Partial confinement in a quantum-link simulator

Confinement is a fundamental property prominently observed in quantum chromodynamics (QCD), where the inter-quark potential increases with their distance [1, 2, 3]. This prevents the existence of isolated quarks due to energetic instability; instead, they prefer to bind together into hadrons, either as mesons (quark-antiquark pairs) or baryons (triplets of quarks). Although the concept originated in QCD, analogous phenomena can also manifest in strongly coupled charges in quantum electrodynamics (QED) [4, 5], i.e., the charge confinement. Dimensional analysis indicates that the dimensionality of the coupling constant is determined by the dimensions of the system. Specifically for (3+1)D, the coupling constant is dimensionless, leading to the deconfined Coulomb potential $\sim 1/r$, where $r$ is the distance between two charges. The deconfinement-confinement phase transition can occur by tuning the coupling strength and the temperature [6]. However, for (1+1)D QED, also known as the Schwinger model, the dimensionality of the coupling constant scales linearly with $r$. Consequently, apart from certain exceptional cases, the confining phase becomes quite prevalent. Furthermore, confinement and deconfinement phenomena also appear in emergent gauge theories from strongly-correlated electrons and recent developed Rydberg atomic arrays [7, 8, 9, 10, 11]. For instance, transition between valence bond solid to spin liquid phase can be understood in a picture of confinement-deconfinement transition of spinons [12, 13]. However, large scale numerical investigation of the real time dynamics of confinement or deconfinement on classical computers is challenging.

Recently, much effort has been made to overcome this barrier through quantum simulation [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24] which leverages systems with discrete degrees of freedom. This includes analog simulations using optical lattices [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46] or trapped ions [47, 48, 49, 50], and digital simulations realized on various quantum computing platforms [51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63].

The quantum link model (QLM) [64] serves as one of the most commonly used approaches to simulate lattice gauge theories, which are based on the Hamiltonian formalism with space discretized while time remains continuous. In QLMs, matter particles are placed on lattice sites, while gauge spins with a finite local Hilbert space are located on the links connecting neighboring sites. The realization of QLM is considered a powerful approach for exploring strongly coupled QED, as strong coupling renders perturbative field theory ineffective, and hence quantum simulation can essentially circumvent the issues encountered by classical simulations, such as the sign problem in quantum Monte Carlo [65]. In these quantum-link simulators, both confinement and deconfinement have been extensively studied, encompassing theoretical [44, 66, 67, 68, 69, 70, 71, 72, 73, 45] and experimental [74, 30] contexts. Particularly for the spin-1/2 QLM, the confinement-deconfinement transition has been experimentally signified in dynamics through tuning the topological angle [30].

In this paper, we delve into an intermediate phenomenon between confinement and deconfinement in (1+1)D QED, called partial confinement, which has hitherto remained unexplored within the context of quantum simulation. It refers to the situation where the confining or deconfining status between charges depends on their relative positions. Our study draws inspiration from the seminal work of S. Coleman in the 1976 [5], which studied half-asymptotic particles in the continuum Schwinger model. Here, we demonstrate that the 1D spin-1 QLM can serve as an excellent platform for observing partial confinement: it retains the essential physics while being simple enough to be realized within the scope of current experimental capabilities. Taking the spin-1 QLM as a background, we introduce the basic concept of partial confinement and discuss the associated emergent physics, both in equilibrium and non-equilibrium dynamics.

It is worth clarifying that the terminology of partial (de)confinement has already been introduced in QCD [75, 76, 77]. Therein, partial confinement refers to a phase where color degrees of freedom split into confined and deconfined sectors, with a subgroup of the SU(N) gauge group becoming deconfined while the remainder stays confined. This is typically characterized by a non-uniform distribution of Polyakov loop phases and the N-dependence of thermodynamic quantities. As such, this notion of partial confinement carries a different physical meaning compared to ours defined above.

The rest of this paper is structured as follows: Sec. II provides a review of the spin-1 quantum link model, detailing its essential physical features. In Sec. III, we delve into the equilibrium properties of partial confinement within the spin-1 QLM. Sec. IV discusses how partial confinement manifests in non-equilibrium dynamics. In Sec. V, we explore the feasibility of experimental realizations using cold atoms trapped in optical super-lattices. A brief summary can be found in Sec. VI.

