# Eigenstate entanglement entropy in $PT$ invariant non-Hermitian system

Ranjan Modak1 and Bhabani Prasad Mandal1 1 Department of Physics, Banaras
Hindu University, Varanasi-221005, India

###### Abstract

Much has been learned about universal properties of the eigenstate
entanglement entropy for one-dimensional lattice models, which is described by
a Hermitian Hamiltonian. While very less of it has been understood for non-
Hermitian systems. In the present work we study a non-Hermitian, non-
interacting model of fermions which is invariant under combined $PT$
transformation. Our models show a phase transition from $PT$ unbroken phase to
broken phase as we tune the hermiticity breaking parameter. Entanglement
entropy of such systems can be defined in two different ways, depending on
whether we consider only right (or equivalently only left) eigenstates or a
combination of both left and right eigenstates which form a complete set of
bi-orthonormal eigenstates. We demonstrate that the entanglement entropy of
the ground state and also of the typical excited states show some unique
features in both of these phases of the system. Most strikingly, entanglement
entropy obtained taking a combination of both left and right eigenstates shows
an exponential divergence with system size at the transition point. While in
the $PT$-unbroken phase, the entanglement entropy obtained from only the right
(or equivalently left) eigenstates shows identical behavior as of an
equivalent Hermitian system which is connected to the non-Hermitian system by
a similarity transformation.

##  I Introduction

Entanglement is a property of the quantum systems, providing unique ways of
characterizing quantum many-body systems Eisert  _et al._ (2010); Amico  _et
al._ (2008). The correlations between two entangled quantum systems that are
in an overall pure state cannot be explained by a local classical theory
Einstein  _et al._ (1935). Studies of entanglement indicators have given
insights into properties of ground states Vidal  _et al._ (2003); Audenaert
_et al._ (2002), quantum phase transitions Osborne and Nielsen (2002);
Osterloh  _et al._ (2002), and highly excited eigenstates that exhibit
eigenstate thermalization Deutsch (1991); Rigol  _et al._ (2008); Srednicki
(1994); D’Alessio  _et al._ (2016). Different measures of entanglement also
have been brought lots of attention in the context of black hole physics
Srednicki (1993); Bombelli  _et al._ (1986), holography Ryu and Takayanagi
(2006); Lewkowycz and Maldacena (2013), and quantum information scrambling in
non-equilibrium quantum dynamics Lewis-Swan  _et al._ (2019); Alba and
Calabrese (2019); Modak  _et al._ (2020). Recently, thanks to the advancements
of ultracold atoms in an optical lattice, the measurements of an entanglement
has been realized even in experiments Islam  _et al._ (2015); Kaufman  _et
al._ (2016).

On the other hand, in recent days the study of non-Hermitian systems, such as
open systems or dissipative systems with gain and loss Xu  _et al._ (2016);
Carmichael (1993); Lee  _et al._ (2019a); Konotop  _et al._ (2016); Kawabata
_et al._ (2017), has revealed various intriguing phenomena that do not exist
in Hermitian systems. For an example, the complex energy spectra of non-
Hermitian systems are theoretically predicted to host bulk Fermi arcs Moors
_et al._ (2019); Carlström and Bergholtz (2018); Budich  _et al._ (2019); Yang
and Hu (2019); Yoshida  _et al._ (2019), which has been also realized in
experiments Ding  _et al._ (2018). Also, there is a growing interest to extend
the idea of topological Bloch theory developed in Hermitian systems to non-
Hermitian Hamiltonian Song  _et al._ (2019); Yao  _et al._ (2018); Imura and
Takane (2019); Lee  _et al._ (2019b, 2018).

Among a large class of non-Hermitian system, if a system is invariant under
combined parity and time-reversal ($PT$ ) operations, they can have purely
real spectra for a finite range of parameters Bender  _et al._ (2002); Bender
and Boettcher (1998). The hermiticity property which is sufficient to ensure
the real spectrum of the Hamiltonian in usual quantum mechanics is replaced by
$PT$ symmetry in the case of non-Hermitian systems. Although the spectrum of
such systems may be completely real, the eigenstates may not form orthonormal
set and may not have positive definite norms. Because of these the
probabilistic interpretation of quantum theories fails and the time-evolution
of the corresponding quantum systems becomes non-unitary. Later a consistent
quantum theory with complete real spectrum, unitary time evolution and
probabilistic interpretation for $PT$ symmetric non-Hermitian systems has been
developed in a modified Hilbert space equipped with a positive definite $CPT$
inner product111 Consistent quantum theories with a bigger class of non-
Hermitian systems described by pseudo-Hermitian Hamiltonians have also been
formulated in the similar fashionMostafazadeh (2010); Bender (2007). $C$ is an
additional symmetry associated with every $PT$ symmetric non-Hermitian
systems. Because of this exciting realization the research in non-Hermitian
systems have received a huge boost over the past two decades Bender (2007).
$PT$ symmetric non-Hermitian systems have found numerous applications in
various branches of physics and interdisciplinary ares Mandal and Ghatak
(2012); Mostafazadeh (2009); Ghatak  _et al._ (2015); Hasan and Mandal (2020);
Berry (2004); Heiss (1994); Gmachl (2010); Wan  _et al._ (2011); Liu  _et al._
(2010); Mostafazadeh and Sarisaman (2012); Hasan and Mandal (2018a, b) and
some of the predictions of non-Hermitian theories are experimentally observed
Musslimani  _et al._ (2008); Rüter  _et al._ (2010); El-Ganainy  _et al._
(2007); Guo  _et al._ (2009). Another important aspects is that such PT
symmetric non-Hermitian systems generally exhibit a phase transition ( or more
appropriately a $PT$ breaking transition ) that separates two parametric
regions (i) region of the unbroken $PT$ symmetry in which the entire spectrum
is real and eigenstates of the system respects $PT$ symmetry and (ii) a region
of broken $PT$ symmetry in which the whole spectrum (or a part of it) appears
as complex conjugate pairs and eigenstates of the Hamiltonian do not respect
$PT$ symmetryZnojil (2003); Mandal  _et al._ (2015); Bender  _et al._ (1999);
Mandal (2005); West  _et al._ (2010); Nanayakkara (2002); Bender  _et al._
(2001a).

One of the most popular example of $PT$ symmetric systems are open systems
with balanced gain and loss Lee  _et al._ (2020); Khare and Mandal (2000);
Mandal  _et al._ (2013); Raval and Mandal (2019); Liu  _et al._ (2020).
Typically, in such a system, the parity denotes a reflection symmetry in its
spatial arrangement, and when balanced gain and loss leads to non-Hermiticity.
Usually for small gain and loss rates, the eigenvalues of the $PT$ -symmetric
Hamiltonian describing such a system remain real, however, when the strength
of the gain (or loss) exceeds a value known as the $PT$ -symmetry breaking
threshold, two or more of its eigenvalues become degenerate and then complex-
conjugate pairs. This emergence of complex conjugate eigenvalues is a
signature of $PT$ symmetry breaking. Recent developments in the fabrication
techniques of optical devices can allow one to create and control arrays of
coupled optical waveguides, and the couplings of these arrays can be tuned to
match the dynamics of a large variety of different tight-binding Hamiltonian
Peschel  _et al._ (1998); Christodoulides  _et al._ (2003) and also controlled
loss and gain can be implemented relatively straightforwardly, allowing the
observable dynamics to extend into the non-Hermitian realm Rüter  _et al._
(2010); El-Ganainy  _et al._ (2007).

In this work, our main aim is to understand entanglement properties of the
eigenstate of non-Hermitian $PT$ symmetric systems. In general quantum
correlation is an extremely useful tool to detect different phases as well as
the phase transition. Specially, for 1D systems on lattice, the scaling of
entanglement entropy with system size gives lots of insights about the system.
In order to distinguish between gapless and gapped phases of a system Biswas
_et al._ (2014) or detect localization-delocalization transition Abanin  _et
al._ (2019), the eigenstate entanglement entropy is one of the most popular
diagnostic. Here, we show that the entanglement entropy also can be used as a
probe to detect different phases of $PT$ invariant system as well as $PT$
transition. Since, the Non-Hermitian systems have two types of eigenvectors
(left and right), we define the entanglement entropy in two different ways
Herviou  _et al._ (2019), depending on whether we consider only right (or
equivalently only left) eigenstates or a combination of both left and right
eigenstates. We find that the entanglement entropy obtained taking a
combination of both left and right eigenstates diverges exponentially with
system size at the transition point. While in the PT-unbroken phase, the
entanglement entropy obtained from only the right (or equivalently left)
eigenstate shows identical behavior as of a Hermitian system.

The paper is organized as follows: In Sec. II, we introduce the non-Hermitian
lattice model which is invariant under $PT$ transformation. Next we discuss
our analytical understanding for the $2\times 2$ model in Sec. III. In Sec. IV
we numerically investigate the $PT$ transition point and Sec. V is devoted for
the analysis of entanglement entropy of the ground state as well as a typical
excited state. Finally, in Sec. VI we summarize our results.

##  II Model

We study non-interacting fermions in 1D lattice with open boundary. The system
is described by the following Hamiltonian,

$\displaystyle{H_{0}}=-\sum_{j=1}^{L-1}(\hat{c}^{{\dagger}}_{j}\hat{c}_{j+1}+\text{H.c.})$
(1)

where $\hat{c}^{{\dagger}}_{j}$ ($\hat{c}_{j}$) is the fermionic creation
(annihilation) operator at site $j$, which satisfies standard anti-commutation
relations. $L$ is the size of the system, which we set to be an even number
for all our calculations ( we choose the lattice spacing as unity).

In order to make the Hamiltonian $PT$ symmetric and non-Hermitian, we add a
local term at site $L/2$ and $L/2+1$. The $PT$ symmetric Hamiltonian reads as,

$\displaystyle H=H_{0}+i\gamma(\hat{n}_{L/2}-\hat{n}_{L/2+1})$ (2)

where, $\hat{n}_{j}=\hat{c}^{{\dagger}}_{j}\hat{c}_{j}$ is the number operator
and $\gamma$ is identified as the hermiticity breaking parameter. While under
Parity transformation $c_{j}\to c_{L-j+1}$, Time reversal symmetry operation
changes $i\to-i$. Hence, $H$ remains invariant under $PT$ transformation,
which implies $[H,PT]=0$.

