Continuum tensor network field states, path integral representations and the encoding of spatial symmetries

A natural way to generalise tensor network variational classes to quantum field systems is via a continuous tensor contraction. This approach is first illustrated for the class of quantum field states known as continuous matrix-product states (cMPS). As a simple example of the path-integral representation we show that the state of a dynamically evolving quantum field admits a natural representation as a cMPS. A completeness argument is also provided that shows that all states in Fock space admit a cMPS representation when the number of variational parameters tends to infinity. Beyond this, we obtain a well-behaved field limit of projected entangled pair states (PEPS) in two dimensions that provide an abstract class of quantum field states with natural symmetries. We demonstrate how symmetries of the physical field state are encoded within the dynamics of an auxiliary field system of one dimension less. In particular, the imposition of Euclidean symmetries on the physical system requires that the auxiliary system involved in the class' definition must be Lorentz-invariant. The physical field states automatically inherit entropy area laws from the PEPS class, and are fully described by the dissipative dynamics of a lower dimensional virtual field system. Our results lie at the intersection many-body physics, quantum field theory and quantum information theory, and facilitate future exchanges of ideas and insights between these disciplines.


Introduction
The quantum states we observe in nature are highly atypical as compared to a state randomly chosen from the full configuration Hilbert space H [16]. Indeed, observable states comprise only a tiny submanifold of H -the physical corner of Hilbert space -whose points exhibit highly nongeneric features such as nontrivial clustering of correlations and entropy areas laws [9,3].It is extremely desirable to develop an efficient parametrisation of this manifold as this would considerably ameliorate the computational costs of solving physical models and provide new analytical tools for the study of quantum field systems.Indeed, even a partial parametrisation of the physical corner provides a powerful tool as it supplies a variational class useful for the description of low-energy physics.
The canonical example of such a class of quantum states appears in the setting of onedimensional lattices.There matrix product states (MPS) [4] have enjoyed remarkable success, not simply for the calculation of physical properties of strongly interacting lattices, but also for such things as the classification of quantum phases, providing a natural foliation of states in terms of entanglement, and the construction of exactly solvable models [23,27,21].It is also well-established that MPS satisfy two important criteria.Firstly they constitute a complete class of quantum states, in the sense that by increasing a "bond dimension" D one can capture any pure quantum state of the system.Secondly the class is efficient in the sense that the computational cost of calculating expectation values scales polynomially in the number of variational parameters.
The MPS class has provided a fruitful basis for generalisations: by understanding the structure of quantum entanglement in such states they have inspired several powerful extensions to higher dimensions and different geometries.The two most prominent examples for Date: May 5, 2014. 1 higher-dimensional lattice systems are the projected entangled-pair states (PEPS) [19] and the multiscale entanglement renormalisation ansatz (MERA) [24,25].Both of these variational classes have proved invaluable in the investigation of strongly correlated physics.So far, however, all of these results have been restricted to the lattice setting; the study of continuous quantum systems using these classes has traditionally proceeded by first discretising the system on the lattice and then employing them as a variational ansatz.
Continuum systems bring considerable difficulties when it comes to variational computations because optimisations can be dominated by UV physics at the expense of infra-red physics which ruins the estimation of observables of physical interest [5].Both of these difficulties have been addressed with the introduction of special continuum versions of the MPS and MERA classes [22,14,7,8].The cMPS class is remarkable in that it requires (in the translation-invariant case) only a finite number of variational parameters to specify, but is expected, by analogy with the discrete case, to be both efficient and complete in the sense already described.Further studies also indicate that cMPS and cMERA are not disrupted by the presence of UV divergences [7,8].
The MPS and cMPS classes are restricted to one spatial dimension.Therefore it is natural to contemplate generalisations of the PEPS class to the continuum.A first step in this direction was taken in [14], where such a generalisation was proposed.Unfortunately the formulation presented there does not make symmetries manifest and, further, has no obvious relationship to the PEPS class.
