Bipartite entanglement entropy of the excited states of the free fermions and the harmonic oscillators

TL;DR

This paper investigates the entanglement properties of excited states in one-dimensional free fermions and harmonic oscillators, revealing conformal field theory descriptions and universal entanglement behaviors, with implications for experiments and related spin chains.

Contribution

It demonstrates that excited states of these systems can be described by conformal field theories and introduces a method to measure universal average entanglement entropy experimentally.

Findings

Excited states exhibit conformal field theory characteristics regardless of the gap.

Even gapless Hamiltonians can have excited states with area-law entanglement due to spectral degeneracy.

A duality-based method is proposed for experimental measurement of universal average entanglement entropy.

Abstract

We study general entanglement properties of the excited states of the one dimensional translational invariant free fermions and coupled harmonic oscillators. In particular, using the integrals of motion, we prove that these Hamiltonians independent of the gap (mass) has infinite excited states that can be described by conformal field theories with integer or half-integer central charges. In the case of free fermions, we also show that because of the huge degeneracy in the spectrum, even a gapless Hamiltonian can have excited states with an area-law. Finally, we study the universal average entanglement entropy over eigenstates of the Hamiltonian of the free fermions introduced recently in L. Vidmar, et al., Phys. Rev. Lett. 119, 020601 (2017) https://link.aps.org/doi/10.1103/PhysRevLett.119.020601 . In particular, using a duality relation, we propose a method which can be useful for experimental measurement of the universal average entanglement entropy. Part of our conclusions can be extended to the quantum spin chains that are associated with the free fermions via Jordan-Wigner(JW) transformation.

Equations

$$S_{vN}=-\text{tr}{\rho }_l\ln {\rho }_l,$$

$$\displaystyle\ S_{vN}=\frac{c}{3}\ln\Big{[}\frac{L}{\pi}\sin[\frac{\pi l}{L}]\Big{]}+\text{const},$$

$$H=\sum _{r=-R}^R\sum _{j\in \Lambda }{a}_r{c}_j^{†}{c}_{j+r}+\frac{b_r}{2}(c_j^{†}{c}_{j+r}^{†}-c_j{c}_{j+r})+\text{const},$$

$$H=\sum _k|f(k)|{\eta }_k^{†}{\eta }_k+\text{const},$$

$$I_n^{+}$$

$$I_m^{-}$$

$$I_m^{-}=\frac{i}{2}\sum _{j\in \Lambda }(c_j^{†}{c}_{j+m}-c_{j+m}^{†}{c}_j),$$

$$I_m^{-}(XY)=$$

$$\displaystyle\sum_{j\in\Lambda}\Big{[}\sigma_{j}^{x}\sigma_{j+1}^{z}...\sigma_{j+m-1}^{z}\sigma^{y}_{j+m}-\sigma_{j}^{y}\sigma_{j+1}^{z}...\sigma_{j+m-1}^{z}\sigma^{x}_{j+m}\Big{]},\hskip 28.45274pt$$

$$H_{XX}=-\sum _{j\in \Lambda }(c_j^{†}{c}_{j+1}+c_{j+1}^{†}{c}_j).$$

$$I_n^{+}(XX)$$

$$I_m^{-}(XX)$$

$$I_n^{+}(XX)$$

$$C_{jk}=\frac{1}{\pi (j-k)}\sum _{m=1}^n(-1{)}^{m+1}\sin \left(\frac{\pi (2m-1)(j-k)}{2n}\right),$$

$$C_{jk}=\frac{-i}{2\pi (j-k)}\times$$

$$\displaystyle\sum_{n=[-m/2]+1}^{[m/2]}\Big{(}e^{\frac{2in\pi}{m}(j-k)}-e^{i\frac{(2n-1)\pi}{m}(j-k)}\Big{)},$$

$$S(A;K)=S(K;A),$$

$$H(\mathbf{K},\mathbf{P})=\frac{1}{2}\sum _{i,j=1}^L({\hat{\pi }}_i{P}_{ij}{\hat{\pi }}_j+{\hat{\varphi }}_i{K}_{ij}{\hat{\varphi }}_j),$$

$$\displaystyle\ \Psi_{g}(\{\phi_{i}\})=\Big{(}\det\frac{K^{1/2}}{\pi}\Big{)}^{1/4}e^{-\frac{1}{4}\langle\phi|K^{1/2}|\phi\rangle}.$$

$$\mathbf{P}.\tilde{\mathbf{K}}-\tilde{\mathbf{P}}.\mathbf{K}=0.$$

$$\displaystyle\ K_{nm}=\frac{1}{L}\sum_{j=0}^{L-1}e^{i\frac{2\pi(m-n)j}{L}}\Big{[}2(1-\cos(\frac{2\pi j}{L})+m^{2}\Big{]}.$$

$${\tilde{P}}_{nm}=\frac{1}{L}\sum _{j=0}^{L-1}{e}^{i\frac{2\pi (m-n)j}{L}}\prod _r(2-2\cos (\frac{2\pi j}{L}+{\beta }_r){)}^{{\alpha }_r},$$