The spin-1 quantum link chain is characterized by the Hamiltonian [44, 64, 78]

where ${\psi}_{j}$ denotes the local matter fields of fermions, and ${S}_{j}^{z,\pm}$ are the spin-1 Pauli operators representing the gauge spins living on the link between two neighboring sites $j$ and $j+1$. The number of sites $N$ must be even, allowing for the division of fermions into electrons and positrons, yielding the particle-antiparticle picture: for $j\in\text{odd}$, the unoccupied and occupied status of electrons are respectively denoted by empty circles and blue disks in Fig. 1; whereas for $j\in\text{even}$, the corresponding occupation status of positrons are illustrated by empty circles and red disks.

The last two terms in $H$ respectively represent the fermion mass with $m\geq 0$ and the electric-field energy $g\sum_{j}E_{j}^{2}$ with $g\geq 0$ and

A local $E_{j}$ consists of two parts: $S_{j}^{z}$ represents the quantized electric field capable of adopting three states $|s_{j}=-1\rangle$, $|0\rangle$, and $|1\rangle$. The factor $(-1)^{j}$ indicates that the electric field $E_{j}$ depends on the gauge spins in an alternating manner: $E_{j}$ is aligning with $S_{j}^{z}$ for $j\in\text{odd}$, while they differ by a minus sign for $j\in\text{even}$. The c-number $\theta$ is called the topological angle [5, 79, 80, 81], which reflects the influence of an external static electric field. Thereby, the eigenvalues of $E_{j}$ can also take three real values, i.e., $\epsilon_{j}=(-1)^{j+1}s_{j}^{z}+\theta/2\pi$. Restricting $E_{j}$ within finite status is a key advantage of the QLM, as it facilitates experimental simulation of electric fields using a finite number of discrete degrees of freedom (such as cold atoms with internal spins). In Fig. 1, the black arrows on links represent the electric-field state $\epsilon_{j}$ in the case of $\theta=0$, where each left(right)-pointing arrow denotes $\epsilon=-1/2(1/2)$. Orange arrows depict the background field of $\pm 1/2$, corresponding to the cases of $\theta=\pm\pi$, respectively. A pair of opposing arrows on the same link can mutually cancel each other out.

The first term in Eq. (1) characterizes the matter-gauge interaction. This term provides the Schwinger mechanism, i.e., a pair of electron and positron merge together accompanied by the emission of gauge photons, as well as its reverse process. Photon creation/annihilation is reflected in the change of spin states via $S_{j}^{\pm}$.

The spin-1 QLM exhibits a U(1) local gauge symmetry generated by the local Gauss operator

satisfying $[G_{j},H]=[G_{j},G_{k\neq j}]=0$. This ensures the invariance of the Hamiltonian under arbitrary U(1) gauge transformations $U_{j}=\exp(i\phi_{j}G_{j})$. As per Eq. (3), $G_{j}$ is defined within a building block consisting of two gauge fields $\{E_{j-1},E_{j}\}$ and a matter field $\psi_{j}$ in the middle [see the box with dashed lines in Fig. 1]. The quantum number of $G_{j}$, denoted by $q_{j}$, is called the static gauge charge, which is apparently a good quantum number. $q_{j}$ characterizes the difference between the net electric flux $E_{j}-E_{j-1}$ and the matter charge $\psi_{j}^{\dagger}\psi_{j}$. The additional factor $(-1)^{j}$ arises from the opposite matter charges carried by electrons and positrons. The U(1) gauge symmetry divides the total Hilbert space into several gauge sectors, each labeled by a unique set of gauge charges $\mathbf{q}=\{q_{1},q_{2},...,q_{N-1}\}$. Notably, the sector with $\mathbf{q}=\mathbf{0}$ is called the physical sector, as now Eq. (3) aligns with the traditional Gauss’s Law in the classical electrodynamics.

