{\footnotesize\sum_{j}}\beta_{j}\phi(x_{j})\rangle_{0}=\left(\frac{\mu}{\rho}\right)^{\left({(\sum_{i}\beta_{i})^{2}/(4\pi)}+\sum_{i\neq
j}(\beta_{i}\beta_{j}/4\pi)\tilde{\Delta}(x_{i}-x_{j})\right)}$ (13)

Since bosons $\phi$ are massless one has to take the $\mu\to 0$ limit, this
implying that the v.e.v. vanishes, except if

$\sum_{i}\beta_{i}=0$ (14)

Hence, only the $i\neq j$ factor remains. Now, taking $\Lambda|x_{i}-x_{j}|\gg
1$ and $\mu|x_{i}-x_{j}|\ll 1$ one has $\Delta(x_{i}-x_{j})$ as given in
eq.(6).

We can now use result (13) taking into account eqs.(4)-(5) and perform a
perturbative expansion in $\alpha_{0}$ [16]. As a consequence of result (14)
only the terms which are even in $\frac{\alpha_{0}}{\beta^{2}}$ contribute to
the series, so that the properly normalized generating functional (1) takes
the form

$\displaystyle Z_{sG}^{{\cal P}{\cal T}}$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\frac{1}{k!^{2}}(\alpha/\beta^{2})^{2k}(1-\epsilon^{2})^{k}$
(15)
$\displaystyle\times\int\frac{\prod_{i>j}^{k}(c^{2}\rho^{2}|x_{i}-x_{j}||y_{i}-y_{j}|)^{\beta^{2}/2\pi}}{\prod_{i,j=1}^{k}(c\rho|x_{i}-y_{j}|)^{\beta^{2}/2\pi}}\prod_{i=1}^{k}d^{2}x^{i}d^{2}y^{i}$

where we have defined the renormalized constant

$\alpha=\frac{\alpha_{0}}{2}\left(\frac{\rho}{\Lambda}\right)^{\beta^{2}/4\pi}$
(16)

## The ${\cal P}{\cal T}$ Thirring model

Let us start from the generating functional $Z_{Th}$ for the ${\cal P}{\cal
T}$ invariant Thirring model introduced in ref.[6],

$Z_{Th}^{{\cal P}{\cal T}}=\int D\bar{\psi}D\psi\exp\left(i\int
d^{2}xL_{Th}^{{\cal P}{\cal T}}\right)\;,$ (17)

with the Thirring ${\cal P}{\cal T}$ invariant Lagrangian given by

$L_{Th}^{{\cal P}{\cal T}}=\bar{\psi}(i\not\\!\partial-
m_{1}-m_{2}\gamma_{5})\psi+\frac{g^{2}}{2}(\bar{\psi}\gamma_{\mu}\psi)(\bar{\psi}\gamma^{\mu}\psi)\;,$
(18)

where we are working in Minkowski space (${\rm diag}\,g_{\mu\nu}={(1,-1)}$)
with gamma matrices taken as

$\gamma_{0}=\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right)\;,\hskip
14.22636pt\gamma_{1}=\left(\begin{array}[]{cc}0&-1\\\
1&0\end{array}\right)\;,\hskip
14.22636pt\gamma_{5}=\gamma_{0}\gamma_{1}\;,\hskip
14.22636pt\gamma^{\mu}\gamma_{5}=\varepsilon^{\mu\nu}\gamma_{\mu}$ (19)

Parity reflection operator ${\cal P}$ and time reversal ${\cal T}$ acting on
fermions satisfy

$\displaystyle{\cal P}\psi(x,t){\cal P}=\gamma^{0}\psi(-x,t)\;,$
$\displaystyle{\cal P}\bar{\psi}(x,t){\cal P}=\bar{\psi}(-x,t)\gamma^{0}$ (20)
$\displaystyle{\cal T}\psi(x,t){\cal P}=\gamma^{0}\psi(x,-t)\;,$
$\displaystyle{\cal T}\bar{\psi}(x,t){\cal T}=\bar{\psi}(x,-t)\gamma^{0}$ (21)

Note that the $m_{2}$ term is non-Hermitian because it changes sign under
Hermitian conjugation. Now, being odd both under parity and time reversal
transformations it is ${\cal P}{\cal T}$-symmetric since $L_{Th}^{{\cal
P}{\cal T}}={\cal P}{\cal T}L_{Th}^{{\cal P}{\cal T}}{\cal P}{\cal T}$.

