Continuous Tensor Network States for Quantum Fields

PHYSICAL REVIEW X 9, 021040 (2019) Featured in Physics

† Antoine Tilloy* and J. Ignacio Cirac Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany

(Received 3 August 2018; revised manuscript received 12 February 2019; published 28 May 2019)

We introduce a new class of states for bosonic quantum fields which extend tensor network states to the continuum and generalize continuous matrix product states to spatial dimensions d ≥ 2.By construction, they are Euclidean invariant and are genuine continuum limits of discrete tensor network states. Admitting both a functional integral and an operator representation, they share the important properties of their discrete counterparts: expressiveness, invariance under gauge transformations, simple rescaling flow, and compact expressions for the N-point functions of local observables. While we discuss mostly the continuous tensor network states extending projected entangled-pair states, we propose a generalization bearing similarities with the continuum multiscale entanglement renormalization ansatz.

DOI: 10.1103/PhysRevX.9.021040 Subject Areas: Particles and Fields, Quantum Physics, Strongly Correlated Materials

I. INTRODUCTION and have helped describe and classify their physical properties. By design, their entanglement obeys the area law Tensor network states (TNSs) provide an efficient para- [17–19], which is a fundamental property of low-energy metrization of physically relevant many-body wave func- states of systems with local interactions. They enable a tions on the lattice [1,2]. Obtained from a contraction of succinct classification of symmetry-protected [20–23] and low-rank tensors on so-called virtual indices, they eco- topological phases of matter [24,25]. TNSs also have a built- nomically approximate the states of systems with local in bulk-boundary correspondence [26], which makes close interactions in thermal equilibrium. Their number of connections to physical phenomena appearing in exotic parameters scales only polynomially with the lattice size materials [27,28]. Finally, they can be used to build toy [3,4], circumventing the exponential growth of the Hilbert models illustrating the holographic principle and the cel- space dimension. TNSs have led to powerful numerical ebrated AdS/CFT correspondence [29–31]. methods to compute the physical properties of complex – 1 For regular spin lattices, PEPSs assign a tensor to each system [5 7], most notably in one spatial dimension d ¼ , lattice site, with 2z virtual and one physical (spin) indices, where matrix product states (MPSs) [8], the simplest where z is the coordination number. The virtual indices are incarnation of TNSs, are at the basis of what is arguably contracted according to the lattice geometry, yielding a the most successful method to describe strongly correlated – ≥ 2 wave function for the spin degrees of freedom. This systems [9 11]. In higher dimensions d , accurate description is particularly useful in translationally invariant results [12] have also been obtained using projected systems, as this symmetry may simply be imposed by entangled-pair states (PEPSs) [13], a natural generalization choosing the same tensor on each site. For MERA [14,32], of MPSs. Another family of TNSs, the multiscale renorm- a treelike structure of two types of tensors is used. In both alization ansatz (MERA) [14], has proved well suited to cases, the whole many-body wave function is determined describe scale-invariant states [15,16] appearing in critical by one or a few tensors, which encode all the physical phenomena. properties. Beyond numerical computations, TNSs provide impor- An important challenge in the theory of TNS is the tant insights into the nature of many-body quantum systems generalization from lattice to continuous systems. Such an extension would allow the direct study of quantum field *[email protected] theories, without the need for a prior breaking of spatial † [email protected] symmetries with a discretization. Furthermore, the continuum provides a whole range of exact and approximate Published by the American Physical Society under the terms of analytic techniques (such as exact Gaussian functional the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to integrals, saddle-point approximations, or diagrammatic the author(s) and the published article’s title, journal citation, expansions) that have no obvious discrete counterparts and and DOI. that could provide useful additions to the TNS toolbox.

2160-3308=19=9(2)=021040(17) 021040-1 Published by the American Physical Society ANTOINE TILLOY and J. IGNACIO CIRAC PHYS. REV. X 9, 021040 (2019)

A natural way to carry out such a program is to simply A. Functional integral representation take the continuum limit of a TNS, by letting the lattice 1. State definition spacing tend to zero while appropriately rescaling the We begin with the functional integral representation. It tensors. In fact, this task has been carried in one spatial will be the most direct to derive from the discrete and dimension, d ¼ 1, where it yields continuous matrix makes Euclidean symmetries manifest. product states (CMPSs) [33,34]. In higher dimensions, Definition 1 (functional integral formulation).—A CTNS however, the situation does not seem trivial. Naive exten- of a bosonic quantum field on a domain Ω ⊂ Rd with sions of CMPSs have a preferred spatial direction and break boundary ∂Ω is a state jV;B;αi parametrized by two Euclidean symmetries [35]. In Ref. [35], a proposal for functions V and α: RD → C and a boundary functional B: CPEPS was put forward to overcome such a limitation, L2 ∂Ω → C defined by the functional integral on an but the resulting state was no longer obtained from the ð Þ auxiliary D-component field ϕ: continuum limit of a TNS. Thus, so far there seems to be no fully satisfactory way of extending TNSs to the continuum Z Z 1 XD in d ≥ 2. d 2 jV;B;αi¼ DϕBðϕj∂ΩÞ exp − d x ½∇ϕkðxÞ In this article, we propose a definition of continuous Ω 2 k¼1 tensor network states (CTNSs) that naturally extends TNSs to the continuum. We obtain them as a genuine continuum þ V½ϕðxÞ − α½ϕðxÞψ †ðxÞ j0i; ð1Þ limit of TNSs but manage to preserve Euclidean invariance. As in previous works [35–37], we exploit the similarity between a tensor contraction over the indices lying on the where j0i is the physical Fock vacuum state, ½ψðxÞ; ψ † δd − ϕ ϕ D α links of a tensor network and a functional integral over a ðyÞ ¼ ðx yÞ, and ¼½ kk¼1. The functions field living on the continuum limit of this mesh. The key and V may depend explicitly on the position. difference lies in the way the continuum limit is taken in The auxiliary D-component field ϕ, which is integrated higher dimensions: As we argue, the d ¼ 1 case of CMPSs over in the functional integral, is the continuous equivalent is too peculiar to be directly extended. of the auxiliary bond indices that are contracted in tensor The first definition of CTNS we propose in Sec. II takes the network states (see Fig. 1). This intuition is made more form of a functional integral over auxiliary scalar fields precise in the next section. For this reason, we call D the as advertised. From this definition, which makes local bond-field dimension. As we see in Sec. IV, the bond-field Euclidean invariance manifest, we derive an operator repre- dimension D bears similarities with the bond dimension χ sentation similar to the one used for CMPSs. Importantly, we of discrete tensor network states. show in Sec. III how this ansatz can be obtained from a If Ω is Rd or a torus with periodic boundary conditions, continuum limit of a discrete TNS. We then study some of its B can simply be set to 1. In that case, the state and its properties reminiscent of the discrete: its ability to approxi- associated properties depend only on V and α, in the same mate (possibly inefficiently) all states (Sec. IVA), its way a TNS depends only on local tensors. If V and α do not redundancy under some so-called gauge transformations depend explicitly on x, the CTNS describes a translation- (Sec. IV B), which play a crucial role for PEPSs, its flow ally invariant state. More generally, if ∂Ω ≠∅, the boun- under scaling transformations (Sec. IV C), and its CMPS dary functional could induce, e.g., approximation in some carefully chosen limit (Sec. IV D). (1) Dirichlet boundary conditions.—Bðϕj∂ΩÞ ∼ δðϕj∂ΩÞ We then propose various methods to carry computations with fixing ϕj∂Ω ¼ 0 in the functional integral; Gaussian and non-Gaussian CTNSs (Sec. V). While most of (2) Neumann boundary conditions.—Bðϕj∂ΩÞ∼δð∇ϕ·nÞ our approach is aimed at the continuum limit of PEPSs, we fixing ∇ϕ · nj∂Ω ¼ 0, where n is normal to ∂Ω;or finally generalize it to MERA-like states (and more exotic TNSs) in arbitrary dimensions by including a metric and restricting physical fields to a boundary (Sec. VI).

