Gauging quantum states: from global to local symmetries in many-body systems

We present an operational procedure to transform global symmetries into local symmetries at the level of individual quantum states, as opposed to typical gauging prescriptions for Hamiltonians or Lagrangians. We then construct a compatible gauging map for operators, which preserves locality and reproduces the minimal coupling scheme for simple operators. By combining this construction with the formalism of projected entangled-pair states (PEPS), we can show that an injective PEPS for the matter fields is gauged into a G-injective PEPS for the combined gauge-matter system, which potentially has topological order. We derive the corresponding parent Hamiltonian, which is a frustration free gauge theory Hamiltonian closely related to the Kogut-Susskind Hamiltonian at zero coupling constant. We can then introduce gauge dynamics at finite values of the coupling constant by applying a local filtering operation. This scheme results in a low-parameter family of gauge invariant states of which we can accurately probe the phase diagram, as we illustrate by studying a Z2 gauge theory with Higgs matter.

The fascinating subject of gauge theories is omnipresent throughout many-body physics. Quantized gauge fields describe the fundamental interactions in the Standard Model, but also emerge as effective degrees of freedom in several models for strongly correlated condensed matter. To this date, the most accurate description of the strongly coupled, nonperturbative behavior of quantum gauge theories comes from Monte Carlo sampling of the path integral corresponding to Wilson's lattice gauge theory (LGT) formulation [1]. The Hamiltonian formulation of Wilson's LGT, originally developed by Kogut and Susskind [2] (but see also [3][4][5]), has been investigated in the context of approximate wave function ansätze [6][7][8][9] and is receiving a renewed interest in the context of cold atom simulators [10][11][12][13] and tensor network approaches [14][15][16][17][18][19][20][21][22].
The representation of quantum many body states as tensor networks [23,24] originates from White's successful density matrix renormalization group [25] and is now well established in the context of one-dimensional quantum chains. However, recent results for the tensor network description of higher dimensional quantum systems [26] (including fermions [27][28][29][30][31]), quantum chemistry models [32][33][34] and even quantum field theories [35,36] look equally promising. In addition, the theoretical underpinning of tensor network states in terms of the area law of entanglement entropy [37,38] makes them suitable for theoretical results as profound as the complete classification of gapped quantum phases [39,40]. The unifying theme in these studies has been a shift in focus from the Hamiltonian of the system towards the quantum state itself.
Following some early results [14,15], the application of tensor network states to systems with gauge symmetry has recently seen a revived interest. Aside from some extremely accurate results for the Schwinger model [17][18][19], there have been first explorations with two-dimensional pure gauge theory [16] and theoretical formulations of tensor network states with explicit gauge invariance [20][21][22]41]. This Letter tackles an orthogonal question of how to understand the concept of gauging a global symmetry at the level of a single quantum state. That is, we consider a quantum many body system (the matter) in a state which is invariant under a global symmetry corresponding to a certain group G. We then introduce other degrees of freedom (the gauge field) and give an operational procedure to obtain a global state which is invariant under local symmetry operations corresponding to the same group. We use the formalism of projected entangled-pair states [26] (PEPS) to derive a corresponding gauging map for operators. This map reproduces the minimal coupling prescription for simple operators such as hopping interactions or correlation functions, but in contrast to the standard technique in terms of Lagrangians [42], is always unambiguous. Furthermore, it allows to derive a parent Hamiltonian of the gauged PEPS, which resembles the Kogut-Susskind hamiltonian at zero coupling. We simultaneously prove that our gauging prescription transforms an injective PEPS [43] into a Ginjective PEPS [44], which reestablishes the close relation between deconfinement and topological order in the case of discrete groups [45,46], or compact groups broken down to discrete subgroups [47][48][49]. We discuss a well known approach for introducing gauge dynamics at nonzero coupling constant and apply this prescription to obtain a lowparameter family of gauge invariant tensor network states that allows for accurate computation of expectation value. We use this strategy to study the phase diagram of a Z 2 gauge theory with Higgs matter.
