PRL 111, 100405 (2013)

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Entanglement Sum Rules Brian Swingle Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA (Received 31 October 2012; published 4 September 2013) We compute the entanglement entropy of a wide class of models that may be characterized as describing matter coupled to gauge fields. Our principle result is an entanglement sum rule that states that the entropy of the full system is the sum of the entropies of the two components. In the context of the models we consider, this result applies to the full entropy, but more generally it is a statement about the additivity of universal terms in the entropy. Our proof simultaneously extends and simplifies previous arguments, with extensions including new models at zero temperature as well as the ability to treat finite temperature crossovers. We emphasize that while the additivity is an exact statement, each term in the sum may still be difficult to compute. Our results apply to a wide variety of phases including Fermi liquids, spin liquids, and some non-Fermi liquid metals. For example, we prove that our model of an interacting Fermi liquid has exactly the log violation of the area law for entanglement entropy predicted by the Widom formula in agreement with earlier arguments. DOI: 10.1103/PhysRevLett.111.100405

PACS numbers: 05.30.d, 03.65.Ud, 71.10.Ay, 71.27.+a

Introduction.—A recent exchange of ideas between quantum many-body physics and quantum information science has led to an increased appreciation for the fundamental role of entanglement in quantum matter. In particular, long-range entanglement underlies many of the more interesting states of matter now known experimentally, including Fermi liquids [1–3], quantum critical points [4–10], and topological phases [11–18]. In some cases, entanglement is essentially the only completely general probe of such states [19,20]. Entanglement considerations have also led to a variety of other results, including a new class of variational states [21–23] and a classification of phases in one dimension [24,25]. The concept of entanglement entropy has played a crucial role in these recent developments, so we first review entanglement entropy. We consider a large quantum system divided into two parts, A and B, such that the whole system is in a pure state j c AB i, e.g., the ground state of a local Hamiltonian. The entanglement entropy of A is defined as the von Neumann entropy of the reduced density matrix of A: SðAÞ ¼ trA ðA lnðA ÞÞ. This definition makes sense in general, but only when the total system is pure does the entanglement entropy truly measure entanglement between A and B. Typically, A and B are spatial regions, but other kinds of Hilbert space partitions can be considered. It is also useful to consider the Renyi (entanglement) entropy, defined as Sn ðAÞ ¼

1 lnðtrðnA ÞÞ; 1n

S ¼ Sm þ Sg ;


as a generalization of entanglement entropy. Knowledge of the Renyi entropy Sn ðAÞ for all n values is equivalent to knowledge of the full spectrum of A . The basic fact about entanglement entropy in local ground states is the area law [26,27]. This law states that the entanglement entropy typically scales like SðAÞLd1 0031-9007=13=111(10)=100405(5)

where L is the linear size of A and d is the spatial dimension. However, it must be emphasized that the area law is not completely universal; exceptions are known in a variety of gapless systems. Conversely, although it is not proven, it is believed that the area law holds for all gapped phases of matter (see Ref. [28] for a renormalization group argument and Ref. [29] for a partial result). The exceptions include conformal field theories in one dimension where S  c lnðLÞ (c is the central charge) [4,6,7] and Fermi liquids in d > 1 dimensions where S  Ld1 lnðLÞ (as well as other systems with a Fermi surface) [1–3,30,31]. Furthermore, although the area law fails for these systems, renormalization group arguments [3,28] correctly capture the leading size dependence. Most other known gapped and gapless phases in d > 1 satisfy the area law. However, despite these many advances, entanglement remains poorly understood in the context of gapless systems. Here we make progress on this issue using what we term entanglement sum rules. An entanglement sum rule provides a way to compute the entanglement properties of an interesting phase by dividing it into more elementary components. A prominent example is the problem of spin liquids where one often has a description involving matter (spinons) coupled to gauge fields (for example, a Z2 spin liquid [32]). In this context, the prototypical entanglement sum rule is the statement that (2)

where Sm and Sg come from matter and gauge fields, respectively. This sum rule is a nonperturbative statement about entanglement entropy that greatly extends our ability to compute entropies and can be very useful for comparing with numerical computations. Such a rule has previously been obtained in the context of various topological phases [9,16] in a certain limit, but our result is much more general.