II. CONTINUOUS TENSOR NETWORK STATES We start by giving two equivalent definitions of continuous tensor network states, leveraging a functional integral and an operator representation. Our objective at FIG. 1. Functional integral representation. In the discrete (left), this stage is only to provide a definition of a class of states a tensor network state is obtained from a contraction of auxiliary for bosonic quantum fields, with only a crude intuition for indices connecting the elementary tensors with each other and why such an object could indeed be a good definition of a with a boundary tensor. In the continuum (right), the contraction CTNS. We forgo the derivation of this CTNS from a class is replaced by a functional integral (1), the auxiliary indices by of discrete tensor networks to the following section. fields ϕ, and the boundary tensor by a boundary functional B.

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(3) something more general, given, e.g., by a quasilocal 3. Correlation functions functional: A state is also fully characterized by its (equal time) I correlation functions. To compute them, we first introduce d−1 j0 j Bðϕj∂ΩÞ¼exp − d xL½ϕðxÞ;∇ϕðxÞ ; ð2Þ the generating functionals for real sources , : ∂Ω R R 0 † hV;B;αjexpð Ω j · ψ Þexpð Ω j ·ψÞjV;B;αi d 1 D Z where L is a function from Rð þ Þ to C. This latter j0;j ¼ ; hV;B;αjV;B;αi option is generated naturally when we discuss gauge R R α ψ 0 ψ † α invariance in Sec. IV B. ˜ hV;B; jexpð Ω j · Þexpð Ω j · ÞjV;B; i Z 0 ¼ : ð8Þ We rewrite expression (1) more explicitly as a sum over j ;j hV;B;αjV;B;αi unnormalized field-coherent states. Introducing the massless free field probability measure dμðϕÞ for the auxiliary They generate the normal-ordered and anti-normal-ordered field: correlation functions, respectively. For example, it is straightforward to verify that Z 1 XD dμðϕÞ¼Dϕ exp − ddx ½∇ϕ ðxÞ2 ð3Þ α ψ † ψ α 2 k † hV;B; j ðxÞ ðyÞjV;B; i Ω 1 hψ ðxÞψðyÞi ≔ ð9Þ k¼ hV;B;αjV;B;αi A ϕ and a complex amplitude Vð Þ: δ δ ¼ Z 0 : ð10Þ Z δ 0 δ j ;j j ðxÞ jðyÞ j;j0¼0 d AVðϕÞ¼Bðϕj∂ΩÞ exp − d xV½ϕðxÞ ð4Þ Ω Using the formula for the overlap of (unnormalized) field- coherent states, yields Z Z hβjαi¼exp dxβðxÞαðxÞ ð11Þ jV;B;αi¼ dμðϕÞAVðϕÞjαðϕÞi; ð5Þ Ω R d † and writing N ¼hV;B;αjV;B;αi, we get where jαðϕÞi ¼ exp f Ω d xα½ϕðxÞψ ðxÞgj0i is an unnor- Z malized field-coherent state. Hence, just like CMPSs in 1 ˜ 0 0 dimension 1, CTNSs are a generalization of field-coherent Z 0 μ ϕ μ ϕ ϕ ϕ j ;j ¼N d ð Þd ð ÞBð ÞB ð Þ states. The latter are obtained, e.g., if AVðϕÞ is only non- Z zero for a given ϕ (for an infinitely deep V) or, in the ×exp − V½ϕ0þV½ϕ−ðα½ϕ0þjÞðα½ϕþj0Þ : homogeneous case, if α is constant. Ω ð12Þ 2. N-particle wave function α A generic state jΨi in the bosonic Fock space We observe an important fact, which is that the function α F½L2ðRd; CÞ can be expanded into a sum of n particle appears squared; hence, an quadratic in the field already brings non-Gaussianities. To compute Zj0;j, one applies the wave functions φn: Baker-Campbell-Hausdorff (BCH) formula to Eq. (8) to Z push annihilation operators to the right and get back to a Xþ∞ φ … nðx1; ;xnÞ † † computation of field-coherent state overlaps. We obtain jΨi¼ dx1…dxn ψ ðx1Þ…ψ ðxnÞj0i; Ωn n! n¼0 Z 1 0 0 ð6Þ Z 0 μ ϕ μ ϕ ϕ ϕ j ;j ¼ N d ð Þd ð ÞBð ÞB ð Þ Z φ where n is a completely symmetric function of its × exp − V½ϕ0ðxÞ þ V½ϕ − α½ϕ0 · α½ϕ coordinates. Simply expanding the exponential of Ω Eq. (1) gives, for the CTNS jV;B;αi, − jα½ϕ − j0 · α½ϕ0 ; ð13Þ Z

φnðx1; …;xnÞ¼ dμðϕÞAVðϕÞα½ϕðx1Þ…α½ϕðxnÞ: ð7Þ hence the same as Eq. (12) but for the removal of the product j · j0, which is responsible for the divergent equal It provides an equivalent definition of the CTNS. point contributions upon functional differentiation.

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B. Operator representation identity in the field basis jϕi at every time step Δτ and 1. State definition writes each resulting overlap in the conjugate momentum basis jπi. Going back to the continuum limit yields a We now provide an equivalent operator representation phase space functional integral which reduces to the of CTNSs. For simplicity, we restrict ourselves to domains formula of Eq. (1) upon Gaussian integration of the Rd Ω of that can be split into Cartesian products ¼ conjugate momenta π. − 2 2 ∂ ∅ ½ T= ;T= × S, where S ¼ . This restriction is not The boundary operator of Eq. (14) is related to the strictly necessary but substantially simplifies the definition. boundary functional of Eq. (1) by Definition 2 (alternative operator formulation).—For a Ω ϕ ϕ ˆ ϕ domain that can be written as a Cartesian product Bð Þ¼h injBj outi; ð15Þ Ω ¼½−T=2;T=2 × S, we write x ¼ðτ; xÞ, where τ ∈ ϕ ϕ ½−T=2;T=2 and x ∈ S. A CTNS is then defined as where j outi and j ini are eigenstates of the auxiliary field operators ϕˆ ðxÞ. The auxiliary field ϕ decomposes into ϕ ϕ ϕ ϕ ϕ jV;B;αi ¼ in þ out, where in (respectively, out) has support on ∂Ω (respectively, ∂Ω ) with ∂Ω ∂Ω ∪ ∂Ω . T=Z 2 Z in out ¼ in out XD πˆ x 2 ∇ϕˆ x 2 As before, we propose a repackaging of formula (14). ˆ T − τ x ½ kð Þ ½ kð Þ ¼ tr B exp d d þ Introducing the Hamiltonian density HðxÞ¼H0ðxÞþ S 2 2 P k¼1 ˆ D 2 ˆ 2 −T=2 V ϕ x H0 x ( πˆ x ∇ϕ x )=2 ½ ð Þ with ð Þ¼ k¼1f ½ kð Þ þ½ kð Þ g yields ˆ ˆ † þ V½ϕðxÞ − α½ϕðxÞψ ðτ; xÞ j0i; ð14Þ Z Z T=2 jV;B;αi¼tr Bˆ T exp − dτ dxHðxÞ ˆ −T=2 S where T is the τ-ordering operator and ϕkðxÞ and πˆ kðxÞ are k-independent canonically conjugated pairs of (auxiliary) − α½ϕˆ ðxÞψ †ðτ; xÞ j0i: ð16Þ ϕˆ x ϕˆ y 0 πˆ x πˆ y 0 field operators: ½ kð Þ; lð Þ ¼ , ½ ð Þk; lð Þ ¼ , and ϕˆ x πˆ y δ δd−1 x − y ½ kð Þ; lð Þ ¼ i k;l ð Þ. These operators act on This equation is a straightforward extension of the CMPS H F 2 D aux ¼ ½L ðSÞ , i.e., D copies of a bosonic Fock space definition [33] (recalled in Sec. IV D) with Qˆ ∼−HðxÞ on a d − 1-dimensional space. The trace is taken over this and Rˆ ∼ α½ϕˆ ðxÞ. auxiliary Hilbert space. As before, V and α may depend on x and τ. ˆ H 2. N-particle wave function The operator B acts on aux and fixes the boundary ˆ φ conditions; e.g., B ¼ 1 encodes periodic boundary con- The N-particle wave function n defined in Eq. (6) can ditions on the coordinate τ. Another natural option is to take also be computed in the operator representation. For ˆ −T=2 < τ1 < < τn