For non-zero values of $\gamma$, $H$ is non-Hermitian. Hence, its left
eigenvectors $|L_{n}\rangle$ and $R_{n}\rangle$ are not the same. However, $H$
is diagonalizable, and $H=\sum_{n}\epsilon_{n}|R_{n}\rangle\langle L_{n}|$
with $\langle L_{n}|R_{m}\rangle=\delta_{mn}$ and $\langle
R_{n}|R_{n}\rangle=1$. $\epsilon_{n}$ can be identified as single-particle
energy eigenvalues of the system, which in general is complex. If the
Hamiltonian $H$ goes through a $PT$ phase transition, then in the $PT$
symmetric phase, the $\epsilon_{n}$ s remain real. On the other hand, in the
broken $PT$ phase $\epsilon_{n}$ s are complex.  We also verified our results
for another model where we have added the hermiticity breaking terms in a $PT$
invariant way in four sites, i.e. described by following Hamiltonian

$\displaystyle\tilde{H}=H_{0}+i\gamma\sum_{j=L/2-(r-1)}^{L/2+r}(-1)^{j}\hat{n}_{j},$
(3)

where we chose $r=2$. We also like to point out that in the absence of
hermiticity breaking term, our model is same as nearest-neighbor fermionic
“tight-binding” model, which is possibly the simplest exactly solvable model
one can write down in the context of condensed matter physics. Also, we
believe that a variant our Hamiltonian with gain and loss is experimentally
realizable in an ultracold fermionic system Yokomizo and Murakami (2019). Note
that for all the many-body calculations, we chose to work at half-filling.

##  III Analytical results: $2\times 2$ matrix example

In this section we restrict the Hamiltonian $H$ to only a lattice of two
sites. Our aim is to analytically solve the $2\times 2$ matrix to gather some
insights about this model. The Hamiltonian $H$ (2) is represented in the
matrix form as

$\displaystyle H^{2\times 2}=\begin{pmatrix}i\gamma&-1\\\
-1&-i\gamma\end{pmatrix}$ (4)

There exists various representation of the parity operator and we define the
parity operator for this model as,

$\displaystyle P=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$ (5)

since $P$ transforms $\begin{pmatrix}1\\\ 0\end{pmatrix}$ to
$\begin{pmatrix}0\\\ 1\end{pmatrix}$ and vice versa. It is an optimal choice
for our case. It is straightforward to check that the Hamiltonian in Eq. 4 is
$PT$ invariant. In the first step, we evaluate the eigenvalues of this
$2\times 2$ matrix, which is,

$\displaystyle E_{\pm}=\pm\sqrt{(1-\gamma^{2})}$ (6)

It implies that for $\gamma>1$ ($\gamma<1$), $E_{\pm}$ is completely complex
(real). Also, it is starightforward to check that for $\gamma<1$,
$|E_{+}\rangle=\frac{1}{\sqrt{2\cos\alpha}}\begin{pmatrix}e^{i\alpha/2}\\\
e^{-i\alpha/2}\end{pmatrix}$ and
$|E_{-}\rangle=\frac{i}{\sqrt{2\cos\alpha}}\begin{pmatrix}e^{-i\alpha/2}\\\
-e^{i\alpha/2}\end{pmatrix}$ are simultaneous eigenstates of $H^{2\times 2}$
Bender  _et al._ (2004) and $PT$, where $\sin\alpha=-\gamma$. Hence, one can
conclude that the $PT$ transition occurs for $H^{2\times 2}$ matrix model at
$\gamma=1$, and $PT$ symmetric phase corresponds to $\gamma<1$.. In the $PT$
broken phase, for $\gamma>1$ $\alpha$ becomes complex and hence the
eigenstates $|E_{+}\rangle$ and $|E_{-}\rangle$ are not eigenstate of $PT$.

Now we go one step ahead, and construct a new linear operator $C$ that
commutes with both $H^{2\times 2}$ and $PT$ . The operator $C$ for $H^{2\times
2}$ matrix turns out to be Bender (2005, 2007),

$\displaystyle C=\frac{1}{\sqrt{1-\gamma^{2}}}\begin{pmatrix}-i\gamma&1\\\
1&i\gamma\end{pmatrix}$ (7)

A more general way to represent the $C$ operator is to express it generically
$C=e^{Q}P$. It has been shown that the square root of the positive operator
$e^{Q}$ can be used to construct a similarity transformation that maps a non-
Hermitian $PT$ -symmetric Hamiltonian $H$ to an equivalent Hermitian
Hamiltonian $h$ Bender  _et al._ (2001b), where

$\displaystyle h=e^{-Q/2}He^{Q/2}$ (8)

For the $H^{2\times 2}$ matrix model, the equivalent Hermitian matrix
$h^{2\times 2}$ will be,

$\displaystyle h^{2\times 2}=\begin{pmatrix}0&-\sqrt{1-\gamma^{2}}\\\
-\sqrt{1-\gamma^{2}}&0\end{pmatrix}$ (9)

Note that $h^{2\times 2}$ is equivalent to $H^{2\times 2}$ because it has the
same eigenvalues as $H^{2\times 2}$. However, the eigenvectors of $H^{2\times
2}$ and $h^{2\times 2}$ are not the same, they are related to each other with
a similarity transformation (see Appendix B for the detail derivation of the
equivalent Hermitian matrix $h^{4\times 4}$ for the $4\times 4$ non-Hermitian
Hamiltonian).

##  IV PT transition

In this section, we investigate the fate of $PT$ transition for the
Hamiltonian (2) and (3) in the thermodynamic limit. In the previous section,
we had showed analytically for the 2 sites version of the Hamiltonian (2), the
$PT$ transition occurs at $\gamma=1$. Here we numerically diagonalize the
Hamiltonian $H$ and $\tilde{H}$ for different values of $L$ to obtain all the
energy eigenvalues. In order to characterize $PT$ transition, we plot the
fraction of complex eigenvalues $I$ as a function of $\gamma$. We expect that
in the $PT$ symmetric (unbroken) phase that ratio should be zero, whereas in
the $PT$ broken phase the value of $I$ should be non-zero. Figure. 1 (main
panel) shows the variation of $I$ as a function of $\gamma$ for different
values of $L$ for the Hamiltonian $H$. We see $I$ jumps from $0$ to $1$ ( it
implies that all eigenvalues become complex in the $PT$ broken phase) at
$\gamma=1$, which concludes a clear signature $PT$ phase transition in this
model, where $PT$ broken (unbroken) phase corresponds to $\gamma>1$
($\gamma<1$). Interestingly, this transition point $\gamma_{th}=1$ obtained
from our numerical results is exactly same as what we obtained by analyzing
the $2\times 2$ matrix in the previous section.

We also like to point out that $PT$ transition we have observed for
Hamiltonian (2) is unique in the sense that here in the $PT$ broken phase all
energy eigenvalues are complex, and hence at $\gamma=1$, the faction of
complex eigenvalues $I$ jumps from $O$ to $1$. However, this feature is just a
manifestation of the fact that we have added the hermiticity breaking
parameter only at sites $L/2$ and $L/2+1$. In the inset of Fig. 1, we have
studied the Hamiltonian $\tilde{H}$ (see Eqn. (3)) where we have added the
hermiticity breaking terms in 4 sites. We show that the $PT$ transition points
$\gamma_{th}\simeq 0.45<1$ and also we find that here $I$ does not jumps
sharply from $0$ to $1$, in contrast there is a parameter regime where $I$
takes values between $[0,1]$. It implies that in that parameter regime, a
fraction of eigenvalues still remains real. However, as we increase the
$\gamma$ all energy eigenvalues become complex. Interestingly, we also like to
point out that for this model near $\gamma=1$, the variation of $I$ with
$\gamma$ is non-monotonic. There is a parameter regime near $\gamma=1$ for
which the number of complex eigenvalues decreases as we increase the
hermiticity breaking parameter. Figure. 2 shows the variation of $I$ with
$\gamma$ for the Hamiltonian Eqn. (3) for different values of $r$ and for
$L=400$. We find indeed the $\gamma_{th}$ becomes much smaller as we increase
$r$. Inset shows that the $\gamma_{th}$ approaches to zero with $r$ as a
powerlaw. Interestingly we find as the number of Hermiticity-breaking terms
increases, the region where real and imaginary eigenvalues co-exist i.e.
$0<I<1$, also increase. Note that it is straightforward to show even
analytically that the $PT$ symmetric phase would not have been stable in the
thermodynamic limit if we had added hermiticity breaking term at all sites
[see Appendix. A for more details].

Figure 1: Variation of the fraction of single-particle complex eigenvalues $I$
as a function of the hermiticity breaking parameter $\gamma$ for $L=100$, 200,
300, and 400 for the Hamiltonian $H$ (2). Inset shows the variation of the
fraction of single-particle complex eigenvalues $I$ as a function $\gamma$,
where the hermiticity breaking terms has been added in 4 sites [see
Hamiltonian $\tilde{H}$ (3).  Figure 2: Variation of the fraction of single-
particle complex eigenvalues $I$ as a function of the hermiticity breaking
parameter $\gamma$ for $L=400$, Hamiltonian $H$ (3) for different values of
$r$. Inset shows $\gamma_{th}$ scales to zero with increasing $r$ as a power-
law. Dashed line corresponds to $~{}1/r$. Figure 3: Comparison of energy
eigenvalues $E_{n}$ between Hamiltonian (2) and Hamiltonian (10) for
$\gamma=0.5$ and $L=400$. Inset shows the differences of the energy
eigenvalues between Hamiltonian (2) and Hamiltonian (10) are of the order of
machine precision.  Figure 4:  Variation of the real part of $S^{R}$ of the
ground state as a function of $\gamma$ for $L=100$, 200, 400. We choose the
subsystem size $L_{A}=L/2$. Solid dashed lines corresponds to ground state
entanglement entropy of the Hermitian model $h$ (10). Inset (a) shows the
variation of the imaginary part of $S^{R}$ of the ground state as a function
of $\gamma$. Inset (b) shows the variation of the real part of $S^{R}$ of the
ground state as a function of $\gamma$ for the subsystem size $L_{A}=L/4$.
Solid dashed lines corresponds to ground state entanglement entropy of the
Hermitian model $h$ (10).