Throughout this Letter we consider a lattice or, more generally, a graph, Λ, with quantum degrees of freedom living on the vertices v, to which we henceforth refer as the matter fields. To every vertex v ∈ Λ, there is an associated Hilbert space H v , such that the total quantum state of the matter fields lives in the Hilbert space H We furthermore decorate Λ with oriented edges e as in arXiv:1407.1025v1 [quant-ph] 3 Jul 2014 Fig. 1(a). With a slight abuse of notation, we denote the edges in Λ as e ∈ Λ, where the only difference with the vertices v ∈ Λ is in the chosen character. For every vertex v, we denote E + v as the set of outgoing edges and E − v as the set of incoming edges. Correspondingly, we define v e± as that vertex for which e ∈ E ± v , such that edge e points from v e+ to v e− .
We start from a quantum many body state |ψ ∈ H (m) for the matter fields, which is invariant under the global action U Λ (g) = v∈Λ U v (g) of elements g in a symmetry group G, i.e. U Λ (g) |ψ = |ψ . Here, U v (g) corresponds to a unitary representation of G on the local Hilbert space H v of site v. In order to transform this state into a new state that is invariant under a local action of G, we introduce new degrees of freedom, the gauge fields. We thereto define on every edge e of the graph Λ a new physical Hilbert space H e = C[G] [50], spanned by the 'position' basis {|g } g∈G . The left and right group action of G on H e is given by L e (h) |g e = |hg e and R e (h) |g e = |gh −1 e . We denote the total gauge field Hilbert space as H . Group averaging using the Haar measure [51] can then be used to build a local projector P v onto the invariant subspace, i.e. the states satisfying 'Gauss law' at vertex v, Note that [P v , P v ] = 0 thanks to [L e (g), R e (h)] = 0, so that the projector onto the gauge-invariant subspace of a region Γ is defined as P Γ = v∈Γ P v . In particular, P = P Λ is the projector onto the gauge-invariant subspace of With these ingredients, we now present our minimal coupling prescription for gauging quantum states. It is given by the linear map G : H Λ as G |ψ = P |ψ e |1 e . We thus construct the direct product of the original state |ψ for the matter field with a state for the gauge field which is a product state of |1 e on every edge e corresponding to the identity element g = 1 of the group. The result is then projected into the gauge invariant subspace H (phys) Λ by P . We can explicitly evaluate G and find From this definition, it is clear that GU (g) = G, since a global transformation g v → g v g will not appear in the configuration of gauge fields on the edges, if every edge is connecting two vertices v e+ and v e− . One can in fact check that G † G is the projector onto the trivial representation of the global symmetry group G in H Λ . This implies that initial states |ψ that transform under a non-trivial representation of the global symmetry G are annihilated by the gauging process G. This is the mathematical equivalent of the well known fact that one cannot have a total net charge in a gauge theory on a closed surface [52]. We will show below that this gauging procedure generates a ground state |Ψ = G |ψ for a zero value of the gauge coupling constant, i.e. only the magnetic energy of the gauge field is minimised. To introduce gauge dynamics for nonzero values of the coupling constant, we could manually add the electric energy term to the Hamiltonian. However, since we are working at the level of quantum states, we follow a different approach. Instead, we apply the well-known local filtering operation [53][54][55] with E e the electric field operator on edge e. It turns out that this gauging procedure is very natural in the framework of PEPS. Let us hereto introduce the PEPS |ψ(A) using tensors A v associated to every vertex v ∈ Λ. These tensors act as a multilinear map from virtual vector spaces V e associated to the incoming edges e ∈ E − v to the virtual vectors spaces V e associated to the outgoing edges e ∈ E + v and the physical Hilbert space H v . We identify V e with C De with D e the bond dimension on edge e. By choosing a canonical basis in all vector spaces, we can write (4) The physical state |ψ(A) is obtained by contracting the corresponding kets and bras of all virtual spaces. This construction is illustrated in Fig. 1(b). We now assume that the PEPS tensors A v satisfy the generic property of injectivity [43], meaning that the there exists a finite region Γ ⊂ Λ such that the map from virtual boundary V ∂Γ = e∈∂Γ V e to physical bulk H (m) Γ = v∈Γ H v is injective. This property guarantees that the PEPS is 'well-behaved', e.g. that it is the unique ground state of a local parent Hamiltonian. If an injective PEPS |ψ(A) is invariant under the global action U Λ (g) for g ∈ G, then it was proven in Ref. 56 that there must exist (projective) representations V e of G on the virtual spaces V e such that A v acts as an intertwiner A slightly different form of this equation is presented in Fig. 1(c). While the representations V e are not required to be unitary, we can in principle perform a 'gauge' transformation [57] on the PEPS tensors to transform any finitedimensional representation to a unitary representation if G is a compact group.