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Exploiting the operator definition of the CTNS (14) and where A−1 and A1 are two χ × χ matrices. These matrices expanding the τ-ordered exponential into an infinite prod- contain the parameters which allow one to vary the state. uct of infinitesimal exponentials, we get Their size χ, called the bond dimension, encodes the depth Z of the variational class and upper bounds the amount of T=2 ˜ spatial entanglement that can be carried by the state. Zj0;j ¼ tr B ⊗ B T exp T j0j ð18Þ −T=2 The two matrices can be be collected into a 3-index k;l k;l¼1…χ tensor ½A 1 written graphically: with the transfer matrix (with sources) i i¼ Z ð24Þ ˆ 0 ˆ T j0j ¼ −H ⊗ 1 − 1 ⊗ H þðα½ϕþj Þ ⊗ ðα½ϕ þ jÞ: S ð19Þ where i is the physical index and k and l are so-called bond indices. Graphically, the corresponding wave function c Using as before the BCH formula yields can be written Z Z T=2 0 Zj0j ¼ tr B ⊗ B T exp T j0j − j · j : ð25Þ −T=2 S ð20Þ where joint legs of the tensor A denote a summation on the The functional derivatives can then be carried explicitly, corresponding index, associated with the matrix multipli- and one obtains e.g., for −T=2 < τ2 < τ1

III. LINK WITH DISCRETE B. Constructing CTNSs TENSOR NETWORK STATES Our objective is to show how CTNSs can be obtained A. (Discrete) tensor network states from a limit of a discrete TNS. To motivate this discrete ansatz, we first provide heuristics for why its main We start with a very brief reminder on TNSs, recalling characteristics seem to be required. Mainly, we aim to only their elementary definition. For an understanding of justify (i) why an infinite bond dimension is needed and their efficiency in representing quantum systems of physi- (ii) why the trivial tensor around which we expand is of the cal interest, we direct the reader to the relevant literature form we postulate. (e.g., Refs. [42,43], and references therein). The first point is a scaling argument. We ask for a strong TNSs are variational ansatz for many-body wave func- notion of continuum limit: We require the discrete tensor to tions that take the form of a contraction of local tensors. be approximately stable by fine graining to the UV. 1 The simplest example, in spatial dimension d ¼ ,is Namely, the discrete ansatz needs to be (at least approx- provided by MPSs. For a translation-invariant quantum imately) expressible as a contraction of tensors with the 1 2 spin- = chain with N sites, a generic state reads same form but different parameters. Each blocking multi- X plies the physical dimension by 2d, which is why in the ψ c … i1 ⊗ ⊗ i ; 22 j i¼ i1; ;iN j i j Ni ð Þ continuum limit one obtains a field theory on the physical … −1 1 N i1; ;iN ¼f ; g degrees of freedom. But each blocking also multiplies the 2d−1 1 N bond dimension by (see Fig. 3). Hence, for d> , the where the wave function c … contains 2 complex i1; ;iN bond dimension is increased when zooming out and parameters. A MPS is an economical ansatz for this wave decreased when zooming in. The only way to make the function: class of states considered approximately stable is for the bond dimension to be infinite. Notice in this respect that c … ¼ tr½A …A ; ð23Þ i1; ;in i1 iN the d ¼ 1 case allows finite bond dimensions even in the

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To obtain a continuum limit, the crucial choice lies in the elementary “trivial tensor,” acting as the identity on the vacuum, and around which to expand:

Tˆ ¼ Tˆ ð0Þ þ εd × corrections: ð30Þ

Indeed, it is natural to want the tensor corresponding to zero particle in an elementary cell of the physical space to dominate. It seems that any other choice would preclude the existence of a continuum limit. In d ¼ 1 dimension, there is only one natural option, which is to take the tensor corresponding to the identity:

ð31Þ

But, in the same way as for the bond dimension, the situation is a little too trivial in d ¼ 1 to give a precise hint FIG. 3. Tensor blocking. In d ¼ 1, blocking does not increase for higher dimensions. In d>1, there are several seem- the bond dimension. In d ¼ 2, going from the UV to the IR ingly natural options which we have to inspect. We discuss doubles the bond dimension at each blocking. Hence, flowing the the d ¼ 2 case, but the reasoning holds for any d ≥ 2.We other way, from IR to UV, one reaches a trivial bond dimension do not aim to prove that the tensor we ultimately expand after a finite number of iterations unless the initial bond dimension is infinite. around is the only option but rather that other seemingly simpler options bring difficulties. (1) A naive option is to generalize the identity on the continuum limit [33,34]. It will be important to see if this ˆ ð0Þ δ auxiliary bond space by taking Tijkl ¼ ijkl, that is, peculiarity can be recovered in some appropriate limit in to take an elementary tensor corresponding to a Sec. IV D. Greenberger-Horne-Zeilinger (GHZ) state on the Note that our argument in favor of an infinite bond bond indices: dimension does not imply that a discrete tensor network state with a finite bond dimension could not behave, at ð32Þ distances sufficiently large compared to the lattice spacing, like a CTNS. Rather, our argument shows that any simple space discretization of a CTNS in d ≥ 2 into a TNS will As will later be manifest, this choice is too brutal and have an infinite bond dimension, even for an arbitrarily would yield a state with a trivial spatial structure. small lattice spacing. As in Ref. [35], it could also be that a (2) Another simple option is to take the identity along a proper choice of boundary conditions would constrain the diagonal, e.g., tensor contraction on a finite-dimensional subspace, despite an apparent infinite bond dimension. ð33Þ We now need to discuss more precisely the form of the elementary tensor. For notational simplicity, we now This is the choice that stays the closest in spirit assume that an elementary tensor Tˆ to be contracted, with the d ¼ 1 case. The problem of such a choice ð27Þ (made, e.g., in Ref. [35]) is that it picks a preferred direction and thus makes Euclidean invariance impossible to obtain directly (that is, without ana- is a vector in its bond indices but an operator acting on the lytic continuation). vacuum in the physical space, namely, (3) We may combine the 2d−1 identity operator along diagonals in a sum, as an attempt to recover the jphysical statei¼contractionfnetwork of Tˆ gj0ið28Þ Euclidean invariance lost with the previous choice:

