Tensor Networks for Lattice Gauge Theories with Continuous Groups

We discuss how to formulate lattice gauge theories in the tensor-network language. In this way, we obtain both a consistent-truncation scheme of the Kogut-Susskind lattice gauge theories and a tensor-network variational ansatz for gauge-invariant states that can be used in actual numerical computations. Our construction is also applied to the simplest realization of the quantum link models or gauge magnets and provides a clear way to understand their microscopic relation with the Kogut-Susskind lattice gauge theories. We also introduce a new set of gauge-invariant operators that modify continuously Rokhsar-Kivelson wave functions and can be used to extend the phase diagrams of known models. As an example, we characterize the transition between the deconfined phase of the $Z_2$ lattice gauge theory and the Rokhsar-Kivelson point of the $U(1)$ gauge magnet in 2D in terms of entanglement entropy. The topological entropy serves as an order parameter for the transition but not the Schmidt gap.

Popular Summary

Physicists have found evidence that three out of the four fundamental interactions (electromagnetic, strong, and weak) are a manifestation of gauge symmetries. These interactions are described by gauge theories that, despite their elegance, still have not been fully solved. As a consequence, we do not understand the role of gauge symmetry in hot and dense nuclear matter, nor do we understand which phases of matter gauge theories can ultimately describe. For instance, it is unknown if gauge theories can explain the onset of high-temperature superconductivity in cuprates, and we do not fully understand the role of local symmetries in the heavy-ion collisions produced in particle accelerators. We extend the formulation of gauge theories to tensor networks, the LEGO building blocks of many-body quantum systems, where quantum states and operators (the equivalent of LEGO figurines) are obtained by contracting small, constituent tensors (the equivalent of LEGO bricks).

Tensor networks can be used both as a numerical tool (as an alternative to Monte Carlo simulations in contexts such as out-of-equilibrium dynamics) and as a theoretical framework. We use tensor networks to uncover new classes of gauge theories designed for quantum simulators whose physics is still not well understood. Our numerical results show that the tensor network approach makes it possible to distinguish two topological phases that cannot be detected by local order parameters.

We anticipate that our approach will foster new insights in gauge symmetries from tabletop experiments with quantum simulators, including quantum magnets in cold-atom experiments and/or numerical simulations based on tensor networks.

Figure 1

The Hilbert space $\mathcal{H}$ of a quantum many-body system (represented here by a 3D box) is exponentially large, since it is the tensor product of the Hilbert spaces of the constituents. Gauge symmetry allows us to identify a smaller space that we call the physical Hilbert space ${\mathcal{H}}_P$. This space is the subspace spanned by those states that fulfill all the local constraints imposed by the gauge symmetry and is represented by a membrane inside $\mathcal{H}$. ${\mathcal{H}}_P$ is smaller than the full $\mathcal{H}$ but it is still exponentially large. Low-energy states of local gauge-invariant Hamiltonians, however, are expected to live in a small corner of ${\mathcal{H}}_P$, in the same way as low-energy states of generic local Hamiltonians live in a small corner of $\mathcal{H}$ [51, 52, 53, 54]. For this reason, we design a variational ansatz based on TNs that allows us to explore this small corner of ${\mathcal{H}}_P$ (orange oval). By increasing the bond dimension of the elementary tensors in the TN ($D\uparrow$), we can explore increasingly large regions of ${\mathcal{H}}_P$, and eventually, for $D\to \infty$, we can cover the whole ${\mathcal{H}}_P$. The projector on ${\mathcal{H}}_P$ and a family of interesting RK states are obtained exactly with a TN with minimal bond dimension that scales as $D\simeq \sqrt{d}$, where $d$ is the dimension of the local Hilbert space.

Figure 2

Graphic representation of tensors and their contractions. Tensors are generalized vertexes (geometric shapes) whose legs are represented by dangling arrows. (a) A tensor with three legs $X_{bc}^a$. The upper indexes are incoming legs while lower indexes are outgoing legs. Complex conjugation is denoted by mirror reflection of the vertex. Different colors denote different tensors. Indexes are ordered clockwise, starting from the solid dot on the vertex. The contraction of two tensors is denoted by an arrow joining them. The dagger operation $X^{†}$ involves both the complex conjugation of the elements of the tensor and the inversion of the arrows attached to its legs. As a consequence, the order of the legs also changes $X^{†}={\overline{X}}_a^{cb}$ [67]. (b) An isometric tensor (or simply isometry) is always represented by a triangular vertex. A tensor $W$ is isometric if there is a specific choice of legs, which identifies two vector spaces $U$ and $V$, such that when $W$ is contracted with $W^{†}$ through the legs in $U$, the result is the identity tensor in $V$. In the figure, $U$ is spanned by the leg $a$ while $V$ is spanned by the two legs $b$ and $c$.

