identification of $\chi>0.3$ as the region for high temperature symmetry
restoration. This does not include the $\chi=0$ limit which returns SM-like
couplings locally; the Standard Model itself presents high temperature
symmetry restoration. The feature then arises in this non-linear theory that
the non-local phenomenology of the SM is not recovered in the local SM
coupling limit. A consequence of this is that the region for high temperature
symmetry restoration in our non-linear theory lies a finite distance away in
parameter space from the SM couplings limit and as such provides a target for
collider experiments. This is illustrated in Fig. 24 where said region for
small $\delta,\epsilon$ is depicted and is partially ruled out by LHC bounds
and would explored in full by future experiments such as the FCC.

Figure 24:  The curvature plane $v^{2}R_{\varphi}(0)$ and $v^{2}R_{h}(0)$ as
defined in Eq. (8.1). In colour is the region for high-temperature symmetry
restoration (see Sec. 4.3), i.e. $\pi/2>\chi>0.3$. We vary $\gamma_{a}^{2}$
independently to account for varying $\epsilon,\delta,\gamma_{4}$. We note for
interest but do not show on the plot for clarity that increasing
$\gamma_{a}^{2}$ decreases the curvatures radially originating from
$(v^{2}R_{h}(0),v^{2}R_{\varphi}(0))=(0.0,1.0)$. The area outside the blue
dashed box is excluded by LHC bounds from Eq. (8.2). The FCC would be expected
to be sensitive, assuming SM-like couplings, up to the small, inner box [81,
82].

  * •

Domain walls. The wall formation scenario required $\epsilon,\delta<10^{-15}$,
far smaller than collider limits can hope to compete with. LHC constraints
therefore will have a relevant impact only in the remaining parameters of the
theory, namely $\gamma_{4},\chi$. In the case that $\chi>0.3$ the theory
exhibits symmetry restoration and produces domain walls. As Fig. 15 shows,
cosmological bounds alone (primarily in the form of wall annihilation before
BBN) allow for potentially detectable GW signals at SKA. To produce a signal
visible at SKA, one requires small $\gamma_{4}<0.1$ given the parametric
dependence in Eq. (7.5). The triple gauge coupling constraint allows very
small and even vanishing $\gamma_{4}$. However, the relevant
$\epsilon,\delta\to 0$ limit implies $\gamma_{a}=\gamma_{4}$, and curvature
bounds of Eq. (8.2) apply. These constrain $\gamma_{4}$ to a neighbourhood
around $1/c_{\chi}^{2}$, so that there is a minimum attainable $\gamma_{4}$
well above 0.1. Plugging this value back into eq (7.5) gives a maximum peak of
the GW power spectrum resulting from domain walls:

$\displaystyle\Omega_{\rm g.w.|peak}^{max}$ $\displaystyle=2.2\times
10^{-17}\left(\frac{10^{-10}\textrm{ Hz}}{f_{peak}}\right)^{4}$ $\displaystyle
f_{\textrm{peak}}$ $\displaystyle>10^{-10}\textrm{ Hz}\,.$ (8.4)

Figure 25(a), shows the impact of LHC bounds and Fig. 25(b) shows the maximal
prediction for the peak spectrum in this non-linear theory versus SKA
sensitivity, see Eq. (8.4). From this figure, it is clear that the parameter
region which sources a domain wall GW signal that SKA is sensitive to is
already ruled out by the LHC while the maximum attainable signal in spectrum
lies five orders of magnitude below projected sensitivity.

(a)

(b)

Figure 25: (a) The ($\epsilon-\gamma_{4}$) plane delineating the wall
formation region in parameter space as in Fig. 15 with the addition of LHC
bounds, Eq. (8.2), in the vertical gray shaded region. (b) Curve for the value
of the spectrum at the peak frequency for the allowed region of parameter
space taking into account LHC bounds and the sensitivity of SKA [46].

  * •

First order phase transition. For first order phase transitions, one requires
couplings sizeably different from the SM, which would naively give collider
constraints a more prominent role than in the walls case. The main LHC
constraint arises again through the bound on $R_{\varphi}$, which constrains
$c_{\chi}^{2}\gamma_{a}$ to lie close to 1. On the other hand we require
largish negative $\delta$ in this region. Recalling

$\displaystyle\delta=\gamma_{4}^{-1}\gamma_{\epsilon}-\gamma_{a}^{-1}$ (8.5)

together with $1<\gamma_{\epsilon}<\sqrt{2}$ for $\epsilon>0$ means $\delta<0$
would require
$\gamma_{4}>\gamma_{\epsilon}\gamma_{a}\simeq\gamma_{\epsilon}/c_{\chi}^{2}>1$.
The effect of greater than one $\gamma_{4}$ is illustrated in Fig. 8; as
$\gamma_{4}$ increases, the extrema of the $T=0$ potential is pushed closer
together, which also facilitates nucleation. We plot $\gamma_{4}=1.6$ in Fig.
26 to provide an illustrative example of a region in parameter space allowed
by current LHC bounds that also predicts a gravitational wave signal
detectable at upcoming experiments LISA and BBO, for the region marked in
orange and cyan respectively. Note that, since $\gamma_{4}\neq 1,\chi\neq 0$,
in the $(\epsilon,\delta)$ plane of Fig. 26 the LHC-allowed region does not go
through the origin.