ð34Þ

ð29Þ An issue is then that the corresponding tensor contraction contains loops

021040-6 CONTINUOUS TENSOR NETWORK STATES FOR … PHYS. REV. X 9, 021040 (2019) Z Y YD ε α ˆ ϕ e ð35Þ jV; i¼ TðxÞ d k x þ 2 1 x∈lattice k¼1 ε ϕ e 0 × d k x þ 2 2 j i; ð40Þ which yield divergent terms as tr½1¼þ∞ for an infinite bond dimension. e e None of the natural options seem to provide a simple where 1 and 2 are unit vectors along the two lattice Euclidean-invariant continuum limit. Our proposal consists directions and the bond fields are indexed by the points on the links of the lattice where they sit. Writing in taking a regularized version of the first possibility, in the pffiffiffi pffiffiffi 1 2 2 1 − 2 2 form of a “soft” delta: u ¼½ðx þ x Þ= and v ¼½ðx x Þ= , we see that the differences in Eq. (38) yield Z Y 2 2 ð36Þ ½∂ ϕ ðxÞ þ½∂ ϕ ðxÞ Tˆ ðxÞ ≃ exp − d2x u k v k ε→0 2 x∈lattice C. Discrete ansatz in d =2 † þ V½ϕðxÞ − α½ϕðxÞψ ðxÞ : ð41Þ The first lesson from the previous sections is that infinite bond dimensions seem to be required. We thus write the bond indices of the elementary tensor as D real numbers. In We recognize the (rotation-invariant) gradient squared term d ¼ 2, this means that an elementary tensor has four bond of the continuum definition II A 1. Defining the path integral “measure” as indices ϕð1Þ, ϕð2Þ, ϕð3Þ, and ϕð4Þ ∈ RD: Y ε ε Dϕ∶≃lim dϕ x þ e1 dϕ x þ e2 ð42Þ ð37Þ ε→0 k 2 k 2 x∈lattice k¼1…D

As we mentioned before, the heart of the problem of the finally yields the continuous tensor network state of Eq. (1) continuum limit lies in defining the proper trivial tensor up to boundary conditions. To get a state on the physical around which to expand. We choose a soft delta: Hilbert space, the auxiliary fields on the boundary just have to be contracted (or integrated) against a boundary functional, which we write B in Eq. (1).

D. Discrete ansatz in general d For d ≥ 3, the derivation is carried along the same way as before. We just note that there is a small peculiarity in the d ¼ 2 case because the auxiliary fields are adimensional. ð38Þ To generalize Eq. (38) to higher spatial dimensions d>2, one naturally extends the prescription of summing all the This ansatz forces the bond indices to remain close to each differences of the squares of the nearest bond indices other and contributes to the generation of the gradient ϕð1Þ; …ϕð2dÞ. But, importantly, to obtain the continuum −2 squared term in the action. Being Gaussian, we also limit, one needs to multiply this expression by εd , where ε naturally expect its form to be stable. To this “trivial” part, is the length of the unit cell: we add local corrections of the order of εd ¼ ε2: d−2 XD 0 ε Tˆ ð Þ ¼ exp − ð43Þ Tˆ ¼ Tˆ ð0Þ exp ½−ε2VðϕÞ1 þ ε2αðϕÞψ †ðxÞ: ð39Þ ϕð1Þ…ϕð2dÞ 2 k¼1

In this expression, ϕ denotes whatever combination of the to obtain the integral of a gradient squared in the continuum bond-field indices ϕð1Þ, ϕð2Þ, ϕð3Þ, and ϕð4Þ. The simplest limit. In d ¼ 2, the εd of the integration measure and ε−2 possibility is to take ϕ as the average of the bond indices, from the gradient squared cancel each other, and this but it does not matter for the continuum limit. The operator scaling factor does not appear. † ψ ðxÞ anticipates the continuum and has commutation In retrospect, it is clear why deriving the continuum limit † 2 2 relations ½ψðxÞ; ψ ðyÞ ¼ ð1=ε Þδx;y ≃ δ ðx − yÞ. by perturbing around the GHZ tensor (32) gives a trivial Ignoring the boundary conditions for now, the contrac- continuum. It corresponds to putting an infinitely large tion of the tensors amounts to integrate over all the bond constant instead of εd−2 in Eq. (43), an infinite “rigidity” indices: that could not be compensated by locally small terms.

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IV. PROPERTIES Xm α ϕ 1 ϕ ½x; ðxÞ ¼ ½j−1=2;jþ1=2½ ðxÞfjðxÞ; ð48Þ We now explore the properties of CTNSs that are j¼1 analogous to those of their discrete counterparts. where 1A½x¼1 if x ∈ A and 0 otherwise. Indeed, when Λ → ∞ A. Stability and expressiveness þ , the auxiliary field is forced to sit on one of the m minima of the potential, which each have a complex weight “ ” − It is first natural to wonder how big the class of CTNSs e aj . To each of these m possible values of the field, the is. It could be, for example, that even for arbitrary large D term in α associates a different coherent state. Hence, one α and arbitrary V, B, and , CTNSs spanned only a small can approximate jΨi with arbitrary precision and, hence, all sector of the Fock space. As in the discrete, can any state be states in the Fock space. approximated, even if inefficiently, by a CTNS? Allowing larger values of D remains useful if V and α are Let us first consider the stability of the CTNS class. The restricted in some way, e.g., to being polynomials with a sum of two CTNSs is still a CTNS, provided we are willing fixed degree. In that case, being able to take a larger bond- to accept singular potentials V. More precisely, let field dimension D substantially increases the expressive- α α jV1;B1; 1i and jV2;B2; 2i be two CTNSs with bond- ness of a CTNS subclass. Gaussian CTNSs (see Sec. VA) field dimension D1 and D2. Then we can easily rewrite provide such an illustration. their sum as a CTNS with bond-field dimension D1 þ 1 D2 þ (although it may, in general, require fewer auxiliary B. Gauge transformation fields). For example, defining the CTNS jWΛ;C;βi with Different choices of V, B,andα can generate the same ˜ ˜ ˜ WΛðϕ1; ϕ2; ϕÞ¼V1ðϕ1ÞθðϕÞþV2ðϕ2Þθð−ϕÞ state. This fact is to be expected: In the discrete, the map between an elementary tensor and a many-body wave ˜ 2 ˜ 2 þ Λðϕ − 1Þ ðϕ þ 1Þ ; ð44Þ function is not injective, either. Understanding the transformations between tensors generating the same state is ˜ ˜ ˜ Cðϕ1; ϕ2; ϕÞ¼B1ðϕ1ÞθðϕÞþB2ðϕ2Þθð−ϕÞ; ð45Þ fundamental in the theory of TNSs, especially for the classification of symmetry-protected and topological phases. ˜ ˜ ˜ βðϕ1; ϕ2; ϕÞ¼α1ðϕ1ÞθðϕÞþα2ðϕ2Þθð−ϕÞ; ð46Þ It is thus natural to ask the same question for CTNSs following the discrete construction. where θ is the Heaviside function, we have jW∞;C;βi ∝ α α Λ jV1;B1; 1iþjV2;B2; 2i. Indeed, when is sent to 1. Intuition from the discrete infinity, the auxiliary field ϕ˜ becomes a “bit” taking values 1 In the discrete, there exists an important subclass of digitally splittingR P the functional integral into two Dϕ˜ ≃ transformations one can apply on the bond indices of an contributions ϕ˜ ≡1, where each term of the elementary tensor and that leave the state invariant. For sum gives the two initial states. example, in d ¼ 2, the transformation (sometimes called The expressiveness of CTNSs is then easy to assess, gauge transformation) following the same technique as for CMPSs [35]. Taking α½x; ϕðxÞ ¼ fðxÞ and V ¼ a=VolðΩÞ, we obtain any ð49Þ field-coherent state with any complex weight e−ajfi. Using the stability result, one can construct arbitrary linear where and gives the same combinations of such field-coherent states which are contracted state up to new boundary terms (that vanish dense in Fock space; hence, one can get arbitrarily close on a torus). Such transformations prove central to classify- to any state in the Fock space. With this construction, the ing topological phases of matter with discrete tensor bond-field dimension grows at each addition of coherent networks. We would thus like to find an analog in the states. continuum. Actually, using larger bond-field dimensions is only For an infinite bond dimension, the equivalent of an a convenience, and, provided V and α are arbitrary, a invertible linear transformation acting on discrete indices is CTNS can approximate any state in the Fock space with a linear operator G acting on functions of D real variables DP¼ 1. Let us consider a sum of coherent states jΨi¼ − (the auxiliary field): m e ai jf i. This sum can be approximated by the CTNS j¼1 i Z jVΛ; 1; αi with G · φðϕÞ¼ dDϕ˜ Gðϕ; ϕ˜ Þφðϕ˜ Þ: ð50Þ ϕ −Λ1 ϕ 2πϕ VΛ½x; ðxÞ ¼ ½−1=2;mþ1=2½ ðxÞ cos½ ðxÞ Xm A G that is too generic typically destroys the continuum 1 ϕ þ aj ½j−1=2;jþ1=2½ ðxÞ; ð47Þ limit when the corresponding gauge transformation (49) is j¼1 applied on the elementary tensor. The main difficulty is to

