# Quantum Hair from Gravity

Xavier Calmet<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of Sussex, Brighton, BN1 9QH, United Kingdom  Roberto Casadio
<EMAIL_ADDRESS>Dipartimento di Fisica e Astronomia, Università di Bologna,
via Irnerio 46, I-40126 Bologna, Italy  I.N.F.N., Sezione di Bologna, IS -
FLAG, via B. Pichat 6/2, I-40127 Bologna, Italy  Stephen D. H. Hsu
<EMAIL_ADDRESS>Department of Physics and Astronomy
Michigan State University
Folkert Kuipers<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of Sussex, Brighton, BN1 9QH, United Kingdom

###### Abstract

We explore the relationship between the quantum state of a compact matter
source and of its asymptotic graviton field. For a matter source in an energy
eigenstate, the graviton state is determined at leading order by the energy
eigenvalue. Insofar as there are no accidental energy degeneracies there is a
one to one map between graviton states on the boundary of spacetime and the
matter source states. Effective field theory allows us to compute a purely
quantum gravitational effect which causes the subleading asymptotic behavior
of the graviton state to depend on the internal structure of the source. This
establishes the existence of ubiquitous quantum hair due to gravitational
effects.

##  I Introduction

Classical no-hair theorems limit the information that can be obtained about
the internal state of a black hole by outside observers Misner:1973prb .
External features (“hair”) of black hole solutions in general relativity are
determined by specific conserved quantities such as mass, angular momentum,
and charge. In this letter we investigate how the situation changes when both
the matter source (black hole interior state) and the gravitational field
itself are quantized. The idea of quantum hair has been discussed in various
approaches to quantum gravity, see e.g. Giddings:1993de ; Moss:1992ne ;
Preskill:1990ty ; Coleman:1992rn .

We begin by showing that the graviton state associated with an energy
eigenstate source is determined, at leading order, by the energy eigenvalue of
the source. These graviton states can be expressed as coherent states of non-
propagating graviton modes, with explicit dependence on the source energy
eigenvalue. Semiclassical matter sources (e.g., a star or black hole) are
superpositions of energy eigenstates with support in some band of energies,
and produce graviton states that are superpositions of the coherent states.
Next, we use effective field theory to compute $r^{-3}$ and $r^{-5}$
corrections to the $r^{-1}$ Newtonian potential. These corrections originate
from non-local terms in the quantum corrections to Einstein’s equations.

We show that the $r^{-5}$ corrections are sensitive to the internal structure
of the matter source. That is, two matter sources with the same semiclassical
mass $M$ can produce different $r^{-5}$ terms in the metric. These
observations imply that information about the interior state of a black hole
exists outside the classical horizon. This could, in principle, affect the
Hawking radiation states produced as the hole evaporates. We discuss
implications for black hole information and holography in the conclusions.

##  II Asymptotic quantum states of the graviton field

General relativity relates the spacetime metric to the energy-momentum
distribution of matter, but only applies when both the metric (equivalently,
the gravitational field) and matter sources are semiclassical. A theory of
quantum gravity is necessary to relate the quantum state of the gravitational
field to the quantum state of the matter source.

A semiclassical matter configuration $S$ is a superposition of energy
eigenstates with support concentrated in some narrow band of energies

$\psi_{S}=\sum_{n}c_{n}\psi_{n}~{}~{},$ (1)

where $\psi_{n}$ are energy eigenstates with eigenvalues $E_{n}$. $S$ produces
a gravitational field (metric) governed by the Einstein equations
$G_{\mu\nu}=8\pi\,G_{\rm N}\,T_{\mu\nu}$, where the energy momentum tensor is
itself semiclassical. Here we assume that $S$ is compact – localized in some
spatial region of an otherwise empty universe – and consider the gravitational
field asymptotically far away.

What can be said about the quantum state of the graviton field given the exact
quantum state of the matter source? This question extends beyond the realm of
classical general relativity, but we show below that the properties of
semiclassical gravity constrain the result in an interesting way.

We find that the quantum state of the asymptotic gravitational field of a
matter source which is an energy eigenstate is controlled by the energy
eigenvalue $E_{n}$. In particular, energy eigenstate sources with different
eigenvalues produce distinct graviton states. This immediately implies that
the asymptotic graviton field of a typical semiclassical matter source is a
superposition state of the form

$\psi_{g}(S)=\sum_{n}c_{n}\psi_{g}(E_{n})~{}~{},$ (2)

where $\psi_{g}(E)\neq\psi_{g}(E^{\prime})$ when $E\neq E^{\prime}$.

It is typically assumed in many body physics that there are no accidental
degeneracies – i.e., that the eigenvalues $E_{n}$ of a complex matter system
are distinct (barring exact symmetries of the Hamiltonian; note even these may
be violated by quantum gravity effects), although energy level splittings
might be exponentially small in the size of the system. If this is the case,
then the above results imply that the state of the matter system can, in
principle, be reconstructed from the asymptotic graviton state. The quantum
information encoded in the matter system is also stored, via entanglement, in
the spacetime metric at infinity.

To obtain the desired result we use the following gedanken construction. In
brief, we want to show that the matter source energy eigenstate $\psi_{n}$
produces a different asymptotic graviton state than another state
$\psi_{n^{\prime}}$ of the system with $E_{n^{\prime}}\neq E_{n}$. The problem
is that the energy splitting $E_{n^{\prime}}-E_{n}$ could be exponentially
small in the size of $S$ and as far as the classical Einstein equations are
concerned the corresponding sources $T$ and $T^{\prime}$ are effectively
identical.

However, we can imagine configurations made of $N$ identical copies of the
original system $S$, which we take to be an energy eigenstate
($\psi_{S}=\psi_{n}$), and the same number $N$ of identical copies of the
system $S^{\prime}$ with the source in the eigenstate
$\psi_{S^{\prime}}=\psi_{n^{\prime}}$. For sufficiently large $N$ the
difference in source terms $T$ and $T^{\prime}$ becomes macroscopic, and the
difference between the corresponding metrics is governed by the classical
Einstein equations. The asymptotic behaviors of these metrics are equivalent
to the additive Newtonian gravitational potentials resulting from each of the
$N$ copies of $S$ and $S^{\prime}$, respectively. Hence the asymptotic
graviton state $\psi_{g}(NE_{n})$ of the system $S$ cannot be identical to
$\psi_{g}(NE_{n^{\prime}})$ of the system $S^{\prime}$, otherwise the
resulting sums would also be identical. See FN  for additional details of this
construction.

This analysis does not determine the graviton states $\psi_{g}(E)$, but does
establish that different energies $E$ correspond to different (albeit possibly
very similar) states $\psi_{g}$.

We can obtain the same result via quantum field theory using the property that
the spin-2 graviton $h_{\mu\nu}$ couples to the operator $T_{\mu\nu}$. The
gravitational potential is generated by graviton exchange between the source
“particle” $S$ and a test mass. At long wavelengths, we can treat the
composite state $S$ as a single particle, analogous to a nucleon which is
composite and has its own complex substructure. The Feynman amplitude for
graviton emission from an incoming source particle $S$ has a vertex factor
which is simply its energy eigenvalue $E$. States $S$ with different energies
$E$ have different graviton emission amplitudes, and hence produce different
asymptotic states of the $h_{\mu\nu}$ field.

The graviton quantum state $\psi_{g}(E)$ is exactly analogous to the quantum
state of the $U(1)$ vector field (Coulomb potential) created by a charge $Q$
Barnich ; Mueck ; Casadio:2021eio . This can be constructed explicitly as a
coherent state

$|0\rangle_{Q}=\exp\,\left[Q\,\int
d^{3}k\,q(k)b^{\dagger}(k)\right]\,|0\rangle_{Q=0}$ (3)

where $b^{\dagger}(k)$ is a linear combination of creation operators of the
non-propagating (temporal and longitudinal, depending on choice of gauge)
modes of the photon. The factor of $Q$ in the exponent shows how the photon
state depends on the source charge. In the gravitational case $Q$ is replaced
by the energy eigenvalue of the source state and the coherent state modes are
temporal and longitudinal graviton modes. In both gauge theory and gravity the
manner in which the charge or energy control the asymptotic quantum state is
determined by the Gauss law via constrained quantization. Note (3) is a formal
expression which avoids some infrared issues: the $q(k)$ are not normalizable
with respect to the Lorentz-invariant 1-particle norm.