In some literature [44, 82, 48, 83], the QLM [Eq. (1)] is presented in an alternative form within the particle picture, described by the Hamiltonian with staggered mass [84, 85]

which relates to the particle-antiparticle Hamiltonian $H$ [Eq. (1)] through a particle-hole transformation on odd sites, i.e. $\psi_{j\in\text{odd}}\mapsto\psi^{\dagger}_{j\in\text{odd}}$, as well as a transformation on even gauge spin $S^{+}_{j\in\text{even}}\mapsto-S^{-}_{j\in\text{even}}$ and $S^{z}_{j\in\text{even}}\mapsto-S^{z}_{j\in\text{even}}$. In this framework, $\psi^{\dagger}_{j}\psi_{j}$ at odd sites represents the occupation below the Dirac sea, thereby exhibiting a negative energy (mass) $-m$. The creation of a hole $\psi_{j}$ in the particle picture is equivalent to the creation of an electron $\psi_{j}^{\dagger}$ in the particle-antiparticle picture. Note that both Hamiltonians, $H$ and $\tilde{H}$, are mathematically equivalent for calculation purposes. Therefore, we proceed with our following analysis using the particle-antiparticle picture.

The confining effects can be clearly demonstrated by the properties of equilibrium states in the physical sector $\mathbf{q}=\mathbf{0}$. We first focus on the simplest case of $J=0$, where the matter and gauge fields are decoupled, thereby $\psi^{\dagger}_{j}\psi_{j}$ and $S_{j}^{z}$ being conserved. We insert a pair of test electron and positron into the vacuum, separated by a distance $d$, as schematically shown in Fig. 2(a). $d$ can be positive or negative, with $d>0$ indicating the electron is to the left of the positron, and vice versa. The case of $d=0$ is excluded by the Pauli exclusion principle. According to Gauss’s Law [Eq. (3)], the system is in the string state

where an electric string exists between the matter charges [see Fig. 2(a)]. The notation convention in a building block is $|_{s_{j-1}},n_{j},~{}_{s_{j}}\rangle$, with $s_{j}$ and $n_{j}$ being the quantum numbers of $S^{z}_{j}$ and ${\psi}^{\dagger}_{j}{\psi}_{j}$, respectively. For a sufficiently large $m$, string state is the ground state, as $m>0$ does not favor matter-particle excitations.

The confining property is determined by the variance of the state energy $\mathcal{E}=\langle\psi|H|\psi\rangle$ on $d$, where the topological angle $\theta$ plays a crucial role. To be more specific, the string state has energy

where $\mathrm{sgn}(x)$ is the sign function being defined as $\mathrm{sgn}(x>0)=1$ and $\mathrm{sgn}(x<0)=-1$. When $\theta=0$, $\mathcal{E}_{\text{str}}=2m+g|d|$ which is linearly proportional to $|d|$, irrelevant to the sign of $d$, which is a typical nature of confinement. The dependence of $\mathcal{E}_{\text{str}}$ on $d$ is numerically confirmed by the line with circles in Fig. 3(a). Pictorially, as depicted in Fig. 2(a), confinement manifests as an increase in the length of the string with local $|\epsilon_{j}|=1$, leading to an increment in the total electric energy. The string tension is defined as $\rho=\partial\mathcal{E}/\partial{|d|}$, which evaluates to $\rho_{\text{str}}=g$ for the string state.

The energy instability of the string state manifests in a possible decay into the meson state with lower energy. The meson configuration is

as illustrated in Fig. 2(a), which results from the binding of test charges to their nearest anti-particles, with the inter-particle electric string being screened. Hence, the decay process is also called the string breaking. The meson state has an energy

which is notably independent of the distance $|d|$. For $\theta=0$, the energy simplifies to $\mathcal{E}_{\text{meson}}=4m+2g$, as indicated by the line with triangles in Fig. 3(a). This implies a transition point

For $|d|<|d_{c}|$, the string state $|\psi_{\text{str}}\rangle$ has lower energy, as it involves fewer matter particles compared to the meson state; however, for $|d|>|d_{c}|$, the meson state becomes more energetically favored. For $\theta=0$, the transition point is where the two curves converge with $|d_{c}|=2+2m/g$, as shown in Fig. 3(a).

