Entanglement Renormalization for Interacting Field Theories

A general method to build the entanglement renormalization (cMERA) for interacting quantum field theories is presented. We improve upon the well-known Gaussian formalism used in free theories through a class of variational non-Gaussian wavefunctionals for which expectation values of local operators can be efficiently calculated analytically and in a closed form. The method consists of a series of scale-dependent nonlinear canonical transformations on the fields of the theory under consideration. Here, we study the $\lambda\, \phi^4$ scalar field theory in arbitrary dimensions and show how non-perturbative effects far beyond the Gaussian approximation are obtained by considering the energy functional and the correlation functions of the theory.

A general method to build the entanglement renormalization (cMERA) for interacting quantum field theories is presented. We improve upon the well-known Gaussian formalism used in free theories through a class of variational non-Gaussian wavefunctionals for which expectation values of local operators can be efficiently calculated analytically and in a closed form. The method consists of a series of scale-dependent nonlinear canonical transformations on the fields of the theory under consideration. Here, we study the λ φ 4 scalar field theory in arbitrary dimensions and show how non-perturbative effects far beyond the Gaussian approximation are obtained by considering the energy functional and the correlation functions of the theory.
In recent years, tensor networks, a new and powerful class of variational states, have proved to be very useful in addressing both static and dynamical aspects of a wide number of interacting many-body systems. They represent a class of systematic variational ansätze which, through the Rayleigh-Ritz variational principle, provide an elegant approximation to the ground state of an interacting theory by systematically identifying those degrees of freedom that are actually relevant for observable physics. These variational ansätze are nonperturbative and can be applied both in the lattice and in the continuum. As an example, the Multiscale Entanglement Renormalization Ansatz (MERA), a variational real-space renormalization scheme on the quantum state, represents the wavefunction of the quantum system at different length scales [1].
A continuous version of MERA, known as cMERA, was proposed in [2] for free field theories. It consists of building a scale-dependent representation of the ground state wavefunctional through a scale-dependent linear canonical transformation of the fields of the theory. Namely, the renormalization in scale is generated by a quadratic operator, and thus, the resulting state is given by a Gaussian wavefunctional. Despite this fact obviously limits the interest of this trial state for interacting QFTs, the Gaussian ansatz has been used in cMERA and correctly reproduces correlation functions and entanglement entropy in free field theories [3,4]. Furthermore, as the Gaussian cMERA is currently studied as a possible realization of holography [5][6][7][8][9][10], it is timely to develop interacting versions of cMERA in order to advance in this program. In [11], the Gaussian cMERA was applied to interacting bosonic and fermionic field theories. In [12], authors developed techniques to build systematic perturbative calculations of cMERA circuits but restricted to the weakly interacting regime.
Our aim here is to provide a non-perturbative method to build truly non-Gaussian cMERA wavefunctionals for interacting QFTs. A justifiable way of doing so would be to formulate a perturbative expansion for which the Gaussian wavefunction appears in its first order [13][14][15][16].
Unfortunately, with these methods, expectation values of operators cannot be calculated exactly and must be approximated by an additional series expansion. On the contrary, our approach clings to the variational method, but using a more elaborated class of trial wavefunctionals. Here, we use a set of nonlinear canonical transformations (NLCT) [17][18][19][20][21] to build a set of scale-dependent extensive functionals which are certainly non-Gaussian. Remarkably, with this prescription, observables can be analytically calculated in a closed form. We illustrate the method by studying the self-interacting λφ 4 scalar theory in (d + 1) dimensions. For d = 1, this theory, does not exhibit any issue when renormalization is considered, and thus the non-Gaussian cMERA lies on a solid ground. In addition, our variational procedure adds up a much larger class of Feynman diagrams than the usual "cactus"-like ones which are captured by the Gaussian approach [22]. Therefore, we are certainly generalizing the variational approach in QFT to non-Gaussian trial states in the canonical formalism.

