Quantum Kibble-Zurek mechanism:

Kink correlations after a quench in the quantum Ising chain

Radosław Nowak
Jagiellonian University, Institute of Theoretical Physics, Łojasiewicza 11, PL-30348 Kraków, Poland
Jacek Dziarmaga
Jagiellonian University, Institute of Theoretical Physics, Łojasiewicza 11, PL-30348 Kraków, Poland

The transverse field in the quantum Ising chain is linearly ramped from the para- to the ferromagnetic phase across the quantum critical point at a rate characterized by a quench time $\tau_Q$.
We calculate a connected kink-kink correlator in the final state at zero transverse field.
The correlator is a sum of two terms:
a negative (anti-bunching) Gaussian that depends on the Kibble-Zurek (KZ) correlation length only
and a positive term that depends on a second longer scale of length.
The second length is made longer by dephasing of the state excited near the critical point during the following ramp across the ferromagnetic phase.
This interpretation is corroborated by considering a linear ramp that is halted in the ferromagnetic phase for a finite waiting time and then continued at the same rate as before the halt.
The extra time available for dephasing increases the second scale of length that asymptotically grows linearly with the waiting time.
The dephasing also suppresses magnitude of the second term making it negligible for waiting times much longer than $\tau_Q$.
The same dephasing can be obtained with a smooth ramp that slows down in the ferromagnetic phase.
Assuming sufficient dephasing we obtain also higher order kink correlators and the ferromagnetic correlation function.

§  INTRODUCTION

Kibble-Zurek mechanism (KZM) originated from a scenario for defect creation in cosmological symmetry-breaking phase transitions <cit.>. As the Universe cools, causally disconnected regions must choose broken symmetry vacua independently resulting in topologically nontrivial configurations that survive as topological defects. In this Kibble scenario it is the speed of light that limits the size of the correlated domains.
In contrast a dynamical theory for the laboratory phase transitions <cit.> employs equilibrium critical exponents of the transition and the quench time to predict the scaling of the resulting density of defects. KZM was successfully tested by numerical simulations <cit.> and laboratory experiments in condensed matter systems <cit.>. More recently, KZM was adapted to quantum phase transitions <cit.>. Theoretical developments <cit.> as well and experimental tests <cit.> of the quantum KZM (QKZM) followed. The recent experiment <cit.>, where a quantum Ising chain in the transverse field is emulated with Rydberg atoms, is consistent with the theoretically predicted scalings <cit.>.

In a cartoon version of QKZM, whose limitations — but also essential correctness — have been discussed in Ref. sonic, the state of the system literally freezes-out in the neighborhood of the critical point due to the closing of the energy gap. In QKZM a system initially prepared in its ground state is smoothly ramped across a quantum critical point. A smooth ramp can be linearized near the critical point:

\begin{equation}
\varepsilon(t)=\frac{t-t_c}{\tau_Q}.
\label{epsilont}
\end{equation}

Here $\varepsilon$ is a dimensionless parameter in the Hamiltonian, whose magnitude measures distance from the critical point, $\tau_Q$ is a quench time, and $t_c$ is the time when the critical point is crossed. Initially, far from the critical point, the evolution is adiabatic and the system follows its adiabatic ground state. The adiabaticity fails at $-\hat t$ when the reaction rate of the system, proportional the the gap $\Delta\propto|\varepsilon|^{z\nu}$, equals instantaneous transition rate $|\dot \varepsilon/\varepsilon| = 1/|t|$. Here $z$ and $\nu$ are, respectively, the dynamical and the correlation length exponent. From this equality we obtain
and the corresponding $\hat\varepsilon=\hat t/\tau_Q\propto\tau_Q^{-1/(1+z\nu)}$. In the cartoon “freeze-out” version of the impulse approximation the ground state at $-\hat\varepsilon$, with a corresponding correlation length,

\begin{equation}
\hat\xi \propto \tau_Q^{\nu/(1+z\nu)},
\label{hatxi}
\end{equation}

is expected to survive until $+\hat t$, when the evolution can restart. In this way, $\hat\xi$ becomes imprinted on the initial state for the final adiabatic stage of the evolution after $+\hat t$. Oversimplified as it is, the adiabatic-impulse-adiabatic approximation predicts correct scaling of the characteristic lengthscale with $\tau_Q$, see Eq. (<ref>), and the timescale
t̂ ∝ξ̂^z.

The post-quench density of excitations is determined by $\hat\xi$ within this scenario.

In the integrable quantum ising chain the excitations are well defined as Bogoliubov quasiparticles <cit.>. They get excited between $t_c-\hat t$ and $t_c+\hat t$. After $t_c+\hat t$, when the evolution of the system crosses over from the impulse to the adiabatic again, their power spectrum $p_k$ becomes frozen. Here $p_k$ is excitation probability for a pair of quasiparticles with opposite quasimomenta: $\pm k$. The excited state after $t_c+\hat t$ is a superposition over many eigenstates. Magnitudes of their amplitudes are determined by $p_k$, and thus remain frozen, but the amplitudes accumulate dynamical phases that depend on $k$. This may eventually lead to dephasing: the $k$-dependent phases become so scrambled that observables that are localized in space can be accurately calculated within an approximation of random phases <cit.>.

Motivated by new experimental opportunities opened by Rydberg atoms <cit.>, in this paper we go beyond the set of observables considered in Ref. QKZteor-e and calculate correlations between ferromagnetic kinks at the end of the quench.
This problem was first considered in Ref. roychowdhury2020dynamics in an elegant dual formulation of the quantum Ising chain. In addition to the probability distribution of the total number of kinks <cit.>, these are experimentally most accessible predictions that go “beyond KZM”, i.e., beyond the most basic average density of kinks. We obtain the kink-kink correlator in a closed analytic form without any random-phase approximation: the dephasing after a generic quench is far insufficient to justify the approximation. However, the dephasing has an impact on the correlator through a second length scale, $l$, that it makes longer than the basic KZ correlation length $\hat\xi$, though for a generic quench the correction is only logarithmic in $\tau_Q$. We demonstrate that $l$ can be made much longer by slowing the ramp after $t_c+\hat t$ to provide more time for dephasing, see Fig. <ref>. When the extra time is long enough then the correlator becomes the same as in the random phase approximation. In this regime we can proceed further and derive a compact formula for higher order kink correlators and the ferromagnetic spin-spin correlation function.

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[color=blue, domain=-25:0,samples=100,smooth,thick] -x node[above,pos=1] ;
[color=green, domain=-7.5:4.51,samples=100,smooth,thick] 7.51 node[above,pos=7.5] ;
[color=green, domain=-25:-7.49,samples=100,smooth,thick] -x+0.1 node[above,pos=7.5] ;
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[black] (7.75,3.15) circle (1.5pt) node[anchor=north west] $g_w$;
[black] (7.75,6.30) circle (1.5pt) node[anchor=north west] $g_c$;
[black] (12.05,0.35) circle (1.5pt) node[anchor=north east] $t_w$;

Transverse field ramp.
Time dependence of the transverse magnetic field $g(t)$.
The blue line is the straight linear ramp that crosses the quantum critical point at $g_c=1$ before it stops at $g(0)=0$ where measurements are made.
The green line is the ramp with a halt at $g_w<g_c$ for a waiting time $t_w$.
The halt allows for extra dephasing of quasiparticles that were excited in the Kibble-Zurek regime near the critical point: within $\pm\hat t$ from the time when the ramp crosses the critical point.

The paper is organized as follows. In section <ref> we recall basic facts about the quantum Ising chain. Linear quench/ramp of the transverse magnetic field is defined in section <ref>. In section <ref> the ramped quantum Ising model is mapped to the Landau-Zener problem. This is where we derive quadratic correlators for Jordan-Wigner fermions, identify the effects of dephasing and introduce the dephasing length $l$. The kink-kink correlator is worked out in section <ref>. It is shown to be a sum of two terms depending either on $\hat\xi$ or $l$. In order to substantiate the discussion of dephasing, in section <ref> we make the linear ramp halt in the ferromagnetic phase for a variable waiting time $t_w$, see Fig. <ref>. A more general analytic formula for the kink-kink correlator is obtained that depends on $t_w$ through a generalized dephasing length $l_w$. For long enough extra waiting time $l_w$ grows linearly with $t_w$ and the $l_w$-dependent term in the correlator decays like $l_w^{-1}$. Eventually the random phase approximation becomes accurate and the dephased correlator depends on the KZ length $\hat \xi$ only. In sections <ref> and <ref> we take advantage of the dephasing and work out higher order kink correlators and the ferromagnetic spin-spin correlator, respectively. Finally, we summarize in section <ref>.