##  V Entanglement entropy

In this section we will discuss the many-body eigenstate entanglement entropy
of the $PT$ invariant Hamiltonian (2). We note that a typical measure of the
entanglement in a quantum system is bipartite von Neumann entanglement entropy
$S$ defined as, $S=-\mbox{Tr}\,_{A}[\rho_{A}\ln\rho_{A}],$ where
$\rho_{A}=\mbox{Tr}\,_{B}\rho$ is the reduced density matrix of a sub-system
$A$ after dividing the system into two adjacent parts $A$ and $B$. $\rho$ is
the total density matrix corresponding to the eigenstate of the system. For
Hermitian system, $\rho=|E_{n}\rangle\langle E_{n}|$, where $E_{n}$s are many-
body energy eigenstates. For non-Hermitian system, since left and right
eigenvectors are not same, we have two choices to define the total many-body
density matrix of the system, i.e. 1) $\rho_{1}=|R_{n}\rangle\langle R_{n}|$
and 2) $\rho_{2}=|R_{n}\rangle\langle L_{n}|$ and corresponding entanglement
entropy are denoted by $S^{R}$ and $S^{LR}$ respectively Gopalakrishnan and
Gullans (2020); Lee (2020). Note that since the Hamiltonian (2) is quadratic,
the ground state (also typical eigenstates) entanglement entropy can be
obtained from the one-body density matrix Herviou  _et al._ (2019) in the
similar spirit as one can do for Hermitian system Vidmar  _et al._ (2018);
Vidmar and Rigol (2017); Vidmar  _et al._ (2017); Hackl  _et al._ (2019);
Peschel and Eisler (2009); Łydżba  _et al._ (2020).

Also, motivated by our analysis for two-sites model, we conjecture that the
Hermitian equivalent model $h$ corresponds to the non-Hermitian Hamiltonian
$H$ is given by,

$\displaystyle
h=-\sum_{j=1}^{L/2-1}(\hat{c}^{{\dagger}}_{j}\hat{c}_{j+1}+\text{H.c.})-\sum_{j=L/2+1}^{L-1}(\hat{c}^{{\dagger}}_{j}\hat{c}_{j+1}+\text{H.c.})$
$\displaystyle-(1-\gamma^{2})^{1/2}(\hat{c}^{{\dagger}}_{L/2}\hat{c}_{L/2+1}+\text{H.c.})$
(10)

Note that if we restrict ourself to $L=2$, the Hamiltonian $h$ becomes
identical to $h^{2\times 2}$ (see Eqn. 9). We also test our conjecture by
comparing the single particle energy eigenvalues $E_{n}$ for the both
Hamiltonians (2) and (10) in the Fig. 3. We find an excellent agreement
between them as shown in Fig. 3.

Next, our goal is to analyze the entanglement entropy of the ground state and
as well as the typical excited states for both the models $H$ (see Eqn. 2) and
$h$ (see Eqn. 10), and compare their results.

###  V.1 Ground state

We first focus on the ground state entanglement entropy of the non-Hermitian
Hamiltonian $H$. Usually the many body ground state of a non-interacting
system is obtained by populating the lowest energy single particle states one
by one. The similar method can also be used for the non-Hermitian system in
the $PT$ symmetric phase given that the energy eigenvalues are purely real. In
case of $PT$ broken phase, given that the eigenvalues can be complex, there is
a bit of ambiguity regarding in which order we should populate the single
particle states to get the desired many-body ground state. However, we arrange
the eigenvalues sorting by its real parts and obtain the many-body ground
state by populating one by one the single-particle states whose real parts of
the eigenvalues are the lowest. Given that we have always restricted ourself
to the even number of particles in the system, this particular choice of the
many-body ground state ensures that many-body ground state energy always
remains real even in the $PT$ broken phase. Moreover, this working definition
of the many-body ground state can also be “analytically continued” in the $PT$
symmetric phase.

Figure. 4 shows the variation of the ground state entanglement entropy
obtained from the right eigenvectors. In the main panel of Fig. 4, we plot the
real part of $S^{R}$ as a function of $\gamma$. Interestingly, we find that
the $S^{R}$ decreases monotonically as a function of $\gamma$ when $\gamma<1$,
i.e. $PT$ symmetric phase. Remarkably, this value goes to zero at the
transition point, i.e. $\gamma=1$. The solid dashed lines in the main panel of
Fig. 4 corresponds to the ground state entanglement entropy of the Hermitian
Hamiltonian $h$. It shows an excellent agreement with the real part of $S^{R}$
in the $PT$ symmetric phase. In the inset (a) of Fig. 4, we also plot the
variation of the imaginary part of the $S^{R}$, we find that while it is zero
in the $PT$ symmetric phase, for $\gamma>1$ (in the broken $PT$ phase) the
imaginary part of $S^{R}$ can have non-zero values. We also like to emphasize
that Re[$S^{R}$] vanishes at $\gamma=1$ is due to the fact that we choose
$L_{A}=L/2$. That becomes even more apparent from our Hermitian equivalent
Hamiltonian (10). At $\gamma=1$, Hamiltonian (10) is just two separate systems
of length $L/2$ who don’t talk to each other. Hence, the entanglement entropy
is bound to be zero if we make a cut in the middile. However, if we choose
$L_{A}=L/4$, at $\gamma=1$, the entanglement will not be zero. In the inset
(b) of Fig. 4, it has been expliclitly shown. However, the variation of the
entanglement entropy in the $PT$ symmetric phase can still be captured by the
Hermitian equivalent Hamiltonian Eqn. 10.

Figure 5:  In the main panel we show the variation of the real part of
$S^{LR}$ of the ground state as a function of $\gamma$ for $L=100$, 200, 400.
We choose the subsystem size $L_{A}=L/2$. Inset shows the variation of the
real part of $S^{LR}$ with $L_{A}$ for $\gamma=0$, 0.25, 0.5, 0.75, 1. We keep
$L=600$ fixed. Red dashed line is the best fit where the fitting function is
$\frac{1}{6}\ln[\sin(\pi L_{A}/L)]$\+ const. Another inset shows the variation
Re[$S^{LR}$] with $L_{A}$ ( for fixed $L=1500$) for $\gamma=1$ in the semi-log
scale. Inset in the right-bottom corner shows the variation of imaginary part
of $S^{LR}$ with $\gamma$.  Figure 6:  Variation of Re$[S^{R}]$ of typical
eigenstates of the Hamiltonian (2) as a function of $\gamma$ for $L=100$, 200,
400. for fixed $L_{A}=L/2$. Black symbols correspond to the entanglement
entropy of a typical state for the Hermitian Hamiltonian $h$. Top right Inset
shows a nice data collapse when we re-scale Re$[S^{R}]$ by Re$[S^{R}]/L_{A}$.
Another inset shows the variation of imaginary part of $S^{R}$ with $\gamma$.
Figure 7:  Variation of Re$[S^{R}]/L$ of typical eigenstates of the
Hamiltonian (3) as a function of $\gamma$ for $L=100$, 200, 400. for fixed
$L_{A}=L/2$, which shows nice data collapse. Inset shows the variation of
Im$[S^{R}]$ with $\gamma$ for the same model.  Figure 8: Variation of
Re$[S^{LR}]/L$ of typical eigenstates as a function of $\gamma$ for $L=100$,
200, 400 for fixed $L_{A}=L/2$ and for the Hamiltonian $\tilde{H}$ (3). Data
collapse is observed for $\gamma<\gamma_{th}\simeq 0.45$. Insets (a) shows the
variation of Re$[S_{typ}^{LR}]$ with $L_{A}$ for $L=500$ and $\gamma$ =
$\gamma_{th}$. Inset (b) shows the variation of the imaginary part of $S^{LR}$
with $\gamma$. Inset (c) shows the data collapse of $S^{LR}/L$ for Hamiltonian
Eqn. 2 in the $PT$ symmetric phase i.e. $\gamma<1$.

Next we investigate how $S^{LR}$ behaves as a function of Hermiticity breaking
parameter $\gamma$. We again focus on the ground state. In the main panel of
Fig. 5, we plot the real part of $S^{LR}$ with $\gamma$ for different values
of $L$, we keep the subsystem size $L_{A}=L/2$ fixed. We find that in the $PT$
symmetric phase and for a fixed value of $\gamma$, the real part of $S^{LR}$
increases with system size. First we focus on $PT$ symmetric phase, i.e.
$\gamma<1$. For $\gamma=0$ we know that the Hamiltonian $H$ is gapless, which
implies that the ground state can be described by a CFT of central charge $c$
. Hence, we expect that the entanglement entropy $S$ should scale as
$S=\frac{c}{6}\ln[\sin(\pi L_{A}/L)]+$const Chang  _et al._ (2020). It is well
known that the central charge corresponding to the Hamiltonian $H_{0}$ is
$c=1$. In the inset of Fig. 5, we show that indeed for $\gamma=0$, $S$ obeys
expected logarithmic scaling. Strikingly we find that the same scaling exists
for $S^{LR}$ in the $PT$ symmetric phase as well. Next, we investigate the
behavior of $S^{LR}$ at the phase transition point i.e. $\gamma=1$. We observe
that at $\gamma=1$ the value of Re[$S^{LR}$] is much larger compared to its
value in the $PT$ symmetric phase. In the inset we show the variation of
Re[$S^{LR}$] with $L_{A}$ for a fixed value of $L$ and we find that $S^{LR}$
actually diverges exponentially with $L_{A}$ at the phase transition point. We
also show the variation of the imaginary part of $S^{LR}$ in the right-bottom
corner inset of Fig. 5 and find that in the $PT$ symmetric phase imaginary
part of $S^{LR}$ is zero, whereas in the $PT$ broken phase it can have non-
zero value.

###  V.2 Typical eigenstates

After investigating the ground state entanglement entropy, now we study the
entanglement entropy of the typical states. Again we investigate separately
both $S^{R}$ and $S^{LR}$. In each configuration, we randomly populate $L/2$
numbers of particles in total $L$ number of single particle states. Then we
average over $1000$ different configurations Vidmar  _et al._ (2018); Vidmar
and Rigol (2017); Vidmar  _et al._ (2017); Modak and Nag (2020). Figure. 6,
shows the variation of the real part of $S^{R}$ of a typical state with
$\gamma$ for the Hamiltonian $H$ (2). We see that in the $PT$ unbroken phase,
the magnitude of $S^{R}$ does not change significantly, it remains almost the
same as the value obtained for $\gamma=0$. However, for $\gamma>1$, the value
monotonically decreases. Also, similar to the ground state, Re[$S^{R}$] in the
$PT$ symmetric phase is almost the same as the entanglement obtained for the
typical eigenstates of Hermitian Hamiltonian $h$ (10). In the inset, we show
the data collapse when we re-scale Re[$S^{R}$] by Re[$S^{R}$]$/L_{A}$, which
indicates the signature of the volume law.We strengthen our claims by
investigating the Hamiltonian Eqn. (3) in Fig. 7 as well.