(a)

(b)

Figure 26: Summary of first order phase transition parameter space for
$\gamma_{4}=1.6$, $\chi^{2}=0.1$. The region in blue shows the combinations of
$(\epsilon,\delta)$ that we found to admit bubble nucleation while tunnelling
from the false vacuum to the true vacuum today. The region in red (dark red)
is excluded by the strong (weak) IR constraints. The pink (dark pink) regions
are excluded by the boundedness from below (perturbativity) constraint. The
light green (green) dashed lines show the LHC constraints on the curvature to
the $1\sigma$ ($2\sigma)$ level. The gray region is unphysical, yielding the
wrong vacuum today. The regions of first order phase transitions that give a
GW signal detectable at LISA (BBO) are shown in orange (cyan). Only a small
sliver of parameter space gives signals observable only at BBO, more visible
in Fig. 26(b).

Figure 27:  Top: Summary of the parameter space that leads to future vacuum
decay (the doom scenario). Combination of Fig. 20 with LHC bounds in light
green ($1\sigma$) and darker green ($2\sigma$). Bottom: Series of plots with
the combination of LHC exlusion regions (delineated with dashed lines) and
perturbativity to illustrate that values of $\gamma_{4}<0.85$ are ruled out by
a combination of LHC and perturbativity bounds, and therefore discarded in our
analysis.

  * •

Doom. Lastly the quadrant $\epsilon<0$, $\delta>0$ can lead to $h_{0}$ as the
single high temperature vacuum while the true vacuum today, $h_{-}$, having
appeared at a distance in field space large enough so that we are currently
trapped in a false vacuum. The requirement of $\delta>0$ in conjunction with
LHC bounds with the same reasoning as the one outlined around Eq. (8.5) but
now in the opposite direction implies $\gamma_{4}<1$. This limit decreases the
Higgs quartic coupling at tree level, increasing the relative contribution of
loop effects so that a stronger perturbativity and boundedness from below
constraint applies now. This is illustrated in the lower panel of Fig. 27
where one can see that the combination of LHC and perturbativity bounds
excludes $\gamma_{4}<0.8$. The reduced range for $\gamma_{4}$ still allows for
a region, partially shown in Fig. 27 upper panel, where the IR constraint, the
LHC and perturbativity constraints are satisfied and we would be currently
trapped in a false vacuum. This doom possibility has been found to lead to a
lifetime for our universe in excess of $10^{2000}$ times the lifetime of the
universe, see Fig. 21.

##  9 Summary

Today the question of whether electroweak symmetry is realised in Nature
linearly or non-linearly is within experimental reach. The distinction is non-
local in field space and thus calls for non-perturbative phenomena as the
unambiguous probe. Such phenomena do arise in cosmology, and this paper has
focused on studying the cosmological phenomenology of the non-linear
realisation, termed here the _non-linear theory space_ , i.e.
HEFT$\backslash$SMEFT which opens qualitatively new features with respect to
SMEFT.

Our study has revealed non-decoupling deviations in the sphaleron energy, the
possibility of domain wall formation, first order phase transitions at the
electroweak scale and vacuum decay in the very distant future, as well as
symmetry non-restoration at high temperature. These can be used to answer the
EW realisation question because processes such as the formation of electroweak
domain walls are, to the best of our current knowledge, exclusive to non-
linear theories.

There is evidence to support the statement that non-linear theories are non-
decoupling, meaning that no limit can be taken in such theories that returns
the SM only. A non-decoupling feature, albeit localised a finite distance away
from our vacuum, has been identified with the non-linear theory used here,
which has wormhole-like space as the scalar manifold. This feature is to be
found in the flat limit that corresponds to the SM locally; the closing of the
throat of the wormhole produces a localised singularity in the effective
potential that prevents symmetry restoration at high temperature and would
likely lead to light new states. Conversely, away from this local SM limit,
there is a region in parameter space that leads to symmetry restoration and
presents local couplings around the vacuum which are a ‘finite distance’ away
from the SM ones. This minimum size of deviations provides a specific target,
and we find that FCC would meet it and rule out the symmetry restoring region.
Another non-decoupling effect has been found in the sphaleron energy, although
this effect seems difficult to probe. The exploration of its implications for
baryogenesis are hitherto unexplored.

The formation of domain walls requires minima that are approximately
degenerate, but not exactly so. This requirement sets upper and lower
cosmological bounds on $\epsilon$, which is related to the triple Higgs
coupling. This window of $\epsilon$ is within a range too small to be probed
directly by the LHC, although indirect LHC constraints on other parameters of
the theory have implications on cosmology. Among them, there is an upper bound
on the strength of gravitational waves from walls which lies a few orders of
magnitude below the sensitivity of SKA.

The case of a strong first order phase transition is realised in this theory
in a way not possible in SMEFT; the extension of the Higgs field range with
the wormhole topology considered here contains naturally a second minimum and
a barrier such that no large potential corrections are needed.. This is
qualitatively different from 1OPT’s in the SMEFT, which require the addition
of higher dimensional operators to generate a barrier. In order to achieve
first order phase transitions in non-linear theories considered here, as
opposed to walls, we require sizeable energy difference between minima and
hence sizeable deviations from SM couplings. In fact, one has that the region
in parameter space that leads to first order transitions is accessible
experimentally via the LHC. However, the combination of LHC and current
cosmological data does only probe this region partially. It is a promising
possibility in fact that non-linear theories could have produced gravitational
waves detectable at LISA and will give rise to new signals at the HL-LHC.

Lastly, a scenario with vacuum decay in the future is difficult to probe with
cosmological observation; the impending annihilation event has been found to
be at least $10^{2000}$ years in the future. The LHC however can probe the
possibility of vacuum decay through the determination of Higgs couplings.

