Entanglement renormalization and gauge symmetry

A lattice gauge theory is described by a redundantly large vector space that is subject to local constraints and can be regarded as the low-energy limit of an extended lattice model with a local symmetry. We propose a numerical coarse-graining scheme to produce low-energy, effective descriptions of lattice models with a local symmetry such that the local symmetry is exactly preserved during coarse-graining. Our approach results in a variational ansatz for the ground state(s) and low-energy excitations of such models and, by extension, of lattice gauge theories. This ansatz incorporates the local symmetry in its structure and exploits it to obtain a significant reduction of computational costs. We test the approach in the context of a $Z_2$ lattice gauge theory formulated as the low-energy theory of a specific regime of the toric code with a magnetic field, for lattices with up to $16\times 16$ sites (${16}^2\times 2=512$ spins) on a torus. We reproduce the well-known ground-state phase diagram of the model, consisting of a deconfined and spin-polarized phases separated by a continuous quantum phase transition, and obtain accurate estimates of energy gaps, ground-state fidelities, Wilson loops, and several other quantities.

Figure 1

Entanglement entropy of half a $4\times 4$ ($32$ spin) lattice $L$ in the ground state $|{\Phi }_{+,+}\rangle$ of $H_{\text{TC}}^{\text{x}}$ as a function of the magnetic field $h_x$. The critical magnetic field $h_x^{\text{crit}}=0.3285(1)$[55] is denoted by a vertical line. The blue (top) curve is a lower bound to the entanglement entropy of the deformed toric code model without $W_{\text{exact}}$. For comparison, the green (middle) curve corresponds to the entanglement entropy of the ground state of the quantum Ising model on a $4\times 4$ ($16$ spin) lattice with periodic boundary conditions and no vacancy. Finally, the red (bottom) curve corresponds to the entanglement left in the ground state of $H_{\text{TC}}^{\text{x}}$ after applying the analytical transformation $W_{\text{exact}}$. Note that the amount of entanglement is very similar to that of the Ising model. Similar reduction of the entanglement entropy was observed by several authors in the context of global symmetries.[56, 57] Thus, by implementing a local version of the duality transformation to the Ising model, the analytical transformation $W_{\text{exact}}$ maps the robustly entangled ground state $|{\Phi }_{+,+}\rangle$ of the deformed toric code model to a significantly less entangled ground state.

Figure 2

Accuracy in the ground-state energy of the deformed toric code in a $4\times 4$ lattice, as a function of the refinement parameter $\chi$ (see Sec. 6), for magnetic field $h_x=0.1$ in $H_{\text{TC}}^{\text{x}}$ of Eq. (11). On the $4\times 4$ lattice we obtain the exact ground-state energy by directly diagonalizing the full Hamiltonian and use it to asses the precision of the results as a function of the computational cost. The computational cost grows monotonically with $\chi$. Both curves correspond to using a tree tensor network (see Sec. 5) as a means to numerically coarse-grain the system. However, the lower curve was obtained by first applying $W_{\text{exact}}$ (in this work we refer as the hybrid ansatz to the combination of $W_{\text{exact}}$ and a tree tensor network, see Sec. 5). For a fixed value of $\chi$, which roughly corresponds to the same computational cost, the hybrid ansatz leads to about four more digits of accuracy in the ground-state energy.

Figure 3

In the toric code model, spin-1/2 degrees of freedom sit at the edges of a square lattice $L$. Here we consider a lattice $L$, with periodic boundary conditions on both directions (torus). A star operator $A_s$ acts on the four spins surrounding site $s\in L$, whereas a plaquette operator $B_p$ acts on the four spins spins surrounding a plaquette $p$. The noncontractible cuts $c_1$ and $c_2$ and the noncontractible loops $l_1$ and $l_2$ are the support of operators $X_1$ and $X_2$ in Eq. (6) and of operators $Z_1$ and $Z_2$ in Eq. (8), respectively.