The direct connection between the gravitational field (Schwarzschild metric)
and the Coulomb potential can also be seen as a consequence of the double copy
relationship oconnell . For our purposes the most important point is that
$\psi_{g}(E)$ depends explicitly on $E$ and for each distinct energy
eigenstate of the compact source there is a different graviton quantum state.

The evaporation of a black hole takes place over a timescale $\sim M^{3}$ so
its evolution from a matter configuration to outgoing radiation is confined to
a finite region of spacetime. Hence the asymptotic gravitational field at
$r\gg M^{3}$ remains unchanged, in the form (2), throughout the entire
process. However, near the horizon the gravitational quantum state presumably
reflects the changing internal state of the hole. The internal state is itself
dependent on the previously emitted Hawking radiation – e.g., due to
conservation of energy, angular momentum, etc. This provides a mechanism
connecting the region just outside the horizon, where the next quantum of
Hawking radiation originates, to the internal state of the black hole and the
radiation quanta emitted in the past. Once we go beyond the semiclassical
approximation the amplitude for radiation emission is a function of
$\psi_{g}(E)$ which itself depends on the internal state of the hole. We
discuss this further in the conclusions.

##  III Leading corrections from quantum gravity

In general relativity, Birkhoff’s theorem states that any spherically
symmetric solution of the vacuum field equations must be static and
asymptotically flat. In other words, the exterior solution must be given by
the Schwarzschild metric. It has been shown that this is not the case in
quantum gravity Calmet:2017qqa ; Calmet:2019eof : the asymptotic gravitational
potential of a compact object received quantum gravitational corrections
Calmet:2019eof ; Calmet:2020tlj  which are not present for an eternal black
hole Calmet:2019eof ; Calmet:2018elv . Quantum gravitational corrections
depend on the composition of the compact object. This quantum memory effect
has also been observed in FLRW cosmology Donoghue:2014yha . In this section,
we show that compact objects are hairy in quantum gravity. We work within the
framework of the effective quantum gravitational action at second order in
curvature Weinberg:1980gg ; Barvinsky:1983vpp ; Barvinsky:1985an ;
Barvinsky:1987uw ; Barvinsky:1990up ; Buchbinder:1992rb ; Donoghue:1994dn ;
Calmet:2018elv : $\Gamma[g]=\Gamma_{\rm L}[g]+\Gamma_{\rm NL}[g]$, where the
local part of the action is given by

$\displaystyle\Gamma_{\rm L}=\int\mbox{${\rm
d}$}^{4}x\,\sqrt{g}\left[\frac{{\cal R}}{16\,\pi\,G_{\rm N}}+c_{1}(\mu)\,{\cal
R}^{2}+c_{2}(\mu)\,{\cal R}_{\mu\nu}\,{\cal R}^{\mu\nu}+c_{3}(\mu)\,{\cal
R}_{\mu\nu\alpha\beta}\,{\cal R}^{\mu\nu\alpha\beta}\right]$ (4)

and the non-local part of the action by

$\displaystyle\Gamma_{\rm NL}=-\int\mbox{${\rm
d}$}^{4}x\,\sqrt{g}\left[\alpha\,{\cal
R}\,\ln{\\!\left(\frac{\Box}{\mu^{2}}\right)}\,{\cal R}+\beta\,{\cal
R}_{\mu\nu}\,\ln\\!{\left(\frac{\Box}{\mu^{2}}\right)}\,{\cal
R}^{\mu\nu}+\gamma\,{\cal
R}_{\mu\nu\alpha\beta}\,\ln\\!{\left(\frac{\Box}{\mu^{2}}\right)}\,{\cal
R}^{\mu\nu\alpha\beta}\right]\ .$ (5)

This effective action is obtained by integrating out the fluctuations of the
graviton and potentially other massless matter fields. The Wilson coefficients
of the local part of the action are not calculable from first principles, as
we do not specify the ultra-violet theory of quantum gravity. However, those
of the non-local part are calculable and model independent quantum
gravitational predictions. These non-local coefficients can be found in e.g.
Calmet:2019eof . The equations of motion obtained from varying the effective
action with respect to the metric are given by

$\displaystyle{\cal R}_{\mu\nu}-\frac{1}{2}\,{\cal
R}\,g_{\mu\nu}+16\,\pi\,G_{\rm N}\left(H_{\mu\nu}^{\rm L}+H_{\mu\nu}^{\rm
NL}\right)=8\,\pi\,G_{N}T_{\mu\nu}\ .$ (6)

The local part of the equation of motion is given by

$\displaystyle H_{\mu\nu}^{\rm L}=$ $\displaystyle\,\bar{c}_{1}\left(2\,{\cal
R}\,{\cal R}_{\mu\nu}-\frac{1}{2}\,g_{\mu\nu}\,{\cal
R}^{2}+2\,g_{\mu\nu}\,\Box{\cal R}-2\nabla_{\mu}\nabla_{\nu}{\cal R}\right)$
(7) $\displaystyle\,+\bar{c}_{2}\left(2\,{\cal R}_{~{}\mu}^{\alpha}\,{\cal
R}_{\nu\alpha}-\frac{1}{2}\,g_{\mu\nu}\,{\cal R}_{\alpha\beta}\,{\cal
R}^{\alpha\beta}+\Box{\cal R}_{\mu\nu}+\frac{1}{2}\,g_{\mu\nu}\,\Box{\cal
R}-\nabla_{\alpha}\nabla_{\mu}{\cal
R}_{~{}\nu}^{\alpha}-\nabla_{\alpha}\nabla_{\nu}{\cal
R}_{~{}\mu}^{\alpha}\right)\ ,$

with $\bar{c}_{1}=c_{1}-c_{3}$ and $\bar{c}_{2}=c_{2}+4\,c_{3}$. Finally, the
non-local part reads

$\displaystyle H_{\mu\nu}^{\rm NL}=$ $\displaystyle\,-2\,\alpha\left({\cal
R}_{\mu\nu}-\frac{1}{4}\,g_{\mu\nu}\,{\cal
R}+g_{\mu\nu}\,\Box-\nabla_{\mu}\nabla_{\nu}\right)\ln\left(\frac{\Box}{\mu^{2}}\right)\,{\cal
R}$ $\displaystyle\,-\beta\bigg{(}2\,\delta_{(\mu}^{\alpha}\,{\cal
R}_{\nu)\beta}-\frac{1}{2}\,g_{\mu\nu}\,{\cal
R}_{~{}\beta}^{\alpha}+\delta_{\mu}^{\alpha}\,g_{\nu\beta}\,\Box+g_{\mu\nu}\,\nabla^{\alpha}\nabla_{\beta}$
$\displaystyle\quad-\delta_{\mu}^{\alpha}\,\nabla_{\beta}\nabla_{\nu}-\delta_{\nu}^{\alpha}\,\nabla_{\beta}\nabla_{\mu}\bigg{)}\ln\left(\frac{\Box}{\mu^{2}}\right)\,{\cal
R}_{~{}\alpha}^{\beta}$
$\displaystyle\,-2\,\gamma\left(\delta_{(\mu}^{\alpha}\,{\cal
R}_{\nu)~{}\sigma\tau}^{~{}\beta}-\frac{1}{4}\,g_{\mu\nu}\,{\cal
R}^{\alpha\beta}_{~{}~{}\sigma\tau}+\left(\delta_{\mu}^{\alpha}\,g_{\nu\sigma}+\delta_{\nu}^{\alpha}\,g_{\mu\sigma}\right)\nabla^{\beta}\nabla_{\tau}\right)\ln\left(\frac{\Box}{\mu^{2}}\right)\,{\cal
R}_{\alpha\beta}^{~{}~{}\sigma\tau}\ .$ (8)

Note that the variation of the $\ln\Box$ term yields terms of higher order in
curvature and can thus safely be ignored at second order in curvature. The
non-local parts of the field equations are responsible for the memory effect.
We can easily illustrate this by considering the corrections to the metric of
a stationary homogeneous and isotropic star with radius $R_{\rm s}$ and
density

