# Composite Spin Approach to the Blockade Effect in Rydberg Atom Arrays

Lei Pan  Institute for Advanced Study, Tsinghua University, Beijing,100084,
China  Hui Zhai<EMAIL_ADDRESS>Institute for Advanced Study, Tsinghua
University, Beijing,100084, China

###### Abstract

The Rydberg blockade induces strongly correlated many-body effects in Rydberg
atom arrays, including rich ground-state phases and many-body scar states in
the excitation spectrum. In this letter, we propose a composite spin
representation that can provide a unified description for major features in
this system. The composite spin combines Rydberg excitation and the auxiliary
fermions, which are introduced to implement the Rydberg blockade constraint
automatically. First, we focus on the PXP model describing one-dimensional
arrays with the Rydberg blocking radius being a lattice spacing. Using
composite spins, the ground state is simply a ferromagnetic product state of
composite spins, and the magnon excitations of these composite spins can
accurately describe the many-body scar states and signal the quantum phase
transition driven by detuning. Then, we show that Rydberg atom arrays with
different blocking radii can share a universal description of the composite
spin representation, and the difference between different blockade radii can
be absorbed in the formation of composite spins.

The Rydberg atom arrays have emerged as a new frontier to study strongly
correlated quantum matters review1 ; review2 ; review3 . Optical tweezers
freeze the spatial motion of atoms, and coherent lasers selectively couple
neutral atoms between its ground state and one of the Rydberg states, making
the system effectively described by a spin-$1/2$ lattice model. The Rydberg
blockade effect introduces strong interactions between spins. In this system,
a series of experiments have realized novel quantum phases, and the critical
behaviors in phase transitions between them exp1 ; exp2 ; exp3 . These
experiments have also discovered intriguing quantum many-body dynamics exp1 ;
exp4 ; exp5 ; exp6 ; exp7 ; exp8 , especially the partial violation of quantum
thermalization phenomenon known as quantum many-body scars exp1 ; PXP_theory ;
NP_papic . These experiments have attracted considerable theoretical attention
to investigating the many-body physics in this model PXP_theory ; NP_papic ;
PRB_Papic ; PXP_theory2 ; PXP_theory3 ; PXP_theory4 ; PXP_theory5 ;
PXP_theory6 ; PXP_theory7 ; PXP_theory8 ; PXP_theory9 ; PXP_theory10 ;
related2 ; PXP_theory11 ; PXP_theory12 ; PPXPP1 ; PPXPP2 ; yao ; Cheng .

The strong blockade effect within the blockade radius makes the model a
strongly correlated one. A key ingredient to treat strongly correlated models
is finding a proper “quasi-particle” that includes the dominative correlation
effect. Then, the system behaves as a weakly interacting one under this quasi-
particle representation. One canonical example is the composite fermions by
attaching magnetic flux to fermions. The fractional quantum Hall effect,
including both the ground state and excitations, then acquires a simple and
unified description with the help of the composite fermions composite-fermion
. Other examples include the slave particles for quantum spin liquids in
frustrated spin models slave_particle  and the dressed impurity cloud for the
Kondo effect Kondo .

Figure 1: (a) Schematic of the Rydberg blockade PXP model and its auxiliary
fermion representation. (b) Schematic of the composite spins. (c) illustrates
the ground state as polarized ferromagnetic state of composite spins, and (d)
illustrates the magnon excitations of composite spins. When $\Delta=0$, the
positive energy states and the negative energy states of the PXP model are
related by a unitary transition. (e) is schematic of the highest energy state,
and (f) lowers energy from (e) and generates state corresponding to (d).

In this letter, we present a composite spin representation for describing the
Rydberg blockade-induced many-body effects in the Rydberg atom arrays. We
introduce auxiliary fermions, because the Pauli exclusion principle of these
fermions can naturally implement the constraints from the Rydberg blockade
Cheng . We combine the fermions with the original spins to form the composite
spins. To demonstrate how it works, we first focus on a one-dimensional array
and consider the Rydberg blockade radius being one lattice spacing, which can
be described by the so-called PXP model. We show that the composite spin
representation provides a simple and unified description of main physics in
the PXP model. The main results are following:

1) The ground state is a polarized ferromagnetic state of the composite spins.