Figure 4

Cut $c_3$ connects two plaquettes $p_0$ and $p_1$ that are as far apart in $L$ as possible. Example of loop $l$ that has a length of eight spins, $\left|l\right|=8$, and encloses an areas of four plaquettes, $\Sigma (l)=4$.

Figure 5

(i) Graphical representation of the Hamiltonian $H_{\text{TC}}$ for a $3\times 3$ lattice $L$. A four-spin star operator $A_s$ acts on each site $s\in L$, and a four-spin plaquette operator $B_p$ acts on each plaquette $p\in L$. (ii) Hamiltonian $H_{\text{TC}}$ can also be defined on more general lattices. The example shows an irregular lattice with a five-spin star operator $A_s$ and a three-spin plaquette operator $B_p$. (iii) Additionally, we represent the magnetic field of the deformed toric code Hamiltonian $H_{\text{TC}}^{\text{x}}$ by means of a yellow shade surrounding each qubit on which $-h_x{\sigma }^x$ acts.

Figure 6

(i) A cnot is represented with an arrow from the control qubit to the target qubit. When applied on two adjacent qubits, it reconnects the target qubit. This affects the two stars $r,s$ to which the control qubits belongs, which become $r^{\text{'}}$ and $s^{\text{'}}$, as well as the two plaquettes $p,q$ to which the target qubit belongs, which become $p^{\text{'}}$ and $q^{\text{'}}$. (ii) A sequence of cnots can also be used to focus a star operator $A_s$ on a single qubit, on which it acts as $A_{s^{\text{'}}}={\sigma }_{s^{\text{'}}}^x$. As a result, the neighboring site $r$ increases the number of qubits connected to it, becoming $r^{\text{'}}$. Note that the single-qubit star can be represented inside any of the plaquettes surrounding site $r^{\text{'}}$. In this work we restrict our attention to the subspace of states that are invariant under star constraints, Eq. (3). The single-qubit star $A_{s^{\text{'}}}$ forces the qubit to be in state $|+\rangle$ and therefore unentangled with respect to the rest of the qubits. We will use this fact to remove it from the effective description of the system at low energies. (iii) A sequence of cnots can be used to focus a plaquette operator $B_p$ on a single qubit, on which it acts as $B_{p^{\text{'}}}={\sigma }_{p^{\text{'}}}^z$. As a result, the neighboring plaquette $q$ expands to $q^{\text{'}}$. Note that the single-qubit plaquette can be represented as connected to any of the sites of plaquette $q^{\text{'}}$.

Figure 7

The magnetic field ${\sigma }^x$ acting on control qubits expands into a multi-qubit interaction ${\sigma }^x\otimes {\sigma }^x\otimes \dots$ including all target qubits. (i) In concentrating a star operator on a single qubit, the magnetic field on that qubit spreads to all the qubits originally involved on that star. In the subspace of locally symmetric states, Eq. (3), we can remove the control qubit (now in state $|+\rangle$ and thus unentangled from the rest of the system, see caption of Fig 6(ii)) from the effective description. (ii) In concentrating a plaquette operator on a single qubit, the magnetic field ${\sigma }^x$ of the rest of qubits on the original plaquette turn into a ${\sigma }^x\otimes {\sigma }_x$ interaction between those qubits and the target qubit.

Figure 8

(i) (Left) Graphical representation of the toric code Hamiltonian $H_{\text{TC}}$, equivalently, $H_{\text{LGT}}$ for $h_x=0$, for a lattice $L$ made of $4\times 4$ sites or $4^2\times 2=16$ spins. (Right) Graphical representation of the dual quantum Ising Hamiltonian $H_{\text{Ising}}$. There are $4\times 4-1=15$ spins with a single-spin plaquette operator, that is, with a transverse magnetic field $-J_m{\mu }^z$. Notice the vacancy and two additional topological spins which are not subject to any star or plaquette constraints. Operators $X_1$ and $X_2$ act each on one of these qubits.