In this paper we make use of a recent representation of one-dimensional cMPS states in terms of path integrals [2] as a guide for the generalisation of the class to higher dimensions.Specifically, we start with a particular sequence of lattice PEPS with decreasing lattice spacing and obtain a continuum limit using a path-integral representation.This generalisation takes the same functional form as the one-dimensional cMPS and manifestly exhibits spatial (i.e., rotational) symmetries.The derivation via a sequence of lattice PEPS means that the resultant class of field states automatically obey entropy area laws.Furthermore, the imposition of spatial symmetries on the physical field state is obtained by encoding the symmetry into the dynamics of an auxiliary boundary system with the novel result that the dynamics of the boundary system is given by the imaginary time evolution of a Lorentz invariant system of one lower spatial dimension.We build directly on [2] and therefore assume the reader is familiar with the contents of that paper.

Path integrals for cMPS
Continuous matrix product states are a variational class of states for one-dimensional quantum fields [22,14,7].The simplest formulation is for the single-species bosonic case on the real line with field annihilation and creation operators ψ A (x) and A cMPS is then defined in terms of the quantum field A and a (bosonic) auxiliary D-level quantum system B by where K is a D × D hermitian matrix and R is D × D complex matrix, |ω L,R are Ddimensional states of the auxiliary system B, ψ A (s) is a bosonic field operator on the physical system A, |Ω A is the Fock vacuum, and P denotes path ordering.We now recall the main result of [2]: we can represent a general cMPS via the path integral over an auxiliary (0 + 1)-dimensional D component complex field φ where the path integral is a coherent-state path integral over all configurations of the Ddimensional complex vector φ, |Φ is a physical field coherent state Φ(s) = φ † Rφ, and the complex action S is given by The case where the auxiliary system is fermionic is identical, except that φ is a vector of Grassmann fields.The expression (2.2) makes equal sense for quantum systems on [0, l] as for the infinite case (−∞, ∞).
The path integral representation of |χ is extremely suggestive of a higher dimensional generalisation, namely, we should have that where the path integral is now over an auxiliary (d − 1 + 1)-dimensional field φ(z, t) with D components, |Φ is now a higher-dimensional field coherent state and S is a local action for a D component auxiliary boundary field φ living on an auxiliary boundary space of one lower dimension.Note the replacement of the i with −1 in the exponential of the action: the euclideanised formulation will prove vital when imposing symmetries on the physical state.

The continuum limit of a PEPS class
While the continuum limit of the one-dimensional MPS class is comparatively straightforward, the two-dimensional equivalent has proved more challenging.It is true that we can simply posit the form of a two-dimensional (or higher-dimensional) cMPS as being generated by the continuous measurement process of a lower-dimensional auxiliary boundary field [14], however this is unsatisfactory for at least two reasons.Firstly, in such a setting it is not clear, a priori, how one might impose certain desirable symmetries, such as rotational symmetry, on the physical quantum state.Any variational class intended for the efficient description of real-world physics should be capable of manifestly exhibiting such symmetries.Secondly, for discrete systems higher-dimensional generalizations of MPS already exist, such as the PEPS class, which have been powerful tools in understanding the physics of local hamiltonians.As such it is also of theoretical importance that we arrive at a continuum limit of PEPS that mirrors the one-dimensional cMPS class.
In [2] a path integral representation was obtained for the one-dimensional cMPS class from the traditional discrete MPS class by taking a well-behaved continuum limit.It is this representation of a cMPS that we use as our guide for constructing higher-dimensional classes with manifest symmetries; in field theory, path integral formulations are ideally suited for the imposition of symmetries that would not be manifest according to, e.g., canonical quantisation of the field.Our strategy is then to develop a continuum limit of of a special class of PEPS such that the physical state is a superposition of field coherent states with amplitudes given by a path integral over an auxiliary system and such that the desired symmetries are manifest.
3.1.The basic Tensor Network setting.A natural higher-dimensional generalization of MPS are the Projected Entangled Pair states (PEPS), which are examples of Tensor Networks [23,12,20].The original formulation of PEPS rested on distributing maximally entangled pairs of D-dimensional quantum systems between neighbouring sites on a graph, and then locally mapping the systems at each point into a single d-dimensional Hilbert space.The PEPS construction for arbitrary D can describe any quantum state, and is naturally suited to systems displaying area laws.A generic PEPS has an expansion in terms of a product basis with expansion coefficients given by a contraction of tensors A r (i•••k) with respect to a particular graph Γ(V, E): where C denotes a complete contraction of the auxiliary indices (i • • • k) according to the graph edge structure E, and r 1 , . . .r N label the (product) configurations of a the discrete physical systems located at each vertex of the graph.