$\rho(r)=\rho_{0}\,\Theta(R_{\rm s}-r)=\begin{cases}\rho_{0}&{\rm if}\
r<R_{\rm s}\\\ 0&{\rm if}\ r>R_{\rm s}\ ,\end{cases}$ (9)

where $\rho_{0}>0$ is a constant and $\Theta(x)$ is Heaviside’s step function.
The solution to the Einstein equation inside this star (for $r\leq R_{\rm s}$)
is the well-known interior Schwarzschild metric

$\displaystyle\mbox{${\rm d}$}s^{2}$
$\displaystyle=\left(3\,\sqrt{1-\frac{2\,G_{\rm N}\,M}{R_{\rm
s}}}-\sqrt{1-\frac{2\,G_{\rm N}\,M\,r^{2}}{R_{\rm
s}^{3}}}\right)^{2}\frac{\mbox{${\rm d}$}t^{2}}{4}-\left(1-\frac{2\,G_{\rm
N}\,Mr^{2}}{R_{\rm s}^{3}}\right)^{-1}\mbox{${\rm d}$}r^{2}-r^{2}\,\mbox{${\rm
d}$}\Omega^{2}$ $\displaystyle\equiv g_{\mu\nu}^{\rm int}\,\mbox{${\rm
d}$}x^{\mu}\,\mbox{${\rm d}$}x^{\nu}\ ,$ (10)

where

$M=4\,\pi\int_{0}^{R_{\rm s}}\rho\,r^{2}\,\mbox{${\rm
d}$}r=\frac{4\,\pi}{3}\,R_{\rm s}^{3}\,\rho_{0}$ (11)

is the total Misner-Sharp mass of the source. The corresponding pressure is of
order $G_{\rm N}$ Calmet:2019eof  in agreement with the fact that the pressure
does not gravitate in Newtonian physics. Of course, the metric outside the
star (for $r>R_{\rm s}$) is the usual vacuum Schwarzschild metric

$\mbox{${\rm d}$}s^{2}=\left(1-\frac{2\,G_{\rm N}\,M}{r}\right)\mbox{${\rm
d}$}t^{2}-\left(1-\frac{2\,G_{\rm N}\,M}{r}\right)^{-1}\mbox{${\rm
d}$}r^{2}-r^{2}\,\mbox{${\rm d}$}\Omega^{2}\equiv g_{\mu\nu}^{\rm
ext}\,\mbox{${\rm d}$}x^{\mu}\,\mbox{${\rm d}$}x^{\nu}\ ,$ (12)

from which one can see that $M$ is also the Arnowitt-Deser-Misner (ADM) mass
of the system.

We now perturb the above metrics:
$\tilde{g}_{\mu\nu}=g_{\mu\nu}+g_{\mu\nu}^{\rm q}$, and take the perturbation
$g_{\mu\nu}^{\rm q}$ to be $\mathcal{O}(G_{\rm N})$. We solve this equation,
imposing the solution to be spherically symmetric and time independent. In
addition we fix the gauge freedom by setting $g_{\theta\theta}^{\rm q}=0$.
Doing so, we obtain the quantum corrections $g_{\mu\nu}^{\rm q}=\delta
g_{\mu\nu}^{\rm ext}$ to the Schwarzschild metric (12) outside the star. The
corrections are given in Calmet:2019eof :

$\displaystyle\delta g_{tt}^{\rm ext}$
$\displaystyle=(\alpha+\beta+3\,\gamma)\,\frac{192\,\pi\,G_{\rm
N}^{2}\,M}{R_{\rm s}^{3}}\left[2\,\frac{R_{\rm
s}}{r}+\ln\\!\left(\frac{r-R_{\rm s}}{r+R_{\rm
s}}\right)\right]+\frac{C_{1}}{r}+C_{2}+\mathcal{O}(G_{\rm N}^{3})$
$\displaystyle\delta g_{rr}^{\rm ext}$
$\displaystyle=(\alpha-\gamma)\,\frac{384\,\pi\,G_{\rm
N}^{2}\,M}{r\,(r^{2}-R_{\rm s}^{2})}+\frac{C_{1}}{r}+\mathcal{O}(G_{\rm
N}^{3})\ ,$ (13)

where $C_{i}$ are integration constants which can be set to zero. This ensures
asymptotic flatness and that the ADM mass is $M$.

We work with the metric with signature $(+---)$, in the signature $(-+++)$
case, the corrections obtain an extra minus sign. Note the two terms in large
brackets, when combined, give rise to the $r^{-3}$ and $r^{-5}$ corrections
mentioned in the introduction. The coefficient of this term is proportional to
$G_{\rm N}^{2}MR_{\rm s}^{-3}$: i.e., it is a quantum gravitational effect
proportional to the density of the source object. Two source objects with the
same mass $M$ but different densities give rise to different metric
perturbations.

Now compare the result to that generated by two nested (one inside the other)
dust balls with densities

$\rho_{i}(r)=\rho_{0,i}\,\Theta(R_{i}-r)=\begin{cases}\rho_{0,i}&{\rm if}\
r<R_{i}\\\ 0&{\rm if}\ r>R_{i}\ ,\end{cases}$ (14)

and masses $M_{1}$ and $M_{2}$

$M_{i}=4\,\pi\int_{0}^{R_{i}}\rho_{0,i}\,r^{2}\,\mbox{${\rm
d}$}r=\frac{4\,\pi}{3}\,R_{i}^{3}\,\rho_{0,i}$ (15)

with $i\in\\{1,2\\}$, such that, e.g., $R_{1}<R_{s}$, $R_{2}=R_{s}$ and
$M=M_{1}+M_{2}$. In other words, the star built from two nested dust balls has
total mass equal to $M$ and the same outer radius $R_{s}$ as the star composed
of only one component.

It is straightforward to show that a solution in general relativity exists. In
the region $r\in[R_{2},\infty)$, the metric is the exterior Schwarzschild
solution with mass $M$. In the region $r\in[0,R_{1})$ (the most inner one),
the metric is the interior Schwarzschild solution with radius $R_{1}$ and mass
$M_{1}+M_{2}(R_{1}/R_{2})^{3}$. In the region $r\in[R_{1},R_{2})$, the metric
is the interior Schwarzschild solution with radius $R_{2}$ and mass $M_{2}$.

In general relativity, an external observer cannot differentiate a star with
radius $R_{s}$ and mass $M$ from the star with two different components but
same external radius and same total mass $M$. However, we will show that the
quantum gravitational corrections are different for the two matter
distributions and there is thus a memory effect. Repeating the same
calculation as in Calmet:2019eof , using the fact that at this order in
$G_{\rm N}$ the equations are linearized, we find a correction

$\displaystyle\delta g_{tt}^{\rm ext}$
$\displaystyle=(\alpha+\beta+3\,\gamma)\,\frac{192\,\pi\,G_{\rm
N}^{2}\,M_{1}}{R_{1}^{3}}\left[2\,\frac{R_{1}}{r}+\ln\\!\left(\frac{r-R_{1}}{r+R_{1}}\right)\right]$
$\displaystyle\quad+(\alpha+\beta+3\,\gamma)\,\frac{192\,\pi\,G_{\rm
N}^{2}\,M_{2}}{R_{2}^{3}}\left[2\,\frac{R_{2}}{r}+\ln\\!\left(\frac{r-R_{2}}{r+R_{2}}\right)\right]+\mathcal{O}(G_{\rm
N}^{3})$ $\displaystyle\delta g_{rr}^{\rm ext}$
$\displaystyle=(\alpha-\gamma)\,\frac{384\,\pi\,G_{\rm
N}^{2}\,M_{1}}{r\,(r^{2}-R_{s,1}^{2})}+(\alpha-\gamma)\,\frac{384\,\pi\,G_{\rm
N}^{2}\,M_{2}}{r\,(r^{2}-R_{s,2}^{2})}+\mathcal{O}(G_{\rm N}^{3})\ .$ (16)

While the classical part of the metric cannot distinguish between the one ball
of dust with mass $M$ and two concentric dust balls with masses $M_{1},M_{2}$
and $M_{1}+M_{2}=M$, the quantum gravitational corrections depend on the
matter distribution of the nested balls.