Figure 3

(a) The orthogonality relation in Eq. (3) allows us to identify a set of isometries $W_r$ projecting $\mathbb{C}(g)$ onto $V_r\otimes {\overline{V}}_r$. If one knows the matrices ${\mathrm{\Gamma }}_r(g)$ for all $g\in G$, $W_r$ is defined by collecting them inside a three-index tensor. The index $r$ identifying the irrep is written on the top of one of the two legs. (b) The isometric property of the tensor $W_r$. (c) Alternatively, if we know the $W_r$ isometry, we can obtain the matrices in the $r$ representation by acting with $W_r$ on a vector proportional to $|g⟩$ (yellow circle in the figure). The resulting tensor has two indexes $i$ and $j$ and is the matrix ${\mathrm{\Gamma }}_r(g{)}_j^i$.

Figure 4

(a) The direct sum of $W_r$ is a unitary tensor. It allows us to change basis from $\mathbb{C}(G)$ to the direct sum of all the irreps times their conjugate ${\oplus }_r(V_r\otimes V_{\overline{r}})$. On the upper part of the figure, we explicitly draw the collection of all the $W_r$’s. A simplified picture is shown on the lower part, where the direct sum is implicit in the absence of the label $r$ attached to the legs of the tensor. We will use this notation in the following. When we want to represent $W_r$, we will attach $r$ to the leg, while we represent $W_G$ by exactly the same drawing but without the $r$. From now on, for simplicity, we will also omit to label the legs with letters. (b) The unitarity of $W_G$ is encoded in the fact that the contraction over the two legs acting on $\oplus (V_r\otimes V_{\overline{r}})$ gives a delta function in $\mathbb{C}(G)$.

Figure 5

A symmetric tensor is left invariant by the simultaneous rotation of all incoming legs by ${\mathrm{\Gamma }}^{†}(g)$ and its outgoing legs by $\mathrm{\Gamma }(g)$ in the appropriate representation. Here, we exemplify the case of a three-leg tensor with an incoming leg that transforms in the irrep $r$ and two out-legs that transform under irreps $r^{\prime }$ and $r^{\prime \prime }$.

Figure 6

Gauge boson constituents are defined on the links of an oriented lattice. Links are represented by dashed lines, and constituents are small solid circles along these lines (which should not be confused with tensors, whose legs are solids lines). (a) In the Kogut-Susskind LGT, each constituent is described by a state $|g⟩$ of the group algebra $\mathbb{C}(G)$. (b) Left and right rotations $L(h^{-1})$ and $R(k)$ are introduced in Eqs. (11) and (12). They transform the state $|g⟩$ into $|h^{-1}gk⟩$. Both operators require an initial change of basis from $\mathbb{C}(G)$ to ${\oplus }_r(V_r\otimes {\overline{V}}_r)$, obtained through the $W_G$ of Eq. (7) and represented in Fig. 4 (the first horizontal triangle). One then applies to each of the legs the corresponding rotation matrix given by ${\oplus }_r[{\mathrm{\Gamma }}_r(h^{-1})\otimes {\mathrm{\Gamma }}_r(k)]$, with the individual ${\mathrm{\Gamma }}_r$ defined in Fig. 3 and represented here by the two vertical triangles. At last, $W_G^{†}$, represented by the inverted triangle, allows us to go back to $\mathbb{C}(G)$.

Figure 7

Graphical representation of the operator $A_s(h)$, the building block of gauge transformations in the Kogut-Susskind LGT. On every link entering a site $s$, it either rotates the state of the link through $R(h)$ or $L(h^{-1})$, depending on whether the link enters or leaves the site. The forms of $L$ and $R$ are given in Fig. 6.