Figure 9

(i) Graphical representation of the Hamiltonian $H_{\text{TC}}^{\text{x}}$, equivalently $H_{\text{LGT}}$, including the magnetic field $-h_x\sum _j{\sigma }_j^x$. (ii) The duality transformation maps the magnetic field to a nearest-neighbor ferromagnetic Ising interaction $-h_x{\mu }^x\otimes {\mu }^x$. For simplicity, transverse magnetic field terms $-J{\mu }^z$ of $H_{\text{Ising}}$ are not depicted here. Note that the four spins surrounding the vacancy, denoted $U$, $D$, $L$, and $R$, as well as all boundary spins, follow a different interaction pattern. On the one hand, $U$ and $L$ have one interaction term less than the rest of the spins but have instead an additional term $-h_x{\mu }^x$ acting on them. (iii) On the other hand, the nearest-neighbor interaction between spins $p$ and $q$ at both sides of a boundary is mediated by a coupling $-h_x{\mu }_p^x\otimes X_{\alpha }\otimes {\mu }_q^x$ that includes one topological spin [here, $X_{\alpha }$ corresponds to one of the nonlocal operators $X_1$ or $X_2$ defined in Eq. (6)]. In the sector $(+,+)$, the topological spins are in state $|+\rangle \otimes |+\rangle$ and boundary spins are coupled with a regular Ising interaction $-h_x{\mu }_p^x\otimes {\mu }_q^x$, corresponding to periodic boundary conditions (BC) in both directions. However, in, e.g., sector $(+,-)$ the topological spins are in state $|+\rangle \otimes |-\rangle$ and the coupling between the spins in the uppermost and lowermost rows becomes antiferromagnetic, $h_x{\mu }_p^x\otimes {\mu }_q^x$, corresponding to antiperiodic BC in the vertical direction, while in the horizontal direction the BC are still periodic. In general, sector $(v_1,v_2)$ the BC in the horizontal (respectively vertical) direction will be periodic/antiperiodic depending on whether $v_1$ (respectively $v_2$) is $+/-$.

Figure 10

(i) The coarse-graining transformation $W$ defines a linear map between the space ${\mathbb{V}}_{L^{\text{'}}}$ of lattice $L^{\text{'}}$ and the space ${\mathbb{V}}_L$ of lattice $L$. (ii) Isometric transformation $W$ that decomposes as a product of disentanglers $u$ and isometries $w$. (iii) Constraints fulfilled by disentanglers and isometries; see Eq. (40).

Figure 11

Coarse-graining of a local operator $o$ acting on three contiguous sites of $L$ into a local operator $o^{\text{'}}$ acting on three contiguous sites of $L^{\text{'}}$. (i) Diagrammatic expression for $o^{\text{'}}=W^{†}\mathit{oW}$, where $W$ decomposes as a product of disentanglers $u$ and isometries $w$. (ii) Using that $u^{†}u=I$, where $I$ is the identity on two sites of $L$, most disentanglers can be removed. (iii) Using that $w^{†}w=I$, where $I$ is the identity on one site of $L^{\text{'}}$, most isometries can be removed so (iv) $o^{\text{'}}$ acts as the identity in all but three sites of $L^{\text{'}}$.

Figure 12

Effective Hamiltonian $H^{\text{'}}\equiv W^{†}HW$ expressed in terms of the original Hamiltonian $H$ and disentanglers $u$ and isometries $w$. If $H$ is a sum of local terms, then $H^{\text{'}}$ can also be expressed as a sum of local terms; see Fig. 11.

Figure 13

By composition, coarse-graining transformations $\{W,W^{\text{'}},W^{\prime },\dots \}$ produce a sequence of increasingly coarse-grained lattices $\{L,L^{\text{'}},L^{\prime },\dots \}$ with effective Hamiltonians $\{H,H^{\text{'}},H^{\prime },\dots \}$.

Figure 14

Example of a state of a lattice $L$ made of 16 sites, $|\Psi \rangle =W{W}^{\text{'}}{W}^{\prime }|{\Psi }^{\text{top}}\rangle$, expressed as a MERA made of three layers of disentanglers and isometries corresponding to three coarse-graining transformations $W$, $W^{\text{'}}$, and $W^{\prime }$.