$${\tilde{K}}_{nm}=\frac{1}{L}\sum _{j=0}^{L-1}{e}^{i\frac{2\pi (m-n)j}{L}}\prod _r(2-2\cos (\frac{2\pi j}{L}+{\beta }_r){)}^{{\alpha }_r}$$

$$\times 2(1-\cos (\frac{2\pi j}{L})+\frac{m^2}{2}),$$

$$c_j=\prod _{l<j}{\sigma }_l^z{\sigma }_j^{-}.$$

$$H=$$

$$\displaystyle+\frac{-a_{r}+b_{r}}{2}\sigma^{y}_{j}\sigma^{z}_{j+1}\cdots\sigma^{z}_{j+r-1}\sigma^{y}_{j+r}\big{]}-\sum_{j=1}^{L}\frac{a_{0}}{2}\big{(}\sigma_{j}^{z}-1\big{)}$$

$$\displaystyle-\hat{\mathcal{N}}\big{(}\frac{-a_{r}-b_{r}}{2}\sigma^{x}_{L}\sigma^{z}_{1}\cdots\sigma^{z}_{r-1}\sigma^{x}_{r}$$

$$\displaystyle+\frac{-a_{r}+b_{r}}{2}\sigma^{y}_{L}\sigma^{z}_{1}\cdots\sigma^{z}_{r-1}\sigma^{y}_{r}\big{)},$$

$$\displaystyle H(\scalebox{0.7}{$\mathcal{N}=-1$})$$

$$\displaystyle+\frac{-a_{r}+b_{r}}{2}\sigma^{y}_{j}\sigma^{z}_{j+1}\cdots\sigma^{z}_{j+r-1}\sigma^{y}_{j+r}\big{]}$$

$$\displaystyle-\sum_{j=1}^{L}\frac{a_{0}}{2}\big{(}\sigma_{j}^{z}-1\big{)}.$$

$$\displaystyle H_{\scalebox{0.5}{$r=0,\pm 1$}}$$

$$\displaystyle+(1-\frac{b_{1}}{a_{1}})\sigma^{y}_{j}\sigma^{y}_{j+1}\big{]}-\frac{a_{0}}{2}\sum_{j=1}^{L}\sigma_{j}^{z}+\frac{La_{0}}{2},$$

$$I_m^{-}=$$

$$\displaystyle+\frac{\hat{\mathcal{N}}}{4}\big{[}\sigma^{x}_{L}\prod_{l<m}\sigma^{z}_{l}\sigma^{y}_{m}-\sigma^{y}_{L}\prod_{l<m}\sigma^{z}_{l}\sigma^{x}_{m}\big{]}.$$

$$I_m^{-}(\mathcal{N}=-1)=$$

Full text

I Introduction

Bipartite entanglement entropy of quantum many-body systems exhibits a wide variety of interesting properties with a myriad of applications in high-energy and condensed matter physics. Some of the earliest studies were the calculation of the entanglement entropy of the ground state (GS) of the coupled harmonic oscillators and the establishment of the area-law in the same models in Sorkin (1983); Bombelli et al. (1986); Srednicki (1993). In Ref Holzhey et al. (1994), the same quantity was calculated for the ground state of the conformal field theories (CFT) and the famous logarithmic-law with the central charge as the coefficient of the logarithm was derived. These studies were followed with many other interesting results which paved the way to better understanding of the bipartite entanglement entropy of the ground state of the free fermions Chung and Peschel (2001); Peschel (2003); Gioev and Klich (2006), coupled harmonic oscillators Audenaert et al. (2002); Osterloh et al. (2002); Osborne and Nielsen (2002), quantum spin chains Vidal et al. (2003); Jin and Korepin (2004); Hastings (2007), CFTs Calabrese and Cardy (2004) and topological systemsKitaev and Preskill (2006); Levin and Wen (2006). To review various applications of the bipartite entanglement entropy of the ground state in many-body quantum systems and quantum field theories see Amico et al. (2008); Affleck et al. (2009); Refael and Moore (2009); Peschel and Eisler (2009); Eisert et al. (2010); Modi et al. (2012); Laflorencie (2016); Chiara and Sanpera (2018) and Calabrese and Cardy (2009); Castro-Alvaredo and Doyon (2009); Casini and Huerta (2009) respectively. Although the investigation of the bipartite entanglement entropy of the ground state of quantum systems has a long history the same is not true for the excited states. The bipartite entanglement entropy of the excited states in the quantum spin chains was first studied with exact methods in Alba et al. (2009), see also Ares et al. (2014). Then the entanglement entropy of the low-lying excited states in CFTs was calculated in Alcaraz et al. (2011); Berganza et al. (2012). For recent numerical calculations regarding the entanglement entropy of the excited states in the quantum spin chains and free fermions see Mölter et al. (2014); Storms and Singh (2014); Beugeling et al. (2015); Lai and Yang (2015); Alba (2015); Nandy et al. (2016); Vidmar et al. (2017); Vidmar and Rigol (2017); Riddell and Müller (2018); Zhang et al. (2018); Liu et al. (2018); Vidmar et al. (2018); Hackl et al. (2019); Miao and Barthel (2019); Lu and Grover (2019); Murthy and Srednicki (2019). For further results on the entanglement entropy of the low-lying excited states in CFTs see Taddia et al. (2013); Herwerth et al. (2015); Pálmai (2014); Taddia et al. (2016); Ruggiero and Calabrese (2017); Zhang et al. (2019a, b). Recently, there have also been analytical calculations regarding the quantum entanglement content of the quasi-particle excitations in massive field theories and integrable chainsCastro-Alvaredo et al. (2018a, b).