Figure 8

(a) The $U_{s_n}$ operator of Eq. (19) in the KS LGT. It acts on the tensor product of the physical space $\mathbb{C}(G)$ of the link $s_n$ and of an auxiliary space that defines the irrep $r_m$: $\mathbb{C}(G)\otimes V_{r_m}$. It is built from the contraction of a copy tensor $\mathcal{C}\equiv |g⟩⟨g|\otimes ⟨g|$ (red circle) and the corresponding rotation matrices ${\mathrm{\Gamma }}_{r_m}(g{)}_j^i$ [defined in Eq. (6)]. (b) The above definition implies that the $U_{s_n}$ transmits the rotation onto the physical index, as induced by conjugation by $L$ and $R$, to the auxiliary indexes. This property allows us to use $U_{s_n}$ as a building block of the gauge-invariant plaquette operator in Eq. (22), where, thanks to the trace, the rotations attached to the auxiliary indexes cancel.

Figure 9

The plaquette operator $B_p$ of Eq. (22) constructed as a matrix-product operator from four $U_{p_n}$’s, acting on the links around a plaquette in the KS LGT. As a consequence of the covariance of the $U_{p_n}$ under conjugation by $L$ and $R$ [see Fig. 8], it transfers the rotation onto the physical legs to rotations onto the auxiliary legs. The operator commutes with the gauge transformations, and, thus, it can be used as a building block of a gauge-invariant Hamiltonian [see Eq. (23)].

Figure 10

The projector on the gauge-invariant states. (a) The corresponding TN is defined through the contraction of several copies of two elementary tensors $\mathcal{C}$, which copies the physical Hilbert space onto the auxiliary Hilbert space, and $\mathcal{G}$, which selects only configurations fulfilling the gauge invariance condition. (b) An example of gauge-constraint decoupling at sites $s-1$ and $s$ obtained through the insertion of the copy tensor $\mathcal{C}$. (c) The projector on the gauge-invariant states for a $4\times 4$ square lattice with periodic boundary conditions.

Figure 11

The definition of the tensors $\mathcal{C}$ and $\mathcal{G}$ for the Kogut-Susskind LGT. The tensor $\mathcal{G}$ is obtained by composition of several copies of unitaries $C_L$ and $C_R$ described in the main text. They are followed by the projection onto the states $|+⟩$ and $\widetilde{|+⟩}$ defined in the main text.

Figure 12

(a) The Hilbert space of a gauge boson in the truncated KS LGT is obtained by truncating $\mathbb{C}(G)$ with the isometry $W_T$ defined in Eq. (28). It is isomorphic to $(V_r\otimes {\overline{V}}_r)\oplus (V_{r^{\prime }}\otimes {\overline{V}}_{r^{\prime }})$. Each term of the direct sum is a tensor product of two constituents, so that we represent it by two solid circles on each link. (b) The operators implementing the left and right rotations defined in Eqs. (29) and (30) are obtained from those of Fig. 6 after conjugation with $W_T$.

Figure 13

The operator $A_s(h)$ that generates the gauge transformations in the truncated KS LGT. It is the result of the truncation of the operator $A_s(h)$ of Fig. 7 with $W_T$ defined in Eq. (28). On each link, the operator acts on the constituent closer to $s$ through a rotation ${\mathrm{\Gamma }}_r(h)$ or ${\mathrm{\Gamma }}_{r^{\prime }}(h)$, which is controlled by the irrep of the neighboring constituent on the same link. It thus performs controlled operations in between the two constituents of a link, as suggested by a dot on the controller leg and by an arrow that joins it with the controlled leg.

Figure 14

(a) The ${U_T}_{s_n}$ operator in the truncated KS LGT as defined in Eq. (32). (b) It has by construction the desired covariance properties, meaning that rotations on the physical legs (upper part of the panel) are transmitted to rotations on the auxiliary legs (lower part of the panel), as described by Eqs. (20) and (21).

Figure 15

(a) The tensor $\mathcal{G}$ that is used to build the projector onto the physical Hilbert space is an invariant tensor under the action of $G$, as defined in Eq. (8). The legs of $\mathcal{G}$ carry an index of the irrep, since the Hilbert space on which they act has a direct sum structure labeled by the irrep $r$. (b) $\mathcal{G}$ can be obtained by using the explicit form of the projector onto the trivial irrep (9). The red circle is the copy tensor in the group algebra. After acting with the projector, we have to take an equal superposition of all vectors in the trivial irrep space, so as to obtain ${\mathcal{P}}_T$. This equal superposition requires finding the image of the projector, taking an equal superposition of vectors spanning such an image state that we represent by the cyan $|+⟩$, and contracting on it all open legs. (c) Alternatively, one can use the construction introduced in Ref. [58] for building symmetric tensors. The tensor is then divided into a piece $Q$ that takes care of correctly matching the various irreps (made by the Clebsch-Gordan coefficients) and a degeneracy tensor $P$, made of free parameters, that assigns a four-leg tensor to each of the irreps which enter in composition of representations. In the example we draw, $P$ depends on the irreps $r_5$ that are contained into the tensor product of $r_1$ and $r_2$. Once more, the projector ${\mathcal{P}}_T$ is obtained by taking a uniform superposition of the possible values of $r_5$, that is, setting $P=|+⟩$.

