Gauge Violation Spectroscopy in Synthetic Gauge Theories

Gauge theories play a central role in modern physics. On one hand, gauge theories provide a unified description of fundamental interactions between elementary particles within the Standard Model [email protected] . On another hand, gauge fields emerge from the low-energy effective theories of strongly correlated condensed matter [email protected] ; [email protected] . For example, the Chern-Simons gauge field can effectively describe the behavior of fractional quantum Hall fluids [email protected] . Gauge fields arise naturally as the slave-particle technique is applied to the quantum magnets. In quantum information theory, Kitaev’s toric code is a $Z_{2}$ lattice gauge model [email protected] . Despite the success of gauge theories, studying the real-time dynamics of gauge fields is a notable challenge due to the limit of the classical computational methods. To overcome these limitations, synthetic gauge fields have been implemented on quantum simulators based on ultracold atoms in optical lattices [email protected] ; [email protected] ; [email protected] ; [email protected] ; [email protected]&critical ; [email protected] , trapped ions  [email protected] , or superconducting qubits  [email protected] ; [email protected] .

The key concept of gauge theories is the local gauge symmetry, $\left[\hat{G}(\mathbf{r}),\hat{H}\right]=0$, where $\hat{G}(\mathbf{r})$ is the local gauge transformation and $\hat{H}$ is the Hamiltonian of the system. Local gauge symmetry separates the Hilbert space into disconnected sectors labelled by local gauge charge $\hat{Q}(\mathbf{r})$, the generator of $\hat{G}(\mathbf{r})$, see Fig.1. In real world, we are living in the so-called physical sector with vanishing gauge charge $\hat{Q}(\mathbf{r})=0$. Projecting into the physical sector enforces an extensive number of local constraints between matter and gauge fields, which is nothing but the Gaussian law. However, in synthetic gauge theories on quantum simulators, local gauge charge $\hat{Q}(\mathbf{r})$ is not restricted to the physical sector. It can even fluctuate due to inter-sector superposed initial states or gauge violation perturbations [email protected] ; [email protected] ; [email protected] . Such fluctuations lead to richer gauge violation physics in synthetic gauge theories. For example, disorder-free localization can emerge in synthetic gauge theories by preparing the initial states as the superposition of several sectors [email protected] ; [email protected] ; [email protected] ; [email protected] . If the Gauss’s law is not imposed, the ground state of the $Z_{2}$ lattice gauge theory forms the charge density wave in the non-physical sector [email protected] . Besides, allowing transitions between different sectors can also lead to exotic phase transition that doesn’t exist in real gauge theories [email protected] .

Spectroscopy measurement is a powerful technique to detect the excitations in both real materials and quantum simulators. It has been widely used to probe the spectrums of single-particle and collective excitations, which determine the phases and dynamics of a given quantum system. For example, the angle-resolved photoemission spectroscopy (ARPES) [email protected] has been applied to study the pseudo gap of high-$T_{c}$ cuprates and edge states of topological insulators. In ultracold atomic gases, radio frequency (RF) spectroscopy [email protected] is used to measure single-particle spectrums of Fermi gases [email protected] ; [email protected] ; [email protected] ; [email protected] ; [email protected] and Fermi/Bose polarons [email protected] ; [email protected] ; [email protected] ; [email protected] ; [email protected] . Several spectroscopy techniques have also been developed in other quantum simulation platforms, such as trapped ions [email protected] ; [email protected] and superconducting qubits [email protected] . However, the spectroscopy study of synthetic gauge theories on quantum simulators is still missing.

In this paper, we propose the gauge violation spectroscopy in synthetic gauge theories on quantum simulators. By gauge violation, we mean that the measurement induces a transition between different gauge sectors, see Fig.1. We demonstrate that the usual single-particle spectroscopy measurement process (such as RF spectroscopy in ultracold atomic gases) of synthetic gauge theories is gauge violation rather than gauge invariant. Besides, the gauge invariant spectroscopy needs highly non-local probes, and is challenging in current simulators. Furthermore, we show that the gauge violation spectroscopy exhibits no dispersion, as it violates the Gauss law. Using three models that possess local $U(1)$ gauge symmetries as examples, we show that the gauge violation spectroscopy can be used to detect the confinement and deconfinement phases in gauge theories. In the confinement phase, gauge violation spectrum is nearly a delta function, while in the deconfinement phase, it exhibits a finite width.