An initial instinct would be to begin with a two-dimensional square lattice, and embed the discrete system into the one-particle sector of a system of bosonic or fermionic auxiliary fields, as was done previously for the 1-D cMPS path integral.If one directly follows this path, passing from the discrete PEPS to a continuum path integral, one finds that the underlying square lattice structure persists in the field, and one does not obtain a rotation invariant physical state (see Appendix A).Here we adopt a slightly more involved strategy to handle this unwanted feature.
We begin with a graph of N 2 points {(x, y) = (nǫ, mǫ) : n, m integers}, with a physical spacing ǫ.We view the y spatial direction as an auxiliary time t.As written (3.1)involves a contraction over N 2 auxiliary subsystems distributed over the graph, with independent couplings to the physical degrees of freedom at each site.Since we wish to view the y direction as an auxiliary time dimension we regard the contraction over the N 2 subsystems as the sum over configurations of N auxiliary subsystems subject to a sequence of N dynamic transformations.The upshot is to replace the contraction along the y direction with a product of N square matrices of dimension D N .This is well-known as a transfer operator approach, i.e., we simply view the contraction from one value of y to the next as multiplication by a transfer operator.The state (3.1) can then be written as where U is an operator on AB given by the (time-ordered) product of transfer operators local with respect to the graph sites and |Ω A is some initial product state of A. Specifically, , where the transfer operator M (t) generates an elementary time-step of size ǫ and is built from local operators on A and B. We then follow the idea of [2] and describe the auxiliary systems at each of the lattice sites of B with the single-particle sector of the Fock space built from C D .
The contraction of indices depends on the particular graph structure being used 2 .However, our goal is to construct cMPS states with symmetries and we follow the key principle that the symmetries of the physical state are encoded in the dynamics of the auxiliary system.For example, a natural symmetry to demand is that of rotation invariance in the spatial coordinates of the physical field state.Assuming a state of the form (2.5) implies that the auxiliary action S is invariant under the induced SO(2) rotation group (assuming the measure is also invariant).By demanding that the auxiliary system is a physical system we deduce that S should be an action describing the completely positive dynamics of an auxiliary system (after we trace out A).However, encoding the symmetry into the dynamics of the auxiliary system means imposing invariance under SO(2).This implies that the dynamics should be viewed as the imaginary-time evolution of a Lorentz -invariant system (which is still a completely positive map of the quantum state).
Our strategy is then to first construct a Lorentz-invariant auxiliary action from the continuum limit of a sequence of discrete PEPS.We then construct an analytic continuation to the euclidean setting and obtain a one-parameter family of discrete PEPS states giving a rotation invariant euclidean action as ǫ → 0. There are clearly different possible choices for a Lorentz-invariant action; motivated by the first-order action (2.4), and certain convenient properties of coherent field states, we derive a Dirac-like action from a specific sequence of 2 It is useful to emphasize that technical subtleties arise when taking the limit of lattice systems.Specifically, we might consider a family of graphs {Λ k } indexed by some variable k = 0, 1, 2, . . ., that converges to some dense subset of a compact spatial region A. To each point x ∈ Λ k we have an associated Hilbert space H(x), which could be a space of finite or countably infinite dimension.The total Hilbert space for the full graph system is then given by H k = ⊗ x∈Λ k H(x), and in the thermodynamic limit k → ∞, the resultant space will have an uncountable dimension.It is common to instead work with a much smaller, separable Fock space F(H) constructed to ensure that every state in F(H) has finite particle expectation value, and splits up into a sum F(H) = ⊕ n H (n) of particle sectors H (n) with finite particle numbers.Central to the formation of this Hilbert space is the identification of a vacuum state, from which the different n-particle spaces H (n) are obtained through the action of creation operators obeying the desired statistics.It is well-known that the Stone-von Neumann theorem fails for these systems, and many unitarily inequivalent Fock spaces may be constructed through the choice of different vacua and creation/annihilation operators.For our analysis of the discrete to continuum limit, we specify the local Hilbert spaces at each point on the graph, but ultimately we make use of a Fock space construction for the state |χ ∈ F(H A ), and work with a particular choice of creation/annihilation operators for both A and B, with the auxiliary system B carrying bosonic or fermionic statistics.