2) The many-body scar states are well described by a set of magnon excitations
of these composite spins.

3) The quantum phase transition driven by detuning is signaled by the magnon
excitation being gapless.

We also show that such a description can be extended to cases in which the
Rydberg blockade radius is multiple lattice spacing.

Composite Spins. For each Rydberg atom, the laser couples the atom from its
ground state (denoted by $\ket{\uparrow}$ here) and a Rydberg state (denoted
by $\ket{\downarrow}$), with the coupling strength $\Omega$ and the detuning
$\Delta$. And the Hamiltonian is given by

$\hat{H}=\sum\limits_{i}\left(\Omega\hat{S}_{i}^{x}+\Delta\hat{S}^{z}_{i}\right).$
(1)

In addition, the Rydberg blockade effect imposes a constraint that there
cannot be more than one Rydberg atom within the Rydberg radius. When the
blockade radius is one lattice spacing, the constraint prevents any two
neighboring spins being both $\ket{\downarrow}$, and it requires that for all
physical states and for all sites $i$,

$\left(\frac{1}{2}-\hat{S}^{z}_{i}\right)\left(\frac{1}{2}-\hat{S}^{z}_{i+1}\right)\ket{\Psi}=0.$
(2)

Eq. 1 and Eq. 2 together gives rise to the PXP model.

Here we present an alternative way to implement the Rydberg blockade
constraint Eq. 2 by introducing spinless fermions sitting at each link Cheng ,
as shown in Fig. 1(a). We write the model as

$\hat{H}=\sum\limits_{i}\left[\frac{\Omega}{2}\left(\hat{S}_{i}^{-}f^{\dagger}_{i-1,i}\hat{f}^{\dagger}_{i,i+1}+\hat{S}^{+}_{i}\hat{f}_{i,i+1}\hat{f}_{i-1,i}\right)+\Delta\hat{S}^{z}_{i}\right],$
(3)

where $\hat{f}^{\dagger}_{i,i+1}$ ($\hat{f}_{i,i+1}$) is the fermion creation
(annihilation) operator at the link between site $i$ and $i+1$. This model
possesses local $U(1)$ gauge symmetry $\hat{f}_{i,i+1}\rightarrow
e^{i\theta_{i}}\hat{f}_{i,i+1}$, $\hat{S}^{+}_{i}\rightarrow
e^{-i\theta_{i}}\hat{S}^{+}_{i}$ and $\hat{S}^{+}_{i+1}\rightarrow
e^{-i\theta_{i}}\hat{S}^{+}_{i+1}$, and it gives rise to local conserved gauge
charge $Q_{i,i+1}=S^{z}_{i}+S^{z}_{i+1}+n_{i,i+1}$, where $n_{i,i+1}$ is the
fermion number. When we focus on the gauge sector with all $Q_{i,i+1}=1$, it
is clear that when the operator $\hat{S}^{-}_{i}$ flips $\ket{\uparrow}$ state
at site $i$ to $\ket{\downarrow}$ state, it simultaneously creates two
fermions at two neighboring links between site $i$ and site $i-1$ (and site
$i+1$). Hence, spin flip can no longer occur in site $i-1$ and site $i+1$
anymore. Hence, we can introduce the composite spins by combining a pair of
fermions with each spin. That is to say, for each site, we define composite
spin $\tau_{i}$ as

$\displaystyle\ket{\Uparrow}=\ket{\uparrow}\otimes|00\rangle,$ (4)
$\displaystyle\ket{\Downarrow}=\ket{\downarrow}\otimes|11\rangle,$ (5)

and we schematically illustrate the composite spins in Fig. 1(b).

Figure 2: The excitation energy of the single magnon excitation as a function
of momentum. The solid blue dots with a solid line is for the PXP model
(blockade radius being one lattice spacing), and the red open dots with a
dashed line is for the PPXPP model (blockade radius being two lattice
spacing). Here the system size of the plot is for $18$ sites.

