Phases of quantum dimers from ensembles of classical stochastic trajectories

Tom Oakes, Stephen Powell, Claudio Castelnovo, Austen Lamacraft, and Juan P. Garrahan

Phys. Rev. B 98, 064302 – Published 8 August 2018

Abstract

We study the connection between the phase behavior of quantum dimers and the dynamics of classical stochastic dimers. At the so-called Rokhsar-Kivelson (RK) point a quantum dimer Hamiltonian is equivalent to the Markov generator of the dynamics of classical dimers. A less well understood fact is that away from the RK point the quantum-classical connection persists: in this case the Hamiltonian corresponds to a nonstochastic “tilted” operator that encodes the statistics of time-integrated observables of the classical stochastic problem. This implies a direct relation between the phase behavior of quantum dimers and properties of ensembles of stochastic trajectories of classical dimers. We make these ideas concrete by studying fully packed dimers on the square lattice. Using transition path sampling—supplemented by trajectory umbrella sampling—we obtain the large deviation statistics of dynamical activity in the classical problem, and show the correspondence between the phase behavior of the classical and quantum systems. The transition at the RK point between quantum phases of distinct order corresponds, in the classical case, to a trajectory phase transition between active and inactive dynamical phases. Furthermore, from the structure of stochastic trajectories in the active dynamical phase we infer that the ground state of quantum dimers has columnar order to one side of the RK point. We discuss how these results relate to those from quantum Monte Carlo, and how our approach may generalize to other problems.

Received 22 March 2018
Revised 10 July 2018

DOI:https://doi.org/10.1103/PhysRevB.98.064302

Physics Subject Headings (PhySH)

Nonequilibrium statistical mechanics

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Tom Oakes^1,*, Stephen Powell¹, Claudio Castelnovo², Austen Lamacraft², and Juan P. Garrahan¹

¹School of Physics and Astronomy and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham NG7 2RD, United Kingdom
²TCM Group, Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom

^*tom.oakes@nottingham.ac.uk

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Vol. 98, Iss. 6 — 1 August 2018