It is widely believed that one expects universality and quantum field theory for the ground state (and low-lying excited states) of the quantum many-body systems at and around quantum phase transition point. This has been one of the reasons that most of the studies were focused on the bipartite entanglement entropy of the ground states. Nevertheless, some typical behaviors (volume-law) have been already observed for the excited states too, see Refs Storms and Singh (2014); Beugeling et al. (2015); Lai and Yang (2015). Some further analytical and numerical results were also obtained for the average of the entanglement entropy in Vidmar et al. (2017, 2018); Hackl et al. (2019) which support some sort of universality. In this paper, we would like to study the entanglement content of the excited states of the generic translational invariant one dimensional free fermions and coupled harmonic oscillators. The quantity of interest is the von Neumann entropy which is defined for a system by partitioning it to $A$ and $\bar{A}$, where in this paper $A$ has $l$ contiguous sites, and $\bar{A}$ has $L-l$ sites where we will occasionally send $L$ to infinity. The von-Neumann entropy is

[TABLE]

where $\rho_{l}$ is the reduced density matrix of the part $A$. Although in this article we will focus on the von Neumann entropy one can almost trivially extend the results to also Rényi entropies too. For the ground state of the gapped systems $S_{vN}$ saturates with the size of the subsystemHastings (2007), this is called an area-law. For the short-range critical one dimensional systems we have logarithmic-law Holzhey et al. (1994); Calabrese and Cardy (2004), i.e.

[TABLE]

where $c$ is the central charge of the underlying CFT. When the $S_{vN}$ changes linearly with $l$ as happens typically for the excited states, we call it a volume-law. In this paper, we will present some exact results that in some cases can be even considered rigorous theorems. Most notably for one dimensional free fermions and coupled harmonic oscillators we prove the followings: $(1)$ All the Hamiltonians (independent of having a gap or not) have a lot of non-low lying excited states that can be described by CFT and have an arbitrary integer central charge. For free fermions sometimes we can also have excited states with half-integer central charges. (2) The degenerate excited states depending on the chosen subspace basis can follow volume-law, logarithmic-law and sometimes even an area-law. (3) For free fermions (and corresponding spin chains) there is a set of Hamiltonians that the ground state of every generic free fermion is one of the eigenstates of these Hamiltonians. In other words, one can find the ground state of a generic Hamiltonian as one of the eigenstates of these Hamiltonians. Excited states with integer central charges have been realized before in the context of the XX chain in Alba et al. (2009), see also Ares et al. (2014). However, our results extend those conclusions to the generic translational invariant free fermions and coupled harmonic oscillators. In addition, our simple method not only explains the existence of these kinds of excited states it also gives a natural way to make some statements regarding the average entanglement entropy studied in Vidmar et al. (2017, 2018); Hackl et al. (2019). In particular, combining the result of Vidmar et al. (2017) with a duality relation introduced in Carrasco et al. (2017) we discover a method to calculate the average bipartite entanglement entropy over all the eigenstates of a generic free fermion Hamiltonian using a single eigenstate of the XX-chain with just a simple hopping coupling. Using this mapping instead of studying an exponential number of states one can just work with a single state which makes the previously considered unmeasurable quantity quite accessible for the experimental study.

The paper is organized as follows: In section II, we first define the Hamiltonian of generic translational invariant free fermions. Then for later arguments, we present some integrals of motion of these Hamiltonians. To clarify the argument, we then consider the XX chain and prove some statements regarding the entanglement content of this model. After that, we extend our arguments to the general free fermions. Then we comment on the average entanglement over the excited states of free fermions, the role of the degeneracies, and how to measure specific averaging introduced in Vidmar et al. (2017). In section III, we will define the Hamiltonian of generic coupled harmonic oscillators and then using the relevant integrals of motion; we will study the bipartite entanglement entropy of the excited states for these models. Finally in the last section we will conclude the paper.

