# Non-Hermitian skin effect in one-dimensional interacting Bose gas

Liang Mao  Institute for Advanced Study, Tsinghua University, Beijing, 100084,
China  Yajiang Hao  Department of Physics, University of Science and
Technology Beijing, Beijing 100083, China  Lei Pan
<EMAIL_ADDRESS>Institute for Advanced Study, Tsinghua University,
Beijing, 100084, China

###### Abstract

Non-Hermitian skin effect (NHSE) is a unique feature studied extensively in
non-interacting non-Hermitian systems. In this work, we extend the NHSE
originally discovered in non-interacting systems to interacting many-body
systems by investigating an exactly solvable non-Hermitian model, i.e., the
prototypical Lieb-Liniger Bose gas with imaginary vector potential. We show
that this non-Hermitian many-body model can also be exactly solved through
generalized Bethe ansatz. By solving the Bethe ansatz equations accurately,
the explicit eigenfunction is obtained, and the model’s density profiles and
momentum distributions are calculated to characterize the NHSE quantitatively.
We find that the NHSE is gradually suppressed on the repulsive side but does
not vanish as the repulsive interaction strength increases. On the attractive
side, the NSHE for bound-state solutions is enhanced as interaction strength
grows. In contrast, for scattering state the NHSE shows a non-monotonic
behavior in the whole interaction range. Our work provides the first example
of the NHSE in exactly solvable many-body systems, and we envision that it can
be extended to other non-Hermitian many-body systems, especially in integrable
models.

##  I Introduction

Open quantum systems are ubiquitous in nature, which is an essential branch of
modern physics and has penetrated into numerous areas, including atomic and
molecule physics, nuclear physics, photonics, biophysics, and mesoscopic
physics, etc. Non-Hermitian Hamiltonians can describe an open quantum system
effectively. The first example was introduced by George Gamow, who derived a
non-Hermitian model to describe the alpha decay of heavy nuclei Gamow . With
the discovery of $\mathcal{PT}$-symmetry Bender  in non-Hermitian Hamiltonian
and associated experimental observation of $\mathcal{PT}$-symmetry breaking,
non-Hermitian physics has attracted intense attention.

In recent years, non-Hermitian physics has roused revived theoretical interest
due to tremendous advances in experimental technology for controlling
dissipation NH_Review1 ; NH_Review2 . Thanks to the great advantage in
manipulating atom-atom interaction and light-matter coupling, ultracold atom
experiments provide an unprecedented opportunity to investigate interacting
non-Hermitian systems Exp1 ; Exp2 ; Exp3 ; Exp4 ; Exp5 ; Exp6 ; Exp7 ; Exp8 ;
Exp9 ; Exp10 . Recent theoretical studies in non-Hermitian many-body systems
have revealed that the interplay between interaction and non-Hermiticity can
alter physical properties and give rise to intriguing phenomena absent in
Hermitian many-body systems NHMB1 ; NHMB2 ; NHMB3 ; NHMB4 ; NHMB405 ; NHMB5 ;
NHMB6 ; NHMB7 ; NHMB8 ; NHMB9 ; NHMB11 ; NHMB12 ; NHMB125 ; NHMB13 ; NHMB14 ;
NHMB15 ; NHMB16 ; NHMB17 ; NHMB18 ; NHMB19 ; NHMB20 ; NHMB21 ; NHMB22 ; NHMB23
; NHMB24 ; NHMB25 ; NHMB26 ; NHMB27 ; NHMB28 ; NHMB29 ; NHMB30 ; NHMB301 ;
NHMB302 ; NHMB303 ; NHMB31 ; NHMB32 ; NHMB33 ; NHMB34 ; NHMB35 ; NHMB355 ;
NHMB356 ; NHMB36 ; NHMB37 ; NHMB38 ; NHMB39 ; NHMB40 ; NHMB41 ; NHMB42 ;
NHMB43 ; NHMB44 ; NHMB45 ; NHMB46 ; NHMB47 ; NHMB48  such as non-Hermitian
superfluidity NHMB4 ; NHMB405 ; NHMB5 , non-Hermitian quantum magnetism NHMB24
; NHMB33 , and non-Hermitian many-body localization NHMB35 ; NHMB355 ; NHMB356
, etc.

A unique feature of non-Hermitian systems in open boundary conditions is the
so-called non-Hermitian skin effect (NHSE) Skin_Theory1 ; Skin_Theory2 ;
Skin_Theory3  which is recognized by eigenfunctions accumulated at a boundary,
akin to the charge distribution over the surface in a conductor. More
recently, non-Hermitian systems featuring the NHSE have been raising an
increasing concern Skin_Theory1 ; Skin_Theory2 ; Skin_Theory3 ; Skin_Theory4 ;
Skin_Theory5 ; Skin_Theory6 ; Skin_Theory7 ; Skin_Theory8 ; Skin_Theory9 ;
Skin_Theory10 ; Skin_Theory11 ; Skin_Theory12 ; Skin_Theory125 ; Skin_Theory13
; Skin_Theory14 ; Skin_Theory15 ; Skin_Theory16 ; Skin_Theory17 ;
Skin_Theory18 ; Skin_Theory185 ; Skin_Theory19 ; Skin_Theory20 ; Skin_Theory21
; Skin_Theory22 ; Skin_Theory23 ; Skin_Theory24  motivated by the experimental
realization of NHSE Skin_Exp1 ; Skin_Exp2 ; Skin_Exp3 ; Skin_Exp4 ; Skin_Exp5
; Skin_Exp6 ; Skin_Exp7 ; Skin_Exp8 ; Skin_Exp9 . Despite extensive
investigations, the current studies of the NHSE discussed in the literature
mainly focus on the single-particle level, such as non-Hermitian topological
bands or non-Hermitian quasicrystals. In contrast, the research on NHSE in
interacting systems is in its infancy Skin_Theory23 ; Skin_Theory24 ;
Skin_Theory25 ; Skin_Theory26 , which just involves few-body
calculationSkin_Theory22 , exact diagonalization study Skin_Theory23 ;
Skin_Theory24 ; Skin_Theory25 , perturbation theory Skin_Theory26 , and hard-
core limit Skin_Theory27 . The NHSE in exactly solvable many-body systems has
not been systematically investigated so far.

In this paper, we theoretically investigate NSHE in an exactly solvable model.
Exactly solvable models play a significant role in sta- tistical physics and
condensed matter physics, such as the verification of Bogoliubov theory in
Lieb-Liniger model Lieb-Liniger  and Wilson’s numerical renormalization group
in exact solutions of Kondo model Kondo1 . With the help of techniques
developed in the integrable model literature, we can obtain exact solutions of
eigen-energies and wavefunctions. The NHSE sensitively depends on boundary
conditions, which is similar as integrability conditions in many-body systems.
So we need to consider a non-Hermitian interacting system with open boundary
condition (OBC), which manifests the NHSE and meanwhile guarantees the
integrability condition. Based on this criterion, we employ the 1D interacting
Bose gas (Lieb-Liniger model) under OBC with an additional imaginary potential
corresponding to the nonreciprocal hopping in the 1D lattice, see (1). This
model has different physical properties as interaction strength varies, and
can form bound states on attractive interaction side. We will investigate NSHE
and its response to interaction in the whole interaction range.

The rest of this paper is organized as follows. In Sec. II, we introduce the
exactly solvable non-Hermitian many-body model and give the exact solutions
obtained by generalized Bethe ansatz, where the explicit eigenfunction and
eigenvalue are derived. In Sec. III, we discuss the NHSE in a repulsive
interaction regime where both density profiles and momentum distributions are
calculated to characterize the degree of NHSE. Then we explore the NHSE in
attractive interaction, including bound states and scattering states in Sec.
IV. Finally, we summarize this paper in Sec. V. For readers who are not
familiar with NSHE, Appendix A provides an introductory example for NSHE in a
tight-binding model.

##  II Model and solution

In this section, we introduce an exactly solvable non-Hermitian many-body
model and then carry out the Bethe ansatz solution. We focus on a 1D system
that consists of $N$ bosons with $\delta$-function interaction and subjects to
an imaginary potential

$\displaystyle\hat{H}=\sum_{j=1}^{N}\Big{[}-i\frac{\partial}{\partial
x_{j}}+i\phi(x_{j})\Big{]}^{2}+2c\sum_{j<l}\delta\left(x_{j}-x_{l}\right),$
(1)

where $c$ denotes the interaction strength and $i\phi(x)$ is the imaginary
potential. Here we set $\hbar=2m=1$. This model is known as Lieb-Liniger model
Lieb-Liniger  when $\phi(x)=0$ and has been realized in ultracold atomic gases
Lieb_Exp1 ; Lieb_Exp2 ; Lieb_Exp3 . The interaction strength $c$ can be tuned
by the confinement induced resonance CIR  or Feshbach resonance FR . In the
following we will show that the non-Hermitian model can be solved exactly both
in periodic boundary condition (PBC) and OBC for uniform potential
$\phi(x)=\phi$.

We start from the Schrödinger equation,
$\hat{H}\Psi\left(x_{1},\ldots,x_{N}\right)=E\Psi\left(x_{1},\ldots,x_{N}\right)$
and then we write the many-body wavefunction
$\Psi\left(x_{1},\ldots,x_{N}\right)$ in the following form

$\displaystyle\Psi\left(x_{1},\ldots,x_{N}\right)$ $\displaystyle=$
$\displaystyle\sum_{{\bf
P}}\psi\left(x_{p_{1}},x_{p_{2}},\ldots,x_{p_{N}}\right)$ (2)
$\displaystyle\times$ $\displaystyle\Theta\left(x_{p_{1}}\leq
x_{p_{2}}\cdots\leq x_{p_{N}}\right)$

where $p_{1},p_{2},\ldots,p_{N}$ presents the one of permutations of the set
$1,\ldots,N$, and $\sum_{\bf P}$ is the summation of all permutations.
$\Theta\left(x_{p_{1}}\leq\cdots\leq
x_{p_{N}}\right)=\theta\left(x_{p_{N}}-x_{p_{N-1}}\right)\cdots\theta\left(x_{p_{2}}-x_{p_{1}}\right)$
where $\theta(x-y)$ is the step function. Since the wavefunction is symmetric
under the interchange of coordinates, one just needs to calculate
$\psi\left(x_{1},x_{{2}},\ldots,x_{N}\right)$ in any region $x_{{1}}\leq
x_{{2}}\leq\cdots\leq x_{{N}}$, which satisfies

$\displaystyle\left(\sum_{j=1}^{N}\Big{[}-i\frac{\partial}{\partial
x_{j}}+i\phi\Big{]}^{2}+2c\sum_{i<j}\delta\left(x_{i}-x_{j}\right)\right)\psi\left(x_{1},\ldots,x_{N}\right)$
$\displaystyle=E\psi\left(x_{1},\ldots,x_{N}\right).~{}~{}~{}~{}~{}$ (3)

The $\delta$-function interaction gives rise to the contact condition

$\displaystyle\left(\frac{\partial}{\partial x_{j+1}}-\frac{\partial}{\partial
x_{j}}\right)\psi\left(\cdots
x_{j},x_{j+1}\cdots\right)\big{|}_{x_{j+1}=x_{j}}$
$\displaystyle=c\psi\left(\cdots
x_{j},x_{j+1}\cdots\right)\big{|}_{x_{j+1}=x_{j}}.$ (4)

We first consider the PBC, i.e.,

$\displaystyle\psi\left(x,x_{2},\ldots,x_{N}\right)=\psi\left(x_{1},x_{2},\ldots,x+L\right),$
(5)

with system size $L$. According to the Bethe ansatz solutions, the
wavefunction takes linear superposition of plane waves

$\displaystyle\psi\left(x_{1},x_{2},\ldots,x_{N}\right)=\sum_{{\bf P}}A_{{\bf
P}}\exp\Big{(}\sum_{j=1}^{N}ik_{p_{j}}x_{j}\Big{)},$ (6)

where $k_{j}$’s denote the quasimomentum of bosons. Combining the wavefunction
(6) into contact condition (4) and boundary condition (5), quasimomenta
satisfy the Bethe ansatz equations (BAEs)

$\displaystyle\exp\left(ik_{j}L\right)=\prod_{l=1(\neq
j)}^{N}\frac{ik_{j}-ik_{l}-c}{ik_{j}-ik_{l}+c},$ (7)

and the eigenenergy is given by $E=\sum_{j=1}^{N}(k_{j}+i\phi)^{2}$. The
solution of quasimomenta and the corresponding eigenfunction are independent
of $\phi$, but the spectrum is complex, which is similar to the single-
particle model in PBC as discussed in appendix A.

We then turn to the OBC

$\displaystyle\psi\left(0,x_{2},\ldots,x_{N}\right)=\psi\left(x_{1},x_{2},\ldots,L\right)=0.$
(8)

It had been shown that the Lieb-Liniger model ($\phi=0$) in OBC can also be
exactly solvedLL_OBC1 ; LL_OBC2 ; LL_OBC3 ; LL_OBC4 . We will show that the
non-Hermitian case can be solved exactly. To solve the non-Hermitian model in
OBC, we require the non-Bloch wavefunction obtained from the single-particle
model in OBC. And then we can construct the many-body wavefunction by means of
the generalized Bethe ansatz form

$\displaystyle\psi\left(x_{1},x_{2},\ldots,x_{N}\right)=\sum_{{\bf
P},r_{1},\ldots,r_{N}}A_{\bf
P}\exp\Big{(}\sum_{j=1}^{N}(ir_{j}k_{p_{j}}x_{j}+\phi x_{j})\Big{)},$ (9)

where $r_{j}=1$ ($r_{j}=-1$) indicates the plane wave of the $j$th boson
moving toward right or left. Based on the wavefunction (6) together Eq. (8)
and Eq. (4), one can derive the BAEs in OBC

$\displaystyle\exp\left(i2k_{j}L\right)=\prod_{l=1(\neq
j)}^{N}\frac{ik_{j}-ik_{l}-c}{ik_{j}-ik_{l}+c}\frac{ik_{j}+ik_{l}-c}{ik_{j}+ik_{l}+c},$
(10)

and the energy eigenvalue is given by $E=\sum_{j=1}^{N}k_{j}^{2}$. Physically,
the second fraction on the right-hand side of Eq. (10) comes from the
reflection at boundaries. Distinct from the situation in PBC, the spectrum in
OBC is independent of $\phi$ and takes real, but the wavefunction depends on
$\phi$. As a consequence, the many-body eigenfunction in OBC exhibits NHSE in
entire area of interaction $c\in(-\infty,\infty)$. The following sections will
elaborate on the properties of NHSE in distinct interaction regions, including
repulsive and attractive interactions.

##  III NHSE in repulsive interaction

In this section, we study the NHSE in repulsive interaction. When $c>0$, the
system is in a scattering state, and the solution of quasimomenta is real and
unique. To analyze the solutions of BAEs, we first take the logarithm of Eq.
(10) that leads to

$\displaystyle k_{j}L=\pi{I}_{j}+\sum_{l=1(\neq
j)}^{N}\left(\arctan\frac{c}{k_{j}-k_{l}}+\arctan\frac{c}{k_{j}+k_{l}}\right),$
(11)

where $\left\\{I_{j}\right\\}$ represents quantum number that takes a set of
integers. For the ground state, we have $I_{j}=1~{}(1\leq j\leq N)$ which can
be found in non-interacting limit where all bosons condense on $k_{j}=\pi/L$.
For a fixed $\left\\{n_{j}\right\\}$, one can solve Eq. (11) to determine
quasimomenta for arbitrary interaction strength.

After some calculations, we can derive the explicit expression of the
eigenfunction as

$\displaystyle\psi\left(x_{1},x_{2},\ldots,x_{N}\right)$
$\displaystyle=\sum_{\bf P}A_{\bf P}\epsilon_{\bf
p}\exp\left[i\left(\sum_{l<j}^{N-1}\Omega_{p_{j}p_{l}}\right)\right]$
$\displaystyle\times\sin\left(k_{p_{1}}x_{1}\right)\exp\left(ik_{p_{N}}L\right)\prod_{j=1}^{N}\exp\left(\phi
x_{j}\right)$ $\displaystyle\times\prod_{1<j<N}\sin$
$\displaystyle\Big{(}k_{p_{j}}x_{j}-\sum_{l<j}\Omega_{p_{l}p_{j}}\Big{)}\sin\Big{[}k_{p_{N}}\left(L-x_{N}\right)\Big{]},$
(12)

where $A_{\bf
P}=\prod_{j<l}^{N}\left(ik_{p_{j}}-ik_{p_{l}}-c\right)\left(ik_{p_{j}}+ik_{p_{l}}-c\right)$
and $\Omega_{jl}=\arctan\frac{c}{k_{j}+k_{l}}-\arctan\frac{c}{k_{j}-k_{l}}$.
Here the sign factor $\epsilon_{\bf p}$ takes $+1$ $(-1)$ relying on even
(odd) permutations of ($p_{1},p_{2},\ldots,p_{N}$). Apparently, the
wavefunction meets boundary conditions (8). Thus, the eigen-energy and
wavefunction can be determined immediately once the BAEs are solved.

To investigate the NHSE, we calculate the density distribution in real space

$\displaystyle\rho(x)=\frac{N\int_{0}^{L}dx_{2}\cdots
dx_{N}\left|\Psi\left(x,x_{2},\ldots,x_{N}\right)\right|^{2}}{\int_{0}^{L}dx_{1}\cdots
dx_{N}\left|\Psi\left(x_{1},x_{2},\ldots,x_{N}\right)\right|^{2}}.$ (13)

In Fig. 1, we plot density distributions for distinct potential $\phi$ at weak
($c=1$), mediate ($c=10$) and strong interaction ($c=100$).

Figure 1: NHSE in repulsive interacting Bosons. Groundstate density
distributions $\rho(x)$ in real space for different $\phi$ at $c=1$ (a),
$c=10$ (b) and $c=100$ (c), respectively. (d) The deviation of mean position
$\delta\langle x\rangle$ as a function of $\phi$ for different interaction
strengths from non-interacting (dashed line) to TG limit (dot-dashed line).
Here we choose $L=1$ as length unit and set $N=4$.

One can see that the density distribution gradually tends to the right
boundary with the increasing of $\phi$. Meanwhile, the repulsive interaction
widens the density profile. In order to characterize the degree of NHSE
qualitatively, we define the deviation of mean position similar to the charge
distribution

$\displaystyle\delta\langle
x\rangle=\frac{1}{N}\int_{0}^{L}x\Big{[}\rho(x)-\rho_{0}(x)\Big{]}dx,$ (14)

where $\rho_{0}(x)$ denotes the density distribution at $\phi=0$. The larger
$\delta\langle x\rangle$ is, the stronger degree of NHSE will be. One can see
from Fig. 1(d) that the NHSE is suppressed as the interaction strength grows.
It is physically reasonable that the repulsive interaction prevents bosons
from clumping together, thereby effectively suppressing NHSE. However, the
NHSE always exists even in the Tonks-Giradeau (TG) limit $c=\infty$. In TG
limit, the ground state solution of the BAEs (11) is $k_{j}=j\pi$
($j=1,2,\cdots,N$) that the bosons look like free spinless fermions, and the
corresponding wavefunction takes the form of

$\displaystyle\Psi\left(x_{1},x_{2},\ldots,x_{N}\right)=$
$\displaystyle\sum_{\bf
P}\theta\left(x_{p_{N}}-x_{p_{N-1}}\right)\cdots\theta\left(x_{p_{2}}-x_{p_{1}}\right)$
$\displaystyle\times$ $\displaystyle\prod_{j=1}^{N}\sin\left(j\pi
x_{p_{j}}/L\right)\exp\left(\phi x_{p_{j}}\right),$ (15)

which leads to finite $\delta\langle x\rangle$ as shown in Fig. 1(d).

We can also characterize the magnitude of NHSE via momentum distribution
$n(k)=\frac{1}{2\pi}\int_{0}^{L}dx\int_{0}^{L}dx^{\prime}\varrho\left(x,x^{\prime}\right)e^{-ik\left(x-x^{\prime}\right)}$
where the single-particle density matrix $\varrho\left(x,x^{\prime}\right)$ is
defined as

$\varrho\left(x,x^{\prime}\right)=\frac{N\int_{0}^{L}dx_{2}\cdots
dx_{N}\Psi^{*}\left(x,x_{2},\ldots,x_{N}\right)\Psi\left(x^{\prime},x_{2},\ldots,x_{N}\right)}{\int_{0}^{L}dx_{1}dx_{2}\cdots
dx_{N}\left|\Psi\left(x_{1},x_{2},\ldots,x_{N}\right)\right|^{2}}.$ (16)

The momentum distribution width is enhanced as $\phi$ increases, as shown in
Fig. 2 (a), Fig. 2 (b) and Fig. 2 (c). The broadening momentum distribution
corresponds to the NHSE in real space, where particles concentrate on the
boundary with a narrower density profile according to the Fourier
transformation. Specifically, Fig. 2(d) plots the deviation of momentum
distribution width
$\delta\overline{k}^{2}=\overline{k^{2}}-\overline{k^{2}}_{\phi=0}$ with
$\overline{k^{2}}=\int_{-\infty}^{\infty}k^{2}n(k)dk$ as a function of $\phi$
at different interaction strengths where the suppression of the momentum
distribution width growth by repulsive interaction can be visualized.

Figure 2: Groundstate momentum distributions $n(k)$ for different $\phi$ at
$c=1$ (a), $c=10$ (b) and $c=50$ (c), respectively. (d) The deviation of
momentum distribution $\delta\langle x\rangle$ as a function of $\phi$ for
different interaction strengths from non-interacting (dashed line) to TG limit
(dot-dashed line). Here we choose $L=1$ as length unit and set $N=4$.