Figure 15

(i) Invariance of a Hamiltonian $H$ under a global symmetry implemented by simultaneously acting with one-site transformation $r$ on all sites of $L$; see Eq. (44). (ii) Disentangler $u$ and isometry $w$ invariant under the action of the single site transformation $r$, represented by a circle, acting on all upper/lower indices of these tensors. Note that the upper index of the isometry $w$ is transformed according to $r^{\text{'}}$, which might be a different representation of the symmetry group.

Figure 16

Sequence of equalities showing that a global symmetry $R=r^{\otimes N}$ of Hamiltonian $H$ is exactly preserved by a coarse-graining transformation $W$ that is the product of disentanglers $u$ and isometries $w$, provided the latter are also invariant under transformation $r$; Fig. 15. Indeed, the effective Hamiltonian $H^{\text{'}}=W^{†}\mathit{HW}$ is invariant under the global transformation $R^{\text{'}}=(r^{\text{'}}){}^{\otimes N^{\text{'}}}$.

Figure 17

Diagrammatic representation of a Hamiltonian with a local symmetry. In this example the local symmetry is implemented by unitary transformations $r\otimes r\otimes r\otimes r$, acting on some blocks of four contigous sites.

Figure 18

(Top) The coarse-graining transformation $W$ transforms the lattice $L$ into the effective lattice $L^{\text{'}}$. Lattice $L$ has spins on its edges. Lattice $L^{\text{'}}$ also has spins on its edges but, in addition, has an effective free spins sitting in the interior of each plaquette. (Bottom) Transformation $W=W_{\text{exact}}{W}_{\text{num}}$ breaks into an exact transformation $W_{\text{exact}}$, which produces an intermediate lattice $\tilde{L}$ with constrained and free spins, and a numerical transformation $W_{\text{num}}$, which coarse-grains the free spins of lattice $\tilde{L}$.

Figure 19

The proposed coarse-graining transformation $W$ breaks into two pieces $W_{\text{exact}}$ and $W_{\text{num}}$. $W$ transforms the (constrained) spins of $L$ into the two types of spins present in the effective lattice $L^{\text{'}}$, namely constrained spins, represented by triple lines, and free spins, represented by single lines. $W_{\text{exact}}$ acts on constrained spins, whereas $W_{\text{num}}$ coarse-grains the free spins and acts on constrained spins diagonally on the ${\sigma }^x$ basis.

Figure 20

Exact coarse-graining transformation $W_{\text{exact}}$. All spins of the original lattice $L$ are equivalent. However, as a guide to the eye, larger circles are used to denote those spins that will remain constrained spins in $\tilde{L}$. (Left) A region of the original lattice $L$ is mapped by $W_{\text{exact}}$ into a region of the intermediate lattice $\tilde{L}$. Blocks of four plaquettes of $L$ are mapped into single square plaquettes of $\tilde{L}$, which contain three free spins and a vacancy in their interior. (Right) Detailed sequence of the cnot gates, denoted by red arrows, applied to each block of four plaquettes in order to modify the support of the star operators $A_s$. For clarity, dashed arrows also show two cnots corresponding to neighboring unit cells. At the end of the sequence, there are three types of spins: free spins (green), unentangled spins (in state $|+\rangle$), and constrained spins. Unentangled spins are removed by single-spin projections and are therefore not present in $\tilde{L}$.

Figure 21

The intermediate lattice $\tilde{L}={\tilde{L}}_{\text{const}}\cup {\tilde{L}}_{\text{free}}$ decomposes as the union of a square sublattice ${\tilde{L}}_{\text{const}}$ with a constrained spin on each of its edges and a square sublattice ${\tilde{L}}_{\text{free}}$ with either a free spin or a vacancies on each of its sites. (Shading in ${\tilde{L}}_{\text{free}}$ is used to indicate position relative to the plaquettes in ${\tilde{L}}_{\text{const}}$).