Following [16], we now proceed to linearize the quartic fermion interaction by
introducing an auxiliary vector field through the identity

$\exp\left(i\frac{g^{2}}{2}\int
d^{2}x(\bar{\psi}\gamma_{\mu}\psi)(\bar{\psi}\gamma^{\mu}\psi\right)\\!\\!=\\!\\!\int\\!\\!DA_{\mu}\exp\left(\\!i\int
d^{2}x\left(-\frac{1}{2}A_{\mu}A^{\mu}-g\bar{\psi}\not\\!\\!A\psi\right)\right)$
(22)

so that the generating functional (17) can be written as

$Z_{Th}^{{\cal P}{\cal T}}\\!=\int\\!D\bar{\psi}D\psi
DA_{\mu}\\!\exp\\!\left(\\!i\\!\int\\!d^{2}x\bar{\psi}(i\not\\!\partial-
m_{1}-m_{2}\gamma_{5})\psi-\frac{1}{2}A_{\mu}A^{\mu}-g\bar{\psi}\not\\!\\!A\psi\right)\\!.$
(23)

Now, in $d=2$ dimensions, the vector field $A_{\mu}$ can be written in the
form

$A_{\mu}=\frac{1}{g}\varepsilon_{\mu\nu}\partial^{\nu}\phi+\frac{1}{g}\partial_{\mu}\eta.$
(24)

This formula implies that while $\eta$ is a scalar field, $\phi$ is a
pseudoscalar in view of the Levi-Civita symbol character, as it was originally
established within the original (Hermitian) Thirring-sine Gordon bosonization
connection [15]. Such behavior under ${\cal P}$ is a general feature in any
dimensions and follows from the bosonization formula which relates the fermion
current to a (pseudo) Kalb-Ramond bosonic dual [17].

Let us now consider the following change in the path-integral variables
$\bar{\psi},\psi\rightarrow\bar{\chi},\chi$ and $A_{\mu}\rightarrow\phi,\eta$
so that fermions change according to

$\displaystyle\psi(x)=\exp(i\gamma_{5}\phi(x)+i\eta(x))\chi(x)$
$\displaystyle\bar{\psi}(x)=\bar{\chi}(x)\exp(i\gamma_{5}\phi(x)-i\eta(x))$
(25)

Now, at the generating functional level we have to take into account the
Fujikawa Jacobian [18] $J[\phi]$ which takes the form [16]

$J[\phi]=\exp\left({-i}\frac{1}{2\pi}\int
d^{2}x(\partial_{\mu}\phi)^{2}\right)$ (26)

The $\eta$ field completely decouples from fermions so that it will play no
role in the Thirring model bosonization [16]. We shall then ignore it in what
follows. Concerning the $A_{\mu}\to(\phi,\eta)$ change of variables, it just
gives a trivial Jacobian $J_{A}\propto(1/g^{2})\det\nabla^{2}$ which can be
absorbed in the generating functional normalization so that $Z_{Th}^{{\cal
P}{\cal T}}$ reads

$Z_{Th}^{{\cal P}{\cal T}}=\int D\bar{\psi}D\psi D\phi\exp\left(i\int
d^{2}xL_{Th}^{{\cal P}{\cal T}}\right)\;,$ (27)

with $L_{Th}^{{\cal P}{\cal T}}[\bar{\chi},\chi,\phi]$ given by

$L_{Th}^{{\cal P}{\cal
T}}=\left(\frac{1}{2g^{2}}-\frac{1}{2\pi}\right)(\partial_{\mu}\phi)^{2}+\bar{\chi}i\not\\!\partial\chi-\bar{\chi}\exp(i\gamma_{5}\phi)(m_{1}+m_{2}\gamma_{5})\exp(i\gamma_{5}\phi)\chi$
(28)

In order to prove the equivalence between the ${\cal P}{\cal T}$ invariant
Thirring and sine-Gordon models we shall make a perturbative expansion of the
former in the “mass” term $\bar{\chi}M(x)\chi$, where $M(x)$ is the matrix

$M(x)\equiv(m_{1}+\gamma_{5}m_{2})\exp(2\gamma_{5}\phi)$ (29)

At this point, in analogy with the procedure followed for the sine-Gordon
model leading to expression (4) , choosing an adequate normalization we write
$Z_{Th}^{{\cal P}{\cal T}}$ as the following vacuum to vacuum expectation
value

$Z_{Th}^{{\cal P}{\cal T}}=\left\langle\exp\left(-i\int
d^{2}x\bar{\chi}M(x)\chi\right)\right\rangle_{0},$ (30)

where the v.e.v. above must be computed in a theory of free fermions and free
bosons, with propagators