Tensor network structure for the transfer operator generating one temporal layer of the PEPS sequence: here H ab denotes the tensor product H a ⊗ H b for the internal degrees of freedom of the auxiliary system H A .
PEPS. (We will discuss general euclidean actions, including second-order actions for bosonic auxiliary fields, in a future paper.)Our first task is to arrive at a Lorentz-invariant situation, a problem for which physical intuition is readily available.Since we are looking for an auxiliary (1 + 1)-dimensional lattice system with locally defined dynamics we assume that each site (x, t) has contraction links to future sites (x, t + ǫ), (x − ǫ, t + ǫ) and (x + ǫ, t + ǫ) and also to past sites (x, t − ǫ), (x − ǫ, t − ǫ) and (x + ǫ, t − ǫ).The simplest such choice is to build the operator M out of quadratic terms involving creation and annihilation operators; to arrive at states with rotational symmetries, we can also make use of spinorial expressions.To generate the spinorial structure we assume that at each site x, in addition to the 'flavour' indices i, j, k, . . ., we have access to two internal degrees of freedom, with annihilation operators a k,x and b k,x at each spacetime point.We also note that the bosonic and fermionic cases can be treated simultaneously by being careful with the ordering of terms.Thus we have [a j,x , a k,y ] ± = [a † j,x , a † k,y ] ± = 0, [a j,x , a † k,y ] ± = δ jk δ x,y , with similar expressions for b, where ± labels the choice of bosonic or fermionic auxiliary system.
3.2.Generation of a kinetic term and flavor doubling.We now construct a transfer operator M ǫ = ½ + ǫ H tot from terms quadratic in a and b, and generate a PEPS via in such a manner that the continuum limit has the desired Lorentz symmetry.It should perhaps be emphasized that the generators that we construct relate entirely to the auxiliary system A, which only forms an abstract device to describe the physical state |χ ǫ , and does not commit us to any particular realisation for the two-dimensional physical field system B.However, by treating the auxiliary system as physical we can make use of natural intuitions of particle interactions when we construct the abstract PEPS class through H tot .Certain assumptions are natural to impose on the terms appearing in H tot , such as left-right symmetry, symmetry between a-particles and b-particles and conservation of total particle number, however the key term in the construction is the a nearest neighbour 'hopping' term, which we take to be where x runs over N spatial points, and J jk (t) measures the strength of the spatial hopping, which for simplicity we take as constant along the spatial direction.
To analyse this, we perform a discrete Fourier transform in the spatial direction to obtain where ãj,p = 1 √ N x e −ipx a j,x is the Fourier-transformed annihilation operator (similarly for bj,p ), and p = 2πn/Nǫ runs over N points in the reciprocal lattice, for n an integer.
Our concern is the continuum limit, ǫ → 0, where the dominant contributions of H h will come from the "low-energy regime" of momenta p near to the zeroes of (1 + 2 cos pǫ).These occur at the points q µ = (−1) µ (2π/3ǫ), for µ = 0, 1.The contributions from the two points give rise to two flavors in a similar manner to fermionic doubling on the lattice, however it must be noted that this doubling only occurs for the auxiliary system, and so the physical system (whether bosonic or fermionic) is unaffected.Overall, in the continuum limit H h splits into H h = H h,0 + H h,1 with the contributions from the two decoupled flavors given by We can redefine J jk → J jk / √ 3 and relabel the mode operators as ã(j,µ);p ≡ ãj,qµ+p and also b(j,µ);p ≡ bj,qµ+p to obtain This term alone would produce 2D massless Lorentz-invariant flavors labeled by the compound index (j, µ), in which the (j, 0)-field is related to the (j, 1)-field through the discrete parity transformation P, given in 1+1 dimensions as P = γ 0 = σ z , which inverts chirality.This is consistent with the Nielsen-Ninomiya theorem [13], which requires doubling in order to achieve a translationally invariant spinor action with chiral symmetry in the continuum limit of a lattice model.

