Role of spatial higher order derivatives in momentum space entanglement

TL;DR

This paper investigates how higher order spatial derivatives influence momentum space entanglement entropy in scalar fields, revealing that Lorentz invariant definitions remove unphysical divergences and that higher derivatives can improve divergence behavior.

Contribution

It introduces a Lorentz invariant approach to compute momentum space entanglement entropy and demonstrates how higher derivative terms affect divergence structures in scalar field theories.

Findings

Lorentz invariant ground state removes spurious power-law divergences.

Logarithmic divergence observed in (1+1)D cubic scalar field.

Higher derivative terms improve divergence behavior of entanglement entropy.

Abstract

We study the momentum space entanglement between different energy modes of interacting scalar fields propagating in general (D + 1)-dimensional flat space-time. As opposed to some of the recent works [1], we use Lorentz invariant normalized ground state to obtain the momentum space entanglement entropy. We show that the Lorenz invariant definition removes the spurious power-law behaviour obtained in the earlier works [1]. More specifically, we show that the cubic interacting scalar field in (1 + 1) dimensions leads to logarithmic divergence of the entanglement entropy and consistent with the results from real space entanglement calculations. We study the effects of the introduction of the Lorentz violating higher derivative terms in the presence of non-linear self inter- acting scalar field potential and show that the divergence structure of the entanglement entropy is improved in the presence of spatial higher derivative terms.

Equations

$$\mathcal{H}={\mathcal{H}}_A\otimes {\mathcal{H}}_B$$

$$H_0=H_A\otimes {\mathcal{I}}_B+{\mathcal{I}}_A\otimes H_B$$

$$|0,0⟩\equiv |0_A⟩\otimes |0_B⟩$$

$$H=H_0+\lambda H_{AB}.$$

$$|\Omega ⟩$$

$${\mathcal{A}}_n$$

$${\mathcal{B}}_N$$

$${\mathcal{C}}_{nN}$$

$$\rho$$

$$=$$

$${\rho }_A$$

$${\rho }_A=\left(\begin{array}{cc}1-{\lambda }^2|\mathcal{C}{|}^2& 0\\ 0& {\lambda }^2\mathcal{C}{\mathcal{C}}^{†}\end{array}\right)+\mathcal{O}({\lambda }^3)$$

$$S_A$$

$$\mathcal{S}=\int d{t}^{\prime }{d}^D\mathbf{y}\left[\frac{1}{2}({\partial }_{\mu }\chi {)}^2-\frac{{ϵ}^{\prime }}{2}({\nabla }_{\mathbf{y}}^2\chi {)}^2-\frac{{\tau }^{\prime }}{2}({\nabla }_{\mathbf{y}}^3\chi {)}^2-\frac{g}{r!}{\chi }^r\right]$$

$$t^{\prime }⟶t=t^{\prime }/L,y⟶x=y/L,$$

$$\chi ⟶\varphi =\frac{\chi }{L^{(1-D)/2}},{ϵ}^{\prime }⟶ϵ={ϵ}^{\prime }{L}^2,$$

$${\tau }^{\prime }⟶\tau ={\tau }^{\prime }{L}^4,$$

$$g⟶\lambda =g{L}^{(1+s/2)+D(1-s/2)}$$

$$H_{AB}$$

$$\varphi (\mathbf{x})=\frac{1}{(2\pi {)}^D}\sum _{\mathbf{p}}\frac{1}{\sqrt{2{\omega }_{\mathbf{p}}}}\left(a_{\mathbf{p}}{e}^{-ip.x}+a_{\mathbf{p}}^{†}{e}^{ip.x}\right)$$

$$|\mathbf{p}⟩=\sqrt{2{\omega }_{\mathbf{p}}}{a}_{\mathbf{p}}^{†}|0⟩$$

$$⟨\mathbf{p}|\mathbf{q}⟩=2{\omega }_{\mathbf{p}}(2\pi {)}^D{\delta }_{\mathbf{p},\mathbf{q}}^D$$

$$|\mathbf{p}⟩=a_{\mathbf{p}}^{†}|0⟩$$

$$⟨\mathbf{p}|\mathbf{q}⟩=(2\pi {)}^D{\delta }_{\mathbf{p},\mathbf{q}}^D$$

$$⟨{\mathbf{p}}_1,{\mathbf{p}}_2\dots ,{\mathbf{p}}_{\mathbf{r}}|H_{AB}|0,\dots ,0⟩=$$

$$=$$

$$S_A$$

$$\sum _{\mathbf{p}}⟶\frac{1}{(2\pi {)}^D}\int d^D\mathbf{p},$$

$$S_A$$

$$S_A/L^D$$

$$S_A$$

$$S_A\propto -{\lambda }^2\log {\lambda }^2\{\begin{array}{l}\log \mu ,\text{mbox}forϵ=\tau =0;b=1\\ \frac{1}{{\mu }^2},\text{mbox}forb=\tau =0;ϵ=1\\ \frac{1}{{\mu }^4},\text{mbox}forb=ϵ=0;\tau =1\end{array}+\mathcal{O}({\lambda }^2)$$

Full text

I Introduction

Quantum entanglement depends on two properties — the superposition principle and the tensor product structure of the quantum states horodecki2009 . Since the same quantum state has different tensor product structure in different Hilbert spaces, entanglement entropy is a partition dependent quantity zanardi2001 ; Thirring2011 . As of now, a large body of literature has investigated the robustness of the entanglement-area relation for the free quantum fields in the real space bombelli86 ; srednicki93 ; cardy2004 ; shanki2006 ; eisert2010 . It is natural to ask whether the entanglement entropy-area relation gets modified due to the presence of self interactions.

In the real space, however, the evaluation of entanglement entropy of self interacting field runs into difficulty: First, the modes functions can not be evaluated exactly and, second the evaluation of entanglement entropy is semi-analytical and the validity of the numerical results, in the perturbative regime, is opaque. Recently, Balasubramanian et al Vijay2012-PRD developed a technique to evaluate the entanglement entropy of the self-interacting scalar fields in the momentum space. The procedure of evaluating momentum space entanglement entropy is similar to the one used in the evaluation of the real-space entanglement entropy, i. e. the modes of different momenta are entangled in the ground state across a particular cut-off which act as an energy partition in the momentum space. More specifically, the low energy IR and high energy UV modes are entangled across the cut-off, say $\mu$ Vijay2012-PRD .

Like the real space entanglement evaluation, there are unsettled issues in evaluating entanglement in the momentum space. First, the entanglement entropy is a cut-off dependent quantity and still we do not have the correct tool to renormalize the entropy. Second, the approach has a close resemblance to the Wilsonian effective low energy action theory as discussed in Refs. WILSON197475 ; zinn2002quantum ; goldenfeld1992lectures . In the Wilsonian renormalization, the UV degrees of freedom are integrated over and IR degrees of freedom are described by an effective density matrix. This is the case for every interacting field theory. Third, it is important to note that the momentum space entanglement is not a universal quantity and depends on what we are integrating out — UV or IR modes. The real space entanglement entropy for a pure bipartite system is symmetric w. r. t. the subsystems Nielsen . However, it is not clear whether the momentum space entanglement entropy satisfies the symmetric property. The breakdown of symmetric property was reported in Ref. Lieb_1963-JMP for the case of Boson-Fermion duality at high energy modes. Hence, the momentum space entanglement entropy is not useful to characterize theories in an invariant way though, real space entropy does Roberts2013-JSTAT . It depends on the partitioning of the UV and IR degrees of freedom. However, in 2-dimensional space-time field theories krishnand2014 , it was shown that the entanglement entropy has UV-IR duality. Our analysis based on the Lorenz invarant definition of the ground state, show that the momentum space entanglement entropy is symmetric for 2-dimensional field theories.

As mentioned earlier, the evaluation of the momentum space entanglement was first reported in Ref. Vijay2012-PRD . While the ground states they used are not Lorentz invariant, the normalization of these states are not consistent with the ones used in the field theory literature peskin . We show that their choice of normalization lead to spurious scaling behaviour for the entanglement entropy. More specifically, we show that the scaling behavior of the entropy $S\propto\mu^{D(r-1)-2-r}$ used in Ref. Vijay2012-PRD go as $S\propto\mu^{D(r-1)-2}$ where $r$ is the index of power indicating the strength of the scalar field interaction and $D$ is the number of space dimensions. We explicitly show that the extra power dependence in the entropy reported in Ref. Vijay2012-PRD is due to their choice of the normalization constant and show that using the standard normalization as in Ref. peskin , the results are consistent with real space entanglement entropy. We study the effects of the introduction of the Lorentz invariance violating higher derivative terms in the presence of non-linear self interacting scalar field potential and show that the divergence structure of the entanglement entropy improves when higher derivative terms are taken into account.

