Extended quantum $U\left(1\right)$-liquid phase in a three-dimensional quantum dimer model

Recently, quantum dimer models have attracted a great deal of interest as a paradigm for the study of exotic quantum phases. Much of this excitement has centered on the claim that a certain class of quantum dimer model can support a quantum $U\left(1\right)$-liquid phase with deconfined fractional excitations in three dimensions. These fractional monomer excitations are quantum analogs of the magnetic monopoles found in spin ice. In this paper, we use extensive quantum Monte Carlo simulations to establish the ground-state phase diagram of the quantum dimer model on the three-dimensional diamond lattice as a function of the ratio $\mu$ of the potential to kinetic-energy terms in the Hamiltonian. We find that, for ${\mu }_c=0.75\pm 0.02$, the model undergoes a first-order quantum phase transition from an ordered “$\mathsf{R}$ state” into an extended quantum $U\left(1\right)$-liquid phase, which terminates in a quantum critical Rokhsar-Kivelson (RK) point for $\mu =1$. This confirms the published field-theoretical scenario. We present detailed evidence for the existence of the $U\left(1\right)$-liquid phase and indirect evidence for the existence of its photon and monopole excitations. Simulations are benchmarked against a variety of exact and perturbative results, and a comparison is made of different variational wave functions. We also explore the ergodicity of the quantum dimer model on a diamond lattice within a given flux sector, identifying a new conserved quantity related to transition graphs of dimer configurations. These results complete and extend the previous analysis of O. Sikora et al. [Phys. Rev. Lett. 103, 247001 (2009)].

Figure 1

Phase diagram of the quantum dimer model on a diamond lattice, as a function of the ratio $\mu$ of potential to kinetic energy, following Refs. 8, 12, and14.

Figure 2

(a) Cubic unit cell of the diamond lattice, with bonds occupied by dimers shown by thick red lines. The dimer configuration shown is the ordered “$\mathsf{R}$ state” found for negative $\mu$. This contains a “flippable” hexagonal plaquette, which is shaded blue. (b) Equivalent cubic unit cell of the pyrochlore lattice, with sites occupied by hard-core bosons shown in red. The boson configuration shown is the same ordered “$\mathsf{R}$ state.”

Figure 3

The subset of diamond-lattice bonds $\Lambda$ (thick, blue, shaded bonds) used in the unitary transformation Eq. (9), which maps $g\to -g$ in Eq. (1).

Figure 4

Construction of the “string” defect (one magnetic flux) configuration from the maximally flippable $\mathsf{R}$-state configuration (assuming periodic boundary conditions): (a) Two adjacent 16-bond unit cells of the diamond lattice within the $\mathsf{R}$ state, with flippable hexagons denoted by blue shading. (b) A pair of monomer excitations created by removing a dimer. (c)–(e) Monomers are separated. The dashed line shows the “string” of displaced dimers that connects them. (f) One of the monomers has traversed the cluster and the dimer is reintroduced. The resulting configuration has a single string excitation and a nonzero magnetic flux.

Figure 5

Numbering convention in the definition of the order parameter $m_{\mathsf{R}}$ for the $\mathsf{R}$ state [Eq. (18)]. Left: cubic cell of a pyrochlore lattice with 16 sites; right: corresponding diamond-lattice cell with 16 bonds.

Figure 6

Collapse of the ordered state: the order parameter $m_{\mathsf{R}}$ in the sector connected to the $\mathsf{R}$ state. The results of VMC (dashed lines) and GFMC (solid lines) simulations are shown for a series of clusters with cubic symmetry, and compared to exact diagonalization results for the smallest, 128-bond cluster. The inset shows finite-size scaling of the order parameter at the RK point.

Figure 7

Estimation of the critical value of ${\mu }_c\left(\infty \right)$ for the transition from the ordered $\mathsf{R}$ state to the quantum $U\left(1\right)$ liquid, taken from GFMC simulation of three different families of clusters. Linear fits to ${\mu }_c\left(L\right)$ as a function of $1/L^4$ are made subject to the constraint that all data sets must converge to the same value ${\mu }_c\left(\infty \right)$ for $1/L\to 0$, regardless of the cluster geometry. We find ${\mu }_c\left(\infty \right)=0.75\pm 0.02$.