3.3.
Coupling and decoupling of the two flavors.The previous analysis shows that the two contributions to H h from the points q 0 and q 1 in momentum space decouple.In position space this tells us that a j,x splits up in the low-energy regime as a j,x = { |p|≪1/ǫ e ipx ãq 0 +p }e iq 0 x + { |p|≪1/ǫ e ipx ãq 1 +p }e iq 1 x .We write this instead as a j,x = a (j,0),x e iq 0 x + a (j,1),x e iq 1 x , in which the operators a (j,µ),x are given by the expressions in the curly brackets of the preceding equation.
In the continuum limit we then have that ǫ −1/2 a (j,µ),x tends to a smooth field Ψ (j,µ) (x), arising from envelopes of plane waves around the point q µ .This real-space description has been useful recently to generate non-trivial field potentials from lattice models [1].For example, one might perturb H h through the addition of an on-site potential such as ǫ x f jk (x)a † j,x a k,x .This term will behave as ǫ µ,ν,x f jk (x)e −i(qµ−qν )x a † (j,µ),x a (k,ν),x , however, if the function f jk (x) does not vary rapidly from site to site, the presence of the highly oscillatory term e −i(qµ−qν )x will ensure that only µ = ν will contribute in the continuum limit, and so the two flavors will decouple.At the other extreme, one can consider functions on the lattice that vary sufficiently rapidly, as say f (x) = g(x(q 0 − q 1 )), which can be used to produce non-trivial couplings of the flavors [1,10], however here we do not consider such scenarios.
3.4.Generation of the full transfer operator.It is straightforward to produce a mass term in the continuum limit, simply by adding the term x m(a † j,x a j,x − b † j,x b j,x ), which then generates the usual dispersion relation E 2 = p 2 + m 2 as ǫ → 0. However, we can more generally use functions {m jk 0 (x, t)} that do not vary too rapidly over the lattice, and perturb H h by the on-site potential term ).This can be written as (3.7) ) + e ±i(q 0 −q 1 )x (flavor coupling terms) where the flavor coupling terms do not contribute in the continuum limit, as explained in the previous section.In addition to the terms H h and H m for the auxiliary system we add a coupling term between the auxiliary and physical systems, which will generate the state |χ ǫ .
For this, we mirror the local coupling in [2] and define the coupling term where again, for simplicity, we assume the functions {R jk (x, t)} vary sufficiently slowly over the lattice so that the µ = 0, 1 flavors decouple once more.
The final transfer operator that generates the PEPS state is finally given by (3.9) where the operator hat serves to specify the non-trivial action on the physical field system.The basic tensor structure of M ǫ (t) is shown in Fig. 1, where there is also an implicit physical index at each site, coupling to the physical field A, which we omit in the diagram for the sake of clarity.
In the next section we analyse the continuum limit, and derive the desired path integral representation from the smooth fields ǫ −1/2 a (j,µ),x and ǫ −1/2 b (j,µ),x .Assuming that m jk 0 and R jk slowly vary on the lattice, the expressions for H m and H int in terms of these smooth fields are obtained by the doubling of flavor index j → (j, µ).The same happens for the indices of the kinetic term H h , but with added feature of a parity flip σ x ↔ σ z σ x σ z relating the two flavors.This discrete transformation is itself a Lorentz symmetry and ensures that the total field state will possess the full symmetry group, as we shall see later.

Construction of the path integral representation.
In this, and the subsequent, subsection we construct the path integral for the continuum limit of the sequence |χ ǫ , ǫ → 0. We follow the one-dimensional prescription, and insert coherent-state resolutions of the identity into the product T l t=0 [ M ǫ (t)].This is first computed for a simple elementary timestep ǫ and the continuum limit taken in the spatial direction.Finally the continuum limit is taken in the timelike direction, to obtain the final path integral representation of the cMPS state.
