Devil’s staircases, quantum dimer models, and stripe formation in strong coupling models of quantum frustration

We construct a two-dimensional microscopic model of interacting quantum dimers that displays an infinite number of periodic striped phases in its $T=0$ phase diagram. The phases form an incomplete devil’s staircase and the period becomes arbitrarily large as the staircase is traversed. The Hamiltonian has purely short-range interactions, does not break any symmetries of the underlying square lattice, and is generic in that it does not involve the fine-tuning of a large number of parameters. Our model, a quantum mechanical analog of the Pokrovsky-Talapov model of fluctuating domain walls in two-dimensional classical statistical mechanics, provides a mechanism by which striped phases with large periods compared to the lattice spacing can, in principle, form in frustrated quantum magnetic systems with only short-ranged interactions and no explicitly broken symmetries.

Figure 1

Examples of striped phases. (a) Stripes of holes separating antiferromagnetic domains. This structure appears in some theories of the high $T_c$ cuprates. (b) Periodically spaced domain walls separating regions where the order parameter takes the uniform value ${\varphi }_1$ or ${\varphi }_2$. (c) Another example where the periodicity is associated with a repeating unit cell. If the repeat distance becomes infinite, then the state is said to be incommensurate with the lattice.

Figure 2

(a) Ground and (b) excited states of the Pokrovsky-Talapov model of fluctuating classical domain walls. This is a two-dimensional anisotropic model where domain walls form along the $y$ direction and separate regions where the order parameter is uniform. While the domain walls cost energy, they are allowed to fluctuate, which carries entropy. For a range of parameters, the domain walls are actually favored by the free energy minimization.

Figure 3

The anisotropic next-nearest-neighbor Ising (ANNNI) model. Ising spins lie on the points of a $d$-dimensional cube. Nearest-neighbor interactions are ferromagnetic $(J_1<0)$. Along one of the directions, we also have antiferromagnetic next-nearest-neighbor interactions $J_2>0$.

Figure 4

Winding numbers: draw a reference line that extends through and around (due to the periodic boundary condition) the system and label the vertical lines of the lattice alternately as $A$ and $B$ lines. For any dimer configuration, we may define, with regard to this reference line, the winding number $W_x=N_A-N_B$, where $N_A$ is the number of $A$ dimers intersecting the line and similarly for $N_B$. We can similarly draw a vertical line and define a similar quantity $W_y$. Note that this particular construction works for a 2D bipartite lattice. For 2D nonbipartite lattices, the construction is simpler: count the total number of dimers intersecting the horizontal and vertical reference lines and there are four sectors corresponding to whether $W_{x,y}$ is even or odd.

Figure 5

Sample dimer configurations with corresponding height mappings. The height mapping involves assigning integers to the squares of the lattice in the following manner. Divide the bipartite lattice into $A$ and $B$ sublattices. Assign zero height to a reference square and then moving clockwise around an $A$ site, the height increases by one if a dimer is not crossed and decreases by three if a dimer is crossed. The same rule applies moving counterclockwise about a $B$ site. The integers correspond to local heights of a crystal whose base lies on the page. In these examples, the lower square in the second column is taken as the reference square. (a) Dimers are arranged in columns corresponding to a surface that is flat on average (though there are fluctuations at the lattice scale). (b) Dimers are staggered and the corresponding surface is maximally tilted. (c) Average tilt is nonzero due to the staggered strip in between the flat columnar regions. (d) Going from left to right, the surface initially falls and then rises giving an average tilt of zero. This is because the two staggered regions have opposite orientation.

Figure 6

A typical domain wall state selected by $H_0$. These states break translational and rotational symmetry. The staggered strips, which are one column wide and may have one of two orientations, are like domain walls separating columnar regions, which may have arbitrary width. When $a=b$ in Eq. (5.2), the set of these states spans the degenerate ground state manifold of $H_0$.