For the one-layer star we obtain

$\displaystyle g_{tt}$ $\displaystyle=$ $\displaystyle 1-\frac{2G_{N}M}{r}$
$\displaystyle-128\pi^{2}(\alpha+\beta+3\gamma)\frac{l_{p}^{2}}{r^{2}}\left[\frac{G_{N}M}{r}\left(1+\frac{3R_{s}^{2}}{5r^{2}}+{\cal
O}(R_{s}/r)^{4}\right)+{\cal O}(G_{N}M/r)^{2}\right]$ $\displaystyle+~{}{\cal
O}(l_{p}/r)^{4}\ ,$

where $l_{p}=\sqrt{\hbar G}$ is the Planck length, and for two layers we
obtain

$\displaystyle g_{tt}$ $\displaystyle=$ $\displaystyle 1-\frac{2G_{N}M}{r}$
$\displaystyle-128\pi^{2}(\alpha+\beta+3\gamma)\frac{l_{p}^{2}}{r^{2}}\left[\frac{G_{N}M}{r}\left(1+\frac{3(M_{1}R_{1}^{2}+M_{2}R_{s}^{2})}{5Mr^{2}}+{\cal
O}(R_{s}/r)^{4}\right)+{\cal O}(G_{N}M/r)^{2}\right]$ $\displaystyle+~{}{\cal
O}(l_{p}/r)^{4}.$

Clearly, the quantum gravitational corrections are different for the two
stars. Here we made explicit the different expansion parameters. The series in
$l_{p}/r$ reflects the truncation of the effective action at second order in
curvature. The series in $G_{N}M/r$ is due to the linearization of the field
equations and the expansion in $R_{s}/r$ corresponds to the asymptotic limit.
In this limit we see that potentials generated by the two stars are
composition dependent at order $r^{-5}$.

In this case we have considered a two-layered star and shown that the result
can differ from a single-layered star. However, the above argument can easily
be extended to show that any $n$\- and $m$-layered stars with $n\neq m$ can be
distinguished by an outside observer due to quantum gravitational effects,
although their classical external gravity fields are identical. The quantum
memory effect leads to hairy stars.

To extend the above discussion, consider two homogeneous stars both with
initial mass $M_{i}$ and radius $R_{i}$. We assume that at a certain time both
stars run out of fuel and collapse towards a new equilibrium state with mass
$M_{f}$ and radius $R_{f}$. Let us furthermore assume that the first star
remains homogeneous, while the second collapses to a two-layered state as
described above. The initial configurations are gravitationally
indistinguishable in terms of classical effects. Moreover, due to Birkhoff’s
theorem the two final states are classically indistinguishable. However, due
the quantum gravitational memory effect the two final states are
distinguishable at the quantum level.

While earlier we assumed a time-independent static star, we could consider a
collapsing dust ball which can form a black hole. We introduce time-dependence
via the radius of the star $R_{s}(t)$. For a distant observer, $r\gg R_{s}(t)$
at all times, we can expand the correction to the metric in Eq. (III), and it
seems likely that the $r^{-5}$ dependence remains during the totality of the
collapse.

Eventually, $R_{s}(t)$ will reach $2G_{N}M$ and a closed trapped surface will
form indicating the formation of a black hole. An observer could in principle
measure the coefficient of the $r^{-5}$ correction to the metric. This
correction contains information about the matter distribution that collapsed
and could thus enable the observer to differentiate between black holes formed
by different matter distributions.

The $r^{-5}$ correction shifts the location of the horizon slightly and
modifies the metric near the horizon. This presumably has an effect on Hawking
radiation. A fully quantum mechanical treatment of the metric $g$, as opposed
to the semiclassical perturbation analysis above, would yield the detailed
quantum state of the graviton field (analogous to (3)) in place of the
$r^{-5}$ correction we obtained footnote1 .

We find that quantum gravity produces a new kind of hair on black holes. While
the corrections described in this section may be very small, with limited
experimental consequences, they can have dramatic consequences for black holes
information Calmet:2021cip .

##  IV Conclusions: holography and black hole information

The existence of a one to one map between the quantum states of compact matter
sources and of their asymptotic gravitational fields is clearly suggestive of
holography and area bounds on entropy. We emphasize that the appearance of the
charge or energy in results like (3) originates in Gauss law constraints which
play an important role in the quantization of gauge theories and gravity. The
recovery of bulk information from asymptotic gravitational fields at the
boundary is also discussed in Marolf:2008mf ; Laddha:2020kvp ;
Chowdhury:2020hse ; Chowdhury:2021nxw .

In a fully quantum mechanical treatment the evolution of the matter source
cannot be considered independently from that of its gravitational field. This
contrasts sharply with the usual approximation of a fixed spacetime background
in which matter fields evolve. For example, Hawking radiation from a black
hole is computed in this approximation, whereas our analysis shows that a
precise treatment (e.g., one which hopes to examine the unitarity of black
hole evaporation) must consider that the metric outside the horizon depends on
the state of the interior. The evaporation process takes the form

$|B_{0},g_{0}\rangle\,\rightarrow\,|B_{1},g_{1},\gamma_{1}\rangle\,\rightarrow\,|B_{2},g_{2},\gamma_{2},\gamma_{1}\rangle\,\rightarrow\,|B_{3},g_{3},\gamma_{3},\gamma_{2},\gamma_{1}\rangle\cdots$
(19)

where $B$ is the black hole internal state, $g$ the quantum state of the
(external) graviton field or metric, and $\gamma$ the emitted radiation which
originates at the horizon. The radiation state $\gamma_{n+1}$ depends on the
metric state $g_{n}$, and each $g_{n}$ depends on, and is entangled with,
$B_{n}$. From this perspective it is clear that the Hawking radiation state is
connected to the internal state of the black hole.

We can give some idea of the complexity of this process through the following
schematic description. Consider the semiclassical superposition state in (2),

$\psi_{g}(S)=\sum_{n}c_{n}\psi_{g}(E_{n})~{}~{},$ (20)

and suppose that each graviton state $\psi_{g}(E_{n})$ (describing the
exterior metric) has amplitude $\alpha(E_{n},\Delta)$ to produce a Hawking
radiation quantum $\gamma$ with energy $\Delta$ (represented by
$\gamma(\Delta)$ in the wavefunction below). Then the exterior state evolves
to

$\psi\approx\sum_{n}c_{n}\left[\psi_{g}(E_{n})~{}+~{}\alpha(E_{n},\Delta)\psi_{g}(E_{n}-\Delta)\gamma(\Delta)~{}+~{}\cdots\right]$
(21)

The state after radiation emission (from second term in the sum, above) is a
different semiclassical state constructed from $\psi_{g}$ corresponding to
energies shifted by $\Delta$. Through $\alpha(E_{n},\Delta)$ and
$\psi_{g}(E_{n}-\Delta)$ the detailed form of this quantum state depends on
the emitted radiation, including on quantum numbers we have suppressed such as
momentum, spin, charge, etc. Even if the deviation of $\alpha(E_{n},\Delta)$
from the semiclassical amplitude is exponentially small, the aggregate effect
on the process of evaporation could be significant. It is plausible that each
initial black hole state, specified by coefficients $c_{n}$, evolves into a
different final quantum state – i.e., the evolution is unitary.

For each history of radiation quanta
$\\{\gamma_{1},\gamma_{2},\cdots,\gamma_{n}\\}$ there is a corresponding
quantum spacetime $\\{g_{1},g_{2},\cdots,g_{n}\\}$. A black hole with entropy
$A$ can produce $\sim\exp A$ distinct evaporation states and corresponding
quantum spacetimes. Schrödinger evolution of the initial state will produce a
superposition of these radiation states and spacetimes Buniy:2020dux ;
Hsu:2009ve ; Hsu:2010sb . It has been conjectured that black hole evaporation
is unitary when all of these branches of the wavefunction are taken into
account Hsu:2013cw ; Hsu:2013fra ; Bao:2017who ; Raju:2020smc .

Acknowledgments: The work of X.C. is supported in part by the Science and
Technology Facilities Council (grants numbers ST/T00102X/1, ST/T006048/1 and
ST/S002227/1). The work of R.C. is partially supported by the INFN grant FLAG,
it has also been carried out in the framework of activities of the National
Group of Mathematical Physics (GNFM, INdAM).

