# Fermionic behavior of ideal anyons

## Abstract

We prove upper and lower bounds on the ground-state energy of the ideal two-dimensional anyon gas. Our bounds are extensive in the particle number, as for fermions, and linear in the statistics parameter \(\alpha \). The lower bounds extend to Lieb–Thirring inequalities for all anyons except bosons.

## Keywords

Intermediate quantum statistics Magnetic interaction Ideal anyon gas Lieb-Thirring inequality## Mathematics Subject Classification

81V70 81Q10 35P15 46N50## 1 Introduction

The behavior of quantum mechanical systems of particles depends sensitively on the geometry of the space in which the particles may move. In particular, dimensionality plays a significant role, and it is a geometric fact that only two fundamental types of identical particles naturally occur in three-dimensional space—bosons and fermions, from whose basic statistical properties many collective quantum phenomena follow. More exotic possibilities of quantum statistics may be realized by confining the particles’ motion and thereby effectively lowering the dimensionality. In two spatial dimensions, which we will be concerned with here, the richer topology allows for a family of hypothetical quantum particles known as *anyons*.

*N*particles is described in terms of a Schrödinger wave function, \(\Psi :(\mathbb {R}^2)^N \rightarrow \mathbb {C}\), whose amplitude \(|\Psi (\text {x})|^2\) represents the probability density of finding the particles at positions \(\text {x}= (\mathbf {x}_1,\ldots ,\mathbf {x}_N)\), \(\mathbf {x}_j \in \mathbb {R}^2\). If the particles are indistinguishable, one must impose that the density is symmetric under particle exchange, i.e.,

*any*phase \(e^{i\alpha \pi } \in U(1)\) or statistics parameter \(\alpha \in \mathbb {R}\), thereby defining a system of

*any*ons.

^{1}Such possibilities have been known since the 1970s and have been studied extensively in the physics literature during the following decades, with notable proposals for concrete realizations and applications, such as for quasi-particles in the fractional quantum Hall effect, rotating cold quantum gases, as well as for future prospects of quantum information storage and computation. We refer to [3, 7, 9, 10, 11, 13, 23, 29, 30, 31, 32, 33] for reviews.

Mathematically, anyons can be realized by viewing \(\Psi \) as a multi-valued function or a section of a complex line bundle over a nontrivial configuration manifold, an approach known in the literature as the anyon gauge picture [4]. Alternatively, one can start with the usual quantum-mechanics setup, taking the familiar bosons or fermions as a reference system, and adding to these magnetic interactions of Aharonov–Bohm type [22, 24, 26]. Here we shall follow this latter approach, known as the magnetic gauge picture.

Many basic questions concerning the behavior of many-particle systems of anyons have remained open since their discovery. This is true even for ideal anyons, i.e., particles without any interactions in addition to the ones forced by statistics. While non-interacting bosons and fermions admit a description solely in terms of the spectrum and eigenstates of the corresponding one-body problem, allowing for the properties of the ideal quantum Bose and Fermi gases to be worked out easily, anyons with \(0< \alpha < 1\) do not admit such a simplification and must be treated within the full many-body context. Even their ground-state properties are thus difficult to determine. In contrast, recall that ideal bosons at zero temperature display complete Bose–Einstein condensation into a single one-body state of lowest energy, while fermions are distributed over the *N* lowest one-body states to satisfy the Pauli exclusion principle, leading in particular to the extensivity of the fermionic ground-state energy.

We show in this work that the ground-state energy of the ideal anyon gas has a similar extensivity as the one for fermions, for all values of \(\alpha \) except for zero (i.e., bosons). In fact, we shall derive upper and lower bounds that interpolate linearly in \(\alpha \) between bosons at \(\alpha =0\) and fermions at \(\alpha =1\). This improves on previous results which only applied to particular rational values of \(\alpha \). Via well-known methods, our new bounds imply that also the celebrated Lieb–Thirring inequality holds for all anyons except for bosons.