021040-8 CONTINUOUS TENSOR NETWORK STATES FOR … PHYS. REV. X 9, 021040 (2019) know what subset of operators to look at. For simplicity, we preserve the continuum limit ultimately give rise to pure restrict ourselves to the case D ¼ 1, from which the general divergence terms as well. In any case, this discussion of the case is easily deduced. discrete setting is but a motivation for the introduction of A first option is to consider the subset of diagonal (some) continuous gauge transformations of CTNSs (see transformations. Let F and G be two operators acting Fig. 4). diagonally: 2. Continuum description G · φðϕÞ¼gðϕÞφðϕÞ; ð51Þ The previous inquiries motivate the following proposi- F · φðϕÞ¼fðϕÞφðϕÞ: ð52Þ tion, which is at the same time a definition of a certain class of gauge transformations for CTNSs. Acting on an elementary discrete tensor in the same way as Proposition 1 (gauge transformation).—Let F½x; ϕðxÞ in Eq. (49) with F and G simply changes the integration be an arbitrary vector field in Ω.IfΩ has no boundary, measure: the CTNS jV;αi is left unchanged by the gauge transformation: f½ϕð1Þg½ϕð4Þ dϕð1Þ…dϕð4Þ → dϕð1Þ…dϕð4Þ: ð53Þ f½ϕð3Þg½ϕð2Þ VðϕÞ → VðϕÞ − ∇ · F½x; ϕðxÞ: ð57Þ

If this change of measure is too general, there will be no The proof is trivial and is just a direct application of continuum limit. A natural choice, preserving the con- Stokes’ theorem. More generally, if Ω has a boundary ∂Ω, tinuum, is to take the gauge transformation (57) adds a boundary term to the measure: −1 fðϕÞ¼exp ½−εd fðϕÞ; ð54Þ I ϕ → ϕ d−1 F ϕ n d−1 Bð Þ Bð Þ exp d x ½x; ðxÞ · ðxÞ ; ð58Þ gðϕÞ¼exp ½−ε gðϕÞ: ð55Þ ∂Ω

In the continuum limit, such a choice put in Eq. (53) where nðxÞ is the unit vector normal to ∂Ω in x. Gauge yields transformations of CTNSs thus have a straightforward Z geometric interpretation. f½ϕðxÞ Dϕ → Dϕ exp d2x∇ · ; ð56Þ g½ϕðxÞ C. Tensor rescaling hence, this choice adds a pure divergence term into the Our objective is now to relate different tensor network CTNS definition which can be transformed into a boundary descriptions of the same state at different scales [14,44]. term thanks to Stokes’ theorem, which is exactly what a More precisely, considering a correlation function for a 1 gauge transformation should do. state parametrized by a tensor Tð Þ in the thermodynamic This very special choice of operators does not exhaust limit: the infinitesimal transformations compatible with the exist- … 1 O …O 1 ence of a continuum limit. However, we conjecture that all Cðx1; ;xnÞ¼hTð Þj ðx1Þ ðxnÞjTð Þi; ð59Þ discrete gauge transformations of the form of Eq. (49) that the objective is to find a tensor TðλÞ of new parameters such that

Cðλx1; …; λxnÞ ∝ hTðλÞjOðx1Þ…OðxnÞjTðλÞi: ð60Þ

Naturally, in the discrete, this relation is at best approximate. For the CTNS of Definition 1, we can write the flow VðλÞ, αðλÞ exactly, following the rather standard dimensional analysis of ordinary QFT. As such, we introduce new FIG. 4. Gauge transformations. In the discrete case (left), creation and annihilation operators ψ˜ †ðxÞ¼λd=2ψ †ðxλÞ transforming the elementary tensor as in Eq. (49) has a nontrivial and ψ˜ ðxÞ¼λd=2ψðxλÞ. They indeed verify the standard result on the boundary only. In the continuum (right), † the transformation of the elementary tensor is equivalent to the commutation relations ½ψ˜ ðxÞ; ψ˜ ðxÞ ¼ δðx − yÞ. These addition of a pure divergence term for the auxiliary fields in the new operators relate correlation functions at different bulk, which can then be integrated into a boundary condition. scales:

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† Cðλx1; …; λxnÞ ≔ hψ ðλx1Þ…ψðλxnÞi ð61Þ 1. Compactification A CMPS of a quantum field defined on a space interval −nd=2 † † χ χ ˆ ˆ ¼ λ hψ˜ ðx1Þ…ψ˜ ðxnÞi; ð62Þ of length T is parametrized by three ( × ) matrices Q, B, and Rˆ and is defined [33,34] as − 2 where the λ nd= factor just comes from the fact that the Z T=2 operators we introduce have a dimension. We just have to ˆ T τ ˆ ˆ ⊗ ψ † τ 0 α jQ; B; Ri¼tr B exp d Q þ R ð Þ j i: rewrite jV; i as a function of the new creation and −T=2 annihilation operators to relate the different scales. To ð68Þ achieve this rescaling, we change the position variable introducing u ¼ x=λ. The free field measure now reads To obtain a CMPS, we could directly instantiate our Z 1 1 CTNS ansatz with d ¼ , e.g., using its functional integral μ ϕ Dϕ − d λd−2∇ϕ λ ∇ϕ λ form (1). However, as we mentioned before, the d 1 case d ð Þ¼ exp 2 d u kðu Þ · kð uÞ ¼ is quite peculiar compared to other dimensions, and so it is ð63Þ nice to see it can also be immediately obtained from a Z general d case where all dimensions but one are taken to be 1 very small. ∝ Dϕ˜ exp − ddu∇ϕ˜ u · ∇ϕ˜ u 64 2 kð Þ kð Þ ð Þ Indeed, consider a domain of the form Ω ¼ ½−T=2;T=2 × S, where S is a d − 1-dimensional torus l − 1 ¼ dμðϕ˜ Þ; ð65Þ of length in all d directions. Expanding the auxiliary fields ϕ of Eq. (1) in Fourier modes on this d − 1 torus S −2 2 and taking the limit l → 0 yields a functional integral in with ϕ˜ ðuÞ¼λ½ðd Þ= ϕðλuÞ. This rescaling gives which only the field zero mode on S survives. Hence, one Z Z obtains a functional integral of the form α μ ϕ˜ − d λd λð2−dÞ=2ϕ˜ jV; iψ ¼ d ð Þ exp d u V½ ðuÞ Z Z T=2 1 XD jV;B;αi¼ DϕBðϕÞ exp − dx ½∂ ϕ ðxÞ2 d=2 ð2−dÞ=2 ˜ † 2 x k − λ α½λ ϕðuÞψ˜ ðuÞ j0ið66Þ −T=2 1 k¼ þ V½ϕðxÞ − α½ϕðxÞψ †ðxÞ j0i; ð69Þ d ð2−dÞ=2 d=2 ð2−dÞ=2 ¼jλ V½λ ·; λ α½λ ·iψ˜ : ð67Þ where the one-dimensional auxiliary field ϕ is the zero This result allows us to discuss the IR behavior of CTNSs mode (on the shrunk torus S) of the initial d-dimensional in terms of relevant, irrelevant, and marginal couplings. To auxiliary field. It is what one would have obtained by fixing this end, we informally expand V and α in powers p of the immediately d 1 in Eq. (1). In operator form, Eq. (69) fields ϕ and analyze the terms of each degree separately. ¼ becomes: The corresponding coupling dimensionality is Δ ¼ d þ 2 − d =2 p for terms in V and Δ d=2 2 − d =2 p Z ½ð Þ ¼ð Þþ½ð Þ XD ˆ 2 α T=2 P for . Consequently, jV;B;αi¼tr Bˆ T exp − dτ k 2 α − 2 2 For d ¼ , all powers of the field in V and yield T= k¼1 relevant couplings. There are no irrelevant couplings. For d ¼ 3, the powers p ¼ 1, 2, 3, 4, 5 of the field in V þ V½Xˆ − α½Xˆ ψ †ðτÞ j0i; ð70Þ yield relevant Δ > 0 couplings. The power p ¼ 6 is α 1 marginal in V. For , the powers p ¼ , 2 are relevant ˆ ˆ and p ¼ 3 is marginal. All other powers are irrelevant. where Pk and Xk are canonically conjugated pairs (D zero- Relevant powers will dominate the behavior of CTNS dimensionalP quantum fields). This is already a CMPS with − ˆ D ˆ 2 2 ˆ ˆ α ˆ correlation functions in the IR and actually be the only ones Q ¼ k¼1ðPk= ÞþV½X and R ¼ ½X. However, the allowed if the CTNS description is aimed to hold at all bond Hilbert space is now that of D particles in one scales in the non-Gaussian case (see Sec. VB). dimension or one particle in D dimensions, and hence χ ¼þ∞. D. Recovering continuous matrix product states Continuous TNSs should reduce to CMPSs in an 2. Bond-dimension quantization appropriate limit, which is an important property to check To obtain a genuine CMPS (with a finite bond dimen- to demonstrate that our ansatz is a natural extension of sion) from a d ¼ 1 CTNS defined by Eq. (70), we need to CMPSs to d ≥ 2. choose a specific potential effectively reducing the Hilbert