This paper has explored a small fraction of the possible cosmological
phenomenology of HEFT$\backslash$SMEFT. It is a fraction which nonetheless
included new phenomena not possible in SMEFT and provides the elementary
results to attempt a more comprehensive survey. Such exploration, to be
complete, should include LHC and cosmological data, but also theory
considerations such as perturbativity and the range of validity of an EFT at
finite temperature and its relation to curvature. Indeed non-linear theories
are constrained from all directions and hence, once these lines are drawn,
they will provide specific targets for experiments both on earth and in space,
to answer the question of what type of electroweak symmetry realisation is
present in Nature.

##  Appendix A UV model for non-linear type A theory.

In this appendix we present a model that presents the geometry of a type A
non-linear theory, i.e. a manifold which does not contain a fixed point, not
even a singular one. This model is not meant to be phenomenologically viable
or representative. In particular it is built out of linear representations and
integration of heavy modes which leaves out the tantalizing but hitherto
unrealised possibility of starting from non linear representations.

The model is built with an $SU(2)$ doublet $\phi$ and a singlet $h$, with
vacuum expectation values given by
$\left\langle|\phi|^{2}\right\rangle=v^{2}/2$ and $\left\langle
h\right\rangle=0$. The Lagrangian is

$\mathcal{L}_{\text{UV}}=\frac{1}{2}(\partial_{\mu}h)^{2}+|\partial_{\mu}\phi|^{2}-V_{h}(h)-\frac{\lambda_{\phi}}{2}\left(|\phi|^{2}-\frac{v^{2}}{2}\right)^{2}+v^{2}G(h)|\phi|^{2}\,,$
(A.1)

with arbitrary functions $V_{h}(h)$ and $G(h)$ of the singlet $h$, with the
minimum for $V_{h}(h)$ at $h=0$ and $G(0)=0$, without loss of generality. We
employ the usual parametrization of the doublet

$\phi=\frac{1}{\sqrt{2}}\;U\begin{pmatrix}0\\\ v+H\end{pmatrix}$ (A.2)

with a radial coordinate $H$ such that $\left\langle H\right\rangle=0$. The
Lagrangian becomes

$\displaystyle\mathcal{L}_{\text{UV}}$
$\displaystyle=\frac{1}{2}(\partial_{\mu}h)^{2}+\frac{1}{2}(\partial_{\mu}H)^{2}+\frac{1}{2}(v+H)^{2}\,\operatorname{Tr}\left[\partial^{\mu}U^{\dagger}\partial^{\mu}U^{\dagger}\right]$
(A.3)
$\displaystyle\phantom{=}-V_{h}(h)-V_{H}(H)+\frac{v^{2}}{2}G(h)(v+H)^{2}.$
(A.4)

The $H$ scalar gets a mass $m_{H}^{2}=\lambda_{\phi}v^{2}$. Assuming a
perturbative $\lambda_{\phi}\lesssim(4\pi)^{2}$, we have
$m_{H}^{2}\lesssim(4\pi)^{2}v^{2}$. Here, we assume that
$m_{H}^{2}\simeq(4\pi)^{2}v^{2}\gg v^{2}$ and integrate out $H$ at tree level.
This can be done by plugging the solution to the equation of motion for $H$,

$H=\frac{1}{m_{H}^{2}}\Big{\\{}v\operatorname{Tr}\left[\partial^{\mu}U^{\dagger}\partial^{\mu}U^{\dagger}\right]+v^{3}G(h)\Big{\\}}+O\left(\frac{1}{m_{H}^{4}}\right),$
(A.5)

into the UV Lagrangian, which gives:

$\displaystyle\mathcal{L}_{\text{eff}}$
$\displaystyle=\frac{1}{2}(\partial_{\mu}h)^{2}+\frac{v^{2}}{2}\operatorname{Tr}\left[\partial^{\mu}U^{\dagger}\partial^{\mu}U^{\dagger}\right]-V_{h}(h)+\frac{1}{2m_{H}^{2}}\Big{\\{}v\operatorname{Tr}\left[\partial^{\mu}U^{\dagger}\partial^{\mu}U^{\dagger}\right]+v^{3}G(h)\Big{\\}}^{2},$
(A.6)

neglecting $O(1/m_{H}^{4})$ terms. In particular, we have

$F(h)^{2}=1+\frac{v^{2}}{m_{H}^{2}}G(h),\qquad
V(h)=V_{h}(h)-\frac{v^{6}}{2m_{H}^{2}}G(h)^{2}.$ (A.7)

This means that non-linear theories with any $F(h)$ can be achieved with this
model. Both type A and B theories can be obtained with an appropriate choice
of $G(h)$. In particular, a renormalizable UV completion with

$\displaystyle G(h)$
$\displaystyle=\frac{c_{\chi}^{2}m_{H}^{2}}{v^{2}}\left(2\frac{h}{v_{\star}}+\frac{h^{2}}{v_{\star}^{2}}\right),$
(A.8) $\displaystyle V_{h}(h)$
$\displaystyle=V(h)+\frac{v^{6}}{m_{H}^{2}}G(h)^{2}$
$\displaystyle=\left(\frac{m_{h}^{2}}{2}+\frac{2c_{\chi}^{4}v^{6}}{m_{H}^{2}v_{\star}^{2}}\right)h^{2}+\left(\frac{m_{h}\sqrt{\lambda}}{2}\gamma_{4}(1-\epsilon)+\frac{2c_{\chi}^{4}v^{6}}{m_{H}^{2}v_{\star}^{3}}\right)h^{3}+\left(\frac{\lambda}{8}\gamma_{4}^{2}+\frac{2c_{\chi}^{4}v^{6}}{2m_{H}^{2}v_{\star}^{4}}\right)h^{4},$
(A.9)

gives the $F(h)$ and $V(h)$ functions we have used in this work, although
additional terms of the form

$\mathcal{L}_{\text{eff}}\supset\frac{v^{2}}{2m_{H}^{2}}\left\\{\operatorname{Tr}\left[\partial^{\mu}U^{\dagger}\partial^{\mu}U^{\dagger}\right]\right\\}^{2}+O\left(\frac{1}{m_{H}^{4}}\right),$
(A.10)

which we have not considered, are present in the Lagrangian. Corrections to
this Lagrangian of order $m_{H}^{-4}$ can be computed systematically with a
functional approach as outlined in [4].