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# Phase diagram and topology of the XXZ chain with alternating bonds and
staggered magnetic field

B. F. Márquez  Centro Atómico Bariloche and Instituto Balseiro, 8400
Bariloche, Argentina  N. Aucar Boidi  Centro Atómico Bariloche and Instituto
Balseiro, 8400 Bariloche, Argentina  K. Hallberg  Centro Atómico Bariloche and
Instituto Balseiro, 8400 Bariloche, Argentina  Instituto de Nanociencia y
Nanotecnología CNEA-CONICET, GAIDI, Centro Atómico Bariloche, 8400 Bariloche,
Argentina  A. A. Aligia  Centro Atómico Bariloche and Instituto Balseiro, 8400
Bariloche, Argentina  Instituto de Nanociencia y Nanotecnología CNEA-CONICET,
GAIDI, Centro Atómico Bariloche, 8400 Bariloche, Argentina

###### Abstract

The XXZ spin-half chain has Heisenberg exchange interactions $J_{z}$
($J_{\perp}$) in the $z$ ($x,y$) direction. The model has a transition from
the spin-fluid phase for $-J_{\perp}<J_{z}<J_{\perp}$ to the Néel phase for
$J_{z}>J_{\perp}>0$. When bond alternation $\delta$ is included, the Néel
phase transitions to the dimer phase for a finite value of $\delta$. We
determine the phase diagram using simple topological indicators related to the
polarization of both spins. When a staggered magnetic field $B$ is included, a
contour plot of these indicators as a function of $\delta$ and $B$ determine
the amount of topological quantized spin pumping around closed circuits in the
$(\delta,B)$ plane.

##  I Introduction

In recent years there has been considerable attention in topological aspects
of matter [1, 2, 3]. In particular, topological quantized charge or spin
pumping can be realized in time-dependent adiabatic evolution in a closed
cycle in a certain space of parameters [typically two-dimensional (2D)]. This
is known as a Thouless pump [4, 5, 6, 7].

Usually, the 2D pump cycle encloses one or more critical points at which a
symmetry protected topological number jumps. Outside the critical points, the
protecting symmetry is lost allowing a continuous variation of charge or spin
Berry phases, the variation of which on the cycle determines the transported
amount of charge or spin. Experimentally, charge transport in Thouless pumps
described by the Rice-Mele chain [8, 9] including interactions in some cases
[10, 11] have been realized in chains of ultracold atoms. Different variants
of the model have been discussed theoretically [12, 13, 14, 15, 16, 7, 17, 18,
19, 20, 21].

Quantum spin pumps have been also realized experimentally [22] and discussed
theoretically [22, 23, 24, 25, 26, 19, 27]. Different realizations of
Heisenberg-like chains have been achieved with ultracold atoms [28, 29, 22,
30] including an extremely anisotropic Heisenberg model [30]. The XXZ chain is
the Heisenberg chain with anisotropic nearest-neighbor (NN) exchange
interactions, $J_{z}$ in the $z$ direction and $J_{\perp}$ in the other two.
While this model has been solved exactly [31], different variants of the model
have been studied using field theoretical [32, 33, 34, 35, 36] and numerical
[37, 38, 39, 40, 27] methods.

The phase diagram of the spin-1/2 model including next NN interactions has
been calculated using the method of crossings of excited energy levels (MCEL),
supported by results of conformal field theory and the renormalization group
[37, 38]. The corresponding diagram for the model with alternating NN
interactions [$H_{0}$ in Eq. (1)] has been calculated using different
numerical techniques [39, 27]. In spite of some quantitative differences,
qualitatively the results of both works coincide. Without alternating
interactions ($\delta=0$), there is a transition at $J_{z}=J_{\perp}$ between
the spin-fluid phase for $-J_{\perp}<J_{z}<J_{\perp}$ to the Néel phase for
large $J_{z}$ [31, 37, 38]. For an infinitesimal alternation, the spin fluid
phase is transformed to a dimer phase, while a finite $|\delta|$ is needed for
the transition between the Néel phase and a dimer phase. The dimer phases are
different for opposite signs of $\delta$. A similar model with experimental
relevance has been studied numerically [40].

Recently, real-time dynamics of Thouless pumps in the XXZ model with
alternating NN interactions including a staggered magnetic field $B$ have been
calculated with the infinite time-evolving block decimation (iTEBD) method
[27]. The authors considered different circuits in the $(\delta,B)$ plane that
touch but do not cross the singular point $\delta=B=0$. The authors find that
quantized pumping takes place only for $J_{z}>J_{\perp}$, when the Néel phase
separates the two dimer phases for positive and negative $\delta$.

The purpose of this paper is twofold: (i) to recalculate the phase diagram of
the model for $B=0$ (which we call $H_{0}$) using topological indicators based
on simple position operators, showing the power of this method. (ii) To show
how a contour plot of expectation values of the position operators in the
$(\delta,B)$ plane allows the prediction of the transported spin in pumping
circuits.

The paper is organized as follows. In Sec. II we present the Hamiltonian of
the model with its known limiting cases. In Sec. III we define the topological
indicators used to determine the phase diagram of $H_{0}$ and discuss some
simple cases in which these indicators characterize the different phases. Sec.
IV presents the results for the phase diagram of $H_{0}$ compared with the
previous results obtained by Tzeng et al. [39] using different sophisticated
numerical methods, showing a quantitative agreement. In Sec. V the contour
plots of the topological indicators in the $(\delta,B)$ plane show that these
can predict the evolution of the spin transport in adiabatic Thouless pump
cycles. At the end, a summary and discussion can be found in Sec. VI.

##  II Model Hamiltonian

The Hamiltonian of the spin-1/2 alternating XXZ chain, including a staggered
magnetic field reads

$\displaystyle H$ $\displaystyle=$ $\displaystyle H_{0}+H_{B},$ $\displaystyle
H_{0}$ $\displaystyle=$
$\displaystyle\sum\limits_{j=1}^{N}\left[1+(-1)^{j}\delta\right][J_{\perp}(S_{j}^{x}S_{j+1}^{x}+S_{j}^{y}S_{j+1}^{y})+$
$\displaystyle\hskip 73.97733pt+\hskip 2.84544ptJ_{z}S_{j}^{z}S_{j+1}^{z}],$
$\displaystyle H_{B}$ $\displaystyle=-$
$\displaystyle\sum\limits_{j=1}^{N}(-1)^{j}BS_{j}^{z},$ (1)

where $\mathbf{S}_{j}=(S_{j}^{x},S_{j}^{y},S_{j}^{z})$ is the spin of the site
$j$ of the chain. Since we use periodic boundary conditions, the number of
sites $N$ should be even for the system to contain an integer number of unit
cells.

The Hamiltonian for $B=0$ ($H_{0}$) has been studied before [39, 41] and can
be experimentally realized with cold atoms, where the bond alternation is
achieved by fine tuning the intensity of the Raman laser beams [42, 28].

Two limits of the model are well known. Without bond alternation ($\delta=0$)
the model has been solved exactly using the Bethe ansatz [31]. If in addition
$J_{z}=0$, the model can be mapped to a simple fermionic tight-binding chain
using a Jordan-Wigner transformation [39]. Thus this limit provides a simple
picture of the spin-fluid phase, where the spins are delocalized in the chain.
Instead, for large positive $J_{z}$ in an infinite system, there is a
spontaneous symmetry breaking to a Néel or anti-Néel state
($\uparrow\downarrow\uparrow\downarrow...$ or
$\downarrow\uparrow\downarrow\uparrow...$ plus fluctuations). For a finite
system, the ground state is a mixture of both. The transition between the
spin-fluid and the Néel phase is exactly at the isotropic point
$J_{z}=J_{\perp}$ [31, 37, 38]. The gap in the Néel phase is exponentially
small near the transition and behaves as $\Delta\sim J_{z}-2J_{\perp}$ for
large $J_{z}$ [31].

Introducing $\delta$ for $J_{z}=0$, the system can still be mapped onto a non-
interacting fermionic chain [39, 43]. A dimerized phase in which the
expectation value of the NN singlets is larger (smaller) at the odd bonds with
respect to the even ones is formed for $\delta<0$ ($\delta>0$). The odd bonds
are those between sites $j$ and $j+1$ with $j$ odd. The difference between
both expectation values as well as the energy gap are both proportional to
$\delta$. Instead, in the isotropic case $J_{z}=J_{\perp}$, it has been shown
by bosonization [44] that the bond-order parameter
$\left\langle(\mathbf{S}_{1}-\mathbf{S}_{3})\cdot\mathbf{S}_{2}\right\rangle\sim\delta^{1/3}$,
while the energy gap $\Delta\sim\delta^{2/3}$.