§  QUANTUM ISING CHAIN

The transverse field quantum Ising chain is
H = -∑_n=1^N ( gσ^x_n + σ^z_nσ^z_n+1 ) .

Here we assume periodic boundary conditions:
\vec\sigma_{N+1}~=~\vec\sigma_1~.
In the thermodynamic limit, $N\to\infty$, the quantum critical point at $g=1$ separates the paramagnetic ($|g|>1$) from the ferromagnetic ($|g|$<1) phase. For definiteness we assume that $N$ is even. After the Jordan-Wigner transformation,

σ^x_n = 1-2 c^†_n  c_n ,

σ^z_n =
-( c_n+ c_n^†)
∏_m<n(1-2 c^†_m c_m) ,

introducing fermionic annihilation operators $c_n$, which satisfy anticommutation relations
$\left\{c_m,c_n^\dagger\right\}=\delta_{mn}$ and
$\left\{ c_m, c_n \right\}=\left\{c_m^\dagger,c_n^\dagger \right\}=0$,
the Hamiltonian (<ref>) becomes <cit.>
H = P^+ H^+ P^+ + P^- H^- P^- .

1/2[1 ± ∏_n=1^N(1-2c_n^†c_n)]

are projectors on the subspaces with even ($+$) and odd ($-$) numbers of $c$-quasiparticles and
H^± =
g c_n^†c_n - c_n^†c_n+1 - c_n+1  c_n - g/2 +h.c.
) .

are the corresponding reduced Hamiltonians. The $c_n$'s in $H^-$ satisfy
periodic boundary conditions, $c_{N+1}=c_1$, but the $c_n$'s in $H^+$
must be anti-periodic: $c_{N+1}=-c_1$.

The parity of the number of $c$-quasiparticles is a good quantum number and the ground state has even parity for any non-zero value of $g$. Assuming that time evolution begins in the ground state, we can confine to the subspace of even parity. $H^+$ is diagonalized by a Fourier transform followed by a Bogoliubov transformation <cit.>. The anti-periodic Fourier transform is
c_n =
∑_k c_k e^ikn ,

where the pseudomomenta $k$ take half-integer values:
k =
±1/2 2π/N,
±N-1/2 2π/N .

It transforms the Hamiltonian into
H^+    =
2(g-cosk) c_k^†c_k + .

c^†_k c^†_-k+
c_-k c_k
} .

Diagonalization of $H^+$ is completed by the Bogoliubov transformation:
c_k =
u_k  γ_k + v_-k^*  γ^†_-k ,

where the Bogoliubov modes $(u_k,v_k)$ are eigenstates of stationary Bogoliubov-de Gennes equations:
ϵ  u_k    =    +2(g-cosk) u_k+2sink v_k ,

ϵ  v_k    =    -2(g-cosk) v_k+2sink u_k .
For each $k$ they have two eigenstates with eigenenergies $\epsilon=\pm\epsilon_k$, where
ϵ_k = 2√((g-cosk)^2+sin^2 k) .

The positive energy eigenstate,
defines a fermionic quasiparticle operator $\gamma_k~=~U_k^* c_k + V_{-k} c_{-k}^\dagger$,
and the negative energy eigenstate,
defines $\gamma_k^-=(U_k^-)^*c_k+V_{-k}^-c_{-k}^\dagger=-\gamma_{-k}^\dagger$.
After the Bogoliubov transformation, up to an additive constant the Hamiltonian is equivalent to
H^+ =
∑_k ϵ_k

but the projector $P^+$ in Eq. (<ref>) implies that only states with even numbers of quasiparticles belong to the spectrum of $H$.

With the quasiparticle dispersion (<ref>) at the critical $g=1$ we obtain a linear dispersion, $\epsilon_k\approx 2|k|$, for small $|k|$ which implies the dynamical exponent $z=1$. On the other hand, for $k=0$ we have $\epsilon_0\propto |g-1|^1$ for a near-critical $g\approx1$ which implies $z\nu=1$ and the correlation length exponent $\nu=1$. Consequently, the KZ scales are
t̂ ∝ξ̂∝√(τ_Q).

§  LINEAR QUENCH

We ramp the Hamiltonian across the quantum critical point by a linear quench
g(t≤0) = -t/τ_Q .

with the characteristic quench time $\tau_Q$. The ramp crosses the critical point at $t_c=-\tau_Q$. For the universal features of the QKZM it is enough to assume that the ramp can be linearized near the critical point, with a slope $-1/\tau_Q$, but here we proceed with a solution of the analytically tractable fully linear ramp. The system is initially in its ground state at large initial value of $g\gg 1$, but as $g$ is ramped down to zero, the system gets excited from its instantaneous ground state and, in general, its final state at $t=0$ has finite number/density of kinks. Comparing the Ising Hamiltonian Eq. (<ref>) at $g=0$ with the Bogoliubov Hamiltonian (<ref>) at $g=0$ we obtain a simple expression for the operator of the total number of kinks
N  =  ∑_n=1^N K_n  =  ∑_k γ_k^†γ_k .

K_n = 1/2 ( 1 - σ_n^zσ_n+1^z )

with eigenvalues $0,1$ is the kink number operator on the bond between sites $n$ and $n+1$. The total number of kinks is equal to the number of quasiparticles excited at $g=0$.

§  LANDAU-ZENER PROBLEM

The initial ground state is the Bogoliubov vacuum $|0\rangle$ annihilated by all quasiparticle operators $\gamma_k$. For the initial $g\gg1$ they are defined by the stationary Bogoliubov modes $(U_k,V_k)\approx(1,0)$. We assume the Heisenberg picture, where Fock states with definite quasiparticle occupations numbers do not change and, therefore, the Bogoliubov quasiparticle operators $\gamma_k$ expressed through these states do not change either but the Jordan-Wigner fermions evolve with the usual Heisenberg equation: $i\frac{d}{dt}c_k=[c_k,H^+]$. With a time-dependent Bogoliubov transformation,
c_k  =  u_k(t)  γ_k + v_-k^*(t)  γ^†_-k,

the Heisenberg equation becomes equivalent to time-dependent Bogoliubov-de Gennes equations (<ref>):
id/dt u_k    =
+2[g(t)-cosk] u_k +
2 sink  v_k ,

id/dt v_k    =
-2[g(t)-cosk] v_k +
2 sink  u_k .

with the initial/asymptotic condition $[u_k(-\infty),v_k(-\infty)]=(1,0)$.

They can be solved exactly by mapping to the Landau-Zener (LZ) problem<cit.>. Indeed, a transformation to a new time variable:
τ = 4τ_Qsink(t/τ_Q+cosk)

brings Eqs. (<ref>) to the standard LZ form:
id/dτ u_k    =
-1/2(τΔ_k) u_k + 1/2 v_k  ,

id/dτ v_k    =
+1/2(τΔ_k) v_k + 1/2 u_k  ,

with an effective transition rate $\Delta_k=(4\tau_Q\sin^2 k)^{-1}$. Here the new time $\tau$ runs from $-\infty$ to $\tau_{\rm final}=2\tau_Q\sin(2k)$ that corresponds to $t=0$.

§.§  Spectrum of excitations and density of kinks

For slow enough transitions only modes with small $k$, which have small gaps at their anti-crossing points, can get excited. For these modes $\tau_{\rm final}$ is much longer than the time when the anti-crossing is completed, $\propto \Delta_k^{-1/2}$, and we can use the LZ excitation probability:
p_k ≈
e^-π/2Δ_k ≈
e^-2πτ_Qk^2 .

The approximations are accurate when $\tau_Q\gg1$. We can calculate the number of kinks in Eq. (<ref>) as
{\cal N}~=~\sum_k~p_k~.
In the thermodynamic limit, $N\to\infty$, the sum can be replaced by an integral and the density of kinks becomes
1/2π∫_-π^πdk p_k=

The density scales like $\hat\xi^{-1}\propto\tau_Q^{-1/2}$ in agreement with QKZM. With kink correlations in mind it is natural to make definition of $\hat\xi$ precise as
ξ̂≡2π√(2τ_Q) = 1/n.

An inverse of this KZ length is equal to average final density of kinks/excitations.

