Stabilizing lattice gauge theories through simplified local pseudogenerators

The postulate of gauge invariance in nature does not lend itself directly to implementations of lattice gauge theories in modern setups of quantum synthetic matter. Unavoidable gauge-breaking errors in such devices require gauge invariance to be enforced for faithful quantum simulation of gauge-theory physics. This poses major experimental challenges, in large part due to the complexity of the gauge-symmetry generators. Here, we show that gauge invariance can be reliably stabilized by employing simplified local pseudogenerators designed such that within the physical sector they act identically to the actual local generator. Dynamically, they give rise to emergent exact gauge theories up to time scales polynomial and even exponential in the protection strength. This obviates the need for implementing often complex multibody full gauge symmetries, thereby further reducing experimental overhead in physical realizations. We showcase our method in the ${\mathbb{Z}}_2$ lattice gauge theory, and discuss experimental considerations for its realization in modern ultracold-atom setups.

Figure 1

Schematic of gauge protection based on the local pseudogenerator (LPG) ${\widehat{W}}_j$ with eigenvalues $w_j$ (yellow boxes), defining a local constraint $j$. (a) When the full generator ${\widehat{G}}_j$ (blue boxes) has an eigenvalue $g_j=g_j^{\text{tar}}$ in the target sector, then $w_j=g_j^{\text{tar}}$ and vice versa (yellow/blue box in the middle). When $g_j\ne g_j^{\text{tar}}, w_j$ can have one or more values, one of which may be equal to $g_j$ (yellow/blue box on the left), but never $g_j^{\text{tar}}$ (forbidden red-dotted regions), or $w_j$ can have no values in the most general case $[{\widehat{G}}_j,{\widehat{W}}_j]\ne 0$. (b) In the presence of gauge-breaking errors at strength $\lambda$, the target sector $(g_1^{\text{tar}},g_2^{\text{tar}},...)$ is energetically isolated by the LPG protection $V{\widehat{H}}_W$, where $c_j\in \mathbb{R};{\sum }_j{c}_j[w_j\left(g_j^{\text{tar}}\right)-g_j^{\text{tar}}]=0\iff w_j\left(g_j^{\text{tar}}\right)=g_j^{\text{tar}},\forall j$. At sufficient strength $V$, LPG protection induces an emergent global symmetry that coincides with the local gauge symmetry within the target sector.

Figure 2

Quench dynamics of an initial state in the target sector under the faulty gauge theory $\widehat{H}={\widehat{H}}_0+\lambda ({\widehat{H}}_1+{\widehat{H}}_1^{\text{nloc}})+V{\widehat{H}}_W$ with experimentally relevant local coherent gauge-breaking errors (6) in addition to the nonlocal error term (7). Protection term (2) is based on the local pseudogenerator ${\widehat{W}}_j$ given in Eq. (5) with a compliant sequence. Results are obtained from ED. (a) Gauge violation (8a) for various values of protection strength $V$ at an error strength of $\lambda /J=0.01$ with $h/J=0.54$. At sufficiently large $V$, the gauge violation settles at a timescale $\propto 1/V$ into a steady state of value $\propto {\lambda }^2/V^2$. (b) The “infinite-time” gauge violation ($t={10}^5/J$ or larger in ED) shows two distinct behaviors for the compliant sequence. At sufficiently large (small) $V$, it enters a controlled (uncontrolled) error regime where it scales $\propto {\lambda }^2/V^2$ (displays chaotic behavior). Noncompliant sequences fail to control nonlocal errors. (c) Staggered magnetization shown for the ideal theory (green), under the faulty gauge theory at zero protection strength (red) and several finite values of $V$ (shades of blue), and under the adjusted gauge theory ${\widehat{H}}_{\text{adj}}={\widehat{H}}_0+\lambda {\widehat{\mathcal{P}}}_0({\widehat{H}}_1+{\widehat{H}}_1^{\text{nloc}}){\widehat{\mathcal{P}}}_0$. At sufficiently large $V$, the dynamics under $\widehat{H}$ is reproduced by ${\widehat{H}}_{\text{adj}}$ within an error $\propto t{V}_0^2{L}^2/V$, i.e., up to a time scale ${\tau }_{\text{adj}}\propto V/{\left(V_{0}L\right)}^2$, with $V_0$ an energy scale dependent on the model parameters (but not $V$), as we analytically predict (see Appendix pp2).

Figure 3

Same as Fig. 2 but with only local error terms given in Eq. (6) and employing only the noncompliant sequence $c_j=[6{(-1)}^j+5]/11$ in the LPG protection of Eq. (2). The qualitative picture is identical to that of Fig. 2 for the case of nonlocal errors and LPG protection with a compliant sequence, meaning that simplified sequences can reliably protect against experimentally relevant local error terms.

Figure 4

Dynamics of the staggered boson number ${\widehat{n}}_{\text{stag}}^{\text{raw}}\left(t\right)={\sum }_j{(-1)}^j\langle \psi \left(t\right)|{\widehat{n}}_j|\psi \left(t\right)\rangle /L$ under the faulty gauge theory $\widehat{H}={\widehat{H}}_0+\lambda {\widehat{H}}_1+V{\sum }_j{\widehat{W}}_j[6{(-1)}^j+5]/11$ demonstrates that LPG protection gives rise to an adjusted gauge theory ${\widehat{H}}_0+\lambda {\widehat{\mathcal{P}}}_0{\widehat{H}}_1{\widehat{\mathcal{P}}}_0$ during all experimentally relevant evolution times already at $V=2J$ and $\lambda =0.1J$, well within the accessible parameter range of state-of-the-art QSM devices.