##  IV NHSE in attractive interaction

Sec. III deals with the situation of repulsive interaction. We now turn to
study the NHSE in attractive interaction. What should be pointed out is that
unlike the case of repulsive interaction, the solution of BAEs in attractive
interaction is not unique, which contains bound state and scattering states.
Physically, there exists a bound state when bosons attract each other,
reflected in the complex solution of quasimomenta $\\{k_{j}\\}$ in BAEs. In
weak interaction region, the ground state solution of quasimomenta consists of
$N/2$ ($N$ is even number) pairs of conjugate complex roots, i.e., $N/2$
dimers $k_{2j-1}=\alpha_{j}-i\beta_{j}$, $k_{2j}=\alpha_{j}+i\beta_{j}$
($j=1,\cdots,N/2$). The state made up of $N/2$ dimers is also called $N/2$
two-string. The $M$-string state is defined by $M$ quasimomenta sharing the
same real part but unequal conjugate imaginary parts, which can be labeled as
$k_{j}=\alpha+\beta_{j}$ ($j=1,\cdots,M$). According to the bound state
solution of BAEs, as the attractive interaction strength increases, the ground
state evolves from a $N/2$ two-string state gradually to an intermediate state
characterized by an $M$-string ($N>M>2$) plus $(N-M)/2$ two-string and finally
turns to $N$-string state.

Figure 3: NHSE in attractive interacting Bosons for bound state. The deviation
of mean position $\delta\langle x\rangle$ as a function of $\phi$ for
different interaction strengths from non-interacting (dashed line) to infinite
attractive limit (dot-dashed line). Here we choose $L=1$ as length unit and
set $N=4$.

The bound state solution of BAEs shows that the NHSE exists in the entire
range of interaction strength $c\leq 0$ as shown in Fig. 3. Moreover, the
deviation $\delta\langle x\rangle$ tends to increase with the growth of
attractive strength. This is because the bosons tend to clump together with
increasing attraction among them. We can also understand the NHSE in
attractive interaction in terms of quantitative analysis in two limits. For
non-interacting limit, all bosons occupy the single-particle ground state
$k_{j}=\pi/L$ and the corresponding groundstate wave function is simply
written as
$\Psi\left(x_{1},x_{2},\ldots,x_{N}\right)=\prod_{j=1}^{N}\sin\left(j\pi
x_{j}/L\right)\exp\left(\phi x_{j}\right)$ which produces the deviation of
mean position $\delta\langle x\rangle=\frac{L}{2}\left[\coth(\phi
L)-\frac{1}{\phi L}-\frac{2\phi L}{\pi^{2}+\phi^{2}L^{2}}\right]$. In strongly
attractive limit $c=-\infty$, the system forms a $N$-body bound state
corresponding to the $N$-string state. At this point, the quasimomentum
distribution is given by $k_{j}=K/N+i(N+1-2j)c/2,(j=1,2,\cdots N)$ where $K$
signifies the total momentum, which is determined by the following
transcendental equation

$KL=\pi I+2\sum_{j=1}^{N-1}\arctan\frac{jcN}{2K}$ (17)

with $I=N$ for the ground state. We can see that in the limit of
$c\rightarrow-\infty$, the total momentum is $K=\pi/L$ which indicates each
boson has the real part $\pi/NL$ in the strong attractive limit and the
corresponding ground state wavefunction to be written as

$\displaystyle\Psi\left(x_{1},x_{2},\ldots,x_{N}\right)=\sin\left(\pi
x/L\right)\exp\left(N\phi x\right),$ (18)

where $x=\frac{x_{1}+x_{2}+\cdots+x_{N}}{N}$ denotes the center-of-mass
coordinate. The wave function (18) gives a clear physical picture that the
system forms a giant molecule of $N$ bosons involving the movement of center-
of-mass. And similarly, the deviation of mean position derived from (18)
yields

$\displaystyle\delta\langle x\rangle=\frac{L}{2}\left[\coth\Phi
L-\frac{1}{\Phi L}-\frac{2\Phi L}{\pi^{2}+\Phi^{2}L^{2}}\right],$ (19)

with $\Phi=N\phi$. We can see that the $\delta\langle x\rangle$ of infinitely
attractive bosons is similar to the one of free bosons in form but contains a
factor $N$. Physically, it comes from the Bose enhancement that all bosons
tend to locate at the same position.

Next, we turn to address the case of scattering states on the attractive side.
In fact, there are real solutions corresponding to scattering states in BAEs
(10) for $c<0$ although its ground state is a string solution (bound state).
In the strong interaction regime, this kind of scattering state is referred to
super-Tonks-Girardeau (STG) gas in the literature STG1 ; STG2 ; STG3 ; STG4 ;
STG5 . For the purpose of analysis, we alternatively rewrite the BAEs (10) as
the following logarithm form

$\displaystyle k_{j}L=\pi I_{j}-\sum_{l=1(\neq
j)}^{N}\left(\arctan\frac{k_{j}-k_{l}}{|c|}+\arctan\frac{k_{j}+k_{l}}{|c|}\right).$
(20)

Here we choose $I_{j}=j$, such that the quasimomentum distribution in
$c\rightarrow-\infty$ determined from Eq. (20) connects to the TG limit from
Eq. (11) where $k_{j}=j\pi/L$ ($j=1,2,\dots,N$). Under this circumstance, one
would obtain the deviation of mean position $\delta\langle x\rangle$ as a
continuous function of $1/c$ as shown in Fig. 4(a). We can recognize that in
contrast to the bound state on the attractive side, the scattering state
exhibits a non-monotonic behavior where $\delta\langle x\rangle$ decreases
first and then increases as the attractive strength weakens. And also the
transition point (the inset of Fig. 4 (a)) locates at the STG regime ($-1\ll
1/c<0$).

Figure 4: Panel (a): The deviation of mean position $\delta\langle x\rangle$
as a function of inverse interaction strength at $\phi=1$. strengths from non-
interacting (dashed line) to infinite attractive limit (dot-dashed line).
Panel (b): Density distributions of the scattering state for different
attractive interaction strengths. Here we choose $L=1$ as length unit and set
$N=4$.

The non-monotonic behavior is closely related to the density distribution on
the attractive side. In the limit of $c\rightarrow-\infty$, the density
profile is identical to the one in the TG limit displaying $N$ peaks with near
equidistant. Then, the density distribution gradually migrates toward
boundaries as the attraction decreases, suppressing the NHSE. However, as the
attraction is getting weaker, the density tends to concentrate in the center
of potential with a large weight which can magnify the NHSE. Finally, in weak
attraction limit, there emerge $2N-1$ peaks in the density distribution as
shown in Fig. 4(b). This is because the bosons occupy $N$ lowest odd single-
particle orbitals $k_{j}=(2j+1)\pi/L$ when $c\rightarrow 0^{-}$.

##  V Summary and Outlook

In summary, through the example of non-Hermitian 1D interacting bose gas, we
investigated the NHSE in a non-Hermitian many-body system. Utilizing
generalized Bethe ansatz, we obtained the exact solutions, including
quasimomenta, eigen-energies, and wavefunctions of the system in OBC. The NHSE
can be characterized by density distribution and momentum distribution. Our
calculations show that the NHSE exists in the entire interaction regime and
displays distinct responses to the interaction effect. The main conclusions
are summarized as follows:

(I) For repulsive interaction $c>0$, the NHSE is gradually suppressed as
interaction strength increases but does not vanish even in the TG limit
($c\rightarrow+\infty$).

(II) For the bound state in attractive interaction $c<0$, the system consists
of $N$ bosons from the 2-string state into the $N$-string state as the
attractive interaction strength grows. And in this process, the NHSE is
enhanced.

(III) For scattering state in attractive interaction $c<0$, we find a non-
monotonic behavior in the deviation of mean position $\delta\langle x\rangle$
where the NHSE is first suppressed and then enhanced during interaction
weakens.

This paper is the first comprehensive investigation of the NHSE in exactly
solvable many-body models, and our results add to the rapidly expanding field
of non-Hermitian physics. Considerably more work will be desired to explore
the interplay between NHSE and other kinds of exactly solvable many-body
systems for further investigation.