Ó 2013 American Physical Society

PRL 111, 100405 (2013)

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Of course, not every spin liquid state admits an entanglement sum rule (for example, those involving continuous gauge fields), but many interesting states do. The notion of an entanglement sum rule is also not restricted to spin liquids; for example, we apply it to a Fermi liquid state below. We prove a general entanglement sum rule for a wide class of special models that include, in various limits, free fermions, the toric code [33] (or equivalently a Z2 spin liquid [32]), and free fermions coupled to the toric code. In the context of the model, our sum rule applies to the full entanglement entropy, but more generally, our sum rule indicates that the universal terms in a variety of phases and phase transitions will be additive in the sense described above. Note that what terms are considered universal depend on the nature of the phase. An important advance over the results in Refs. [9,16] is that we can also vary the gauge field dynamics (previous computations were down in an extreme deconfined limit where the gauge field does not fluctuate). We also demonstrate that the sum rule holds for the Renyi entropy and for finite temperature entanglement-thermal crossover functions. For concreteness, we focus on a particular model, introduced in Ref. [34] in the context of non-Fermi liquid metals, and prove the entanglement sum rule for this model. More generally, as briefly discussed in the Supplemental Material [35], our sum rule applies to a variety of physical states including Fermi liquids, some spin liquids, some non-Fermi liquid metals, and more. Before proceeding, let us emphasize a few important points. We do consider special models that are not completely generic; for example, the Fermi liquid state we consider has nontrivial quasiparticle residue, which can even be tuned to zero, but all Landau parameters are zero. The entanglement sum rule, as a statement about the full entropy, only applies to these special models. However, as long as one of our factorizable models lies within the same universality class as the phase or phase transition in question, the entanglement sum rule will still apply to universal terms in the entanglement entropy since such terms depend only on the universality class. Furthermore, although claims of universality are nontrivial physical statements based on the renormalization group, we are able to partially check these claims in that our models have many (actually infinite) parameters that we can prove do not affect various putative universal terms. Thus, our sum rule is useful because, as we demonstrate below and in the Supplemental Material [35], models of the type we consider are in the same universality class as many phases of physical interest, including Fermi liquids, certain kinds of spin liquids (e.g., Ref. [16]), and a variety of critical points (e.g., Ref. [9]). With this perspective in mind, we now turn to the details of our model and results. Model.—Consider a square lattice with spinless fermions cr on sites and Ising spins zrr0 on links. Following Ref. [34], the Hamiltonian is taken to be

H ¼ w


X X cyr zrr0 cr0   cyr cr  gJ zrr0

hrr0 i

hrr0 i


X X Y z y Y  J ð1Þcr cr xrr0  V rr0 : hr0 ri



p hrr0 i2p

The J term is a product over all links sharing a vertex r whereas the V term is a product over all links in given plaquette p. We will see that this Hamiltonian describes a (highly fine tuned) transition from a Fermi liquid to a metal with vanishing quasiparticle residue as a function of g. This Hamiltonian can actually be decoupled, in a sense made precise below, for all values of the parameters, and can be partially solved. However, let us first get a sense of the physics as a function of the coupling g. When g  1 the spins want to polarize with zrr0 ¼ 1. The fermions then decouple from the spins and are simply described by a free Fermi gas with hopping w and chemical potential . In the opposite limit, when g  1, the spin Hamiltonian is that of the toric code or of Z2 gauge theory with gapped matter, and the fermions cr couple minimally to the gauge field. Thus, in this limit we have again a free Fermi surface coupled to the tensionless limit of a Z2 topological phase. The topological structure guarantees a transition between the g  1 and g  1 phases. The V term commutes with everything else in the Hamiltonian, andQhence we may consider only the subspace in which p ¼ hrr0 i2p zrr0 ¼ 1 for all p. In our interpretation above, p is the Z2 flux through the plaquette p. Within this subspace, the  variables are constrained, so we can introduce new variables zrr0 ¼ xr xr0 and



xrr0 ¼ zr :


hr0 ri

This transformation is the usual duality transformation of Wegner [36]. The commutation relations of these operators are preserved by this identification, provided the  variables also obey the standard Pauli commutation relations. If we now define fr ¼ xr cr ;


~xr ¼ xr ;


and y


~zr ¼ ð1Þcr cr zr ¼ ð1Þfr fr zr ;


then the ~ and f variables all commute and the Hamiltonian is X X X X H ¼ w fry fr0   fry fr  gJ ~xr ~xr0  J ~zr :


hrr0 i


hrr0 i



PRL 111, 100405 (2013)

Thus, we have decoupled fermion (f) and transverse field Ising (~ ) systems. We have also dropped the V term since it is simply an additive constant within the constrained Hilbert space. We see immediately that ff correlations are identical to that of a free fermion model for all g values. In particular, since nr ¼ cyr cr ¼ fry fr , it follows that the physical density-density correlator is given by the free fermion result for all g. