Gapless symmetry-protected topological phases and generalized deconfined critical points from gauging a finite subgroup (2024)

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  Figure 1

  (a)Schematic diagram for the 1D Bose-Hubbard model (blue sites) coupled to Ising spins (violet bonds). (b)Schematic phase diagram. The original BKT transition between the superfluid and the insulator phase is enriched to a gauged BKT transition between a gapless SPT superfluid phase, protected by $\text{U}\left(1\right)$ and $W$, and the insulator phase with $W$ spontaneously broken. (c)Degeneracies in the superfluid and the insulator phase with PBC and OBC, respectively. [(d)and (e)] Finite size scaling of the gap ${\mathrm{\Delta }}_{\text{bulk}}$ in both the superfluid ($t=1.0,U=1.0$) and the insulator phase ($t=0.1,U=1.0$). ${\mathrm{\Delta }}_{\text{bulk}}$ is defined to be the gap in spectrum above the (possibly) degenerate ground states. OBC is used in both cases.

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  Figure 1

  (a)Schematic diagram for the 1D Bose-Hubbard model (blue sites) coupled to Ising spins (violet bonds). (b)Schematic phase diagram. The original BKT transition between the superfluid and the insulator phase is enriched to a gauged BKT transition between a gapless SPT superfluid phase, protected by $\text{U}\left(1\right)$ and $W$, and the insulator phase with $W$ spontaneously broken. (c)Degeneracies in the superfluid and the insulator phase with PBC and OBC, respectively. [(d)and (e)] Finite size scaling of the gap ${\mathrm{\Delta }}_{\text{bulk}}$ in both the superfluid ($t=1.0,U=1.0$) and the insulator phase ($t=0.1,U=1.0$). ${\mathrm{\Delta }}_{\text{bulk}}$ is defined to be the gap in spectrum above the (possibly) degenerate ground states. OBC is used in both cases.

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  Figure 2

  (a)Boson pair correlation function, which is shown to follow a power law $\langle b_i^{†}{b}_i^{†}{b}_j{b}_j\rangle \sim r^{-{\eta }_{bb}}$ in the superfluid phase. The inset shows the extrapolation of the exponent ${\eta }_{bb}$ to the thermodynamic limit using finite-size scaling. Error bars are obtained from the upper and the lower bound of the extrapolation. (b)Gauge-invariant boson correlation, which also follows power law $\langle b_i^{†}{\sigma }^z\cdots {\sigma }^z{b}_j\rangle \sim r^{-{\eta }_b}$. The inset shows the extrapolation of ${\eta }_b$. (c)The $\tilde{\mathrm{U}}\left(1\right)$ disorder operator $|\langle {\tilde{X}}_R\left(\theta \right)\rangle |$ decays as power law $r^{-\alpha \left(\theta \right)}$ in $r$. The main plot shows the $\theta =2\pi$ case. The inset shows the $\theta$ dependence of the exponent $\alpha$, which is $4\pi$-periodic. $\alpha$ is symmetric about $2\pi$, and a quadratic fit (dashed line) is performed for the segment from 0 to $2\pi$. (d)Subsystem von Neumann entanglement entropy $S_E$ as a function of subsystem size $l$. The inset shows the linear dependence of $S_E$ on $\lambda \left(l\right)\equiv \frac{1}{6}ln(\frac{2L}{\pi }sin\left(\frac{\pi l}{L}\right))$. The central charge $c$, which is given by the slope, is shown to be almost exactly 1. Recently, the entanglement spectrum of gapless SPT phases has also been studied and the entanglement spectrum also has degeneracy [36]. All the main plots are for $L=100,t=0.5,$ and $U=1.0$.

