Higgs mechanism in higher-rank symmetric U(1) gauge theories

We use the Higgs mechanism to investigate connections between higher-rank symmetric $\text{U}\left(1\right)$ gauge theories and gapped fracton phases. We define two classes of rank-2 symmetric $\text{U}\left(1\right)$ gauge theories: the $(m,n)$ scalar and vector charge theories, for integer $m$ and $n$, which respect the symmetry of the square (cubic) lattice in two (three) spatial dimensions. We further provide local lattice rotor models whose low-energy dynamics are described by these theories. We then describe in detail the Higgs phases obtained when the $\text{U}\left(1\right)$ gauge symmetry is spontaneously broken to a discrete subgroup. A subset of the scalar charge theories indeed have X-cube fracton order as their Higgs phase, although we find that this can only occur if the continuum higher-rank gauge theory breaks continuous spatial rotational symmetry. However, not all higher-rank gauge theories have fractonic Higgs phases; other Higgs phases possess conventional topological order. Nevertheless, they yield interesting novel exactly solvable models of conventional topological order, somewhat reminiscent of the color code models in both two and three spatial dimensions. We also investigate phase transitions in these models and find a possible direct phase transition between four copies of ${\mathbb{Z}}_2$ gauge theory in three spatial dimensions and X-cube fracton order.

Figure 1

Arrangement of the gauge field degrees of freedom for a $\text{U}\left(1\right)$ symmetric lattice gauge theory in (a) $d=2$ and (b) $d=3$. Each dot is a rotor variable, and the circles indicate that $d$ spins live on each site.

Figure 2

Charge configurations created in the $(m,n)$ scalar charge theory by (a) $e^{i{A}_{xx}}$, the raising operator for $E_{xx}$, acting on the center site (b) $e^{i{A}_{xy}}$, the raising operator for $E_{xy}$, acting on the black plaquette.

Figure 3

Transverse hopping operator for a unit dipole in the (1,2) scalar charge theory. The $e^{i{A}_{ij}}$ operators act on the black rotors and create the charge configuration shown. Because $e^{i{A}_{xy}}$ only creates charge-2 objects in the (1,2) theory, it cannot cause a unit dipole to hop, unlike the (1,1) theory.

Figure 4

Charge configurations created in the $(m,n)$ vector charge theory by (a) $e^{i{A}_{xx}}$, the raising operator for $E_{xx}$, acting on the black site (b) $e^{i{A}_{xy}}$, the raising operator for $E_{xy}$, acting on the black plaquette.

Figure 5

Effect of the Higgs mechanism on the local operator $e^{i{A}_{xx}}$ in the (1,1) scalar charge theory. The original charge configuration (top) is modified when charge-2 particles are condensed; the charge-2 particle is absorbed into the condensate, and charges $+1$ and $-1$ become equivalent.

Figure 6

Operator generating a ${\mathbb{Z}}_3$ charge hopping operator in the $p=3$ Higgsed (1,1) scalar charge theory.

Figure 7

Operator in the (1,1) vector charge theory that, after $p=2$ Higgsing, allows ${\mathbb{Z}}_2$ charges on $x$-directed links to hop in the $y$ direction.

Figure 8

Terms in the $d=2$ Higgsed (1,1) scalar charge theory Hamiltonian (a) in isolation and (b) in the context of the larger lattice. $a$ is associated with the center site and is the product of the eight indicated Pauli $X$ operators, which act on the green spins. The $b_{zi}$ operators are associated with the center links and are products of four Pauli $Z$ operators acting on the orange spins. The sites/plaquettes on which operators act (in addition to the spins themselves) have been shaded for easier visibility.

Figure 9

Schematic phase diagram at $V=\infty$ for the $d=2(1,1)$ scalar charge theory. The phases at large and small $h$ are accurate, but the phase boundaries and behavior at intermediate coupling are schematic. The direct transition from ${\mathbb{Z}}_2^3$ to paramagnetism is protected by $C_4$ rotation symmetry.

Figure 10

(a) String operators and the anyons they create in the Higgsed (1,1) scalar charge model. Electric excitations, shown on the left, consist of (green-boxed) sites where the eigenvalue of $a$ is $-1$ and are created by a string of $Z$ operators acting on the orange spins. Magnetic excitations, shown on the right, consist of a set of (heavy orange) links where the eigenvalue of $b_{zi}$ is $-1$ and are created by a string of $X$ operators acting on the green spins. For the purposes of checking braiding, all the pictures should be regarded as representing the same portion of the square lattice. The sites/plaquettes on which an operator acts have been shaded (light orange/green) for easier visibility. (b) The local action of $Z_{xy}$ on a bound state of $e_1, e_2$, and $e_3$ converts the bound state to a single-charge excitation on the top-left site.