For clarity we shall write j µ to denote the compound index (j, µ) with j 0 = 1, . . ., D and j 1 = 1, . . ., D for the two flavor sectors.This notation is useful since the actual PEPS parameters m jk 0 and R jk , that we can control do not have any µ dependence, and so the µ label simply doubles up the auxiliary fields, without playing an independent variational role.In what follows use coherent-state resolutions of the identity which, in both the fermionic and bosonic cases, are given by where we have the usual coherent state displacement operator, given for a single mode with annihilation operator c as D(α) = exp[αc † − cα * ], the label s = a, b labels the particle type, and in the fermionic case φ jµ,s (x, t) are Grassmann numbers.We also use bold-faced φ(x, t) := {φ jµ,s (x, t)} to suppress indices when the contractions are clear.The identity contribution is easily calculated via the overlap formulae for coherent states (see e.g.[2] for details) and gives φ(x, however the H h and H m terms require more attention. It is simplest to work in momentum space, for which (3.11) where mjk 0 (p, t) = 1 √ N x e −ipx m jk 0 (x, t) and the indices on m jk 0 and J jk are explicitly independent of µ.
Instead of expanding in terms of fermionic/bosonic coherent states of a jµ,x and b jµ,x we expand in terms of fermionic/bosonic coherent states of ãjµ,p and bjµ,p .Insertion of the above is a two-component auxiliary field with flavour index j µ , and where j µ = 1, 2, . . ., D for each of the two separate µ = 0, 1 sectors.
The interaction term can be evaluated in the same way, and we obtain where the hat on ψ † A is again to emphasize that the term is an operator on the physical system, as opposed to Ψ jµ (x, t) which is an auxiliary two component (Grassmann) spinor, with flavor index j µ .
The total sum in the spatial direction can now be evaluated, and becomes an integral over x, which provides us the single time-step amplitude coming from M ǫ (t).Once done we can then sum over the time direction to obtain the final expression for the field state.However the process requires care, and in particular a field rescaling to preserve finiteness, and so we discuss this in the next section.
3.6.Field rescaling and the continuum limit.For the graph used in the previous section, and also the square-lattice model in the appendix, the two-dimensional contraction across the graph requires a passage to the continuum in two independent directions, and so must be handled carefully.In this section we briefly spell out the technical details showing that we obtain a well-defined two-dimensional action, and we temporarily suppress the flavor-doubling label µ to reduce our index load.
Recall the basic form of the 2D PEPS state: where (i M • • • k M ) are a set of contraction indices dependent on the particular graph structure of the state, and |r M is basis state at lattice site M.As we have already explained, the contraction can be rewritten as a dynamical process involving the product of transfer operators generating infinitesimal steps For clarity, we restrict to a finite set of points {(x, t)} where x runs over N x points, separated by a distance ǫ x and t runs over N t points, in timesteps of ǫ, and we make explicit all indices.
Once we have introduced coherent state resolutions of the identity at each graph point we have at each value of t a total of 4DN x complex-valued functions to integrate over, coming from Inserting N t + 1 such resolutions into (3.14)gives with the measure for the integral given by (3.17) The amplitudes in the integrand of (3.16) have been calculated in the low-energy sector and in this regime we obtain the expression for the tensor contraction where the matrices m 0 and R are allowed to vary smoothly in both x and t.
We now take the spatial ǫ x → 0 limit followed by the temporal ǫ → 0 limit.However, to keep things well-behaved, we must first rescale the integration variables Indeed, this rescaling was to be expected in order to respect the correct commutation/anticommutation relations: in 1+1 dimensions we have Ψ kµ ∼ a kµ / √ ǫ x , while in 2+1 dimensions we should instead have Ψ kµ ∼ a kµ / √ ǫ x ǫ y , where ǫ x and ǫ y are the two spatial lattice scales.Once this rescaling is performed, we find in the ǫ x , ǫ → 0 continuum limit that the resultant cMPS field state becomes and we obtain a well-defined two-dimensional action.

3.7.
The two-dimensional variational class of field states.Once the rescaling has been performed we may directly integrate by parts, to obtain the class of two-dimensional field states (3.19) where we have introduced the coherent field state , and there is an implicit sum over µ, j and k.