Concept. We know that the single-particle absorbing and emission spectrum function can be obtained from the Fourier transformation of the following Green’s functions

These Green’s functions are defined as

where $\hat{\Psi}(\mathbf{r},t)=e^{iHt}\hat{\Psi}(\mathbf{r)}e^{-iHt}$ is the matter field operator in Heisenberg picture, and $\varepsilon=+1(-1)$ for Bosonic (Fermionic) matter field. Here we choose $\left|\psi_{0}\right\rangle$ to be the ground state in the physical sector, i.e. the sector with vanishing gauge charge $\hat{Q}(\mathbf{r})=0$. These Green functions describe the process that adding one particle (hole) to the system at position $\mathbf{r}_{1}$, and removing one particle (hole) at position $\mathbf{r}_{2}$ after evolution time $t$. In synthetic gauge theories, the gauge field can not adjust to follow the charge we add (remove). Thus this process violates the Gaussian law, and excites the system away from the physical sector. More specifically, the matter field operator is not invariant under a local gauge transformation, thus $\left[\hat{\Psi}\left(\mathbf{r}^{\prime}\right),\hat{Q}(\mathbf{r})\right]\neq 0$. One finds $\hat{Q}(\mathbf{r})\hat{\Psi}^{\left({\dagger}\right)}\left(\mathbf{r}^{\prime}\right)\left|\psi_{0}\right\rangle=\pm e\delta(\mathbf{r}-\mathbf{r}^{\prime})\hat{\Psi}^{\left({\dagger}\right)}\left(\mathbf{r}^{\prime}\right)\left|\psi_{0}\right\rangle$. Note that the state $\hat{\Psi}^{\left({\dagger}\right)}\left(\mathbf{r}^{\prime}\right)\left|\psi_{0}\right\rangle$ is no longer in the physical sector as shown in Fig.1(b). Then the Green’s functions can be simplified into

where $\hat{H}_{\xi}\equiv\hat{H}\left[\hat{Q}(\mathbf{r})=\xi e\delta(\mathbf{r-r}_{1})\right]$, $\xi=\pm$, is the Hamiltonian in the non-physical sectors, and energy of $\left|\psi_{0}\right\rangle$ is set to be zero. Here the delta-function $\delta(\mathbf{r}_{2}-\mathbf{r}_{1}\mathbf{)}$ is due to the fact that if $\mathbf{r}_{2}\neq\mathbf{r}_{1}$, the state will not go back to the physical sector, $\hat{Q}(\mathbf{x})\hat{\Psi}\left(\mathbf{r}_{2}\right)\hat{\Psi}^{{\dagger}}\left(\mathbf{r}_{1}\right)\left|\psi_{0}\right\rangle=\left\{\delta\left(\mathbf{x-r}_{2}\right)-\delta\left(\mathbf{x-r}_{1}\right)\right\}\hat{\Psi}\left(\mathbf{r}_{2}\right)\hat{\Psi}^{{\dagger}}\left(\mathbf{r}_{1}\right)\left|\psi_{0}\right\rangle$. As a result, the gauge violation spectrum exhibits no dispersion $\mathcal{A}_{\mathrm{ab/em}}(\mathbf{k},\omega)=$ $\mathcal{A}_{\mathrm{ab/em}}(\omega)$.

