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

We can reformulate the cMPS state (6) so that expectation values for the auxiliary system are recast as path-integral expressions, using standard techniques. The motivation for this is two-fold: firstly, to facilitate the passage to higher-dimensional cMPS states; and secondly, to make manifest the symmetries of the physical state in terms of symmetries of an action for the auxiliary system. Our discussion is centred on the case of a single bosonic field in (1+1) dimensions; the generalization to spinor and vector fields follows easily, and we only comment on the modifications required.

Write a basis for the Hilbert space ${{\mathcal{H}}}_{{\mathcal{B}}}$ of ${\mathcal{B}}$ as $\{| j\rangle \;| \;j=0,1,\ldots ,D-1\}$. We enlarge this space via second quantization, and introduce bosonic annihilation and creation operators b_j and ${b}_{j}^{\dagger }$ according to the canonical commutation relations $[{b}_{j},{b}_{k}^{\dagger }]={\delta }_{j,k},\quad j=0,1,\ldots ,D-1$, with all other commutators vanishing, or fermionic annihilation and creation operators c_j and ${c}_{j}^{\dagger }$ according to the canonical anticommutation relations $\{{c}_{j},{c}_{k}^{\dagger }\}={\delta }_{j,k},\quad j=0,1,\ldots ,D-1$, with all other anticommutators vanishing. The Hilbert space for our enlarged auxiliary system is that of the Fock space ${{\mathfrak{F}}}_{\pm }({{\mathcal{H}}}_{{\mathcal{B}}})$, where the ± subscript indicates either bosonic or fermionic Fock space.

The connection between ${{\mathcal{H}}}_{{\mathcal{B}}}$ and our enlarged Fock space ${{\mathfrak{F}}}_{\pm }({{\mathcal{H}}}_{{\mathcal{B}}})$ is made, in the bosonic case, by identifying ${{\mathcal{H}}}_{{\mathcal{B}}}$ with the single-particle sector via $| j{\rangle }_{{\mathcal{B}}}\equiv {b}_{j}^{\dagger }| {\scr{\Omega}} {\rangle }_{{\mathcal{B}}}$, or, in the fermionic case, $| j{\rangle }_{{\mathcal{B}}}\equiv {c}_{j}^{\dagger }| {\scr{\Omega}} {\rangle }_{{\mathcal{B}}}$, where $| {\scr{\Omega}} {\rangle }_{{\mathcal{B}}}$ is the Fock vacuum. We identify, whenever clear from the context, states $| \omega \rangle \in {{\mathcal{H}}}_{{\mathcal{B}}}$ with their single-particle counterparts in ${{\mathfrak{F}}}_{\pm }({{\mathcal{H}}}_{{\mathcal{B}}})$.

Using this embedding, a cMPS (6) is equivalent, in the bosonic case, to

$$\begin{eqnarray}&&| \chi \rangle =\langle {\omega }_{L}| U(l,0)| {\omega }_{R}\rangle | {{\scr{\Omega}} }_{{\mathcal{A}}}\rangle =\langle {\omega }_{L}| {\mathcal{P}}\mathrm{exp}\left[-{\rm{i}}{\displaystyle \int }_{0}^{l}F(s){\rm{d}}s\right]| {\omega }_{R}\rangle | {{\scr{\Omega}} }_{{\mathcal{A}}}\rangle ,\end{eqnarray} \tag{ 8 }$$

where F is a one-parameter set of field operators on ${\mathcal{A}}{\mathcal{B}}$, generated by $U(l,0)$ and given is by

$$\begin{eqnarray*}&&F(s)=\underset{j,k=1}{\overset{D}{\displaystyle \sum }}{K}^{{jk}}(s){b}_{j}^{\dagger }{b}_{k}\;\otimes \;{\mathbb{1}}+{\rm{i}}{R}^{{jk}}(s){b}_{j}^{\dagger }{b}_{k}\otimes {\hat{\psi }}_{{\mathcal{A}}}^{\dagger }(s)-{\rm{i}}{R}^{*\;{kj}}(s){b}_{j}^{\dagger }{b}_{k}\otimes {\hat{\psi }}_{{\mathcal{A}}}(s),\end{eqnarray*}$$

This equivalence of definitions follows from the fact that F(s) is particle-number conserving on system ${\mathcal{B}}$ (i.e. its action on ${\mathcal{B}}$ is through terms of the form ${b}_{j}^{\dagger }{b}_{k}$ only), and so we remain in the single-particle sector throughout. The fermionic version is identical except that b_j operators are replaced with c_js.

The parameter s can be regarded as a time coordinate for the auxiliary system. We then obtain a path integral by dividing $[0,l]$ into small intervals ${s}_{0}=0,{s}_{1},{s}_{2},...{s}_{N}=l$ with uniform spacing ${s}_{k+1}-{s}_{k}=\epsilon$, so that $U(l,0)=U(l,l-\epsilon )U(l-\epsilon ,l-2\epsilon )\cdots U(\epsilon ,0)$, and then inserting resolutions of the identity between each term. Our choice of resolution is, in the bosonic case, in terms of coherent states of the auxiliary system, defined as $| {\phi }_{k}\rangle =\mathrm{exp}[{\phi }_{k}{b}_{k}^{\dagger }-{\phi }_{k}^{*}{b}_{k}]| {{\scr{\Omega}} }_{{\mathcal{B}}}\rangle$:

$$\begin{eqnarray}&&{\mathbb{1}}=\displaystyle \frac{1}{{\pi }^{N}}\int \underset{k}{\displaystyle \prod }{{\rm{d}}}^{2}{\phi }_{k}\;| {\otimes }_{k}\;{\phi }_{k}\rangle \langle {\otimes }_{k}\;{\phi }_{k}| ,\end{eqnarray} \tag{ 9 }$$

where $N=l/\epsilon$. In the fermionic case we exploit fermion coherent states of the form $| {\phi }_{k}\rangle =\mathrm{exp}[{c}_{k}^{\dagger }{\phi }_{k}-{\phi }_{k}^{*}{c}_{k}]| {{\scr{\Omega}} }_{{\mathcal{B}}}\rangle$, where ${\phi }_{k}$ are now Grassmann-valued. Apart from the use of anticommuting Grassmann numbers the fermionic calculation mirrors the bosonic case in essentially all other respects; we thus focus on the details of the bosonic calculation and write out the fermionic case at the end.

After the resolution (9) has been placed between each term we end up with a product of transition amplitudes of the form $\langle {\otimes }_{k}\;{\phi }_{k}(s+\epsilon )| U(s+\epsilon ,s)| {\otimes }_{{k}^{\prime }}{\phi }_{{k}^{\prime }}(s)\rangle$ ≈ $\langle {\otimes }_{k}\;{\phi }_{k}(s+\epsilon )| {\mathbb{1}}-{\rm{i}}\epsilon F(s)| {\otimes }_{{k}^{\prime }}{\phi }_{{k}^{\prime }}(s)\rangle$. We then make use of the expression

$$\begin{eqnarray}&&\langle {\otimes }_{k}\;{\phi }_{k}(s+\epsilon )| {\otimes }_{{k}^{\prime }}{\phi }_{{k}^{\prime }}(s)\rangle =\mathrm{exp}\left[-\displaystyle \frac{1}{2}\underset{k=1}{\overset{D}{\displaystyle \sum }}{| {\phi }_{k}(s+\epsilon )| }^{2}+{| \phi (s)| }^{2}-2{\phi }_{k}^{*}(s+\epsilon ){\phi }_{k}(s)\right],\end{eqnarray} \tag{ 10 }$$

and the assumption that only smooth variations of ${\phi }_{k}(s)$ contribute, which allows us to expand the terms in the exponential and obtain, in the continuum limit $\epsilon \to 0$,

$$\begin{eqnarray}&&| \chi \rangle =\int \underset{k}{\displaystyle \prod }{{\mathcal{D}}}^{2}{\phi }_{k}\;{\mathcal{P}}\mathrm{exp}\left[{\rm{i}}\hat{S}({\phi }_{k},{\phi }_{k}^{*})\right]| {{\scr{\Omega}} }_{{\mathcal{A}}}\rangle ,\end{eqnarray} \tag{ 11 }$$

where the path integral is over D complex fields and $\hat{S}$ is an operator-valued action given by

$$\begin{eqnarray}&&\hat{S}=\int {\rm{d}}s\left({\rm{i}}{\phi }^{\dagger }{\partial }_{s}\phi -{\phi }^{\dagger }K\phi -{\rm{i}}({\phi }^{\dagger }R\phi ){\hat{\psi }}_{{\mathcal{A}}}^{\dagger }+{\rm{i}}({\phi }^{\dagger }{R}^{\dagger }\phi ){\hat{\psi }}_{{\mathcal{A}}}\right),\end{eqnarray} \tag{ 12 }$$

where we abbreviate $\{{\phi }_{k}\}$ as a vector ϕ. However, since the field operator ${\hat{\psi }}_{{\mathcal{A}}}^{\dagger }(s)$ commutes with ${\hat{\psi }}_{{\mathcal{A}}}(s\prime )$ and ${\hat{\psi }}_{{\mathcal{A}}}^{\dagger }(s\prime )$ at all other points s' the ordering over the auxiliary time variable is trivial and we can simply write the path integral as

$$\begin{eqnarray}&&| \chi \rangle =\int {{\mathcal{D}}}^{2}\phi \;\mathrm{exp}\left[{\rm{i}}S(\phi ,{\phi }^{\dagger })\right]| {\scr{\Phi}} \rangle ,\end{eqnarray} \tag{ 13 }$$

where $| {\scr{\Phi}} \rangle$ is a physical field coherent state

$$\begin{eqnarray}&&| {\scr{\Phi}} \rangle \equiv \mathrm{exp}\left[\int {\scr{\Phi}} (s){\hat{\psi }}_{{\mathcal{A}}}^{\dagger }(s)-{{\scr{\Phi}} }^{*}(s){\hat{\psi }}_{{\mathcal{A}}}(s)\;{\rm{d}}s\right]| {{\scr{\Omega}} }_{{\mathcal{A}}}\rangle ,\end{eqnarray} \tag{ 14 }$$

${\scr{\Phi}} (s)={\phi }^{\dagger }R\phi$, and the complex action S is given by

$$\begin{eqnarray}&&S(\phi ,{\phi }^{\dagger })=\int {\rm{d}}s\left({\rm{i}}{\phi }^{\dagger }{\partial }_{s}\phi -{\phi }^{\dagger }K\phi \right).\end{eqnarray} \tag{ 15 }$$

This formulation (13) of the one-dimensional cMPS state as a path integral is a natural guiding expression for the generalization to higher-dimensional scenarios which we describe later (The fermionic case is identical, except that ϕ is now a vector of Grassmann fields. Of course, both the bosonic and fermionic cases yield exactly the same physical state $| \chi \rangle$, since they coincide on the 1-particle sector). Notice that we've dropped the limits from the integrals; the expression (13) makes equal sense for quantum systems on $[0,l]$ as for the infinite case $(-\infty ,\infty )$.