## Acknowledgements

We would like to thank Pengfei Zhang and Tianshu Deng for helpful discussions.
We also thank Hui Zhai for his valuable comment. This work is supported by
Beijing Outstanding Young Scientist Program held by Hui Zhai. YH was supported
by NSF of China under Grants No. 11774026. LP acknowledges support from the
project funded by the China Postdoctoral Science Foundation (Grant No.
2020M680496).

##  Appendix A Single-particle non-Hermitian skin effect in tight-binding
model

In this appendix we give a brief introduction to the non-Hermitian skin effect
at a single-particle level. We consider a one-dimensional tight-binding model
with unequal hopping, whose Hamiltonian is written as

$\displaystyle\hat{H}=t_{1}\sum_{n}|n+1\rangle\langle
n|+t_{2}\sum_{n}|n\rangle\langle n+1|,$ (21)

where $t_{1}>t_{2}$ denote the unequal hopping amplitudes. This non-Hermitian
model with disorder potential was first introduced by Hatano and Nelson
HN_Model  to investigate localization transition and is investigated in
ultracold atoms XuYong ; RenZhang . Now we solve its eigenvalues and
eigenvectors. Under periodic boundary condition (PBC), the momentum is a good
quantum number ($k=$ $0,2\pi/L,\ldots,2\pi(L-1)/L$). Then the eigenenergies
and eigenstates are given by

$\displaystyle
E_{k}=t_{1}e^{-ik}+t_{2}e^{ik},\quad\left|E_{k}\right\rangle=\frac{1}{\sqrt{L}}\sum_{n}e^{ikn}|n\rangle,$
(22)

from which we can see the eigenvector is nothing but the Bloch state which is
identical to Hermitian case but the spectrum becomes complex.

Under open boundary condition (OBC), the momentum is not good quantum number
anymore. Expanding the eigenvector in real space
$|E\rangle=\sum_{n}\psi_{n}|n\rangle$, we have

$\displaystyle E\psi_{n}=t_{1}\psi_{n-1}+t_{2}\psi_{n+1},$ (23)

with OBC $\quad\psi_{0}=\psi_{L+1}=0$. We introduce a non-Bloch wavefunction
$\psi_{n}=A_{1}\beta_{1}^{n}+A_{2}\beta_{2}^{n}$, and then the eigenvalues are
determined by

$\displaystyle E(\beta)=t_{1}/\beta_{a}+t_{2}\beta_{a},$ (24)

where $a=1,2$. The boundary condition $\quad\psi_{0}=0,~{}\psi_{L+1}=0$ give

$\displaystyle A_{1}+A_{2}=0,\quad
A_{1}\beta_{1}^{L+1}+A_{2}\beta_{2}^{L+1}=0,$ (25)

which will lead to the condition $\beta_{1}^{L+1}=\beta_{2}^{L+1}$. Together
with $\beta_{1}\beta_{2}=t_{1}/t_{2}$, we can obtain

$\displaystyle\beta_{1}=\sqrt{t_{1}/t_{2}}e^{i\theta_{m}}=\beta_{2}^{*},\quad\theta_{m}=\pi
m/(L+1),$ (26)

with $m=1,2,\ldots,L$. Therefore, we can derive the eigenenergies in OBC

$\displaystyle E_{m}=2\sqrt{t_{1}t_{2}}\cos\theta_{m},$ (27)

and associated eigenstates

$\displaystyle\left|E_{m}\right\rangle=\sum_{n}e^{n\phi}\sin
n\theta_{m}|n\rangle,$ (28)

where $\phi=\ln\sqrt{t_{1}/t_{2}}$. Note that the spectrum in OBC is always
real. Meanwhile, the non-Bloch wavefunction consists of the standing wave
$\sin(n\theta_{m})$ with an amplification factor $e^{n\phi}$ exhibiting
exponential accumulation at the boundary. This feature is called the non-
Hermitian skin effect. The non-Bloch wavefunction in the eigenstate (28) will
be the starting point for solving non-Hermitian many-body systems.