Figure 7

Ground state phase diagram of the parent Hamiltonian $H_0$ as a function of the parameter $a-b$ [see Eq. (5.1)]. In these states, the dimers may only have two attractive bonds. When $a-b=0$, the states of Fig. 6 are degenerate ground states. Away from this point, the system enters a state where dimers either maximize or minimize the number of staggered interactions. The maximally staggered configuration is commonly called the “herringbone” state.

Figure 8

(Color online) One of the terms in the operator of Eq. (4.2) will flip the circled cluster as shown.

Figure 9

Examples of ideal tilted states. In these states, the domain walls have the same orientations and are uniformly spaced. The notation $\left[1n\right]$ denotes the state where one staggered strip is followed by $n$ columnar strips and so on. It is understood that $\left[1n\right]$ collectively refers to the above states and those related by translational, rotational, and reflection (i.e., switching the orientation of the staggering) symmetries. The examples drawn here, where it is understood that what we are seeing is part of a larger lattice, are (a) [11], (b) [12], (c) [13], (d) [14], (e) [15], and (f) [16].

Figure 10

(Color online) Ground state phase diagram of $H=H_0+tV$ from second order perturbation theory.

Figure 11

(Color online) The excited state on the right is obtained from the initial state by acting twice with the perturbation in Eq. (4.2).

Figure 12

(Color online) Ground state phase diagram of $H=H_0+tV$ from fourth order perturbation theory. The width of the [12] phase is order $t^4$.

Figure 13

(Color online) Ground state phase diagram of $H=H_0+tV$ from $n$th order perturbation theory. The width of the $\left[1n\right]$ phase is of order $t^{2n}$.

Figure 14

(Color online) The boundaries of the $\left[1n\right]$ sequence will typically open into finer phases and subsequently the fine boundaries can themselves open. While the detailed structure depends on parameters in the model, generically we expect an incomplete devil’s staircase to be realized. In the figure, we have explicitly drawn the opening of the [11]-[12] boundary but the other boundaries will behave similarly.

Figure 15

(Color online) (a) and (b) The two ways in which a dimer can participate in three attractive bonds. (c) The one way in which a dimer can participate in four attractive bonds. The attractive bonds are shown by the blue (dark gray) arrows. However, these configurations also involve repulsive interactions, which are shown in red (light gray), from terms $c$ and $d$ in the Hamiltonian. (d) An example of a state where every dimer has only two attractive bonds but with some dimers the two bonds are of different types. These “kinks” in the staggered domain walls involve an energy cost from term $d$ as indicated by the red (light gray) arrows.

Figure 16

(Color online) The three types of intermediate states obtained by acting once with the perturbation (5.3) on the domain wall states. The blue/dark gray (red/light gray) arrows denote attractive (repulsive) interactions that are present in the initial (left) or excited (right) state but not both. The circled cluster of the excited state of (3) is another such interaction. (1) is a staggered line next to a columnar region of greater than unit width. (2) and (3) are staggered lines separated by one columnar strip from another staggered line of the same and opposite orientation, respectively. Notice that relative to the excited state of (1), the excited state of (2) has two additional attractive $a$ bonds and two additional repulsive $c$ bonds. Similarly, excited state (3) has one additional attractive $b$ bond, two additional repulsive $d$ bonds, and a repulsive $p$ interaction. Since $c,d\ge a,b$, cases (2) and (3) involve higher energy intermediate states than (1).

Figure 17

(Color online) Second order phase diagram when $c>a$. The [11] phase has width $\sim t^2$ and the $A_2$ phase has width $\sim (c-a)t^2$. On the [11]-$A_2$ and the $A_2$-columnar boundaries, intermediate domain wall states are degenerate as described in the text.