Partial confinement occurs at $\theta=-\pi$, where the string-state energy simplifies to

Accordingly, the string tension is $\rho_{\text{str}}=2g$ for $d>0$, and $\rho_{\text{str}}=0$ otherwise. It is clearly indicated that the confining effect only occurs for $d>0$. For $d<0$, $\mathcal{E}_{\text{str}}$ is independent of $d$, suggesting a deconfinement with $d_{c}=\infty$. This phenomenon, that the confining property depends on the relative positions of the opposite charges, is termed the partial confinement. Visually, as illustrated in Fig. 2(b), the string states for $d>0$ and $d<0$ are subjected to different electric potentials. For the former, the energy density within the string is $g(3/2)^{2}$, while that of the vacuum is $g(1/2)^{2}$, resulting in a non-vanishing string tension $\rho_{\text{str}}=2g$, which is twice the value corresponding to the $\theta=0$ case. For the latter, the electric fields inside and outside the string have opposite signs but with the same energy density $g(1/2)^{2}$, thus making $\mathcal{E}$ being $|d|$-invariant. In Fig. 3(b), the line with squares depicts the variation of the string state energy $\mathcal{E}_{\text{str}}^{\theta=-\pi}$ as a function of $d$, while the line with left-pointing triangles represents the meson state energy $\mathcal{E}_{\text{meson}}^{\theta=-\pi}$. It is evident that the string breaking can only occur in the confined regime $d>0$, with $d_{c}=2+m/g$. The value of $d_{c}$ is smaller compared to the $\theta=0$ case, as shown in Fig. 2(b), due to the larger string tension.

Partial confinement also occurs at $\theta=\pi$, but the dependence of quantities (such as $\mathcal{E}$ and $\rho$) on the sign of $d$ is opposite to the case of $\theta=-\pi$ case, as illustrated in Fig. 2(c) and Fig. 3(b) [see lines with diamonds and squares]. The underlying mechanism can be understood in the following way. For the Hamiltonian (1), $\theta\rightarrow-\theta$ is equivalent to $S_{j}^{+}\rightarrow S_{j+1}^{+},S_{j}^{z}\rightarrow S_{j+1}^{z}$ and $\psi_{j}\rightarrow\psi_{j+1}$, the latter corresponds to the charge conjugation $\mathcal{C}$. As a result, the physics under $\theta=-\pi$ with a given $d$ is reproduced by $\theta=\pi$ with $-d$. This suggests that reversing $\theta$ alone can switch between confinement and deconfinement scenarios without the need to adjust the spatial ordering of charges. It would facilitate experimental observation of partial confinement since tuning the topological angle (namely tuning the external field) is generally more accessible than manipulating the particle positions in practice.

For other cases with $\theta\in(-\pi,\pi)$ and $\theta\neq|\pi|$, the string state is generally confined according to Eq. (6), with $\mathcal{E}_{\text{str}}$ for various $\theta$ being shown in Fig. 3(c). $\mathcal{E}_{\text{str}}$ is asymmetric about $d=0$, with the corresponding string tension being given by $\rho_{\text{str}}=g[1-(\theta/\pi)\mathrm{sgn}(d)]$. The asymmetry of $\mathcal{E}(d)$ and $\rho_{\text{str}}(d)$ for $\theta\neq 0$ originates from the breaking of both $\mathcal{C}$ and $\mathcal{P}$ symmetry due to the topological angle $\theta$, where $\mathcal{P}$ is the parity operator acting as $S_{j}^{+}\rightarrow S_{-j-1}^{+},S_{j}^{z}\rightarrow S_{-j-1}^{z},\psi_{j}\rightarrow(-1)^{j}\psi_{-j}$ on the Hamiltonian (1). In Fig. 3(d), we fix $d=\pm 12$ and display the dependence of $\rho$ on $\theta$. One can clearly observe that the partial confinement begins to occur at $\theta=\pm\pi$, where $\rho_{\text{str}}=0$. For a larger $\theta$, i.e., $|\theta|>\pi$, the string state becomes deconfined with a negative string tension. This is also intuitive, as a strong background electric field would polarize the charging pair and yield a large dipole moment.