Figure 16

(a) Gauge boson constituents in gauge magnets are states of the direct sum of two irreps $V_r\oplus V_r$, which can be identified as the left and the right constituents at the ends of each link. Graphically, we represent this local Hilbert space as a single circle embracing the two copies of $V_r$, each of them being a smaller circle identified by a different color and position inside the bigger circle. (b) The legs of the tensors acting on this Hilbert space are represented by bands rather than lines so that we can specify operators that only act on one sector as acting on half of the band. Each subsector is colored differently. (c),(d) The left and right rotations in the gauge magnets only act on half of the direct sum, as depicted here and discussed in Eq. (35).

Figure 17

The operator that generates gauge transformations in the gauge magnet LGT defined in Eq. (35). It rotates both incoming and outgoing links by the same group elements. On every link, it only acts on the constituent that is closer to the site.

Figure 18

(a) The $U_{s_n}^G$ operator in the gauge magnet LGT defined in Eq. (37). It can be defined, once more, as the contraction of three ${\mathrm{\Gamma }}_r(g)$’s and a copy tensor in $\mathbb{C}(G)$ (red circle). (b),(c) From the definition, one can check that it allows us to pass the $L$ and $R$ rotations on the physical legs, induced by the gauge transformation of Fig. 17, to analogous rotations onto the auxiliary legs. The covariance of $U_{s_n}^G$ allows us to build a gauge-invariant plaquette operator, defined as in Eq. (22), where the rotation on the auxiliary legs cancels by taking the trace onto the auxiliary legs.

Figure 19

(a) In gauge magnet LGT, the tensor $\mathcal{G}$ is an invariant $G$ tensor; i.e., it is left invariant under multiplying all its legs by ${\mathrm{\Gamma }}_r(g)$. (b) $\mathcal{G}$ is obtained, for a finite group, by summing all the representation matrices through the projector on the trivial representation. After projecting, the uniform superposition of all the states of the trivial representation is obtained by closing the free legs with the product of $|+⟩$. (c) For continuous gauge groups, the projector can either be written as an integral over the group elements with the appropriate invariant measure or it can be obtained by using the Clebsch-Gordan coefficients, which transform the product of the four representations into a direct sum of irreps, and then by projecting onto to the trivial irrep.

Figure 20

(a) The $\mathcal{C}$ tensor used to build a variational ansatz for a gauge-invariant state. The tensor acts on a link $s_n$ and embeds its state onto the auxiliary space. It is composed by two parts, a “symmetry part” that does not contain any variational parameter (the two lower lines) and a degeneracy part that contains the variational parameters (the cyan tensors). The tensor has several blocks labeled by the irrep $r$. (b) The $\mathcal{G}$ tensor only acts on the auxiliary space, and it has again a block structure. It is divided in two pieces. The first is responsible for the correct symmetry properties of $\mathcal{G}$ [see Fig. 15]. This part does not contain any free parameters and is given in terms of the Clebsch-Gordan coefficients of the group (small black dots). The second part is a degeneracy part that is formed by a sum of several tensors (one for each allowed value of $r_5$) acting on $D_{r_1}\otimes D_{r_2}\otimes D_{r_3}\otimes D_{r_4}$. These tensors store the variational parameters of the $\mathcal{G}$ tensor in the appropriate symmetry blocks. (c) Variational ansatz for gauge-invariant states on a lattice of $4\times 4$ sites and periodic boundary conditions. The network contains one $\mathcal{C}$ per link of the lattice and one $\mathcal{G}$ per site. The double lines connecting the tensors are used to remind readers that each of the elementary tensors has a double structure, one part dictated by the symmetry and the other one containing the actual variational parameters. In the figure, we are assuming the sum over all the irreps $r$ on every bond of the TN, so that the specific irrep label is omitted.

