Quantum robustness of fracton phases

The quantum robustness of fracton phases is investigated by studying the influence of quantum fluctuations on the X-Cube model and Haah's code, which realize a type-I and type-II fracton phase, respectively. To this end, a finite uniform magnetic field is applied to induce quantum fluctuations in the fracton phase, resulting in zero-temperature phase transitions between fracton phases and polarized phases. Using high-order series expansions and a variational approach, all phase transitions are classified as strongly first order, which turns out to be a consequence of the (partial) immobility of fracton excitations. Indeed, single fractons as well as few-fracton composites can hardly lower their excitation energy by delocalization due to the intriguing properties of fracton phases, as demonstrated in this work explicitly in terms of fracton quasiparticles.

Figure 1

Illustration of the ${\widehat{A}}_c$ and ${\widehat{B}}_c$ operators that define Haah's code. The cube on the left shows the chosen labeling of the sites of cube $c$ and the chosen coordinate system used in this work. Two spins $\frac{1}{2}$ called $\sigma$ and $\mu$ are located on each vertex of the lattice. The cube in the middle illustrates the action of ${\widehat{A}}_c$ on the spins of cube $c$. Here, IZ represents the tensor product $\mathbb{1}\otimes {\mu }^z$, where the operator on the left acts on the $\sigma$ spin and the operator on the right on the $\mu$ spin. Analogously, the right cube illustrates the action of ${\widehat{B}}_c$ on the spins of cube $c$.

Figure 2

Illustration of the fractal character of the operators ${\widehat{B}}_c$ in Haah's code. The action of a single ${\widehat{B}}_c$ operator on the $\sigma$ spins can geometrically be illustrated as a tetrahedron. This is shown on the left side. The figure in the center illustrates the action of a single ${\widehat{B}}_c$ operator on the $\mu$ spins, again, forming a tetrahedron. The fractal character can be seen when putting four tetrahedrons together. These form a larger version of the same tetrahedron in a self-similar way, which is shown on the right. The same is true for the ${\widehat{A}}_c$ operators by symmetry.

Figure 3

The spin-$\frac{1}{2}$ degrees of freedom of the X-Cube model are located on the edges of a cubic lattice and are illustrated as filled circles. The ${\widehat{A}}_c$ operator is the product of the $12{\sigma }^x$ operators acting on the 12 edges of the cube $c$. The three types of ${\widehat{B}}_s^{\left(\kappa \right)}$ operators are given by ${\widehat{B}}_s^{\left(xy\right)}={\sigma }_{(\mathit{\nu },2)}^z{\sigma }_{(\mathit{\nu },1)}^z{\sigma }_{(\mathit{\nu }-{\mathit{e}}_x,1)}^z{\sigma }_{(\mathit{\nu }-{\mathit{e}}_y,2)}^z,{\widehat{B}}_s^{\left(xz\right)}={\sigma }_{(\mathit{\nu },3)}^z{\sigma }_{(\mathit{\nu },1)}^z{\sigma }_{(\mathit{\nu }-{\mathit{e}}_x,1)}^z{\sigma }_{(\mathit{\nu }-{\mathit{e}}_z,3)}^z$, and ${\widehat{B}}_s^{\left(yz\right)}={\sigma }_{(\mathit{\nu },2)}^z{\sigma }_{(\mathit{\nu },3)}^z{\sigma }_{(\mathit{\nu }-{\mathit{e}}_z,3)}^z{\sigma }_{(\mathit{\nu }-{\mathit{e}}_y,2)}^z$.

Figure 4

From top left to bottom right: first a fractonic four-cube excitation is created by acting with one ${\sigma }^z$ operator. The action of ${\sigma }^z$ operators is indicated by red lines. In the following, a planon hops to the right by the action of a ${\sigma }^z$ operator on the adjacent link. Then, by acting with several such Pauli operators on adjacent links the membrane is extended. In the end, one of the edges of the membrane is moved around the corner.