While the use of the auxiliary Fock space and its 1-particle sector to encode the virtual process might seem initially excessive in the one-dimensional scenario, it turns out to be much more flexible in the higher dimensional generalizations. There the auxiliary field system has genuine spatial extent, and permits generalizations that are not simply 1-particle sector restrictions. The degree to which extending off the 1-particle sector in these models brings new physics and deviates from the discrete tensor network description is at present unclear, and demands further investigation.

The expression (6) admits a simple, yet useful interpretation. A cMPS is a superposition of coherent states $| {\scr{\Phi}} \rangle$ with some weighting e^iS determined by the virtual dynamics. The standard intuition concerning coherent states is that they are the 'most classical' states of a quantum system due to their saturation of the Heisenberg uncertainty relation. Thus, (13) tells us that a cMPS is a superposition of 'classical' field states centred around classical field configurations ${\scr{\Phi}} :{\mathbb{R}}\to {\mathbb{C}}$ in phase space. These field configurations Φ themselves are scalar functions of a vector of auxiliary classical fields $\phi :{\mathbb{R}}\to {\mathbb{C}}$. By interpreting that spatial variable s as a temporal variable one can understand the action S for these auxiliary fields as that of a $(0+1)$-dimensional quantum field, i.e., ordinary quantum mechanics.

One therefore has the picture of an auxiliary system undergoing a classical trajectory of its discrete variables, however to gain information (by measurement) about a dynamically evolving quantum system we inevitably disturb it because of the back-action of the quantum measurement. The closest representation of the dynamics in this quantum setting is to continuously monitor the evolving auxiliary system with a sequence of infinitesimally weak measurements [5]. By exploiting von Neumann's prescription for quantum measurement this process is then understood as entangling the auxiliary system and an infinite collection of meter systems. The combined auxiliary system+meter collection undergoes completely positive dynamics. In the continuum limit the meter systems constitute a quantum field with one extra spatial dimension, the reduced state of the meters alone is a quantum state. The cMPS coherent field state is then an imprint of the discrete trajectory, and is as classical a record as possible. The stength and manner of this imprint is entirely contained in the particular coupling R(t). Each trajectory for the auxiliary system contributes a coherent field state, and the cMPS is simply a superposition of 'classical' trajectories with the according weighting by the action S (see figure 1).

In this section we show that the cMPS is a complete class: an arbitrary quantum field state can be approximated with arbitrary accuracy, by allowing the bond dimension D to become arbitrarily large. The argument we present here is for the case of bosonic Fock space ${{\mathfrak{F}}}_{-}({L}^{2}([0,l]))$ on a finite interval $[0,l]$—one expects this to hold in the case of the interval $(-\infty ,\infty )$.

The argument is rather simple and relies on three facts. The first is that an arbitrary quantum field coherent state

$$\begin{eqnarray}&&| {\scr{\Phi}} \rangle \equiv \mathrm{exp}\left[{\displaystyle \int }_{0}^{l}{\scr{\Phi}} (s){\hat{\psi }}_{{\mathcal{A}}}^{\dagger }(s)-{{\scr{\Phi}} }^{*}(s){\hat{\psi }}_{{\mathcal{A}}}(s)\;{\rm{d}}s\right]| {{\scr{\Omega}} }_{{\mathcal{A}}}\rangle ,\end{eqnarray} \tag{ 16 }$$

is exactly representible as a cMPS $| \chi ({\scr{\Phi}} )\rangle$ with bond dimension D = 1. This follows upon taking K and R to be the one-dimensional matrices $K(s)=0$ and $R(s)={\scr{\Phi}} (s)$. The boundary vectors $| {\omega }_{L}\rangle$ and $| {\omega }_{R}\rangle$ are simply taken to be equal to 1. The next fact we require is that the span of all field coherent states is dense in Fock space, meaning that an arbitrary field state $| {\scr{\Psi}} \rangle \in {{\mathfrak{F}}}_{-}({L}^{2}([0,l]))$ may be approximated arbitrarily well by an increasing linear combination of field coherent states:

$$\begin{eqnarray}&&\underset{l=0}{\overset{N}{\displaystyle \sum }}{c}_{j}| {{\scr{\Phi}} }_{j}\rangle \stackrel{N\to \infty }{\longrightarrow }| {\scr{\Psi}} \rangle .\end{eqnarray} \tag{ 17 }$$

This property follows from the over-completeness of coherent states in spanning the Hilbert space [11]. The final fact we need is that a linear combination $| \chi \rangle ={c}_{1}| {\chi }_{1}\rangle +{c}_{2}| {\chi }_{2}\rangle$ of two cMPS $| {\chi }_{1}\rangle$ and $| {\chi }_{2}\rangle$ with bond dimensions D₁ and D₂, respectively, is again a cMPS with bond dimension $D={D}_{1}+{D}_{2}$ and parameters $K={K}_{1}\oplus {K}_{2}$, $R={R}_{1}\oplus {R}_{2}$, $\langle {\omega }_{L}| =({c}_{1}\langle {\omega }_{L,1}| \oplus {c}_{2}\langle {\omega }_{L,2}| )$, and $| {\omega }_{R}\rangle =| {\omega }_{R,1}\rangle \oplus | {\omega }_{R,2}\rangle$.

Putting these facts together allows us to deduce that ${\{| {\chi }_{N}\rangle \equiv {\displaystyle \sum }_{l=0}^{N}{c}_{j}| {{\scr{\Phi}} }_{j}\rangle \}}_{N}$ is a sequence of cMPS with bond dimensions ${D}_{N}=N$ that tend, in the limit, to an arbitrary state $| {\scr{\Psi}} \rangle$ in Fock space. Thus we have confirmed the completeness or expressiveness property of the cMPS variational class in one dimension. It is worth noting that the argument we present here is by no means the most economical: there are, exploiting gauge invariance, almost certainly more efficient sequences of representations tending to the state $| {\scr{\Psi}} \rangle$ using lower bond dimensions. Indeed, as we argue in the next section, a more economical representation of a physical field state is strongly suggested by the path integral representation.

It is worth noting that in the previous subsection we showed that an arbitrary cMPS is a superposition of field coherent states. Here we've shown the converse: an arbitrary superposition of field coherent states is also a cMPS.

A natural higher-dimensional generalization of MPS are the PEPS, which are examples of Tensor Networks [19, 31, 34]. 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 ${A}_{(i\cdots \;k)}^{r}$ with respect to a particular graph $\Gamma (V,E)$:

$$\begin{eqnarray}&&| \chi \rangle ={\mathcal{C}}[{A}_{({i}_{1}\cdots {k}_{1})}^{{r}_{1}}\cdots \;A\cdots \;{A}_{({i}_{N}\cdots \;{k}_{N})}^{{r}_{N}}]| {r}_{1}\cdots \;{r}_{N}\rangle ,\end{eqnarray} \tag{ 32 }$$

where ${\mathcal{C}}$ denotes a complete contraction of the auxiliary indices $(i\cdots \;k)$ 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). Here we adopt a slightly more involved strategy to handle this unwanted feature.

We begin with a graph of N² points $\{(x,y)=(n\epsilon ,m\epsilon ):n,m\ \mathrm{integers}\}$, with a physical spacing . We view the y spatial direction as an auxiliary time t. As written (32) involves a contraction over N² 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 N² 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 particular transfer operator. The state (32) can then be written as

$$\begin{eqnarray}&&| \chi \rangle ={}_{{\mathcal{B}}}\langle {\omega }_{L}| \hat{U}(t=l,t=0){| {\omega }_{R}\rangle }_{{\mathcal{B}}}{| {\scr{\Omega}} \rangle }_{{\mathcal{A}}},\end{eqnarray} \tag{ 33 }$$

where $\hat{U}$ is an operator on ${\mathcal{A}}{\mathcal{B}}$ given by the (time-ordered) product of transfer operators local with respect to the graph sites and $| {\scr{\Omega}} {\rangle }_{{\mathcal{A}}}$ is some initial product state of ${\mathcal{A}}$. Specifically, $\hat{U}={\text{}}{\displaystyle \prod }_{t=0}^{l}[\hat{M}(t)]\equiv \hat{M}(l)\hat{M}(l-\epsilon )\hat{M}(l-2\epsilon )\cdots \hat{M}(0)$, where the transfer operator $\hat{M}(t)$ generates an elementary time-step of size and is built from local operators on ${\mathcal{A}}$ and ${\mathcal{B}}$. We then follow the idea used for the 1-dimensional case and regard each the auxiliary system at lattice site of ${\mathcal{B}}$ as the single-particle space of the Fock space built from ${{\mathbb{C}}}^{D}$.