The rest of the paper is organized as follows: Sec. (II) discusses the approach used to evaluate entanglement entropy in momentum space and gives explicit formula for calculating the entanglement entropy of a scalar field in any dimension. In Sec. (III), we discuss the model action in any $(D+1)$- dimensional space-time using the perturbative expansion. In Sec. (IV), we discuss the results for two specific cases and generalize to any space dimensions. We show that the entropy relation derived here is different from the ones obtained in Ref. Vijay2012-PRD by an extra power factor which has quite compelling implications in the renormalization. It is shown that the divergence in the entropy is tunable by changing the dimension of the space-time and the power of the self interaction. Sec. (V), concludes with the discussion about our results and its possible connection to the renormalization of entanglement entropy. In this work, we set $c=\hbar=1$.

II Approach to evaluate momentum space entanglement entropy

Let us start with the first protocol in the pure state entanglement — partition the total system into two parts. Let $H_{A}$ and $H_{B}$ be the Hamiltonian corresponding to two parts $A$ and $B$ with associated Hilbert spaces $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$, respectively. The total Hilbert space of the system is given by,

[TABLE]

Let $|n\rangle$, $|N\rangle$ correspond to the complete energy eigen basis of the subsystem $A$ and $B$, respectively. Furthermore, these represent the occupation numbers of the particle state in the Fock space. Before switching on the interaction, the Hamiltonian of the total system is given by,

[TABLE]

where $\mathcal{I}$ is the unit operator and the energy eigenvalues are represented by $E_{n}$ and $\tilde{E}_{N}$. The ground state is the tensor product of the individual ground states of the Hamiltonian $H_{A}$ and $H_{B}$. Mathematically,

[TABLE]

After turning on the interaction term, the total Hamiltonian can be written as,

[TABLE]

For the present discussion, we assume that strength of the interaction term $H_{AB}$ is weak in order to calculate the interacting ground state ($|\Omega\rangle$) perturbatively. Up to first order in $\lambda$, we have,

[TABLE]

where

[TABLE]

are the first order coefficients in the perturbative expansion and in general we can treat them as matrices and $\mathcal{N}$ is the normalization constant.

The next protocol is the calculation of total density matrix, the matrix entries are written in the basis of $|0,0\rangle,|n,0\rangle,|0,N\rangle$ and $|n,N\rangle$ are given by111For simplicity, we drop the summation, in the coefficients $\mathcal{A,B}$, $\mathcal{C}$, and different product combinations.

[TABLE]

where we fix the normalization constant as, $\mathcal{N}=\left(1+|\mathcal{A}|^{2}+|\mathcal{B}|^{2}+|\mathcal{C}|^{2}\right)^{-1/2}$ and here onwards drop the subscripts attached to the coefficients $\mathcal{A,B}$ and $\mathcal{C}$.

The reduced density matrix of the $A$ subsystem which is in the basis of $|0\rangle$ and $|n\rangle$ is obtained by tracing over the degrees of freedom of subsystem $B$. i. e.,

[TABLE]

The above matrix can be diagonalized and using the unit trace property of the reduced density matrix, gives the diagonal form of the matrix as,

[TABLE]

The entanglement entropy of the $A$ subsystem is,

[TABLE]

where in the last step we have assumed that $\lambda\ll 1$. The final expression for the entanglement entropy is,

[TABLE]

The leading contribution to entropy is $\lambda^{2}\,(\log\lambda^{2})$ and vanishes as $\lambda\to 0$. This is consistent with the fact that the momentum space entanglement entropy of a free field is zero. In the next section, we apply the above procedure to calculate the momentum space entanglement entropy for massless interacting scalar field in different dimensions in the leading order of $\lambda^{2}\,(\log\lambda^{2})$.