For $J_{z}>J_{\perp}$, the Néel phase competes with the dimer phases [39], as
described in detail in Sec. IV. Including magnetic field, pumping circuits in
the $(\delta,B)$ plane were studied, which are discussed in Sec. V.

##  III Methods

Our calculations for the model Eq. (1) are based on two phases for the model
defined in a ring

$\alpha_{s}=\alpha(1,-1),\hskip 14.22636pt\alpha_{\uparrow}=\alpha(1,0),$ (2)

where

$\alpha(m_{{}_{\uparrow}},m_{\downarrow})=\text{Im}\text{ ln}\left\langle
U(m_{\uparrow},m_{\downarrow})\right\rangle\text{ mod }2\pi,$ (3)

is the phase of the expectation value of the following operator

$\displaystyle
U(m_{{}_{\uparrow}},m_{\downarrow})=\exp\left[i(2\pi/L)\sum_{j}x_{j}\left(m_{\uparrow}\hat{n}_{j\uparrow}+m_{\downarrow}\hat{n}_{j\downarrow}\right)\right],$
(4)

where $x_{j}$ is the position of the site $j$,
$\hat{n}_{j\uparrow}=1/2+S_{j}^{z}$, $\hat{n}_{j\downarrow}=1/2-S_{j}^{z}$,
$m_{s}$ (where $s=\uparrow,\downarrow$) can take the values $-1$, 0 or 1 and
$L=Na/2$ is the length of the system, where $a$ is the lattice parameter for
$\delta\neq 0$ (two times the NN distance) and $N$ is even.

For fermionic systems it has been shown that changes in $\alpha(1,1)$ [45, 46,
47] (or more appropriately $\alpha(l,l)$ with suitably chosen $l$[48, 19]) are
proportional to the average displacement of the particles and, in the
thermodynamic limit, coincide with the charge Berry phase (the phase obtained
changing the twisted boundary conditions from 0 to $2\pi$), which in turn
gives information on the polarization of the system (changes in polarization
are proportional to the corresponding changes in the charge Berry phase) [49,
47]. Similarly, $\alpha(1,-1)$ is an approximation to the spin Berry phase
[50] which is proportional to the difference of polarizations between
electrons with spin up and down [51]. Performing the Jordan-Wigner
transformation mentioned in Sec. II, it is easy to show that
$\alpha_{\uparrow}$ is related to the charge Berry phase of the equivalent
spinless fermionic model. In recent years, similar expectation values been
used in different cases [34, 52, 35, 53, 54, 55, 56, 21].

Figure 1: (Color online) Schematic representation of simple states and the
corresponding values of $\alpha_{\uparrow}$ and $\alpha_{s}$. The grey circles
represent the sites, arrows represent the spin degree of freedom and dashed
ellipses indicate dimerization between sites.

To gain some intuition we discuss $\alpha_{\uparrow}$ in simple cases which
are schematically represented in Fig. 1. Assume for the moment that $N$ is
multiple of four and that the origin of coordinates is chosen so that
$x_{j}=ja/2$. Assume that the ground state is a pure Néel one with spins up
occupying the odd sites (represented in the second row of Fig. 1). Then the
factor entering in the exponent of Eq. (4) becomes
$(2\pi/N)\Sigma_{j}(2j+1)=\pi N/2\equiv 0\text{ mod }2\pi$ and
$\alpha_{\uparrow}=0$. The anti-Néel state is obtained shifting all spins half
a lattice parameter ($a/2$) to the left or to the right (fourth row of Fig.
1). This amounts to replacing $2j+1$ by $2j$ or $2j+2$ above and in both cases
$\alpha_{\uparrow}=\pi$. More generally, it is easy to realize that the change
in $\alpha_{\uparrow}$ under a displacement $d$ of all spins up is
$\Delta\alpha_{\uparrow}=(d/a)2\pi$. Therefore for a completely dimerized
state in which the spins up are on average at the center of the odd bonds
(third row of Fig. 1), displaced $a/4$ with respect to the Néel state, one
expects $\alpha_{\uparrow}=\pi/2$. For the other dimer state (fifth row of
Fig. 1) one expects $\alpha_{\uparrow}=3\pi/2\equiv-\pi/2$. Explicit
calculations done in Ref. [19] confirm these values.

Extending this argument to $\alpha_{s}$ is straightforward and one obtains
$\alpha_{s}=0$ for both extremely dimerized states and $\alpha_{s}=\pi$ for
both pure Néel states. Shifting the origin of coordinates or choosing $N$ not
multiple of 4 does not affect the differences between the $\alpha$ for
different states [19].

For a ring, if $\delta=0$, $H$ is invariant under the reflection $R$ that
passes through two opposite sites. In addition, for any value of $\delta$,
$H_{0}$ is invariant under the reflection $\tilde{R}$ that passes through the
middle point of any two opposite NN bonds. This has important consequences on
the $\alpha$. The reflection $\tilde{R}$ is equivalent to a change of sign of
$x_{j}$ plus a translation in a NN distance $a/2$. This change of sign is
equivalent to a complex conjugation in Eq. (4), while the translation, as
explained above, modifies the exponent of $U(1,0)$ ($U(1,-1)$) by $\pi$
($2\pi\equiv 0$). This implies

$\displaystyle\tilde{R}U(1,0)\tilde{R}^{\dagger}$ $\displaystyle=$
$\displaystyle-\overline{U}(1,0),$
$\displaystyle\tilde{R}U(1,-1)\tilde{R}^{\dagger}$ $\displaystyle=$
$\displaystyle\overline{U}(1,-1).$ (5)

Since the ground state for any finite ring is unique, both $\alpha$ should
remain invariant under inversion and this implies that $\alpha_{\uparrow}$ can
only by either $\pi/2$ or $-\pi/2$ mod $2\pi$, while $\alpha_{s}$ can only be
0 or $\pi$ mod $2\pi$. Thus both $\alpha$ become $Z_{2}$ topological numbers.

As explained above, the two extremely dimerized phases are characterized by
$\alpha_{\uparrow}=\pm\pi/2$ with the sign depending if the singlets occupy
odd or even NN bonds. However, by continuity, for each dimer phase, the same
topological number characterizes the whole dimer phase until a phase
transition or level crossing takes place. In fact, for $|J_{z}|<J_{\perp}$ it
jumps at $\delta=0$ signaling the transition between both dimer phases. This
jump is related to the jump in the winding number and charge Berry phase (Zak
phase for $J_{z}=0$) [12, 27] that takes place in the fermionic Su-Schrieffer-
Hegger model obtained using a Jordan-Wigner transformation[39].

In contrast, $\alpha_{\uparrow}$ is unable to detect the transition to the
Néel phase. As explained with the simple arguments above, the extreme Néel and
anti-Néel states have $\alpha_{\uparrow}=$ 0 or $\pi$. For a finite system the
unique ground state contains a measure of both states and
$\alpha_{\uparrow}=\pm\pi/2$ as required by the topological protection by the
reflection symmetry. Instead, as also discussed above, $\alpha_{s}=0$ for both
extreme dimer phases and $\alpha_{s}=\pi$ for both pure Néel states. By
continuity, the same values extend to the whole dimer and Néel phases and
therefore the jump in $\alpha_{s}$ denotes the dimer-Néel phase transition. In
fact, in the thermodynamic limit, this jump coincides with the jump in the
spin Berry phase [50]. In turn, the latter jump corresponds to a crossing of
excited energy levels used by the MCEL to determine the transition [37, 38].

The MCEL coincides with jumps of the Berry phases also in other models [57].
Furthermore, it has been shown that for a general extended Hubbard model
including NN repulsion, density-dependent hopping, pair hopping and exchange,
in a wide range of parameters with weak interactions, for which continuum-
limit field theory techniques are expected to be quantitatively reliable, the
bosonization results for the phase diagram coincide with those obtained
numerically from jumps in charge and spin Berry phases [58].