§.§  Exact solution

The kink correlations will require more than just the excitation spectrum (<ref>). A general solution of equations (<ref>) is<cit.>:
v_k(τ)   =
-[ a D_-s-1(-iz) + b D_-s-1(iz) ]  ,

u_k(τ)   =
v_k(τ)  ,

with free complex parameters $a,b$. Here $D_m(x)$ is a Weber function,
$s=\frac{1}{4i\Delta_k}$, and $z=\sqrt{\Delta_k}\tau e^{i\pi/4}$.
The free parameters can be fixed by the asymptotic conditions:
$u_k(-\infty)=1$ and $v_k(-\infty)=0$. Using the asymptotes of the Weber
functions when $\tau\to-\infty$, we obtain $a=0$ and
|b|^2=e^-π/8Δ_k/4Δ_k .
The exact solution of the linear quench problem is then
v_k(τ)   =
- b D_-s-1(iz)  ,

u_k(τ)   =
v_k(τ)  ,

At the end of the quench for
$t=0$ and when $\tau=\tau_k=2\tau_Q \sin(2k)$, the argument of the Weber function
$iz=\sqrt{\Delta_k}\tau e^{i\pi/4}=2\sqrt{\tau_Q}e^{i\pi/4}\cos(k){\rm sign}(k)$.

§.§  Fermionic correlators

For the considered $\tau_Q\gg1$ the magnitude of $iz$ is large for most $k$, except the neighborhoods of $k=\pm\frac{\pi}{2}$, and we can again use the asymptotes of the Weber functions to obtain <cit.>
|u_k|^2   =
1/2(1-cosk)+p_k ,

u_k v_k^*    =
e^iφ_k ,

φ_k    =

k^2 τ_Qlnτ_Q-
[Γ(1+ik^2τ_Q)] .

Here $\Gamma(x)$ is the gamma function and $\varphi_k$ is a dynamical phase acquired by a pair of excited quasiparticles with quasimomenta $(k,-k)$. These formulas depend on $k$ and $\tau_Q$ through two combinations: $\tau_Q k^2$, which implies the usual KZ correlation length, $\hat\xi\propto\tau_Q^{1/2}$, and $k^2 \tau_Q\ln\tau_Q$ which implies a second scale of length $\propto\sqrt{\tau_Q\ln\tau_Q}$. The final quantum state at $g=0$ cannot be fully characterized by a single scale of length. Physically, this reflects a combination of two processes: KZM that sets up the post-transition spectrum of excitations, $p_k$, and subsequent dephasing of the excited quasiparticle modes that manifests through the dynamical phase $\varphi_k$.

In order to make the phase more intelligible we can approximate ${\rm arg} [\Gamma \left(1+i\tau_Qk^2\right)]\approx -\gamma_E \tau_Qk^2$ for small enough $\tau_Qk^2$, where $\gamma_E$ is the Euler gamma constant. Given that excited quasiparticles have at most $\tau_Qk^2\approx 1/2\pi$, see (<ref>), this is an accurate approximation that renders $\varphi_k$ quadratic in $k$:
φ_k-φ_0    =    ( lnτ_Q+ln4-2+γ_E ) k^2τ_Q

=    ( lnτ_Q-0.036 ) k^2τ_Q

≈    k^2τ_Qlnτ_Q.

It also makes manifest that the dynamical phase is characterized solely by the second scale $\propto\sqrt{\tau_Q\ln\tau_Q}$.

xlabel = $\sqrt{\tau_Q k^2}$,
ylabel = Equation (<ref>),

legend pos=north west,
exp(-pi*x^2)*sqrt(1 - exp(-2*pi*x^2));
$e^{-\pi\tau_Q k^2}
\sqrt{1-e^{-2\pi\tau_Q k^2}}$

$ e^{-1.361\pi\tau_Q k^2} 0.964 \sqrt{2\pi} \left(\tau_Q k^2\right)^{1/2}  $

$ e^{-\frac43\pi\tau_Q k^2} (19/20) \sqrt{2\pi} \left(\tau_Q k^2\right)^{1/2}  $

Approximation in Eq. (<ref>).
Comparison between the exact formula in Eq. (<ref>) (black) and the approximate one with either the raw parameters from minimization of the quadratic error: $A=0.964,a=1.361$ (green) or the adjusted ones: $A=19/20,a=4/3$ (red).

The Gaussian state can be fully characterized by its two quadratic fermionic correlators:
α_R    ≡    ⟨c_n+R c_n^†⟩=
1/2π∫_-π^πdk |u_k|^2 e^ikR

ξ̂^-1 e^-π(R/ξ̂)^2 ,

β_R ≡ ⟨c_n+R c_n ⟩=
1/π∫_0^πdk u_kv_k^* sinkR.

With (<ref>) we obtain
β_R=1/4 sign(R)δ_|R|,1+δβ_R,

where the first term is a ground state contribution while the second one comes solely from the excitations:
e^-πτ_Q k^2
√(1-e^-2πτ_Q k^2)

In order to make the integral analytically tractable we make an approximation:
e^-πτ_Q k^2
√(1-e^-2πτ_Q k^2) ≈e^-aπτ_Q k^2
A √(2 π)
(τ_Qk^2)^1/2  .

With $A=1$ and $a=1$ this would be just the leading term in a series expansion in powers of $\left(\tau_Qk^2\right)^{1/2}$. However, for greater accuracy we treat $A$ and $a$ as variational parameters. Minimization of the quadratic error of the approximation yields $A=0.964$ and $a=1.361$. Within its broad minimum we slightly adjust these numbers to
A=19/20,     a=4/3.
Comparisons between these two approximations and the exact formula are made in Fig. <ref>.

Putting together all the approximations Eq. (<ref>) becomes:
∫_0^π√(τ_Q) qdq
e^-aπq^2+i q^2 lnτ_Q
sinq R/√(τ_Q).

Here $q=\sqrt{\tau_Q}k$.
After the upper limit of the integral is safely extended to infinity we obtain
δβ_R    =
e^ -2π/a (R/l)^2

e^ -3π/2 (R/l)^2

where $\phi$ is a phase and the correlation range is
l  =
ξ̂  √( 1 + (lnτ_Q/aπ)^2 ) =
ξ̂  √( 1 + (3lnτ_Q/4π)^2 ).

For very slow quenches, when $\ln\tau_Q\gg4\pi/3$, the range of this correlator becomes much longer than $\hat\xi$.

[name=plot1,grid=major,height=6cm,width=8cm,legend pos=south east,xlabel = $n R$,
ylabel = $n^{-2}C_{R}^{KK}$]
[color=black,mark =*, mark size =.7]table[] correlator_data/DS128;
[color=red,mark =*, mark size =.7]table[] correlator_data/AS128;
[domain=0:1.12403, samples=100, color = blue, style=ultra thick]
[domain=0:1.12403, samples=100, color = green, style=ultra thick]
[fill=white,draw] at (10,100) $\tau_Q = 128$;
exact (<ref>), approximate (<ref>), analytic (<ref>), dephased (<ref>)

[name=plot3,grid=major,at=($(plot1.south)-(0,2cm)$),anchor=north,height=6cm,width=8cm,xlabel = $n R$,
ylabel = $n^{-2}C_{R}^{KK}$]
[color=black,mark =*, mark size =.7]table[] correlator_data/DS32;
[color=red,mark =*, mark size =.7]table[] correlator_data/AS32;
[domain=0:1.12403, samples=100, color = blue, style=ultra thick]
[domain=0:1.12403, samples=100, color = green, style=ultra thick]
[fill=white,draw] at (10,100) $\tau_Q = 32$;

[name=plot5,grid=major,at=($(plot3.south)-(0,2cm)$),anchor=north,height=6cm,width=8cm,xlabel = $n R$,
ylabel = $n^{-2}C_{R}^{KK}$]
[color=black,mark =*, mark size =.7]table[] correlator_data/DS8;
[color=red,mark =*, mark size =.7]table[] correlator_data/AS8;
[domain=0:1.12403, samples=100, color = blue, style=ultra thick]
[domain=0:1.12403, samples=100, color = green, style=ultra thick]
[fill=white,draw] at (10,100) $\tau_Q = 8$;

Kink-kink correlator.
Comparison between (scaled) exact correlator (<ref>) and (scaled) approximate one (<ref>), both plotted in function of scaled distance $n R$, demonstrating that the approximate one is accurate for long enough $\tau_Q$. We also plot the analytic formula in Eq. (<ref>) to demonstrate that it becomes accurate for long enough $\tau_Q$.
The dephased correlator (<ref>), after the halted ramp with long enough waiting time, is also shown for comparison.