Figure 22

Depiction of the terms included in the transformed Hamiltonian $\mathrm{H̃}$. From the structure of lattice $\tilde{L}$, one can readily read the new star and plaquette operators of $\mathrm{H̃}$ [as explained in Fig. 5(ii)]. In addition, the magnetic field $-h_x\sum _i{\sigma }_j^x$ in $H_{\text{TC}}^{\text{x}}$ gives rise to other interacting terms $-h_x{\mathrm{K̃}}_{\text{int}}$, which are depicted as yellow shades. (i) Two-spin interaction $-h_x{\sigma }^x\otimes {\sigma }^x$ between nearest-neighbor free spins contained in the same plaquette. Note that in the presence of a vacancy, some of these terms become a magnetic field $-h_x{\sigma }^x$ acting on a free spin. (ii) Three-spin interaction $-h_x{\sigma }^x\otimes {\sigma }^x\otimes {\sigma }^x$ between two free spins on neighboring plaquettes and a constrained spin sitting at the boundary between those plaquettes. Again, the presence of a vacancy reduces the support of some of the interaction operators, which end up coupling one free spin and one constrained spin according to $-h_x{\sigma }^x\otimes {\sigma }^x$.

Figure 23

This simplified choice of the numerical transformation $W_{\text{num}}$ consist of a tensor product of isometries $w$ that coarse-grain the three free spins inside a square plaquette of $\tilde{L}$ into an effective free spin of $L^{\text{'}}$; see Eq. (71).

Figure 24

The effective lattice $L^{\text{'}}=L_{\text{const}}^{\text{'}}\cup L_{\text{free}}^{\text{'}}$ decomposes as the union of a square sublattice $L_{\text{const}}^{\text{'}}$ with a constrained spin on each of its edges and a square sublattice $L_{\text{free}}^{\text{'}}$ with an effective free spin on each of its sites. [Shading in $L_{\text{free}}^{\text{'}}$ is used to indicate position relative to the plaquettes in $L_{\text{const}}^{\text{'}}$].

Figure 25

Terms that appear in the effective Hamiltonian $H^{\text{'}}$, Eqs. (72) and (73). (i) Star operator $A_{s^{\text{'}}}$ involving four constrained spins. (ii) Plaquette operator $B_{p^{\text{'}}}$ involving four constrained spins and an effective free spin. (iii) Operators ${\Sigma }^z$ [Eq. (74)] and ${\Sigma }^x$ [Eq. (75)], acting on an effective free spin. (iv) Interactions between effective free spins on neighboring plaquettes of $L^{\text{'}}$, controlled by the a constrained spin.

Figure 26

The coarse-graining transformation $W^{\text{'}}$ transforms a block of four plaquettes of the effective lattice $L^{\text{'}}$ into a single plaquette of a new effective lattice $L^{\prime }$. Note that lattice $L^{\prime }$ has the same local composition of constrained and free spins as $L^{\text{'}}$. Transformation $W^{\text{'}}$ again breaks into an exact transformation $W_{\text{exact}}^{\text{'}}$, which produces an intermediate lattice $\widetilde{L^{\text{'}}}$ and a numerical transformation $W_{\text{num}}$.

Figure 27

The coarse-graining transformation $W^{\text{'}}$ is composed of an exact transformation $W_{\text{exact}}$ that acts only on constrained spins and a numerical transformation $W_{\text{num}}$ that coarse-grains the fresh free spins obtained from $W_{\text{exact}}$ together with the free spins in $L^{\text{'}}$ to produce the free spins of $L^{\prime }$ while acting on the constrained spins diagonally in the ${\sigma }^x$ basis.

Figure 28

First round of the cnot gates that constitute $W_{\text{exact}}$. Note that the free spins are simply ignored by the cnot gates. (In the first step, we have repositioned the free spins for notational convenience.) The rest of the sequence of cnot gates proceeds as in Fig. 20.