The contraction of indices depends on the particular graph structure being used. 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 (30) implies that the auxiliary action S is invariant under the induced ${\text{}}{SO}(2)$ 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 auxiliary system (after we trace out ${\mathcal{A}}$). However, encoding the symmetry into the dynamics of the auxiliary system means imposing invariance under ${\text{}}{SO}(2)$. 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). It is also useful to emphasize that technical subtleties arise when taking the limit of lattice systems. Specifically, we might consider a family of graphs $\{{\Lambda }_{k}\}$ indexed by some variable $k=0,1,2,...$, that converges to some dense subset of a compact spatial region A. To each point ${\bf{x}}\in {\Lambda }_{k}$ we have an associated Hilbert space ${\mathcal{H}}({\bf{x}})$, which could be a space of finite or countably infinite dimension. The total Hilbert space for the full graph system is then given by ${{\mathcal{H}}}_{k}={\otimes }_{{\bf{x}}\in {\Lambda }_{k}}{\mathcal{H}}({\bf{x}})$, and in the thermodynamic limit $k\to \infty$, the resultant space will have an uncountable dimension. One instead works with a much smaller, separable Fock space ${\mathfrak{F}}({\mathcal{H}})$ constructed to ensure that every state in ${\mathfrak{F}}({\mathcal{H}})$ has finite particle expectation value, and splits up into a sum ${\mathfrak{F}}({\mathcal{H}})={\oplus }_{n}{{\mathcal{H}}}^{(n)}$ of particle sectors ${{\mathcal{H}}}^{(n)}$ 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 ${{\mathcal{H}}}^{(n)}$ 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 $| \chi \rangle \in {\mathfrak{F}}({{\mathcal{H}}}_{{\mathcal{A}}})$, and work with a particular choice of creation/annihilation operators for both ${\mathcal{A}}$ and ${\mathcal{B}},$ with the auxiliary system ${\mathcal{B}}$ carrying bosonic or fermionic statistics.

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 $\epsilon \to 0$. There are clearly different possible choices for a Lorentz-invariant action; motivated by the first-order action (15), and certain convenient properties of coherent field states, we derive a Dirac-like action from a specific sequence of PEPS. One might question why we bother going via a Lorentz-invariant setting. The reason is that if we begin with SO(2) symmetry as our target then we do not have ready access to the intuition that the physical field state is generated by the virtual dynamics of a lower dimensional system.

The 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 $(x,t+\epsilon ),(x-\epsilon ,t+\epsilon )$ and $(x+\epsilon ,t+\epsilon )$ and also to past sites $(x,t-\epsilon ),(x-\epsilon ,t-\epsilon )$ and $(x+\epsilon ,t-\epsilon )$. The simplest such choice is to build the operator $\hat{M}$ 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 ${[{a}_{j,x},{a}_{k,y}]}_{\pm }={[{a}_{j,x}^{\dagger },{a}_{k,y}^{\dagger }]}_{\pm }=0,$ ${[{a}_{j,x},{a}_{k,y}^{\dagger }]}_{\pm }={\delta }_{{jk}}{\delta }_{x,y}$, with similar expressions for b, where ± labels the choice of bosonic or fermionic auxiliary system.

We now construct a transfer operator ${\hat{M}}_{\epsilon }={\mathbb{1}}+\epsilon {\hat{H}}_{\mathrm{tot}}$ from terms quadratic in a and b, and generate a PEPS via

$$\begin{eqnarray}&&| {\chi }_{\epsilon }\rangle ={}_{{\mathcal{B}}}\langle {\omega }_{L}| {\hat{M}}_{\epsilon }(l){\hat{M}}_{\epsilon }(l-\epsilon ){\hat{M}}_{\epsilon }(l-2\epsilon )\cdots {\hat{M}}_{\epsilon }(0){| {\omega }_{R}\rangle }_{{\mathcal{B}}}{| {\scr{\Omega}} \rangle }_{{\mathcal{A}}},\end{eqnarray} \tag{ 34 }$$

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 ${\mathcal{B}},$ which only forms an abstract device to describe the physical state $| {\chi }_{\epsilon }\rangle$, and does not commit us to any particular realization for the two-dimensional physical field system ${\mathcal{A}}$. 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 ${\hat{H}}_{\mathrm{tot}}$. Certain assumptions are natural to impose on the terms appearing in ${\hat{H}}_{\mathrm{tot}}$, 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

$$\begin{eqnarray}&&{H}_{{\rm{h}}}(t)=\displaystyle \frac{1}{\epsilon }\underset{x}{\displaystyle \sum }{J}^{{jk}}(t)({a}_{j,x}^{\dagger }{b}_{k,x-\epsilon }+{a}_{j,x}^{\dagger }{b}_{k,x}+{a}_{j,x}^{\dagger }{b}_{k,x+\epsilon }+{\rm{h}}.{\rm{c}}.),\end{eqnarray} \tag{ 35 }$$

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

$$\begin{eqnarray}&&{H}_{h}=\displaystyle \frac{1}{\epsilon }\underset{p,j,k}{\displaystyle \sum }{J}^{{jk}}(t)(1+2\mathrm{cos}p\epsilon )({\tilde{a}}_{j,p}^{\dagger }{\tilde{b}}_{k,p}+{\tilde{b}}_{j,p}^{\dagger }{\tilde{a}}_{k,p}),\end{eqnarray} \tag{ 36 }$$

where ${\tilde{a}}_{j,p}=\frac{1}{\sqrt{N}}{\displaystyle \sum }_{x}{{\rm{e}}}^{-{\rm{i}}{px}}{a}_{j,x}$ is the Fourier-transformed annihilation operator (similarly for ${\tilde{b}}_{j,p}$), and $p=2\pi n/N\epsilon$ runs over N points in the reciprocal lattice, for n an integer.

Our concern is the continuum limit, $\epsilon \to 0$, where the dominant contributions of H_h will come from the 'low-energy regime' of momenta p near to the zeroes of $(1+2\mathrm{cos}p\epsilon )$. These occur at the points ${q}_{\mu }={(-1)}^{\mu }(2\pi /3\epsilon )$, for $\mu =0,1$. 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

$$\begin{eqnarray}&&{H}_{h,\mu }(t)=\sqrt{3}\underset{| p| \ll 1/\epsilon }{\displaystyle \sum }{(-1)}^{\mu }{J}^{{jk}}(t)\left({\tilde{a}}_{j,{q}_{\mu }+p}^{\dagger }{\tilde{b}}_{j,{q}_{\mu }+p}^{\dagger }\right)p{\sigma }_{x}\left(\begin{array}{c}{\tilde{a}}_{j,{q}_{\mu }+p}\\ {\tilde{b}}_{j,{q}_{\mu }+p}\end{array}\right).\end{eqnarray} \tag{ 37 }$$

We can redefine ${J}^{{jk}}\to {J}^{{jk}}/\sqrt{3}$ and relabel the mode operators as ${\tilde{a}}_{(j,\mu );p}\equiv {\tilde{a}}_{j,{q}_{\mu }+p}$ and also ${\tilde{b}}_{(j,\mu );p}\equiv {\tilde{b}}_{j,{q}_{\mu }+p}$ to obtain

$$\begin{eqnarray}&&{H}_{h}(t)=\underset{| p| \ll 1/\epsilon ;\mu =0,1}{\displaystyle \sum }{J}^{{jk}}(t)\left({\tilde{a}}_{(j,\mu ),p}^{\dagger }{\tilde{b}}_{(j,\mu ),p}^{\dagger }\right)p{\sigma }_{z}^{\mu }{\sigma }_{x}{\sigma }_{z}^{\mu }\left(\begin{array}{c}{\tilde{a}}_{(j,\mu ),p}\\ {\tilde{b}}_{(j,\mu ),p}\end{array}\right).\end{eqnarray} \tag{ 38 }$$

This term alone would produce $2D$ massless Lorentz-invariant flavors labeled by the compound index $(j,\mu )$, in which the $(j,0)$-field is related to the $(j,1)$-field through the discrete parity transformation ${\mathcal{P}},$ given in 1+1 dimensions as ${\mathcal{P}}={\gamma }^{0}={\sigma }_{z}$, which inverts chirality. This is consistent with the Nielsen–Ninomiya theorem [20], which requires doubling in order to achieve a translationally invariant spinor action with chiral symmetry in the continuum limit of a lattice model.

The previous analysis shows that the two contributions to H_h from the points q₀ and q₁ in momentum space decouple. In position space this tells us that ${a}_{j,x}$ splits up in the low-energy regime as ${a}_{j,x}=\{{\sum }_{| p| \ll 1/\epsilon }{{\rm{e}}}^{{\rm{i}}{px}}{\tilde{a}}_{{q}_{0}+p}\}{{\rm{e}}}^{{\rm{i}}{q}_{0}x}$ $+\{{\sum }_{| p| \ll 1/\epsilon }{{\rm{e}}}^{{\rm{i}}{px}}{\tilde{a}}_{{q}_{1}+p}\}{{\rm{e}}}^{{\rm{i}}{q}_{1}x}$. We write this instead as ${a}_{j,x}={a}_{(j,0),x}{{\rm{e}}}^{{\rm{i}}{q}_{0}x}+$ ${a}_{(j,1),x}{{\rm{e}}}^{{\rm{i}}{q}_{1}x}$, in which the operators ${a}_{(j,\mu ),x}$ are given by the expressions in the curly brackets of the preceding equation.