## References

  * (1) G. Gamow, Z. Phys. 51, 204 (1928).
  * (2) C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 524 (1998).
  * (3) E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Rev. Mod. Phys. 93, 015005 (2021).
  * (4) Y. Ashida, Z. Gong, and M. Ueda, Adv. Phys. 69, 249 (2020).
  * (5) G. Barontini, R. Labouvie, F. Stubenrauch, A. Vogler, V. Guarrera, and H. Ott, Phys. Rev. Lett. 110, 035302 (2013).
  * (6) Y. S. Patil, S. Chakram, and M. Vengalattore, Phys. Rev. Lett. 115, 140402 (2015).
  * (7) R. Labouvie, B. Santra, S. Heun, and H. Ott, Phys. Rev. Lett. 116, 235302 (2016).
  * (8) H. P. Lüschen, P. Bordia, S. S. Hodgman, M. Schreiber, S. Sarkar, A. J. Daley, M. H. Fischer, E. Altman, I. Bloch, and U. Schneider, Phys. Rev. X 7, 011034 (2017).
  * (9) T. Tomita, S. Nakajima, Y. Takasu, and Y. Takahashi, Phys. Rev. A 99, 031601(R) (2019).
  * (10) R. Bouganne, M. B. Aguilera, A. Ghermaoui, J. Beugnon, and F. Gerbier, Nat. Phys. 16, 21 (2020).
  * (11) T. Tomita, S. Nakajima, I. Danshita, Y. Takasu, and Y. Takahashi, Sci. Adv. 3, e1701513 (2017).
  * (12) K. Sponselee, L. Freystatzky, B. Abeln, M. Diem, B. Hundt, A. Kochanke, T. Ponath, B. Santra, L. Mathey, K. Sengstock, and C. Becker, Quantum Sci. Technol. 4, 014002 (2018).
  * (13) Y. Takasu, T. Yagami, Y. Ashida, R. Hamazaki, Y. Kuno, and Y. Takahashi, Prog. Theor. Exp. Phys. 2020, 12A110 (2020).
  * (14) B. Yan, S. A. Moses, B. Gadway, J. P. Covey, K. R. Hazzard, A. M. Rey, D. S. Jin, and J. Ye, Nature (London) 501, 521 (2013).
  * (15) F. Schäfer, T. Fukuhara, S. Sugawa, Y. Takasu, and Y. Takahashi,  Nat. Rev. Phys. 2, 411 (2020).
  * (16) K. Kawabata, Y. Ashida, and M. Ueda, Phys. Rev. Lett. 119, 190401 (2017).
  * (17) Y. Ashida, S. Furukawa, and M. Ueda, Nat. Commun. 8, 15791 (2017).
  * (18) L. Pan, S. Chen, and X. Cui, Phys. Rev. A 99, 011601(R) (2019).
  * (19) L. Pan, S. Chen, and X. Cui, Phys. Rev. A 99, 063616 (2019).
  * (20) Z. Zhou and Z. Yu, Phys. Rev. A 99, 043412 (2019).
  * (21) K. Yamamoto, M. Nakagawa, K. Adachi, K. Takasan, M. Ueda, and N. Kawakami, Phys. Rev. Lett. 123, 123601 (2019).
  * (22) N. Okuma and M. Sato, Phys. Rev. Lett. 123, 097701 (2019).
  * (23) L. Zhou and X. Cui, iScience 14, 257 (2019).
  * (24) L. Pan, X. Chen, Y. Chen, H. Zhai, Nat. Phys. 16, 767 (2020).
  * (25) T. Yoshida, K. Kudo, and Y. Hatsugai, Sci. Rep. 9, 16895 (2019).
  * (26) T. Liu, J. J. He, T. Yoshida, Z.-L. Xiang, and F. Nori, Phys. Rev. B 102, 235151 (2020).
  * (27) Z. H. Xu and S. Chen, Phys. Rev. B 102, 035153 (2020).
  * (28) K. Kawabata, K. Shiozaki, and S. Ryu, Phys. Rev. B 105, 165137 (2022).
  * (29) J. Carlström, Phys. Rev. Research 2, 013078 (2020).
  * (30) Z. Cai and T. Barthel, Phys. Rev. Lett. 111, 150403 (2013).
  * (31) T.-S. Deng, L. Pan, Y. Chen, H. Zhai, Phys. Rev. Lett. 127, 086801 (2021).
  * (32) Z. Wang, Q. Li, W. Li, Z. Cai, Phys. Rev. Lett. 126, 237201 (2021).
  * (33) T.-S. Deng and L. Pan, Phys. Rev. B 104, 094306 (2021).
  * (34) B. Buča, C. Booker, M. Medenjak, and D. Jaksch, New J. Phys. 22, 123040 (2020).
  * (35) K. Yamamoto, Y. Ashida, and N. Kawakami, Phys. Rev. Research 2, 043343 (2020).
  * (36) K. Yamamoto, M. Nakagawa, N. Tsuji, M. Ueda, and N. Kawakami, Phys. Rev. Lett. 127, 055301 (2021).
  * (37) R. Hanai, A. Edelman, Y. Ohashi, and P. B. Littlewood, Phys. Rev. Lett. 122, 185301 (2019).
  * (38) K. L. Zhang and Z. Song, Phys. Rev. Lett. 126, 116401 (2021).
  * (39) X. Z. Zhang and Z. Song, Phys. Rev. B 104, 094301 (2021).
  * (40) X. Z. Zhang, L. Jin, and Z. Song, Phys. Rev. B 101, 224301 (2020).
  * (41) K. L. Zhang and Z. Song, Phys. Rev. B 104, 245140 (2020).
  * (42) D. Sticlet, B. Dóra, and C. P. Moca, Phys. Rev. Lett. 128, 016802 (2022).
  * (43) B. Dóra, and C. P. Moca, Quantum Quench in PT-Symmetric Luttinger Liquid, Phys. Rev. Lett. 124, 136802 (2020).
  * (44) Á. Bácsi, C. P. Moca, and B. Dóra, Phys. Rev. Lett. 124, 136401 (2020).
  * (45) J.-S. Bernier, R. Tan, C. Guo, C. Kollath, and D. Poletti, Phys. Rev. B 102, 115156 (2020).
  * (46) L. Pan, X. Wang, X. Cui, and S. Chen, Phys. Rev. A 102, 023306 (2020).
  * (47) X. Z. Zhang and Z. Song, Phys. Rev. B 102, 174303 (2020).
  * (48) X. Z. Zhang and Z. Song, Phys. Rev. B 103, 235153 (2021).
  * (49) T. Hayata and A. Yamamoto, Phys. Rev. B 104, 125102 (2021).
  * (50) X. Cui, Phys. Rev. Research 4, 013047 (2022).
  * (51) M. Nakagawa, N. Kawakami, and M. Ueda, Phys. Rev. Lett. 126, 110404 (2021).
  * (52) M. Nakagawa, N. Tsuji, N. Kawakami, and M. Ueda, Phys. Rev. Lett. 124, 147203 (2020).
  * (53) K. Yamamoto, M. Nakagawa, N. Tsuji, M. Ueda, and N. Kawakami, Phys. Rev. Lett. 127, 055301 (2021).
  * (54) R. Hamazaki, K. Kawabata, and M. Ueda, Phys. Rev. Lett. 123, 090603 (2019).
  * (55) L.-J. Zhai, S. Yin, and G.-Y. Huang, Phys. Rev. B 102, 064206 (2020).
  * (56) K. Suthar, Y.-C. Wang, Y.-P. Huang, H.-H. Jen, and J.-S. You, arXiv:2202.12675 (2022).
  * (57) N. Matsumoto, K. Kawabata, Y. Ashida, S. Furukawa, and M. Ueda, Phys. Rev. Lett. 125, 260601 (2020).
  * (58) S. Sarkar, Sci. Rep. 11, 5510 (2021).
  * (59) M. Nakagawa, N. Kawakami, and M. Ueda, Phys. Rev. Lett. 121, 203001 (2018).
  * (60) J. A. S. Lourenço, R. L. Eneias, and R. G. Pereira, Phys. Rev. B 98, 085126 (2018).
  * (61) R. Hanai and P. B. Littlewood, Critical fluctuations at a many-body exceptional point, Phys. Rev. Research 2, 033018 (2020).
  * (62) K. Yamamoto, M. Nakagawa, M. Tezuka, M. Ueda, and N. Kawakami, arXiv:2112.12467 (2022).
  * (63) Y.-N. Zhou, L. Mao, and H. Zhai, Phys. Rev. Research 3, 043060 (2021).
  * (64) P. Zhang, S.-K. Jian, C. Liu, and X. Chen, Quantum 5, 579 (2021).
  * (65) L. Sá, P. Ribeiro, and T. Prosen, Phys. Rev. X 10, 021019 (2020).
  * (66) J. Li, T. Prosen, and A. Chan, Phys. Rev. Lett. 127, 170602 (2021).
  * (67) G. Sun, J.-C. Tang, and S.-P. Kou, Front. Phys. 17, 33502 (2022).
  * (68) S. Mu, C. H. Lee, L. Li, and J. Gong, Phys. Rev. B 102, 081115(R) (2020).
  * (69) F. Qin, R. Shen, C. H. Lee, arXiv:2202.10481 (2022).
  * (70) S. Yao, and Z. Wang, Phys. Rev. Lett. 121, 086803 (2018).
  * (71) S. Yao, F. Song, and Z. Wang, Phys. Rev. Lett. 121, 136802 (2018).
  * (72) F. Song, S. Yao, and Z. Wang, Phys. Rev. Lett. 123, 170401 (2019).
  * (73) F. Song, S. Yao, and Z. Wang, Phys. Rev. Lett. 123, 246801 (2019).
  * (74) H. Jiang, L.-J. Lang, C. Yang, S.-L. Zhu, and S. Chen, Phys. Rev. B 100, 054301 (2019).
  * (75) C.-X. Guo, C.-H. Liu, X.-M. Zhao, Y. Liu, and S. Chen, Phys. Rev. Lett. 127, 116801 (2021).
  * (76) L. Jin and Z. Song, Phys. Rev. B 99, 081103(R) (2019).
  * (77) D. S. Borgnia, A. J. Kruchkov, and R.-J. Slager, Phys. Rev. Lett. 124, 056802 (2020).
  * (78) S. Longhi, Phys. Rev. Research 1, 023013 (2019).
  * (79) K. Zhang, Z. Yang, and C. Fang, Phys. Rev. Lett. 125, 126402 (2020).
  * (80) C. H. Lee, L. Li, and J. Gong, Phys. Rev. Lett. 123, 016805 (2019).
  * (81) L. Li, C. H. Lee, and J. Gong, Phys. Rev. Lett. 124, 250402 (2020).
  * (82) L. Li, C. H. Lee, S. Mu, and J. Gong, Nat. Commun. 11, 5491 (2020).
  * (83) C.-H. Liu, K. Zhang, Z. Yang, and S. Chen, Phys. Rev. Research 2, 043167 (2020).
  * (84) Z. Yang, K. Zhang, C. Fang, and J. Hu, Phys. Rev. Lett. 125, 226402 (2020).
  * (85) T. Yoshida, T. Mizoguchi, and Y. Hatsugai, Phys. Rev. Research 2, 022062(R) (2020).
  * (86) Y. Yi and Z. Yang, Phys. Rev. Lett. 125, 186802 (2020).
  * (87) N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato Phys. Rev. Lett. 124, 086801 (2020).
  * (88) S. Longhi, Phys. Rev. Lett. 124, 066602 (2020).
  * (89) Z. Zhou, and Z. Yu, arXiv:2110.09874 (2021).
  * (90) S. Guo, C. Dong, F. Zhang, J. Hu, and Z. Yang, arXiv:2111.04220 (2021).
  * (91) L. Zhou, H. Li, W. Yi, X. Cui, arXiv:2111.04196 (2021).
  * (92) K. Zhang, Z. Yang, and C. Fang, Nat. Commun. 13, 2496 (2022).
  * (93) C. H. Lee, Phys.Rev. B 104, 195102 (2021).
  * (94) R. Shen and C. H. Lee, arXiv:2107.03414 (2021).
  * (95) D.-W. Zhang, Y.-L. Chen, G.-Q. Zhang, L.-J. Lang, Z. Li, and S.-L. Zhu, Phys. Rev. B 101, 235150 (2020).
  * (96) Z. Wang, L.-J. Lang, and L. He, Phys. Rev. B 105, 054315 (2022).
  * (97) N. Okuma and M. Sato, Phys. Rev. Lett. 126, 176601 (2021).
  * (98) Y.-N. Wang, W.-L. You, G. Sun, arXiv:2206.11827 (2022).
  * (99) T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. W. Molenkamp, C. H. Lee, A. Szameit, M. Greiter, and R. Thomale, Nat. Phys. 16, 747 (2020).
  * (100) T. Hofmann, T. Helbig, F. Schindler, N. Salgo, M. Brzezinska, M. Greiter, T. Kiessling, D. Wolf, A. Vollhardt, A. Kabasi, C. H. Lee, A. Bilusic, R. Thomale, and T. Neupert, Phys. Rev. Research 2, 023265 (2020).
  * (101) L. Xiao, T. Deng, K. Wang, G. Zhu, Z. Wang, W. Yi, and P. Xue, Nat. Phys. 16, 761 (2020).
  * (102) A. Ghatak, M. Brandenbourger, J. van Wezel, and C. Coulais, Proc. Natl. Acad. Sci. U.S.A. 117, 29561 (2020).
  * (103) S. Weidemann, M. Kremer, T. Helbig, T. Hofmann, A. Stegmaier, M. Greiter, R. Thomale, and A. Szameit, Science 368, 311 (2020).
  * (104) D. Zou, T. Chen, W. He, J. Bao, C. H. Lee, H. Sun, and X. Zhang, Nat. Commun. 12, 7201 (2021).
  * (105) L. Zhang, Y. Yang, Y. Ge, Y.-J. Guan, Q. Chen, Q. Yan, F. Chen, R. Xi, Y. Li, D. Jia, S.-Q. Yuan, H.-X. Sun, H. Chen, and B. Zhang, Nat. Commun. 12, 6297 (2021).
  * (106) X. Zhang, Y. Tian, J.-H. Jiang, M.-H. Lu, and Y.-F. Chen, Nat. Commun. 12, 5377 (2021).
  * (107) Q. Liang, D. Xie, Z. Dong, H. Li, H. Li, B. Gadway, W. Yi, and B. Yan, arXiv:2201.09478 (2022).
  * (108) E. H. Lieb and W. Liniger, Phys. Rev. 130, 1605 (1963).
  * (109) N. Andrei, K. Furuya, and J. H. Lowenstein, Rev. Mod. Phys. 55, 331, (1983).
  * (110) T. Kinoshita, T. Wenger and D. S. Weiss, Science 305, 1125 (2004).
  * (111) B. Paredes, A. Widera, V. Murg, O. Mandel, S. Fölling, I. Cirac, G. V. Shlyapnikov, T. W. Hänsch, and I. Bloch, Nature 429, 277 (2004).
  * (112) T. Kinoshita, T. Wenger and D. S. Weiss, Phys. Rev. Lett. 95 190406 (2005).
  * (113) M. Olshanii, Phys. Rev. Lett. 81, 938 (1998).
  * (114) C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev. Mod. Phys. 82, 1225 (2010).
  * (115) M. T. Batchelor, X. W. Guan, N. Oelkers, and C. Lee, J. Phys. A 38, 7787 (2005).
  * (116) N. Oelkers, M. T. Batchelor, M. Bortz, and X. W. Guan, J. Phys. A 39, 1073 (2006).
  * (117) A. del Campo and J. G. Muga, Europhys. Lett. 74, 965 (2006).
  * (118) Y. Hao, Y. Zhang, J. Q. Liang, and S. Chen, Phys. Rev. A 73, 063617 (2006).
  * (119) G. E. Astrakharchik, J. Boronat, J. Casulleras and S. Giorgini, Phys. Rev. Lett. 95, 190407 (2005).
  * (120) M. T. Batchelor, M. Bortz, X. W. Guan, and N. Oelkers, J. Stat. Mech. (2005) L10001.
  * (121) S. Chen, L. Guan, X. Yin, Y. Hao, and X.-W. Guan, Phys. Rev. A 81, 031609(R) (2010).
  * (122) M. Panfil, J. D. Nardis, and J.-S. Caux, Phys. Rev. Lett. 110 125302 (2013).
  * (123) M. Kormos, G. Mussardo, and A. Trombettoni, Phys. Rev. A 83 013617 (2011).
  * (124) N. Hatano and D. R. Nelson, Phys. Rev. Lett. 77, 570 (1996).
  * (125) K. Li, and Y. Xu, arXiv:2201.09469 (2022).
  * (126) C. Lv, R. Zhang, Z. Zhai, and Q. Zhou, Nat Commun. 13, 2184 (2022).