The projector onto the gauge invariant subspace of also has a simple tensor network description by introducing virtual spaces V e ≡ C[G] on every edge e and contracting all virtual bonds of vertex tensors X v : as sketched in Fig 1(d), and edge tensors X e : For the case of continuous groups, the virtual dimensions of this tensor network are infinite and this representation is not amenable to numerical computations. Similar constructions of P appeared in the context of spin networks [58], and recently in the context of tensor networks [20,21], where it was also discussed how to compress the bond dimension to finite values. Applying this gauging procedure G to the symmetric PEPS |ψ(A) ∈ H (m) Λ , whose tensors A satisfy Eq. (5), results in a new PEPS with virtual spaces given by W e = V e ⊗ V e . Indeed, we can write the new state G |ψ(A) as a PEPS |Ψ(B) sketched in Fig. 1(g) with vertex tensors and edge tensors given by where the first ket corresponds to the physical state, the second factor to the action on V e and the last factor to the action on the V e . Using the intertwining property of A v in Eq. (5) or Fig. 1(c) and the symmetry property of X v sketched in Fig. 1(g), one can check where all factors act on the virtual level W e = V e ⊗ V e . In particular, R e (g) corresponds to the right group action of G on the virtual space V e . We similarly have that [V e (g −1 ) ⊗ R e (g −1 )]B e [V e (g) ⊗ R e (g)] = B e . This implies that the resulting PEPS cannot be injective, but we can prove for it to be G-injective [44] instead, meaning that the map from virtual boundary to physical bulk is only invertible up to the action of group G, whose representation on W e = V e ⊗ V e is given by V e ⊗ R e .
Thereto, consider a region Γ on which the PEPS tensors act as an injective map. The range of this map is denoted as R ⊂ H The symmetry under G follows from the fact that U Γ (g) |φ i = j u j,i (g) |φ j with u(g) a unitary representation whose matrix elements are given by u j,i (g) = (φ j |V ∂Γ (g)|φ i ), where V ∂Γ is the tensor product representation of the different V e representations on the virtual boundary V ∂Γ .
For the gauged PEPS with tensors B, we choose Γ such that it excludes the physical spaces of the gauge fields on the edges e ∈ ∂Γ. We denote by Γ • the set of vertices in the interior of Γ, i.e. those vertices for which all edges and neighbouring vertices are also contained in Γ. The vertices in the set ∆Γ = Γ \ Γ • are on the (inside) boundary and have one edge e ∈ ∂Γ [59]. If we now define the state |Φ i,{gv} ) = |φ i ) |g ve ) e on the boundary W ∂Γ = e∈∂Γ V e ⊗ V e , where every edge e has a one to one correspondence with a vertex v e ∈ ∆Γ, the resulting state in the bulk H (g,m) Γ is given by The reason for this is that the edges along the boundary, between two vertices in ∆Γ, allow to resolve the elements g v up to the global transformation g v → g v g. Having resolved g v up to a factor g, the edges connecting ∆Γ to Γ • will act as a rough boundary, so that the inner product of the edge degrees of freedom will force all interior gauge transformations in ket and bra to be equal up to the global transformation g, Hence, the preimage of every bulk state |Φ i,{gv} is the set of states { e∈∂Γ V e (g) ⊗ R e (g)|Φ i,{gv} ), ∀g ∈ G}, in line with the concept of G-injectivity [44]. Even though the set of {|Φ i,{gv} , ∀i, ∀g v ∈ G, ∀v ∈ ∆Γ} is overcomplete, we can still check that is a projector that annihilates the PEPS |Ψ(B) . We can now further reformulate this term so as to obtain a general prescription for gauging an arbitrary matter operator with non-trivial support on a region Γ. For any Γ ⊂ Λ containing both vertices v e± of its edges, but not necessarily all edges of its vertices, we first introduce the operator map for any v ∈ Λ, so that P Γ produces gauge-invariant operators, even though it does not include an explicit projector onto the gauge-invariant subspace, i.e. it does not necessarily annihilate states which are not gauge-invariant. In particular, P Γ [1] = 1. We can then also define the gauging map G Γ : . Note that G Γ is invertible onto the space of symmetric operators, since We now return to the parent hamiltonian of the gauged PEPS. Note that contains an explicit energy penalty for all non gaugeinvariant states. Since the physical Hilbert space H (phys) Γ of a gauge theory is restricted to gauge invariant states satisfying the 'Gauss law' constraint on every vertex, we can safely omit this additional factor and instead write We can recognise the first term as the gauged matter Hamiltonian, whereas the second