In the continuum limit we then have that ${\epsilon }^{-1/2}{a}_{(j,\mu ),x}$ tends to a smooth field ${{\scr{\Psi}} }_{(j,\mu )}(x)$, arising from envelopes of plane waves around the point ${q}_{\mu }$. 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 $\epsilon {\sum }_{x}{f}^{{jk}}(x){a}_{j,x}^{\dagger }{a}_{k,x}$. This term will behave as $\epsilon {\sum }_{\mu ,\nu ,x}{f}^{{jk}}(x){{\rm{e}}}^{-{\rm{i}}({q}_{\mu }-{q}_{\nu })x}{a}_{(j,\mu ),x}^{\dagger }{a}_{(k,\nu ),x}$, however, if the function ${f}^{{jk}}(x)$ does not vary rapidly from site to site, the presence of the highly oscillatory term ${{\rm{e}}}^{-{\rm{i}}({q}_{\mu }-{q}_{\nu })x}$ will ensure that only $\mu =\nu$ 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, 17], however here we do not consider such scenarios.

It is straightforward to produce a mass term in the continuum limit, simply by adding the term ${\sum }_{x}m({a}_{j,x}^{\dagger }{a}_{j,x}-{b}_{j,x}^{\dagger }{b}_{j,x})$, which then generates the usual dispersion relation ${E}^{2}={p}^{2}+{m}^{2}$ as $\epsilon \to 0$. However, we can more generally use functions $\{{m}_{0}^{{jk}}(x,t)\}$ that do not vary too rapidly over the lattice, and perturb H_h by the on-site potential term ${H}_{m}(t)={\sum }_{x}{m}_{0}^{{jk}}(x,t)({a}_{j,x}^{\dagger }{a}_{k,x}-{b}_{j,x}^{\dagger }{b}_{k,x})$. This can be written as

$$\begin{eqnarray*}&&{H}_{m}(t)=\underset{x,j,k,\mu }{\displaystyle \sum }\left({m}_{0}^{{jk}}(x,t)({a}_{(j,\mu ),x}^{\dagger }{a}_{(k,\mu ),x}-{b}_{(j,\mu ),x}^{\dagger }{b}_{(k,\mu ),x})+{{\rm{e}}}^{\pm {\rm{i}}({q}_{0}-{q}_{1})x}({\rm{flavor}}\;{\rm{coupling}}\;{\rm{terms}})\right)\end{eqnarray*}$$

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 $| {\chi }_{\epsilon }\rangle$. For this, we mirror the local coupling used for the 1-dimensional system and define the interaction term contribution is easily calculated

$$\begin{eqnarray}&&{\hat{H}}_{\mathrm{int}}(t)=\underset{x}{\displaystyle \sum }{R}^{{jk}}(x,t)\left[({a}_{j,x}^{\dagger }{a}_{k,x}+{b}_{j,x}^{\dagger }{b}_{k,x})\otimes {\hat{\psi }}_{{\mathcal{A}}}^{\dagger }(x,y=t)\right],\end{eqnarray} \tag{ 39 }$$

where again, for simplicity, we assume the functions $\{{R}^{{jk}}(x,t)\}$ vary sufficiently slowly over the lattice so that the $\mu =0,1$ flavors decouple once more.

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The final transfer operator that generates the PEPS state is finally given by

$$\begin{eqnarray}&&{\hat{M}}_{\epsilon }={\mathbb{1}}+\epsilon ({H}_{{\rm{m}}}+{H}_{{\rm{h}}}+{\hat{H}}_{\mathrm{int}}),\end{eqnarray} \tag{ 40 }$$

where the operator hat serves to specify the non-trivial action on the physical field system. The basic tensor structure of ${\hat{M}}_{\epsilon }(t)$ is shown in figure 2, where there is also an implicit physical index at each site, coupling to the physical field ${\mathcal{A}},$ which we omit in the diagram for the sake of clarity.

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In the next section we analyse the continuum limit, and derive the desired path integral representation from the smooth fields ${\epsilon }^{-1/2}{a}_{(j,\mu ),x}$ and ${\epsilon }^{-1/2}{b}_{(j,\mu ),x}$. On the assumption that m₀^jk and R^jk slowly vary on the lattice, the expressions for H_m and ${\hat{H}}_{\mathrm{int}}$ in terms of these smooth fields are obtained by the doubling of flavor index $j\to (j,\mu )$. The same happens for the indices of the kinetic term H_h, but with added feature of a parity flip ${\sigma }_{x}\leftrightarrow {\sigma }_{z}{\sigma }_{x}{\sigma }_{z}$ relating the two flavors. As we shall see later, since this discrete symmetry is itself a Lorentz symmetry the total field state that arises will possess the full symmetry group that we require.

In this, and the subsequent, subsection we construct the path integral for the continuum limit of the sequence $| {\chi }_{\epsilon }\rangle$, $\epsilon \to 0$. We follow the one-dimensional prescription, and insert coherent-state resolutions of the identity into the product ${\text{}}{\displaystyle \prod }_{t=0}^{l}[{\hat{M}}_{\epsilon }(t)]$. This is first computed for a simple elementary timestep 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 ${j}_{\mu }$ to denote the compound index $(j,\mu )$ 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^jk₀ 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 will use coherent-state resolutions of the identity which, in both the fermionic and bosonic cases, are given by

$$\begin{eqnarray}&&| {\boldsymbol{\phi }}(x,t)\rangle \equiv | {\otimes }_{{j}_{\mu },x,s}{\phi }_{{j}_{\mu },s}(x,t)\rangle := D(\{{\phi }_{{j}_{\mu },s}(x,t)\})| {{\scr{\Omega}} }_{{\mathcal{B}}}\rangle ,\end{eqnarray} \tag{ 41 }$$

where we have the usual coherent state displacement operator, given for a single mode with annihilation operator c as $D(\alpha )=\mathrm{exp}[\alpha {c}^{\dagger }-c{\alpha }^{*}]$, the label $s=a,b$ labels the particle type, and in the fermionic case ${\phi }_{{j}_{\mu },s}(x,t)$ are Grassmann numbers. We also use bold-faced ${\boldsymbol{\phi }}:= \{{\phi }_{{j}_{\mu },s}(x,t)\}$ to suppress indices when the contractions are clear. The identity contribution is easily calculated via the overlap formulae for coherent states and gives $\langle {\boldsymbol{\phi }}(x,t+\epsilon )| {\boldsymbol{\phi }}(x\prime ,t)\rangle =\mathrm{exp}[-\frac{\epsilon }{2}{\sum }_{x}({{\boldsymbol{\phi }}}^{\dagger }(x,t){\partial }_{t}{\boldsymbol{\phi }}(x,t)-{\partial }_{t}{\boldsymbol{\phi }}{(x,t)}^{\dagger }{\boldsymbol{\phi }}(x,t))]$, however the H_h and H_m terms require more attention.

It is simplest to work in momentum space, for which

$$\begin{eqnarray*}&&{H}_{h}+{H}_{m}=\underset{p,p\prime ,{j}_{\mu },{k}_{\mu }}{\displaystyle \sum }\left({\tilde{a}}_{{j}_{\mu },p}^{\dagger }{\tilde{b}}_{{j}_{\mu },p}^{\dagger }\right)\left(\displaystyle \frac{{\tilde{m}}_{0}^{{jk}}(p-p\prime ,t)}{\sqrt{N}}{\sigma }_{z}+{J}^{{jk}}(t)\delta (p-p\prime )p\prime {\sigma }_{z}^{\mu }{\sigma }_{x}{\sigma }_{z}^{\mu }\right)\left(\begin{array}{c}{\tilde{a}}_{{k}_{\mu },p\prime }\\ {\tilde{b}}_{{k}_{\mu },p\prime }\end{array}\right),\end{eqnarray*}$$

where ${\tilde{m}}_{0}^{{jk}}(p,t)=\frac{1}{\sqrt{N}}{\displaystyle \sum }_{x}{{\rm{e}}}^{-{\rm{i}}{px}}{m}_{0}^{{jk}}(x,t)$ and the indices on m₀^jk and J^jk are explicitly independent of μ.

Instead of expanding in terms of fermionic/bosonic coherent states of ${a}_{{j}_{\mu },x}$ and ${b}_{{j}_{\mu },x}$ we expand in terms of fermionic/bosonic coherent states of ${\tilde{a}}_{{j}_{\mu },p}$ and ${\tilde{b}}_{{j}_{\mu },p}$. Insertion of the above ${H}_{h}+{H}_{m}$ into $\langle \tilde{{\boldsymbol{\phi }}}(p,t)| {H}_{h}+{H}_{m}| \tilde{{\boldsymbol{\phi }}}(p\prime ,t)\rangle$ gives, to $O(\epsilon )$,