021040-10 CONTINUOUS TENSOR NETWORK STATES FOR … PHYS. REV. X 9, 021040 (2019) space dimensionality. The intuition is quite clear: Take a V. COMPUTATIONS potential with deep minima. To carry computations with CTNSs, one could of course Let us take a potential with D deep minima m on the k rediscretize them and use the standard TNS algorithms. vertices closest to 0 of an hypercube, i.e., m1 ¼ð1; 0; …; 0Þ, We now mention techniques relying only on the con- m2 ¼ð0; 1; 0; …; 0Þ; …,andm ¼ð0; …; 0; 1Þ. The effec- D tinuum limit. tive dynamics is now restricted to a D-dimensional Hilbert space spanned by jmki corresponding to wave packets localized around each minima. In this reduced Hilbert space, A. Gaussian states the minima are coupled by tunneling. Because of the There exists a subclass of CTNSs for which all quantities geometrical configuration we consider, the minima can all of interest can be computed exactly: Gaussian CTNSs. be connected by independent saddle points, and, hence, the Definition 3 (Gaussian CTNSs).—A CTNS is said to be effective coupling between the minima can be chosen freely. Gaussian if the functions V and α are, respectively, at most With this freedom, we can obtain any D × D complex matrix quadratic and affine in the auxiliary field: Q of standard CMPSs by adjusting the value of the D2 saddle 1 points of V. ð0Þ ð1Þ ð2Þ Vðx; ϕÞ¼V ðxÞþV ðxÞϕ þ V ðxÞϕ ϕl; ð71Þ The R matrix is fixed in the same way. The values of α½X k k 2 kl k on the minima jmki of the potential fix the diagonal α ϕ αð0Þ αð1Þ ϕ coefficients of R, and the value on the saddle point connect- ðx; Þ¼ ðxÞþ k ðxÞ k: ð72Þ ing jmk1 i and jmk2 i fixes the nondiagonal terms Rk1;k2 . Hence, not only can CTNSs reduce to CMPSs when Naturally, a Gaussian CTNS is also a Gaussian state in d ¼ 1 (or when d − 1 dimensions are small) for a specific the usual sense of the term. More precisely, for a Gaussian Z Z˜ choice of potential, but actually all (bosonic) CMPSs can be CTNS, j0;j and j0;j are manifestly Gaussian functionals. Z obtained this way. In this context, the bond-field dimension Let us compute j0;j in the translation-invariant case D reduces to the usual bond dimension χ. Ω ¼ Rd. Inserting Definition 3 into Eq. (13) yields

Z Z 2 ð1Þ ð1Þ 1 ϕ T −△þVð Þ −α ⊗α ϕ 0 d 2 2 Zj0;j ¼ D½ϕD½ϕ exp − d x · · N ϕ0 −αð1Þ⊗αð1Þ −△þVð2Þ ϕ0 2 2 Vð1Þ − αð1Þðj þ αð0ÞÞ T ϕ þ · − jαð0Þ − j0αð0Þ − αð0Þαð0Þ þ Vð0Þ þ Vð0Þ : ð73Þ Vð1Þ − αð1Þðj0 þ αð0ÞÞ ϕ0

Carrying the Gaussian integration, we then obtain Z 1 1 d d 0 T 0 ð0Þ 0 ð0Þ Z 0 ¼ exp d xd y Λðj; j Þ ðxÞ · Kðx; yÞ · Λðj; j ÞðyÞþδðx − yÞðjα þ j α ÞðyÞ ; ð74Þ j;j N˜ 2 where Vð1Þ − αð1Þðj þ αð0ÞÞ −△ þ Vð2Þ −αð1Þ ⊗ αð1Þ Λ 0 1 δ − ðj; j Þ¼ and Kðx; yÞ¼ 2D×2D ðx yÞ: ð75Þ Vð1Þ − αð1Þðj0 þ αð0ÞÞ −αð1Þ ⊗ αð1Þ −△ þ Vð2Þ

Because of translation invariance, Kðx; yÞ¼Kðx − yÞ, which can be written in Fourier space: Z Kðx − yÞ¼ ddpeip·ðx−yÞKðpÞ: ð76Þ

Inserting this expression into Eq. (75) and integrating over the variable u ¼ðx − yÞ yields 1 p2 þ Vð2Þ −αð1Þ ⊗ αð1Þ −1 KðpÞ¼ ; ð77Þ ð2πÞd −αð1Þ ⊗ αð1Þ p2 þ Vð2Þ which is difficult to make more explicit but could be computed exactly for given α and V. To get an intuition of the behavior of the two point functions, we may instantiate this expression on a simple example where the bond-field dimension D equals 1 and αð0Þ ¼ Vð1Þ ¼ 0 for simplicity. In that case, we have

021040-11 ANTOINE TILLOY and J. IGNACIO CIRAC PHYS. REV. X 9, 021040 (2019) 1 1 p2 þ Vð2Þ jαð1Þj2 KðpÞ¼ 2 2 2 2 1 4 : ð78Þ ð2πÞd ðp þ Vð ÞÞðp þ Vð ÞÞ − jαð Þj jαð1Þj2 p2 þ Vð2Þ

Using Eq. (9) gives the correlation function:

αð1Þ 2 − ≔ ψ † ψ j j 1 0 0 1 T 0 1 1 0 T Cðx yÞ h ðxÞ ðyÞi ¼ 2 ½ð ; ÞKðx; yÞð ; Þ þð ; ÞKðx; yÞð ; Þ ð79Þ