Ground State. In term of the composite $\tau$-spin, the Hamiltonian simply
reads

$\hat{H}=\sum\limits_{i}\left(\Omega\hat{\tau}_{i}^{x}+\Delta\hat{\tau}^{z}_{i}\right).$
(6)

Note that we do not need to add the Rydberg blockade constraint when the model
is written in terms of $\tau$-spin, because the Fermi exclusion principle
automatically forbids the composite spins in two neighboring sites being both
$\ket{\Downarrow}$, implementing the constraint from Rydberg blockade effect.
Motivated by Eq. 6, it is natural to speculate that the ground state is a
product state of composite spins polarized by an external field, i.e.

$\ket{\text{GS}}=\frac{1}{\mathcal{N}}\prod_{i}\big{(}u\ket{\Uparrow}_{i}+v\ket{\Downarrow}_{i}\big{)},$
(7)

where $\mathcal{N}$ is the normalization factor. This state is illustrated by
Fig. 1(c). Below we will set $u=1$, and then $v$ is the unique parameter in
the wave function.

We first consider the situation $\Delta=0$. The parameter $v$ is determined by
minimizing $E_{\text{GS}}=\bra{\text{GS}}\hat{H}\ket{\text{GS}}$ and we obtain
$v=-0.636$. We also compare the wave function with the ground state obtained
from exact diagonalization (ED), where the overlap between them is as high as
$0.99$. This shows that the wave function Eq. 7 provides a very precise
description of the ground state. Moreover, note that when $\Delta=0$, we can
define a unitary transformation $\hat{U}=\prod_{i}\sigma^{z}_{i}$, and
$\hat{U}\hat{H}\hat{U}=-\hat{H}$. Thus, if $\ket{\Psi}$ is an eigenstate of
$\hat{H}$ with negative eigen-energy $E$, $\hat{U}\ket{\Psi}$ generates an
eigenstate with positive eigen-energy $-E$. Thus,
$\ket{\widetilde{\text{GS}}}=\hat{U}\ket{\text{GS}}$ generates the highest
energy state, which takes the same form as Eq. 7 but $v=0.636$. This state is
illustrated by Fig. 1(e).

Figure 3: (a) Overlap between the many-body scar states and the quantum states
constructed from the $\pi$-magnon excitations. These states are marked in (b)
and (c) with the same colors. (b-c) The spin polarization
$-\langle\hat{S}_{z}\rangle$ along $\hat{z}$ and the entanglement entropy
$\mathcal{S}_{\text{v}}$ between the left and the right half systems. The dots
represent results from ED with system size $N_{\text{s}}=18$. The crosses with
corresponding colors represent results predicted by $\pi$-magnon states. Here
we consider $\Delta=0$ and energy is in unit of $\Omega$.

Magnon Excitation. Since we have obtained a very accurate wave function for
the ground state, now we consider excitations above it. The most natural
excitation is the magnon or spin wave. We define the operator $\hat{\eta}_{i}$
that only acts on $\tau$-spin at site $i$ as

$\hat{\eta}^{\dagger}_{i}(u\ket{\Uparrow}_{i}+v\ket{\Downarrow}_{i})=v\ket{\Uparrow}_{i}-u\ket{\Downarrow}_{i},$
(8)

which is illustrated in Fig. 1(d) and satisfies the condition
$(\hat{\eta}^{\dagger}_{i})^{2}=0$. Then, we consider the magnon excitation
with momentum $k$, and up to a normalization factor, it is defined as

$\ket{k}\propto\hat{\eta}^{\dagger}_{k}\ket{\text{GS}}=\frac{1}{N_{\text{s}}}\sum\limits_{i}e^{ikR_{i}}\hat{\eta}^{\dagger}_{i}\ket{\text{GS}},$
(9)

where $N_{\text{s}}$ is the number of total sites. We compute the energy
$E_{k}=\bra{k}\hat{H}\ket{k}$, and plot $E_{k}-E_{\text{GS}}$ as the single
magnon dispersion in Fig. 2. We can see that for the PXP model, the single
magnon excitation displays a minimum at momentum $\pi$ and acquires a rather
large gap.