The key point for what follows is that $\alpha_{\uparrow}$ and $\alpha_{s}$
complement each other and characterize the three different topological phases
of $H_{0}$. In Sec. IV we determine the phase diagram of $H_{0}$ using these
topological indicators.

When a staggered magnetic field $B$ is included, the reflection symmetry
$\tilde{R}$ is broken and the $\alpha$ lose their topological protection for
$\delta\neq 0$, but they continue to give information on the change in the
position of the spins under changes in the parameters. In particular, a
contour plot of them in a two-dimensional parameter space, allows us to
predict these changes, under an adiabatic cycle. This is shown in Sec. V.

For $\delta=0$, the reflection $R$ is a symmetry of the Hamiltonian for any
$B$, and an argument similar but simpler to that used to analyze the
consequences of $\tilde{R}$, shows that both $\alpha_{\uparrow}$ and
$\alpha_{s}$ can only be either 0 or $\pi$ mod $2\pi$.

At this point, we explain why we use periodic boundary conditions (PBC). When
open boundary conditions (OBC) are used, in general both reflection symmetries
$R$ and $\tilde{R}$ are lost. If either $\delta$ or $B$ are different from
zero, the unit cell contains two sites and in order to have an integer number
of unit cells, the number of sites $N$ should be odd. Therefore the only
possible reflection symmetry is $R$ through the site $(N+1)/2$ at the middle
of the chain. However the exchange interactions are different at the left and
the right of this point. Therefore, the topological protection is lost. One
might expect that for a long enough chain similar results are obtained.
However for the charge transition in the ionic Hubbard model, results for the
charge gap near the charge transition with $N\sim 30$ using PBC seem superior
to $N\sim 400$ using OBC [18]. Also, as discussed at the beginning of the next
Section, for our system $\alpha_{s}$ becomes smooth and does not jump for
$N\sim 60$ using OBC [27].

While the information above is enough to understand the results presented
below, in the rest of this Section we discuss more technical points, in
particular in relation with continuum-limit field theory. Using bosonization
techniques, it has been shown that the operator
$U(m_{\uparrow},m_{\downarrow})$ takes the form

$U=\exp\left[-ic\phi_{a}\right]\text{,~{}~{}~{} }\phi_{a}=\frac{1}{L}\int
dx\phi(x),$ (6)

where $\phi(x)$ is a charge or spin field, $\phi_{a}$ its average and $c$,
like $\phi(x)$, depends on the particular $m_{\sigma}$ [59, 60]. A similar
expression that differs in an irrelevant constant was derived more recently
[35].

The scaling of the expectation value of $U$ in gapless phases of the XXZ chain
[34, 35] and ’t Hooft anomalies [36] of the model have been studied using
conformal field theory and renormalization group (RG). In addition, similar
models were studied with RG [32, 33].

##  IV Phase diagram of $H_{0}$

The phase diagram of $H_{0}$ has been calculated by Tzeng et al. using Rényi
entropies and the second derivative of the ground state energy obtained with
density matrix renormalization group (DMRG) in systems with up to $N\sim 120$
[39]. Recently a calculation of the phase diagram also using DMRG with $N\sim
60$ has been reported [27].

From these works, as well as the jump of $\alpha_{\uparrow}$, it is clear that
the boundary between both dimer phases for $|J_{z}|\leq J_{\perp}$ is at
$\delta=0$. In Ref. [27], a calculation of the Zak Berry phase $\varphi$
(which in principle should give the same information as $\alpha_{\uparrow}$
except for a constant [19, 21]) is reported. Instead of a jump in $\pi$ at
$\delta=0$, $\varphi$ shows a continuous evolution with a total change of
$\pi$ as $\delta$ is varied. This is likely due to the use of open boundary
conditions in Ref. [27], under which the protecting symmetry $\tilde{R}$ is
lost [see Eq. (5)].

For $J_{z}>J_{\perp}$, the Néel phase appears at $\delta=0$ and the boundary
between this phase and any of the dimer phases is not trivial to determine. In
particular, there is a quantitative discrepancy between Refs. [39] and [27]
for small $J_{z}-J_{\perp}$. The extent of the Néel phase is larger in the
latter.

Here we use $\alpha_{s}$ as a topological indicator to detect the transition,
extrapolating the value of $\delta$ at which $\alpha_{s}$ jumps in rings of
$N$ sites with even $N$ in the range $12\leq N\leq 24$ using exact
diagonalization. For $J_{z}<1.5J_{\perp}$ we also include $N=26$. The
extrapolation was done using a quadratic function in $1/N$, as in Ref. [18].
To estimate the error, we repeated the extrapolation taking out the point with
largest $N$. The difference between both values of $\delta$ is always less
than $1.5\times 10^{-3}$ and, in general, less than $5\times 10^{-4}$ for
$J_{z}/J_{\perp}\geq 2$.

Figure 2: (Color online) Phase diagram of $H_{0}$ obtained from the jump
$\alpha_{s}$ (blue triangles) and compared with previous results (red circles)
[39]. Dimer I (II) corresponds to larger
$\langle\mathbf{S}_{j}\cdot\mathbf{S}_{j+1}\rangle$ in the odd (even) bonds.

The resulting boundaries of the Néel phase are shown in Fig. 2, and compared
with previous ones obtained using finite-size scaling of the Rényi entropies
$S_{2}$ and the second derivative of the ground state energy, obtained with
DMRG in systems with up to $N\sim 120$ sites [39]. The agreement is
noticeable. This shows the power of our method which is computationally much
less expensive. However, we believe that the results so far are not accurate
enough in the region near $J_{z}/J_{\perp}=1$.

A characteristic of the phase diagram is that the region of the Néel phase is
very narrow for small $J_{z}/J_{\perp}-1$. This is expected, since at the
point $J_{z}/J_{\perp}-1=\delta=0$, the effect of a small increase in $J_{z}$
opens a gap exponentially [31], while an increase in $\delta$ opens a gap
proportional to $\delta^{2/3}$ [44]. However, the detailed behavior of the
boundary in this zone is difficult to predict. Our value for the transition at
$J_{z}/J_{\perp}=1.2$, $|\delta|\sim 5\times 10^{-4}$ is less than the above
estimated error $1.5\times 10^{-3}$. Instead, the value that we obtain for
$J_{z}/J_{\perp}=1.4$, $|\delta|=8.17\times 10^{-3}$ (slightly smaller than
that of Tzeng et al.) is more reliable.

Figure 3: (Color online) Critical value of $J_{z}/J_{\perp}-1$ at the Néel-
dimer transition as a function of $\delta$ (black triangles) and compared with
previous results (red circles) [39].

In Fig. 3 we display the results of Fig. 2 for $J_{z}/J_{\perp}\geq 1.4$ and
$\delta>0$ using a log10 scale for both axis. Our results and those of Tzeng
et al. [39] suggest that approximately the boundary for $1.4\leq
J_{z}/J_{\perp}\leq 2.4$ ($0.0082\leq\delta\leq 0.124$) corresponds to
$J_{z}/J_{\perp}-1\sim\delta^{0.46}$. Instead, the results of Ref. [27]
suggest $J_{z}/J_{\perp}-1\sim\delta$.

##  V Contour plots of the phases

Figure 4: (Color online) $\alpha_{s}$ (top) and $\alpha_{\uparrow}$ (bottom)
as a function of dimerization parameter $\delta$ and staggered magnetic field
for $J_{z}/J_{\perp}=2$. C1 and C2 curves represent the pumping cycles used.

In Fig. 4 we show a contour plot of both $\alpha$ for $J_{z}>J_{\perp}$. The
sharp transitions between the Néel and the dimer phases are evident in
$\alpha_{s}$ but imperceptible in $\alpha_{\uparrow}$. To understand this
behavior, let us analyze the change in both position indicators as the system
changes slowly from the point $(\delta,B/J_{\perp})=(-1,0)$ to
$(0,B/J_{\perp})$ with small positive $B$. The first point corresponds to a
perfectly dimerized phase with singlets at the odd bonds. As discussed in Sec.
III, this corresponds to $\alpha_{\uparrow}=\pi/2$. Since the down spins in
this phase are localized at the same places as the up spins, their difference
vanishes and therefore $\alpha_{s}=0$. As $\delta$ increases beyond the
transition to the Néel phase, the ground state is a mixture of the Néel and
anti-Néel states for small $B$. As a consequence, the spin up continues to be
mostly centered at the odd bonds. Near the critical value of $\delta$, only
for large positive $B$ the up spins move to the right (increasing $x_{j}$)
occupying the even sites forming an anti-Néel state with
$\alpha_{\uparrow}=\pi$. Instead, $\alpha_{s}$ jumps abruptly at the
transition for $B=0$, since the difference between positions of up and down
electrons moves from 0 to the NN distance as the system enters the Néel phase.
When $\delta=0$ is reached even for small positive $B$ the anti-Néel phase is
favored with $\alpha_{\uparrow}=\alpha_{s}=\pi$.