Figure 8

Monopole string tension ${\kappa }_1$ [Eq. (21)] for a range of $\mu$ spanning the ordered $\mathsf{R}$ state and proposed quantum $U\left(1\right)$-liquid phase. Results are taken from VMC (dashed lines) and GFMC (solid lines) simulations for a series of clusters with cubic symmetry, and compared to exact diagonalization results for the smallest, 128-bond, cluster.

Figure 9

Reduction in the average number of flippable plaquettes $\Delta N_{\phi }$, as calculated in simulations at the RK point. (a) $-\Delta N_{\phi }$ plotted as a function of ${\phi }^2$ for cubic clusters of linear dimension $L=4$ (432 bonds) to $L=16$ (8192 bonds). (b) The same data set plotted as $-\Delta N_{\phi }/{\phi }^2$ vs ${\phi }^2$ to extract the leading dependence on $\phi$. (c) Finite-size scaling of the coefficient of ${\phi }^2$ found from fits to $-\Delta N_{\phi }$ for clusters of linear dimension $L$ [cf. intercept to ordinate axis in (b), above]. The linear collapse of the coefficient of ${\phi }^2$ with $1/L$ is consistent with the prediction of the effective field theory for a quantum $U\left(1\right)$ liquid, Eq. (14).

Figure 10

Comparison of the energy gaps ${\Delta }_{\phi }$ found in GFMC simulations with the predictions of a perturbation theory about the RK point [Eq. (22)] for a range of $\mu$ within the proposed quantum $U\left(1\right)$-liquid phase. (a) Energy gap ${\Delta }_{\phi }$ normalized to $-\delta \mu =\mu -1$ to extract the leading dependence on $\mu$ and plotted as a function of ${\phi }^2$ [cf. Eq. (14)]. (b) Energy gap ${\Delta }_{\phi }$ normalized to $-\delta \mu \times {\phi }^2$ to extract the leading dependence on both $\mu$ and $\phi$ and plotted as a function of ${\phi }^2$ [cf. Eq. (14)]. All results are for a 1024-bond cluster with cubic symmetry.

Figure 11

First-order quantum phase transition from the ordered $\mathsf{R}$ state into the quantum $U\left(1\right)$ liquid, as imaged in the finite-size spectrum of a 1024-bond cluster. GFMC simulation results for the energy gap ${\Delta }_{\phi }$ are plotted as ${\Delta }_{\phi }/\left[{\phi }^2(1-\mu )\right]$ so as to collapse the spectra for the $U\left(1\right)$-liquid phase. The phase transition is visible as an abrupt qualitative change in the form of the spectra for $\mu =0.7$. The results of perturbation theory about the RK point (black line and points) are shown for comparison.

Figure 12

(a) Block structure of the Hamiltonian for the quantum dimer model on a diamond lattice, for a 128-bond cluster with a fixed value of the flux $\vec{\phi }=(0,0,0)$, following Table . Connected dimer configurations break up into different subensembles. These can be divided into two groups, according to whether their transition graphs contain an odd or even number of loops. Both even and odd subsectors contain isolated configurations that do not participate in the dynamics of the model and do not contribute to the ground state for $\mu <1$. (b) Contrasting block structure of the Hamiltonian for the quantum dimer model on a square lattice for a finite-size cluster with a fixed value of the flux $\vec{\phi }=(0,0)$. All dimer configurations that contribute to the ground state for $\mu <1$ belong to a single connected block. The remaining, isolated, dimer configurations do not participate in the dynamics of the model.

Figure 13

An 8-link “loop” used to connect subsectors A and B of the Hilbert space of the QDM on a diamond lattice [cf. Table ]. (a) An 8-link loop of bonds within the cubic unit cell of the diamond lattice. Three distinct 8-link loops of this type are possible, each of which appears octagonal when projected into an appropriate [100] plane. Where exactly four bonds are occupied by dimers, this loop is “flippable,” i.e., a new state obeying the dimer constraint can be obtained by cyclicly permuting those dimers around the loop. (b) Equivalent 8-link loop of sites within the cubic unit cell of the pyrochlore lattice.