## Acknowledgments

The authors would like to thank Djuna Croon, Philipp Schicho and Dave
Sutherland for helpful comments. R. A., R. H. and M. W. are supported by the
STFC under Grant No. ST/T001011/1. J.C.C. is supported by grant
RYC2021-030842-I funded by
MCIN/AEI/10.13039/501100011033 and by the European Union
NextGenerationEU/PRTR, and grant PID2022-139466NB-C22 funded by
MCIN/AEI/10.13039/501100011033 and by ERDF A way of making Europe.

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# Entanglement and Subsystem Particle Numbers in Free Fermion Systems

Y. F. Zhang1 Huichao Li1 L. Sheng1<EMAIL_ADDRESS>R. Shen1 D. Y. Xing1
1National Laboratory of Solid State Microstructures and Department of Physics,
Nanjing University, Nanjing 210093, China

###### Abstract

We study the relationship between bipartite entanglement and subsystem
particle number in a half-filled free fermion system. It is proposed that the
spin-projected particle numbers can distinguish the quantum spin Hall state
from other states, and be linked to the topological invariant of the system.
It is also shown that the subsystem particle number fluctuation displays
behavior very similar to the entanglement entropy. It provides a lower-bound
estimation for the entanglement entropy, which can be measured experimentally.

###### pacs:

73.22.Pr, 03.65.Vf, 03.65.Ud, 65.40.gd, 05.40.Ca

##  I Introduction

Topological phases of matter are usually distinguished by using some global
topological properties, such as topological invariants and topologically
protected gapless edge modes, rather than certain local order parameters. The
integer quantum Hall effect iqhe , fractional quantum Hall effect fqhe , and
band Chern insulators ci  can be characterized by Chern numbers or Berry
phases tknn . The quantum spin Hall (QSH) effect qshe1 ; qshe2  and the three-
dimensional topological insulators 3d1 ; 3d2  are characterized by the $Z_{2}$
invariant z2  or spin Chern number spinch1 ; spinch2 . Recently, quantum
entanglement entangle1 , which reveals the phase information of the quantum-
mechanical ground-state wavefunction, has been used as a tool to characterize
the topological phases. As shown by Levin and Wen wen  and also by Kitaev and
Preskill kitaev , the existence of topological entanglement entropy in a fully
gapped system, such as fractional quantum Hall and the gapped integer spin
systems teefqh ; teeqsl , indicates existence of long-range quantum
entanglement (topological order toporder  in equivalent parlance). Another
important progress is the demonstration that the entanglement spectrum (ES)
es1  reveals the gapless edge spectrum for fractional quantum Hall systems,
Chern insulators, and topological insulators es1 ; es2 ; es3 .

Supposing $A$ and $B$ to be two blocks of a large system in a pure quantum
state, the reduced density matrix (RDM) $\rho_{A}$ can be obtained by tracing
over degrees of freedom of $B$. Then the Von Neumann entanglement entropy (EE)
can be computed

$S_{ent}=-\mbox{tr}(\rho_{A}\ln\rho_{A})=-\mbox{tr}(\rho_{B}\ln\rho_{B})\ .$
(1)

It has been shown that for bipartite subsystems $A$ and $B$ with a smooth
boundary, $S_{ent}$ has the form of $S_{ent}=\alpha L-S_{top}$, where $L$ is
the length of the boundary, $\alpha$ is a non universal coefficient, and
$-S_{top}$ is a universal constant called the _topological entanglement
entropy_ wen ; kitaev . Moreover, if we write the RDM in the form of
$\rho_{A}=\exp(-H_{ent})/Z$, where $Z$ is a normalization constant, and
$H_{ent}$ is known as the _entanglement Hamiltonian_ , the eigenvalue spectrum
$\\{\varepsilon_{i}\\}$ of $H_{ent}$ is called the ES, which stores more
information about the quantum entanglement than the EE es1 .

(a) cylinder geometry

(b) torus geometry

Figure 1: (Color online) Schematic view of a cylinder and a torus. The
entanglement cuts divide the system into two equal parts $A$ and $B$. For the
cylinder geometry, the entanglement cut leads to one interface (a); and for
the torus geometry, the cuts lead to two interfaces (b).

In this paper, we study the relationship between bipartite entanglement and
subsystem particle number in half-filled free fermion systems. It was proposed
in Ref. trace , for systems with translational invariance in one dimension,
the discontinuity in the subsystem particle number as a function of the
conserved momentum indicates whether or not the ES has a spectral flow, which
is determined by the topological invariant of the system es3 . Nevertheless,
this approach has an exceptional case for a half-filled QSH system with two-
dimensional inversion symmetry. To overcome the inadequacy, we define spin-
projected particle numbers, based on which spin trace indices can be well
defined, for the QSH system with or without $s_{z}$ conservation. Spin trace
indices are univocally related to the topological invariant of QSH system,
i.e., the $Z_{2}$ index. We further investigate the relationship between the
EE and subsystem particle number fluctuation. The latter is also dominated by
the boundary excitations of the system, and satisfies a similar area law as
the EE. It gives a lower-bound estimation of the EE, and can be measured
experimentally.