II Generic free fermions

In this section we discuss some generic properties of the entanglement entropy of the excited states of the one dimensional free fermions. The Hamiltonian of a translational invariant (periodic) free fermions with time-reversal symmetry can be written as

[TABLE]

where $\Lambda=\{1,2,\cdots,L\}$. Using the Majorana operators $\gamma_{j}=c_{j}+c_{j}^{\dagger}$ and $\bar{\gamma}_{j}=i(c_{j}^{\dagger}-c_{j})$ one can write $H=\frac{i}{2}\sum_{r=-R}^{R}\sum_{j\in\Lambda}t_{r}\bar{\gamma}_{j}\gamma_{j+r}$, where $t_{r}=-(a_{r}+b_{r})$ and $t_{-r}=-(a_{r}-b_{r})$. It is very useful to put the coupling constants as the coefficients of the following holomorphic function $f(z)=\sum_{r}t_{r}z^{r}$, seeVerresen et al. (2018): Then the Hamiltonian can be diagonalized by going to the Fourier space and then Bogoliubov transformation as follows:

[TABLE]

where $\eta_{k}=\frac{1}{2}(1+\frac{f(k)}{|f(k)|})c_{k}^{\dagger}+\frac{1}{2}(1-\frac{f(k)}{|f(k)|})c_{-k}$ with $f(k):=f(e^{ik})$, where $k:=\frac{2\pi}{L}j$ with $j=1,2,...,L$. When the system is critical it is well-known that the number of zeros of the complex function $f(k)$ on the unit circle is twice the central charge of the underlying CFT. This will be our guiding principle in most of the upcoming discussions. The local mutually commuting integrals of motions can be written as follows:

[TABLE]

The interesting and crucial fact is that the second set of integrals of motions in the real space can be written as

[TABLE]

which is independent of the parameters of the Hamiltonian. The above considerations are also correct if one uses Jordan-Wigner transformation and find the quantum spin chain equivalent of the above Hamiltonians and integrals of motion, see Appendix A. For example, all the periodic quantum spin chains that can be mapped to the free fermions commute with the following integrals of motion

[TABLE]

which is independent of the parameters of the Hamiltonian.

To introduce the main idea we first start with the XX-chain with the following Hamiltonian

[TABLE]

We are interested in the structure of particular excited states in the spectrum of the above Hamiltonian. The local commuting integrals of motions after a bit of rearrangement are

[TABLE]

In the real space $I_{n}^{-}(XX)$ is (7) and $I_{n}^{+}(XX)$ has the following form:

[TABLE]

The Hamiltonian and the integrals of motion share a common eigenbasis. That means, for example, the ground state of $I_{n}^{+}(XX)$ appears as the excited state of $H_{XX}$. We use this basic fact to prove our statements. Following Verresen et al. (2018), and references therein, it is easy to see that for $I_{n}^{+}(XX)$ we have $f(z)=-(z^{n}+z^{-n})$ which have $2n$ zeros on the unit circle and so its ground state is critical and in the limit of large $L$ it can be described by a CFT with the central charge $c=n$. This can be also checked by calculating the entanglement entropy analytically (using the FH theorem Alba et al. (2009)) and numerically (using the Peschel method Peschel (2003), see appendix B) by hiring the correlation matrix $C_{jk}=\langle c_{j}^{\dagger}c_{k}\rangle$. For the ground state of infinite size $I_{n}^{+}(XX)$ we have 111We note that this sum can be simplified formally but for numerical calculations the above form is more useful

[TABLE]

and $C_{jj}=\frac{1}{2}$. The above $\mathbf{C}$ matrix can be also found as the correlation matrix of one of the excited states with energy zero of the Hamiltonian (9). The correlation matrix of the ground state of the infinite size $I_{m}^{-}(XX)$ can be also calculated easily and it is 11footnotemark: 1

[TABLE]

and $C_{jj}=\frac{1}{2}$. The central charge of the underlying CFT is $c=m$. In the Figure 1 we show the logarithmic behavior (and the corresponding coefficient) of the entanglement entropy of the ground state of $I_{m}^{-}(XX)$ for $m=1,2,3$222We note that although the two correlation matrices (13) and (II) are different they have the same set of eigenvalues.

Based on the above arguments one can conclude that there are infinite conformal excited states with logarithmic scaling of the entanglement in the spectrum of $H_{XX}$ with the central charge $c=m$, where $m$ can be any integer number. In the above, we looked to the finite $m$ when $L$ goes to infinity. However, it is clear that if $m$ is comparable to $L$, then one would expect for large $L$ probably a volume-law instead of a logarithmic behavior. This is indeed correct as it was shown in the Ref Ares et al. (2014) for $I_{n}^{+}(XX)$. The entanglement entropy of these excited states follows a volume-law with a subleading term which is logarithmic. In the Figure 1 we show the linear behavior of the entanglement entropy of the ground state of $I_{m}^{-}(XX)$ for $m=\frac{L}{8},\frac{L}{4},\frac{L}{2}$22footnotemark: 2. In the $L\to\infty$ there are an infinite number of this kind of energy excited states too. It is just enough to take $n=\alpha L$ where $0<\alpha<1$. Note that in these cases the corresponding Hamiltonians $I_{n}^{+}(XX)$ are not local Hamiltonians. We note that there are also excited states that follow an area-law. For example, the states $|11...1\rangle$ and $|00...0\rangle$ have energy zero and trivially follow an area-law. In the case of $I_{m}^{-}$ as we will discuss soon this observation is more pronounced.