## 2 Model and main results

*N*ideal (i.e., point-like) anyons in \(\mathbb {R}^2\) with statistics parameter \(\alpha \in \mathbb {R}\) is given by

^{2}

*N*-particle Hilbert space to be \(\mathcal {H}= L^2_\mathrm {sym}(\mathbb {R}^{2N})\), the permutation-symmetric square-integrable functions. The operators \(\text {D}_\alpha = (D_j)_{j=1}^N\) and \(\hat{T}_\alpha \) then act as unbounded operators on \(\mathcal {H}\) and, because of the singular nature of the vector potentials \(\mathbf {A}_j \notin L^2_\mathrm {loc}\), some care is needed to properly define their domains. One can in fact show [26, Theorem 5] that on \(\mathbb {R}^{2N}\) the minimal and maximal realizations of \(\text {D}_\alpha \) coincide and hence induce a natural form domain \(\mathscr {D}^N_\alpha = {{\mathrm{\mathrm {dom}}}}(\text {D}_\alpha ) \subset \mathcal {H}\) for the kinetic energy \(\hat{T}_\alpha \). This choice is then taken to model ideal anyons. Indeed \(\alpha =0\) yields free bosons, while \(\alpha =1\) corresponds to fermions, with their domains being the Sobolev spaces \(\mathscr {D}^N_0 = H^1_\mathrm {sym}\), \({{\mathrm{\mathrm {dom}}}}(\hat{T}_0) = H^2_{\mathrm {sym}}\) and \(\mathscr {D}^N_1 = U^{-1}H^1_\mathrm {asym}\), \({{\mathrm{\mathrm {dom}}}}(\hat{T}_1) = U^{-1}H^2_{\mathrm {asym}}\), respectively. Here, the unitary map \(U:L^2_{\mathrm {sym}/\mathrm {asym}} \rightarrow L^2_{\mathrm {asym}/\mathrm {sym}}\),

*N*anyons on a domain \(\Omega \subset \mathbb {R}^2\) as

*N*lowest eigenvalues of the one-body operator, i.e., the Laplacian \(-\Delta _{Q_0}^{\mathcal {N}/\mathcal {D}}\). From the Weyl asymptotics, one obtains

*N*times the infimum of the spectrum of the Laplacian \(-\Delta _{Q_0}^{\mathcal {N}/\mathcal {D}}\). In the case \(0<\alpha < 1\) of proper anyons, there is no simplification to a one-body problem, however; the system must be treated as a fully interacting many-body system.

Our main result is to show that for anyons with \(0< \alpha < 1\) and confined to the unit square, \(E^{\mathcal {N}/\mathcal {D}}_N(\alpha ) \sim N^2\), as in the fermionic case, with a prefactor that is of order \(\alpha \) both in the upper and lower bounds. In this sense, the ideal anyon gas behaves fermionic, for any \(\alpha > 0\). Since \(E^{\mathcal {D}}_N(\alpha ) \ge E^\mathcal {N}_N(\alpha )\), it is natural to derive an upper bound on \(E^{\mathcal {D}}_N(\alpha )\) and a lower bound on \(E^{\mathcal {N}}_N(\alpha )\).

Our main result is as follows:

### Theorem 2.1

*all*\(\alpha \), not just odd-numerator rationals.

Theorem 2.1 answers a question raised in [24, 25] whether for \(\alpha _* = 0\) (and \(\alpha \ne 0\)) the energy \(E^{\mathcal {N}/\mathcal {D}}_N(\alpha )\) could be of lower order in *N* than the one for fermions or anyons with \(\alpha _* > 0\). It shows that the behavior of the ground-state energy is fermionic, for any \(\alpha \ne 0\). However, it still leaves open the possibility that the exact energy in the thermodynamic limit may be smaller around even-numerator rational \(\alpha \) than around \(\alpha \) with relatively large \(\alpha _*\), i.e., odd-numerator rationals with small denominator. In particular, it is not known whether it depends smoothly, or even continuously, on \(\alpha \). We refer to [1, 2, 20, 21, 25] for further discussion on the \(\alpha \)-dependence of the ground-state energy.

### Theorem 2.2

*N*, where \(V_{-} := \max \{- V, 0\}\) denotes the negative part of

*V*. Applying this, e.g., to \(V(\mathbf {x})=|\mathbf {x}|^2 - \mu \) and optimizing over \(\mu >0\) gives the lower bound \(\frac{4}{3} N^{3/2} \sqrt{C\alpha /\pi }\) on the ground-state energy of the ideal anyon gas in a harmonic oscillator potential.