Figure 18

(Color online) Most of the fourth order terms in Eq. (5.8) involve “disconnected” clusters of dimers. In this figure, the perturbation connects the initial state (0) to excited states, labeled (1) and (2), depending on whether cluster 1 or 2 has been flipped. Acting again with the perturbation connects to an excited state, labeled (12), where both of these clusters are flipped. Acting two more times with the perturbation brings us back to the initial state (0) via either of excited states (1) or (2). Such terms are called disconnected because there are no interactions [in Eq. (5.1)] between the dimers of clusters 1 and 2. The figure depicts a particular resonance from the energy term in Eq. (5.8). There is an analogous contribution from the wave-function term which is a product of the second order processes $\left(0\right)\to \left(1\right)\to \left(0\right)$ and $\left(0\right)\to \left(2\right)\to \left(0\right)$. While the number of such disconnected terms scales as $N^2$, where $N=L_x{L}_y$ is the system size, these resonances do not contribute to the energy because the contributions from the energy and wave-function terms exactly cancel.

Figure 19

(Color online) These are examples of fourth order self-energy resonances. Each resonance is confined to a single line and the number of resonances in the system is proportional to the number of lines. In resonances such as (a), there are interactions connecting dimers of two flipped clusters on the same line. Terms such as (b) arise from the wave-function term in Eq. (5.8) but have no analogous processes in the energy term to cancel against.

Figure 20

(Color online) These are examples of fourth order resonances which are effective interactions between lines. At fourth order, such interactions are possible only when lines are separated by two or fewer columns. On the $A_2$-columnar boundary, resonances (a) and (b) are the only processes to consider. These involve terms $p$ and $q$ in the Hamiltonian, as indicated by the circles.

Figure 21

(Color online) These are the long fourth order resonances which occur in states where lines are separated by two or more columnar strips. These processes are always stabilizing though the amount depends on the environment of the line as in the second order (see Fig. 16). The resonance in (a) is strongest because it involves the lowest energy intermediate state but (b) and (c) allow for a denser packing of lines. Resonance (c) is especially suppressed due to term $p$ in Eq. (5.1).

Figure 22

(Color online) Fourth order phase diagram. The new [13] state has width $\sim t^4$.

Figure 23

(Color online) These are the 12th order resonances which drive the transition between the [16] and [17] states. Process (a) selects the [16] state from the others in the $A_6$ manifold, which were stabilized equally at tenth order. Process (b) destabilizes the [16] state near its columnar boundary in favor of the [17] state, which maximizes the number of most favorable resonances (c).

Figure 24

(Color online) These are fourth order processes available between two staggered lines separated by a single columnar strip.

Figure 25

(Color online) These are fourth order processes which can occur if an [11] region is next to (a) another [11] region or (b) a [12] region. Both of these resonances contribute to the energy with a negative sign because the intermediate state involves two fewer repulsive $c$ bonds than if the flipped clusters were farther apart.

Figure 26

(Color online) These are eighth order processes which can lead to a new phase between the [11-12] and [12] phases. Resonance (a) stabilizes the [11-12] phase while (b) stabilizes the $\left[11\text{−}\right(12{)}^2]$ phase. In the limit where $c$ is large, resonance (b) is preferable. The easiest way to see this is by setting $c=\infty$. Then the energy term of resonance (a) and all of the wave-function terms in resonances (a) and (b) will give zero because the intermediate states cannot form without creating $c$ bonds. However, the intermediate state in resonance (b) does not involve $c$ bonds and can be obtained without creating $c$ bonds (i.e., in an eighth order process where the first and last two actions involve flipping the cluster on the right and the first cluster in the middle). Therefore the energy term of resonance (b) will give a stabilizing contribution.

Figure 27

(Color online) One of the four possible $\left[11\right]$ configurations in terms of Ising spins on the fully frustrated square lattice. The hard-core dimer constraint corresponds to the requirement that the FFSI ground state has one “unsatisfied” bond per plaquette. The columnar-dimer regions correspond to ferromagnetic-Ising domains. The staggered-dimer strips correspond to the Ising domain walls, depicted by the red-colored (dashed) curved lines, which separate ferromagnetic domains of different orientation. Clearly, in the other equivalent configurations, even though they share the same principle of domain-wall competition, the separated regions are not ferromagnetic but one of several metamagnetic choices (Ref. 63).