Deconfined quantum critical point in one dimension

We perform a numerical study of a spin-1/2 model with ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ symmetry in one dimension which demonstrates an interesting similarity to the physics of two-dimensional deconfined quantum critical points (DQCP). Specifically, we investigate the quantum phase transition between Ising ferromagnetic and valence bond solid (VBS) symmetry-breaking phases. Working directly in the thermodynamic limit using uniform matrix product states, we find evidence for a direct continuous phase transition that lies outside of the Landau-Ginzburg-Wilson paradigm. In our model, the continuous transition is found everywhere on the phase boundary. We find that the magnetic and VBS correlations show very close power-law exponents, which is expected from the self-duality of the parton description of this DQCP. Critical exponents vary continuously along the phase boundary in a manner consistent with the predictions of the field theory for this transition. We also find a regime where the phase boundary splits, as suggested by the theory, introducing an intermediate phase of coexisting ferromagnetic and VBS order parameters. Interestingly, we discover a transition involving this coexistence phase which is similar to the DQCP, being also disallowed by the Landau-Ginzburg-Wilson symmetry-breaking theory.

Figure 1

The phase diagram in the $K_2\text{−}\delta$ plane includes the $zFM$, $VBS-I$, $VBS-II$, and $x\mathrm{UUDD}$ phases. Inset shows a centered view of the coexistence region, denoted “C”, appearing between the $zFM$ and $VBS-I$ or $VBS-II$ phases for $\delta$ close to 1. While the distinction between the $VBS-I$ and $VBS-II$ phases is protected by the on-site symmetries, the $VBS-I+zFM$ and $VBS-II+zFM$ coexistence phases are not distinct and there is no transition inside the C region. The cut indicated at $\delta =0.5$ will be investigated in detail in Sec. 3a as an example case.

Figure 2

Transition between the $zFM$ and VBS phases, as detected by the corresponding order parameters. This scan is taken at fixed $\delta =0.5$ using bond dimension $\chi =192$. We observe that at fixed $\chi$, the MPS ground state shows a first-order transition; the discontinuities in the order parameters decrease towards zero with increasing $\chi$, as studied in detail in Fig. 7.

Figure 3

The divergence of the correlation length in the exact ground state at the critical point manifests as a $\chi$-dependent cusp in the MPS correlation length $\xi \left(\chi \right)$; specifically, the height grows as a power law with $\chi$, as studied in detail in Fig. 6. This feature is indicative of a continuous phase transition.

Figure 4

The scaling of the critical entanglement entropy $S[\chi ;K_{2c}\left(\chi \right)]$ is nearly linear in $ln\xi [\chi ;K_{2c}\left(\chi \right)]$, with the slope in good agreement with predicted central charge $c=1$. Data shown are taken at parameter $\delta =0.5$, and the dashed line is provided as a guide to the eye. The pseudocritical point $K_{2c}\left(\chi \right)$ is defined later in the text and included here only for specificity; it is important insofar as it is particular to the MPS of bond dimension $\chi$.

Figure 5

Illustration of the process of locating the critical point from finite-entanglement scaling at $\delta =0.5$. (a) The energies of both trial wave functions from the $zFM$ and $VBS$ phases (fully optimized at each $K_2$) follow smooth curves, which determine the level crossing for a given bond dimension $\chi$ to a finer resolution than the scan in parameter $K_2$ via interpolation. Due to hysteresis, in many cases we directly observe the crossing using the adiabatic protocol described in the text. (b) Using the finite-entanglement scaling form Eq. (19), we extrapolate from the extracted pseudocritical $K_{2c}\left(\chi \right)$ to estimate the location of the critical point at $\chi \to \infty$. The scatter in data points is not noise from the variational algorithm, but rather may be a consequence of the uneven spacing of the entanglement spectrum.