Figure 11

(a) Terms in the Hamiltonian at large $h_p$. The operators are products of the indicated spins; the plaquette spins are frozen out. (b) Sublattice of spins (dark) which form one of four decoupled copies of the toric code at large $h_p$. The large-$h_p$ effective Hamiltonian (36) does not couple the dark spins to any others. Some links of the lattice have been lightened to make it easier to see the relation to the toric code.

Figure 12

(a) Terms in the Hamiltonian and (b) phase diagram of the $d=2$ Higgsed $(2r,2s+1)$ vector charge Hamiltonian. $a_i$ are associated with the center links and are the product of the two indicated Pauli $X$ operators, which act on the green spins. The $b$ operator is associated with the center site and is a product of the indicated Pauli $Z$ operators on the four orange spins. The direct transition in (b) is protected by $C_4$ rotation symmetry.

Figure 13

(a) Terms in the Hamiltonian and (b) schematic phase diagram of the $d=2$ Higgsed $(2r+1,2s+1)$ vector charge theory Hamiltonian. $a_i$ are associated with the center bonds and are the product of the four indicated Pauli $X$ operators, which act on the green spins. The $b$ operator is associated with the center site and is a product of the indicated Pauli $Z$ operators on the eight orange spins. The direct transition to a paramagnet in (b) is protected by $C_4$ rotational symmetry.

Figure 14

Terms in the Hamiltonian (38) for the Higgsed (0,1) scalar charge model in $d=3$. $a$ is a product of Pauli $X$ operators on the 12 green spins, while the $b_{ii}$ are products of 4 Pauli $Z$ operators on the orange spins.

Figure 15

Terms in the Hamiltonian for the Higgsed $(2r,2s+1)$ scalar charge theory in $d=3$. The $a$ operators are associated with the sites and are products of 12 $X_{ij}$ operators acting on the green spins. The $b_{ij}$ operators are products of two (off-diagonal terms, associated with links) or four (diagonal terms, associated with cubes) Pauli $Z_{ij}$ operators acting on the orange spins.

Figure 16

(a) Terms in the Hamiltonian and (b) schematic phase diagram of the $d=3$ Higgsed $(2r+1,2s+1)$ scalar charge theory Hamiltonian. $a$ are associated with the sites and are the product of 18 Pauli $X$ operators, which act on the green spins. The $b_{ij}$ operators are products of 4 Pauli $Z$ operators acting on the orange spins. The diagonal components $b_{ii}$ are associated with cubes and the off-diagonal components are associated with links. The direct transitions in (b) are protected by cubic rotation symmetry.

Figure 17

String and membrane operators and excitations in the Higgsed $d=3(2r+1,2s+1)$ scalar charge model. The bold link labeled $l_0$ is the same link in all images so that braiding statistics can be checked; $e_i$ excitations braid nontrivially with $m_i$ strings but no others. Left: pairs of electric particles (green boxes) are created by a string of $Z$ operators acting on the orange spins (relevant sites shaded in orange), and have $a=-1$. Middle: excitation patterns of magnetic strings. The orange links have either $b_{yx}$ or $b_{zx}=-1$, as appropriate for the link orientation. Pink cubes have $b_{xx}=b_{zz}=-1$, and blue cubes have $b_{xx}=b_{yy}=-1$. Right: membrane operators, consisting of products of $X_{ij}$ on the green spins, for producing the magnetic strings shown in the middle. The membrane operators for $m_1$ and $m_2$ include all $X_{xz}$ operators within the membrane, while those for $m_2$ and $m_3$ include all $X_{xy}$ operators within the membrane.

Figure 18

(a) Terms in the effective Hamiltonian (40) for the $d=3(2r+1,2s+1)$ scalar charge model at large $h_f$. (b) One of eight sublattices (dark) of site spins; the terms in (a) do not couple different sublattices. Some bonds of the lattice have been lightened to make the relation to the $d=3$ toric code clearer.