In the next section, we introduce the model Hamiltonian, and explain the
procedure to calculate the ES and EE. In Sec. III, numerical calculation of
the ES is carried out, and the connection between the subsystem spin-projected
particle numbers and the topological invariants in different phases is
established. In Sec. IV, the relationship between the EE and subsystem
particle number fluctuation is discussed. The final section is a summary.

##  II  MODEL HAMILTONIAN

We begin with the tight-binding model Hamiltonian for the QSH system
introduced by Kane and Mele qshe1 ; z2 , plus an additional exchange field
yang

$\begin{split}H&=-\sum\limits_{\langle\bm{i},\bm{j}\rangle}c_{\bm{i}}^{\dagger}c_{\bm{j}}+iv_{so}\sum\limits_{\ll\bm{i},\bm{j}\gg}c_{\bm{i}}^{\dagger}\sigma_{z}v_{ij}c_{\bm{j}}\\\
&+iv_{r}\sum\limits_{\langle\bm{i},\bm{j}\rangle}c_{\bm{i}}^{\dagger}(\bm{\sigma}\times\bm{d}_{\bm{i}\bm{j}})_{z}c_{\bm{j}}+\sum_{\bm{i}}m_{i}c_{\bm{i}}^{\dagger}c_{\bm{i}}+g\sum\limits_{\bm{i}}c_{\bm{i}}^{\dagger}\sigma_{z}c_{\bm{i}}\
.\end{split}$ (2)

Here, the first term is the usual nearest neighbor hopping term with
$c_{\bm{i}}^{\dagger}=({c^{\dagger}_{\bm{i},\uparrow}},{c^{\dagger}_{\bm{i},\downarrow}})$
as the electron creation operator on site $\bm{i}$, where the hopping integral
is set to be unity. The second term is the intrinsic spin-orbit coupling (SOC)
with
$v_{ij}=(\bm{d}_{kj}\times\bm{d}_{ik})_{z}/|(\bm{d}_{kj}\times\bm{d}_{ik})_{z}|$,
where $\bm{k}$ is the common nearest neighbor of $\bm{i}$ and $\bm{j}$, and
vector $\bm{d}_{ik}$ points from $\bm{k}$ to $\bm{i}$. The third term stands
for the Rashba SOC with $\bm{\sigma}$ the Pauli matrix. The fourth term stands
for a staggered sublattice potential $(m_{i}=\pm m)$, and the last term
represents a uniform exchange field with strength $g$.

We consider systems with cylinder or torus boundary conditions, consisting of
$N_{x}$ ($N_{x}$ to be even) zigzag chains along the circumferential direction
($y$ direction). The size of the sample will be denoted as $N=N_{x}\times
N_{y}$ with $N_{y}$ as the number of atomic sites on each chain. We perform
the entanglement cut along the $y$ direction, which results in one or two
interfaces between the two equal parts $A$ and $B$, respectively, for the
cylinder or torus geometry, as shown in Fig. 1. In order to examine the EE and
ES, an Schmidt decomposition on the ground-state wavefunction or calculation
of the RDM is usually needed. For non-interacting fermion systems, however,
the necessary information of the entanglement can also be obtained from the
following two-point correlators dmrg

$c_{\tau_{1},\tau_{2}}(\bm{i},\bm{j})=\langle
c^{\dagger}_{\bm{i},\tau_{1}}{c_{\bm{j},\tau_{2}}}\rangle\ .$ (3)

Here, $\langle\cdot\rangle$ means the ground-state expectation of an operator.
$\tau$ can be an index of spin, pseudospin, or orbital degree of freedom.

Using the Fourier transformation (FT) along the $y$ direction, the Hamiltonian
can be rewritten as
$H=\sum_{k_{y},i,j}c^{\dagger}_{i}(k_{y})h_{i,j}(k_{y})c_{j}(ky)$, where
$c^{\dagger}_{i}(k_{y})=(c^{\dagger}_{i,\uparrow}(k_{y}),c^{\dagger}_{i,\downarrow}(k_{y}))$
are the electron creation operators. After performing the entanglement cut, we
treat part $A$ as the subsystem, and trace out the degrees of freedom of $B$.
It should be noted that any of the correlators
$c_{\tau_{1},\tau_{2}}(\bm{i},\bm{j})$ with $\bm{i}$ and $\bm{j}$ confined in
$A$ is unchanged by the tracing. When carrying out the FT on the correlators,
we can get

$c_{\tau_{1},\tau_{2}}(\bm{i},\bm{j})=\frac{1}{N_{y}}\sum_{k_{y}}e^{ik_{y}\cdot(i_{y}-j_{y})}\langle
c^{\dagger}_{i,\tau_{1}}(k_{y})c_{j,\tau_{2}}(k_{y})\rangle\ ,$ (4)

where $i$ and $j$ discriminate the zigzag chains. We use $\langle
c^{\dagger}_{i,\tau_{1}}(k_{y})c_{j,\tau_{2}}(k_{y})\rangle$ to form a
hermitian matrix ${\cal C}(k_{y})$. Then the entanglement Hamiltonian is given
by dmrg

$H_{ent}=\ln({\cal C}^{-1}-1)\ .$ (5)