We have obtained this state through the continuum limit of a PEPS contraction, and so (3.19) describes a class of field states that inherits the desirable properties of PEPS, such as entropy area laws.The general action given above is not necessarily invariant under SO(1, 1).However, Lorentz symmetry for the field can be achieved for a subclass of states in which J jk (t) = iδ jk .This allows the momentum terms to respect the desired symmetry and we obtain where we have also let m jk 0 = −im jk .For the situation where m is diagonal and constant over the auxiliary spacetime, the coherent field amplitudes are then recognised as Grassmann/complex number path integrals for a set of 2D uncoupled Lorentz invariant spinor fields.The spinors have the associated gamma matrices γ 0 = σ z and γ 1 = iσ y obeying the Clifford algebra relations {γ µ , γ ν } = 2η µν ½ with η µν = diag(1, −1).Also for µ = 1 flavor sector we also perform a parity transformation on Ψ k 1 → γ 0 Ψ k 1 , which is given by γ 0 = σ z for the 1+1 dimensional case.
Note that, in obtaining this symmetrical state, we were forced to take both m 0 and J to be purely imaginary, which means the term H h + H m corresponds to a generator of a unitary transformation on the auxiliary system.In particular ] generates the cMPS state, where H(t) is built out of second-order and fourth-order combinations of creation and annihilation operators for the auxiliary and system fields.Furthermore, the distinctively two-dimensional term is the contribution from J(t), which couples the two spinor degrees of freedom.By setting J(t) = 0 we reduce to a diagonal scenario, of the same form as obtained for the one-dimensional cMPS.
Of course j µ = Ψγ µ Ψ is a conserved current for the free Dirac field, and in particular Ψ † Ψ is its charge density.Thus, in the case where the auxiliary state has manifest Lorentz symmetry, and where the different flavors decouple, we have the appealing interpretation that the matrix R, which is allowed to vary in both x and t, couples the densities of the different flavors and dynamically generates the physical field state.
Of course, we could now weaken the conditions and allow more general J(x, t) and m(x, t) to obtain a Dirac action on a 1+1 dimensional spacetime with non-trivial metric.To do so in general would additionally require the modification of the identity term ½ to include a well-behaved function T (x, t).Looking back at the analysis, the key feature involved in the derivation is that the auxiliary particles have two degrees of internal freedom and are allowed to hop left or right with some amplitude or remain stationary while flipping an internal (spin) freedom.This is reminiscent of Feynman's 'checkerboard derivation' [6,17] of the Dirac propagator in 1+1 dimensions from a discrete lattice model.There, an electron moves along infinitesimal lightlike trajectories, while jointly flipping direction and spin under a Poisson process with an imaginary rate 1/im.In light of this, it is not so surprising that we have obtained a Dirac-like action in our continuum limit, although for us a key component is that an expansion in terms of coherent states in the auxiliary time direction either side of an operator M ǫ = ½ + ǫ H tot effectively generates a Legendre transformation of a 'Hamiltonian' For a coherent state-expansion, we obtain a term iφ † ∂ t φ from the coherent state overlaps which can be viewed as π∂ t φ where π = iφ † is the momentum conjugate to φ.
3.8.Analytic continuation to the euclidean sector.We now have a manifestly Lorentzinvariant auxiliary action, however the resultant physical state for A will have non-trivial entanglement structure in general.As such we would like to analytically continue to the euclidean sector and arrive at a rotation invariant auxiliary action.For second-order actions we can achieve this analytic continuation simply via t → it, but for spinors subtleties arise.This coordinate transformation, when carried over to the Lorentz transformation, does provide the correct rotation group, but when acting on the spinors themselves results in ΨΨ no longer transforming as a scalar.A direct euclideanizing of fermion fields results in a number of problems, such as a loss of hermiticity within the euclidean propagator.However, these difficulties were overcome by Osterwalder and Schrader [15] by making use of a construction that involves fermion doubling where the number of degrees of freedom are doubled so that the spinor and conjugate spinor are independent of each other and transform appropriately under the Lorentz group.However, there exist alternative approaches to euclideanizing the field that do not require this.Instead of analytically continuing the coordinates it is possible to analytically continue the metric itself η µν → η µν (θ) so that it forms a one-complex-parameter family of metrics interpolating between the Minkowski and the euclidean one [11].A more abstract formulation can be achieved by using vielbeins, but for our purpose we work directly with the spacetime metric.