§ KINK-KINK CORRELATOR AFTER A LINEAR RAMP

The connected kink-kink correlator is
C_R^KK    =    ⟨K_n K_n+R ⟩_c

⟨K_n K_n+R ⟩- ⟨K_n ⟩⟨K_n+R ⟩ ,
where $K_n$ is the kink number operator on the bond between sites $n$ and $n+1$, see (<ref>). In terms of the fermionic correlators it becomes
1/2|β_R|^2 +
1/2Re(β_R+1β_R-1^*) -
α_R+1α_R-1 +

1/2Re ( β_R+1β_R-1 - β_R^2 ) +

α_R-1Reβ_R+1  -α_R+1Reβ_R-1  .

When $R\pm 1$ is approximated by $R$, which should be accurate for the assumed $\hat\xi\gg1$, the correlator reduces to:
C_R^KK = |β_R |^2 - α_R^2 .

Interestingly, this is $1/4$ of the connected transverse correlator:
\langle \sigma^x_n \sigma^x_{n+R} \rangle -
\langle \sigma^x_n     \rangle
\langle \sigma^x_{n+R} \rangle.
In order to properly assess the strength of kink-kink correlations, the correlator (<ref>) should be normalized by the square of the average density of kinks in (<ref>). After this normalization the exact formula in (<ref>) and the approximate one in (<ref>) are compared in Fig. <ref>. As expected, they become the same for large enough $\tau_Q$.

With equations (<ref>), (<ref>), (<ref>), and (<ref>) we obtain a compact analytic formula:
n^-2 C^KK_R =
e^ -3π(R/l)^2
- e^ - 2π(R/ξ̂)^2 .

Here $\alpha=\frac{ 9747 \pi }{3200}=9.57$ is a numerical prefactor. In figure <ref> we compare the normalized correlator in (<ref>) with the analytic formula in (<ref>) finding good agreement that is improving with increasing $\tau_Q$. The normalized correlator is of the order of $1$ implying strong correlation effects. Especially its second negative term implies strong anti-bunching. The kinks can hardly approach one another closer than a half of $\hat\xi$, i.e., half of the typical distance between them.

[name = plot1,grid = major, legend pos = north east, ylabel = $p_k$,xlabel = $k$]
[mark size = 0.2, mark = *] table[] ps_data/pl0;

[mark size = 0.2, mark = *,color = blue] table[] ps_data/pl1;

[mark size = 0.2, mark = *,color = red] table[] ps_data/pl2;
[mark size = 0.2, mark = *,color = gray] table[] ps_data/ppp3;
[mark size = 0.2, mark = *,color = cyan] table[] ps_data/ppp5;
[mark size = 0.2, mark = *,color = green] table[] ps_data/ppp8;
[mark size = 0.2, mark = *,color = brown] table[] ps_data/ppp15;
$t_w = 0$, $t_w = 1$, $t_w = 2$,$t_w = 3$, $t_w = 5$, $t_w = 8$, $t_w = 15$;
[fill=white,draw] at (100,1000) $\tau_Q = 2$;

[name = plot2,at=($(plot1.south)-(3.46cm,7cm)$),grid = major, legend pos = north east, ylabel = $p_k$,xlabel = $k$]
[mark size = 0.2, mark = *] table[] ps_data/pp40;

[mark size = 0.2, mark = *,color = blue] table[] ps_data/pp41;

[mark size = 0.2, mark = *,color = red] table[] ps_data/pp42;
[mark size = 0.2, mark = *,color = gray] table[] ps_data/pp43;
[mark size = 0.2, mark = *,color = cyan] table[] ps_data/pp45;
[mark size = 0.2, mark = *,color = green] table[] ps_data/pp48;
[mark size = 0.2, mark = *,color = brown] table[] ps_data/pp415;
$t_w = 0$, $t_w = 1$, $t_w = 2$,$t_w = 3$, $t_w = 5$, $t_w = 8$, $t_w = 15$;
[fill=white,draw] at (70,1000) $\tau_Q = 4$;

Power spectrum after the halt.
Power spectrum $p_k=\langle \gamma_k^\dag \gamma_k \rangle$ at the final transverse field $g=0$ after a linear ramp with a halt at $g_w=1/2$ for a waiting time $t_w$, see Fig. <ref>. As predicted, with increasing $\tau_Q$ the influence of discontinuous time derivative at the beginning and the end of the halt on the power spectrum quickly becomes negligible, compare $\tau_Q=2$ in the top panel with $\tau_Q=4$ in the bottom one.

§  KINK-KINK CORRELATOR AFTER A LINEAR RAMP WITH A HALT

We have seen that there is an interplay between the KZ mechanism and the dephasing after $+\hat t$. Its manifestation are the two scales of length, $\hat\xi$ and $l$, that show up in the kink-kink correlator (<ref>). In order to make the distinction between the two effects even sharper, here we consider the same linear ramp as before but with an additional halt at $g_w\in(0,1)$ for a waiting time $t_w$, see Fig. <ref>. We expect that for long enough waiting time the nontrivial quasiparticle dispersion (<ref>) will completely dephase excited quasiparticles with different quasimomenta. The dynamical phase $\varphi_k$ in (<ref>) will depend on $k$ strongly enough for the magnitude of $\delta\beta_R$ to be suppressed and the kink-kink correlator to become
n^-2 C^KK_R =
-e^ -2π(R/ξ̂)^2  .

This purely negative dephased correlator demonstrates strong anti-bunching effect.

Formula (<ref>) coincides with the one advocated in Ref. roychowdhury2020dynamics but, contrary to Ref. roychowdhury2020dynamics, waiting at the final $g_w=0$ cannot dephase the kink correlator because the dispersion (<ref>) at $g=0$ is flat: $\epsilon_k = 2$. Even if it were not, the kink number operator (<ref>) commutes with the Hamiltonian at $g=0$ and the correlator must remain constant there.

The halt at $g_w>0$ is convenient analytically but it has a disadvantage that the discontinuous time derivative of the function $g(t)$ at the beginning and the end of the halt results in additional excitations on top of the KZ quasiparticles already excited near the critical point. However, the additional excitation energy is proportional to $\tau_Q^{-2}$, hence for large $\tau_Q$ it quickly becomes negligible when compared to the KZ excitation energy which is proportional to the density of excited quasiparticles and decays like $\tau_Q^{-1/2}$. Exact power spectra with and without the halt are compared in Fig. <ref> and they confirm this expectation. However, in experiment — where $\tau_Q$ is limited — instead of the sharp halt it may be more practical to avoid the discontinuities by performing a smooth ramp that simply takes longer to reach $g=0$ than the straight linear ramp. The ramp should just remain linear between $t_c\pm\hat t$ for the KZ scaling to remain unaffected. All of that being said, in the following we continue with the analytically convenient halt.

An exact solution between initial $g=\infty$ and $g_w$ is the same as the one for the linear ramp, see (<ref>). Then for time $t_w$ the Bogoliubov modes continue their evolution with a static Bogoliubov-de Gennes Hamiltonian at $g_w$. This stage is described by Eq. (<ref>) with constant $g(t)=g_w$. Further evolution after the end of the halt is described by the general solution in (<ref>) but with time $t$ replaced by $t-t_w$. Its coefficients $a$ and $b$ are determined by matching this general solution with the $u_k,v_k$ at the end of the halt. This solution is continued until $g=0$, which is arrived at $t=t_w$, where the kink-kink correlator (<ref>) is measured.

The dephasing time can be estimated based on the quasiparticle dispersion relation (<ref>). According to the excitation probability (<ref>) quasiparticles are excited up to small quasimomenta with
hence at $g=g_w$ the dispersion (<ref>) can be approximated by
ϵ_k    ≈    2√((g_w-1)^2+g_w k^2)

≈    2|1-g_w| +g_wk^2/|1-g_w|.
The last form is accurate when $2\pi\tau_Q\gg g_w (1-g_w)^{-2}$, i. e., either for slow enough quenches or deep enough in the ferromagnetic phase. A difference between dynamical phases, $2\epsilon_k t_w$, for $k=0$ and $k=\hat k$ becomes ${\cal O}(1)$ after a dephasing time
t_D ≈|1-g_w|/g_w τ_Q.