Figure 19

(a) Terms in the Hamiltonian and (b) phase diagram for the $d=3$ Higgsed $(2r+1,2s+1)$ vector charge theory. The labels for the Pauli operators are omitted to avoid clutter. The $a_i$ are associated with the center bonds and are the product of Pauli $X$ operators on the six green spins. The diagonal (off-diagonal) $b_{ij}$ operators are associated with the center site (face) and are products of Pauli $Z$ operators on the 8 (10) orange spins. The direct transitions from the ${\mathbb{Z}}_2^7$ phase in (b) are protected by the rotational symmetries of the cubic lattice.

Figure 20

(a) Elementary membrane operator ($X_{xx}$ acting on green site spins) and associated string of excitations of $b_{yy}$ and $b_{zz}$ terms (orange squares) for the Higgsed $(2r+1,2s+1)$ vector charge model in $d=3$. This membrane can live on eight possible sublattices of lattice constant 2. (b) Action of a string of $X_{ij}$ operators on green face spins, creating excitations of $b_{yy}$ and $b_{zz}$ on sites (orange spheres). This collection of excitations is equal to a bound state of all eight of the strings discussed in part (a); such a bound state is topologically trivial since it can be destroyed by a string operator.

Figure 21

Two types of string operators ($Z$ operators acting on orange spins) and their associated electric particles (green links) in the Higgsed $d=3(2r+1,2s+1)$ vector charge model. Eight strings total are obtained by shifting these operators by zero or one unit in one or both lattice directions perpendicular to the green links, but only seven are topologically distinct because bound states of eight excitations can be destroyed with a local operator.

Figure 22

Open string of $X_{xy}$ acting on the green faces and its associated excitations in the Higgs phase of the $d=3$ vector charge $(2r+1,2s+1)$ model. The blue faces have magnetic excitations; off-diagonal $b_{ij}$ terms have eigenvalue $-1$ on those phases. The orange spheres are site excitations ($b_{ii}$ which have eigenvalue $-1$) and give the string tension; at large $h_s$, they condense, deconfining the collection of blue face excitations.

Figure 23

Terms in the $d=3$ Higgsed $(4r+2,2s+1)$ vector charge theory Hamiltonian. The $a_i$ are associated with the center bonds and are the product of Pauli $X$ operators on the four green spins. The diagonal (off-diagonal) $b_{ij}$ operators are associated with the center site (face) and are products of Pauli $Z$ operators on the 4 (14) orange spins.

Figure 24

$a$ operator for the Higgsed (0,1) scalar charge model in $d=2$.

Figure 25

(a) Terms in the Hamiltonian and (b) phase diagram for the $d=2$ Higgsed $(2r+1,2s)$ scalar charge theory. The electric term $a$ is a four-spin interaction on the sites and the magnetic terms $b_{zi}$ are Ising interactions on the $i$-directed bonds.

Figure 26

Terms in the Hamiltonian of the Higgsed $(2r+1,2s+2)$ scalar charge theory. The $a$ operators are associated with the sites and are products of six $X_{ii}$ operators acting on the green spins. The $b_{ij}$ operators are products of two (off-diagonal terms, associated with links) or four (diagonal terms, associated with cubes) Pauli $Z_{ij}$ operators acting on the orange spins.

Figure 27

Terms in the Hamiltonian and for the $d=2$ Higgsed $(2r+2,2s+1)$ scalar charge theory. The electric term $a$ is a four-spin interaction on the sites and the magnetic terms $b_{zi}$ are Ising interactions on the $i$-directed bonds.

Figure 28

Terms in the Hamiltonian of the $d=2$ Higgsed $(2r+1,2s+2$) vector charge theory Hamiltonian. $a_i$ are associated with the bonds and form Ising interactions acting on the green spins. The $b$ operator is associated with the center site and is a product of the indicated Pauli $Z$ operators on the four orange plaquette spins.

Figure 29

Terms in the Hamiltonian of the Higgsed $(4r+4,2s+1)$ vector charge theory. The $a$ operators are associated with the links and are products of four $X_{ij}$ operators acting on the green spins. The $b_{ij}$ operators are products of four Pauli $Z_{ij}$ operators acting on the orange spins.

Figure 30

Terms in the Hamiltonian of the Higgsed $(2r+1,2s+2)$ vector charge theory. The $a$ operators are associated with the links and are Ising-type products of $X_{ii}$ operators acting on the green spins. The $b_{ij}$ operators are products of 10 (off-diagonal terms, associated with faces) or 4 (diagonal terms, associated with sites) Pauli $Z_{ij}$ operators acting on the orange spins.