Figure 4: (a): The optimal $v$ for our magnon wave function. (b): The overlap
between our magnon wave function and the ground state obtained by ED with
system size $N_{\text{s}}=18$. (c) The gap predicted by our magnon wave
function. (d) The domain wall density and variance for the ground state
predicted by our magnon wave function. Arrows label the locations of expected
quantum phase transition. $\Delta$ is in unit of $\Omega$. Here all data shown
are for $N_{\text{s}}=18$.

We can then define another operator $\hat{\xi}_{i}$ that only acts on
$\tau$-spin at site $i$ as

$\hat{\xi}_{i}(u\ket{\Uparrow}_{i}-v\ket{\Downarrow}_{i})=v\ket{\Uparrow}_{i}+u\ket{\Downarrow}_{i},$
(10)

which is illustrated in Fig. 1(f) and satisfies the condition
$\hat{\xi}_{i}^{2}=0$. We can also introduce

$\ket{\tilde{k}}\propto\hat{\xi}_{k}\ket{\widetilde{\text{GS}}}=\frac{1}{N_{\text{s}}}\sum\limits_{i}e^{ikR_{i}}\hat{\xi}_{i}\ket{\widetilde{\text{GS}}},$
(11)

and it is easy to see that $\ket{\tilde{k}}=\hat{U}\ket{k}$ with the unitary
transformation $\hat{U}$ defined above.

Many-body Scar States. Because of the composite nature, the $\tau$ spins are
not independent because two neighboring $\tau$ spins do not commute. The
residual interactions between magnons can lead to thermalization of these
excitations. For instance, Beliaev damping can occur for magnons away from
momentum-$\pi$. Therefore, we focus on the magnon with momentum-$\pi$. Because
this magnon sits at the bottom of excitation spectrum and possesses a gap from
the ground state, it is more immune to quasi-particle interactions and is more
likely to violate thermalization.

Motivated by this insight, we construct a set of orthogonal states composed of

$\\{\ket{\text{GS}},\hat{\eta}^{\dagger}_{\pi}\ket{\text{GS}},(\hat{\eta}^{\dagger}_{\pi})^{2}\ket{\text{GS}},\dots,(\hat{\eta}^{\dagger}_{\pi})^{N_{\text{s}}/2}\ket{\text{GS}}\\},$
(12)

where $\ket{\text{GS}}$ is the ground state Eq. 7 with $u=1$ and $v=-0.636$
determined by energy minimization. This set of states includes a different
number of magnons from zero to $N_{\text{s}}/2$, resembling the nearly equal
energy spacing between different many-body scar states in the PXP model.
Therefore, we can establish a one-to-one correspondence between this set of
states and the negative energy scar states. For positive energy scar states,
we construct another set of orthogonal states composed of

$\left\\{\ket{\widetilde{\text{GS}}},\hat{\xi}_{\pi}\ket{\widetilde{\text{GS}}},\hat{\xi}_{\pi}^{2}\ket{\widetilde{\text{GS}}},\dots,\hat{\xi}_{\pi}^{N_{\text{s}}/2}\ket{\widetilde{\text{GS}}}\right\\},$
(13)

which are respectively related to states in Eq. 12 by $\hat{U}$ discussed
above.

Remarkably, as shown in Fig. 3, we find high overlaps for all these states
without using any fitting parameter. In Fig. 3(b) and (c), we plot the average
of spin polarization $\langle\hat{S}^{z}\rangle$ and the bipartite
entanglement entropy $\mathcal{S}_{\text{v}}$, where
$\hat{S}^{z}=\frac{1}{N_{s}}\sum_{i}\hat{S}^{z}_{i}$ is the average of spin
along $\hat{z}$, and the bipartite entanglement entropy is obtained by
dividing the system equally into the left and right halves. We compare our
prediction of the $\pi$-magnon wave functions with the ED results. Fairly good
agreements are reached, especially for these states near the bottom and the
top of the energy spectrum. The overlaps slightly decrease, and the degrees of
agreement for $\langle\hat{S}^{z}\rangle$ are slightly reduced when energy
increases (or decreases) toward the middle of the spectrum. This is because
the total magnon number increases when energy increases, and the effect of
residual interactions between magnons also becomes pronounced.