The bound (2.5) may for example be applied in a physically relevant setting involving several species of charged particles subject to Coulomb interactions and confined to a very thin two-dimensional layer. Taking one of the species of particles in the layer to be anyons, as was previously considered in [26, Theorem 21], our result proves that such a system is thermodynamically stable for any type of anyon except for bosons. Our method of proof also clarifies that, at least in two dimensions, stability is a consequence solely of the local two-particle repulsive properties of any of the component species, in the sense that all that is required is a strictly positive energy \(E_2^\mathcal {N}(\alpha )\), generalizing the Pauli exclusion principle.

## 3 Upper bounds

A key tool for obtaining upper bounds is to use the fact that interactions between particles with wave functions supported on disjoint sets can be gauged away, as described in [25]. In fact, we have the following subadditivity property for the Dirichlet energy \(E_N^{\mathcal {D}}(\alpha ;\Omega )\) on a general domain \(\Omega \subset \mathbb {R}^2\).

### Lemma 3.1

### Proof

The following lemma gives an upper bound on \(E^\mathcal {D}_N(\alpha )\) that is linear in \(\alpha \) for small \(\alpha \). It is restricted to small particle number, however. The bound follows from a calculation using a trial state similar to the one introduced by Dyson in [5] to obtain an upper bound on the ground-state energy of the hard-sphere Bose gas.

### Lemma 3.2

### Proof

*N*-body Bose gas with two- and three-body interactions of the form

*all*functions \(\Phi \) is the same as the one over only bosonic \(\Phi \) (see, e.g., [14, Corollary 3.1]), hence we may choose a \(\Phi \) that is not permutation-symmetric. In particular, we can use a Dyson ansatz [5, 17, 18] of the form

*f*is a nonnegative radial function bounded by 1, and \(\mathbf {y}_j(\mathbf {x}_j;\mathbf {x}_1,\ldots ,\mathbf {x}_{j-1})\) denotes the nearest neighbor of \(\mathbf {x}_j\) among the points \(\{\mathbf {x}_1,\ldots ,\mathbf {x}_{j-1}\}\). A straightforward generalization of the calculation in [5, 17, 18] leads to the upper bound

*B*denotes the ball of radius \(\sqrt{2}\) centered at the origin. Note that \(\Vert \varphi \Vert _4^4 = 9/4\) and \(\Vert \varphi \Vert _\infty ^2 = 4\). We shall choose

The same strategy can be used to obtain the upper bound (3.3) on the Neumann energy \(E^\mathcal {N}_N(\alpha )\). In this case, one simply chooses \(\varphi =1\) in (3.5). \(\square \)

A combination of Lemmas 3.1 and 3.2 leads to the following result, which immediately implies the upper bound claimed in (2.2) in Theorem 2.1.

### Proposition 3.3

### Proof

*n*particles into as many boxes as possible, and fewer than

*n*in the remaining ones, if necessary. Denoting the number of particles in \(Q_q\) by \(n_q\), and using the subadditivity in Lemma 3.1 as well as the scaling property (2.1), we obtain

*n*such that \(8\pi \alpha n < 1\), in which case we can apply the bound of Lemma 3.2 to \(E^\mathcal {D}_{n_q}(\alpha )\), and obtain

## 4 Lower bounds

As in [12, 24, 25, 26], the key ingredient in the strategy to obtain lower bounds is to first prove a lower bound for the local Neumann energy that is *linear* in the particle number *N*. By splitting the original domain suitably, one may then lift such a bound to one that is *quadratic* in *N*. This method and local bound, referred to as a “local exclusion principle”, goes back to the way Dyson and Lenard incorporated the Pauli exclusion principle for fermions in their original proof of stability of matter [6], and was further developed in [24, 26, 27, 28] for interacting bosonic gases and in [8] for a model of fermions with point interactions.

### 4.1 Preliminaries

We start by recalling some of the previously obtained lower bounds which shall also turn out to be useful in deriving the new bounds. The simplest one is the usual diamagnetic inequality which is also valid for anyons [26, Lemma 4] and tells us that their kinetic energy is always at least as big as the one of bosons:

### Lemma 4.1

Next we consider a certain analog of Lemma 3.1 for the Neumann energy, where subadditivity becomes superadditivity.