$$\begin{eqnarray}&&{N}^{-1/2}\underset{p,p\prime ,{j}_{\mu },{k}_{\mu }}{\displaystyle \sum }{\tilde{m}}_{0}^{{jk}}(p-p\prime ,t)\left[\tilde{\phi }{*}_{{j}_{\mu },a}(p,t){\tilde{\phi }}_{{k}_{\mu },a}(p\prime ,t)-{\tilde{\phi }}_{{j}_{\mu },b}^{*}(p,t){\tilde{\phi }}_{{k}_{\mu },b}(p\prime ,t)\right)\\ &&\qquad +\sqrt{N}\delta (p-p\prime ){J}^{{jk}}p\prime \left({\tilde{\phi }}_{{j}_{\mu },a}^{*}(p,t){\tilde{\phi }}_{{k}_{\mu },b}(p\prime ,t)+{\tilde{\phi }}_{{j}_{\mu },b}^{*}(p,t){\tilde{\phi }}_{{k}_{\mu },a}(p\prime ,t)\right]\\ &&\quad ={N}^{-1/2}\underset{p,p\prime ,{j}_{\mu },{k}_{\mu }}{\displaystyle \sum }{\tilde{m}}_{0}^{{jk}}(p-p\prime ,t){\tilde{{\scr{\Psi}} }}_{{j}_{\mu }}^{\dagger }(p,t){\sigma }_{z}{\tilde{{\scr{\Psi}} }}_{{k}_{\mu }}(p\prime ,t)+\sqrt{N}\delta (p-p\prime ){J}^{{jk}}(t)({\tilde{{\scr{\Psi}} }}_{{j}_{\mu }}^{\dagger }(p,t)p\prime {\sigma }_{x}{\tilde{{\scr{\Psi}} }}_{{k}_{\mu }}(p\prime ,t))\\ &&\quad =\underset{x,{j}_{\mu },{k}_{\mu }}{\displaystyle \sum }{m}_{0}^{{jk}}(x,t){{\scr{\Psi}} }_{{j}_{\mu }}^{\dagger }(x,t){\sigma }_{z}{{\scr{\Psi}} }_{{k}_{\mu }}(x,t)+{J}^{{jk}}(t){{\scr{\Psi}} }_{{j}_{\mu }}^{\dagger }(x,t){\rm{i}}{\partial }_{x}{\sigma }_{z}^{\mu }{\sigma }_{x}{\sigma }_{z}^{\mu }{{\scr{\Psi}} }_{{k}_{\mu }}(x,t),\end{eqnarray} \tag{ 42 }$$

where ${{\scr{\Psi}} }_{{j}_{\mu }}(x,t)=\left({\phi }_{{j}_{\mu },a}(x,t),{\phi }_{{j}_{\mu },b}(x,t)\right)$ is a two-component auxiliary field with flavour index ${j}_{\mu }$, and where ${j}_{\mu }=1,2,\ldots ,D$ for each of the two separate $\mu =0,1$ sectors.

The interaction term can be evaluated in the same way, and we obtain

$$\begin{eqnarray*}&&\langle {\boldsymbol{\phi }}(x,t+\epsilon )| {\hat{H}}_{\mathrm{int}}| {\boldsymbol{\phi }}(x\prime ,t)\rangle =\underset{x,{j}_{\mu },{k}_{\mu }}{\displaystyle \sum }\left({R}^{{jk}}(x,t){{\scr{\Psi}} }_{{j}_{\mu }}^{\dagger }(x,t){{\scr{\Psi}} }_{{k}_{\mu }}(x,t)\right){\hat{{\scr{\Psi}} }}_{{\mathcal{A}}}^{\dagger }(x,y=t),\end{eqnarray*}$$

where the hat on ${\hat{{\scr{\Psi}} }}_{{\mathcal{A}}}^{\dagger }$ is again to emphasize that the term is an operator on the physical system, as opposed to ${{\scr{\Psi}} }_{{j}_{\mu }}(x,t)$ which is an auxiliary two component (Grassmann) spinor, with flavor index ${j}_{\mu }$.

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 ${\hat{M}}_{\epsilon }(t)$. 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.

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:

$$\begin{eqnarray}| \chi \rangle & = & {\mathcal{C}}[{A}_{({i}_{1}\cdots \;{k}_{1})}^{{r}_{1}}\cdots \;{A}_{({i}_{M}\cdots \;{k}_{M})}^{{r}_{M}}\cdots ]| {r}_{1}\cdots \;{r}_{M}\cdots \;\rangle \\ & = & \hat{{\mathcal{C}}}[\cdots ]| {{\scr{\Omega}} }_{{\mathcal{A}}}\rangle ,\end{eqnarray} \tag{ 43 }$$

where $({i}_{M}\cdots \;{k}_{M})$ are a set of contraction indices dependent on the particular graph structure of the state, and $| {r}_{M}\rangle$ 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

$$\begin{eqnarray}&&\hat{{\mathcal{C}}}[\cdots ]=\langle {\omega }_{L}| {\hat{M}}_{\epsilon }(l){\hat{M}}_{\epsilon }(l-\epsilon )\cdots {\hat{M}}_{\epsilon }(\epsilon ){\hat{M}}_{\epsilon }(0)| {\omega }_{R}\rangle .\end{eqnarray} \tag{ 44 }$$

For clarity, we restrict to a finite set of points $\{(x,t)\}$ where x runs over N_x points, separated by a distance ${\epsilon }_{x}$ and t runs over N_t points, in timesteps of , 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

$$\begin{eqnarray}&&\displaystyle \frac{1}{{\pi }^{8{{DN}}_{x}}}\int \underset{k,s,x}{\displaystyle \prod }{{\rm{d}}}^{2}{\phi }_{k,s}(x,t)| {\otimes }_{k,s,x}{\phi }_{k,s}(x,t)\rangle \langle {\otimes }_{k,s,x}{\phi }_{k,s}(x,t)| ={\mathbb{1}}.\end{eqnarray} \tag{ 45 }$$

Inserting ${N}_{t}+1$ such resolutions into (44) gives

$$\begin{eqnarray}\hat{{\mathcal{C}}}[\cdots ] & = & \displaystyle \int {\rm{d}}\mu \left[\underset{t=0}{\overset{l+\epsilon }{\displaystyle \prod }}\langle {\otimes }_{{k}_{t+\epsilon },{s}_{t+\epsilon },{x}_{t+\epsilon }}{\phi }_{{k}_{t+\epsilon },{s}_{t+\epsilon }}({x}_{t+\epsilon },t+\epsilon )| \hat{M}(t)| {\otimes }_{{k}_{t},{s}_{t},{x}_{t}}{\phi }_{{k}_{t},{s}_{t}}({x}_{t},t)\rangle \right]\\ & & \times \langle {\omega }_{L}| {\phi }_{{k}_{l+\epsilon },{s}_{l+\epsilon }}({x}_{l+\epsilon },l+\epsilon )\rangle \langle {\phi }_{{k}_{0},{s}_{0}}({x}_{0},0)| {\omega }_{R}\rangle ,\end{eqnarray} \tag{ 46 }$$

with the measure for the integral given by

$$\begin{eqnarray}&&{\rm{d}}\mu =\displaystyle \frac{1}{{\pi }^{8{{DN}}_{x}({N}_{t}+1)}}\underset{t=0}{\overset{l+\epsilon }{\displaystyle \prod }}\left[\underset{{k}_{t},{s}_{t},{x}_{t}}{\displaystyle \prod }{{\rm{d}}}^{2}{\phi }_{{k}_{t},{s}_{t}}({x}_{t},t)\right].\end{eqnarray} \tag{ 47 }$$

The amplitudes in the integrand of (46) have been calculated in the low-energy sector and in this regime we obtain the expression

$$\begin{eqnarray*}&&\int {\rm{d}}\mu \;{{\rm{e}}}^{{\sum }_{x,t}\epsilon [-\frac{1}{2}({\phi }_{k,s}^{*}(x,t){\partial }_{t}{\phi }_{k,s}(x,t)-{\phi }_{k,s}(x,t){\partial }_{t}{\phi }_{k,s}^{*}(x,t))+{{\scr{\Psi}} }_{j}^{\dagger }(x,t)({J}^{{jk}}{\sigma }_{z}^{\mu }{\sigma }_{x}{\sigma }_{z}^{\mu }i{\partial }_{x}+{m}_{0}^{{jk}}{\sigma }_{z}){{\scr{\Psi}} }_{k}(x,t)+{R}^{{jk}}{{\scr{\Psi}} }_{j}^{\dagger }(x,t){{\scr{\Psi}} }_{k}(x,t)\otimes {\hat{\psi }}_{{\mathcal{A}}}^{\dagger }(x,y=t)]},\end{eqnarray*}$$

for the tensor contraction $\hat{{\mathcal{C}}}[\cdots ]$, where the matrices m₀ and R are allowed to vary smoothly in both x and t.

We now take the spatial ${\epsilon }_{x}\to 0$ limit followed by the temporal $\epsilon \to 0$ limit. However, to keep things well-behaved, we must first rescale the integration variables

$$\begin{eqnarray}&&{\phi }_{{k}_{\mu },s}(x,t)\to \displaystyle \frac{1}{\sqrt{{\epsilon }_{x}}}{\phi }_{{k}_{\mu },s}(x,t).\end{eqnarray} \tag{ 48 }$$

Indeed, this rescaling was to be expected since in order to respect the correct commutation/anti-commutation relations: in 1 + 1 dimensions we have ${{\scr{\Psi}} }_{{k}_{\mu }}\sim {a}_{{k}_{\mu }}/\sqrt{{\epsilon }_{x}}$, while in 2 + 1 dimensions we should instead have ${{\scr{\Psi}} }_{{k}_{\mu }}\sim {a}_{{k}_{\mu }}/\sqrt{{\epsilon }_{x}{\epsilon }_{y}}$, where ${\epsilon }_{x}$ and ${\epsilon }_{y}$ are the two spatial lattice scales.