The spectrum $\\{\zeta_{i}\\}$ of ${\cal C}$ is related to spectrum
$\\{\varepsilon_{i}\\}$ of $H_{ent}$ by $\zeta_{i}=1/(e^{\varepsilon_{i}}+1)$,
where $\zeta_{i}$ acts as the average fermion number in the entanglement
energy level $\varepsilon_{i}$ at “temperature” $T=1$. By using the spectrum
of ${\cal C}$, the EE at each $k_{y}$ sector is given by
$s_{ent}(k_{y})=\sum_{i}s_{i}$, with

$s_{i}=-\zeta_{i}\ln\zeta_{i}-(1-\zeta_{i})\ln(1-\zeta_{i})\ .$ (6)

Figure 2:  (a-c) Entanglement spectrum in the cylinder geometry (left panels)
and torus geometry (right panels) for the QSH phase with $v_{so}=m=0.2$,
$v_{r}=0.1$, $g=0$ (upper row), the insulator phase with $v_{r}=-0.3$,
$m=0.3$, $v_{so}=g=0$ (middle row), and the quantum anomalous Hall phase with
$v_{r}=g=-0.3$, $v_{so}=m=0$ (lower row). (g) Phase diagram in the $m$ versus
$g$ plane for $v_{so}=0$ and $v_{r}\neq 0$. Points $A$ and $B$ correspond to
the parameter values used (c,d) and (e,f), respectively.

From the viewpoint of probability theory, $s_{i}$ in Eq. (6) can be regarded
as the Shannon (information) entropy of the Bernoulli distribution, i.e., the
$i$-th entanglement level $\varepsilon_{i}$ has probability $\zeta_{i}$ of
being occupied while $(1-\zeta_{i})$ of being unoccupied. As a result,
$S_{ent}$ is the Shannon entropy of a series of such independent Bernoulli
distributions. In the following, we will perform systematic numerical
simulations to study various phases of Hamiltonian (2) in terms of the ES and
the subsystem particle number.

##  III  Entanglement spectrum and subsystem particle number

At $g=0$, Hamiltonian (2) is the standard Kane-Mele model qshe1 , which is
invariant under time reversal symmetry. The system is in a QSH phase when
$|m/v_{so}|<[9-\frac{3}{4}(v_{r}/v_{so})^{2}]$, and is an insulator when
$|m/v_{so}|>[9-\frac{3}{4}(v_{r}/v_{so})^{2}]$. On the other hand, if we set
$v_{so}=0$, $v_{r}$ and $g$ nonzero, a middle band gap opens when
$|g|\neq|m|$. The system is in a quantum anomalous Hall phase with Chern
number $C=\pm 2$ yang  for $|g|<|m|$, and is an insulator for $|g|>|m|$. The
band gap closes at the transition point $|g|=|m|$. The phase diagram for
$v_{so}=0$ and $v_{r}\neq 0$ is plotted in Fig. 2(g).

Figs. 2(a) and (b) show the ES for the QSH phase, Figs. 2(c) and (d) for the
insulator phase, and Figs. 2(e) and (f) for the quantum anomalous Hall phase.
Here, it should be emphasized that the nontrivial topological phases exhibit
gapless ES [Figs. 2(a), (b), (e), and (f)], corresponding to physical gapless
edge modes, and this property is named as the _spectral flow_ es3 . However,
the spectral flow is broken for the topologically trivial phase [Figs. 2(c)
and (d)], which is also consistent with the property of the correspondent edge
states.

In a recent work trace , the authors proposed a new characteristic quantity
called the ”trace index” to describe topological invariants, which is defined
through a subsystem particle number operator $N_{A}(k_{y})=\sum_{i\in
A}c^{\dagger}_{i,k_{y}}c_{i,k_{y}}$. The expectation of $N_{A}(k_{y})$ is
given by

$\displaystyle\langle N_{A}(k_{y})\rangle=\langle GS|\sum_{i\in
A}c^{\dagger}_{i}(k_{y})c_{i}(k_{y})|GS\rangle=\mbox{Tr}{\cal C}\ .$ (7)

In Fig. 3, we plot the expectation of $N_{A}(k_{y})$ for the three different
phases mentioned above. In the cylinder geometry, $N_{A}(k_{y})$ is
discontinuous at some discrete momenta in the nontrivial topological phases,
as shown in Figs. 3(a) and (c). This is in contrast to the normal insulator
phase [see Fig. 3(b)], where $N_{A}(k_{y})$ is a continuous function of
$k_{y}$. In the torus geometry, $N_{A}(k_{y})$ is exactly equal to half of the
total particle number in the $k_{y}$ sector, without showing any
discontinuity, because the change of the particle number in $A$ around
interface $I$ is just canceled by that around interface $II$ due to the
rotation invariance of the torus. In the cylinder geometry, the _trace index_
was defined as the total discontinuities of $\langle N_{A}(k_{y})\rangle$ with
varying momentum. Alexandradinata, Hughes, and Bernevig trace  presented a
detailed analysis and proved that the trace index is equivalent to the Chern
number (or $Z_{2}$ invariant) for the Chern ($Z_{2}$) insulators. Therefore,
the subsystem particle number provides a new alternative tool to reveal the
topological invariants.

However, as mentioned in Ref. trace , there is an exceptional case in which
the subspace of the occupied bands at the symmetric momenta is not closed
under time reversal in the ground state. If at the symmetric momenta the
Kramers’ doublet that extends along the edge of $A$ is singly-occupied,
$\langle N_{A}(k_{y})\rangle$ is continuous, even when the system is in a
nontrivial topological phase. For the half-filled system under consideration,
an exception still happens. While the two-dimensional inversion symmetry
remains unchanged ($m=0$), $N_{A}(k_{y})$ becomes continuous, as shown in Fig.
3(d). This is because the Kramers’ partners extending along the edge
simultaneously cross the Fermi level at the symmetric momentum ($k_{y}=\pi$)
and have opposite contributions to the discontinuities of $\langle
N_{A}(k_{y})\rangle$.