For generic free fermions, the arguments are similar: the Hamiltonian $H=I_{0}^{+}$ commutes with $I_{m}^{-}$ which means that the ground state of these integrals of motion should appear in the excited states of $H$. This means that even if the Hamiltonian is gapped (such as the gapped XY-chain) with the area-law property for the GS, there are still infinite excited states in the spectrum that are conformal invariant with the central charge $c=m$ and follow the logarithmic behavior. Of course, there are also an infinite number of excited states that follow the volume-law too. Interestingly, one can also argue that although the Hamiltonians $I_{m}^{-}$ have critical ground states, they also have an infinite number of excited states that follow an area-law. This simply because independent of the parameters in $H=I_{0}^{+}$ the Hamiltonians $I_{m}^{-}$ commute with it. Clearly, we have infinite possibilities to produce Hamiltonians $I_{n}^{+}$ with gapped ground states that follow an area-law. If one starts with a critical Hamiltonian $H$ with half-integer central charge (for example an Ising critical chain) then it is easy to see that the Hamiltonians $I_{m}^{+}$ will have ground states with all the possible half integer numbers, i.e. $c=n+\frac{1}{2}$. That means, for example, the Hamiltonian of the critical Ising chain has excited states with all the possible integer and half-integer numbers. In the Figure 2 we show the logarithmic behavior (and the corresponding coefficients) of the entanglement entropy of the ground state of the $I_{n}^{+}$, associated to the critical transverse field Ising chain, for $n=1,2,3$.

II.1 VHBR averaging and a proposal for measurement

In the case of the periodic free fermions, we expect a lot of degeneracy in the excited states. As it was discussed in more detail in the Appendix C the number of independent energies over the size of the Hilbert space decays exponentially with respect to the size of the system which indicates an enormous number of degeneracies.

The way that one defines the excited state might be significant in getting area, logarithmic or volume law for the entanglement entropy. Recently in Vidmar et al. (2017, 2018) it was observed that if one takes all the eigenstates produced by the creation operators $\eta_{k}^{\dagger}$ and calculates the bipartite entanglement entropy of a connected region the result is universal and independent of the Hamiltonian. Here we would like to comment that the result can be different if one takes into account the degeneracies. For this purpose in Figure 3 we did the averaging for the XX chain in four different ways: The first one is the averaging proposed in Vidmar et al. (2017, 2018) (VHBR). Second (third) we identify the degenerate states and pick the one with the minimum (maximum) entanglement entropy, then we do the averaging over the selected independent states. We call this MIN (MAX) averaging. Finally, in the last averaging we first identify the degenerate states and average over the entanglement entropy inside the subspace and then average over all the averaged entanglement entropies (WA averaging), for more details see Appendix C. In Figure 3, one can see that the VHBR and the WA averagings converge to the same value but the MIN and the MAX averagings converge to the different values. This clearly shows that the averaging done in the Vidmar et al. (2017, 2018) is special. Remarkably, the averaging and universality proposed in these papers have a good chance to be measured experimentally. This is because of a recent duality proposed in Carrasco et al. (2017). In this work, it was shown that there is a one to one correspondence between bipartite entanglement entropy of the excited states (produced by the $\eta^{\dagger}_{k}$) of the XX-chain and the entanglement entropy of one eigenstate (the ground state for the case of $l=\frac{L}{2}$) but with different partition. There are $2^{L}$ possible multi interval bipartitions for the eigenstate of the XX-chain which is identically equal to the one interval bipartite entanglement entropy of $2^{L}$ excited states. Using this method it is possible to regenerate the universal figure proposed in Vidmar et al. (2017), see Figure 3. Consider the XX-chain Hamiltonian with the eigenstates $|K\rangle$, where $K=\{k_{1},k_{2},...,k_{M}\}\subset\{0,...,L-1\}$ are the excited modes. Then consider a subset of sites $A=\{x_{1},x_{2},...,x_{M}\}\subset\{0,...,L-1\}$. Based on Carrasco et al. (2017) if we consider $S(A;\psi)$ as the entanglement of the sites $A$ with respect to the rest for the pure state $|\psi\rangle$, then we have

[TABLE]

for any entropy functional. We now discuss a few examples of the above equality. For the half-filling to get the ground state, we need to fill all the negative modes which make half of the modes that are adjacent in the set $K$. This means the entanglement entropy of the ground state for the set $A$ is equal to the entanglement entropy of the half of the system for an eigenstate with the modes $A$ excited. Of course, if one averages over the entanglement entropy of all the possible sets $A$ one recover the average entanglement entropy VHBR for half of the system. Now consider we are interested in the averaging of VHBR for one site, in another word, $A=\{x_{1}\}$. This can be calculated by finding the average of the entanglement entropy of all the multi interval bipartitions of the state $K=\{k_{1}\}$ with just one mode excited. The rest of the VHBR graph can be produced similarly by calculating the average of the entanglement entropy of all the multi interval bipartitions of one eigenstate.