Phase Transition. It is known that the PXP model displays a quantum phase
transition to a $Z_{2}$ symmetry breaking phase as $\Delta$ increases Sachdev
; Fendley ; Zoller . When $\Delta=0$, the magnon excitation acquires a sizable
gap, and mixing between states with different magnon numbers is negligible.
However, as $\Delta$ increases toward the phase transition, the magnon gap is
significantly reduced and approaches zero, and the coupling between states
within the set Eq. 12 can no longer be ignored. Thus, we will slightly modify
our scheme. We still focus on the subspace spanned by Eq. 12. However, instead
of fixing $v=-0.636$ to minimize $\bra{\text{GS}}\hat{H}\ket{\text{GS}}$, we
first keep $v$ as an undetermined parameter and diagonalize the Hamiltonian in
this subspace. The diagonalization procedure mixes states with different
magnon numbers, and generates a new ground state $\ket{\text{GS}^{\star}}$
with energy $E_{\text{GS}^{\star}}(v)$. The optimum $v$ is determined by
minimizing $E_{\text{GS}^{\star}}(v)$ for any given $\Delta$.

Figure 5: (a-b) Schematic of the auxiliary fermion representation and the
composite spins for the PPXPP model. (c-d) The overlap (c) and comparison of
$-\langle\hat{S}_{z}\rangle$ (d) between a set of magnon excitation states and
eigenstates obtained by ED. In (d), dots are results from ED, and crosses are
obtained by magnon wave function. The same color is used for same states
between (c) and (d). Here we keep the total number of magnons up to $4$. All
plots are for $N_{\text{s}}=18$.

Fig. 4(a) shows the optimal $v$ determined in this way from which we can see
that $v$ is always negative, and $|v|$ keeps increasing as $\Delta$ increases.
It displays a kink around the expected critical point
$\Delta_{\text{c}}=0.655$. With the optimal $v$, $\ket{\text{GS}^{\star}}$
always acquires a large overlap ($>0.97$) with the ground state obtained by ED
for all range of $\Delta$, as shown in Fig. 4(b). Most remarkably, we find
that the excitation gap within this subspace becomes vanishing small when
$\Delta>\Delta_{\text{c}}$, signaling the symmetry-breaking nature of the
quantum phase transition. Note that both many-body scars states and the
quantum phase transition in the PXP model can be well captured by the same set
of $\pi$-magnon excitations, which is consistent with the previous study
revealing the connection between scar states at $\Delta=0$ and the low-energy
critical states around $\Delta_{\text{c}}$ yao .

Ref. exp1  measured the so-called “domain-wall” density and its variance. When
two neighboring atoms are both in the ground states, it is counted as a
domain-wall of the $Z_{2}$ symmetry breaking state. We compute these two
quantities using the wave function $\ket{\text{GS}^{\star}}$, as shown in Fig.
4(d). We find that the domain-wall density monotonically decreases as $\Delta$
increases, and its variance displays a peak around $\Delta_{\text{c}}$. This
feature also qualitatively agrees with the experimental observation exp1 .

PPXPP Model. The composite spin picture can be easily generalized to atom
arrays with different Rydberg blockade radii and geometries. For instance, we
consider a one-dimensional array with the Rydberg blockade radius being two
lattice spacing. That is to say, not only two nearest spins but also two next
nearest spins cannot be both in $\ket{\downarrow}$ state. In addition to the
constraint Eq. 2, we need to apply another constraint

$\left(\frac{1}{2}-\hat{S}^{z}_{i}\right)\left(\frac{1}{2}-\hat{S}^{z}_{i+2}\right)\ket{\Psi}=0.$
(14)