Figure 5

The rhombs correspond to $b_s^{\left(\kappa \right)}=-1$ where the orientation $\kappa$ corresponds to the orientation of the rhombs. The green lines symbolize the action of ${\sigma }^x$ operators in order to move these excitations. From top left to bottom right: first two lineons (each corresponding to a two-vertex excitation) are created as nearest neighbors in $y$ direction by acting with one ${\sigma }^x$ operator. By acting with a straight string of ${\sigma }^x$ operators on the adjacent links in the same direction, one of the two lineons is moved to the right without creating any further excitations (upper right figure). In contrast, an additional lineon excitation is always created at a finite-energy cost when moving a lineon around a corner (bottom left figure) demonstrating the restricted one-dimensional mobility of lineon excitations. However, a pair of parallel lineons has a two-dimensional mobility since the pair is allowed to move around corners without exciting additional excitations (bottom right).

Figure 6

Construction of a wireframe operator. In the figure on the top left, a pair of lineons moves around a corner. This creates two additional excitations. These can be annihilated by a further ${\sigma }^x$ operator as depicted on the top right. On the bottom left, these excitations move back to the left. Then, they annihilate with the initial excitations. On the bottom right, no excitations are left and the wireframe operator as a product of ${\sigma }^x$ operators is visible.

Figure 7

Illustration of the winding of a planon around a lineon with flavor $x$ in the $xy$ plane (left). On the right, this process is depicted by projecting on the $xy$ plane, thus the only visible parts of the planon loop are the red links where ${\sigma }^z$ operators have acted on. The lineon is located at the end of a string of ${\sigma }^x$ operators (depicted in green). The string and the loop share a single spin, here depicted as red-green dashed string at the bottom. The commutation relation of Pauli operators results in an additional minus sign as phase for the wave function showing the semionic statistics of planon and lineon excitations.

Figure 8

The figures on the left and in the center illustrate the action of ${\sum }_{\mathrm{tetrahedra},a=(a_1,a_2,a_3,a_4)}{\tau }_{a_1}^x{\tau }_{a_2}^x{\tau }_{a_3}^x{\tau }_{a_4}^x$ and ${\sum }_{\mathrm{tetrahedra},b=(b_1,b_2,b_3,b_4)}{\tau }_{b_1}^x{\tau }_{b_2}^x{\tau }_{b_3}^x{\tau }_{b_4}^x$ on the pseudospins on the dual cubic lattice. The figure on the right shows the action of a ${\widehat{B}}_c$ operator on the $\sigma$ spins in the original formulation of Haah's code (the operator is depicted as ${\widehat{B}}_c^{\left(\sigma \right)})$. Note that the action of a ${\widehat{B}}_c$ operator on the $\sigma$ spins and the action of a ${\sigma }^x$ operator on the $\tau$ pseudospins are identical up to an inversion about the center of the cube. As these two actions correspond to the perturbation of the high- and low-field limits in the single-type case, respectively, this case is self-dual.

Figure 9

Ground-state energy per site $\epsilon /J$ of Haah's code in the single-type case as a function of ${\lambda }_x=h_x^{\sigma }/J$ obtained by pCUT (solid lines) and from the variational ansatz (dashed lines). The high- and low-field expansions for pCUT intersect exactly at ${\lambda }_{x,\text{c}}^{\mathrm{pCUT}}=1$ due to the self-duality. The variational phase transition is located at ${\lambda }_{x,\text{c}}^{\mathrm{var}}\approx 0.844$. The variational phase-transition point is at smaller values of ${\lambda }_x$ since the variational energy of the fracton phase does not depend on the magnetic field.

Figure 10

Ground-state energy per site $\epsilon /J$ of Haah's code in the two-type parallel case as a function of ${\lambda }_x=h_x/J$ obtained by pCUT (solid lines) and from the variational ansatz (dashed lines). The high- and low-field expansions for pCUT intersect at ${\lambda }_{x,\mathrm{c}}^{\mathrm{pCUT}}=0.456\pm 0.006$. The uncertainty represents the standard deviation of the energies at which the relevant Padé extrapolants of the low- and high-field expansions ${\epsilon }_{\mathrm{lf}}$ and ${\epsilon }_{\mathrm{hf}}$ intersect. The variational phase transition point is located at ${\lambda }_{\text{c,var}}\approx 0.422$.

Figure 11

The exact ground-state phase diagram of Haah's code in the two-type orthogonal case as a function of ${\lambda }_x$ and ${\lambda }_z$.