Next, we investigate the real part of $S^{LR}/L$ in Fig. 8 for the Hamiltonian
$\tilde{H}$ (3). We again find a nice data collapse in the $PT$ symmetric
phase i.e. $\gamma<0.45$, which indicates the volume entanglement. However, at
the transition point $\gamma\simeq 0.45$, the volume law scaling breaks down.
At the point of phase transition even after we rescale Re$[S^{LR}]$ by
Re$[S^{LR}]/L$, the value increases with $L$, that suggests the scaling is
much faster than the volume law. In the inset (a) of Fig. 8, we plot the
variation Re$[S^{LR}]$ for the Hamiltonian $\tilde{H}$ at the $PT$ transition
point with $L_{A}$ for a fixed $L=500$, and we confirm that the scaling of
Re$[S^{LR}]$ with system size is much faster than the volume law scaling
observed in the $PT$ symmetric phase.We also show the imaginary part of
$S^{LR}$ for typical state in the inset (b), which shows that similar to our
previous findings in the $PT$ symmetric (broken) phase Im[$S^{LR}$] is zero
(non-zero). Note that the similar behavior has been observed even for the
Hamiltonian $H$ (2). In the inset (c) of Fig. 8, we show the nice data
collapse for $S^{LR}/L$ in $PT$ symmetric phase i.e. $\gamma<1$.

##  VI Conclusions

In this paper, we investigate the entanglement properties of the ground state
and a typical excited state of a non-interacting non-Hermitian lattice model
which is invariant under $PT$ transformation. The model we had looked into,
has two parts. The Hermitian part is described by a fermionic system having
just nearest-neighbor hopping, while we add the Hermiticity breaking terms
(also known as gain-loss terms) only at the two sites (four sites for the
Hamiltonian $\tilde{H}$) which are situated in the middle of the lattice.
Usually in Hermitian systems if one adds some local pertubations, it does not
alter significantly the extensive quantities like energy of the system,
strikingly here we find that even though we have added the hermicity breaking
terms only in a finite number of sites (in 2 sites for Hamiltonian (2) and in
4 sites for Hamiltonian (3)), but it significantly modifies the properties of
the system. Most importantly, we find that the non-Hermitian model shows $PT$
phase transition as we change the Hermiticity breaking parameter.

Next, we analyze the entanglement properties of different phases of this
model. We find that in the $PT$ symmetric phase, the entanglement entropy
obtained from only the right eigenvetors, are same as the entanglement entropy
obtained for the equivalent Hermitian model. We find this result extremely
non-trivial, given that the equivalence between the non-Hermitian and
Hermitian model implies they have same set of energy eigenvalues, but it does
not imply that the eigenvectors of these two models are the same. Hence, It is
not at all obvious that entanglement entropy of the eigenstates of these two
models should be the same.

Another interesting finding of our work is that the entanglement entropy
obtained by considering both left and right vectors, diverges exponentially
with system size at the $PT$ phase transition point. On the other hand, in the
$PT$ symmetric phase, the ground state entanglement entropy scales
logarithmically with subsystem size, which is an evidence that presumably low
energy states of this $PT$ symmetric non-Hermitian system, might also be
described by CFT. We also find that the typical excited states are volume law
entangled. We would also like to point out that we have repeated all our
calculations for the models where we have added the hermiticity breaking
parameters even at more than $4$ sites. We find that even though the the
results changes quantitatively, qualitative features of the entanglement
entropy remains un-altered. Interestingly the $PT$ phase transition point
$\gamma_{th}$ approach to zero as we increase the number of non-Hermitian
sites, hence we only show the results for the Hamiltonian $H$ (2) and
$\tilde{H}$ (3).

Our future plan is to understand the effect of interaction and disorder in
such systems and study how they modify the $PT$ phase transitions. Recently,
there have been efforts to investigate non-Hermitian Many-body localized phase
Hamazaki  _et al._ (2019), it will be interesting to investigate the $PT$
symmetric system in the same shade of lights.

##  VII Acknowledgements

RM acknowledges the support of DST-Inspire fellowship, by the Department of
Science and Technology, Government of India. The authors would like to thank
Debraj Rakshit for many stimulating discussions.

## Appendix A

Here we investigate a similar model as (2), but we add complex on-site
potential in each site Joglekar and Barnett (2011). The model is described by
the following Hamiltonian,

$\displaystyle\tilde{H}=H_{0}+i\gamma\sum_{j}(-)^{j}\hat{n}_{j}$ (11)

It is straightforward to verify that the Hamiltonian (11) also invariant under
$PT$ transformation, i.e. $[\tilde{H},PT]=0$

Thus, we arrive at the stationary discrete Schrodinger equation,

$\displaystyle E\psi_{j}=\psi_{j+1}+\psi_{j-1}+i\gamma(-1)^{j}\psi_{j}$ (12)

We assume a trial solution $\psi_{j}=Ae^{ijk}+Be^{-ijk}$. Using the boundary
condition $\psi_{0}=\psi_{L+1}=0$, one obtains $A=-B$ and $k=s\pi/(L+1)$, with
$s=0,1$, $,\cdots$ $L-1$. Inserting the trial solution into the Eqn. (12), it
is straightforward to obtain the energy eigenvalues, which is given by

$\displaystyle E^{2}=4\cos^{2}k-\gamma^{2}.$ (13)

From Eq. (13), one gets that all eigenvalues are real for $\gamma<2\cos k$ for
any value of $s$. Since the smallest value of $\cos k$ occurs for $s=L/2$, the
condition of a completely real spectrum is

$\displaystyle\gamma<\gamma_{th}=2\cos\bigg{[}\frac{L\pi}{2(L+1)}\bigg{]}\simeq\pi/L$
(14)

Hence, in the thermodynamic limit the Hamiltonian (11) does not have a true
$PT$ unbroken phase.

## Appendix B

In this section we restrict the Hamiltonian $\tilde{H}$ Eqn. (3) to only a
lattice of four sites. We show the details of the numerical calculation, which
one can use to obtain the Hermitian equivalent Hamiltonian for this system.
The outline of the calculation is already presented in the main text in the
section sec. III. The Hamiltonian $4\times 4$ is represented in the matrix
form as

$\displaystyle H^{4\times 4}=\begin{pmatrix}i\gamma&-1&0&0\\\
-1&-i\gamma&-1&0\\\ 0&-1&i\gamma&-1\\\ 0&0&-1&-i\gamma\\\ \end{pmatrix}$ (15)

We choose $\gamma=0.2$, for which all energy eigenvalues are completely real
i.e. $E_{1}=-1.60563$, $E_{2}=1.60563$, $E_{3}=0.584779$,and
$E_{4}=-0.584779$. Hence, it belongs to the $PT$ symmetric phase. Simultaneous
eigenvectors of $H^{4\times 4}$ and $PT$ are
$|E_{1}\rangle=(0.373182,0.59919+0.07463i,0.603821,0.37032+0.0461278i)^{T}$,
$|E_{2}\rangle=(0.373182,-0.59919+0.07463i,0.603821,-0.37032+0.0461278i)^{T}$,
$|E_{3}\rangle=(0.585094+0.200108i,-0.382172,-0.361608-0.123674i,0.618368)^{T}$,
and
$|E_{4}\rangle=(0.618368,0.361608+0.123674i,-0.382172,-0.585094-0.200108i)^{T}$,
where $PT|E_{1,3}\rangle=|E_{1,3}\rangle$, and
$PT|E_{2,4}\rangle=-|E_{2,4}\rangle$. Now, it is straightforward to obtain $C$
operator, where $C^{2}=I$, and $[C,H^{4\times 4}]=0$, it reads as,

$\displaystyle C=\begin{pmatrix}-0.2819i&0.92331&0.097245i&-0.486228\\\
0.92331&0.184663i&0.437088&-0.097245i\\\
0.097245i&0.437088&-0.184663i&0.92331\\\
-0.486228&-0.097245i&0.923316&0.2819i\end{pmatrix}$

Given that $C=e^{Q}P$ and the parity operator reads as,

$\displaystyle P=\begin{pmatrix}0&0&0&1\\\ 0&0&1&0\\\ 0&1&0&0\\\ 1&0&0&0\\\
\end{pmatrix}$ (16)

it is trivial to obtain $h^{4\times 4}=e^{-Q/2}H^{4\times 4}e^{Q/2}$, which is
represented as,

$\displaystyle h^{4\times 4}=\begin{pmatrix}0&-0.979579&0&-0.0206343\\\
-0.979579&0&-1.00021&0\\\ 0&-1.00021&0&-0.979579\\\
-0.0206343&0&-0.979579&0\end{pmatrix}$

where, $h^{4\times 4}$ is a Hermitian matrix and it is straight forward to
check that it’s eigenvalues are same as the eigenvalues of $H^{4\times 4}$,
i.e. $E_{1}$, $E_{2}$, $E_{3}$,and $E_{4}$. Note that one can perform the
similar calculation for any $PT$ symmetric non-Hermitian system, and obtain
the the Hermitian equivalent Hamiltonian.