Z 1 jαð1Þj4eip·ðx−yÞ ¼ ddp : ð80Þ ð2πÞd ðp2 þ Vð2ÞÞðp2 þ Vð2ÞÞ − jαð1Þj4

Importantly here, the correlation function in momentum unproblematic. Indeed, carrying an optimization on bare space CðpÞ ∝ p−4 when p → þ∞. Hence, the integral is or renormalized parameters is equivalent, and the fact that not UV divergent for x ¼ y so long as d ≤ 3, in which case some properties break above a cutoff momentum is anyhow the particle density hψ †ðxÞψðxÞi is finite. imposed by the physical QFT being approximated. In this approach, there is no restriction on the powers of the α B. Non-Gaussian states auxiliary field appearing in V and . One may also be interested in the class of CTNSs for For a non-Gaussian CTNS, it is no longer possible to their properties and not necessarily to approximate the compute the correlation functions exactly, in general. ground state of a given system. In that case, going Furthermore, the definitions we provided for the CTNSs beyond regularization and renormalizing the state with in Eqs. (1) or (14) are generically divergent. Nonetheless, proper counterterms and renormalization conditions one can use approximations or numerical techniques seems necessary to preserve the locality of the underlying coming from the quantum field theory and tensor network tensor network. In the general case, this procedure is toolboxes. equivalent to renormalizing a relativistic open quantum field theory, a problem which has received interest 1. Regularization and renormalization recently [45]. At the level of dimensional analysis, this procedure restricts powers of the auxiliary field in V and In the general case, the ansatz we put forward suffers α to renormalizable interactions and, hence, to the from the same UV divergences that plague quantum field relevant and marginally relevant powers obtained in theories. As in QFTs, these divergences are in a way 2 Sec. IV C.Ind ¼ 3, it restricts the parameters to a finite inevitable: The gradient squared ð∇ϕÞ in the path integral number of tensors appearing in the finite polynomial insufficiently penalizes high momenta in d ≥ 1 (note that, expansion of V and α. Allowing for more auxiliary fields again, the d ¼ 1 case is trivial). On the other hand, the is thus necessary to make the CTNS class arbitrarily large locality of the underlying tensor network forbids higher and expressive in d ¼ 3. derivatives. Hence, as in QFTs, the divergences are tied to Finally, a natural regularization may be provided by the very property (locality) that we require. restricting the class of quantum states in d − 1 on which the Given this state of affairs, there are essentially three transfer matrix T acts. As we see, for special cases of T, one options to deal with divergences, depending on what one can indeed recover finite results in d ¼ 2. In that case, the needs the state for. state itself is implicitly defined by the approximate method The first option is simply to regularize the state with a used to contract it. momentum cutoff Λ, either directly in the path integral— which breaks locality and destroys the operator representation—or in the operator representation—which generi- 2. Dimensional reduction cally breaks Euclidean invariance. In both cases, the scale Λ We now discuss the last option. To compute physical is reminiscent of the inverse lattice spacing of discrete correlation functions in the general case, one can exploit tensor networks. The parameters appearing in the expan- their operator expression (20) given by the exponential of a sion of V and α are then the equivalent of the bare transfer matrix acting in a space of one dimension less. In parameters in QFT Lagrangians. As long as the state is the d ¼ 2 case, the theory one needs to solve is thus simply used as a variational ansatz, e.g., to minimize the energy of a one-dimensional QFT. The latter is solvable with CMPSs an anyway regularized QFT Hamiltonian, this fact is (i.e., CTNSs in one dimension less), which, as a bonus,

021040-12 CONTINUOUS TENSOR NETWORK STATES FOR … PHYS. REV. X 9, 021040 (2019) have a built-in UV regulator [46] and bring the computation CMPS for two species of bosons. The dominant eigenvector back to a zero-dimensional problem [47]. We outline the can then be approximated by choosing steps of such a computation on a simple example. We consider a CTNS on a torus ðτ;xÞ, τ ∈ ½0;T, hQ; R1;R2jTjQ; R1;R2i ∈ 0 Q; R1;R2 ¼ argmax : ð85Þ x ½ ;L (as in Fig. 2), which reads, in the operator Q;R1;R2 hQ; R1;R2jQ; R1;R2i representation, Z Z The right-hand side of Eq. (85) can be computed explicitly as T L a function of Q, R1,andR2. Indeed, in the same way as we jV;αi¼tr T exp − dτ dxHðxÞ 0 0 compute the normal ordered correlation functions for a CTNS in Eq. (20), one can compute the normal ordered ˆ † − α½ϕðxÞψ ðτ;xÞ j0i: ð81Þ correlation functions for a CMPS, replacing −HðxÞ by Q and αðxÞ by R1, R2 [34], e.g., for x ≥ y: τ Correlation functions for this state at a fixed take a † ðL−xÞT ðx−yÞT yT hψ ðxÞψ 1ðyÞi ¼ tr½e ð1 ⊗ R Þe ðR1 ⊗ 1Þe particularly simple form, with the propagator expðτTÞ 1 1 appearing only once. For example, the two-point function ð86Þ reads with the (zero-dimensional) transfer matrix: † TT hψ ðτ;x1Þψðτ;x2Þi ¼ tr½1 ⊗ αˆ ðx1Þ · αˆðx2Þ ⊗ 1 · e : T ¼ Q ⊗ 1 þ 1 ⊗ Q þ R1 ⊗ R þ R2 ⊗ R : ð87Þ ð82Þ 1 2

One then just has to express T as a function of ψ 1 and ψ 2 Such N-point functions at equal τ contain useful informa- → ∞ instead of the field and conjugate momenta, which requires a tion about the state in the thermodynamic limit T þ , choice, e.g., L → þ∞. Indeed, because of Euclidean invariance, they give access to all correlation functions of aligned points and ψ ψ † −1 2 ½d−1ðxÞþ d−1 ðxÞ a fortiori to all possible two-point functions, which is ϕˆ ðxÞ¼Λ = pffiffiffi ½ ; ð88Þ 0 2 sufficient to compute the expectation values of most homogeneous and isotropic quasilocal Hamiltonians. † ψ −1 x − ψ x 1=2 ½d ð Þ ½d−1ð Þ To simplify the discussion, we now consider the special πˆðxÞ¼Λ0 pffiffiffi ; ð89Þ case of an Hermitian T (obtained, e.g., when all the 2i coefficients of V and α are real). In the T → þ∞ limit, Λ expðTTÞ is dominated by the projector on the eigenvector for some 0. Taking the expectation value of products of jssi of T with the largest eigenvalue. The correlation local operators on the CMPS yields divergent contributions. H α functions then simplify, e.g., They can be removed, e.g., by normal ordering and [48] in the operator representation of Eq. (14) or by adding a † counterterm in the Hamiltonian as in Ref. [49]. In the end, the hψ ðτ;x1Þψðτ;x2Þi ∝ hssj1 ⊗ αˆ ðx1Þ·αˆðx2Þ ⊗ 1jssi: ð83Þ expectation value to maximize can be written

The right-hand side is the correlation function for (2D LT hQ; R1;R2jTjQ; R1;R2i tr½MðQ; R1;R2Þe copies of) a one-dimensional bosonic field theory, which ¼ LT ; ð90Þ motivates the use of a CMPS. More precisely, we use a hQ; R1;R2jQ; R1;R2i tr½e CMPS defined on two copies of the auxiliary quantum where MðQ; R1;R2Þ is some polynomial of Q, R1,andR2 fields to approximate the dominant eigenvector jssi. explicitly calculable from V and α. This expression can be 1 Assuming only D ¼ auxiliary field, we can write simplified in the thermodynamic limit and then be maxi- Z mized, e.g., by gradient ascent [34]. In practice, for transfer L H jQ; R1;R2i¼tr Px exp dxQ ⊗ 1 matrices with relativistic like the ones we consider, this 0 maximization has to be carried over matrices Q, R1, and R2 with a fixed maximum norm (or with a soft penalization of ⊗ ψ † ⊗ ψ † 0 þ R1 1ðxÞþR2 2ðxÞ j i; ð84Þ large norms). This constraint is necessary to prevent the CMPS and its finite entanglement from capturing only the † † UV features of the stationary state [46]. For sufficiently large where Q, R1,andR2 are (χ × χ) matrices, ψ 1 ¼ ψ ⊗ 1 ½d−1 bond dimension χ, and taking into account this subtlety, we ψ † 1 ⊗ ψ † and 2 ¼ ½d−1 are the creation operators associated to expect to get a good estimate of the stationary state. Once Q, each copy of the Fock space on which T acts, j0i is the Fock R1,andR2 are fixed this way, physical correlation functions vacuum of these two copies, and the trace is taken over the can be computed analytically using Eq. (83). For example, if matrices. This object is nothing but a translation-invariant α is linear αðϕÞ ∝ ϕ,weget,forx ≥ y,