The above remarkable observation means that one can calculate the average entanglement entropy by just calculating the entanglement entropy of all the multi interval bipartitions for a single eigenstate. The entanglement entropy of different partitions in the spin chains has been already studied theoretically and experimentally in Elben et al. (2018); Brydges et al. (2019). The method proposed and implemented in these works gives access to the entanglement entropy of all the $2^{L}$ different partitions for a particular state. Extension of this technique to free fermions can make it an outstanding method to measure the universal average entanglement entropy proposed in Vidmar et al. (2017, 2018). It is important to mention that because of the non-local nature of the JW transformation the average over the entanglement of all the possible bipartitions of the GS in the free fermions and the equivalent spin chains are not equal Iglói and Peschel (2010).

III Coupled harmonic oscillators (CHO)

In this section we generalize the ideas of the previous section to coupled harmonic oscillators. The Hamiltonian of a periodic CHO can be written as

[TABLE]

where $\mathbf{P}$ and $\mathbf{K}$ are symmetric positive definite matrices. For translational invariant cases, these matrices are circulant matrices. The ground state of the above Hamiltonian can be written as

[TABLE]

The ground state does not depend on the matrix $\mathbf{P}$ which clearly also means that the entanglement entropy of the ground state is also independent of the matrix $\mathbf{P}$. To find a set of integrals of motion for the above Hamiltonian we consider the operator, $\tilde{H}=H(\tilde{\mathbf{K}},\tilde{\mathbf{P}})$. This operator commutes with the Hamiltonian as far as

[TABLE]

Without loosing generality now consider $\mathbf{P}=\mathbf{I}$ then we have $\tilde{\mathbf{K}}=\tilde{\mathbf{P}}.\mathbf{K}$. Since the $\tilde{\mathbf{P}}$ can be chosen arbitrarily one can have very generic circulant matrix $\tilde{\mathbf{K}}$. The ground state of $\tilde{H}$ which is the excited state of $H$ can now have very generic form with quite general entanglement content. As the most natural example we first consider discrete massive Klein-Gordon Hamiltonian with the following $\mathbf{K}$ matrix

[TABLE]

The above matrix is just a massive discrete Laplacian on a circle. In the limit of large $L$ when $m=0$ the underlying conformal field theory has the central charge $c=1$. Consequently the entanglement entropy grows logarithmically in this case. However, for the massive case we have an area-law Bombelli et al. (1986); Srednicki (1993); Plenio et al. (2005); Cramer et al. (2006). These results can be easily checked numerically using the method described in Plenio et al. (2005); Cramer et al. (2006), see Appendix B. Now consider the Hamiltonian $\tilde{H}$ with the following $\tilde{\mathbf{P}}$ and $\tilde{\mathbf{K}}$ matrices

[TABLE]

where $\beta_{r}$’s are different real numbers 333Similar couplings for harmonic oscillators have been also discussed in Unany2005. For example, we can take $\beta_{r}=2\pi\frac{r}{L}$ with $r\in\{1,2,...,L-1\}$. Following Nezhadhaghighi and Rajabpour (2014), consider $e^{i\beta_{r}}$’s appear in complex conjugate form and we have $2n$ of them which $n$ is not proportional to the size of the system and $\alpha_{r}=1$. Then we expect to have a free field theory $\tilde{H}$ with a ground state that have the central charge $n+1$, see Nezhadhaghighi and Rajabpour (2014). When $n$ is proportional to the size of the system we expect to get a volume-law as we had in the case of the free fermions. Numerical calculations presented in the Figures 4 confirm the above expectation. We note that since the $\tilde{\mathbf{K}}$ matrix always has a singular part (with this method) it is not possible to generate an excited state which follows the area-law. When $m\neq 0$ the ground state and the low-laying states follow an area-law. However, if we choose the $\tilde{\mathbf{P}}$ as (20) then one can generate a $\tilde{\mathbf{K}}$ matrix which is singular and follows the logarithmic-law or the volume-law. In other words, as the case of free fermions even for massive cases we have a lot of excited states that can be described by CFT.

IV Conclusions

We showed that independent of the gap a generic translational invariant free-fermion Hamiltonian in one dimension has infinite eigenstates that follow logarithmic-law of entanglement and can be described by CFT. We argued that because of the huge degeneracy, even a Hamiltonian with a critical GS can in principle have a lot of excited states with an area-law behavior. Similar conclusions are valid also for the excited states of the corresponding spin chains. We also proposed a method to measure the recently proposed universal average entanglement entropy over all the exponential number of eigenstates of a generic free fermion Hamiltonian by averaging over multi interval bipartite EE of just a single eigenstate of the XX-chain. We finally extended our discussion to the generic HOs in one dimension. It will be interesting to explore in more detail the averaging over the multi interval bipartite EE of the eigenstates of the generic free fermions and also spin chains.

Acknowledgements.