$G_{F}(x)=-\frac{i}{2\pi}\frac{\gamma^{\mu}x_{\mu}}{x^{2}},$ (31)

and

$\Delta_{\mu}(x;\mu)=-\lambda^{2}\frac{1}{2\pi}\log(c\mu|x|),$ (32)

respectively, with

$\lambda^{2}=\frac{g^{2}}{1-g^{2}/\pi}.$ (33)

In order to get insight into the structure of the perturbative expansion, we
shall display terms up to order $M^{2}$ explicitly. To this end we express the
fermionic bispinors as

$\chi=\left(\begin{array}[]{c}\chi_{1}\\\ \chi_{2}\end{array}\right)$ (34)

and $\bar{\chi}=\chi^{\dagger}\gamma_{0}$. The first order term reads

$\bar{\chi}M\chi=(m_{1}+m_{2})\exp(2i\phi)\bar{\chi}_{1}\chi_{1}++(m_{1}-m_{2})\exp(-2i\phi)\bar{\chi}_{2}\chi_{2}.$
(35)

Concerning second order one has, after some work,

$\displaystyle\bar{\chi}M\chi(x)\bar{\chi}M\chi(y)$ $\displaystyle=$
$\displaystyle(m_{1}^{2}-m_{2}^{2})\left(\vphantom{\frac{1}{2}}A_{+}(x)A_{-}(y)\exp(2i(\phi(x)-\phi(y))\right.$
(36)
$\displaystyle+\left.\vphantom{\frac{1}{2}(m_{1}^{2}-m_{2}^{2})}A_{-}(x)A_{+}(y)\exp(-2i(\phi(x)-\phi(y))\right),$

where we have defined

$A_{\pm}\equiv\bar{\chi}\frac{1}{2}(1\pm\gamma_{5})\chi=\bar{\chi}_{j}\chi_{j},$
(37)

with $j=1(2)$ for upper (lower) sign of $A_{\pm}$ (repeated indices do not
imply sum in this formula).

Then, each term in this expansion can be written as the product of a fermionic
and a bosonic factor. In particular one can easily see that the expectation
value of the first order term (35) vanishes. Indeed, in this case, the
v.e.v.’s of both fermionic and bosonic factors are zero when an adequate
normal ordering that removes Wick contractions of fields at the same point, is
adopted. Concerning the evaluation of the bosonic piece, it is completely
similar to the one performed for the sine-Gordon model, which led to equation
(14), the only difference being that in the present case one has
$\beta_{i}=\pm 2$.

In this way one can calculate all orders in $M(x)$ powers. One finds that the
above arguments leading to the vanishing of the first order term can be
generalized to all odd orders. Concerning the even ones, all of them
contribute to (30), and can be readily computed using (31) and (32). By
carefully analyzing the combinatorics of each term, and writing $m_{1}=m$ and
$m_{2}=\epsilon m$ we finally obtain

$Z_{Th}^{{\cal P}{\cal
T}}=\sum_{k=0}^{\infty}\frac{1}{k!^{2}}(m^{2}(1-\epsilon^{2}))^{k}\int\frac{\prod_{i>j}^{k}(c^{2}\rho^{2}|x_{i}-x_{j}||y_{i}-y_{j}|)^{\beta^{2}/2\pi}\prod_{i=1}^{k}}{\prod_{i,j=1}^{k}(c\rho|x_{i}-y_{j}|)^{\beta^{2}/2\pi}}d^{2}x_{i}d^{2}y_{i}$
(38)

Comparing (15) and (38) one can see that both generating functionals are
identical provided the following identification hold,

$\displaystyle\frac{\beta^{2}}{4\pi}=\frac{1}{1-g^{2}/\pi}$ (39)
$\displaystyle\frac{\alpha}{\beta^{2}}\cos(\beta\phi)=m\sqrt{1-\epsilon^{2}}\bar{\psi}\psi$
(40)

For $\epsilon=0$ the generating function $Z_{Th}^{{\cal P}{\cal T}}$ just
coincides with the generating functional for the Hermitian Thirring model with
fermion mass $m$. Hence as originally found in [6] one should impose
$\epsilon^{2}<1$ in order to have a consistent ${\cal P}{\cal T}$ Thirring
massive model with a real fermion mass $m(1-\epsilon^{2})^{1/2}$.