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  Figure 2

  (a)Boson pair correlation function, which is shown to follow a power law $\langle b_i^{†}{b}_i^{†}{b}_j{b}_j\rangle \sim r^{-{\eta }_{bb}}$ in the superfluid phase. The inset shows the extrapolation of the exponent ${\eta }_{bb}$ to the thermodynamic limit using finite-size scaling. Error bars are obtained from the upper and the lower bound of the extrapolation. (b)Gauge-invariant boson correlation, which also follows power law $\langle b_i^{†}{\sigma }^z\cdots {\sigma }^z{b}_j\rangle \sim r^{-{\eta }_b}$. The inset shows the extrapolation of ${\eta }_b$. (c)The $\tilde{\mathrm{U}}\left(1\right)$ disorder operator $|\langle {\tilde{X}}_R\left(\theta \right)\rangle |$ decays as power law $r^{-\alpha \left(\theta \right)}$ in $r$. The main plot shows the $\theta =2\pi$ case. The inset shows the $\theta$ dependence of the exponent $\alpha$, which is $4\pi$-periodic. $\alpha$ is symmetric about $2\pi$, and a quadratic fit (dashed line) is performed for the segment from 0 to $2\pi$. (d)Subsystem von Neumann entanglement entropy $S_E$ as a function of subsystem size $l$. The inset shows the linear dependence of $S_E$ on $\lambda \left(l\right)\equiv \frac{1}{6}ln(\frac{2L}{\pi }sin\left(\frac{\pi l}{L}\right))$. The central charge $c$, which is given by the slope, is shown to be almost exactly 1. Recently, the entanglement spectrum of gapless SPT phases has also been studied and the entanglement spectrum also has degeneracy [36]. All the main plots are for $L=100,t=0.5,$ and $U=1.0$.

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  Figure 3

  (a)The bulk spin-spin correlation in the superfluid phase with a power-law fit (dash line) $|\langle {\sigma }_i^x{\sigma }_j^x\rangle |\sim r^{-{\eta }_s}$. The inset shows the extrapolation of the critical exponent ${\eta }_s$ to the thermodynamic limit using finite-size scaling. (b)The magnetization $|\langle {\sigma }_i^x\rangle |$ in the superfluid phase across the system with OBC. The spins on the edges are perfectly polarized and the magnetization decays to a small value to the bulk. The inset shows the center spin magnetization $|\langle {\sigma }_{L/2}^x\rangle |$ follows a power law decay as system size increases, compatible with the fact that magnetization vanishes when PBC is used. $L=100,t=0.5,$ and $U=1.0$ for both plots.

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  Figure 3

  (a)The bulk spin-spin correlation in the superfluid phase with a power-law fit (dash line) $|\langle {\sigma }_i^x{\sigma }_j^x\rangle |\sim r^{-{\eta }_s}$. The inset shows the extrapolation of the critical exponent ${\eta }_s$ to the thermodynamic limit using finite-size scaling. (b)The magnetization $|\langle {\sigma }_i^x\rangle |$ in the superfluid phase across the system with OBC. The spins on the edges are perfectly polarized and the magnetization decays to a small value to the bulk. The inset shows the center spin magnetization $|\langle {\sigma }_{L/2}^x\rangle |$ follows a power law decay as system size increases, compatible with the fact that magnetization vanishes when PBC is used. $L=100,t=0.5,$ and $U=1.0$ for both plots.

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  Figure 4

  (a)Schematic diagram for the 2D Bose-Hubbard model (blue sites) coupled to Ising spins (violet bonds). A star operator related to the gauge constraints in Eq.(18) and a plaquette operator are highlighted. (b)Schematic phase diagram. The gapless SPT in the superfluid phase where $\tilde{\mathrm{U}}\left(1\right)$ is spontaneously broken and the SET enriched by $\tilde{\mathrm{U}}\left(1\right)$ in the insulator phase are separated by a generalized DQCP.

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  Figure 4

  (a)Schematic diagram for the 2D Bose-Hubbard model (blue sites) coupled to Ising spins (violet bonds). A star operator related to the gauge constraints in Eq.(18) and a plaquette operator are highlighted. (b)Schematic phase diagram. The gapless SPT in the superfluid phase where $\tilde{\mathrm{U}}\left(1\right)$ is spontaneously broken and the SET enriched by $\tilde{\mathrm{U}}\left(1\right)$ in the insulator phase are separated by a generalized DQCP.