Figure 29

Examples of interactions contained in the term $(W_{\text{exact}}^{\text{'}}){}^{†}(-h_x{K}_{\text{int}}^{\text{'}})W_{\text{exact}}^{\text{'}}$ of the Hamiltonian $(W_{\text{exact}}^{\text{'}}){}^{†}{H}^{\text{'}}{W}_{\text{exact}}^{\text{'}}$. (i) The interaction between two effective free spins is mediated by two fresh spins, on which it acts as ${\sigma }^x\otimes {\sigma }^x$. (ii) Analogous interaction across the boundary of a square plaquette, which involves a constrained spin. (iii)–(iv) In the presence of a vacancy, the interaction simply acts on one spin less.

Figure 30

Graphical representation of $L^{\text{top}}$, the intermediate lattice ${\tilde{L}}^{\text{top}}$, and the final lattice $L^{*}$ made of just two topological spins and a free spin. Note the toric boundary conditions.

Figure 31

Transformation $W_{\text{exact}}^{\text{top}}$ produces two topological spins and three fresh new free spins. Transformation $W_{\text{num}}^{\text{top}}$ coarse-grains seven free spins into a single spin, while acting diagonally in a ${\sigma }^x$ basis on the two topological spins. The topological sector $(v_1,v_2)$ can be specified by setting the topological spins in the state $|v_1\rangle \otimes |v_2\rangle$. Index $\alpha$ labels an orthonormal basis $|\alpha \rangle \in {\mathbb{C}}_{\chi }$ of states of the final free spin. Each choice of topological sector $(v_1,v_2)$ and of state $|\alpha \rangle$ labels one of $4\chi$ orthonormal states $|{\Psi }_{(v_1,v_2)}^{\alpha }\rangle \in {\mathbb{V}}_L$ of the original lattice $L$, which can be reconstructed by undoing the sequence of coarse-graining transformations.

Figure 32

Lattices $L^{\text{top}}$ and ${\tilde{L}}^{\text{top}}$, with toric boundary conditions, and sequence of cnot gates and single-spin projections included in $W_{\text{exact}}^{\text{top}}$. For simplicity, the four free spins of $L^{\text{top}}$ have not been depicted. Note that due to the boundary conditions, in occasions two cnot gates (denoted in lighter red) are applied twice on the same control and target spins, in which case the do not need to be applied, since the square of a cnot gate is the identity operator.

Figure 33

Lowest eigenvalues of $H_{\text{TC}}^{\text{x}}$ as a function of the magnetic field $h_x$, for a system of linear size $L=8$, corresponding to $8^2\times 2=128$ spins. The two lowest energies in each topological sector $(v_1,v_2)$ are plotted. The lowest energy in each sector is represented with a solid line while the second lowest with a dashed line. Different colors refer to different sectors. The energies in sector ($-,+$) are identical to the energies in sector ($+,-$). The inset shows the gap between the energy of the ground states in sector $(v_1,v_2)$ and in the sector $(+,+)$. These gaps are very small for values of the magnetic field $h_x$ smaller than some critical value $h_x^{\text{crit}}=0.3285$[55] (denoted by a vertical line) and much larger for larger magnetic field $h_x$, see Fig. 34. These results were obtained with the hybrid ansatz with $\chi =100$.

Figure 34

(Top) Energy gap $\Delta E$ between the ground states of the topological sectors $(+,-)$ and $(+,+)$ [see Eq. (87)] as a function of the linear size $L$ for $h_x=0.2$ and $0.3$, corresponding to the deconfined phase of the deformed toric code model. In accordance with Eq. (12), $\Delta E$ decays exponentially with $L$. These results were obtained with the hybrid ansatz with $\chi =100$. (Bottom) Energy gap $\Delta E$ between the ground states of the topological sectors $(+,-)$ and $(+,+)$ [see Eq. (87)] as a function of the linear size $L$ for $h_x=0.5$ and $0.55$, corresponding to the spin-polarized phase of the deformed toric code model. In accordance with Eq. (18), $\Delta E$ grows linearly with $L$. These results were obtained with the hybrid ansatz with $\chi =100$.