Once this rescaling is performed, we find in the ${\epsilon }_{x},\epsilon \to 0$ continuum limit that the resultant cMPS field state becomes

$$\begin{eqnarray*}&&| \chi \rangle =\int {\mathcal{D}}{{\scr{\Psi}} }_{{k}_{\mu }}{\mathcal{D}}{{\scr{\Psi}} }_{{k}_{\mu }}^{\dagger }{{\rm{e}}}^{\int {\rm{d}}x{\rm{d}}t(-{{\scr{\Psi}} }_{{k}_{\mu }}^{\dagger }(x,t){\partial }_{t}{{\scr{\Psi}} }_{{k}_{\mu }}(x,t)+{{\scr{\Psi}} }_{{j}_{\mu }}^{\dagger }(x,t)({J}^{{jk}}{\sigma }_{z}^{\mu }{\sigma }_{x}{\sigma }_{z}^{\mu }{\rm{i}}{\partial }_{x}+{m}_{0}^{{jk}}{\sigma }_{z}){{\scr{\Psi}} }_{{k}_{\mu }}(x,t)+({R}^{{jk}}{{\scr{\Psi}} }_{{j}_{\mu }}^{\dagger }(x,t){{\scr{\Psi}} }_{{k}_{\mu }}(x,t)){\hat{\psi }}_{{\mathcal{A}}}^{\dagger }(x,y=t))}| {{\scr{\Omega}} }_{{\mathcal{A}}}\rangle ,\end{eqnarray*}$$

and we obtain a well-defined two-dimensional action.

Once the rescaling has been performed we may directly integrate by parts, to obtain the class of two-dimensional field states

$$\begin{eqnarray}&&| \chi (J,{m}_{0},R)\rangle =\int {\mathcal{D}}{{\scr{\Psi}} }_{{k}_{\mu }}{\mathcal{D}}{{\scr{\Psi}} }_{{k}_{\mu }}^{\dagger }{{\rm{e}}}^{\int {\rm{d}}x{\rm{d}}t({{\scr{\Psi}} }_{{j}_{\mu }}^{\dagger }(x,t)(-{\delta }^{{jk}}{\partial }_{t}+{J}^{{jk}}(t){\sigma }_{z}^{\mu }{\sigma }_{x}{\sigma }_{z}^{\mu }{\rm{i}}{\partial }_{x}+{m}_{0}^{{jk}}(x,t){\sigma }_{z}){{\scr{\Psi}} }_{{k}_{\mu }}(x,t))}| {\scr{\Phi}} \rangle ,\end{eqnarray} \tag{ 49 }$$

where we have introduced the coherent field state $| {\scr{\Phi}} (x,y)\rangle =\mathrm{exp}[\int {\rm{d}}x{\rm{d}}t({\scr{\Phi}} (x,t){\hat{\psi }}_{{\mathcal{A}}}^{\dagger }(x,y=t)-{\scr{\Phi}} (x,t){\hat{\psi }}_{{\mathcal{A}}}(x,y=t))]| {{\scr{\Omega}} }_{{\mathcal{A}}}\rangle$ with ${\scr{\Phi}} (x,t)={R}^{{jk}}(x,t){{\scr{\Psi}} }_{{j}_{\mu }}^{\dagger }(x,t){{\scr{\Psi}} }_{{k}_{\mu }}(x,t)$, 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 (49) 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 ${J}^{{jk}}(t)={\rm{i}}{\delta }^{{jk}}$. This allows the momentum terms to respect the desired symmetry and we obtain

$$\begin{eqnarray}&&| \chi (m,R)\rangle =\int {\mathcal{D}}{{\scr{\Psi}} }_{{k}_{\mu }}{\mathcal{D}}{\bar{{\scr{\Psi}} }}_{{k}_{\mu }}{{\rm{e}}}^{{\rm{i}}\int {\rm{d}}x{\rm{d}}t({\bar{{\scr{\Psi}} }}_{{j}_{\mu }}({\rm{i}}{\delta }^{{jk}}/\partial -{m}^{{jk}}(x,t)){{\scr{\Psi}} }_{{k}_{\mu }}(x,t)}| {\scr{\Phi}} (x,y=t)\rangle ,\end{eqnarray} \tag{ 50 }$$

where we have also let ${m}_{0}^{{jk}}=-{\rm{i}}{m}^{{jk}}$. For the situation where m is diagonal and constant over the auxiliary spacetime, the coherent field amplitudes are then recognized as Grassmann/complex number path integrals for a set of $2D$ uncoupled Lorentz invariant spinor fields. The spinors have the associated gamma matrices ${\gamma }^{0}={\sigma }_{z}$ and ${\gamma }^{1}={\rm{i}}{\sigma }_{y}$ obeying the Clifford algebra relations $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2{\eta }^{\mu \nu }{\mathbb{1}}$ with ${\eta }^{\mu \nu }=\mathrm{diag}(1,-1)$. Also for $\mu =1$ flavor sector we also perform a parity transformation on ${{\scr{\Psi}} }_{{k}_{1}}\to {\gamma }^{0}{{\scr{\Psi}} }_{{k}_{1}}$, which is given by ${\gamma }^{0}={\sigma }_{z}$ for the 1 + 1 dimensional case.

Note that, in obtaining this symmetrical state, we were forced to take both m₀ 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 $U(l,0)=({\mathbb{1}}-{\rm{i}}\epsilon H(l))({\mathbb{1}}-{\rm{i}}\epsilon H(l-\epsilon ))\cdots ={\text{}}$ $\mathrm{exp}[-{\rm{i}}{\int }_{0}^{l}H(t)]$ 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 ${j}^{\mu }=\bar{\psi }{\gamma }^{\mu }\psi$ is a conserved current for the free Dirac field, and in particular ${{\scr{\Psi}} }^{\dagger }{\scr{\Psi}}$ 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 ${\mathbb{1}}$ 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' [10, 29] 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 $1/{\rm{i}}m$. 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 ${\hat{M}}_{\epsilon }={\mathbb{1}}+\epsilon {\hat{H}}_{\mathrm{tot}}$ generates a Legendre transformation of a Hamiltonian ${\hat{H}}_{\mathrm{tot}}[\pi ,\phi ]$. For a coherent state-expansion, we obtain a Berry-phase term ${\rm{i}}{\phi }^{\dagger }{\partial }_{t}\phi$ from the coherent state overlaps which can be taken as $\pi {\partial }_{t}\phi$ where $\pi ={\rm{i}}{\phi }^{\dagger }$ is the momentum conjugate to ϕ.

Another point perhaps worth emphasizing is the physical interpretation of the auxiliary degrees of freedom that emerge in the preceding analysis. This might seem reminiscent of statistical mechanical settings involving interacting lattice models, where the computation of the partition function is often facilitated by the introduction of auxiliary degrees of freedoms that reduce the computation to non-interacting systems. The partition function can often be computed through some form of approximation, such as a mean-field assumption or method of steepest decent. Such auxiliary fields typically have the interpretation of an effective local external field, or order parameter, which captures the essential local physics of the system. Here, in contrast, the auxiliary degrees of freedom should not be viewed in the same way. We need not specify the physical interactions present in order to define the auxiliary system, but instead the auxiliary system acts as an 'entanglement regulator', and through the virtual dynamics that sweeps over the whole system, ensures that the physical field state automatically obeys entanglement area laws.

We now have a manifestly Lorentz-invariant auxiliary action, however the resultant physical state for ${\mathcal{A}}$ 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 $t\to {it}$, 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 $\bar{\psi }\psi$ 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 [23] 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 ${\eta }^{\mu \nu }\to {\eta }^{\mu \nu }(\theta )$ so that it forms a one-complex-parameter family of metrics interpolating between the Minkowski and the Euclidean one [18]. A more abstract formulation can be achieved by using vielbeins, but for our purpose we work directly with the spacetime metric.

The appropriate metric is ${\eta }_{\mu \nu }(\theta )=(\mathrm{cos}2\theta /| \mathrm{cos}2\theta | ,-1)$ defined for $0\leqslant \theta \leqslant \pi /2$ except for the singularity at $\theta =\pi /4$, which can be circumvented via extending θ to be complex. To carry the symmetry group over we also demand that the Clifford Algebra relation $\{{\gamma }_{\mu }(\theta ),{\gamma }_{\nu }(\theta )\}=2{\eta }_{\mu \nu }(\theta ){\mathbb{1}}$ holds for all θ, and introduce the ${\gamma }_{5}(\theta )$ matrix obeying ${\gamma }_{5}{(\theta )}^{\dagger }={\gamma }_{5}(\theta )$, $\{{\gamma }_{5}(\theta ),{\gamma }_{\mu }(\theta )\}=0$ and ${\gamma }_{5}{(\theta )}^{2}={\mathbb{1}}$. A particular parameterized set of gamma-matrices [18] that allow the continuation are then given by

$$\begin{eqnarray}{\gamma }_{1}(\theta ) & = & {\gamma }_{1}\\ {\gamma }_{0}(\theta ) & = & \displaystyle \frac{1}{{| \mathrm{cos}2\theta | }^{1/2}}({\gamma }_{0}\mathrm{cos}\theta +i{\gamma }_{5}\mathrm{sin}\theta )\\ {\gamma }_{5}(\theta ) & = & \displaystyle \frac{1}{{| \mathrm{cos}2\theta | }^{1/2}}({\gamma }_{5}\mathrm{cos}\theta -i{\gamma }_{0}\mathrm{sin}\theta ),\end{eqnarray} \tag{ 51 }$$

where ${\gamma }_{5}={\sigma }_{x}$ for our 1 + 1 dimensional case.

This family of gamma matrices obeys the correct anti-commutation relations, and more importantly, generates an interpolation from the ${SO}(1,1)$ Lorentz transformations to the SO(2) rotations via $\Lambda (\omega ;\theta )=\mathrm{exp}[{\omega }^{\mu \nu }{\Sigma }_{\mu \nu }(\theta )]$, where ω parametrizes the group and ${\Sigma }_{\mu \nu }(\theta )=\frac{1}{4}[{\gamma }_{\mu }(\theta ),{\gamma }_{\nu }(\theta )]$ are its generators. The intuition behind this choice of parameterization is that to construct scalars under the Lorentz symmetry ${SO}(1,1)$, fermions in the $(0,\frac{1}{2})$ representation are contracted with ones in $(\frac{1}{2},0)$, whereas for SO(2) we form contractions within the same representation [30]. Since ${\gamma }_{5}$ is reducible over the different helicities, while ${\gamma }_{0}$ is not, the intuition is to rotate ${\gamma }_{0}\leftrightarrow {\gamma }_{5}$ to pass from ${SO}(1,1)$ to SO(2).