arXiv:1904.07241v1 [hep-th] 15 Apr 2019
Gaussian cMERA.-cMERA [2,3] is a real-space renormalization group procedure on the quantum state that builds a scale-dependent wavefunctional Ψ[φ, u], where u parametrizes the scale of the renormalization. Eq. (3) contains the path-ordered exponential of the dilatation operator L and the generating operator K(u ). The renormalization scale parameter u in cMERA is usually taken to be in the interval [u IR , u U V ] = (−∞, 0]. u U V = u is the scale at the UV cutoff , and the corresponding momentum space UV cutoff is Λ = 1/ . u IR = u ξ is the scale in the IR limit, where ξ is a longwavelength correlation length. The state |Ψ U V is the ground state of a quantum field theory. The L-invariant state |Ω IR is a Gaussian state with no entanglement between spatial regions. The cMERA Hamiltonian evolution generates translations along the cMERA parameter u. The term K(u) in the cMERA-Hamiltonian is called the entangler operator and the only variational parameters of the ansatz are those which parametrize it. In the case of free scalar theories, K(u) is the quadratic operator given by [2,3] with g 0 (p, u) being the only variational parameter to optimize. This function factorizes as g 0 (p; u) = g 0 (u) Γ(p/Λ) where Γ(x) ≡ Θ(1 − |x|) and Θ(x) is the Heaviside step function; g 0 (u) is a real valued function and Γ(p/Λ) implements a high-frequency cut-off such that p ≡ Λ p . Choosing |Ω IR as [3] √ for all p , with ω Λ = √ Λ 2 + m 2 , it is possible to show that the cMERA ansatz with a quadratic entangler is equivalent to the Gaussian wavefunctional given by where the relation between the scale-dependent Gaussian kernel F (p; u) and the variational cMERA parameter g 0 (p, u) is given by [11] We note that Ψ In addition, the shifting operator is given by Finally, we remark that the Gaussian cMERA ansatz may be also understood as the set of scale-dependent linear transformation of the fields given by with f (p, u) = u 0 du g 0 (pe −u , u ).
Non-Gaussian cMERA.-In QFT, trial states created by introducing polynomial corrections to a Gaussian state correspond to a finite number of particles and those are suppressed in the thermodynamic limit. Thus, in going beyond the Gaussian ansatz, it is necessary to use a class of variational extensive states for which the energy density does not depend on the volume. Following [17,18,21], we build extensive non-Gaussian trial states considering wavefunctionals of the form where Ψ G [φ] is a normalized Gaussian wavefunctional and U N G = exp(B), with B † = −B, an anti-Hermitian operator that, for the moment, it may add new variational parameters, in addition to those in the Gaussian wavefunctional. The expectation value of any operator O(φ, π) in these states amounts to the calculation of a Gaussian expectation value for the transformed operator The transformed operator O is straightforwardly built once the transformations are known. The transformation on the operator O generated by B is given by the Hadamard's lemma in terms of a series of nested commutators [26] It can be seen that a suitable choice of B, while leading to a non-Gaussian trial state, can indeed truncate the commutator expansion, thus reducing the calculation of expectation values of functionals to a finite number of Gaussian expectation values [27]. The exponential form of the transformation ensures the correct extensive volume dependence of observables such as the energy of the system. In addition, as U N G is unitary, the normalization of the state is preserved. The operator B consists of a product of π's and φ's, which is given by where h(p, q 1 , . . . , q l ) = g(p, q 1 , ..., q l )δ(p+q 1 +· · ·+q l ), s is a variational parameter, g(p, q 1 , . . . , q l ) is a variational function that must be optimized upon energy minimization and l ∈ N The other variational parameter is the kernel F (p) entering the Gaussian wavefunctional. The function g(p, q 1 , . . . , q l ) is symmetric under the exchange of q i 's, it must ensure the anti-Hermiticity of B and is constrained to satisfy g(p, p, q 2 , . . . , q l ) = 0 and g(p, q 1 , . . . , q l )g(q i , k 1 , . . . , k l ) = 0, for i = 1, . . . , l. This constraint ensures that the multiple commutator series in (12) terminates after the first non-trivial term. Such procedure yields a variational approximation to the calculation of observables in an interacting theory which improves upon the Gaussian ansatz. The parameter s is a truly non-Gaussian tracking parameter which shows the deviation of any observable from the Gaussian case. The action of U N G on the canonical field operators is given by The canonical commutation relations (CCR) still hold under the unitary, albeit non-linear, transformation of the fields (13), [ φ(p), π(q)] = iδ(p + q) . Noticing that the Gaussian cMERA is generated by the quadratic operator (4), it is clear that operators B which are linear or quadratic in π's and φ's do not yield any improvement upon the Gaussian ansatz. Therefore, in going beyond, one must consider operators B that at least are cubic in the products of these fields. Hence, our proposal to build non-perturbative cMERA states for interacting field theories is based on the idea of defining the set of scale-dependent non-linear transformations where U N G (u) ≡ U N G U SG (u) and U SG (u) ≡ U S U G (u, u IR ). As commented above, in going beyond the Gaussian approach, for U N G one must consider operators B