III The model: Massless self interacting scalar field

Let us consider the action for a massless scalar field $(\chi)$ propagating in $(D+1)$-dimensional flat space-time with linear, higher spatial derivative terms:

[TABLE]

where $\epsilon^{\prime}$ and $\tau^{\prime}$ are the constants with dimensions [Length]-2 and [Length]-4 respectively, $\nabla^{2}_{\bf y}$ and $\nabla^{3}_{\bf y}$ are the higher order spatial derivatives, $g$ is a dimensionfull tunable constant and $r$ refers to the index of interaction. The importance of the higher derivative spatial terms were studied in Refs. shanki2012 ; shanki2016 and can be used to understand some quantum phase transitions. Unlike the Wilsonian type renormalization WILSON197475 ; zinn2002quantum ; goldenfeld1992lectures , the higher derivative terms introduce Next-to-Next-to-Next interaction in the lattice. It is interesting to note that these higher derivative terms appear in the effective Hamiltonian description of certain high temperature superconductors Arturo .

Rescaling these fields, coupling constants using the following scaling:

[TABLE]

the Hamiltonian $H_{AB}$ corresponding to the action in eq. (17) is given by

[TABLE]

where, $\pi$ is the canonical conjugate momentum corresponding to scalar field $\phi$ and satisfies the equal time commutation relation $\left[\phi(\bf x),\pi(x^{\prime})\right]=i\delta^{D}(\bf x-x^{\prime})$.

Expanding the field in terms of the bosonic creation and annihilation operators

[TABLE]

the Lorentz invariant one particle excited states are peskin ; Yu-2004-PRD :

[TABLE]

and satisfy Lorentz invariant orthogonality relation peskin ,

[TABLE]

where $\omega_{\bf p}=\displaystyle\sqrt{{\bf p}^{2}+\epsilon\,{\bf p}^{4}+\tau\,{\bf p}^{6}}$.

We would like to compare and contrast the above relations (24, 25) to the ones used by Balasubramanian et al in Ref. Vijay2012-PRD :

[TABLE]

It is important to note that authors have not used the normalization factor, $\sqrt{2{\bf\omega_{p}}}$ and this, as we will show below, leads to interesting results for the momentum space entanglement entropy. To understand the effect of the normalization constant, let us calculate the expectation value of $H_{AB}$, i. e.,

[TABLE]

where in the last step we have used equations (24) and (25) simultaneously. The above expression is different from Eq. (29) of Ref. Vijay2012-PRD , importantly, it does not contain the inverse frequency terms.

The leading term in the momentum space entanglement entropy is then given by:

[TABLE]

where the summation is over the momentum such that at least one momentum is below the cut off $\mu$ and at least one momentum above the cut off Vijay2012-PRD . More specifically, we study the entanglement between the high and low energy modes across the cut-off $\mu$. Here, we are interested to understand the momentum space entropy of the low energy mode by tracing over the high energy modes above the cut-off like in the Wilsonian effective action. Taking the continuum limit, i.e,

[TABLE]

Eq. (29) becomes:

[TABLE]

Let us compare and contrast Eq. (31) with Eq. (31) of Ref. Vijay2012-PRD that is reproduced here for easy comparison:

[TABLE]

First, the entanglement entropy is a dimensionless quantity. However, Eq. (31) of Ref. Vijay2012-PRD is a dimensionfull quantity. Second, the consideration of the Lorentz invariant ground state removes the factors, $\omega_{\bf p_{1}}\ldots.\omega_{\bf p_{r}}$ from the denominator of Eq. (32). In the next subsection, we obtain the momentum space entanglement of the interacting fields with higher spatial derivative terms.

IV Role of spatial higher derivatives in momentum space entanglement

Having shown the importance of the normalization constant in the evaluation of the momentum space entanglement entropy, in this section, we discuss the role the spatial higher derivatives play in the divergence of the entanglement entropy. We evaluate the entanglement entropy for two specific examples — $\phi^{3}$ theory in $(1+1)$ and $\phi^{4}$ theory in $(2+1)$ dimensions — and generalize to generic interaction in arbitrary dimensional space-time.