### Lemma 4.2

### Proof

*q*, all the particles with labels in \(A_q\) are located in \(\Omega _q\), while the others are located in \(\Omega {\setminus }\Omega _q\). The interaction of particles inside and outside \(\Omega _q\) can then be gauged away, as in the proof of Lemma 3.1, explicitly by writing \(\tilde{\Psi }= \prod _{j \in A_q, k \in A_q^c} e^{i\alpha \phi _{jk}} \Psi \), with \(\phi _{jk}\) defined in (3.1):

With the aid of the previous two lemmas, we can obtain the following bound, which is an adaptation of [19, Proposition 2].

### Lemma 4.3

### Proof

*W*defined in (4.1) (with \(K=4\) and \(\Omega _q = Q_q\) for \(1\le q\le 4\)). If we keep only the terms in (4.1) where \(n_q=2\), we obtain the lower bound

*P*is true and \((P)=0\) otherwise, and used the scaling property \(E_2^\mathcal {N}(\alpha ;Q_q) = 4 E_2^\mathcal {N}(\alpha )\) for \(1\le q\le 4\). The average value of \(W_2\) can be computed to be

### Remark 4.4

*N*-dependent constant times \(E_2^\mathcal {N}(\alpha )\). In fact, by localizing the two particles in different halves of the unit square \(Q_0\) (following the proof of Lemma 3.1), one readily checks that \(E_2^\mathcal {N}(\alpha ) \le 2 \pi ^2\) independently of \(\alpha \). Using this in the denominator in (4.4) leads to the simpler (but worse) bound

As a final step in this subsection, we shall give a lower bound on \(E_2^\mathcal {N}(\alpha )\). The following bound is actually contained in [12, Lemma 5.3].

### Lemma 4.5

*f*in Lemma 4.5 is defined as

*r*centered at the origin, and

### Proposition 4.6

### Remark 4.7

*N*.

### 4.2 New bounds

Our improved lower bounds are due to the following lemma, which utilizes the scale invariance of the problem:

### Lemma 4.8

*N*-linear bound in terms of few-particle energies) For any \(0 \le \alpha \le 1\) and \(N \ge 2\) we have

### Proof

*N*particles into the four squares there must be at least one square with at least

*N*/ 4 particles. Dropping the other terms, we thus obtain the recursive bound

*k*, we have

The previous lemma gives a lower bound on \(E^\mathcal {N}_N(\alpha )\) that is linear in *N*, at least for \(N\ge 2\), for all \(\alpha >0\). The following bound (which also appeared in slightly different formulations in the earlier works; see [8, 12, 19]) lifts any linear growth in the particle number *N* to a quadratic one.

### Lemma 4.9

### Proof

*n*particles on a square \(Q_q\). Lemma 4.2 implies that

*K*would be such as to make \(\rho _Q = 2k\), in which case the desired bound would be obtained exactly. However, we have to take into account the constraint that \(\rho _Q = N/K^2\) with \(K \in \mathbb {N}\). Thus, taking \(K := \lceil \sqrt{N/(2k)}\rceil \) we obtain

*N*.

### 4.3 Lieb–Thirring inequality

*N*-anyon wave function \(\Psi \in \mathscr {D}_\alpha ^N\) and domain \(\Omega \subseteq \mathbb {R}^2\) the local kinetic energy on \(\Omega \)

### Lemma 4.10

By applying the method of [24] (see also [20] for a more detailed exposition), replacing [24, Lemma 8] by the above bound and using (4.10) and Lemma 4.5, one directly obtains the Lieb–Thirring inequality of Theorem 2.2 for some universal constant \(C>0\).

## Footnotes

## Notes

### Acknowledgements

D. L. would like to thank Simon Larson for discussions. Financial support from the Swedish Research Council, Grant No. 2013-4734 (D. L.), the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No 694227, R. S.), and by the Austrian Science Fund (FWF), project Nr. P 27533-N27 (R. S.), is gratefully acknowledged.