Figure 3: (Color online) The subsystem particle number in the cylinder
geometry and torus geometry for the QSH phase (a), the insulator phase (b),
and the quantum anomalous Hall phase (c) with all the parameters same as those
in Fig. 2. (d) is also for the QSH phase with $v_{so}=v_{r}=0.2$, where the
two-dimensional inversion symmetry is retained. Discontinuities in the
expectation of the particle number can be observed only in the cylinder
geometry.

To overcome this difficulty, enlightened by the definition of the spin Chern
number spinch2 , we define spin trace indices. Considering that $s_{z}$ is not
necessarily conserved, it is an adaptable way that we choose operator
$Ps_{z}P$ to split the fiber bundle of the occupied states into two bundles
with nontrivial Chern numbers, where $P$ is the ground state projector. At
half filling and in the presence of time reversal symmetry ($g=0$), $Ps_{z}P$
is always a time-odd operator ($TPs_{z}PT^{-1}=-Ps_{z}P$), so that the
spectrum of $Ps_{z}P$ is symmetric in respect to the origin. As a result, the
spectral spaces of $Ps_{z}P$ can provide a splitting of the Hilbert space
spanned by the wave functions of the occupied states, resulting in a smooth
decomposition $P(k_{y})$ into

$P(k_{y})=P^{+}(k_{y})\oplus P^{-}(k_{y})\ ,$ (8)

with $\alpha=\pm$ corresponding to the positive and negative sectors of the
spectrum of $Ps_{z}P$ for all $k_{y}\in(0,2\pi]$. Straightforwardly, the two-
point correlator matrix can also be decomposed into ${\cal C}(k_{y})={\cal
C}^{+}(k_{y})\oplus{\cal C}^{-}(k_{y})$. In this way, the Chern numbers
$C_{\pm}$ of the two spin sectors can be well defined on these new wave
functions. It has been proved that $C_{\pm}$ are topological invariants
protected by the energy and spin spectrum gaps spinch2 . It will be shown
below that the traces of ${\cal C}^{\pm}$, called the _spin-projected
subsystem particle numbers_ , are related to the topological invariants.

Figure 4: (Color online) Subsystem spin-projected particle numbers in the
cylinder geometry for the QSH phase (a-c) with $g=0$ and different parameters:
(a) $v_{so}=0.2$ and $v_{r}=m=0$, (b) $v_{so}=v_{r}=0.2$ and $m=0$, (c)
$v_{so}=m=0.2$ and $v_{r}=0.1$, and for the insulator phase (d) with
$v_{so}=v_{r}=0.05$, $m=0.5$ and $g=0$.

We plot $\mbox{Tr}{\cal C}^{\alpha}$ $(\alpha=\pm)$ as functions of $k_{y}$ in
Fig. 4. Both $\mbox{Tr}{\cal C}^{\alpha}(k_{y})$ are discontinuous in the QSH
phase [Figs. 4(a-c)], in contrast to the continuous functions in the normal
insulator phase [Fig. 4(d)]. If $\mbox{Tr}{\cal C}^{\alpha}(k_{y})$ is
discontinuous at some discrete momenta $\\{k_{dis}\\}$ with
$k_{dis}\in(0,2\pi]$, we can define the _spin trace indices_ as the total
discontinuity, i.e., difference between the limits of $\mbox{Tr}{\cal
C}^{\alpha}(k_{y})$ from right and left,

$A^{\alpha}\equiv\sum_{k_{dis}}(\lim_{k\rightarrow k_{dis+}}\mbox{Tr}{\cal
C}^{\alpha}(k)-\lim_{k\rightarrow k_{dis-}}\mbox{Tr}{\cal C}^{\alpha}(k))\ ,$
(9)

in the thermodynamic limit. As shown in Figs. 4(a) and (b), no matter whether
$s_{z}$ is conserved, both $\mbox{Tr}{\cal C}^{+}(k_{y})$ and $\mbox{Tr}{\cal
C}^{-}(k_{y})$ show discontinuities at momentum $k_{y}=\pi$ with $A^{+}=1$ and
$A^{-}=-1$ in the QSH phase, where the two-dimensional inversion symmetry is
present ($m=0$). Figure 4(c) shows the discontinuities of $\mbox{Tr}{\cal
C}^{+}(k_{y})$ and $\mbox{Tr}{\cal C}^{-}(k_{y})$ in the QSH phase in which
$s_{z}$ is not conserved ($v_{r}\neq 0$) and the two-dimensional inversion
symmetry is broken ($m\neq 0$). In this case, the spin trace indices are equal
to 1 and $-1$, respectively, contributed by two different momentum points. It
is noteworthy that in analogy with the Laughlin gauge experiment, $A^{\alpha}$
can be regarded as the number of particles with spin $\alpha$ pumped from one
edge to the other when a unit flux is inserted adiabatically, and so
$A^{\alpha}$ is equivalent to the Chern numbers $C_{\alpha}$. On the other
hand, the $Z_{2}$ index can be defined as the parity of $A^{\alpha}$,

$A_{Z_{2}}\equiv A^{\alpha}mod\ 2,$ (10)

for any $\alpha$. As shown in Figs. 4, $A_{Z_{2}}=1$ for QSH phase [Figs.
4(a-c)] and $A_{Z_{2}}=0$ for insulator phase [Figs. 4(d)]. Therefore, the
subsystem particle number expectation can be used to characterize the
topological invariants. Especially, for the QSH systems, the spin trace
indices are well-defined quantities that can reveal the $Z_{2}$ invariant and
distinguish different quantum phases.