IV.1 $\phi^{3}$ theory in (1+1)-dimensions

Setting $r=3$ and $D=1$ in Eq. (31), entanglement entropy is given by:

[TABLE]

where $\omega_{\bf p}=\sqrt{b\,{\bf p}^{2}+\epsilon\,{\bf p}^{4}+\tau\,{\bf p}^{6}}$ and $b$ takes values [math] or $1$. Evaluating the above integrals, such that at least one momentum is below the cut off $\mu$ and at least one momentum is above the cut off Vijay2012-PRD and using the Lorentz invariant orthogonality relation (25), the leading order term in the entanglement entropy (in the large $\mu$ limit) is

[TABLE]

This is one of the key results of this work regarding which we would to stress the following points: First, in the absence of the higher derivative spatial terms $\epsilon=\tau=0$, the entanglement entropy depends on the cutoff logarithmically and not a power-law. This is consistent with the analysis in Ref. krishnand2014 . It was shown that the divergence of the entanglement entropy of $(1+1)-$ dimensional field theory can be mapped to IR problem and should be valid for small values of the coupling constant $\lambda$. Thus, the results obtained here for the momentum space entanglement entropy are consistent. Second, the UV-IR mapping of 2-dimensional field theories krishnand2014 also shows that the momentum space entanglement entropy with the correct normalization constant is symmetric w.r.t. partitioning. Third, introduction of the higher derivative terms improve the divergence structure of the entanglement entropy i. e. $\nabla^{2}$ term leads to entanglement entropy decaying as $\mu^{2}$. The divergence structure of quantum field theory is expected to vastly improve when higher derivative terms are taken into account. In particular, introducing $\Box^{2}$ term to the scalar field theory leads to logarithmic divergence instead of power-law divergence of the two-point function 1983-Barth.Christensen-PRD . Thus, our analysis is consistent with these results.

IV.2 $\phi^{4}$ theory in (2+1)-dimensions

Let us now consider $\phi^{4}$ interacting, massless scalar field propagating in $(2+1)$-dimension. Substituting $D=2$ and $r=4$, in Eq. (31), the momentum space entanglement entropy in the large $\mu$ limit is given by

[TABLE]

As in the previous case, our results show that the introduction of the higher derivative makes the entanglement entropy less divergent and is consistent with the analysis of Ref. 1983-Barth.Christensen-PRD .

In general for any self interacting scalar model of the type in Eq. (22) in D -dimensional space, the momentum space entanglement entropy, in the large $\mu$ limit, is given by

[TABLE]

The above results show that the interaction terms do not improve the divergence problem of the entanglement entropy, however, the higher derivative terms improve the divergence structure of the entanglement entropy.

V Conclusions

In this work, we have evaluated the momentum space entanglement entropy of the interacting scalar fields in the presence of spatial higher derivative terms. We have explicitly shown that the correct choice of the normalization of the Lorenz invariant states does not improve the divergence problem of the entanglement entropy, however, the presence of higher derivative terms improves the divergence of the entanglement entropy.

Our analysis should be contrasted to the analysis reported in Ref. Vijay2012-PRD where the authors claimed that the interaction terms help to improve the divergence problem of entanglement entropy. As we have shown here, this is due to the wrong choice of the normalization constant. Taking two specific examples, scalar fields in $(1+1)-$ dimensions and $(2+1)-$ dimensions, we have shown that the divergence structure of the entanglement entropy is not improved due to the presence of the interaction terms.

Real space entanglement entropy is symmetric, however, the momentum space entanglement defined in Ref. Vijay2012-PRD is not and hence can not be considered as an universal quantity Roberts2013-JSTAT . Our analysis in Sec. (IV-A), in the light of Ref. krishnand2014 , shows that at least for the $(1+1)-$dimensional field theory, the momentum space entanglement entropy can indeed be considered as an universal quantity. More specifically, since the UV and IR are related by a simple rescaling of the variables in $(1+1)-$dimensions krishnand2014 , the momentum space entanglement entropy evaluated by integrating over the UV modes or IR modes is identical. Our aim is to extend the analysis for higher dimensions. This is currently under investigation.

VI Acknowledgments

SSK is financially supported by Senior Research Fellowship of the Council of Scientific & Industrial Research (CSIR), Government of India. SSK is also thankful to the Condensed Matter and Statistical Physics section at ICTP, Italy for the hospitality during the final stages of the manuscript. This work is supported by Max Planck-India Partner Group on Gravity and Cosmology.