Figure 14

Schematic representation of all possible changes in the topology of a transition graph that follow from the cyclic permutation of dimers on a hexagonal 6-link loop. Red lines denote the loop segment with a single dimer. Blue lines denote a loop segment with an odd number of dimers. All loop segments, whether red or blue, start and finish on sites belonging to different sublattices of the (bipartite) diamond lattice, denoted here by filled or empty circles. The number of loops associated with each section of the transition graph is shown in the middle of the hexagonal plaquette. This number changes by either 0 or 2.

Figure 15

Schematic representation of all possible changes in the topology of a transition graph that follow from the cyclic permutation of dimers on an 8-link loop. The number of loops of the transition graphs changes by either 1 or 3.

Figure 16

Histogram of the number of flippable plaquettes in different sets of dimer configurations, calculated at the RK point. (a) Results for a 128-bond cubic cluster, calculated within the entire zero-flux sector (blue, shaded columns) and in the subsector connected to the $\mathsf{R}$ state (unshaded columns). (b) Equivalent results for a 1024-bond cluster.

Figure 17

$\mu$ dependence of the order parameter $m_{\mathsf{R}}$ for the $\mathsf{R}$ state, within two different subsectors of the Hilbert space: starting from the $\mathsf{R}$ state in flux $\phi =0$ subsector (open symbols) and in the subsector that is connected to it by an 8-site loop (solid symbols). Results are calculated using the variational Monte Carlo simulation for cubic clusters with 432 (triangles) and 1024 (circles) diamond-lattice bonds.

Figure 18

The difference in the average number of flippable plaquettes $\Delta N_{\phi }$ [Eq. (24)], calculated at the RK point for dimer configurations in different subsectors of the Hilbert space. Green circles denote results for configurations connected to the $\mathsf{R}$ state by matrix elements of Eq. (1) and string defects. Maroon diamonds denote results for configurations containing an additional 8-link “loop” defect. All results are for a cubic 1024-bond cluster.

Figure 19

Labeling of the sites of the pyrochlore lattice by enumerating one of the two types of tetrahedra and four sublattices on each tetrahedron. This labeling allows us to show that the Hamiltonian $H_{\text{QDM}}$ has no fermionic sign problem even if the underlying particles are fermions.

Figure 20

Value of $\mu$ for which the one-parameter variational wave function $|{\psi }_{\alpha }^{\mathsf{var}}\rangle$ [cf. Eq. (C1)] has the lowest energy, as a function of the variational parameter $\alpha$. Results are plotted for cubic clusters ranging in size from 128 to 2000 bonds.

Figure 21

Comparison of different variational wave functions for 1024-bond cluster: (a) order parameter and (b) energy per site: green, $|{\psi }_{\alpha }^{\mathsf{var}}\rangle$ [cf. Eq. (C1)]; orange, $|{\psi }_{\beta }^{\mathsf{var}}\rangle$ [cf. Eq. (C3)]; red, $|{\psi }_{\alpha \beta }^{\mathsf{var}}\rangle$[cf. Eq. (C4)]; black, $|{\psi }_{\alpha \beta \gamma }^{\mathsf{var}}\rangle$ [cf. Eq. (C5)].

Figure 22

Quantum hysteresis—variational Monte Carlo results for $|{\psi }_{\alpha \beta \gamma }^{\mathsf{var}}\rangle$ [cf. Eq. (20)] in a 1024-bond cluster. Left: energy per site as a function of the order parameter $m_{\mathsf{R}}$. Right: order parameter as a function of $\mu$. Simulations converge to different local minima for a range of values of $0.3<\mu <0.42$ according to whether they are started from an ordered or a disordered state.

Figure 23

Energy per site as a function of $\mu$. GFMC results for 1024-bond cluster (points) are compared to Padé approximants for series expansion around an ordered state (solid lines) and perturbation expansion in the vicinity of the RK point (dashed lines).

Figure 24

Monopole string tension ${\kappa }_1={\Delta }_1/L$, plotted for a range of $\mu$ spanning the ordered $\mathsf{R}$-state phase. Solid lines denote different Padé approximants to the series expansion about an $\mathsf{R}$-state configuration [cf. Eq. (D2)]. Points denote the results of GFMC simulations of 432-bond (red) and 1024-bond (green) cubic clusters.