In contrast to gauge violation spectroscopy, to calculate the gauge invariant spectrum, one needs bound the matter field operator to the gauge fields $\hat{\Psi}(\mathbf{r)\longrightarrow}\hat{\Psi}(\mathbf{r)}e^{i\int d^{d}x\mathbf{E}_{\mathrm{cl}}(\mathbf{x})\cdot\mathbf{\hat{A}}(\mathbf{x})}$, where $\mathbf{E}_{\mathrm{cl}}(\mathbf{x})$ is a classical electric field satisfying $\nabla\cdot\mathbf{E}_{\mathrm{cl}}(\mathbf{x})=\delta\left(\mathbf{x-r}\right)$. It is invariant under gauge transformation, and commutes with the gauge charge $\left[\hat{Q}(\mathbf{r}),\hat{\Psi}(\mathbf{r}^{\prime}\mathbf{)}e^{i\int d^{d}x\mathbf{E}_{\mathrm{cl}}(\mathbf{x})\cdot\mathbf{\hat{A}}(\mathbf{x})}\right]=0$. We note that in most quantum simulators of synthetic gauge theories, it is hard to perform the gauge invariant spectroscopy. Since it is challenging to excite such highly non-local excitations. Therefore the commonly used techniques, such as RF spectroscopy, probes the gauge violation spectrum rather than gauge invariant spectrum. Furthermore, we will show that the gauge violation spectroscopy can be used to detect the confinement and deconfinement phases in synthetic gauge theories.

Two Schwinger-like Models. In the following, we will present two one-dimensional models with U(1) local gauge symmetry, which can be both exactly solved via the Bosonization method. One is the celebrated Schwinger model. Its Hamiltonian is given by

where $\hat{H}_{\mathrm{mat}}=\int dx\hat{\Psi}^{{\dagger}}(x)\sigma_{z}[-i\partial_{x}-e\hat{A}(x)]\hat{\Psi}(x)$ and the fermion fields have two components $\hat{\Psi}=(\hat{\Psi}_{R},\hat{\Psi}_{L})^{T}$. The Schwinger model describes the (1+1) dimensional QED [email protected] ; [email protected] , and exhibits charge confinement phenomena. Another is similar to the Schwinger model. Its Hamiltonian is given by

Note that it possesses a modified Maxwell term. The corresponding energy density of gauge field is proportional to the square of electrical field gradient rather than the square of electrical field. This modified Maxwell leads to the deconfinement of charges in this model.

These two models possess the local $U(1)$ gauge symmetries. The corresponding gauge transformation operators are both $\hat{G}=\exp\left\{i\int dx\left[e\hat{\rho}(x)\theta(x)+\hat{E}(x)\partial_{x}\theta\right]\right\}$, where $\theta(x)$ is an arbitrary phase distribution function, and $\hat{\rho}(x)=\hat{\Psi}^{{\dagger}}(x)\hat{\Psi}(x)$ is the particle density operator. Since $\left[\hat{G},\hat{H}^{\mathrm{c(d)}}\right]=0$, the generator of this gauge transformation, $\hat{Q}(x)=\partial_{x}\hat{E}/e-\hat{\rho}(x)$ is a conserved quantity, which is also called gauge charge. In the physical sector, $\hat{Q}(x)=0$, the conservation leads to the Gaussian law in one dimension, $\partial_{x}\hat{E}=e\hat{\rho}(x)$.

We use the Bosonization method to deal with these two models. The Bosonization method maps one dimensional fermions to a problem of bosonic fields [email protected] . Here both $\hat{H}^{\mathrm{c}}$ and $\hat{H}^{\mathrm{d}}$ can be bosonized in arbitrary gauge sector. As discussed above we only focus on the sector with gauge charge $\hat{Q}(x)=\xi\delta\left(x-x^{\prime}\right)$. The corresponding Bosonized Hamiltonian in these sectors is given by,

where $\hat{H}_{\mathrm{mat}}=\frac{1}{2}\int dx\left[\Pi^{2}+\left(\partial_{x}\phi\right)^{2}\right]$, $g=um^{2}$ and $m^{2}=e^{2}/\pi$. Note that all of these Hamiltonians are quadratic, thus can be exactly diagonalized. Then it is straightforward to calculate the gauge violation correlations defined in Eq. (5,6). For the deconfinement model, one obtains

where $v=\sqrt{1+g}$ and $\gamma=\frac{v^{2}+1}{2v^{3}}$. The ultraviolet cutoff $a$ is introduced by the bosonization procedure, and $E_{Q}$ is an unimportant constant energy. For the Schwinger model, it is hard to obtain a compact analytic form (see supplementary material). Then performing the Fourier transformation one obtains the gauge violation spectral functions.