term is a pure gauge term. One can verify that it acts as a projector giving an energy penalty 1 to states with nonzero magnetic flux through any plaquette contained in Γ, which has to contain at least a single plaquette for the injectivity construction. Hence, for a single plaquette p, we can then write the second term as with χ (reg) (g) = δ(g − 1) the character of the regular representation, where the product in its argument is ordered in the way the edges e appear along the boundary ∂p of the plaquette p, and all edges are assumed to be oriented similarly. This term corresponds exactly to the magnetic term of the quantum double models [45]. The typical magnetic term from the Kogut-Susskind lattice gauge Hamiltonian would replace χ (reg) with Re χ (l) , with l the fundamental representation in the case of a Lie group G, but has the same ground state subspace. Indeed, one can check that throughout our gauging construction, by initialising the gauge fields in the |1 configuration, we are effectively working at zero coupling constant for the gauge field and the magnetic energy term is automatically minimised. Finally, the filtering operation in Eq. (3) can be applied to the PEPS without increasing the bond dimension or changing the G-injectivity property. If H (g,m) is a frustration free Hamiltonian with terms h (g,m) that annihilate the ground state, then the filtered PEPS is the ground state of a parent Hamiltonian built of terms where the terms in . . . can be expected to become irrelevant under renormalization for small β, as they correspond to higher-dimensional operators in the continuum theory. We now apply this gauging procedure for quantum states and operators to a number of examples. Consider as a first consistency check a nearest neighbor pair of vertices i.e. it is the operator that extracts the (j, k) element of the representation φ. We thus recover the 'minimal coupling' rule for e.g. a hopping term. This example trivially generalises to the case where Γ contains a path between Figure 2. Trace of the metric g(β) with β = (βz, βx), as obtained from the fidelity | Ψ β |Ψ β+δβ | = exp(−N δβ T g(β)δβ) where N is the (infinite) number of sites, as defined in Ref. 65. An analytic expression for g along the coordinate axes (βz = 0 or βx = 0) was obtained in Ref. 66. The red lines on the right panel indicate the slices studied in Fig. 3 and Fig. 4.
two distant vertices, as would be the case for a correlation function. The map G Γ then creates a gauge-invariant correlation function by inserting a Wilson line along the path. For more complex matter Hamiltonians with e.g. plaquette interactions, one can check that our prescription exactly reproduces the gauging construction used in Ref. 60 to establish the relation between symmetry protected topological order [60] and the twisted quantum double models [61].
In a forthcoming paper [62], we investigate the corresponding effect at the level of the PEPS which, after applying the gauging prescription, acquires the property of twisted Ginjectivity [63] or, more generally, MPO-injectivity [64]. As a more elaborate example, we now consider the phase diagram of a gauge theory with Higgs matter, i.e. scalar bosonic matter transforming non-trivially under the gauge group, for the specific case of G = Z 2 = {1, −1} in (2 + 1) dimensions. Using a basis |1 , |−1 for both H v and H e we have L e (−1) = R e (−1) = τ x and we also choose to have U v (−1) = σ x , with σ x , τ x the Pauli operators for matter and gauge fields respectively. We start from the ground state of the Ising model This matches the Ising Hamiltonian H I at lowest order in β z . We now apply the gauging procedure |Ψ βz,0 = G |ψ βz and switch on an electric field E 2 = (1 − τ x )/2 with coupling constant β x using local filtering, to obtain the state   written as a PEPS with bond dimension 4. However, we can easily 'disentangle' the matter fields by applying a CNOT gate with the matter field as control and the gauge field as target for every pair of (v, e ∈ E v ). Since all these gates commute, this is a finite depth quantum circuit which transforms the Gauss law (P v − 1) |Ψ = 0 into (P v − 1) |Ψ = 0, withP v = (1 + σ x v )/2 and |Ψ the transformed state. Hence, gauge invariance in this transformed frame requires all matter fields to be in the |+ v state so that we are left with unconstrained degrees of freedom on the edges. Applying the CNOT transformation to |Ψ βz,βx results in the state with |Ψ TC the toric code ground state [45] for the edge degrees of freedom. This is equivalent to the well-known correspondence between the