##  IV  Entanglement entropy and subsystem particle number fluctuation

We have shown that topological properties of the ground state can be extracted
from the expectation of subsystem particle number. Now we turn to the variance
of $N_{A}(k_{y})$. In the past two years, extensive works have been devoted to
the study of the relation between the EE and subsystem particle fluctuation
for non-topological systems fluc1 . In this section, we will show that the
relation is rather general, it does apply to non-interacting electron systems
with a nontrivial band topology. We start from the definition of the variance

$\displaystyle\triangle N^{2}_{A}(k_{y})=\langle
N^{2}_{A}(k_{y})\rangle-\langle N_{A}(k_{y})\rangle^{2}\ .$ (11)

Substituting Eq. (7) into Eq. (11) and using the Wick’s theorem to expand all
the four-point correlators, one can obtain

$\displaystyle\triangle N^{2}_{A}(k_{y})$ $\displaystyle=\sum_{i,j\in
A}\langle c^{\dagger}_{i,k_{y}}c_{j,k_{y}}\rangle\langle
c_{j,k_{y}}c^{\dagger}_{i,k_{y}}\rangle$ $\displaystyle=\mbox{Tr}[{\cal
C}(1-{\cal C})]\ ,$ (12)

yielding $\triangle N^{2}_{A}(k_{y})=\sum_{i}\zeta_{i}(1-\zeta_{i})$, which is
in keeping with the variance formula of the Bernoulli distributions. It then
follows that the variance is also dominated by the low-energy boundary
excitations ($0<\zeta_{i}<1$). Moreover, each maximally entangled state with
$\varepsilon_{m}=0$ $(\zeta_{m}=1/2)$ contributes a maximal value to the
subsystem particle number fluctuation and the EE, which cannot be eliminated
by adiabatic continuous deformation.

Figure 5: (Color online) Entanglement entropy in comparison with subsystem
particle number fluctuation for the QSH phase (upper row), the insulator phase
(middle row), and the quantum anomalous Hall phase (lower row), in the
cylinder geometry (left panels) and torus geometry (right panels). All the
parameters are same as those in Fig. 2.

In order to find a definite relationship between the EE and the variance, one
can construct a concave function $f(x)=-\ln x/(1-x)$ for $x\in[0,1]$, and
apply the Jensen’s inequality

$-x\ln x-(1-x)\ln(1-x)\geqslant(4\ln 2)\cdot x(1-x)\ .$ (13)

The equality is taken if and only if $x=1/2$. Equation (13) enables us to make
a lower-bound estimation of the EE

$s_{ent}(k_{y})\geqslant(4\ln 2)\cdot\triangle N^{2}_{A}(k_{y})\ .$ (14)

Thus a lower bound of the EE is given by $s_{0}(k_{y})\equiv(4\ln
2)\cdot\triangle N^{2}_{A}(k_{y})$, which is directly proportional to the
particle number fluctuation of subsystem. In Fig. 5 we plot $s_{ent}(k_{y})$
and $s_{0}(k_{y})$ in the QSH phase, insulator phase, and quantum anomalous
Hall phase. In all the cases, the curves for the particle number fluctuation
behave somewhat similarly, and are very close to the corresponding EE. This
similarity was observed in the non-topological systems lately fluc1 , and here
we find that the similarity remains to hold for the topologically nontrivial
system.

Furthermore, one can use $N_{A}(k_{y})=\sum_{i\in
A}c^{\dagger}_{i,k_{y}}c_{i,k_{y}}$ to verify $\triangle
N^{2}_{A}=\sum_{k_{y}}\triangle
N^{2}_{A}(k_{y})\rightarrow\frac{L_{y}}{2\pi}\int dk_{y}N^{2}_{A}(k_{y})$,
indicating that $\triangle N^{2}_{A}(k_{y})$ satisfies a area law area ,
similar to the EE,
$S_{ent}=\sum_{k_{y}}s_{ent}(k_{y})\rightarrow\frac{L_{y}}{2\pi}\int
dk_{y}s_{ent}(k_{y})$. Therefore, the subsystem particle number fluctuation
shares several common characteristics with the EE, and so can be utilized to
detect the EE experimentally.

##  V  Summary

To conclude, we have investigated the relationship between the quantum
entanglement and subsystem particle number. The spin trace indices can reveal
the topological invariants and be used to classify different phases in QSH
systems. This new tool always works well even though $s_{z}$ is not conserved.
As to the subsystem particle number fluctuation, it shares several common
properties with the EE. They both satisfy the same area law, and are dominated
by the boundary excitations with each zero mode having a maximal contribution.
The connection between the two quantities is universal, regardless of whether
the system has a nontrivial band topology. As a result, the subsystem particle
number fluctuation, as an observable quantity, can be used to detect the EE
experimentally fluc1 .

##  VI ACKNOWLEDGMENTS

This work is supported by the State Key Program for Basic Researches of China
under Grants Nos. 2009CB929504 (LS), 2011CB922103, and 2010CB923400 (DYX), the
National Natural Science Foundation of China under Grant Nos. 11074110 (LS),
11074111 (RS), 11174125, 11074109, 91021003 (DYX), and a project funded by the
PAPD of Jiangsu Higher Education Institutions.

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