that at least are cubic in the products of these fields. Here we will focus in the simplest one, i.e., the case l = 2 which explicitly reads where, from a cMERA point of view, g(p, q 1 , q 2 ) can be interpreted as a variational coupling-dependent momentum cut-off function [24]. With this choice for B, the transformed fields result where we have made the change of variables in momenta p ≡ e up . In addition, we have defined Σ (±) (p; u) ≡ e ±f (p,u) e − d 2 u and where the scale-transformed non-Gaussian variational cut-off is given bỹ That is to say, as it occurs in the standard cMERA formulation, the variational parameters explicitly depend on the scale transformation. Hence, the cMERA scale- where we have assumed, for simplicity, that χ 0 = 0 . Regarding the solution of the Gaussian variational parameter f (p; u) given in [2,3], it is straightforward to see that Σ (±) (pe u ; u)| u→0 = 1 and thus, Eq. (18) reduces to Eq. (13) and Non Gaussian Correlation Functions.-As in the Gaussian case, the non-Gaussian cMERA presented here is specially well suited to analyze correlation functions. These observables distinguish the ground states of interacting theories from those of noninteracting ones: that is, while for Gaussian states the connected correlation functions of order higher than two vanish, those of interacting systems are generally non zero. In addition, the multiscale approach provides a procedure to gain an understanding of the non-perturbative effects that take place at different scales.
From Eq. (18), we write the following structure of the n-point correlators at scale u in real space The correlation functions break up into interaction-less disconnected functions and connected ones containing information about the interaction. The first four connected functions are where we use the notation ab ≡ x ab ≡ x a − x b .D(ab) ≡ D(ab; u) is the scale-dependent propagator and F (p) is the kernel. When optimizing, we can variate either with respect to F (p) or with respect to a variational mass parameter µ, if we assume F (p) ≡ (p 2 + µ 2 ) −1/2 . The loop integralsχ i (x; u), i = 1, · · · 4 depend both on the positions and the scale u and their explicit expressions and bracketed quantities involving them can be found in [24].
Connected functions show how the non-Gaussian cMERA procedure goes beyond the Gaussian approximation and captures scale-dependent non-perturbative contributions, which are arranged in powers of the variational parameter s. If we focus on quantities that usually measure the non-Gaussianity of a system, we notice that the skewness, whicrelated with the 3-point function, is given by In the limit of large s (s → ∞), the skewness achieves the limiting value γ 2 1,∞ ∼χ 2 4 (12, 23, 31)/[χ 2χ2χ2 ] + O(s −2 ). Usually, the quantities that can be measured in the experiments are the full and connected 2-point and 4-point correlation functions, as well as the point-dependent excess kurtosis over a Gaussian model [25]. For the latter, we obtain, c (23). In the limit of strong non-Gaussianity, s → ∞, the excess kurtosis goes to a limiting value Equations for the variational parameters.-We remark that to fully evaluate the previous expressions we need to obtain the optimal values for the variational parameters F (p), g(p, q 1 , q 2 ) and s by minimizing the energy expectation value Ψ N G |H|Ψ N G = Ψ G |U † N G H U N G |Ψ G at some length scale u. We choose the UV limit, (i.e., u → 0), to compute the expectation value of the Hamil-tonian, which yields [17,18] with φ c = χ 0 + sχ 1 . The notation χ i means that the loop integrals are evaluated at the same spatial point x, i.e., χ i ≡χ i (x ab = 0; u = 0). In addition, ∆ N = I N (µ) − I N (m), with [22] The equations for the optimal values of the variational parameters s, F (p) and g(p, q 1 , q 2 ) may be obtained by derivation of H w.r.t. them and then equating to zero. This yields a set of coupled equations that must be selfconsistently solved. In order to simplify the process, we restrict the integral kernel F (p) to have the Gaussian form F (p) = (p 2 + µ 2 ) −1/2 , thus leaving the variational mass µ, s and g(p, q 1 , q 2 ) as the parameters to be solved. While the full set of equations obtained from this procedure can be found in [24], here, we only present the result for the variational mass parameter µ (gap equation), which yields where µ 2 G = m 2 + λ 2 ∆ 0 + χ 2 0 amounts to the variational mass parameter that would be obtained through the Gaussian cMERA ansatz [11]. The term proportional to s χ 3 in (24) constitutes the major contribution in order to improve the value of the energy with respect to the Gaussian estimate [17,18]. Indeed, the optimal χ 3 (given in terms of the optimal g(p, q 1 , q 2 ), solution of the integral equation resulting from δ H /δg = 0 ) might be seen to contain an infinite series of diagrammatic contributions to the two-point function that go far beyond the "cactus"-diagrams resummation [20]. This highlights to what extent, the trial wavefunctionals of the non-Gaussian cMERA, may produce approximations that go far beyond the Gaussian approximation. Remarkably, the NLCT procedure in d = 1 includes more physics but no further infinities than those posed by the cactus-diagrams. However, the renormalization of the non-Gaussian variational calculations in d > 1 is shown to be much more involved and the contributions generated by the NLCT need infinite rescalings of the bare parameters [18].