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  Figure 5

  (a)Action of boson parity $P$, effectively a 't Hooft line (purple), and a Wilson line $W$ (orange) terminating on the boundary (top) of a semi-infinite system. The other end of $W$ terminates either in the bulk or at infinity. The anticommutativity of $P$ and $W$ forces a SSB on the boundary. (b)Disorder operator ${\tilde{X}}_R(\theta =2\pi )$ supported on sites (highlighted in red) inside a region $R$, which is the same with with the 't Hooft loop operator, ${\prod }_{e\in \partial R}{\sigma }_e^x$, supported on the (dual lattice) boundary of $R$. (c)Gauge invariant Wilson line $W$ attached to boson operators (Higgs order operator), and a 't Hooft line connecting two $\pi$ vortices (which is suppressed by the zero-flux condition). (d)Example of two topologically distinct phase modes with winding number 1, ${\phi }_{1,0}$ (red), and winding number 0, ${\phi }_{0,0}$ (blue) along the $x$ direction. 0 and $L_x$ are identified.

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  Figure 5

  (a)Action of boson parity $P$, effectively a 't Hooft line (purple), and a Wilson line $W$ (orange) terminating on the boundary (top) of a semi-infinite system. The other end of $W$ terminates either in the bulk or at infinity. The anticommutativity of $P$ and $W$ forces a SSB on the boundary. (b)Disorder operator ${\tilde{X}}_R(\theta =2\pi )$ supported on sites (highlighted in red) inside a region $R$, which is the same with with the 't Hooft loop operator, ${\prod }_{e\in \partial R}{\sigma }_e^x$, supported on the (dual lattice) boundary of $R$. (c)Gauge invariant Wilson line $W$ attached to boson operators (Higgs order operator), and a 't Hooft line connecting two $\pi$ vortices (which is suppressed by the zero-flux condition). (d)Example of two topologically distinct phase modes with winding number 1, ${\phi }_{1,0}$ (red), and winding number 0, ${\phi }_{0,0}$ (blue) along the $x$ direction. 0 and $L_x$ are identified.

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  Figure 6

  Ground state degeneracy in different phases of the 1D ${\mathbb{Z}}_8$-clock model with its ${\mathbb{Z}}_2$ subgroup gauged, under PBC (a)and under OBC (b).

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  Figure 6

  Ground state degeneracy in different phases of the 1D ${\mathbb{Z}}_8$-clock model with its ${\mathbb{Z}}_2$ subgroup gauged, under PBC (a)and under OBC (b).

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  Figure 7

  (a)Schematic diagram for the energy levels in the grand canonical ensemble, where parity even and parity odd states coexist. ${\mathrm{\Delta }}_{\text{GCE}}^{P=\pm 1}$ is the gap in the parity even/odd sector and ${\mathrm{\Delta }}_{\mathrm{GCE}}$ is the true gap above the doubly degenerate ground state. (b)The upper bound ${\mu }_U$ and the lower bound ${\mu }_L$ for the chemical potential in order to have unit filling at different system sizes. The inset shows the difference between the two bounds. The parameters used are $t=0.5\text{and}U=1.0$.

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  Figure 7

  (a)Schematic diagram for the energy levels in the grand canonical ensemble, where parity even and parity odd states coexist. ${\mathrm{\Delta }}_{\text{GCE}}^{P=\pm 1}$ is the gap in the parity even/odd sector and ${\mathrm{\Delta }}_{\mathrm{GCE}}$ is the true gap above the doubly degenerate ground state. (b)The upper bound ${\mu }_U$ and the lower bound ${\mu }_L$ for the chemical potential in order to have unit filling at different system sizes. The inset shows the difference between the two bounds. The parameters used are $t=0.5\text{and}U=1.0$.

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  Figure 8

  Gap scaling with system size when the ${\sigma }^x$ perturbation is added. The $1/L$ behavior already shows up for relatively small system sizes. Parameters used: $t=0.5,U=1,h_x=0.1,\text{and}K=10$.

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  Figure 9

  Gap scaling with system size when the ${\sigma }^z$ perturbation is added. It shows an exponential decay. Parameters used: $t=0.5,U=1,h_z=5,\text{and}K=10$. Here we called $h_z$ perturbation, but in fact it has to be of the same order with $K$ for a sizable gap to show up even for small system sizes.

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