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Images

Figure 1
Possible ordered phases of the quantum dimer model on the square lattice: (a) columnar phase; (b) plaquette or mixed phase, depending on the relative amplitude between the horizontal and vertical orientations; (c) staggered.
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Figure 2
An illustration of the TPS shifting method. The current trajectory, of total time extension $τ$ , is shown on the left. A cut time $τ_{cut}$ is chosen randomly and uniformly between initial time 0 and final time $τ$ . With equal probabilities, a new trajectory is proposed via the shift backward (center, top) or via the shift forward (center, bottom) procedures. For a shift backward (center, top), the portion of the original trajectory after $τ_{cut}$ is kept (black) while the portion before is discarded (gray). From the end of the retained trajectory segment a new segment of extent $τ_{cut}$ is generated (red) with the usual dynamics. The proposed new trajectory is formed of the old black segment and the new red segment (all shifted in time backward by $τ_{cut}$ ). For a shift forward (center, bottom), the portion of the original trajectory before $τ_{cut}$ is kept (black) while the portion after is discarded (gray). Starting from the initial condition of the retained segment a new segment of extent $τ - τ_{cut}$ is generated (red) with the usual dynamics and time reversed (which is possible—and efficient—in our case due to detailed balance in the CDM dynamics). The proposed new trajectory is formed of the new red segment and the old black segment (all shifted in time forward by $τ - τ_{cut}$ ). The proposed new trajectories are then accepted or rejected according to a Metropolis criterion as described in the main text, as sketched on the right.
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Figure 3
Comparison of TPS acceptance rates for $L = 6$ as a function of $s$ . The orange curve corresponds to the TPS acceptance when using the original dynamics. The green curve is for the reference dynamics with $D = D_{s}$ from the $2 \times 2$ approximation of Appendix pp1. The blue curve corresponds to the optimal value of $D$ found from exploring the acceptance rate landscape. The data shown are for trajectories of length $τ = 50$ for $s > 0$ , and $τ = 5$ for $s \leq 0$ (convergence in time is much faster on the active side $s \leq 0$ ). Each point shown corresponds to $5 \times 10^{6}$ attempted TPS moves. The optimized $D$ values used are $D = 0.25 s$ for $s < 0$ , and $D = s$ for $s > 0$ . Inset: Acceptance rates for $L = 12$ in the region where the active-inactive transition occurs for this system size (see Fig. 4).
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Figure 4
(a) Activity rate $〈 k 〉 = 〈 K 〉 / τ$ as a function of $s$ , for various system sizes $L$ . Symbols show MC results, while the solid black line is from exact diagonalization for $L = 6$ . The activity converges quickly for $s \leq 0$ , but is strongly system-size dependent for $s > 0$ . For larger sizes, there is an increasingly sharp drop in $〈 k 〉$ at $s = 0$ , suggesting a first-order transition in the thermodynamic limit. Inset: Example (for an $8 \times 8$ lattice) of the minimally flippable zero-flux configurations, which dominate the dynamics at large positive $s$ . The colored squares show domains within which the dimers (stadium shapes) form a staggered arrangement with maximal local contribution to the flux $\vec{Φ}$ . The eight domains, two of each orientation, have equal size, and so the total flux of the configuration is zero. There are four flippable plaquettes, indicated with stars ( $★$ ), the minimum possible number for a zero-flux configuration [44]. (b) Distribution of activity rate for $L = 24$ and $s = 0$ (solid green curve) compared with a Gaussian distribution of the same curvature at the maximum (dashed black). The broadening, particularly on the low-activity side, is a reflection of the sudden drop in $〈 k 〉$ at $s = 0^{+}$ .
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Figure 5
(a)–(d) Distribution $p (\vec{N})$ of magnetization $\vec{N}$ in the quantum dimer model, for various values of the parameter $s = ln (v / t)$ , as indicated on the central $s$ axis, and system size $L = 16$ . (Note that the magnetization, like the flux $\vec{Φ}$ , obeys $| N_{x} | + | N_{y} | \leq \frac{1}{2} L^{2}$ .) For all $s < 0$ , the distribution has approximate circular symmetry, but with peaks along the square axes, corresponding to columnar order. The prominence of the peaks and the magnitude $| \vec{N} |$ of the ring decrease as one approaches the RK point at $s = 0$ , where the distribution is Gaussian around $\vec{N} = \vec{0}$ . (e), (f) Anisotropy measure $〈 cos (4 ϕ) 〉$ , where $tan ϕ = N_{y} / N_{x}$ , evaluated in the ground state $| gs 〉$ , versus $s$ . Panel (e) shows the dependence on system size $L$ , while in panel (f) the trajectory time $τ$ used in the simulations is varied with $L = 12$ fixed. Positive values correspond to a distribution peaked along the square axes, confirming that the ordering is columnar and that it becomes more pronounced as $| s |$ increases. (The small negative value at $s = 0$ is, we believe, a consequence of the discrete values taken by $\vec{N}$ .) (g) Root-mean-square magnetization magnitude $〈 | \vec{N} {|^{2} 〉}^{1 / 2}$ , corresponding roughly to the radius of the ring in the distribution of $\vec{N}$ , as a function of $s$ for $L = 6$ and 16. The dashed line shows a power-law fit to the data for $L = 16$ and $- 0.4 \leq s \leq - 0.1$ with fitted exponent $β_{eff} = 0.254$ , while the dotted line shows an approximate fit to the $L = 6$ data, with $β_{eff} = 0.1$ .
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Figure 6
Dimer configurations of a $2 \times 2$ region with open boundary conditions. There are seven configurations, which can be divided into three classes; one example of each is shown. (a) One of the two configurations with two dimers inside the included region. (b) One of the four configurations with a single dimer inside. (c) The single configuration with no dimers inside. In (b) and (c), the red circles represent sites whose dimers point out of the region.
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