A one-parameter family of actions, symmetric under this group action, can then be constructed, where θ = 0 is the Lorentz-invariant Dirac action and θ = π/2 is the desired rotation-invariant euclidean action.It is found to take the form By inspection, we can see that this action could be obtained as the continuum limit of a one-parameter family of discrete tensor networks described by M ǫ (t; θ) of the form The parametrization of the metric has been transferred to the tensor contraction across the discrete graph, and we can smoothly transform from θ = 0 to θ = π/2 to obtain, in the continuum limit, the rotation-invariant euclidean action, and the cMPS class where is the rotation-invariant action with γ E 5 = −iγ 0 , and the flavor indices are implicit, and are the same as in equation (3.20).
The above analysis works independent of whether we have used bosonic fields or fermionic fields, however in the latter case it is possible to adopt a related approach that does not require the modification of the gamma matrices.Instead the Grassmann spinor Ψ(x) and its conjugate Ψ(x) are considered independent of each other, and instead of working with the full metric η µν one can analytically continue vielbeins e m µ → e m µ (θ) = e iθ δ m µ and use these to construct a Dirac action in the way one would for curved spacetimes [26].
3.9.Area law properties.The two-dimensional field state |χ given by (3.24) naturally inherits local properties from the discrete state.In particular, the resultant state is necessarily local in its entanglement structure and obeys an area law.For example, we could consider a finite region A of points on the graph described above.For this region we can define its boundary ∂A as the set of points in A within a distance ǫ of points not in A, and |∂A| as the number of points in this set.For any pure quantum state |ψ AB there exists a unique measure of entanglement, namely the von Neumann entropy of entanglement, or simply the entanglement entropy, defined as S A = − Tr[ρ A log ρ A ] where ρ A is the reduced state on system A.
For discrete states the dimension D of the tensor labels within the contraction places an upper bound on the rank of the reduced state on A. The local definition of the state means that only systems in ∂A are entangled with the region B and so the entanglement entropy is upper bounded as S A ≤ c|∂A| where c is a constant dependent on D. For the continuum limit we need only impose that the region A is of fixed area with boundary of fixed length |∂A|.Since the number of points in ∂A will scale linearly in 1/ǫ, we then deduce that S A ≤ c|∂A|/ǫ and so it is clear that the resultant field state also obeys an area law.

Conclusions
In this paper we have constructed an abstract class of physically natural higher-dimensional quantum field states emerging from an underlying discrete PEPS state.Central to the construction was the recent path-integral representation introduced for one-dimensional cMPS states in [2].The decomposition of the one-dimensional states into a superposition of coherent field states, together with an action for the auxiliary system, provided a useful guide in extending the class of field states to higher dimensions.
The desired symmetries of the physical states are encoded in the auxiliary dynamics, and thus to obtain rotation-invariant states we may considered the imaginary time evolution of a Lorentz-invariant auxiliary field theory in (1 + 1) dimensions.To obtain this from first principles we began with a discrete PEPS on a particular graph, and demonstrated that the low-energy sector gave rise to a class of states which manifestly included rotation-invariant states.The continuum limit of the lattice state was shown to produce a doubling of flavors for which the two sectors can be made to simply decouple for PEPS data that varies sufficiently slowly with respect to the scale of the network.Significantly, this doubling only occurs for the auxiliary degrees of freedom while the physical fields are left unaltered.The rotation invariance of the resultant field state arises from the symmetries of an auxiliary spinor action realised as the analytic continuation of a Lorentz-invariant action to imaginary time.Since PEPS states automatically obey area laws, we deduce that so too do the constructed quantum field states.
There are no restrictions on the statistics of the physical fields, and since variational methods evade the sign problem afflicting Monte Carlo techniques, it would be of interest to see how the cMPS variational class performs in finite fermion density scenarios.
Finally, while in principle there is no obstacle to obtaining a continuum limit from a PEPS class with a bosonic auxiliary action with the desired symmetries, we leave this avenue to future work.acknowledgements