Figure 35

(Top) Gap $\Delta E_{+,+}$ between the ground state and first excited state of the topological sector $(+,+)$ as a function of the magnetic field $h_x$ and for system sizes $L=4$, 6, and $8$. (Bottom) Product of the systems size $L$ times the energy gap $\Delta E_{+,+}$, plotted as a function of the magnetic field $h_x$. An estimate of $0.3267(5)$ for the critical magnetic field $h_x^{\text{crit}}=0.3285(1)$[55] (denoted by a solid vertical line) is obtained by extrapolating, using the phenomenological renormalization group,[94] the sequence of intersections among curves obtained from systems with consecutive sizes (2–4, 4–6, and 6–8) identified by dotted vertical lines in the plot using the known $L$ dependence for the sequence.[55]

Figure 36

Disorder parameter $\langle X_3\rangle$ as a function of the magnetic field $h_x$, for several lattice sizes. The critical magnetic field $h_x^{\text{crit}}=0.3285(1)$[55] is denoted by a vertical line. The inset shows a log-log plot of $\langle X_3\rangle$ at $h_x=h_x^{\text{crit}}$ as a function of $L$, from which we obtain an estimate of $0.96(5)$ for $2\beta /\nu$, whose current estimates are around $1.05$.[77]

Figure 37

(Top) Wilson loop $\langle Z[l]\rangle$ as a function of the perimeter $p(l)$ of loop $l$, on a $16\times 16$ lattice with magnetic field $h_x=0.1$, corresponding to the deconfined phase. Note the exponential decay of $\langle Z[l]\rangle$ as a function of the perimeter $p(l)$ of loop $l$, or perimeter law; see Eq. (17). (Bottom) Wilson loop $\langle Z[l]\rangle$ as a function of the area $a(l)$ of loop $l$, on a $16\times 16$ lattice with magnetic field $h_x=1$, corresponding to the spin-polarized phase. Note the exponential decay of $\langle Z[l]\rangle$ as a function of the area $a(l)$ of loop $l$, or area law; see Eq. (20).

Figure 38

Logarithm of the intensive fidelities $f_0(h_x)$ and $f_{\infty (h_x)}$ as a function of the magnetic field $h_x$ for different values $L=4$, 6, and 8 of the system size. The critical magnetic field $h_x^{\text{crit}}=0.3285(1)$[55] is denoted by a vertical line.

Figure 39

Logarithm of the intensive topological fidelity $f_{+,+}^{+,-}$ as a function of the magnetic field $h_x$. In the deconfined phase, $f_{+,+}^{+,-}$ remains large (its logarithm remains close to zero), indicating that the ground states $|{\Phi }_{+,+}\rangle$ and $|{\Phi }_{+,-}\rangle$ differ mostly on their topological spins but are otherwise very similar. This trend changes for larger magnetic fields $h_x$. As we enter the spin-polarized phase, the ground states $|{\Phi }_{+,+}\rangle$ and $|{\Phi }_{+,-}\rangle$ differ structurally, and modifying the topological spins of, e.g., $|{\Phi }_{+,-}\rangle$ no longer produces a good approximation to $|{\Phi }_{+,+}\rangle$. (Left) Data collapse occurs in the deconfined phase when we plot $L\log (f_{+,+}^{+,-})$. The critical magnetic field $h_x^{\text{crit}}=0.3285(1)$[55] is denoted by a vertical line. (Right) Instead, plotting $\log (f_{+,+}^{+,-})$ leads to data collapse in the spin-polarized phase. This behavior is expected as explained in the main text below Eq. (94).

Figure 40

Duality transformation between the $Z_2$ lattice gauge theory and the quantum Ising model, broken into six stages [(i)–(vi)].

Figure 41

cnot gates that transform stage (i) into stage (ii). The sequence proceeds from top to bottom. Only two of the three required steps are depicted. In a $L\times L$ lattice, $L-1$ such steps are required.

Figure 42

cnot gates that transform stage (ii) into stage (iii). The sequence proceeds from top to bottom. Only two of the three required steps are depicted. In a $L\times L$ lattice, $L-1$ such steps are required.