A one-parameter family of actions, symmetric under this group action, can then be constructed, where $\theta =0$ is the Lorentz-invariant Dirac action and $\theta =\pi /2$ is the desired rotation-invariant Euclidean action. It is found to take the form

$$\begin{eqnarray}&&S[\psi ,\bar{\psi };\theta ]=\int {{\rm{d}}}^{2}x\sqrt{-\mathrm{det}(\eta (\theta ))}{\psi }^{\dagger }{\gamma }_{0}[{\rm{i}}{\eta }^{\mu \nu }(\theta ){\gamma }_{\mu }(\theta ){\partial }_{\nu }-m]\psi .\end{eqnarray} \tag{ 52 }$$

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 ${\hat{M}}_{\epsilon }(t;\theta )$ of the form

$$\begin{eqnarray}&&{\hat{M}}_{\epsilon }(t;\theta )={\mathbb{1}}\displaystyle \frac{\mathrm{cos}\theta }{\sqrt{\mathrm{cos}2\theta }}-{\sigma }_{y}\displaystyle \frac{\mathrm{sin}\theta }{\sqrt{\mathrm{cos}2\theta }}+\displaystyle \frac{\sqrt{\mathrm{cos}2\theta }}{\sqrt{| \mathrm{cos}2\theta | }}({H}_{m}+{H}_{{\rm{h}}}+{\hat{H}}_{\mathrm{int}}).\end{eqnarray} \tag{ 53 }$$

The parametrization of the metric has been transferred to the tensor contraction across the discrete graph, and we can smoothly transform from $\theta =0$ to $\theta =\pi /2$ to obtain, in the continuum limit, the rotation-invariant Euclidean action, and the cMPS class

$$\begin{eqnarray}&&| \chi (m,R)\rangle =\int {\mathcal{D}}\psi {\mathcal{D}}{\psi }^{\dagger }\mathrm{exp}[-{S}_{E}[\psi ,{\psi }^{\dagger }]]| {\scr{\Phi}} \rangle ,\end{eqnarray} \tag{ 54 }$$

where ${S}_{E}=\int {{\rm{d}}}^{2}x{\psi }^{\dagger }({\rm{i}}{\gamma }_{5}^{E})({\rm{i}}/{\partial }^{E}-m)\psi$ is the rotation-invariant action with ${\gamma }_{5}^{E}=-{\rm{i}}{\gamma }_{0}$, and the flavor indices are implicit, and are the same as in equation (50).

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 $\psi (x)$ and its conjugate $\bar{\psi }(x)$ are taken independent of each other, and instead of working with the full metric ${\eta }^{\mu \nu }$ one can analytically continue vielbeins ${{\rm{e}}}_{\mu }^{m}\to {{\rm{e}}}_{\mu }^{m}(\theta )={{\rm{e}}}^{{\rm{i}}\theta }{\delta }_{\mu }^{m}$ and use these to construct a Dirac action in the way one would for curved spacetimes [38].

The two-dimensional field state $| \chi \rangle$ given by (54) 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 $\partial A$ as the set of points in A within a distance of points not in A, and $| \partial A|$ as the number of points in this set. For any pure quantum state $| {\psi }_{{AB}}\rangle$ there exists a unique measure of entanglement, namely the von Neumann entropy of entanglement, or simply the entanglement entropy, defined as ${S}_{A}=-\mathrm{Tr}[{\rho }_{A}\mathrm{log}{\rho }_{A}]$ where ${\rho }_{A}$ 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 $\partial A$ are entangled with the region B and so the entanglement entropy is upper bounded as ${S}_{A}\leqslant c| \partial A|$ 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 $| \partial A|$. Since the number of points in $\partial A$ will scale linearly in $1/\epsilon$, we then deduce that ${S}_{A}\leqslant c| \partial A| /\epsilon$ and so it is clear that the resultant field state also obeys an area law.

For any fixed diagonal $v={v}_{0}$ we can split the tensor contraction ${C}[\cdots ]$ into two parts and write the result as an inner product ${\mathcal{C}}[\cdots ]=\langle {\omega }^{L}({v}_{0})| {\omega }^{R}({v}_{0})\rangle (B)$

Where $\langle {\omega }^{L}({v}_{0})|$ denotes a 'forward pointing' tensor and $| {\omega }^{R}({v}_{0})\rangle$ denotes a 'backward pointing' tensor (see figure A.3). Furthermore, we have that

$$\begin{eqnarray}&&| {\omega }^{R}({v}_{0})\rangle \in {{\mathcal{H}}}_{\mathrm{aux}}({v}_{0}),\\ &&\langle {\omega }^{L}({v}_{0})| \in {{\mathcal{H}}}_{\mathrm{aux}}^{*}({v}_{0}).\end{eqnarray} \tag{ A.4 }$$

We capture the dependence of ${\mathcal{C}}[\cdots ]$ on ${A}_{({ijkl})}^{r}$ by writing $| {\omega }^{R}({v}_{0})\rangle =U[{\boldsymbol{r}}({v}_{0})]| {\omega }^{R}({v}_{1})\rangle$. Here we have advanced the cut from $v={v}_{0}$ to $v={v}_{0}+\epsilon /2$ and have the new 'frontier' vector $| {\omega }^{R}({v}_{1})\rangle \in {{\mathcal{H}}}_{\mathrm{aux}}({v}_{0}+\epsilon /2)$ and a transition operator

$$\begin{eqnarray}&&U[{\boldsymbol{r}}({v}_{0})]:{{\mathcal{H}}}_{\mathrm{aux}}({v}_{0}+\epsilon /2)\to {{\mathcal{H}}}_{\mathrm{aux}}({v}_{0}),\end{eqnarray} \tag{ A.5 }$$

being a linear operator that describes the advancing contraction cut (or equivalently the infinitesimal evolution of the state $| {\omega }^{R}({v}_{1})\rangle$), and ${\boldsymbol{r}}({v}_{0})=({r}_{1}({v}_{0}),{r}_{2}({v}_{0}),...)$ denotes a list of physical indices along the $v={v}_{0}$ timeslice.

The operator $U[{\boldsymbol{r}}({v}_{0})]$ is a string of tensor terms built from an operator $\hat{M}$ (different from the $\hat{M}$ defined in the paper). In particular

$$\begin{eqnarray}&&U[{\boldsymbol{r}}({v}_{0})]=[\hat{M}[{r}_{1}({v}_{0})]\otimes \hat{M}[{r}_{2}({v}_{0})]\otimes \hat{M}[{r}_{3}({v}_{0})]\cdots ],\end{eqnarray} \tag{ A.6 }$$

where crucially, $\hat{M}[{r}_{u}({v}_{0})]$ is a mapping between the Hilbert spaces

$$\begin{eqnarray*}&&\hat{M}[{r}_{u}({v}_{0})]:{{\mathcal{H}}}_{a}(u+\epsilon /2,{v}_{0}+\epsilon /2)\otimes {{\mathcal{H}}}_{b}(u-\epsilon /2,{v}_{0}+\epsilon /2)\to {{\mathcal{H}}}_{a}(u,{v}_{0})\otimes {{\mathcal{H}}}_{b}(u,{v}_{0}),\end{eqnarray*}$$

and is given in terms of the PEPS tensors as

$$\begin{eqnarray}&&\hat{M}[{r}_{u}({v}_{0})]={\mathbb{1}}+{A}_{({ijkl})}^{{r}_{u}({v}_{0})}| {\rm{i}}j\rangle \langle {kl}| .\end{eqnarray} \tag{ A.7 }$$

We may iterate this expansion of $| {\omega }^{R}({v}_{0})\rangle$ to obtain the expression

$$\begin{eqnarray*}&&{\mathcal{C}}[\cdots ]=\langle {\omega }^{L}({v}_{0})| U[{\boldsymbol{r}}({v}_{0})]U[{\boldsymbol{r}}({v}_{1})]U[{\boldsymbol{r}}({v}_{2})]U[{\boldsymbol{r}}({v}_{2})]\cdots U[{\boldsymbol{r}}({v}_{N})]| {\omega }^{R}({v}_{N})\rangle ,\end{eqnarray*}$$

where ${v}_{n}={v}_{0}+n\epsilon /2$.

We assume that at each point (u, v) on the square lattice we have the index state space ${{\mathcal{H}}}_{a}(u,v)\otimes {{\mathcal{H}}}_{b}(u,v)$, and the resolution of the identity operator for this space ${{\mathbb{1}}}_{(u,v)}$ given by

$$\begin{eqnarray}&&{{\mathbb{1}}}_{(u,v)}=\displaystyle \frac{1}{{\pi }^{2D}}\int \underset{k,s}{\displaystyle \prod }{{\rm{d}}}^{2}{\phi }_{k,s}\;| {\otimes }_{k,s}{\phi }_{k,s}(u,v)\rangle \langle {\otimes }_{k,s}{\phi }_{k,s}(u,v)| ,\end{eqnarray} \tag{ A.8 }$$

where $| {\phi }_{k,s}\rangle =\mathrm{exp}[{\phi }_{k,s}{\hat{a}}_{k,s}^{\dagger }-{\phi }_{k,s}^{*}{\hat{a}}_{k,s}]| {{\scr{\Omega}} }_{{\mathcal{B}}}\rangle =D({\phi }_{k,s})| {{\scr{\Omega}} }_{B}\rangle$. We also have the identity operator for ${{\mathcal{H}}}_{\mathrm{aux}}(v)$ as ${{\mathbb{1}}}_{(v)}={\otimes }_{u}{{\mathbb{1}}}_{(u,v)}$. Here, we do not consider a single tensor product of Hilbert spaces at every point on the lattice, but instead view things 'dynamically' as a tensor product defined along a line $v\;=$ constant, and so operators like $U({\boldsymbol{r}}(v))$ and ${{\mathbb{1}}}_{v}$ are linear mappings from ${{\mathcal{H}}}_{\mathrm{aux}}(v)$ to ${{\mathcal{H}}}_{\mathrm{aux}}(v+\epsilon /2)$.