## References

  * Eisert  _et al._ (2010) J. Eisert, M. Cramer, and M. B. Plenio, Rev. Mod. Phys. 82, 277 (2010).
  * Amico  _et al._ (2008) L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod. Phys. 80, 517 (2008).
  * Einstein  _et al._ (1935) A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935).
  * Vidal  _et al._ (2003) G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev. Lett. 90, 227902 (2003).
  * Audenaert  _et al._ (2002) K. Audenaert, J. Eisert, M. B. Plenio, and R. F. Werner, Phys. Rev. A 66, 042327 (2002).
  * Osborne and Nielsen (2002) T. J. Osborne and M. A. Nielsen, Phys. Rev. A 66, 032110 (2002).
  * Osterloh  _et al._ (2002) A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature 416, 608 (2002).
  * Deutsch (1991) J. M. Deutsch, Phys. Rev. A 43, 2046 (1991).
  * Rigol  _et al._ (2008) M. Rigol, V. Dunjko, and O. M., Nature 45, 854 (2008).
  * Srednicki (1994) M. Srednicki, Phys. Rev. E 50, 888 (1994).
  * D’Alessio  _et al._ (2016) L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, Adv. Phys. 65, 239 (2016).
  * Srednicki (1993) M. Srednicki, Phys. Rev. Lett. 71, 666 (1993).
  * Bombelli  _et al._ (1986) L. Bombelli, R. K. Koul, J. Lee, and R. D. Sorkin, Phys. Rev. D 34, 373 (1986).
  * Ryu and Takayanagi (2006) S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96, 181602 (2006).
  * Lewkowycz and Maldacena (2013) A. Lewkowycz and J. Maldacena, Journal of High Energy Physics 2013, 90 (2013).
  * Lewis-Swan  _et al._ (2019) R. Lewis-Swan, A. Safavi-Naini, J. J. Bollinger, and A. M. Rey, Nature communications 10, 1 (2019).
  * Alba and Calabrese (2019) V. Alba and P. Calabrese, Physical Review B 100, 115150 (2019).
  * Modak  _et al._ (2020) R. Modak, V. Alba, and P. Calabrese, Journal of Statistical Mechanics: Theory and Experiment 2020, 083110 (2020).
  * Islam  _et al._ (2015) R. Islam, R. Ma, P. M. Preiss, M. E. Tai, A. Lukin, M. Rispoli, and M. Greiner, arXiv preprint arXiv:1509.01160 (2015).
  * Kaufman  _et al._ (2016) A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, P. M. Preiss, and M. Greiner, Science 353, 794 (2016).
  * Xu  _et al._ (2016) H. Xu, D. Mason, L. Jiang, and J. Harris, Nature 537, 80 (2016).
  * Carmichael (1993) H. J. Carmichael, Phys. Rev. Lett. 70, 2273 (1993).
  * Lee  _et al._ (2019a) J. Y. Lee, J. Ahn, H. Zhou, and A. Vishwanath, Phys. Rev. Lett. 123, 206404 (2019a).
  * Konotop  _et al._ (2016) V. V. Konotop, J. Yang, and D. A. Zezyulin, Rev. Mod. Phys. 88, 035002 (2016).
  * Kawabata  _et al._ (2017) K. Kawabata, Y. Ashida, and M. Ueda, Phys. Rev. Lett. 119, 190401 (2017).
  * Moors  _et al._ (2019) K. Moors, A. A. Zyuzin, A. Y. Zyuzin, R. P. Tiwari, and T. L. Schmidt, Phys. Rev. B 99, 041116 (2019).
  * Carlström and Bergholtz (2018) J. Carlström and E. J. Bergholtz, Phys. Rev. A 98, 042114 (2018).
  * Budich  _et al._ (2019) J. C. Budich, J. Carlström, F. K. Kunst, and E. J. Bergholtz, Phys. Rev. B 99, 041406 (2019).
  * Yang and Hu (2019) Z. Yang and J. Hu, Phys. Rev. B 99, 081102 (2019).
  * Yoshida  _et al._ (2019) T. Yoshida, R. Peters, N. Kawakami, and Y. Hatsugai, Phys. Rev. B 99, 121101 (2019).
  * Ding  _et al._ (2018) K. Ding, G. Ma, Z. Q. Zhang, and C. T. Chan, Phys. Rev. Lett. 121, 085702 (2018).
  * Song  _et al._ (2019) F. Song, S. Yao, and Z. Wang, Phys. Rev. Lett. 123, 246801 (2019).
  * Yao  _et al._ (2018) S. Yao, F. Song, and Z. Wang, Phys. Rev. Lett. 121, 136802 (2018).
  * Imura and Takane (2019) K.-I. Imura and Y. Takane, Phys. Rev. B 100, 165430 (2019).
  * Lee  _et al._ (2019b) C. H. Lee, L. Li, and J. Gong, Phys. Rev. Lett. 123, 016805 (2019b).
  * Lee  _et al._ (2018) C. H. Lee, G. Li, Y. Liu, T. Tai, R. Thomale, and X. Zhang, arXiv preprint arXiv:1812.02011 (2018).
  * Bender  _et al._ (2002) C. M. Bender, D. C. Brody, and H. F. Jones, Physical Review Letters 89, 270401 (2002).
  * Bender and Boettcher (1998) C. M. Bender and S. Boettcher, Physical Review Letters 80, 5243 (1998).
  * Note (1) Consistent quantum theories with a bigger class of non-Hermitian systems described by pseudo-Hermitian Hamiltonians have also been formulated in the similar fashionMostafazadeh (2010); Bender (2007).
  * Bender (2007) C. M. Bender, Reports on Progress in Physics 70, 947 (2007).
  * Mandal and Ghatak (2012) B. P. Mandal and A. Ghatak, Journal of Physics A: Mathematical and Theoretical 45, 444022 (2012).
  * Mostafazadeh (2009) A. Mostafazadeh, Physical review letters 102, 220402 (2009).
  * Ghatak  _et al._ (2015) A. Ghatak, M. Hasan, and B. P. Mandal, Physics Letters A 379, 1326 (2015).
  * Hasan and Mandal (2020) M. Hasan and B. P. Mandal, The European Physical Journal Plus 135, 1 (2020).
  * Berry (2004) M. V. Berry, Czechoslovak journal of physics 54, 1039 (2004).
  * Heiss (1994) W. Heiss, Physics Reports 242, 443 (1994).
  * Gmachl (2010) C. F. Gmachl, Nature 467, 37 (2010).
  * Wan  _et al._ (2011) W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, Science 331, 889 (2011).
  * Liu  _et al._ (2010) N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, Nano letters 10, 2342 (2010).
  * Mostafazadeh and Sarisaman (2012) A. Mostafazadeh and M. Sarisaman, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 468, 3224 (2012).
  * Hasan and Mandal (2018a) M. Hasan and B. P. Mandal, Annals of Physics 396, 371 (2018a).
  * Hasan and Mandal (2018b) M. Hasan and B. P. Mandal, Annals of Physics 391, 240 (2018b).
  * Musslimani  _et al._ (2008) Z. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Physical Review Letters 100, 030402 (2008).
  * Rüter  _et al._ (2010) C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nature physics 6, 192 (2010).
  * El-Ganainy  _et al._ (2007) R. El-Ganainy, K. Makris, D. Christodoulides, and Z. H. Musslimani, Optics letters 32, 2632 (2007).
  * Guo  _et al._ (2009) A. Guo, G. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. Siviloglou, and D. Christodoulides, Physical Review Letters 103, 093902 (2009).
  * Znojil (2003) M. Znojil, Journal of Physics A: Mathematical and General 36, 7825 (2003).
  * Mandal  _et al._ (2015) B. P. Mandal, B. K. Mourya, K. Ali, and A. Ghatak, Annals of Physics 363, 185 (2015).
  * Bender  _et al._ (1999) C. M. Bender, S. Boettcher, and P. N. Meisinger, Journal of Mathematical Physics 40, 2201 (1999).
  * Mandal (2005) B. P. Mandal, Modern Physics Letters A 20, 655 (2005).
  * West  _et al._ (2010) C. T. West, T. Kottos, and T. Prosen, Physical review letters 104, 054102 (2010).
  * Nanayakkara (2002) A. Nanayakkara, Physics Letters A 304, 67 (2002).
  * Bender  _et al._ (2001a) C. M. Bender, G. V. Dunne, P. N. Meisinger, and M. Şimşek, Physics Letters A 281, 311 (2001a).
  * Lee  _et al._ (2020) E. Lee, H. Lee, and B.-J. Yang, Physical Review B 101, 121109 (2020).
  * Khare and Mandal (2000) A. Khare and B. P. Mandal, Physics Letters A 272, 53 (2000).
  * Mandal  _et al._ (2013) B. P. Mandal, B. K. Mourya, and R. K. Yadav, Physics Letters A 377, 1043 (2013).
  * Raval and Mandal (2019) H. Raval and B. P. Mandal, Nuclear Physics B 946, 114699 (2019).
  * Liu  _et al._ (2020) Y. Liu, Y. Wang, X.-J. Liu, Q. Zhou, and S. Chen, arXiv preprint arXiv:2009.02012 (2020).
  * Peschel  _et al._ (1998) U. Peschel, T. Pertsch, and F. Lederer, Optics letters 23, 1701 (1998).
  * Christodoulides  _et al._ (2003) D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
  * Biswas  _et al._ (2014) A. Biswas, R. Prabhu, A. Sen(De), and U. Sen, Phys. Rev. A 90, 032301 (2014).
  * Abanin  _et al._ (2019) D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Rev. Mod. Phys. 91, 021001 (2019).
  * Herviou  _et al._ (2019) L. Herviou, N. Regnault, and J. H. Bardarson, SciPost Physics 7 (2019).
  * Yokomizo and Murakami (2019) K. Yokomizo and S. Murakami, Phys. Rev. Lett. 123, 066404 (2019).
  * Bender  _et al._ (2004) C. M. Bender, J. Brod, A. Refig, and M. E. Reuter, Journal of Physics A: Mathematical and General 37, 10139 (2004).
  * Bender (2005) C. M. Bender, Contemporary physics 46, 277 (2005).
  * Bender  _et al._ (2001b) C. M. Bender, G. V. Dunne, P. N. Meisinger, and M. Şimşek, Physics Letters A 281, 311 (2001b).
  * Gopalakrishnan and Gullans (2020) S. Gopalakrishnan and M. J. Gullans, arXiv preprint arXiv:2012.01435 (2020).
  * Lee (2020) C. H. Lee, arXiv preprint arXiv:2011.09505 (2020).
  * Vidmar  _et al._ (2018) L. Vidmar, L. Hackl, E. Bianchi, and M. Rigol, Phys. Rev. Lett. 121, 220602 (2018).
  * Vidmar and Rigol (2017) L. Vidmar and M. Rigol, Phys. Rev. Lett. 119, 220603 (2017).
  * Vidmar  _et al._ (2017) L. Vidmar, L. Hackl, E. Bianchi, and M. Rigol, Phys. Rev. Lett. 119, 020601 (2017).
  * Hackl  _et al._ (2019) L. Hackl, L. Vidmar, M. Rigol, and E. Bianchi, Phys. Rev. B 99, 075123 (2019).
  * Peschel and Eisler (2009) I. Peschel and V. Eisler, Journal of physics a: mathematical and theoretical 42, 504003 (2009).
  * Łydżba  _et al._ (2020) P. Łydżba, M. Rigol, and L. Vidmar, Phys. Rev. Lett. 125, 180604 (2020).
  * Chang  _et al._ (2020) P.-Y. Chang, J.-S. You, X. Wen, and S. Ryu, Physical Review Research 2, 033069 (2020).
  * Modak and Nag (2020) R. Modak and T. Nag, Phys. Rev. Research 2, 012074 (2020).
  * Hamazaki  _et al._ (2019) R. Hamazaki, K. Kawabata, and M. Ueda, Phys. Rev. Lett. 123, 090603 (2019).
  * Joglekar and Barnett (2011) Y. N. Joglekar and J. L. Barnett, Phys. Rev. A 84, 024103 (2011).
  * Mostafazadeh (2010) A. Mostafazadeh, International Journal of Geometric Methods in Modern Physics 7, 1191 (2010).