Increasing the waiting time $t_w$ beyond $t_D$ should begin to have a noticeable effect on the anomalous correlator $\delta\beta_R$ eventually suppressing its magnitude to zero.

This rough estimate can be elevated to an accurate prediction. In Fig. <ref> we have shown that quasiparticle spectrum, $p_k$, does not depend on the waiting time. The dynamical phase $\varphi_k$ in (<ref>) acquires an extra term, $\delta\varphi_k$, such that
δφ_k -δφ_0 =
2ϵ_k t_w -2ϵ_0 t_w =
2g_w/|1-g_w| k^2 t_w .
Just as the bare $\varphi_k$ in (<ref>) the extra term is quadratic in $k$. Consequently, new $\vert\delta\beta_R\vert$ for the ramp with a halt is obtained from the old $\vert\delta\beta_R\vert$ without a halt by a simple replacement:
\tau_Q \ln\tau_Q   \to  \tau_Q \ln\tau_Q + \frac{2g_w}{|1-g_w|} t_w.
Therefore, sole effect of the halt on the kink-kink correlator (<ref>) is to replace the length scale $l$ in (<ref>) with
l_w =
ξ̂√(1+( 3lnτ_Q+6g_w/|1-g_w| t_w/τ_Q/4π  )^2).

With the replacement the correlator after the halt becomes
n^-2 C^KK_R =
e^ -3π(R/l_w)^2
- e^ - 2π(R/ξ̂)^2 .

Here $l_w$ is longer than the bare $l$ corresponding to zero waiting time. Comparing (<ref>) with (<ref>) we can infer that the waiting time begins to have noticeable effect when
6g_w/|1-g_w| t_w/τ_Q = 4πor, equivalently, for $t_w$ longer than a dephasing time
This is when not only the correlation range, $l_w$, begins to increase but also the maximal value of the first term in (<ref>), proportional to $\hat\xi/l_w$ and achieved at $R=l_w/\sqrt{\pi}$, begins to shrink. For $t_w\gg t_D$ this magnitude becomes negligible and the kink-kink correlator simplifies to the single anti-bunching term in (<ref>).

§  HIGHER ORDER KINK CORRELATORS

The dephasing makes higher order correlators tractable. A connected $(M+1)$-point correlator reads
⟨K_0 K_R_1 …K_R_M
⟩_c .
Thanks to permutation symmetry and translational invariance, we can assume $0=R_0<R_1<\dots<R_M$ without loss of generality. Expressing the kink number operators (<ref>) with the Jordan-Wigner fermions (<ref>) allows us to write

C_R_1,…,R_M =

(-2)^-(M+1) ×

⟨(b_0 a_1)   (b_R_1 a_R_1+1) …(b_R_M a_R_M+1)

Here $b_n=c_n^\dag-c_n$ and $a_n=c_n^\dag+c_n$ are Majorana fermions.

Their quadratic correlators are
⟨b_m a_n ⟩   =
δ_m,n-2α_n-m+2 Re β_n-m,

⟨a_m a_n ⟩   =
⟨b_m b_n ⟩=
δ_m,n+2i Im β_m-n.
After the dephasing $\beta_R\approx \frac14 {\rm sign}(R) \delta_{|R|,1}$ and, when $R_i$'s differ from each other by more than $2$, the relevant correlators simplify to
⟨b_m a_n ⟩= -2α_n-m,
⟨a_m a_n ⟩= ⟨b_m b_n ⟩= 0.
Given that $\alpha_R$ is even in $R$, the correlator becomes

C_R_1,R_2,…,R_M ≈

∑_{ i_1,…,i_M  }
ϵ^ i_0 i_1 …i_M _ i_1 …i_M i_0
α_R_i_0-R_i_1 α_R_i_1-R_i_2 …α_R_i_M-R_i_0 =

∑_{ i_1,…,i_M  }
α_R_i_0-R_i_1 α_R_i_1-R_i_2 …α_R_i_M-R_i_0

Here $i_0=0$ is fixed and the sum runs over all permutations of the set $\{ 1,\dots,M  \}$. For the assumed $\hat\xi\gg1$ we used $\alpha_{R\pm1}\approx\alpha_R$ as usual.

For $M=1$ we recover (<ref>) as expected. For $M=2$ we obtain a connected $3$-kink correlator:

n^-3 C_R_1,R_2 ≈

2 n^-3 α_R_0-R_1 α_R_1-R_2 α_R_2-R_0 =

2 e^-π[(R_0-R_1)/ξ̂]^2 e^-π[(R_1-R_2)/ξ̂]^2 e^-π[(R_2-R_0)/ξ̂]^2.
While the negative kink-kink correlator (<ref>) keeps pairs of kinks apart, the genuine three-body correlations turn out to be attractive. In general, odd/even kink correlations are attractive/repulsive.

§  SPIN-SPIN CORRELATOR

The dephasing also simplifies the ferromagnetic spin-spin correlator
C^zz_R =
⟨σ^z_n σ^z_n+R ⟩-
⟨σ^z_n ⟩⟨σ^z_n+R ⟩.
Given that $\langle \sigma^z_n \rangle=0$ for symmetry reasons, expressing the spin operators with the Jordan-Wigner fermions (<ref>) allows us to write
C^zz_R =
In a similar way as for the higher order kink correlators, after dephasing the correlator becomes
C^zz_R = Det T.
{\cal T}_{ij}=\langle b_ia_{j+1} \rangle\approx \delta_{1+j-i,0}-2\alpha_{1+j-i}
for $i,j=1,\dots,R$. As ${\cal T}$ is a Toeplitz matrix, asymptotic behavior of the determinant for large $R$ can be obtained using standard methods <cit.> as
C^zz_R ∝exp( -1.54 R/ξ̂ )
cos( √( 4πln2 ) R/ξ̂ - φ_0 ).
This formula is the same as in Ref. QKZteor-e but here its range of applicability is much wider thanks to the extra dephasing during the waiting time $t_w$: $\ln\tau_Q\gg1$, that was assumed in Ref. QKZteor-e, is no longer required to make $l_w$ long enough, see (<ref>). The correlator exhibits damped oscillations in function of $R/\hat\xi$. At short distance the ...-kink-kink-... train appears to have crystalline order. This is consistent with the anti-bunching seen in the kink-kink correlator: subsequent kinks keep safe distance from each other.

§ SUMMARY

The connected kink-kink correlator at the end of the linear ramp is a sum of two terms. One of them is universal. It depends only on the spectrum of quasiparticles excited in the KZ regime, between $t_c\mp\hat t$ near the critical point, that depends only on the slope of the ramp at the critical point and the universal critical exponents. The other term is non-universal. It depends on how long the system is dragged across the ferromagnetic phase where the excited quasiparticles are dephasing by accumulation of quasimomentum-dependent dynamical phases. When the ramp is slowed mid way between the critical point and the zero transverse field for a waiting time much longer than $\tau_Q$ then the magnitude of the non-universal term begins to be suppressed by dephasing.

The dephased kink-kink correlator exhibits strong anti-bunching of kinks. Subsequent kinks along the chain are not allowed to approach each other closer than half of the typical distance between kinks. With dephasing it is possible to obtain also higher order kink correlators. The odd/even correlators turn out to be attractive/repulsive. The same dephasing makes the ferromagnetic spin-spin correlator tractable. In function of a distance it exhibits exponentially damped oscillations. The oscillations are consistent with the anti-bunching seen in the kink-kink correlator.

This research was supported in part by the National Science Centre (NCN), Poland together with the European Union through QuantERA ERA NET program No. 2017/25/Z/ST2/03028.


[table]capposition=top 0.0pt0.0pt

# A Linked Data Scalability Challenge:
Concept Reuse Leads to Semantic Decay

Paolo Pareti

<EMAIL_ADDRESS>Ewan Klein

<EMAIL_ADDRESS>Adam Barker

<EMAIL_ADDRESS>University of Edinburgh
United Kingdom  University of Edinburgh
United Kingdom  University of St Andrews
United Kingdom

###### Abstract

The increasing amount of available Linked Data resources is laying the
foundations for more advanced Semantic Web applications. One of their main
limitations, however, remains the general low level of data quality. In this
paper we focus on a measure of quality which is negatively affected by the
increase of the available resources. We propose a measure of semantic richness
of Linked Data concepts and we demonstrate our hypothesis that the more a
concept is reused, the less semantically rich it becomes. This is a
significant scalability issue, as one of the core aspects of Linked Data is
the propagation of semantic information on the Web by reusing common terms. We
prove our hypothesis with respect to our measure of semantic richness and we
validate our model empirically. Finally, we suggest possible future directions
to address this scalability problem.