Figure 12

The ground-state energy $\epsilon /J$ of the X-Cube model per link as a function of the parameter ${\lambda }_x$. Upper panel: different bare series of the low-field (darker lines) and high-field (lighter lines) expansions in orders 4, 6, and 8. Lower panel: comparison of the order-8 bare series from the low- and high-field expansions with the variational energy (dashed line) and the QMC data (black circles) from Ref. [66]. The vertical solid line in both panels indicates the phase-transition point ${\lambda }_{x,\mathrm{c}}^{\mathrm{pCUT}}=0.9196\pm 0.0012$ according to the Padé analysis of the pCUT results. Insets represent zooms close to ${\lambda }_{x,\mathrm{c}}$.

Figure 13

The ground-state energy $\epsilon /J$ of the X-Cube model per link as a function of ${\lambda }_z$. Upper panel: different bare series of the low-field (darker lines) and high-field (lighter lines) expansions in orders 4, 6, and 8. Lower panel: comparison of the order 6 (8) bare series from the low-field (high-field) expansion with the variational energy (dashed line) and the QMC data (black circles) from Ref. [66]. The vertical solid line in both panels indicates the phase-transition point ${\lambda }_{z,\mathrm{c}}^{\mathrm{pCUT}}=0.29364\pm 0.00017$ according to the Padé analysis of the pCUT results. Insets represent zooms close to ${\lambda }_{z,\mathrm{c}}$.

Figure 14

Relevant excitation energies of the one-, two-, and four-fracton sectors in the fracton phase of Haah's code for the single-type case as a function of ${\lambda }_x=\frac{h_x}{J}$. Shown are bare series in order 6 in ${\lambda }_x$ for all sectors.

Figure 15

Bare series as well as Padé extrapolants of the one-fracton gap ${\mathrm{\Delta }}_{\mathrm{lf}}^{1\mathrm{qp}}/J$ as a function of ${\lambda }_x$ for the single-type case of Haah's code. The vertical solid line indicates the exactly known phase-transition point ${\lambda }_{x,\mathrm{c}}=1$.

Figure 16

Illustration of the most relevant unperturbed two-fracton states at small fields for the low-energy physics in the two-fracton sector of Haah's code. A cube $c$ colored in blue indicates that its eigenvalue is $a_c=+1$ and hence represents a fracton excitation. For the single-type case, an ${\sigma }^x$ operator acting on the spin at the right red vertex changes the state shown in the center into the state shown at the right. Hence, we have a hopping element between these states in first-order perturbation. Additionally, for the two-type parallel case, a ${\mu }^x$ acting on the left red spin changes the state depicted in the center to the one depicted on the left.

Figure 17

Bare series as well as Padé extrapolants of the two-fracton gap ${\mathrm{\Delta }}_{\mathrm{lf}}^{2\mathrm{qp}}/J$ as a function of ${\lambda }_x$ for the single-type case of Haah's code. The vertical solid line indicates the exactly known phase-transition point ${\lambda }_{x,\mathrm{c}}=1$.

Figure 18

Illustration of the most relevant unperturbed four-fracton states at small fields for the low-energy physics in the four-fracton sector of Haah's code. A cube $c$ colored in blue indicates that its eigenvalue $a_c=-1$ and hence represents a fracton excitation. Left: the action of a ${\sigma }^x$ operator at the red vertex on the bare fracton ground state creates the four-fracton state depicted on the left. This four-fracton configuration can hop in second-order perturbation theory and is therefore most relevant for the low-energy physics in the four-fracton sector. Right: an operator ${\mu }^x$ acting on the spin at the red vertex creates the state shown on the right. Again, this state is mobile. Furthermore, the left and the right configurations can interact with each other in the two-type parallel case.

Figure 19

Dispersion ${\omega }^{4\mathrm{qp}}\left(\mathit{k}\right)/J$ of the lowest mode of the four-fracton sector of Haah's code in the single-type case for different ${\lambda }_x$.