normal Z 2 gauge theory with matter, whose phase diagram was first considered by Fradkin and Shenker [68], and the toric code Hamiltonian with magnetic fields H hz,hx = H TC − h z e τ z − h x e τ x e as studied in Refs. [69][70][71]. Note that the definition of the state |Ψ βz,βx depends on the order of applying the filtering in τ z and in τ x . Since the motivation for these filtering operations comes from the lowest order in β, at which level they do commute, we can also opt for a more symmetric definition The PEPS representation of |Ψ TC has bond dimension 2 [67], which is not increased by the local filtering. We can probe the phase diagram on this Z 2 gauge theory as function of β = (β z , β x ) by studying the ground state fidelities in Fig. 2 and Fig. 3. Fidelities were computed as described in Ref. 72. The fidelities illustrate that our ansatz qualitatively reproduces the phase diagram of H hx,hz and describes the same gapped phases in weak-and strong coupling limits. There is the deconfined phase with topological order around β = 0, and the trivial Higgs phase and confined phase which are connected by a local unitary spin rotation at β → ∞, where |Ψ βz,βx is just a product state of eigenvectors of β z τ z +β x τ x corresponding to the largest eigenvalue. The critical behaviour, however, is not exactly reproduced. It is known, for example, that the topological phase transition from the deconfined to the trivial phase along the coordinate axis β z = 0 (and by τ x ↔ τ z duality also along the coordinate axis β x ) is in the 2D Ising universality class [55,73], whereas the corresponding phase transition of H hx,hz is in the 3D Ising universality class. Also, we obtain a second order phase transition along the duality line β x = β z which does not seem to vanish until β = ∞, even though the singularities in the fidelity (Figs. 2,3) and expectation values (Fig. 4) become very quickly small. The corresponding transition of H hx,hz is conjectured to be first order with a discontinuity in the expectation values of the star and plaquette operator, and to end at a finite value of β [69][70][71]. In contrast, the state |Ψ βz,βx seems to have continuous expectation values but a diverging derivative along a slice passing through the phase transition at the duality line β x = β z (Fig. 4). Note that one could also use the state |Ψ βz,βx or a slightly more general class as variational ansatz for the actual Hamiltonian, as was done for the case β z = 0 in Ref. 74 and quickly led to accurate results. See also Ref. 75, where the state |Ψ βz,βx at the limiting values β = 0 and β = ∞ was used as variational ansatz, as well as other Hamiltonian studies using large coupling expansions [76,77]. A more detailed analysis of our results for Z 2 and higher Z N groups is presented elsewhere [78].
In conclusion, we have proposed an operational, nonambiguous procedure for gauging global symmetries at the level of individual quantum states. This procedure combines naturally with the framework of PEPS. When gauging an injective PEPS, we have shown to obtain a G-injective PEPS and we have derived a parent hamiltonian, which automatically contains a projector version of the Kogut-Susskind hamiltonian for lattice gauge theory at zero coupling constant. By introduce gauge dynamics for nonzero values of the coupling constant using a local filtering operation, this construction results in low-parameter family of PEPS for which the phase diagram can accurately be probed, as we have illustrated by studying the phase diagram of a Z 2 gauge theory with Higgs matter. Tensor networks are indeed promising candidates to avoid Feynman's objections [79] regarding non-Gaussian states that allow to efficiently compute expectation values [36]. Similar strategies are perfectly feasible for studying gauge theories with fermionic matter, using the framework of fermionic PEPS [27][28][29][30][31], or even gauge theories (with or without matter) in three spatial dimensions, since no variational optimization is required [80].
This work was initiated during the program on 'Quantum Hamiltonian Complexity' held at the Simons Institute for the Theory of Computing. We acknowledge fruitful discussions with Bela Bauer, Luca Tagliacozzo, Ashley Milsted, David Dudal and Henri Verschelde. Work supported by the Alexander von Humboldt foundation (NS), the EU grants QUERG, SIQS, and by the Austrian FWF SFB grants FoQus and ViCoM. During the final stage of this work, Ref. 21 appeared which considers the related question of constructing gauge-invariant tensor network states for pure gauge theories.