## References

- 1.Correggi, M., Lundholm, D., Rougerie, N.: Local density approximation for almost-bosonic anyons. In: Proceedings of QMath13, Atlanta, October 8–11, 2016 (2017). arXiv:1705.03203 (
**to appear**) - 2.Correggi, M., Lundholm, D., Rougerie, N.: Local density approximation for the almost-bosonic anyon gas. Anal. PDE
**10**, 1169–1200 (2017). https://doi.org/10.2140/apde.2017.10.1169 MathSciNetCrossRefzbMATHGoogle Scholar - 3.Date, G., Murthy, M.V.N., Vathsan, R.: Classical and quantum mechanics of anyons, arXiv e-prints (2003). arXiv:cond-mat/0302019
- 4.Dell’Antonio, G., Figari, R., Teta, A.: Statistics in space dimension two. Lett. Math. Phys.
**40**(3), 235–256 (1997). https://doi.org/10.1023/A:1007361832622 MathSciNetCrossRefzbMATHGoogle Scholar - 5.Dyson, F.J.: Ground-state energy of a hard-sphere gas. Phys. Rev.
**106**(1), 20–26 (1957). https://doi.org/10.1103/PhysRev.106.20 ADSCrossRefzbMATHGoogle Scholar - 6.Dyson, F.J., Lenard, A.: Stability of matter. I. J. Math. Phys.
**8**(3), 423–434 (1967). https://doi.org/10.1063/1.1705209 ADSMathSciNetCrossRefzbMATHGoogle Scholar - 7.Forte, S.: Quantum mechanics and field theory with fractional spin and statistics. Rev. Mod. Phys.
**64**, 193–236 (1992). https://doi.org/10.1103/RevModPhys.64.193 ADSMathSciNetCrossRefGoogle Scholar - 8.Frank, R.L., Seiringer, R.: Lieb–Thirring inequality for a model of particles with point interactions. J. Math. Phys
**53**(9), 095201, 11 (2012). https://doi.org/10.1063/1.3697416 MathSciNetCrossRefzbMATHGoogle Scholar - 9.Fröhlich, J.: Quantum statistics and locality. In: Proceedings of the Gibbs Symposium (New Haven, CT, 1989), pp. 89–142. American Mathematical Society, Providence (1990)Google Scholar
- 10.Iengo, R., Lechner, K.: Anyon quantum mechanics and Chern–Simons theory. Phys. Rep.
**213**, 179–269 (1992). https://doi.org/10.1016/0370-1573(92)90039-3 ADSMathSciNetCrossRefGoogle Scholar - 11.Khare, A.: Fractional Statistics and Quantum Theory, 2nd edn. World Scientific, Singapore (2005)CrossRefGoogle Scholar
- 12.Larson, S., Lundholm, D.: Exclusion bounds for extended anyons. Arch. Ration. Mech. Anal.
**227**, 309–365 (2017). https://doi.org/10.1007/s00205-017-1161-9 MathSciNetCrossRefzbMATHGoogle Scholar - 13.Lerda, A.: Anyons. Springer, Berlin (1992)zbMATHGoogle Scholar
- 14.Lieb, E.H., Seiringer, R.: The Stability of Matter in Quantum Mechanics. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
- 15.Lieb, E.H., Thirring, W.E.: Bound for the kinetic energy of fermions which proves the stability of matter. Phys. Rev. Lett.
**35**, 687–689 (1975). https://doi.org/10.1103/PhysRevLett.35.687 ADSCrossRefGoogle Scholar - 16.Lieb, E.H., Thirring, W.E.: Inequalities for the moments of the eigenvalues of the Schrödinger hamiltonian and their relation to Sobolev inequalities. Studies in Mathematical Physics. Princeton University Press, Princeton (1976)zbMATHGoogle Scholar
- 17.Lieb, E.H., Yngvason, J.: The ground state energy of a dilute two-dimensional Bose gas. J. Stat. Phys.