Discussion.-In this work, a rather general method for building non-Gaussian generalisations of the cMERA has been presented. The method uses a class of non-linear canonical transformations which are then applied to a Gaussian wavefunctional. These transformations shift the field modes by non-linear functions of modes with non-overlapping domains in momentum space. We have shown how to obtain non-perturbative effects on the correlation functions far beyond the Gaussian approximation in the λ φ 4 scalar field theory. Furthermore, our method shows how the cMERA formalism could provide a systematic UV regularization scheme for generic interacting QFTs. In this sense, the approach here is suitable to be used both with fermionic and gauge field theories. In particular, we propose the following fermionic transformation acting on a spinor ψ(k): where Greek indices denote spinor components, g is a variational (non-)Grassmannian function and π α (p) ≡ δ/δψ α (−p) is the conjugate momentum. Despite this transformation also truncates, a model-dependent analysis, which is beyond the scope of this paper, would impose additional restrictions on the indices β i . We expect this transformation to be useful in addressing relevant physical phenomena in strongly coupled theories including chiral field theories.
Regarding dynamical settings such as quantum quenches, the method promises to be useful as for the moment, all studies with the Gaussian cMERA, assume that the time-evolved state after the quench remains Gaussian along the evolution. Finally, it is worth to explore what geometrical interpretation can be found for the non-Gaussian cMERA ansatz presented in this work.

II. Optimized variational parameters
Being cMERA a variational method, we can ask for the minimization of the energy expectation value H with respect to the variational parameters s, F (a) and f (a, b), where f (a, b) ≡ g(|a + b|, a, b) .
In this section we show the optimization conditions for our variational prescription. We will do it in the UV regime.
Finally, if we assume that the variational parameter F (p) has the form F (p) = (p 2 +µ 2 ) −1/2 , the following expression for µ is obtained by varying H w.r.t. I 0 (µ): III. Correlator scale dependence The correlators are modulated by the scale transformations. As an example, let us consider the 2-point function G (2) c (12) in (20). In Fig. 1 we show the termχ 2 (12) = s −2 G (2) c (12) −D(12) as a function of the position |x 12 | at a given scale u = log σ Λ . As expected, the non-Gaussian contributions showed in the figure, vanish when σ → 0. The integralχ 2 depends on the functions f, g and the kernel F . While restricting the form of the latter to F (p) = (p 2 + µ 2 ) −1/2 , we need to find a solution for the constrained function g(p, q, r) in terms of some variational parameters [17,18]. Thus, we have chosen g(p, q, r) = Γ((p/C 1 ) 2 ) Γ((C 1 /q) 2 ) − Γ((C 2 /q) 2 ) Γ((C 1 /r) 2 ) − Γ((C 2 /r) 2 ) , where Γ(x) is the cMERA momentum cut-off function and C 1,2 are variationally optimized coupling-dependent momentum cut-offs, with |C i | ≤ Λ. With this choice, the optimal function g(p, q, r) must be found self-consistently by determining the cut-offs C 1,2 which are coupling-dependent. The same applies for the remaining variational parameters, µ and s. That is to say, from a cMERA point of view, the equation above strongly suggests that g(p, q, r) might be understood as a variational coupling-dependent momentum cut-off function. Upon minimization, this function potentially exhibits the non-trivial interaction effects of the theory, which turn out essential in the case in which the Gaussian quasi-particle picture is no longer valid.