## The ${\cal P}{\cal T}$ Gross-Neveu model and its bosonization

We start from the path-integral defining the generating functional $Z_{GN}$
for the Hermitian $d=2$ Gross-Neveu fermion model

$Z_{GN}=\int D\bar{\psi}D\psi\exp\left(i\int
d^{2}xL_{GN}[\bar{\psi},\psi]\right)\;,$ (41)

with $\psi$ a massless fermion and the Lagrangian $L_{GN}$ given by

$L_{GN}[\bar{\psi},\psi]=\bar{\psi}i\\!\\!\not\\!\partial\psi+\frac{1}{2}g^{\prime
2}(\bar{\psi}\psi)^{2}$ (42)

Introducing an auxiliary scalar field $\sigma(x)$, $Z_{GN}$ can be written in
the form

$Z_{GN}\\!=\\!\int D\sigma\exp\left(\\!-i\int
d^{2}x\frac{\sigma^{2}}{2}\right)\int D\bar{\psi}D\psi\exp\left(i\int
d^{2}x\left(\bar{\psi}i\\!\\!\not\\!\partial\psi-g^{\prime}\bar{\psi}\psi\sigma\right)\right)$
(43)

or

$Z_{GN}=\int D\sigma\exp\left(\\!-i\int
d^{2}x\frac{\sigma^{2}}{2}\right)\det(i\\!\\!\not\\!\partial-g^{\prime}\sigma)$
(44)

with the fermion determinant given by

$\det(i\\!\\!\not\\!\partial-g^{\prime}\sigma)=\left\langle\exp\left(-ig^{\prime}\int
d^{2}x\sigma\bar{\psi}\psi\rangle\right)\right\rangle_{0}$ (45)

where $\langle~{}\rangle_{0}$ indicates the v.e.v. for the case of free
fermions. Inspired in the mass expansion that we performed in the case of the
Thirring model we shall now proceed, starting from the r.h.s. of eq.(45) to
expand the exponential in powers of $g^{\prime}$ taking into account that no
odd $g^{\prime}$ powers will contribute after an appropriate tadpole
treatment. Moreover, a careful comparison with the Hermitian Thirring model
allows to verify that the combinatorial factors of each term coincides with
the Thirring case. Then, it becomes apparent that the above fermionic
determinant can be expressed as the same expansion in the Hermitian Thirring
coupling constant limit $g\to 0$ and $m\to m(x)\to g^{\prime}\sigma(x)$. Thus,
one gets

$\det(i\\!\\!\not\\!\partial-g^{\prime}\\!\sigma)=\\!\\!\sum_{k=0}^{\infty}\\!\\!\frac{1}{k!^{2}}{g^{\prime}}^{2k}\\!\\!\int\\!\\!\frac{\prod_{i>j}^{k}(c^{2}\rho^{2}|x_{i}-x_{j}||y_{i}-y_{j}|)^{2}\prod_{i=1}^{k}}{\prod_{i,j=1}^{k}(c\rho|x_{i}-y_{j}|)^{2}}\sigma(x_{i})\sigma(y_{i})d^{2}x_{i}d^{2}y_{i}$
(46)

We shall now show that the same expansion in eq.(46) can be obtained for a
sine-Gordon like bosonic model provided the cosine coefficient is
$x$-dependent. Indeed, consider the following vacuum functional

$\displaystyle Z_{sG_{x}}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\int\\!\\!D\phi\exp\left(i\int
d^{2}x\frac{1}{2}\partial_{\mu}\phi\,\partial^{\mu}\phi\right)\exp\left(i\frac{\alpha_{0}(x)}{\beta^{2}}\cos(\beta\phi)\right)$
(47) $\displaystyle=$ $\displaystyle\left\langle\exp\left(i\int
d^{2}x\frac{\alpha_{0}(x)}{\beta^{2}}\cos(\beta\phi)\right)\right\rangle_{0}$

where now $\langle~{}\rangle_{0}$ is the v.e.v. for free bosons.