Acknowledgments: We thank P. Calabrese and B. Vermersch for useful discussions and M. Sarandy for support through the computational resources. M. A. R. thanks ICTP for the financial support and hospitality during the final stages of the work. This work was supported in parts by CNPq.

Appendices

In the next three Appendices for the sake of completeness, we summarize first the spin chain version of the fermionic integrals of motion that we introduced in the main text. Then we introduce the method of correlations that can be used to calculate the entanglement entropy in the free fermions and the harmonic oscillators. In the third subsection, we comment on the degeneracies in the XX-chain and various averagings on the entanglement entropy of the excited states.

Appendix A: Integrals of motion: spin chains

This section serves to pay more attention to the spin version of the integrals of motion and the fermionic Hamiltonian defined in the main text. In most of the cases, the integrals of motion do not have a simple form. However, one can still claim that the Hamiltonian $I_{m}^{-}$ which is parameter independent in the spin version, commutes with all the possible Hamiltonians and integrals of motion and so it provides bases which in those bases the average of entanglement entropy is equal for the eigenstates of $I_{m}^{-}$ and the Hamiltonian. We begin by writing the spin form of fermionic operators using the Jordan-Wigner transformation

[TABLE]

where $\sigma_{i}^{\pm}=\frac{\sigma_{i}^{x}\pm\sigma_{i}^{y}}{2}$. Substituting the above in the fermionic Hamiltonian we get

[TABLE]

where $\hat{\mathcal{N}}=\prod_{l=1}^{L}\sigma_{l}^{z}$, is the parity of the spins down, i.e. the parity of the number of fermions. Because of the string of $\sigma^{z}$, the JW transformation is non-local. The non-locality of the JW transformation affects the boundary conditions through appearance of the operator $\hat{\mathcal{N}}$. The eigenvalues: $\mathcal{N}=+1\>\text{and}\>-1$ correspond to anti-periodic and periodic boundary conditions respectively. The operator $\hat{\mathcal{N}}$ should appear every time we go from spin format to fermion format or vice versa. For instance, in the periodic boundary condition, one gets:

[TABLE]

The above Hamiltonian when $a_{r}$ and $b_{r}$ are nonzero just for $r=0,1$ becomes:

[TABLE]

which is the Hamiltonian of the anisotropic periodic XY spin chain with $-a_{1}\mp b_{1}=\frac{J(1\pm\gamma)}{2}$ and $a_{0}=h$.

For Integrals of motion $I_{n}^{+}$ and $I_{m}^{-}$, we can find the spin form likewise. Using the inverse Fourier transformation, we write them in the $c$-fermion form. The $c$-fermion form of $I^{-}_{m}$ is already written in the main text; therefore, the spin form of this operator is

[TABLE]

Putting $\mathcal{N}=-1$, the boundary term absorbs in the sum and $I^{-}_{m}$ reads as:

[TABLE]

For $I^{+}_{n}$, the fermionic form would be

[TABLE]

In the XY spin chain case this integral of motion can be written as

[TABLE]

where we have used the fact that $a_{r}=a_{-r}$ and $b_{r}=-b_{-r}$. $I^{+}_{n}$ commutes with both the Hamiltonian (A4) and $I_{m}^{-}$. To write the above expression in the spin form, we need to separate two possible cases: $n>1$ and $n=1$. Consequently, using the transformation (A1), the spin form of integrals of motion is:

[TABLE]

[TABLE]

These quantities commute with each other and the Hamiltonian. Note that these operators commute with each other after fixing the $\mathcal{N}$ too. For example, with $R=1$, we have:

[TABLE]

In a general case of $R$, we write for $I^{+}_{n}$:

[TABLE]

for the case which $n>R$. In the case of $n\leq R$ one gets

[TABLE]

There is also the possibility to write the spin form of our quantities of interest in a way that operator $\hat{\mathcal{N}}$ does not appear explicitly. As it can be seen, whenever a product of two fermionic operators at sites $i$ and $j$ is written in the Pauli spin operators, there is a string of $\sigma^{z}$ between these two sites. As in equations (Appendix A: Integrals of motion: spin chains) and (Appendix A: Integrals of motion: spin chains), we placed the string of spin-$z$ operator in the smallest path between our lattice sites. However, In the boundary terms, this smallest path is passing through the boundary. Therefore, we face terms like $\cdots\sigma^{z}_{L}\sigma^{z}_{1}\cdots$ accompanied with operator $\hat{\mathcal{N}}$. If before the JW-transformation, we rearrange our fermionic operators $c_{i}^{(\dagger)}$ and $c_{j}^{(\dagger)}$ in a way that the operator with smaller index comes on the left of the operator with a bigger index. In particular, one can write:

[TABLE]

where $n,m<\frac{L}{2}$. This way of writing the integrals of motion eliminates the operator $\hat{\mathcal{N}}$ and both of these methods are equivalent. For example for $I^{+}_{n}$ with $n>R$ can be written as

[TABLE]

which $j=i\pm r+n$, and since $n>r$ for all $r$ then $j>i$ for all r.