###### category:

I.2.4 Artificial Intelligence Knowledge Representation Formalisms and Methods

###### keywords:

Semantic networks

###### keywords:

Linked Data; Semantic Richness; Information Content

##  1 Introduction

Recent years have seen a constant increase in the amount of available Linked
Data resources. While the problem of data availability is gradually reducing,
data quality remains one of the main limiting factors of Linked Data
applications. At the root of this problem lies the open nature of the Web,
where knowledge can be erroneous, incomplete, and where concepts can be used
in different ways between different sources. In this paper we focus on a
particular measure of quality, namely the semantic richness of Linked Data
concepts. Following a number-of-features approach [10], we define our measure
of semantic richness as the amount of facts that we can expect to infer from a
concept.

Web-scale reuse of semantic formalisations, such as concepts and relations, is
an essential aspect of Linked Data [2]. However the openness of Linked Data
results in the paradox that, while the reuse of terms is considered a good
practice, it also progressively decreases the semantic richness of the reused
terms. This trend of diminishing semantics as the usage of a term increases is
a severe limitation to the scalability of Linked Data. In fact, the less
semantically rich a concept becomes the less assumptions can be safely made
about its entities, thus reducing its potential applications.

For example, the concept of _Person_ in a restricted domain, such as an
individual organisation, might follow some particular conventions, such as
having an office number and an address. However, if we extend the usage of
this concept to more and more entities outside the organisation, then it might
not be possible to assume that a _Person_ has an office number, or maybe even
an address. In the Semantic Web, by virtue of the fact that _anybody can say
anything about anything_ , an entity of type _Person_ could be anything. This
problem affects any Linked Data term, including relations. A notable example
is the _owl:sameAs_ 111http://www.w3.org/2002/07/owl#sameAs relation,
originally intended to represent strict equality between two entities, which
is frequently misused to represent a wide range of weaker similarity relations
[5].

The main contributions of this work are two. The first is the validation of
our hypothesis that the more datasets reuse a Linked Data concept, the less
semantically rich it becomes. The second is a measure of semantic richness
that can be used to compare the usage of Linked Data concepts between
different datasets. This problem will be discussed in more detail in Section 2
and it will be contextualised with previous research in Section 3. In Section
4 we define our measure of semantic richness and we prove our hypothesis with
respect to this measure. In Section 5 we validate our hypothesis empirically
to provide insights on the extent of this problem. We will then suggest
possible future directions to address this problem in Section 6.

##  2 Problem Definition

We define a measure of semantic richness $\Gamma$ as a function that takes a
concept $\alpha$, such as _foaf:Person_ 222http://xmlns.com/foaf/0.1/Person
and a Linked Data graph $S$, such as the DBpedia graph [1], and returns a
positive real number. This number quantifies the amount of information
conveyed by the concept $\alpha$ within the graph $S$. In other words, it is a
measure of how many facts we can expect to infer about the instances of that
concept that are found in the graph. If
$\Gamma(\alpha,S_{1})>\Gamma(\alpha,S_{2})$ then knowing that an entity is of
type $\alpha$ entails more facts in the context of the graph $S_{1}$ rather
than $S_{2}$.

When creating such measure it is important to consider scalability and
robustness, as there is large variability in the size and quality of Linked
Data resources. It should be possible to compute this measure effectively over
large amounts of Linked Data. Moreover, this measure should be applicable to
any dataset, even when taxonomic information is not available or when all the
available entities are of a single type. These requirements prevent the
applicability of existing semantic measures such as the Information Content.

Being based on the usage of a concept, rather than on its position within a
conceptual framework such as a taxonomy, this notion differs from the notion
of semantic specificity. For example, while a concept is less semantically
specific (or less informative) than its sub-concepts, it might be more
semantically rich. This situation occurs when the probability of an entity
being a member of a sub-concept depends on the other sub-concepts it is a
member of, such as in case of mutual exclusivity of sub-concepts. For example,
in a dataset $S$ a semantically rich concept $\alpha$ might contain a small
set of entities $s$ of which nothing is known. If we consider those entities
as members of concept $\beta$, we will have that
$\Gamma(\beta,S)<\Gamma(\alpha,S)$ although $\beta$ is a sub-concept of
$\alpha$.

We define $S^{\alpha_{1}}$ and $S^{\alpha_{2}}$ as the sets of entities of
type $\alpha$ contained, respectively, in datasets $S_{1}$ and $S_{2}$. Our
hypothesis is that the reuse of a Linked Data concept results in a decrease of
its semantic richness. Assuming that the sets of entities of type $\alpha$ in
two different datasets are disjoint ($S^{\alpha_{1}}\cap
S^{\alpha_{2}}=\emptyset$), we express this hypothesis as follows:

$\Gamma(\alpha,S_{1}\cup
S_{2})\leq\frac{|S^{\alpha_{1}}|\Gamma(\alpha,S_{1})+|S^{\alpha_{2}}|\Gamma(\alpha,S_{2})}{|S^{\alpha_{1}}|+|S^{\alpha_{2}}|}$
(1)

Intuitively, this inequation states that the semantic richness of a concept
with respect to the union of two graphs is at most equal to the average
semantic richness of the graphs. The difference between these values
represents the amount of information about a concept that is lost when merging
two datasets. In Section 5 we will show how, in practice, the union of two
graphs typically results in a significant decrease of semantic richness.

##  3 Background

The notion of Information Content (IC) has been proposed to measure the
informativeness of concepts within a taxonomy in the field of Information
Theory [11]. This measure has been shown to be effective not only with respect
to concepts, but also to measure the informativeness of resources and
relations within a knowledge base [8]. The IC of a concept $\alpha$ within a
knowledge base is defined as $IC(\alpha)=-log(p(\alpha))$, where $p(\alpha)$
is the probability of an entity belonging to concept $\alpha$. One limitation
to the applicability of this measure is that the probability $p(\alpha)$ might
not be meaningful when considering domain specific datasets. For example,
several existing datasets contain only instances of type _foaf:Person_. The IC
of this concept computed over these datasets will always be equal to $0$ since
$p(\text{\emph{foaf:Person}})=1$. The IC measure would not distinguish between
the different datasets although they might have a different semantic
interpretation of the concept _foaf:Person_. A different measure is therefore
necessary to compare the amount of information conveyed by the concept in each
dataset.

In the field of Cognitive Neuroscience several measures of semantic richness
have been proposed [10]. These measures have been used to explain why certain
concepts can be processed better or faster by humans to perform certain tasks.
We adopt one such measure, called number-of-features (NF), by measuring the
number of facts that characterise a Linked Data concept. The basic intuition
behind the NF measure is that semantically rich concepts tend to be
characterised with more attributes than less semantically rich ones.

Our metric to compute the semantic richness of a concept does not assume the
availability of pre-existing inference rules. In order to calculate how many
facts we can infer about the entities of a given concept we adopt an Inductive
Learning approach. For example, if all the entities of concept $\alpha$ from a
graph $S$ share property $i$, then we can induce the rule that if an entity
belongs to $\alpha$ then it will have property $i$. The applications of
Inductive Learning for Linked Data have received considerable attention in the
recent years thanks to an increasing availability of Linked Data resources.
The main advantages of Inductive Learning approaches are scalability and
robustness to errors and uncertainty [4].

Schema information can be inferred from a knowledge base by considering the
frequency of certain patterns in the data, such as the common properties
shared by the entities of the same type [3]. This schema information can then
be used to evaluate the quality of the knowledge base. For example, it is
possible to evaluate how well a Linked Data graph matches a set of schema
rules by using SPARQL query patterns [7].

When considering data from multiple sources, the decrease of semantic richness
of a concept can be alleviated by annotating data with contextual information.
Several approaches have been proposed to represent contextual information
about Linked Data, in particular with respect to Provenance [9]. For example,
we might be able to infer more knowledge about a set of entities of a concept
if we know that they all originate from DBpedia. One limitation of this
approach is the difficulty of creating and maintaining contextual information.
Moreover, contextual information does not explicitly describe how a concept is
used within a dataset.