Figure 21

Left: The isometry $\tilde{\mathcal{G}}$ can be used to go to the gauge-invariant states. Now, a diagonal operator in this space commutes with the gauge transformations and is thus gauge invariant. Right: The vertex operator described in Eq. (45) is built by concatenating the tensor $\tilde{\mathcal{G}}$, a diagonal tensor $\mathrm{\Sigma }$ (a red circle) [acting on $\mathcal{C}(G{)}^3$], and then ${\tilde{\mathcal{G}}}^{†}$. It acts as a potential for different gauge-invariant configurations and allows us to favor one with respect to another.

Figure 22

(a) The setup used in our numerical calculations. Each of the elementary tensors $\mathcal{C}$ (cyan) and $\mathcal{G}$ (orange) has bond dimension $D=2$, and its matrix elements are those described in the main text. The elementary tensors are contracted, so as to provide the quantum states $|\psi ⟩$ of a 2D infinitely large cylinder of circumference $L$. In the drawing, one has to assume periodic boundary conditions along the vertical direction. Here, we represent the norm of the state that defines the partition function $Z$ of a 2D classical model. (b) We compute the spectrum of the TM across the cylinder (sketched on the left), which characterizes the decay of the correlations addressed in Sec. 11a, and the spectrum of the reduced density matrix of half of the cylinder ${\rho }_{1/2}$ (shown on the right), which gives access to the entanglement characterization of the states addressed in Sec. 11b. Both calculations are performed via sparse exact diagonalization, and their costs increase exponentially with $L$, so that we can address at most systems with $L=20$.

Figure 23

Upper panel: The second gap of the TM represented in Fig. 22 in the ${\mathbb{Z}}_2$ spin-liquid phase closes as $\theta$ approaches $\pi /2$, where there is the transition from the ${\mathbb{Z}}_2$ spin-liquid phase to the $U(1)$ algebraic spin-liquid phase. The collapse of the data for the ${\mathrm{\Delta }}_{2}L$ close to $\pi /2$, for the values of $L$ in the range $L=10,\dots ,16$, confirms that the gap at the transition closes as $1/L$ as expected. Thus, what we are studying is a transition from a gapped phase with an exponential decay of correlations to a gapless phase governed by an algebraic decay of correlations. Lower panel: On the other hand, the first gap of the TM ${\mathrm{\Delta }}_1$, representing the gap between the two different topological sectors appearing on the cylinder, opens exponentially with $L$ as $\theta$ tends to $\pi /2$. This behavior is again confirmed by the collapse of our numerical data for $\log ({\mathrm{\Delta }}_1)/L$, with $L$ in the range $L=10,\dots ,16$.

Figure 24

The topological entropy ${\gamma }_T$ defined in Eq. (49) and extracted from the scaling of the entropy of half infinite cylinders with respect to their circumference $L$. The red dots are obtained by considering the scaling of the entropy in the interval $L=4,\dots ,10$, blue up-facing triangles $L=10,\dots ,16$, green squares $L=12,\dots ,18$, and the yellow triangles pointing to the right $L=14,\dots ,20$. The solid orange line represents the exact value of ${\gamma }_T$ for the ${\mathbb{Z}}_2$ spin liquid in Eq. (50), while the cyan dashed line represents its value for the $U(1)$ spin liquid from Eq. (52) with $R$ associated with $\mathrm{\Delta }=-1/2$. We see that for small $\theta$, ${\gamma }_T$ coincides, independently of the size of the cylinder considered, with the expected exact value ${\mathbb{Z}}_2$ (left inset). As we move toward the transition, ${\gamma }_T$ shows a transient oscillation that tends to become sharper and deeper and move toward $\pi /2$ for larger $L$. At $\pi /2$, it attains again the expected analytical value with very small corrections induced by considering the two different set of data (right inset). In the main text, we provide a discussion of these results.

Figure 25

The behavior of the first 100 eigenvalues ${\lambda }_n$ ($n$ is on the $y$ axis) of the reduced density matrix of half infinite cylinders ${\rho }_{1/2}$ defined in Fig. 22 with $L=16$. We plot them as a function of $\theta$ ($x$ axis). $\theta$ varies in the crossover range $1\le \theta \le \pi /2$ identified during the analysis of the topological entropy in Fig. 24. The entanglement spectrum presents clear plateaux, footprints of RK wave functions. Nevertheless, its lower part does not seem to detect the phase’s transition. The first Schmidt gap is identically 0 everywhere (there are two degenerate eigenvalues), and the second Schmidt gap increases while approaching the transition (inset in the figure) for all the $L=10,\dots ,16$.