Figure 6

The MPS correlation length $\xi \left(\chi \right)$ exhibits power-law behavior in an intermediate region around $K_{2c}\left(\chi \right)$, here shown in the $zFM$ phase in the top panel and $VBS$ in the bottom. Close to the pseudocritical point, the correlation length saturates to a maximum value dependent on the bond dimension, whereas farther away it approaches a constant in the gapped phase. In the case of the $VBS$ phase, a nearby critical point (the transition to the $x\mathrm{UUDD}$ phase) affects the behavior of $\xi \left(\chi \right)$. In the insets, we show the dependence of the maximum correlation length $\xi [\chi ;K_{2c}\left(\chi \right)]$ extrapolated to the pseudocritical point $K_{2c}\left(\chi \right)$ as a function of $\chi$. A fit to the scaling form Eq. (17) is shown (note that the axes are logarithmic), along with extracted values of $\kappa$.

Figure 7

Expectation values of the $zFM$ and VBS order parameters on the slice $\delta =0.5$ show a region of power-law dependence in an intermediate range near the critical point which extends closer to the transition with increasing bond dimension. Far from the critical point, the order parameters approach their maximal values, whereas very close to $K_{2c}\left(\chi \right)$ at fixed $\chi$, they saturate due to the discontinuous mean-field description of the transition. The top panel shows $K_2$ in the $zFM$ phase, and the bottom panel $K_2$ in the VBS phase; in both panels, we give $K_2$ relative to the pseudocritical $K_{2c}\left(\chi \right)$ determined for each bond dimension as in Fig. 5. The dashed line in each panel shows the fitted power-law onset form with exponent ${\beta }_{zFM}$ or ${\beta }_{VBS}$ in this intermediate range, using the largest bond dimension data, which is essentially already converged to the infinite-$\chi$ values. Insets show the limiting values of the corresponding order parameters at $K_{2c}\left(\chi \right)$ as a function of the limiting $\xi [\chi ,K_{2c}\left(\chi \right)]$, and a power-law fit to Eq. (20). Note that inset axes use logarithmic scaling.

Figure 8

Top panel shows measurements of important correlation functions (for example, $\langle M_z^{\text{FM}}{M}_z^{\text{FM}}\rangle \sim \langle {\sigma }_0^z{\sigma }_r^z\rangle$) in the $zFM$ ansatz state tuned to the pseudocritical point $K_{2c}(\chi =192)$. These data are taken at $\delta =0.5$. All correlators show a region of critical power-law behavior before reaching a constant or decaying exponentially, as they eventually must in a finitely entangled state. The correlation length in this state is $\sim 425$. The bottom plot shows the same correlation functions, but measured in the $VBS$ ansatz MPS tuned to the pseudocritical point.

Figure 9

The measured power-law decay exponents are in good agreement with the predicted behavior $p_{zFM}=p_{VBS}$ and $p_{xFM}=p_{yAFM}$, as well as with Eq. (21). At the larger values of $\delta$, the state begins to feel the tricritical point, which affects the more quickly decaying $M_x^{\text{FM}}$ and $M_y^{\text{AFM}}$ correlation functions.

Figure 10

We find good agreement with Eq. (22), particularly for the larger values of $\delta$ on the critical line. The states at small $\delta$ are near to the $\delta =0$ QLRO phase and thus are relatively highly entangled, which makes it difficult to reach the limit $\chi \to \infty$ near enough to the critical point to extract the $\nu$ and $\beta$ critical exponents.

Figure 11

The slice at $\delta =0.9$ clearly exhibits a region of coexistence of order parameters (measurements shown use the bond dimension $\chi =144$ MPS), and the correlation length displays $\chi$-dependent cusps at both boundaries. However, we do not have good $\chi$-converged properties inside of this phase, as the correlation length does not saturate for the bond dimensions shown here.

Figure 12

Illustration of the process of locating the critical point for each order parameter from finite-entanglement scaling in the coexistence regime at $\delta =0.9$. Here the data points are found via fits to the power-law onset behavior shown in Fig. 11. Again, using the finite-entanglement scaling form Eq. (19), we extrapolate from pseudocritical $K_{2c,VBS}\left(\chi \right)$ and $K_{2c,zFM}\left(\chi \right)$ to estimate the width of the coexistence region in the limit $\chi \to \infty$.