Figure 20

Bare series as well as Padé extrapolants of the four-fracton gap ${\mathrm{\Delta }}_{\mathrm{lf}}^{4\mathrm{qp}}/J$ at $\mathit{k}=(0,0,0)$ as a function of ${\lambda }_x$ for the single-type case of Haah's code. The vertical solid line indicates the exactly known phase-transition point ${\lambda }_{x,\mathrm{c}}=1$.

Figure 21

Relevant excitation energies of the one-, two-, and four-fracton sectors in the fracton phase of Haah's code for the two-type parallel case as a function of ${\lambda }_x=h_x/J$. Shown are bare series in order 6 in ${\lambda }_x$ for the one- and two-fracton sectors while order 4 is displayed for the four-fracton sector.

Figure 22

Bare series as well as Padé extrapolants of the one-fracton gap ${\mathrm{\Delta }}_{\mathrm{lf}}^{1\mathrm{qp}}/J$ as a function of ${\lambda }_x$ for the two-type parallel case of Haah's code. The vertical solid line indicates the phase-transition point ${\lambda }_{x,\mathrm{c}}^{\mathrm{pCUT}}=0.456$.

Figure 23

Bare series as well as Padé extrapolants of the two-fracton gap ${\mathrm{\Delta }}_{\mathrm{lf}}^{2\mathrm{qp}}/J$ as a function of ${\lambda }_x$ for the two-type parallel case of Haah's code. The vertical solid line indicates the phase-transition point ${\lambda }_{x,\mathrm{c}}^{\mathrm{pCUT}}=0.456$.

Figure 24

Dispersion ${\omega }^{4\mathrm{qp}}\left(\mathit{k}\right)/J$ of the lowest mode in the four-fracton sector for the two-type parallel case of Haah's code along a high-symmetry path in the three-dimensional Brillouin zone for different fixed ${\lambda }_x$ using the bare order-6 series.

Figure 25

Bare series as well as Padé extrapolants of the four-fracton gap ${\mathrm{\Delta }}_{\mathrm{lf}}^{4\mathrm{qp}}/J$ at $\mathit{k}=(0,0,0)$ as a function of ${\lambda }_x$ for the two-type parallel case of Haah's code. The vertical solid line indicates the phase-transition point ${\lambda }_{x,\mathrm{c}}^{\mathrm{pCUT}}=0.456$.

Figure 26

Dispersion ${\omega }_{\mathrm{lineon}}^x\left(\mathit{k}\right)/J$ of the $x$-type lineon excitation of the X-Cube model in a single $x$ field as a function of $k_x$ for different values of ${\lambda }_x$ using the bare pCUT series of order 10.

Figure 27

Bare series (solid lines) as well as Padé extrapolants (dashed lines) of the lineon gap ${\mathrm{\Delta }}_{\mathrm{lineon}}/J$ of the X-Cube model in a $x$ field as a function of ${\lambda }_x$. The vertical solid line indicates the phase-transition point at ${\lambda }_{x,\mathrm{c}}^{\mathrm{pCUT}}=0.9196\pm 0.0012$ obtained from the pCUT analysis of the ground-state energy.

Figure 28

Bare series (solid lines) as well as Padé extrapolants (dashed lines) of the one-fracton gap ${\mathrm{\Delta }}_{\text{fracton}}/J$ of the X-Cube model in a $z$ field. The vertical solid line indicates the phase transition at ${\lambda }_{z,\mathrm{c}}^{\mathrm{pCUT}}=0.29364\pm 0.00017$ obtained from the pCUT analysis of the ground-state energy.

Figure 29

Dispersion of the planon excitation ${\omega }_{\mathrm{planon}}^{xy}\left(\mathit{k}\right)/J$ of the X-Cube model in a $z$ field along a high-symmetry path in the Brillouin zone for different values of ${\lambda }_z$ using the bare order-7 pCUT series.

Figure 30

Bare series (solid lines) as well as Padé extrapolants (dashed lines) of the planon gap ${\mathrm{\Delta }}_{\mathrm{planon}}/J$ as a function of ${\lambda }_z$ for the $z$-field case of the X-Cube model. As before, the vertical solid line indicates the phase-transition point ${\lambda }_{z,\mathrm{c}}^{\mathrm{pCUT}}=0.29364\pm 0.00017$ obtained from the pCUT analysis of the ground-state energy.