Figure 43

cnot gates that transform stage (iii) into stage (iv). Note that stage (iii) contains four columns of single-spin plaquettes, with three spins on each column. Instead, stage (iv) contains four columns of single-spin plaquettes with four spins on each column, except for the rightmost column, which only contains three spins and a vacancy. The sequence proceeds from left to right. Only two of the three required steps are depicted. In a $L\times L$ lattice, $L-1$ such steps are required.

Figure 44

cnot gates that transform stage (v) into stage (vi). Three required steps are depicted. In a $L\times L$ lattice, $L-1$ such steps are required.

Figure 45

Ground-state energy $E_0$ for the deformed toric code on a $8\times 8$ torus as a function of the refinement parameter $1/\chi$ for different values of the magnetic field, (i) $h_x=0.1$ and $h_x=0.2$ in the topological phase, (ii) $h_x^{\mathrm{crit}}=0.3285$ at the critical point, and (iii) $h_x=0.5$ and $h_x=0.6$ in the spin-polarized phase. Results of the simulations in the range of refinement parameter $50⩽\chi ⩽200$ are shown with blue circles. We estimate upper and lower bounds for the ground-state energy using the analysis of Ref. 78. The upper bound indicated by a black horizontal line is the value of $E_0$ at $\chi =200$ while the lower bound is shown by a green line. In the plots we also write the exact numerical values for both upper and lower bounds. They provide narrow windows (in general around $0.01\%$ or less of the exact value) in which the exact ground-state energy is contained.

Figure 46

Scaling of the gap $\Delta E_{++}$ for the deformed toric code on the $8\times 8$ torus as a function of $1/\chi$ for different values of the magnetic field in the topological phase, $h_x=0.1$ and $h_x=0.2$ (top), at the critical point, $h_x^{\mathrm{crit}}=0.3285$ (middle), and in the spin-polarized phase, $h_x=0.5$ and $h_x=0.6$ (bottom). The dependence of the gap as a function of $1/\chi$ is not uniform as in the case of the ground-state energy and thus we cannot extract rigid bounds on its value. However, due to the small variations of the gap as a function of $\chi$ in the range $50⩽\chi ⩽200$ (in the worst case of the critical point of about $1\%$ of its value) compared with its large variation as a function of the magnetic field $h_x$, we can neglect the dependence on $\chi$ of the gap provided we accept an uncertainty of about $1\%$ on the actual gap value.

Figure 47

Scaling of the disorder parameter $\langle X_3\rangle$ for the deformed toric code on the $8\times 8$ torus as a function of $1/\chi$ for different values of the magnetic field in the topological phase, $h_x=0.1$ and $h_x=0.2$ (top), at the critical point, $h_x^{\mathrm{crit}}=0.3285$ (middle), and in the spin polarized phase, $h_x=0.5$ and $h_x=0.6$ (bottom). On the one hand, the values in the deconfined phase (top) are compatible with the theoretical expectation that the disorder parameter vanishes. However, they fluctuate substantially as a function of $\chi$ (we appreciate variations of around $30\%$ around the value of disorder parameter at $h_x=0.2$). We can still distinguish the results at $h_x=0.1$, where the disorder parameter fluctuates around ${10}^{-5}$, from those at $h_x=0.2$ where it fluctuates around ${10}^{-4}$. We believe that the origin of these large fluctuations is due not as much to the finite size of $\chi$ as to a lack of convergence of the iterative procedure used to compute the ground state, combined with the small values of the disordered parameter. Our iterative optimization procedure is stopped when the ground-state energy has converged to about 10 digits. This criterion could be insufficient for a disorder parameter if the relevant critical exponents imply a much slower convergence than the ground-state energy.[112] On the other hand, both at the critical point (central panel) and in the spin-polarized phase (lower panel), we extract again very accurate results and we can neglect the dependence on $\chi$ of the disorder parameter by introducing a systematic error lower than $1\%$ of its actual value.