# Rate of Adaptive Evolution under Blending Inheritance

Alan R. Rogers Dept. of Anthropology, 270 S 1400 E Rm 102, University of Utah,
Salt Lake City, UT 84112\<EMAIL_ADDRESS>
In a population of size $N$, adaptive evolution is $2N$ times faster under
Mendelian inheritance than under the 19th-century theory of blending
inheritance.

> Unfortunately we have no means of determining, according to the standard of
> years, how long a period it takes to modify a species.
>  Charles Darwin (1872, p. 270)

In the latter half of the 19th century, evolutionary time seemed limited.
Victorian science suggested that the earth was not much older than 100 million
years (Thomson, 1862; Jackson, 2001). Was it really plausible that so much
evolution could have happened in this interval? Evolutionists had no answer.
In part, this was because the question was quantitative—involving rates and
intervals—yet evolutionary theory was at that time merely qualitative. It was
impossible to say how much time was needed, because (as Darwin laments in the
quote above), no one had a clear understanding of the rate of evolution.

Well, almost no one. In 1871, A.S. Davis published the first coherent,
quantitative model of evolution (Davis, 1871). But no one noticed for over a
century, until the article was finally discovered by Morris (1994) and Bulmer
(2004). In the paragraphs that follow, I ask what Davis might have concluded
about the rate of adaptive evolution, had he pursued the matter.

Victorians assumed that heredity involved blending the hereditary material of
the parents, and there were at least two theories of blending (Bulmer, 2004).
For some, blending was like mixing paint. For others, it was consistent with
an underlying particulate heredity:

> Skin-colour is therfore a good example of what I call blended inheritance.
> It need be none the less “particulate” in origin, but the result may be
> regarded as a fine mosaic too minute for its elements to be distinguished in
> a general view.
>  Francis Galton (1889, p. 12)
>

Following Davis (1871), I consider the paint-pot model of blending inheritance
rather than Galton’s particulate version.