Figure 5

$(2+1)-\mathrm{D}{\mathbb{Z}}_2$ gauge theory on a triangular lattice with $L=6$ matter sites and $L_{\ell }=5$ gauge links. Circles indicate matter sites, with red circles denoting single occupation of hard-core bosons, while white circles are empty matter sites. The electric fields on the links between matter sites are initialized at one of their eigenvalues $\pm 1$ (yellow). Note how the link between matter sites 2 and 3 is the same link as that between matter sites 4 and 5, i.e., ${\widehat{\tau }}_{2,3}^{\{x,y,z\}}={\widehat{\tau }}_{4,5}^{\{x,y,z\}}$.

Figure 6

$(2+1)-\mathrm{D}{\mathbb{Z}}_2$ LGT on a triangular lattice with gauge-breaking terms ${\widehat{H}}_{\text{err}}={\widehat{H}}_1+{\widehat{H}}_1^{\text{nloc}}$ given in Eqs. (12) and (13). LPG protection with a noncompliant sequence, see Eq. (14), is used to stabilize gauge invariance. (a) Gauge-violation dynamics at sufficiently large $V$ settles into a plateau $\propto {\lambda }^2/V^2$ that begins at a time scale $\propto 1/V$ and lasts up to all accessible times in ED. It is remarkable that this occurs despite the LPG-protection sequence being noncompliant, which seems unable to protect against such extremely nonlocal errors in $(1+1)-\mathrm{D}$, see Fig. 2. (b) A two-regime picture emerges, with an uncontrolled long-time violation at small enough values of $V$, while at sufficiently large values of $V$ the long-time violation enters a regime of controlled error $\propto {\lambda }^2/V^2$. (c) LPG protection gives rise to the adjusted gauge theory ${\widehat{H}}_{\text{adj}}={\widehat{H}}_0+\lambda {\widehat{\mathcal{P}}}_0{\widehat{H}}_{\text{err}}{\widehat{\mathcal{P}}}_0$, which faithfully reproduces the dynamics of the electric field under the faulty theory up to a time scale ${\tau }_{\text{adj}}\propto V/{\left(V_{0}L\right)}^2$. As predicted analytically, the corresponding error grows linear in time and is suppressed as $\propto 1/V$.

Figure 7

Comparing LPG protection to full protection in the case of experimentally relevant local errors. The LPG protection sequence here is noncompliant, with $c_j=[6{(-1)}^j+5]/11$, but as shown in the main text, this is sufficient to protect against local gauge errors. Here we have chosen $\lambda /J=0.01$ and $h/J=0.54$, but we have checked that our results hold for other values of these parameters. Even though the full protection shows slightly better protection quantitatively, the LPG protection exhibits similar qualitative performance, with the transition from an uncontrolled-error to a controlled-error $\propto {\lambda }^2/V^2$ regime occurring at $V\gtrsim 5J$ compared to $V\gtrsim 3J$ for the full protection. In the case of LPG protection, certain resonances between the target sector and other gauge-invariant sectors are not fully controlled at certain values of $V$ within the controlled-error regime (see small “spike” at $J/V\approx 3.7\times {10}^{-3}$), but nevertheless they are still reliably suppressed.

Figure 8

Initial states used in our ED calculations. Circles represent matter sites, where red circles denote single hard-core boson occupation and white circles are empty matter sites. The yellow arrows on the links between matter sites denote the eigenvalue of the electric field ${\widehat{\tau }}_{j,j+1}^x$ as $\pm 1$ when pointing right (left). All three initial states are in the target sector $g_j^{\text{tar}}=+1,\forall j$. In the main text, we have focused on $|{\psi }_0\rangle$, although LPG protection offers reliable stabilization of gauge invariance independently of the initial state, as shown in Fig. 9.

Figure 9

Same as Fig. 3 but for (a) different initial states, (b) different model-parameter values, and (c) different error strengths. In all cases, the qualitative picture of two regimes persists, where at values of $V$ that are too small the long-time violation cannot be directly determined from the value of $V$, but whereas at sufficiently large $V$ the long-time violation enters a regime of controlled error $\propto {\lambda }^2/V^2$.

Figure 10

Same as Fig. 2 but the nonlocal gauge-breaking error now reads ${\widehat{H}}_1^{\text{nloc}}={\sum }_{\xi =\pm 1}{\prod }_j[\mathbb{1}+\xi {\widehat{\tau }}_{j,j+1}^z][\mathbb{1}+\xi ({\widehat{a}}_j+{\widehat{a}}_j^{†}\left)\right]$, which also violates boson-number conservation. Here we restrict to a system of only $L=4$ matter sites to reduce numerical overhead in light of the large evolution times we access. The initial state is the staggered-matter product state $|{\psi }_0\rangle$ of Fig. 8 but with only $L=4$ matter sites and $L=4$ gauge links. The qualitative picture is identical to that of Fig. 2.