Figure 13

Our study of the $VBS$ ordering transition in the coexistence region is impeded by the narrow width of the phase, here shown at $\delta =0.9$. The top panel shows correlation length along with our best fit, though the MPS results are not reflective of the $\chi \to \infty$ limit and the exponent is far from the Ising $\nu =1$. The feature seen near $3\times {10}^{-4}$ on the $x$ axis is the $zFM$ order transition on the boundary of the coexistence region with the $VBS-I$ phase (this transition is studied in Fig. 14). Further from the critical point, one sees the effect of the transition to the $x\mathrm{UUDD}$ phase. The bottom panel shows the onset of the $VBS$ order parameter, which is roughly consistent with a continuous phase transition but does not agree with the Ising $\beta =1/8$. Data here do not use the adiabatic protocol; every point is independent.

Figure 14

Similarly to Fig. 13, we provide only a rough study of the $zFM$ ordering transition in the coexistence region, here shown at $\delta =0.9$. The top panel shows correlation length along with our best fit, though the MPS results are not reflective of the $\chi \to \infty$ limit and the exponent is far from the Ising $\nu =1$. The feature seen near $3\times {10}^{-4}$ on the $x$ axis is the $VBS$ order transition on the boundary of the coexistence region with the $zFM$ phase (this transition is studied in Fig. 13). The bottom panel shows the onset of the $zFM$ order parameter, which is roughly consistent with a continuous phase transition but does not agree with the Ising $\beta =1/8$. Data here do not use the adiabatic protocol; every point is independent.

Figure 15

Illustration of determining the critical point between $zFM+VBS$ coexistence and $x\mathrm{UUDD}$ from finite-entanglement scaling at $\delta =1$, performed using the same procedure used in Fig. 5. (a) The energies of both the coexistence and $x\mathrm{UUDD}$ phases follow smooth curves which determine the level crossing, but here happen to not display hysteresis. This does not present a problem, as the smoothness of the evolution of the trial state energies permits extrapolation. (b) Using a fit to the finite-entanglement scaling form Eq. (19), we extrapolate the pseudocritical $K_{2c,\text{zFM}}\left(\chi \right)$ to estimate the location of the critical point at $\chi \to \infty$. As was the case for the DQCP, the scatter in data points is again not noise from the variational algorithm, but a reproducible feature of the ground state at each bond dimension.

Figure 16

Analysis of the phase transition from the $zFM+VBS$ coexistence phase to the $x\mathrm{UUDD}$ phase along the cut $\delta =1$, using the discontinuous VUMPS (generalized mean field) procedure. Because we cannot make an accurate determination of ${\beta }_{zFM}$ using measurements arising from inside the coexistence phase, we consider only measurements of $M_x^{\text{UUDD}}$, within the $x\mathrm{UUDD}$ phase. However we can perform finite-entanglement scaling analysis of $M_z^{\text{FM}}$ extrapolated to the pseudocritical points. Doing so, we find a value ${\beta }_{zFM}/\nu =0.167$ which is quite similar to ${\beta }_{x\text{UUDD}}/\nu$ shown in the inset.

Figure 17

Comparing the energies of the separable trial wave functions of Appendix pp1-s1 results in a phase diagram which is broadly similar to the actual behavior of the model, but renders all phase transitions as first-order.

Figure 18

The phase diagram of the improved mean-field trial states described in Appendixes pp1-s2 and pp1-s3 provides a somewhat more realistic picture, in particular with continuous phase transitions for all boundaries of the $VBS-I$ and $VBS-II$ phases. There is an extended boundary between the coexistence and $x\mathrm{UUDD}$ phases, which is first-order, as well as a first-order transition between $x{\mathrm{UUDD}}^{\prime }$ and $x{\mathrm{UUDD}}^{″}$. Along the dotted line at $\delta =1$ the $VBS+zFM$ ansatz coincides with the simple product $zFM$ state from Appendix pp1-s1; however, the wider $zFM$ phase is not represented, as away from this special line the simple $zFM$ wave function is always energetically unfavorable. In Appendix pp2, we show how some of the unphysical features can be fixed using more entangled wave functions.