# Haag as a How-To Theorem

David<EMAIL_ADDRESS>
Marian<EMAIL_ADDRESS>
Chris<EMAIL_ADDRESS>444This work was produced through equal
contribution from each author.

###### Abstract

Haag’s theorem is a classic no-go theorem. It rigorously demonstrates there is
a logical problem with the interaction picture (IP), one of the most widely
used modeling tools in quantum field theory (QFT). The significance of the
theorem for the use of the IP in QFT has been the subject of long-running
debate, focused around how “worried” we should be. In this paper, we argue for
an alternative and opposite perspective on Haag’s theorem, rejecting the
‘worry’ framing in favor of emphasizing the no-go theorem’s implications for
model development.

##  1 Introduction

Haag’s theorem is a classic no-go theorem. It rigorously demonstrates there is
a logical problem with the interaction picture (IP), one of the most widely
used modeling tools in quantum field theory (QFT). The significance of the
theorem for the use of the IP in QFT has been the subject of long-running
debate, focused around how “worried” we should be. One prominent slogan for
the theorem is that ‘the IP doesn’t exist.’ Others argue that the IP remains
sound despite Haag’s theorem due to technicalities of its
implementation.555See [Earman and Fraser, 2006, Fraser, 2006] for crucial and
seminal philosophical treatments of the significance of Haag’s theorem. See
[Freeborn, Gilton, and Mitsch, 2023] for a recent and detailed set of mappings
of the prominent viewpoints on the significance of Haag’s theorem for the
theoretical status of the IP as well as many other issues in philosophy of
QFT.  Either we take the no-go theorem at face value and accept that one of
contemporary physics’s mainstay techniques is mathematically bankrupt, or else
we accept that at least one of the fundamental assumptions of QFT must be at
the very least bent out o shape if not outright abandoned in order to salve
the IP. On this score, the debate has paralleled that around Bell’s theorem,
which seems to force us into a “pick your poison” trilemma. Call this the
what-not perspective on no-go theorems.

But this what-not perspective is not always correct. In this paper, we argue
for an alternative and opposite perspective on Haag’s theorem. On this how-to
view, Haag’s theorem is significant for the positive value it contributes for
model building and for theory development. In this way, Haag’s theorem is not
comparable to Bell’s theorem. We therefore reject the antagonism of the
‘worry’ framing in favor of emphasizing the no-go theorem’s implications for
model development. We call implications of the latter sort how-to
implications.

The argument proceeds by analogy with Arrow’s theorem, extracting two lessons:
(1) treating assumptions as sacrosanct could hinder theory development, and
(2) ignoring an existing multiplicity of modeling purposes could conceal the
model-development affordances of no-go theorems. We apply these lessons to
Haag’s theorem to highlight its how-to implications. We conclude that a how-to
view of Haag’s theorem better captures its significance.

##  2 The how-to implications of Arrow’s theorem

To underscore the significance of Arrow’s theorem, let’s consider a real-world
example: sustainable forest management. In the mid-90s, the U.S.D.A. Forest
Service was tasked with planning land use in parts of the Shoshone National
Forest for a proposed oil and gas lease. In decades past, the Forest Service
made decisions unilaterally [Martin et al., 1996, 603–4]. Yet, by the mid-90s,
new legal frameworks and increasing public scrutiny had led to new decision-
making processes under the rubric of “sustainable forest management”
(SFM)—sustainable in the sense that it generates land management decisions
that can be executed indefinitely, i.e., with less difficulty of
implementation.

The Forest Service now needed to involve stakeholders (including the oil
company, local industries, and government units) in its decision-making
procedure. It identified several alternative plans [Martin et al., 1996, 617].
Naturally, the stakeholders disagreed about their preferred plan. The Forest
Service needed a framework to implement a participatory planning that
accounted for the stakeholders’ varying preferences.

Happily, social choice theory provides just such a framework. Given its
presumptive goal of “combin[ing] individual preferences into collective
choice” [MacKay, 1980, vii], it seems well-positioned to render SFM for the
Forest Service. All the standard social choice ingredients are in place:
several alternative land-use plans—call them alternatives; multiple
stakeholders whom we can model as voters with individual preference schedules:
what the Forest Service needs is then an aggregation procedure for (fairly)
determining a collective choice.

Yet, social choice theory delivers troubling news for our Forest Service team
in the form of Arrow’s theorem, a no-go theorem proven by Kenneth Arrow in
1951 [Arrow, 1950]. This theorem purports to show that, given at least three
voters and two alternatives, no (fair) aggregation procedure exists. The case
at hand meets the conditions, so, we begrudgingly conclude, the Forest Service
cannot fairly determine what to do with the land.

Nevertheless, the Forest Service has persisted in using participatory
planning. Moreover, the processes implemented have led to more sustainable
plans. This raises a question: if we can apparently just ignore it here, what
is the significance of Arrow’s theorem?

As has long been recognized, the significance of Arrow’s theorem resides not
in its directly telling us which goals are impossible or, contrapositively,
which theorems can be ignored by dint of a goal’s achievement in practice.
Instead, its significance has been in guiding the subsequent development of
social choice theory. In other words, the how-to—rather than what-
not—implications of Arrow’s theorem are most significant.

###  2.1 Malleability and Combinatorial Salvation

The first mistake invited by comparison to Bell’s theorem concerns the
sacrosanctity of no-go assumptions. It is generally agreed that Bell’s theorem
undercuts any search for a (non-superdeterministic) local hidden variable
theory of quantum phenomena (see [Clauser and Shimony, 1978, Maudlin, 2014]).
Much of this assessment turns on the presumed sacrosanctity of its
assumptions. Yet, to the degree that the assumptions are model-laden, this
presumption is methodologically harmful.

We can see the value of treating assumptions as malleable (as apposed to
sacrosanct) when we return to SFM and Arrow’s theorem. Let’s make our
assumptions explicit. First, stakeholders are rational in the sense that their
preferences exhibit a weak ordering. Second, stakeholders may rank the
alternatives in any possible order (unrestricted domain). Third, what the
Forest Service needs is an aggregation procedure, a function from complete
lists of stakeholder preferences to the set of social orderings. A democratic
aggregation procedure requires that: social preferences are weakly ordered; if
everyone prefers $A$ to $B$, then so do social preferences; no single voter
can dictate the outcome; and modifications to the order of irrelevant
alternatives cannot change the social preference profile. The Forest Service’s
goal is to find a democratic aggregation procedure; however, Arrow’s theorem
tells us there is no such procedure [Morreau, 2019].

While the goal may yet be impossible, the clear articulation of the
assumptions and goal already points to the potential salvation. Picture the
assumptions as faces of the dials of a combination lock, and “unlocking” it as
finding an aggregation procedure. Arrow’s theorem merely tells us that the
chosen combination doesn’t open the lock. The Forest Service could simply
change the dials by revising assumptions that don’t capture their target
system, suss out further background assumptions,666Three are worth
mentioning—aggregation procedures: are decisive in the sense that they always
generate a social ordering; are determinate in the sense that the social
ordering they generate are determined solely by their inputs; and deliver a
ranking that is utility-maximizing. or even modify the specific goal they
meant to achieve. Merely recognizing the number of possible combinations might
suffice to quell Arrow-fueled worry.

This “combinatorial” understanding was fruitful in both revealing dials (i.e.,
seeing Arrow’s assumptions as malleable), and leading to their turning,
spurring social choice theory’s development.777For this reason [Dardashti,
2021, 47] calls no-go theorems “methodological tools in theory development.”
For example, many ([Hansson, 1973, Pazner, 1979, Sen, 1993, Le Breton and
Weymark, 2011, Fleurbaey, 2003]) have argued that several assumptions are too
strong in any application. More radically, [Balinski and Laraki, 2007] have
shown that alternative aggregation procedures (scoring or grading rules) can
escape Arrow’s theorem. Finally, Arrow’s theorem prompted [Sen, 1999] to
question the merits of the social choice framework itself.

In the case at hand, salvation was to be had. As for all proposals, the Forest
Service had to prepare an Environmental Impact Statement, the legal framework
for which invariably produces a natural ordering of alternatives [Martin et
al., 1996, 614,617]. Since the Forest Service can always double-count itself
to ensure an odd number of voters, an aggregation procedure exists (Condorcet
winner). For another, the independence of irrelevant alternatives may be too
strong, too: given the natural ordering on alternatives and a relatively
equitable distribution of information, strategic voting is easier to detect
and address. Again, Arrow’s theorem is evaded.

What is clear is that the no-go theorem led to a blossoming of responses which
have guided exploration of numerous methodological pathways forward in social
choice theory. Notably, pessimism regarding the original goal of developing
fair aggregation procedures has been the least popular path in this
literature. Thus, while a no-go theorem does show that a specific goal is
impossible to attain given certain assumptions, practitioners, like the Forest
Service, need not despair. While each no-go theorem deserves a native-tongue
analysis, the combinatorial analogy highlights the pathways (dials) one could,
in principle, explore to achieve the same goal.

So, what lesson does Arrow’s theorem teach about no-go theorems that Bell’s
does not? We have seen that treating Arrow’s assumptions as malleable opened
new conceptual and mathematical paths to the original goal, relevant to social
choice theory’s applications. Arrow’s theorem served as an impetus for
substantial development toward democratic aggregation procedures. A sacrosanct
treatment of the assumptions would have harmed such theory development. In a
slogan, the first lesson is this: viewing your assumptions as sacrosanct could
endanger the development of your theory.

###  2.2 Modeling Goal Iteration and SFM’s Purpose

Yet however far the first lesson might take us toward a no-go theorem’s goal,
Bell’s theorem fails to teach us an even more critical lesson. With Bell’s
theorem, it is taken for granted that a would-be quantum theory’s purpose is
to generate probabilities corresponding to those predicted by standard theory.
Bell’s theorem seems to rule out the goal of a local hidden variable theory
generating probabilities that violate Bell’s inequality. There is a seemingly-
inviolable connection between the general purpose of a would-be quantum theory
and the technical goal for achieving that purpose, the latter of which is
ruled out by the no-go theorem. But this connection is not a general feature
of no-go theorems and assuming as much obscures the way such theorems bridge
concrete scientific modeling and abstract theory development.