Appendix B: Entanglement entropy in the free fermions and the harmonic oscillators: the method of correlations

For the free fermions/bosons one can use the matrix of correlations to calculate the entanglement entropy for relatively large sizes. In this subsection we provide the well-known exact formulas that can be found in the reviewsPeschel and Eisler (2009); Casini and Huerta (2009). The entanglement entropy for the eigenstates of the free fermions (XX-chain) can be found using the following formula:

[TABLE]

where $\lambda_{j}$’s are the eigenvalues of the correlation matrix $\mathbf{C}_{A}$ restricted to the subsystem $A$ with the elements $C_{ij}=\langle c_{i}^{\dagger}c_{j}\rangle$.

In the case of Harmonic oscillators the von Neumann entanglement entropy has the following form:

[TABLE]

where $\nu_{j}$’s are the eigenvalues of the matrix $(\mathbf{X}_{A}\mathbf{P}_{A})^{\frac{1}{2}}$, where $\mathbf{X}_{A}=\frac{1}{2}(\mathbf{K})^{-\frac{1}{2}}|_{A}$ and $\mathbf{P}_{A}=\frac{1}{2}(\mathbf{K})^{\frac{1}{2}}|_{A}$ are the two point correlations of the position and the momentum restricted to the subsystem $A$.

Appendix C: Structure of degeneracies and average entanglement over excited states of the XX-chain

In this appendix, we tend to present a detailed study of various averagings of entanglement entropy for a periodic free fermion system in excited states. By excited states, we mean eigenstates of Hamiltonian produced by the action of $\eta^{\dagger}_{k}$ on the vacuum state of the XX-chain; Among these states, there are sets of degenerate states. In Vidmar et al. (2017), the authors calculated the entanglement entropy for all the states produced as explained above, and then they took the average of all the entropies calculated without counting for degenerate states. Here, we revisit the work of Vidmar et al. (2017) by taking into account the degeneracies. First, we identify the degenerate states and then preform various averagings on the entanglement entropies.

Degenerate Energy States: The XX-chain can be solved exactly by diagonalizing the Hamiltonian using the Fourier transformation, as written in the main text. Each excited state can be produced by acting on the vacuum with a set of creation operators with different modes. Each mode $\eta_{k}$ has energy

[TABLE]

where $L$ is the size of the system. For instance, states $|\psi^{m}\rangle$ and $|{\psi^{n}}\rangle$ can be degenerate ($E^{m}=E^{n}$) but they are made of different combinations of $\eta^{\dagger}$’s.

[TABLE]

in a way that

[TABLE]

For instance in $XX$ model, the energy of the modes with $k=l$ and $k=L-l$ is the same, therefore the degeneracy in the eigenstates is expected. To unravel these groups of degenerate eigenstates as accurate as possible (machine precision), we first calculated the minimum energy gap ($\Delta E_{min}$) of the spectrum. Since in these types of Hamiltonians the energy levels are not equally spaced, the $\Delta E_{min}$ helps us to have an idea for the required precision value to decide whether two energies are equal or not. Not surprisingly, as shown in the inset of the figure 5, this gap decreases exponentially with the size of the system which makes the decision that two states are degenerate or not more difficult by increasing the size of the system. After finding all the degeneracies, we sum over all the non-degenerate states and call the number $N_{ind}$. This number which we loosely call the number of independent states is exponentially smaller than the dimension of the Hilbert space. In other words, as shown in the figure 5, the ratio $\frac{N_{ind}}{N_{T}}$ decays exponentially with the size of the system $L$. This indicates the presence of an enormous amount of degeneracies in the spectrum of the XX-chain.

Averaging Types: Because of the huge degeneracy in the spectrum of periodic free fermions it is clear that one has a lot of freedom in choosing an eigenstate with different amount of entanglement. The more degeneracy means that there is more freedom in picking a state with very large or very small entanglement content. The sum of two maximmaly entangled states can be in principle a product state with zero entanglement. The reverse is also true, the sum of two product states can be maximmaly entangled. When there is a degeneracy in the spectrum it is not feasible to look for a state with max/min entanglement. We take another approach by focusing on the eigenstates that one can find by the method of previous section. After creating all the eigenstates, we calculated the entanglement entropy using the Peschel formula Peschel (2003) which is valid also for the excited eigenstates. Note that this is not the case for an arbitrary state due to the lack of Wick’s theorem. After getting the entanglement entropy for all the eigenstates, we averaged these entanglements for $L_{A}$ from $1$ to $L-1$ in four different ways.

• •

Averaging over all the entanglement entropies without counting for degeneracies (VHBR mean), as in Vidmar et al. (2017).

• •

Taking the average of entropies of degenerate eigenstates, and then calculating the average of resulting entropies (WA mean).

• •

First, finding the minimum value of the entropy in any set of the degenerate eigenstates, and then calculating the average of this minimum entropies (MIN mean).

• •

Finding the maximum value of entropy in any set of degenerate eigenstates, and then calculating the average of this maximum entropies (MAX mean).