This leads to a so-called PPXPP model PPXPP1 ; PPXPP2 . These constraints can
also be taken care of automatically by the Pauli exclusion principle by
introducing auxiliary fermions sitting at sites (instead of links in the case
of the PXP model), as shown in Fig. 5(a). And we rewrite the model as

$\hat{H}=\sum\limits_{i}\left[\frac{\Omega}{2}\left(\hat{S}_{i}^{-}f^{\dagger}_{i-1}\hat{f}^{\dagger}_{i}\hat{f}^{\dagger}_{i+1}+\hat{S}^{+}_{i}\hat{f}_{i+1}\hat{f}_{i}\hat{f}_{i-1}\right)+\Delta\hat{S}^{z}_{i}\right].$
(15)

It is easy to see that the spin flip at site $i$ simultaneously creates three
fermions, which forbids further spin flips at the nearest and the next nearest
sites. Hence, as shown in Fig. 5(b), we modify the definition of the composite
spin $\tau_{i}$ into

$\displaystyle\ket{\Uparrow}=\ket{\uparrow}\otimes|000\rangle,$ (16)
$\displaystyle\ket{\Downarrow}=\ket{\downarrow}\otimes|111\rangle.$ (17)

Such an auxiliary fermions and composite spin description can also be
generalized to a one-dimensional array with even larger blockade radius.

Under the composite spin representation, the Hamiltonian still reads as Eq. 6,
and the ground state ansatz still reads as Eq. 7. When $\Delta=0$, we obtain
$v=-0.498$ by minimizing $\bra{\text{GS}}\hat{H}\ket{\text{GS}}$, and the
overlap between this state and the ground state obtained by ED is also as high
as $0.99$. The magnon excitation is also generated by Eq. 8 and Eq. 9. The
single magnon dispersion is also shown in Fig. 2, which shows the minima of
single magnon dispersion located nearby momentum $\approx 2\pi/3$ and $\approx
4\pi/3$. Following the same stratagem discussed in the PXP model, we consider
a subspace spanned by

$\\{\ket{\text{GS}},\hat{\eta}^{\dagger}_{\frac{2\pi}{3}}\ket{\text{GS}},\hat{\eta}^{\dagger}_{\frac{4\pi}{3}}\ket{\text{GS}},\hat{\eta}^{\dagger}_{\frac{2\pi}{3}}\hat{\eta}^{\dagger}_{\frac{4\pi}{3}}\ket{\text{GS}},\dots\\},$
(18)

and we construct a set of orthogonal states in this subspace and compare them
with the eigenstates obtained by ED. The wave function overlaps are shown in
Fig. 5(c), and the corresponding eigenstates are marked by the same colors in
Fig. 5(d). We also compare the prediction of spin polarization between these
magnon states and the results from ED. These states also show the most
significant deviation from the thermal value in a finite-size system. Like the
PXP model, the agreements also decrease as energy increases toward the middle
of the spectrum because the effect of the quasi-particle interaction becomes
stronger as the magnon number increases. Overall, the agreements are not as
perfect as the PXP model because the PPXPP model confronts more constraints
that induce stronger interaction between quasi-particle.

Summary. The analytical wave function ansatz presented by this work can
provide a unified description of both the ground state and the non-thermal
excited states in the PXP model and can also provide a universal description
of models with different blockade radii. We acknowledge that certain aspects
of our wave function have also been discussed in previous works. The same
ground state has been studied in Ref. related ; related2 ; PRB_Papic , and a
similar but different $\pi$-magnon wave function has been proposed for many-
body scar states PXP_theory6 . However, our work provides a so-far the most
systematical and unified view for Rydberg blockade-induced many-body
phenomena, and the validity of our approach is supported by the high overlaps
between our wave function ansatz and the wave functions obtained by ED. This
wave function ansatz can be extended to study high-dimension models and the
non-equilibrium dynamical processes. The correlation effects imposed by the
Rydberg blockade are automatically taken care of by combining fermions with
spins. Therefore, the composite spins serve as a natural building block,
resulting in an intuitive physical picture.

Acknowledgment. The project is supported by Beijing Outstanding Young Scholar
Program, NSFC Grant No. 11734010 and the XPLORER Prize.

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