##  4 The Semantic Richness Measure

Given a concept $\alpha$, such as foaf:Person, we define a measure of its
semantic richness $G$ which quantifies the amount of information conveyed by
the concept. Unlike IC-based measures [6], our measure is not proportional to
the probability of an entity being a member of the concept. This probability
cannot be estimated reliably for Linked Data due to its open nature and
partial observability. When evaluating the semantic richness of a concept we
only assume the availability of a representative subset of entities of that
concept obtained from a finite number of sources. Taxonomical information
about the concept is not required.

Our measure of semantic richness $G$ is defined as a function of the number of
facts that we can infer about its instances, and about their probability of
being correct. These facts are represented as graph patterns. For example, the
fact that persons have a birth date can be represented by the graph pattern: {
?person a foaf:Person . ?person foaf:birthday ?date . } here represented as a
SPARQL333http://www.w3.org/TR/rdf-sparql-query/ graph pattern using the
FOAF444http://xmlns.com/foaf/spec/ ontology. We define the probability of an
entity of type $\alpha$ from dataset $S$ matching pattern $i$ as
$p(i,\alpha,S)$. The average number of patterns that we can observe in the
entities of type $\alpha$ is the expected value $\mu$ of a Poisson binomial
distribution of the probabilities $p$. We only need to compute this value over
the finite set of patterns $I$ that have been observed among the entities of
the dataset, namely when $p(i,\alpha,S)>0$. The expected value of this
distribution can then be computed as follows:

$\mu=\sum_{i\in I}p(i,\alpha,S)$ (2)

Equation 2 gives us a measure of how many patterns, in average, we can observe
from the entities of type $\alpha$ in dataset $S$. This measure, however, does
not tell us how well we can characterize a concept with a set of common
features. In fact, a high value of $\mu$ could be equally a result of a large
number of infrequent patterns or of a small number of frequent patterns.
However, frequent patterns are more useful than infrequent ones for the
purpose of characterising a concept, for example to learn schema information
using Inductive Learning. Taking this into consideration, we define our
measure of semantic richness with respect to a set of characteristic patterns
$Y\subseteq I$ that define which patterns we can expect entities of type
$\alpha$ to have.

If a pattern $i$ belongs to $Y$, then whenever we observe an entity $\epsilon$
of type $\alpha$ we can infer that it would match pattern $i$. We base our
measure of semantic richness on the correctness of such inferences. If an
entity $\epsilon$ really matches pattern $i$ then we count it as a correct
inference, we count it as incorrect otherwise. For example, we might consider
including the pattern “has a dedicated Wikipedia page" in the set of
characteristic patterns of concept _foaf:Person_. Whenever we observe an
entity of type _foaf:Person_ which has a dedicated Wikipedia page we say that
inferring this pattern is correct; we say it is incorrect otherwise.

Our measure of semantic richness $G(\alpha,S)$ of a concept $\alpha$ with
respect to a dataset $S$ and a set of characteristic patterns $Y$ is the
difference between the expected number of correct inferences and the expected
number of incorrect ones that we can make about an entity:

$\begin{array}[]{l}G(\alpha,S)=\sum_{i\in Y}p(i,\alpha,S)-\sum_{i\in
Y}(1-p(i,\alpha,S))\\\\[5.69054pt] G(\alpha,S)=\sum_{i\in
Y}2p(i,\alpha,S)-1\end{array}$ (3)

From equation 3 it follows that, in order to maximize the value of
$G(\alpha,S)$, $Y$ should be the set of patterns which occur in the majority
of the entities ($\forall i\in Y.p(i,\alpha,S)>0.5$). Therefore equation 4 can
be rewritten as follows:

$\begin{array}[]{l}G(\alpha,S)=\sum_{i\in
I}\begin{cases}2p(i,\alpha,S)-1&\text{if }p(i,\alpha,S)>0.5\\\
0&\text{otherwise }\end{cases}\end{array}$ (4)

The particular threshold of $0.5$ is a result of the fact that in this
measure, which aims at being generic, correct and wrong inferences carry the
same (absolute) weight. Similarly, all patterns carry equal weight. This
formulation of semantic richness is not meant to be unique and variations are
possible. For example, equation 4 could be modified to penalize wrong
inferences more. Also, different patterns could be given different weights to
represent how important or informative they are for a specific application.

In this work we use use our generic measure of semantic richness to analyse
our hypothesis, namely that as we consider more entities as members of a
concept, the semantic richness of that concept decreases. In fact, assuming
that the two sets of entities are disjoint ($S^{\alpha_{1}}\cap
S^{\alpha_{2}}=\emptyset$) the semantic richness of the union of two sources
of data is always inferior or equal to the average individual semantic
richness of the sources:

$G(\alpha,S_{1}\cup
S_{2})\leq\frac{|S^{\alpha_{1}}|G(\alpha,S_{1})+|S^{\alpha_{2}}|G(\alpha,S_{2})}{|S^{\alpha_{1}}|+|S^{\alpha_{2}}|}$
(5)

###  4.1 Proof of Semantic Richness Decrease

Formula 5 can be proved according to our definition of semantic richness
defined in Formula 4. It can be observed that the semantic richness of a
concept is the sum of the individual semantic richness of each pattern
$G^{\prime}(\alpha,S,i)=(2p(i,\alpha,S)-1)[p(i,\alpha,S)>0.5]$. Therefore
Formula 5 can be proved by proving that the inequation holds for every
individual pattern:

$G^{\prime}(\alpha,S_{1}\cup
S_{2},i)\leq\frac{|S^{\alpha_{1}}|G^{\prime}(\alpha,S_{1},i)+|S^{\alpha_{2}}|G^{\prime}(\alpha,S_{2},i)}{|S^{\alpha_{1}}|+|S^{\alpha_{2}}|}$
(6)

Having defined $w_{1}=p(i,\alpha,S_{1})$ and $w_{2}=p(i,\alpha,S_{2})$, then
the probability $p(i,\alpha,S_{1}\cup S_{2})$ can be defined as ratio between
the number of entities of each set which match pattern $i$ and the number of
all the entities in the union of the sets:

$p(i,\alpha,S_{1}\cup
S_{2})=\frac{|S^{\alpha_{1}}|w_{i}+|S^{\alpha_{2}}|w_{2}}{|S^{\alpha_{1}}|+|S^{\alpha_{2}}|}$
(7)

Formula 6 can then be written as follows:

$\begin{array}[]{l}(2\frac{|S^{\alpha_{1}}|w_{i}+|S^{\alpha_{2}}|w_{2}}{|S^{\alpha_{1}}|+|S^{\alpha_{2}}|}-1)[\frac{|S^{\alpha_{1}}|w_{i}+|S^{\alpha_{2}}|w_{2}}{|S^{\alpha_{1}}|+|S^{\alpha_{2}}|}>0.5]\leq\\\\[5.69054pt]
\frac{|S^{\alpha_{1}}|(2w_{1}-1)[w_{1}>0.5]+|S^{\alpha_{2}}|(2w_{2}-1)[w_{2}>0.5]}{|S^{\alpha_{1}}|+|S^{\alpha_{2}}|}\end{array}$
(8)

From Formula 8 it follows that in case $w_{1}\leq 0.5$ and $w_{1}\leq 0.5$
then $p(i,\alpha,S_{1}\cup S_{2})<0.5$ and therefore Formula 8 is reduced to
the tautology $0\leq 0$. This matches the intuition that if a pattern was not
frequent in any of the original datasets, it will not be frequent also in the
union of the datasets.

In case $w_{1}>0.5$ and $w_{1}>0.5$ then $p(i,\alpha,S_{1}\cup S_{2})>0.5$ and
therefore equation 8 becomes:

$\begin{array}[]{l}2\frac{|S^{\alpha_{1}}|w_{i}+|S^{\alpha_{2}}|w_{2}}{|S^{\alpha_{1}}|+|S^{\alpha_{2}}|}-1\leq\frac{|S^{\alpha_{1}}|(2w_{1}-1)+|S^{\alpha_{2}}|(2w_{2}-1)}{|S^{\alpha_{1}}|+|S^{\alpha_{2}}|}\end{array}$
(9)

This equation can be simplified to the tautology $0\leq 0$. This matches the
intuition that a pattern which was frequent in both of the original datasets
will still be frequent in the union of the datasets.