To measure the rate of adaptive evolution, I focus on the rate of increase in
the mean Darwinian fitness of a population. This requires comment, because it
is not clear that mean fitness does increase. Each adaptive improvement may
generate an increase in population size, which increases competition for
resources and leaves the population no better off than before. We therefore
study the marginal effect on mean fitness, ignoring compensatory effects
(Price, 1972; Frank and Slatkin, 1992; Plutynski, 2006; Grafen, 2003).

Suppose that individuals vary in Darwinian fitness, interpreted as the
probability of surviving from birth to reproductive maturity. As a cohort
matures, its mean fitness rises as selection culls individuals with low
fitness. To measure this effect, let $W$ denote an individual’s absolute
fitness. The mean fitness of adults exceeds that of newborns by111If $p_{W}$
is the fraction of newborns with fitness $W$, then the corresponding fraction
among adults is $Wp_{W}/\bar{W}$, where $\bar{W}=\sum Wp_{W}$ is the mean
fitness among newborns. The mean fitness of adults is $\sum
W^{2}p_{W}/\bar{W}=V/\bar{W}+\bar{W}$, where $V$ is the variance in fitness
among newborns. Subtracting $\bar{W}$ gives Eqn. 1, the fitness difference
between newborns and adults (Price, 1970).

$\Delta\bar{W}=V/\bar{W}$ (1)

where where $\bar{W}$ and $V$ are the mean and variance of fitness among
newborns. This describes changes within a generation, so it does not depend on
the mechanism of inheritance. It works as well with blending as with Mendelian
inheritance.

Let us approximate Eqn. 1 as a differential equation:

$d\bar{W}(t)/dt=V(t)/\bar{W}(t)$ (2)

Under blending inheritance, the variance is halved each generation (Fisher,
1930):

$dV(t)/dt=-V(t)/2$ (3)

This equation captures the effect of blending on variance but not that of
selection. Unless selection is very strong, however, its effect on variance
will be negligible compared with that of blending. Eqn. 3 implies that

$V(t)=V_{0}e^{-t/2}$ (4)

where $V_{0}$ is the variance in fitness immediately after the mutation.

Consider the effect of a single new mutation, whose fitness is $1+s$ times
$W_{0}$, the fitness of a normal individual. The initial population consists
of a single mutant and $N-1$ normal individuals. The initial variance in
fitness is

$V_{0}=W_{0}^{2}s^{2}\left(\frac{1}{N}\right)\left(1-\frac{1}{N}\right)\approx
W_{0}^{2}s^{2}/N$ (5)

ignoring terms of order $N^{-2}$.

Equations 2–5 imply that mean fitness will evolve toward an asymptote at which
$\bar{W}_{\infty}^{2}=W_{0}^{2}+4V_{0}=W_{0}^{2}(1+4s^{2}/N)$. This implies
that $\bar{W}_{\infty}/W_{0}\approx 1+2s^{2}/N$. A single mutation causes a
proportional increase,

$\frac{\bar{W}_{\infty}-W_{0}}{W_{0}}\approx 2s^{2}/N$ (6)

in mean fitness. Using a different argument, Davis derived this result in 1871
for the special case in which $s=1$.

Let us define mutation rates so as to equalize mutational inputs under
blending and Mendelian inheritance. To this end, assume for blending that each
of the $N$ individuals mutates with probability $2u$, so that $2Nu$ new
mutants arise per generation, each with selective advantage $s$. The rate of
proportional increase in mean fitness is $2Nu\times 2s^{2}/N=4us^{2}$ under
blending inheritance.

Under Mendelian inheritance, the corresponding theory is well known. Let $u$
represent the probability per generation that a single gene mutates to a new
allele, whose heterozygous carriers have fitness $1+s$ times that of normal
individuals. There are on average $2Nu$ such mutations per generation, of
which a fraction $2s$ is eventually fixed (Haldane, 1927). If gene effects are
additive, so that homozygous mutants have fitness $1+2s$, then each fixed
mutation increases mean fitness by $2s$. The product, $8Nus^{2}$, of these
quantities, is the rate of proportional increase in mean fitness under
Mendelian inheritance.

This is $2N$ times the rate under blending inheritance. In a population of
10,000 individuals, adaptive evolution would be 20,000 times slower under
blending than under Mendelian inheritance, given equal mutational inputs.

Late in the 19th century, two misconceptions undermined the debate about
evolutionary time. One of these—the age of the earth—has been widely
discussed. Yet by comparison, its effect was minor. Victorians underestimated
the age of the earth by 1 or 2 orders of magnitude. The error implied by their
theory of heredity was much larger.

Evolutionists have long known that Mendelian inheritance solved a fundemental
evolutionary problem—the maintenance of variation—and also that evolutionary
rates depend on variance. We have not, however, fully understood the
overwhelming advantage in speed that Mendelism gave to evolution.

## Acknowledgements

I am grateful for comments from Elizabeth Cashdan, Stephen Downes, and Steven
Gaulin.

## References

  * Bulmer (2004) Bulmer, M. 2004.  Did Jenkin’s swamping argument invalidate Darwin’s theory of natural selection?  The British Journal for the History of Science 37:281–297.
  * Darwin (1872) Darwin, C. 1872.  On the Origin of Species by Means of Natural Selection, or the Preservation of Favoured Races in the Struggle for Life.  6th ed. John Murray, London.
  * Davis (1871) Davis, A. S. 1871.  The “North British Review” and the Origin of Species.  Nature 5:161.
  * Fisher (1930) Fisher, R. A. 1930.  The Genetical Theory of Natural Selection.  Clarendon Press, Oxford.
  * Frank and Slatkin (1992) Frank, S. A., and M. Slatkin. 1992.  Fisher’s fundamental theorem of natural selection.  Trends in Ecology and Evolution 7:92–95.
  * Galton (1889) Galton, F. 1889.  Natural Inheritance.  Macmillan, London.
  * Grafen (2003) Grafen, A. 2003.  Fisher the evolutionary biologist.  Journal of the Royal Statistical Society, Series D 52:319–329.
  * Haldane (1927) Haldane, J. B. S. 1927.  A mathematical theory of natural and artificial selection, part V: Selection and mutation.  Mathematical Proceedings of the Cambridge Philosophical Society 23:838–844.
  * Jackson (2001) Jackson, P. N. W. 2001.  John Joly (1857–1933) and his determinations of the age of the earth.  Pages 107–119 _in_ C. Lewis and S. Knell, eds. The Age of the Earth: from 4004 BC to AD 2002, vol. 190 of _Special Publications_. Geological Society of London, London.
  * Morris (1994) Morris, S. W. 1994.  Fleeming Jenkin and _The Origin of Species_ : A reassessment.  The British Journal for the History of Science 27:313–343.
  * Plutynski (2006) Plutynski, A. 2006.  What was Fisher’s fundamental theorem of natural selection and what was it for?  Studies in History and Philosophy of Science Part C: Studies in History and Philosophy of Biological and Biomedical Sciences 37:59–82.
  * Price (1970) Price, G. R. 1970.  Selection and covariance.  Nature 227:520–521.
  * Price (1972) ———. 1972.  Fisher’s “fundamental theorem” made clear.  Annals of Human Genetics 36:129–140.
  * Thomson (1862) Thomson, W. 1862.  On the secular cooling of the earth.  Transactions of the Royal Society of Edinburgh 23:157–170.