**103**(3–4), 509–526 (2001). https://doi.org/10.1023/A:1010337215241 MathSciNetCrossRefzbMATHGoogle Scholar - 18.Lieb, E.H., Seiringer, R., Yngvason, J.: Bosons in a trap: a rigorous derivation of the Gross–Pitaevskii energy functional. Phys. Rev. A
**61**(4), 043602 (2000). https://doi.org/10.1103/PhysRevA.61.043602 ADSCrossRefGoogle Scholar - 19.Lundholm, D.: Anyon wave functions and probability distributions, IHÉS preprint, IHES/P/13/25, (2013). http://preprints.ihes.fr/2013/P/P-13-25.pdf
- 20.Lundholm, D.: Methods of modern mathematical physics: uncertainty and exclusion principles in quantum mechanics, Lecture notes, KTH (2017). arXiv:1805.03063
- 21.Lundholm, D.: Many-anyon trial states. Phys. Rev. A
**96**, 012116 (2017). https://doi.org/10.1103/PhysRevA.96.012116 ADSCrossRefGoogle Scholar - 22.Lundholm, D., Rougerie, N.: The average field approximation for almost bosonic extended anyons. J. Stat. Phys.
**161**(5), 1236–1267 (2015). https://doi.org/10.1007/s10955-015-1382-y ADSMathSciNetCrossRefzbMATHGoogle Scholar - 23.Lundholm, D., Rougerie, N.: Emergence of fractional statistics for tracer particles in a Laughlin liquid. Phys. Rev. Lett.
**116**, 170401 (2016). https://doi.org/10.1103/PhysRevLett.116.170401 ADSCrossRefGoogle Scholar - 24.Lundholm, D., Solovej, J.P.: Hardy and Lieb–Thirring inequalities for anyons. Commun. Math. Phys.
**322**, 883–908 (2013). https://doi.org/10.1007/s00220-013-1748-4 ADSMathSciNetCrossRefzbMATHGoogle Scholar - 25.Lundholm, D., Solovej, J.P.: Local exclusion principle for identical particles obeying intermediate and fractional statistics. Phys. Rev. A
**88**, 062106 (2013). https://doi.org/10.1103/PhysRevA.88.062106 ADSCrossRefGoogle Scholar - 26.Lundholm, D., Solovej, J.P.: Local exclusion and Lieb–Thirring inequalities for intermediate and fractional statistics. Ann. Henri Poincaré
**15**, 1061–1107 (2014). https://doi.org/10.1007/s00023-013-0273-5 ADSMathSciNetCrossRefzbMATHGoogle Scholar - 27.Lundholm, D., Portmann, F., Solovej, J.P.: Lieb–Thirring bounds for interacting Bose gases. Commun. Math. Phys.
**335**(2), 1019–1056 (2015). https://doi.org/10.1007/s00220-014-2278-4 ADSMathSciNetCrossRefzbMATHGoogle Scholar - 28.Lundholm, D., Nam, P.T., Portmann, F.: Fractional Hardy–Lieb–Thirring and related inequalities for interacting systems. Arch. Ration. Mech. Anal.
**219**(3), 1343–1382 (2016). https://doi.org/10.1007/s00205-015-0923-5 MathSciNetCrossRefzbMATHGoogle Scholar - 29.Myrheim, J.: Topological aspects of low dimensional systems. In: Comtet, A., Jolicœur, T., Ouvry, S., David, F. (eds.) Les Houches—Ecole d’Ete de Physique Theorique, Anyons, vol. 69, pp. 265–413. Springer, Berlin (1999). https://doi.org/10.1007/3-540-46637-1_4 CrossRefzbMATHGoogle Scholar
- 30.Nayak, C., Simon, S.H., Stern, A., Freedman, M., Sarma, S.D.: Non-abelian anyons and topological quantum computation. Rev. Mod. Phys.
**80**, 1083–1159 (2008). https://doi.org/10.1103/RevModPhys.80.1083 ADSMathSciNetCrossRefzbMATHGoogle Scholar - 31.Ouvry, S.: Anyons and lowest Landau level anyons. Séminaire Poincaré
**11**, 77–107 (2007). https://doi.org/10.1007/978-3-7643-8799-0_3 CrossRefzbMATHGoogle Scholar - 32.Stern, A.: Anyons and the quantum Hall effect—a pedagogical review. Ann. Phys.
**323**(1), 204–249 (2008). https://doi.org/10.1016/j.aop.2007.10.008. (**January Special Issue 2008**)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 33.Wilczek, F.: Fractional Statistics and Anyon Superconductivity. World Scientific, Singapore (1990)CrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.