The above results for $J=0$ will not be qualitatively altered when we turn on the matter-gauge interaction $J$. When $J$ is finite, the system lacks integrability, causing a resort to numerical calculations. By setting $J=g=1$ and $m=6g$, we numerically calculate the energy spectrum of the system. Although quantum fluctuations render $s_{j}^{z}$ and $n_{j}$ no longer good quantum numbers, we can still identify low-energy string-like and meson-like states, with the averaged local observables, such as $\langle S_{j}^{z}\rangle$ and $\langle\psi_{j}^{\dagger}\psi_{j}\rangle$, resembling the configurations of string and meson states shown in Fig. 3(b). Intuitively, the string remains but is thickened by quantum fluctuations. In Fig. 4(a), we present the energy variance of these two eigenstates $\mathcal{E}$ on $d$ for various $\theta$; Additionally, Fig. 4(b) shows the corresponding string tension $\rho$ as a function of $\theta$ for a fixed $d=\pm 12$. A comparison between panels Fig. 3(b) and Fig. 4(a), as well as Fig. 3(d) and Fig. 4(b), clearly demonstrates that the main physical results, such as partial confinement and string breaking, are qualitatively preserved for a non-vanishing $J$.

It may also be necessary to elucidate the differences between the spin-1 QLM discussed here and the spin-1/2 QLM which has been extensively studied both theoretically and experimentally [27, 28, 29, 66, 67, 30, 44, 10, 86, 87]. The Hamiltonian of the spin-1/2 QLM has the same form as Eq. (1), but with the spin operators $S_{j}$ being spin-1/2 Pauli operators (up to a constant factor). In this case, even without an external electric field, the string state is no longer well-defined, as there always exist electric strings between the charges (inner string) and outside the charges (outer string). Commonly, the spin-1/2 QLM with the outer string pointing to the right (left) is considered to have an inherent topological angle $\theta=\pi$ ($\theta=-\pi$) [10, 66, 67, 30]. In contrast to the spin-1 case, the charge conjugation $\mathcal{C}$ now would simultaneously change the order of the charges and the sign of $\theta$, rendering the total energy $\mathcal{E}$ irrelevant to the sign of $d$. In this sense, the spin-1 QLM may serve as a better platform for the study of partial confinement, as it allows for independently changing the charge ordering and $\theta$.

We finally discuss the potential experimental realization using ultracold atoms. The spin-1 QLM has been theoretically proposed to be engineered from a generalized Bose-Hubbard model (BHM) [88], which can be implemented with ultracold bosonic gases confined in a superlattice, as illustrated in Fig. 9(a). Specifically, the atomic gas is governed by the Hamiltonian

where $L=2N$ is the total number of lattice sites, $b_{l}$ and $b_{l}^{\dagger}$ are local bosonic operators satisfying $[b_{k},b_{l}^{\dagger}]=\delta_{k,l}$, and $n_{l}=b_{l}^{\dagger}b_{l}$. $\tilde{J}$ is the hopping between neighboring sites, $\delta$ creates energy offsets between matter sites and gauge spins, and $\gamma$ serves as a tilted potential. $\delta$ and $\gamma$ help to eliminate gauge-breaking hoppings. $\chi_{l}$ is a four-site periodic term employed to realize the topological angle [67], i.e.,

where $\chi=g{\theta}/{\pi}$. In the second line of Eq. (16), $U$ is the on-site interaction, $V$ and $W$ are respectively the nearest-neighbor and next-nearest-neighbor interactions.

The generalized Bose-Hubbard model [Eq. (16)] serves as the foundation for realizing the spin-1 QLM [Eq. (1)] of length $N$. In this setup, the even lattice sites with $l\in\text{even}$ can only be singly occupied or empty, representing matter particles; whereas the occupancies at odd sites are restricted to $n_{l\in\text{odd}}=\{0,2,4\}$ to realize the three gauge spin states. The detailed mapping relations between the two models are presented in Fig. 9(b). To realize the gauge invariance in the $\mathbf{q}=\mathbf{0}$ sector, we focus on the following three configurations:

where the notation $|n_{l},n_{l+1},n_{l+2},n_{l+3}\rangle_{\text{BHM}}$ denotes the particle number representation of the BHM occupation status of four consecutive sites for $l\in\text{even}$, and the $|n_{j},~{}_{s_{j+1}},n_{j+2},~{}_{s_{j+3}}\rangle_{\text{QLM}}$ represents the corresponding matter and gauge configuration in the QLM, where $j=l/2$. When $\tilde{J}=0$, the bare energy of the three configurations $|C_{1,2,3}\rangle$ are

where the $\pm$ determined by the bosonic lattice site index $l$ according to Eq. (17). Based on this, all head-to-tail combinations of the three configurations span the entire Hilbert space in the physical sector.