To see this connection we make an expansion in powers of $\alpha_{0}(x)$ in
the r.h.s. of (47). Again, odd powers vanish so that, for the v.e.v. (47) we
get an expression similar to the one corresponding to the usual, Hermitian
sine-Gordon model, but with the coefficient $\alpha$ promoted to be a function
of $x$:

$\displaystyle Z_{sG_{x}}$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\frac{1}{k!^{2}}(1/\beta^{2})^{2k}\times$
(48)
$\displaystyle\int\frac{\prod_{i>j}^{k}(c^{2}\rho^{2}|x_{i}-x_{j}||y_{i}-y_{j}|)^{\beta^{2}/2\pi}}{\prod_{i,j=1}^{k}(c\rho|x_{i}-y_{j}|)^{\beta^{2}/2\pi}}\prod_{i=1}^{k}\alpha(x_{i})\alpha(y_{i})d^{2}x_{i}d^{2}y_{i}$

where we have defined the renormalized function

$\alpha(x)=\frac{\alpha_{0}(x)}{2}\left(\frac{\rho}{\Lambda}\right)^{\beta^{2}/4\pi}.$
(49)

Comparing this result with that in (46) for the fermion model, we see that
both expansions coincide provided the following identities hold:

$\displaystyle\frac{\beta^{2}}{2\pi}$ $\displaystyle=$ $\displaystyle 2$
$\displaystyle g\sigma(x)$ $\displaystyle=$
$\displaystyle\frac{\alpha(x)}{\rho}.$ (50)

We have then that the GN fermion model can be bosonized becoming the sGx boson
model since their generating functionals have been identified

$Z_{GS_{x}}[\bar{\psi},\psi,\sigma]=Z_{GN}[\phi]$ (51)

Hence, together with identities (50) the bosonization rule for the interaction
terms reads

$g\sigma(x)\bar{\psi}\psi=\frac{\alpha(x)}{4\pi}\cos(2\sqrt{\pi}\ \phi)$ (52)

Up to now we have considered the path-integral bosonization of the usual,
Hermitian Gross-Neveu model, which was not previously discussed in the
literature. Concerning the connection for the case of ${\cal P}{\cal T}$
symmetric models, from what we learnt for the case of ${\cal P}{\cal T}$ sine-
Gordon-Thirring equivalence, it is evident that for the case of the sine-
Gordon model with an $\alpha(x)$ dependent coefficient, in order to define a
${\cal P}{\cal T}$ symmetric model one should extend (47) as

$Z_{sG_{x}}^{{\cal P}{\cal T}}=\\!\\!\int D\phi\exp\left(\\!i\int
d^{2}x\left(\frac{1}{2}\partial_{\mu}\phi\,\partial^{\mu}\phi+\frac{\alpha(x)}{4\pi}(\cos(\beta\phi(x))+i\epsilon\sin(\beta\phi(x))\right)\right)$
(53)

with $\beta=2\sqrt{\pi}$. Then, it is easy to show that the bosonic partition
function $Z_{sG_{x}}^{{\cal P}{\cal T}}$ is equivalent to a fermionic ${\cal
P}{\cal T}$ symmetric Gross-Neveu model that reads

$Z_{GN}^{{\cal P}{\cal
T}}\\!\\!=\\!\\!\int\\!\\!D\sigma\exp\left(\\!\\!-i\int\\!\\!d^{2}x\frac{\sigma^{2}}{2}\\!\right)\\!\\!\int\\!\\!D\bar{\psi}D\psi\exp\left(\\!\\!i\int\\!\\!\\!d^{2}x\\!\\!\left(\bar{\psi}i\\!\\!\not\\!\partial\psi-g^{\prime}\bar{\psi}\psi\sigma\sqrt{1-\epsilon^{2}}\,\right)\\!\right)\\!.$
(54)

This result completes our discussion on the equivalence between non Hermitian
fermionic and bosonic 2D models with ${\cal P}{\cal T}$ symmetry, through
path-integral methods.

In summary, we have presented a path-integal approach to bosonization of
${\cal P}{\cal T}$ models complementary to the one originally discussed in
[6]. Concerning the sine-Gordon-Thirring connection we have explicitly shown
the identity between the resulting generating functionals $Z_{Sg}^{{\cal
P}{\cal T}}$ and $Z_{Th}^{{\cal P}{\cal T}}$, the path-integral analog of the
well-honored Coleman perturbation series results for the Hermitian models
[14]. Using the same approach we have also discussed bosonization in the case
of the Gross-Neveu model finding the corresponding generating functionals of
the dual boson and fermion ${\cal P}{\cal T}$ symmetric models.

In view of the recent applications in condensed matter physics discussed in
the introduction we expect to extend our path integral bosonization approach
to $d=3$ dimensional fermionic models along the lines of the Hermitian case
described in [17], in view to describe ${\cal P}{\cal T}$ planar models.

## Acknowledgments

We would like to thank Eduardo Fradkin for discussions and Daniel Cabra and
Aníbal Iucci for helpful suggestions. F.A.S research was supported by PIP
02229 and UNLP X910 grants. C.M.N. research was supported by UNLP X917 grant.

## References

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