Summarizing, in the displacement from $(\delta,B/J_{\perp})=(-1,0)$ to
$(0,0^{+})$, one starts with $\alpha_{\uparrow}=\pi/2$ and $\alpha_{s}=0$
reflecting the fact that both spins are centered at the even bonds, and at the
end $\alpha_{\uparrow}$ ($\alpha_{s}$) has increased by $\pi/2$ ($\pi$) as a
consequence of the displacement of the spins up (down) by a quarter of a unit
cell to the right (left).

One can continue the evolution to a point $(0.6,0)$ for example to the other
dimerized phase, implying a displacement of the spins another quarter of a
unit cell in the same direction, reaching the values
$\alpha_{\uparrow}=3\pi/2$ and $\alpha_{s}=2\pi\equiv 0$ mod $2\pi$. Finally,
one can close the circuit with any path with $B<0$ with a total change of
$2\pi$ in $\alpha_{\uparrow}$, which means a quantized transport of a spin up
to the next unit cell to the right, and a total change of $\alpha_{s}$ in
$4\pi$ indicating that the spins down moved in the opposite direction. In
fact, all closed circuits that enclose once the critical segment between
critical points $(-\delta_{c},0)$ and $(\delta_{c},0)$, without crossing it,
in the same direction, are topologically equivalent and lead to the same spin
transport in an adiabatic time cycle. This is also the case for a circuit
studied recently in which this critical segment is touched but not crossed
[27].

As examples, in Fig. 5 we show the change in $\alpha_{s}$ and
$\alpha_{\uparrow}$ for pumping circuits defined by

$\begin{split}\delta(\theta)&=-0.9\sin{(\theta)}\\\
B(\theta)&=-B_{0}\cos{(\theta)},\end{split}$ (7)

as $\theta$ is increased adiabatically from 0 to $2\pi$ closing the circuit.
We have chosen $B_{0}/J_{\perp}=0.9$ (0.02) for the circuits C1 (C2) indicated
in Fig. 4.

Figure 5: (Color online) $\alpha_{\uparrow}$ and $\alpha_{s}$ as a function of
the adiabatic parameter $\theta$ through the pumping circuits defined by Eqs.
(7) for $J_{z}/J_{\perp}=2$. Dashed (full) lines correspond to C1 (C2).

For the circuit C1, the indicators change smoothly passing through
topologically protected points when $\theta$ is multiple of $\pi/2$.
Specifically for $\theta=0$, the system is at the point
$(\delta,B/J_{\perp})=(0,-0.9)$, where the ground state is the Néel state plus
fluctuations, with $\alpha_{\uparrow}=0$ and $\alpha_{s}=\pi$, as described in
Sec. III. Increasing successively $\theta$ in $\pi/2$, the changes in
thermodynamic phase and the values of $\alpha_{\uparrow}$ correspond to those
represented in Fig. 1 jumping to the next row: odd dimers with
$\alpha_{\uparrow}=\pi/2$ at $(-0.9,0)$, anti-Néel with
$\alpha_{\uparrow}=\pi$ at $(0,0.9)$ and even dimers with
$\alpha_{\uparrow}=\pi/2$ at $(0.9,0)$. At the end of the cycle a spin up has
been transported to the next unit cell, corresponding to a total change in
$\alpha_{\uparrow}$ by $2\pi$. The spin down is displaced in the opposite
direction with a total change of $\alpha_{s}$ by $4\pi$.

For the circuit C2, the total spin transported and the values of both $\alpha$
at the topologically protected points are the same as for C1. There are
however, important differences at the intermediate points. The small value of
$B_{0}/J_{\perp}$ implies that moderate values of $|\delta|$ dimerize the
system more easily and this fact is reflected in the values of both $\alpha$.
In addition, there is a large slope in $\alpha_{s}$ at the points where
$\alpha_{s}=\pi/2$ and $\alpha_{s}=3\pi/2$. This is reminiscent of the jump in
$\alpha_{s}$ at the Néel-dimer transition for $B=0$ between $\alpha_{s}=\pi$
and $\alpha_{s}=2\pi\equiv 0$ mod $2\pi$.

For $J_{z}<J_{\perp}$, the contour plots are similar, but the segment between
critical points collapses to the origin. An example is shown in Fig. 6 for
$\alpha_{\uparrow}$, which gives more information than $\alpha_{s}$ because
the former is able to distinguish between both dimer phases and between the
Néel and anti-Néel states. Adiabatic pump cycles that enclose the origin have
the same topological properties as the ones discussed above, and again the
contour plot allows to predict the spin transport along a time dependent path.

However, a time-dependent calculation in a circuit that touches but does not
cross the origin has shown recently that quantized spin transport is lost in
this case [27]. We believe that the difference is that for $J_{z}<J_{\perp}$,
the gap is closed at the origin, and pumping with finite velocity necessary
leads to spin excitations losing the adiabatic condition. In fact,
oscillations in the transported spin indicate that this is the case. In
contrast in the Néel phase, the Néel and anti-Néel states are separated from
the rest by a finite gap [31]. One might wonder if the quasi degeneracy
between these two states (degeneracy in the thermodynamic limit) could also
affect quantized pumping. However, the mixture between both states decreases
exponentially with system size, and one can expect that even crossing the
segment between critical points $(-\delta_{c},0)$ and $(\delta_{c},0)$ by a
small amount does not spoil the quantized pumping. Time-dependent calculations
in the interacting Rice-Mele model also obtained that the addition of an Ising
term ($J_{z}$) to the model stabilizes pumping (of charge in this case)
despite the quasi degeneracy of the ground state [16].

Figure 6: (Color online) $\alpha_{\uparrow}$ as a function of dimerization
parameter $\delta$ and staggered magnetic field for $J_{z}=0$.

##  VI Summary and discussion

We have used two position indicators $\alpha_{\uparrow}$ and $\alpha_{s}$, to
study the XXZ ring with interactions with alternation proportional to $\delta$
and staggered magnetic field $B$. The value of $\alpha_{\uparrow}$ provides
information on the shift in the position of the spins $\uparrow$ under changes
in the parameters, while $\alpha_{s}$ accounts for the difference in
displacements between both spins.

For either $\delta=0$ or $B=0$, both indicators become topological $Z_{2}$
numbers protected by a reflection symmetry. For $B=0$ we have used
$\alpha_{s}$ to calculate the phase diagram of the system, obtaining results
that agree quantitatively with those of one of the groups which calculated the
diagram before [39], but with a smaller computational cost. The topological
indicators $\alpha_{\uparrow}$ and $\alpha_{s}$ provide complementary
information on the thermodynamic phases of the model.

We have calculated contour plots of both $\alpha_{\uparrow}$ and $\alpha_{s}$
in the $(\delta,B/J_{\perp})$ plane, which permit to predict the evolution of
the spin transport in adiabatic Thouless pump cycles. The transported spin is
quantized in the cycle and only the intermediate values depend on the specific
cycle for topologically equivalent cycles. Again $\alpha_{\uparrow}$ and
$\alpha_{s}$ provide complementary information. Since experimentally the
occupation numbers in different sites are measured simultaneously [22], it is
in principle possible to observe differences in the behavior of
$\alpha_{\uparrow}$ and $\alpha_{s}$ for pumping circuits that pass near
critical points and for small number of particles, for which the entanglement
is larger.

## Acknowledgments

We thank Daniel Cabra for helpful discussions. AAA and KH acknowledges
financial support provided by PICT 2020A 03661 and PICT 2018-01546 of the
ANPCyT, Argentina.

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