Notably, there exist configurations that do not preserve gauge invariance. To circumvent these states, it is necessary to ensure that the bare energies of gauge-invariant configurations are nearly resonant, while those of the gauge-violating configurations are far-detuned from resonance. In the case of $U,V,W,\gamma\gg\tilde{J}$, it requires that

By applying the Schrieffer-Wolff transformation, we can derive the effective Hamiltonian of the Bose-Hubbard model within the gauge-invariant subspace, which takes the form of Eq. (1). The detailed coefficient relations are given by

where we have omitted the term $\propto\tilde{J}^{2}$ in Eq. (21a) and Eq. (21b) due to the perturbative nature of $J$. Additionally, $\chi$ is considered negligible in the expression of Eq. (21c) as $|\delta\pm\gamma|\gg|\chi|$ is satisfied [67].

The extreme vacuum state $|\psi_{\text{ev}}\rangle$ forms the basis for preparing string states, which is defined by

This state, as shown in Fig. 9(c1), can be prepared systematically following a well-defined protocol [67, 26, 89]. The protocol begins with the system in a uniform superfluid (SF) state. The lattice potential is then gradually modified to establish the desired staggered structure by ramping the parameters $\gamma$ and $\chi$. Following this, the ratio $U/\tilde{J}$ is tuned to induce a phase transition from the SF to a Mott insulator state, where deep lattice sites achieve a four-particle occupancy, shallow sites remain vacant, and even sites have an average occupancy satisfying $0<\langle n_{l\in\text{even}}\rangle<4$. Thereafter, spin-selective techniques [89] are applied to selectively remove particles from the even sites, resulting in the formation of the extreme vacuum state.

The string state $|\psi_{\text{str}}\rangle$ [Eq. (11)] is distinguished from $|\psi_{\text{ev}}\rangle$ by modifications only at the boundaries. Employing single-site addressing techniques [67, 26, 90] enables the generation of particle-antiparticle pairs at the boundaries of the vacuum state, as depicted in Fig. 9(c2). Subsequently, by locally removing the outer particles through a laser-induced resonant excitation, the string state can ultimately be obtained, as shown in Fig. 9(c3). At the left end of the chain, the left outer gauge sector (enclosed by the red box), with configuration $|012\cdots\rangle_{\text{BHM}}\leftrightarrow|_{-1}0_{0}\cdots\rangle_{\text{QLM}}$, does not belong to the physical sector $q=0$. This ensures the rest of the chain (marked by the blue frame) operates consistently within the physical sector and experiences a hard-wall boundary. A similar strategy is also employed at the right end of the chain to achieve the complete particle distribution and boundary condition required for the string state.

To conclude, we have presented a comprehensive investigation of partial confinement on the platform of the spin-1 quantum link model, a promising platform realizable with cold atoms in optical superlattices. The partial confinement is characterized by a dependency of confinement properties on the spatial arrangement of charged particles, manifested by the asymmetry of the equilibrium energy and the string tension of the string state with respect to the charge ordering. In the non-equilibrium dynamics, both string and meson states exhibit strikingly distinct dynamical features depending on their (de)confining status, as reflected in such quantities as local fermion occupations, bipartite entanglement entropy, and edge charge correlations. We have also elucidated that manipulating the topological angle can be an effective proxy for controlling charge ordering, thereby simplifying experimental procedures by obviating the need for direct charge manipulation. Given that the quantum link model is amenable to current experimental capabilities, our study offers a strategic avenue for exploring novel physics in gauge theories using state-of-the-art quantum simulators. Furthermore, it may also be interesting to discuss the partial confinement in other forms of lattice gauge theories such as the improved Hamiltonian [91, 92].