The role of Arrow’s theorem in SFM makes this second mistake apparent, too.
Begin by asking: what makes social choice theory a good model of decision-
making? Following [Parker, 2020], we assume that models are adequate when they
serve their intended purpose. Importantly, adequacy does not turn only on
representational merits; rather, models must also meet pragmatic constraints
to be adequate. This can separate a model’s achieving the specific goal ruled
out by a no-go theorem—e.g., a democratic aggregation procedure—and its
adequacy for a modeler’s—e.g., the Forest Service—purpose. We can view
adequacy as the finding of solutions within a multi-dimensional purpose space,
indexed by the mathematical model and aspects of how the model is to be
implemented. Some regions will be ruled out by model non-existence (e.g., no-
go theorems), while others will be ruled out for impracticalities of
implementation.

Recall that the Forest Service aims to deliver a sustainable plan. Effective
participatory planning is expected to do this [Kangas et al., 2006, 78]. While
many models are usable in-principle, the pragmatics of the case at hand
constrain which are going to be adequate. In particular, the stakeholders are
likely to withhold challenges to the decision only if it is perceived to be
fair and transparent. But this does not force the Forest Service to use social
choice theory, let alone find a democratic aggregation procedure. That is, the
Forest Service does not need to satisfy the goal that Arrow’s theorem rules
out (modulo its assumptions) in order for a model to be adequate for its
purpose. Indeed, if the stakeholders perceive the decision procedure as fair,
does the Forest Service even need it to be decisive, determinate, or utility-
maximizing?

The geometry of the purpose space helps us understand the development and
deployment of adequate models in SFM. First, the decision procedures do not
need to be democratic. Second, since the legal framework generates
alternatives with a natural order, the original assumptions of Arrow’s theorem
were too restrictive. Third, the Forest Service may not require even the most
basic of properties of the aggregation procedure, depending on the
circumstances. The purpose space has room for many more adequate models,
beyond those excluded by Arrow’s Theorem.

Importantly, some of this space has been filled in with decision procedures
whose models are not strictly captured by social choice theory, (for instance,
see [Sheppard and Meitner, 2005]). Conversely, compilation of knowledge about
various models’ satisfaction of Arrow’s theorem’s goal (and like theorems) has
facilitated more perspicuous selection of models in application (e.g., see
[Kangas et al., 2006, 89]). Thus, Arrow’s theorem has served as a link between
abstract theory development and concrete model deployment.

Elucidating these “how-to” implications paints a richer picture of the
significance of no-go theorems. In particular, the goals of no-go theorems
often simplify the purposes for which models are built to the point that the
former are neither necessary nor sufficient for the latter’s adequacy.
Nevertheless, insofar as a no-go theorem’s goal can be modified so as to
better capture a modeler’s purpose, it affords the development of better
models. Conversely, insofar as a no-go theorem guides the selection of models
more likely to be adequate for purpose, it also affords the development of
more perspicuous and efficient modeling practices. In a slogan, the second
lesson is this: recognizing the liminal space between no-go goals and modeling
purposes can encourage model development and improve model deployment.

##  3 How-to for Haag’s theorem

We now turn to applying our two lessons from Arrow’s theorem to Haag’s
theorem. First, the assumptions of Haag’s theorem are not sacrosanct. Second,
Haag’s theorem can still be epistemically valuable outside of its original
modeling context. Taken together, these two lessons constitute a novel
position on the significance of Haag’s theorem. On this “how-to” view, Haag’s
theorem is scientifically valuable for its positive contributions to the
development of mainstay techniques for calculating scattering amplitudes, and
not, as the what-not view would have it, as an inconvenient impediment for
such techniques.

###  3.1 The Malleability of Haag’s Theorem’s Assumptions

The central purpose of scattering theory in QFT is to generate predictions
about interactions that we can compare against observations, e.g. for
scattering experiments. This involves specifying how the interacting system
behaves and “evolving” the inputs to the outputs. Unfortunately, the
complexity of the relevant interacting systems presents numerous mathematical
obstacles to generating exact predictions. Various modeling techniques pursue
the purpose of scattering with different concrete goals. A sizable portion of
theoretical particle physics proceeds, broadly, with the following modeling
goal: idealize the “in” and “out” multi-particle states (resp.
$\langle\beta\lvert$, $\rvert\alpha\rangle$) as free of interactions,
occurring at $-\infty$ and $+\infty$, respectively; let an interpolating field
between free “in” and “out” states handle the interaction; and consider only
the amplitudes of particular scatterings rather than the full dynamics of
interaction. Call this the “interpolating field” approach to scattering
theory.

Worriers sometimes suppose Haag’s theorem undermines any such use of an
interpolating field approach to rigorously derive (non-trivial) scattering
amplitudes (e.g., [Redhead, 1990, Sklar, 2000]). Such reactions betray an
understanding of the theorem’s assumptions as sacrosanct and model-
independent. There are three main classes of assumptions (cf. [Duncan, 2012,
253-268]). The _state axioms_ require that: (S1) states form a separable
Hilbert space carrying a unitary representation of the Poincar’e group; (S2)
the infinitesimal generators of the translation subgroup have a spectrum
restricted to the forward light-cone; and (S3) there is a unique vacuum state.
The _field axioms_ require that: (F1) (interacting/interpolating) fields are
represented as unbounded operators defined on a dense subset of a Hilbert
space such that their (finite) products are also in the dense subset; (F2) the
fields transform under the unitary representation of the Poincar’e group as
expected; (F3) the commutator vanishes for fields defined on space-like
separated regions; (F4) all states can be approximated arbitrarily well by
linear combinations of field operators applied to the vacuum. The _duality
axioms_ require that: (D1) the interpolating field is given in the interaction
picture; and (D2) the “in” and “out” state spaces are unitarily equivalent.

Focus on (D1), which tells us how systems time-evolve. Quantum mechanics
usually works within the Schrödinger picture, where states, not operators,
time-evolve. Relativistic symmetries make this difficult in QFT. Equivalently,
we could let operators time-evolve but not states (Heisenberg picture). This
works better with relativistic symmetries, making it preferable in QFT.
However, calculating in the Heisenberg picture is difficult. Finally (and
equivalently), we can use the interaction picture (IP), in which we split
fields into their free and interacting components ($H_{0}$ and $V=H-H_{0}$,
resp.), which capitalizes on the strength of both.

This works as follows. The scattering amplitude of a multi-particle state
$\rvert\alpha\rangle$ time-evolving to $\langle\beta\lvert$ by some $\Phi(x)$
has the form

$\underbrace{\langle\beta\lvert e^{iHt}e^{-iH_{0}t}}_{\text{$V$-dep. out-
state}}\underbrace{e^{iH_{0}t}\Phi(x)e^{-iH_{0}t}}_{\text{$H_{0}$-dep.
operator}}\underbrace{e^{iH_{0}t}e^{-iHt}\rvert\alpha\rangle}_{\text{$V$-dep.
in-state}}.$

The IP’s neat trick involves two steps. First, the interaction-picture field
$\Phi(x,t)=e^{iH_{0}t}\Phi(x)e^{-iH_{0}t}$ evolves according to the free
dynamics, while the states carry the interaction. But by (F4), we can
approximate any state by application of field operators to the vacuum, so
$\rvert\alpha\rangle$ actually has the form
$e^{iVt}\Psi_{\alpha}(x)e^{-iVt}\rvert 0\rangle_{V}$, where $\Psi$ is the
field representing interactions, $\Psi_{\alpha}(x)$ is a polynomial of
$\Psi$-field operators approximating $\rvert\alpha\rangle$, and $\rvert
0\rangle_{V}$ is the interacting vacuum; call this the Heisenberg field. The
second step uses the fact that the various pictures coincide at $t=0$. This
allows us to relate the interacting ($V$) and free ($H_{0}$) states by a
unitary transformation. The existence of such a transformation entails that
the free and interacting states occupy the same state space. This ostensibly
accomplishes the goal of the interpolating field approach.

Unfortunately, the IP’s neat trick undermines its use when working with a
unitary representation of the Poincaré group. Our two fields must coincide and
are both free. This is the essence of Haag’s theorem, and it directly
contradicts our specified goal.

Yet on inspection, (D1) does not seem to be model-independent. Rather, it is a
way to (i) simplify interpolating field dynamics and (ii) ensure the
coincidence of the “in” and “out” state spaces with the full state space. But
(i) is obviously not necessary, and (ii) might be achieved by other means.
Haag-Ruelle scattering does just this by weakening (D1): now the interpolating
field is given by a Heisenberg field for the full, interacting-field. Then one
can rigorously construct “in” and “out” state spaces using products of full
field operators and, thereby, derive our (non-trivial) scattering amplitudes
(Haag Asymptotic Theorem). Assuming the Haag Asymptotic Condition, wherein
amplitudes built from the interpolating field converge to those built from
free in-out fields, we can even recover S-matrix amplitudes in terms of
expectation values of time-ordered Heisenberg fields (i.e., the LSZ theory).

LSZ is thus a natural iteration of the IP because free in-out states are still
related by an interpolating field. It differs in that full, interacting-field
states do not need to converge to free in-out states. Instead, they need only
generate the same scattering amplitudes (i.e., preserve inner product
structure). Consequently, while the S-matrix is unique, the _interpolating
field_ is not. Thus, the how-to view emphasizes Haag’s theorem’s role in
guiding the iterative modeling process from the naive IP to the sophisticated
LSZ formalism for accomplishing the specific goal of the interpolating field
approach. This is exactly opposite to the standard what-not view, which takes
Haag’s theorem to primarily be a stumbling block to achieving that modeling
goal.

Nevertheless, one might still worry that Haag’s theorem should threaten
perturbative calculations, which use the IP. However, in practice we _also_
fully regularize our theories which, by breaking Poincaré covariance or
otherwise ensuring a finite-dimensional state space, evades the theorem (see
[Duncan, 2012, Miller, 2018]). What about when we remove cutoff dependence
during renormalization, and purportedly restore Poincaré covariance? Even
then, we only satisfy ourselves with the Poincaré covariance of the subset of
terms used in the calculation, which does not restore Poincaré covariance in
the (S1) and (F2) sense. Thus, lesson one applies when the IP is deployed:
Haag’s theorem’s assumptions are not sacrosanct.