The results are shown in Figs.2 and 3. Note that for both models, the gauge violation spectrums exhibit no dispersion. For the model with charge confinement, the gauge violation spectrum is a delta function. By contrast, for the deconfined model, the gauge violation spectrum has a finite width. This behavior can be understood as follows: The gauge violation spectroscopy measurement adds one particle with charge $+e$ into the system, add create a gauge charge $-e$ at the same position, which can not move. The interaction between this particle and the gauge charge is governed by electrical field. In the Schwinger model, this interaction energy is proportional to the length of separation between the added particle and the gauge charge, i.e. it is in the confinement phase. Thus the added particle can not move far away from the original position. However, in the deconfinement model, the interaction energy is nearly constant. Then the added particle can move away from the original position. For comparison, we also have calculated the gauge invariance spectroscopy in supplementary material, which exhibit linear dispersion in momentum space.

Quantum Link Model. Recently, the one-dimensional quantum link model, a U(1) lattice gauge theory, has been realized on a quantum simulator based on ultracold bosons in optical lattices [email protected] . The Gauss law, as well as the thermalization of this model have been observed [email protected] . By further engineering a tunable topological theta term, the confinement-deconfinement transition has been observed [email protected] . The Hamiltonian of the quantum link model is given by [email protected] ; [email protected]

Here the gauge field is represented by spin-1/2 operators $\hat{S}_{i+1;i}^{+}$ on links. $J$ is the gauge-matter coupling strength and $M$ is the mass of the matter field. $\chi$ can tune the topological theta angle $\theta$. When $\chi=0$, $\theta=\pi$, as $\chi\neq 0$, $\theta$ is tuned away from $\pi$. When $\chi\neq 0$, i.e. $\theta$ is away from $\pi$, it is in the confined phase, and no single charge can be observed. When $\chi=0$, i.e. $\theta=\pi$, there is a transition from confined phase to deconfined phase by tuning the matter mass from negative to positive.

The local gauge transformation operator is $\hat{G}_{j}=e^{i\phi\hat{Q}_{j}}$, and the conserved gauge charge is $\hat{Q}_{j}=\hat{S}_{j+1;j}^{z}+\hat{S}_{j;j-1}^{z}+\hat{\Psi}_{j}^{{\dagger}}\hat{\Psi}_{j}$. We calculate the gauge violation spectrum by numerical exact diagonalization. The results of numerical simulation are shown in Fig.4. In Figs.4(a-b), we observe the deconfinement behavior with $M=10J$ and the topological angle $\theta=\pi$. The spectrum has a finite width. Instead, Fig.4(c-d) shows the confinement behavior with $M=-10J,\theta=\pi$. It shows the delta function behavior. This difference clearly distinguishes confinement from deconfinement phase. In Fig.4(c-d), we also observe the same delta function behavior with $M=\pm 10J,\theta\neq\pi$, because there is only a confinement phase when $\theta\neq\pi$.

Summary. We propose the gauge violation spectroscopy in synthetic gauge theories on quantum simulators, which could be used to detect the confinement phase or deconfinement phase. In most single-particle spectroscopy of synthetic gauge theories, such as RF spectroscopy in ultracold quantum gases, one measured the gauge violation spectrum, rather than the gauge invariant spectrum. Since later spectroscopy needs highly non-local perturbations. We used three one-dimensional models with local U(1) gauge symmetry to show that in the confined phase, gauge violation spectrum is nearly a delta-function, while in the deconfinement phase the spectrum has a finite width. However, our conclusions are not limited to one-dimensional models with U(1) gauge symmetry. It can be applied to higher dimensions or non-Abelian gauge. In addition, we would like to point out that, for some simulators, people realize the projected Hamiltonian instead of the Hamiltonian in the full Hilbert space [email protected] . In this situation, there is no room for fluctuation of gauge charge. Thus the gauge violation spectroscopy can not be applied in these simulators.

Acknowledgements. We thank Hui Zhai and Yanting Cheng for discussion. This work is supported by Innovation Program for Quantum Science and Technology (Grant No. 2021ZD0302000).