To construct the path integral, we begin with

$$\begin{eqnarray*}&&{\mathcal{C}}[\cdots ]=\langle {\omega }^{L}({v}_{0})| U[{\boldsymbol{r}}({v}_{0})]U[{\boldsymbol{r}}({v}_{1})]U[{\boldsymbol{r}}({v}_{2})]U[{\boldsymbol{r}}({v}_{2})]\cdots U[{\boldsymbol{r}}({v}_{N})]| {\omega }^{R}({v}_{N})\rangle \end{eqnarray*}$$

and as before, we insert a resolution of the identity either side of $U[r(v)]$. That is, we consider ${{\mathbb{1}}}_{(v+\epsilon /2)}U[{\boldsymbol{r}}(v)]{{\mathbb{1}}}_{(v)}$. This provides us with amplitudes

$$\begin{eqnarray*}&&\langle {\otimes }_{i}{\phi }_{i,a}(u,v)| \otimes \langle {\otimes }_{j}{\phi }_{j,b}(u,v)| M[{r}_{u}(v)]| {\otimes }_{k}\;{\phi }_{k,a}u+\epsilon /2,v+\epsilon /2)\rangle | {\otimes }_{l}{\phi }_{l,b}(u-\epsilon /2,v+\epsilon /2)\rangle .\end{eqnarray*}$$

The central amplitude is then

$$\begin{eqnarray*}&&\langle {\otimes }_{k}\;\;{\phi }_{k,a}(u,v)| {\otimes }_{k}\;{\phi }_{k,a}(u+\epsilon /2,v+\epsilon /2)\rangle \langle {\otimes }_{j}\;{\phi }_{j,b}(u,v)| {\otimes }_{j}{\phi }_{j,b}(u-\epsilon /2,v+\epsilon /2)\rangle .\\ &&\qquad +{A}_{({ik};{jl})}^{{r}_{u}(v)}\langle {\otimes }_{i}{\phi }_{i,a}(u,v)| {1}_{i,a}\rangle \langle {\otimes }_{j}{\phi }_{j,b}(u,v)| {1}_{j,b}\rangle \\ &&\qquad \times \langle {1}_{k,a}| \otimes {\phi }_{k,a}(u+\epsilon /2,v+\epsilon /2)\rangle \langle {1}_{l,b}| \otimes {\phi }_{l,b}(u-\epsilon /2,v+\epsilon /2)\rangle .\end{eqnarray*}$$

From the formula for the overlap of coherent states, the identity term gives the amplitude

$$\begin{eqnarray*}&&\mathrm{exp}[-\epsilon /4({\phi }_{k,a}({\partial }_{u}+{\partial }_{v}){\phi }_{k,a}^{*}+{\phi }_{k,b}(-{\partial }_{u}+{\partial }_{v}){\phi }_{k,b}^{*}-({\phi }_{k,a}^{*}({\partial }_{u}+{\partial }_{v}){\phi }_{k,a}+{\phi }_{k,b}^{*}(-{\partial }_{u}+{\partial }_{v}){\phi }_{k,b}))]\\ &&\quad +{A}_{({ik};{jl})}^{{r}_{u}(v)}\langle {\phi }_{i,a}(u,v)| {1}_{i,a}\rangle \langle {\phi }_{j,b}(u,v)| {1}_{j,b}\rangle \langle {1}_{k,a}| {\phi }_{k,a}(u+\epsilon /2,v+\epsilon /2)\rangle \langle {1}_{l,b}| {\phi }_{l,b}(u-\epsilon /2,v+\epsilon /2)\rangle \end{eqnarray*}$$

which can be expressed more compactly by defining ${{\scr{\Psi}} }_{k}:= ({\phi }_{k,a},{\phi }_{k,b})$, to give

$$\begin{eqnarray*}&&\mathrm{exp}[-\displaystyle \frac{\epsilon }{4}({({{\scr{\Psi}} }_{k})}^{\dagger }({\sigma }_{z}{\partial }_{u}+{\mathbb{1}}{\partial }_{v}){{\scr{\Psi}} }_{k}^{*}-{({{\scr{\Psi}} }_{k})}^{\dagger }({\sigma }_{z}{\partial }_{u}+{\mathbb{1}}{\partial }_{v}){{\scr{\Psi}} }_{k})]\\ &&\quad +{A}_{({ik};{jl})}^{{r}_{u}(v)}\langle {\phi }_{i,a}(u,v)| {1}_{i,a}\rangle \langle {\phi }_{j,b}(u,v)| {1}_{j,b}\rangle \langle {1}_{k,a}| {\phi }_{k,a}(u+\epsilon /2,v+\epsilon /2)\rangle \langle {1}_{l,b}| {\phi }_{l,b}(u-\epsilon /2,v+\epsilon /2)\rangle \end{eqnarray*}$$

Note that transforming to the original coordinates, $x=u+v$ and $y=v-u$ we then have the first term contributing the amplitude

$$\begin{eqnarray}&&\mathrm{exp}[{\rm{i}}\delta S[{\scr{\Psi}} ,{{\scr{\Psi}} }^{*}]]\equiv \mathrm{exp}[-\displaystyle \frac{\epsilon }{4}({{\scr{\Psi}} }_{k}^{T}\nabla \cdot {{\scr{\Psi}} }_{k}^{*}-{{\scr{\Psi}} }_{k}^{\dagger }\nabla \cdot {{\scr{\Psi}} }_{k})],\end{eqnarray} \tag{ A.9 }$$

defining an action S in either the (u, v) coordinates or in the original (x, y) coordinate system. It is clear that this action is not rotationally invariant. This is not particular to PEPS states, but would generically be expected from a continuum construction based on an underlying regular lattice.

The above term is for the identity contribution in U. A particular TNS on the square lattice is described by $\{{A}_{{ijkl}}^{r}\}$, and so by following the above prescription, we can create the appropriate operators within the one-particle sector of the Fock space.

A simple and natural candidate is to take $\hat{M}$ of the form $\hat{M}={\mathbb{I}}+\frac{\epsilon }{2}({Q}_{a}^{{ij}}(u,v){\hat{a}}_{i}^{\dagger }{\hat{b}}_{j}\otimes {{\mathbb{I}}}_{b}+{{\mathbb{I}}}_{a}\otimes {Q}_{b}^{{ij}}(u,v){\hat{b}}_{i}^{\dagger }{\hat{b}}_{j})$, where for example ${a}_{j},{b}_{k}$ are single particle annihilation operators for mode-j in ${{\mathcal{H}}}_{a}$ and mode-k in ${{\mathcal{H}}}_{b}$, and ${\{{Q}_{\alpha }^{{ij}}(u,v)\}}_{\alpha ,i,j}$ is a set of functions defined over the lattice points. The interpretation of this Q-term is that it generates scattering of particles within the internal bond degrees of freedom, without any scattering in the spatial degrees of freedom.

Then for example, if we assume ${Q}_{\alpha }^{{ij}}=Q(u,v)$ we automatically obtain the term

$$\begin{eqnarray}&&\displaystyle \frac{\epsilon }{2}Q(u,v)\underset{i}{\sum }({| {\phi }_{a,i}| }^{2}+{| {\phi }_{b,i}| }^{2})=\displaystyle \frac{\epsilon }{2}Q(u,v){{\scr{\Psi}} }^{\dagger }{\scr{\Psi}} ,\end{eqnarray} \tag{ A.10 }$$

in the auxiliary system action after the above expansion in terms of coherent field states.

Of course other options exist for the local contractions. For example, we can consider mixing the spatial modes with a term such as

$$\begin{eqnarray}\hat{M}(u,v) & = & {\mathbb{1}}+\epsilon {\hat{R}}^{{IJ}}{\hat{a}}_{I}^{\dagger }{\hat{a}}_{J},\\ & = & {\mathbb{1}}+\epsilon {\hat{R}}^{({i}_{a},{i}_{b})({j}_{a}{j}_{b})}({\hat{a}}_{{i}_{a}}^{\dagger }\otimes {\hat{b}}_{{i}_{b}}^{\dagger })({\hat{a}}_{{j}_{a}}\otimes {\hat{b}}_{{j}_{b}}),\end{eqnarray} \tag{ A.11 }$$

where ${\hat{R}}^{{IJ}}(u,v)$ can be taken to be a creation operator for particles in the physical field state. Clearly the addition of such a term generates quartic terms in the action of $({\phi }_{a},{\phi }_{b})$.

The above analysis provides us with an amplitude contribution along a spatial slice, however to then sum over the auxiliary time and obtain the full amplitude requires a re-scaling of the field. The result of this is to yield an action

$$\begin{eqnarray}&&S=\int {\rm{d}}x{\rm{d}}y\displaystyle \frac{1}{2}({{\scr{\Psi}} }_{k}^{\dagger }\nabla \cdot {{\scr{\Psi}} }_{k}-Q(x,y){{\scr{\Psi}} }^{\dagger }{\scr{\Psi}} ).\end{eqnarray} \tag{ A.12 }$$

We have shown the explicit details of this rescaling in section (5.6), but there for the case of the rotationally invariant action.