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† ðL−xþyÞT hψ ðτ;xÞψðτ;yÞi ∝ tr½e ðR2 ⊗ 1 þ 1 ⊗ R2Þ VI. GENERALIZATIONS ðx−yÞT × e ðR1 ⊗ 1 þ 1 ⊗ R1Þ: ð91Þ A. General metric and anisotropy The main difficulty to overcome in order to construct More complicated cases could be treated in a similar way. CTNSs lies in preserving local Euclidean symmetries. We Through two successive dimensional reductions, we can thus now wish to relax this constraint by allowing a general 2 compute certain correlation functions of a CTNS in d ¼ metric and anisotropic terms in the functional integral with an expression involving only matrices with a finite definition (1). Namely, it is natural to consider the follow- 0 number of entries (d ¼ ). There is a priori no objection, in ing generalization. 3 principle, to contracting a d ¼ CTNS this way, but each Definition 4 (general functional integral formulation).— additional dimensional reduction is done at the price of a A CTNS of a bosonic quantum field on a smooth variational optimization. For numerical purposes, the opti- Riemannian manifold M with boundary ∂M and metric mization of the CMPS is the crucial step. While current g is a state jV;B;αi parametrized by two functions V and α: – methods [46,49 51] can be used, the prospect to use CMPSs RDþdD → C and a boundary functional B: L2ð∂MÞ → C to solve field theories in more than one spatial dimension defined by the functional integral on an auxiliary provides a strong additional motivation to make them more D-component field ϕ: efficient. Z Z ffiffiffi d p 3. Perturbation theory jV;B;αi¼ DϕBðϕj∂MÞexp − d x g M Given that it is possible to compute correlation functions μν g ∂μϕ ∂νϕ for Gaussian CTNSs, it is natural to compute correlation × k k þV½ϕ;∇ϕ−α½ϕ;∇ϕψ † j0i; functions for more general states by carrying a perturbative 2 expansion around Gaussian states. One simply Dyson ð92Þ expands the non-Gaussian part of the exponential in the expression for the generating functional (13). It generically where all functions depend explicitly on the position and yields an expansion in terms of Feynman diagrams, similar summation on k is assumed. to that of QFT. For example, if α½ϕ is linear in ϕ with a correction B. Specialization: CMERA ∝ λklϕkϕl, the expansion contains diagrams composed of vertices with three and four legs, corresponding to the A natural specialization of the previous generalization α½ϕα½ϕ0 term in Eq. (13), connected by Gaussian consists in having an auxiliary field living on an hyperbolic propagators. As previously mentioned, unless cancellations manifold M coupled to a physical field restricted to the between different auxiliary fields occur, loop diagrams are boundary ∂M (see Fig. 5). Namely, we have in mind a state UV divergent and a regularization is needed. We leave the of the form derivation of the general Feynman rules, including a Z Z renormalization scheme, to future work. Notice that, in μν pffiffiffi g ∂μϕk∂νϕk this approach, it is not the state itself that is defined through jV;αi ∼ Dϕ exp − g þ V½ϕ M 2 a perturbative expansion, but rather the correlation func- I tions computed with it. † × exp α½ϕψ j0i; ð93Þ ∂M 4. Others There are, of course, many other ways one could compute correlation functions. As we mentioned before, one could rediscretize the CTNS to go back to a tensor network description, truncate the bond dimension, and use existing algorithms to contract it. However, this approach would seem to partially defeat the purpose of introducing the continuum in the first place. An interesting avenue is to explore known approximations or tools of the quantum FIG. 5. Physical states on a boundary. In the discrete (left), the field theory (besides the perturbation theory) that would not MERA is a (special case of a) tensor network state with a be obvious in the discrete, like saddle point approxima- hierarchical structure, with physical indices only at the boundary. tions, large D limits, or functional renormalization. Finally, Tensor network states with such a structure can also be extended direct Monte Carlo sampling of the auxiliary field, although to the continuum with our CTNS ansatz, by restricting the it yields oscillating terms harming convergence in the physical field to the boundary and choosing an appropriate general case, is a last-resort option. metric (here hyperbolic) for the bulk auxiliary fields.

021040-14 CONTINUOUS TENSOR NETWORK STATES FOR … PHYS. REV. X 9, 021040 (2019) where the integral on the boundary may have to be taken as in canonical form? Are there important gauge transforma- some appropriately rescaled limit of a bulk integral. Such tions our discussion in Sec. IV B ignores? How do V and α states could provide a natural generalization of the MERA encode topological order and (local and global) gauge [32] in the continuum and for an arbitrary number of symmetries? Can this approach be combined with tech- physical dimensions. They could provide a natural con- niques developed on the lattice to study gauge theories with tinuum versions of tensor network toy models of the tensor networks [58–61]? Are there nontrivial non- AdS=CFT correspondence [29–31]. The form (93) is also Gaussian CTNSs for which correlation functions can be reminiscent of field-theory toy models of the AdS=CFT computed exactly? Do (possibly regularized) CTNSs correspondence [52], where scalar field theories on a fixed generically obey the area law like their discrete counter- AdS background are related to conformal field theories on parts? Can CTNSs be used to construct interesting toy the boundary. models of the AdS/CFT correspondence? To what extent Note that this approach is different in spirit from that of does the bond-field dimension D quantify entanglement for the standard entanglement renormalization approach to (possibly only some) classes of CTNSs? quantum fields [53–57], constructed as a unitary trans- Tackling these questions is an important goal for future formation applied on a QFT ground state. In the proposal work, to fully extend the success of tensor networks from (93), there is a straightforward lattice discretization and a the lattice to the continuum. natural “bulk” description in terms of auxiliary fields. However, the isometry property, characteristic of the ACKNOWLEDGMENTS MERA, is less straightforward to implement. We are grateful to Denis Bernard, Adrián Franco-Rubio, Giacomo Giudice, Anne Nielsen, German Sierra, Guifr´e C. Fermions Vidal, and Erez Zohar for helpful discussions. We thank We define our ansatz for bosonic quantum fields, two anonymous referees for valuable suggestions and because functional integrals and field-coherent states are comments. A. T. was supported by the Alexander von more natural in this context. To extend our proposal to Humboldt Foundation and the Agence Nationale de la fermions, one would have to introduce quite peculiar Recherche (ANR) Contract No. ANR-14-CE25-0003-01. Grassmannian integrals with even kinetic and potential The research of J. I. C. is partially supported by the ERC V terms but a Grassmann odd term α in front of the creation Advanced Grant QENOCOBA under the EU Horizon 2020 † operator ψ . For fermions, it may be more convenient to program (Grant Agreement No. 742102). start with an operator representation like that of Eq. (14), where Euclidean invariance is less natural, to subsequently derive the functional integral formulation. [1] J. I. Cirac and F. Verstraete, Renormalization and Tensor D. Conformal field theory Product States in Spin Chains and Lattices, J. Phys. A 42, We define CTNSs with the help of D auxiliary free 504004 (2009). [2] G. Evenbly and G. Vidal, Algorithms for Entanglement massless scalar fields of measure dμ. A natural generali- Renormalization: Boundaries, Impurities and Interfaces, zation would be to consider more general CFTs for the J. Stat. Phys. 157, 931 (2014). auxiliary space, in the spirit of what has been proposed in [3] A. Molnar, N. Schuch, F. Verstraete, and J. I. Cirac, the context of matrix product states with infinite bond Approximating Gibbs States of Local Hamiltonians Effi- dimensions [41]. 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