Since the assumptions are not sacrosanct, we should reject the what-not view
of Haag’s theorem. By adjusting the duality assumptions, Haag-Ruelle and LSZ
demonstrate the interpolating field approach is a coherent modeling strategy.
Likewise, tracking the theorem’s assumptions across IP calculations reveals
the shifted meaning of Poincaré covariance. On the what-not view, the central
question is whether Haag’s theorem is a source of serious worry that our
modeling goal is not achievable since either the stated goal or at least one
of the sacrosanct assumptions must be sacrificed in order to evade the theorem
and its dire consequences. When we apply lesson one of the how-to view, we
arrive at the opposite assessment: Haag’s theorem helps us in the iterative
modeling practice by suggesting opportunities for changing modeling
assumptions. This is cause for optimism rather than worry since our
assumptions were never meant to be sacrosanct, set in stone for all eternity.

###  3.2 Iterations On QFT’s Purposes and Modeling Goals

Outside the context of the specific goal associated with a no-go theorem, one
might have thought the theorem has no further how-to implications to offer for
the pursuit of other goals or purposes. But Arrow’s theorem showed us that
this is not always the case. As we saw in section 2.2, the how-to implications
of Arrow’s theorem encompass both concrete model deployment and more abstract
theory development. In this section, we argue for the analogous lesson in the
case of Haag’s theorem.

Recall the central purpose of the area of QFT research to which Haag’s theorem
most immediately applies, namely, generating predictions about interactions to
which we can compare observations. We can pursue this same purpose with
alternative technical goals to the interpolating field approach to accomplish.
One such radical change of goal is argued for by [Kastner, 2015]. Kastner
suggests that the central problem of QFT, leading to Haag’s theorem, is
precisely the assumption that interpolating fields mediate between in-out
states. Instead, Kastner proposes the elimination of these field states by
moving to a direct-action theory. Curiously, a related observation fuels
Seidewitz’s rehabilitation of the IP [Seidewitz, 2017]. However, his version
of the IP does not rely on the interpolating field approach per se since his
field evolves according to a time-invariant parameter encoding paths in
spacetime. His proposal recovers the perturbative expansions for scattering
amplitudes term-by-term. Both Kastner’s and Seidewitz’s proposals for
substantive theory development make explicit use of Haag’s theorem in
identifying and motivating their respective avenues of research. Though both
have given up on the original interpolating field goal in their own way, the
no-go theorem still has methodological value to import from this model-
building arena to the new arena of abstract theory development.

Or perhaps we are seeking a successor theory by building upon aspects of QFT.
The inability to generate usable predictions is not necessarily damning for
this purpose. For instance, we might need to understand the limitations that
Haag’s theorem places on non-perturbative renormalization techniques. Here we
are concerned with precisely the models susceptible to Haag’s theorem. For
example, it has been shown that the ambiguities in the full perturbative
expansion arising from Haag’s theorem can be reduced to a single one [Maiezza
and Vasquez, 2020, 8-10]. Alternatively, suppose we wish to countenance
gravity in QFT. To do this, [Hollands and Wald, 2010, 2] suggest, it is
“essential” to abandon the interpolating field approach because “there will
not, in general, even be an asymptotic notion of particle states.” Thus not
only is the goal of the last section insufficient, even the broader purpose of
modeling the behavior of particle-like objects is not held fixed. Despite
this, the new framework provides insight into the non-analytic dependencies
related to Haag’s theorem [Hollands and Wald, 2010, 37]. In each of these
pursuits, Haag’s theorem helpfully organizes the new insights; this
organization can itself have how-to implications for the purposes of theory
development.

Thus, as with Arrow’s theorem, the how-to reading provides a richer picture of
the connection between the mathematical result and its implications for
modeling. Haag’s theorem does not capture a singular purpose as directly as
does Bell’s theorem. The implications of Haag’s theorem are no less
significant for this; but they have a more _methodological_ , and less
_metaphysical_ character. Haag’s theorem plays a critical role in the task of
QFT model construction. Unlike Bell’s theorem, there is enough room between
the specific goal Haag’s theorem rules out and relevant broader purposes that
it can serve as a useful organizational and exploratory fulcrum for relating
various goals and purposes. We contend that to ignore this how-to potential is
to ignore the very things that make Haag’s theorem significant.

##  4 Conclusion

Discussions of Haag’s theorem in the philosophy literature tend to fall into
one of two camps. There are those who take the theorem at face-value: it says
that we have no means of modeling interactions, or that conventional QFT
derives scattering amplitudes from unsound methods. In response, a second camp
emphasizes the simple logic behind how some calculational techniques of
conventional QFT at some point negate at least one of the key assumptions
needed to prove Haag’s theorem: the ‘worry’ is thus assuaged. We have argued
that this dialectic suffers from a misunderstanding of the epistemic value of
a no-go theorem such as Haag’s. The epistemic value of a no-go theorem does
not boil down to a straightforward adjudication of the possibility of
achieving a specified modeling goal. Rather, many no-go theorems are
epistemically valuable for their how-to implications.

By drawing attention to a no-go theorem’s how-to implications, we do not mean
to detract from their logical identity as no-go’s. The theorems under
consideration are all recognized as yielding a technical result on the basis
of a valid argument. Thus, this highly specific epistemic value of validity is
not in question. But once we have such a proven no-go theorem in hand, there
is the subsequent question of what further implications we should draw from
it. We have argued that an important and overlooked epistemic value of no-go
theorems such as Haag’s arises from the concrete and systematic guidance they
provide to the modeler for generating alternative ways to achieve her goal, as
well as in the bridge these theorems provide between practical model-building
and mathematical theory development.

## Bibliography

  * [Arrow, 1950] Arrow, K. J. (1950).  A difficulty in the concept of social welfare.  Journal of Political Economy, 58(4):328–346.
  * [Balinski and Laraki, 2007] Balinski, M. and Laraki, R. (2007).  A theory of measuring, electing, and ranking.  Proceedings of the National Academy of Sciences, 104(21):8720–8725.
  * [Clauser and Shimony, 1978] Clauser, J. F. and Shimony, A. (1978).  Bell’s theorem. experimental tests and implications.  Reports on Progress in Physics, 41(12):1881.
  * [Dardashti, 2021] Dardashti, R. (2021).  No-go theorems: What are they good for?  Studies in History and Philosophy of Science Part A, 86:47–55.
  * [Duncan, 2012] Duncan, A. (2012).  The conceptual framework of quantum field theory.  Oxford University Press.
  * [Earman and Fraser, 2006] Earman, J. and Fraser, D. (2006).  Haag’s theorem and its implications for the foundations of quantum field theory.  Erkenntnis, 64(3):305.
  * [Fleurbaey, 2003] Fleurbaey, M. (2003).  On the informational basis of social choice.  Social Choice and Welfare, 21(2):347–384.
  * [Fraser, 2006] Fraser, D. L. (2006).  Haag’s Theorem and the Interpretation of Quantum Field Theories with Interactions.  PhD Thesis.
  * [Freeborn et al., 2022] Freeborn, D., Gilton, M. J., and Mitsch, C. (2022).  Haag’s theorem: Assessment, repair, renovation.  in circulation. 2.
  * [Hansson, 1973] Hansson, B. (1973).  The independence condition in the theory of social choice.  Theory and Decision, 4(1):25–49.
  * [Hollands and Wald, 2010] Hollands, S. and Wald, R. M. (2010).  Axiomatic quantum field theory in curved spacetime.  Communications in Mathematical Physics, 293:85–125.
  * [Kangas et al., 2006] Kangas, A., Laukkanen, S., and Kangas, J. (2006).  Social choice theory and its applications in sustainable forest management—a review.  Forest Policy and Economics, 9(1):77–92.
  * [Kastner, 2015] Kastner, R. E. (2015).  Haag’s theorem as a reason to reconsider direct-action theories.  International Journal of Quantum Foundations, 1:56–64.
  * [Le Breton and Weymark, 2011] Le Breton, M. and Weymark, J. A. (2011).  Chapter seventeen - arrovian social choice theory on economic domains.  In Arrow, K. J., Sen, A., and Suzumura, K., editors, Handbook of Social Choice and Welfare, volume 2 of Handbook of Social Choice and Welfare, pages 191–299. Elsevier.
  * [MacKay, 1980] MacKay, A. (1980).  Arrow’s Theorem: The Paradox of Social Choice.  Yale University Press.
  * [Maiezza and Vasquez, 2020] Maiezza, A. and Vasquez, J. C. (2020).  On haag’s theorem and renormalization ambiguities.  arXiv preprint arXiv:2011.08875.
  * [Martin et al., 1996] Martin, W. E., Shields, D. J., Tolwinski, B., and Kent, B. (1996).  An application of social choice theory to u.s.d.a. forest service decision making.  Journal of Policy Modeling, 18(6):603–621.
  * [Maudlin, 2014] Maudlin, T. (2014).  What bell did.  Journal of Physics A: Mathematical and Theoretical, 47(42):424010.
  * [Miller, 2018] Miller, M. E. (2018).  Haag’s theorem, apparent inconsistency, and the empirical adequacy of quantum field theory.  The British Journal for the Philosophy of Science, 69(3):801–820.
  * [Morreau, 2019] Morreau, M. (2019).  Arrow’s Theorem.  In Zalta, E. N., editor, The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, Winter 2019 edition.
  * [Parker, 2020] Parker, W. S. (2020).  Model evaluation: An adequacy-for-purpose view.  Philosophy of Science, 87(3):457–477.
  * [Pazner, 1979] Pazner, E. A. (1979).  Equity, nonfeasible alternatives and social choice : a reconsideration of the concept of social welfare.
  * [Redhead, 1990] Redhead, M. (1990).  A philosopher looks at quantum field theory.  In Brown, H. R. and Harr, R., editors, Philosophical Foundations of Quantum Field Theory. Clarendon Press.
  * [Seidewitz, 2017] Seidewitz, E. (2017).  Avoiding haag’s theorem with parameterized quantum field theory.  Foundations of Physics, 47(3):355–374.
  * [Sen, 1993] Sen, A. (1993).  Internal consistency of choice.  Econometrica, 61(3):495–521.
  * [Sen, 1999] Sen, A. (1999).  The possibility of social choice.  The American Economic Review, 89(3):349–378.
  * [Sheppard and Meitner, 2005] Sheppard, S. R. and Meitner, M. (2005).  Using multi-criteria analysis and visualisation for sustainable forest management planning with stakeholder groups.  Forest Ecology and Management, 207(1):171–187.  Decision Support in Multi Purpose Forestry.
  * [Sklar, 2000] Sklar, L. (2000).  Theory and truth: Philosophical critique within foundational science.  Oxford University Press, USA.