We then consider the last case in which a pattern is frequent only in one of
the original datasets: when $w_{1}>0.5$ and $w_{2}\leq 0.5$. This requires us
to consider two sub-cases, depending on the value of $p(i,\alpha,S_{1}\cup
S_{2})$. If $p(i,\alpha,S_{1}\cup S_{2})\leq 0.5$, then Formula 8 can be
simplified to:

$\begin{array}[]{l}0\leq\frac{|S^{\alpha_{1}}|(2w_{1}-1)}{|S^{\alpha_{1}}|+|S^{\alpha_{2}}|}\end{array}$
(10)

This formula can be further simplified to $w_{1}>0.5$ which is true by
assumption. Finally, in case $w_{1}>0.5$ and $w_{2}\leq 0.5$ and
$p(i,\alpha,S_{1}\cup S_{2})>0.5$ Formula 8 can be simplified to:

$\begin{array}[]{l}2\frac{|S^{\alpha_{1}}|w_{i}+|S^{\alpha_{2}}|w_{2}}{|S^{\alpha_{1}}|+|S^{\alpha_{2}}|}-1\leq\frac{|S^{\alpha_{1}}|(2w_{1}-1)}{|S^{\alpha_{1}}|+|S^{\alpha_{2}}|}\end{array}$
(11)

This equation can be simplified to $w_{2}\leq 0.5$ which is true by
assumption. Having proved Formula 6, and therefore Formula 5, we have proved
our hypothesis (Formula 1) with respect to our measure of semantic richness.

##  5 Experimental Evaluation

Having defined our model to measure semantic richness we perform an empirical
validation of this measure and of our hypothesis. This experimental
evaluation, performed over several large datasets, also provides evidence of
the scalability of our measure. These experiments are based on an
implementation of a system that can sample entities from a dataset and
calculate the semantic richness of a concept within the dataset.

Figure 1: The DBpedia ontology tree represented in function of the semantic
richness of each concept. Each segment represents a subclass relation.

###  5.1 Validation of the Semantic Measure

Our approach uses a numerical measure of the semantic richness of a concept
using equation 4. But how well does this measure actually capture the concept
of semantic richness? We evaluate this measure according to the intuition that
concepts are, in average, less semantically rich than their sub-concepts. A
similar property is also found in the IC measure, as the IC of a concept is,
by definition, always inferior or equal to the IC of its sub-concepts.

We have evaluated how well our measure matches this intuition on the concepts
of the DBpedia ontology555http://dbpedia.org/ontology/ by calculating the
semantic richness of the entities of each concept.666The entities were
extracted from DBpedia version 3.5.1 To reduce the computational complexity,
all concepts with over 1000 entities have been replaced by a randomly sampled
set of 1000 entities. Also, we have considered only simple graph patterns
which involve a single triple. More specifically, we have considered patterns
as pairs $<p,o>$ which match an entity $\epsilon$ if the triple
$<\epsilon,p,o>$ is asserted in the dataset. Improving the quality of this
measure by considering more complex patterns is left for further research.

Figure 1 shows the change of semantic richness between each concept and its
sub-concepts. This graph shows a significant tendency of concepts to be less
semantically rich than their sub-concepts. In fact, 90% of the subclass
relations resulted in an increase of semantic richness from a concept to its
sub-concepts. Also, in all cases concepts resulted in a lower semantic
richness than the average semantic richness of their sub-concepts, thus
supporting our hypothesis. Only in 10% of the cases a subclass relation
resulted in a decrease of semantic richness from a concept to a sub-concept.
This situation occurred because in the DBpedia ontology, sibling concepts are
mutually exclusive. Therefore, the knowledge that an entity belongs to a
certain concept affects the probability of it being a member of another,
potentially more semantically rich, concept.

###  5.2 Validation of the Hypothesis

We have run an experiment to quantify the decrease in semantic richness as
concepts are reused. In this experiment we have considered the concept
_foaf:Person_ as it is used in ten different datasets. The list of the
endpoints used to access the datasets is shown in Table 1. From each endpoint
we have extracted information about 1000 randomly selected entities.

For each dataset, we have calculated its semantic richness at different
states. Starting from the semantic richness of the original dataset, we have
recalculated this measure after adding progressively more entities from the
other datasets. Figure 2 shows the progressive decay of semantic richness as
we add new entities of the same type from other sources. The semantic richness
of each dataset quickly converges to a value which is significantly inferior
to the average measure. While the average semantic richness of all the
individual sources is $9.6$, the semantic richness of the merged set results
is $1.3$. This experiment validates our hypothesis since the semantic richness
of the union of a number of sources is inferior to their average individual
semantic richness. Another fact that can be observed is that different
datasets represent the same concept with significantly different levels of
semantic richness. In this case, the values ranged from $0.7$ of
services.data.gov.uk to $35.2$ of dbpedia.org.

Figure 2: Decrease of semantic richness of the concept _foaf:Person_ of ten
different datasets as we add more entities from the other datasets.
http://dbpedia.org/sparql
---
http://data.nobelprize.org/sparql
http://data.archiveshub.ac.uk/sparql
http://dati.camera.it/sparql
http://data.utpl.edu.ec/ecuadorresearch/lod/sparql
http://lod.sztaki.hu/sparql
http://semantic.eea.europa.eu/sparql
http://data.open.ac.uk/query
http://semanticweb.cs.vu.nl/europeana/sparql
http://services.data.gov.uk/reference/sparql
Table 1: SPARQL endpoints used in the experiment (accessed on 7/3/2015)

##  6 Retaining Semantic Richness

Since Linked Data does not allow negation, it is not possible to prevent an
entity from becoming a member of a concept. It is however possible to predict
the effect that adding an entity to a dataset would have on the semantic
richness of a concept. We define $E$ as the set of patterns that are matched
by entity $\epsilon$. The degree of typicality of an entity with respect to a
concept $\alpha$ can then be defined as $\delta^{\epsilon}=|E\cap Y|/|Y|$
where $Y$ is the set of frequent patterns of $\alpha$. Given two entities
$\epsilon$ and $\epsilon^{\prime}$, we can say that $\epsilon$ is more typical
than $\epsilon^{\prime}$ if $\delta^{\epsilon}>\delta^{\epsilon^{\prime}}$. We
define an entity to be _atypical_ , meaning that including it into a concept
would result in a decrease of its semantic richness, if $\delta<0.5$. If
$\delta\geq 0.5$ we say that the entity is _typical_ , meaning that its
inclusion in the concept would maintain or increase its semantic richness.

This typicality measure is scalable and easily verifiable because it can be
computed automatically over a subset of the data. In fact, any agent can
calculate whether an entity is a typical or an atypical member of a concept.
In a closed domain, this measure could be used to determine which entities
should be considered as members of a concept, and which should not. In Linked
Data, however, it is not possible, nor desirable, to prevent entities from
becoming members of a concept. It is however possible to add more knowledge
about the typical entities to preserve their semantic richness. A possible
approach to do this is to create a sub-concept which groups together the
typical entities.

For example, the particular way in which the concept _foaf:Person_ is used in
DBpedia results in the property that the majority of those entities have an
associated Wikipedia page. If we merge these entities with another source
about persons which are not related with a Wikipedia page, then this property
will no longer hold, thus reducing the semantic richness of the concept. In
order to preserve this property, we can create a new sub-concept of
_foaf:Person_ that captures the particular interpretation of this concept
adopted by DBpedia.

##  7 Conclusion

In this paper we have proposed a measure of semantic richness of Linked Data
concepts that quantifies the amount of facts that can be inferred from a
concept within the scope a particular dataset. This measure differs from
previous semantic measures, such as Information Content, as it is meant to
compare the difference in semantic richness of the same concept between
different datasets. We have validated this measure with respect to the DBpedia
ontology by showing that it captures the expected intuition that sub-concepts
are, in average, more semantically rich than their super-concepts. We have
used the mathematical formulation of this measure to prove our hypothesis that
the more a concept is reused, the less semantically rich it becomes. To verify
the extent of this problem we have compared the semantic richness of the
concept _foaf:Person_ from ten different existing datasets. Our preliminary
results show that the semantic richness of a concept quickly decreases when
merging entities of multiple datasets together. We have also shown that the
same concept can be used in different datasets with significantly different
levels of semantic richness. Since it is not desirable to prevent concepts
from being reused we propose to preserve their semantic richness by organising
their heterogeneous set of entities in a number of sub-concepts. These sub-
concepts can be generated and populated automatically by considering the
effect on their semantic richness that would result after the inclusion of a
new entity.

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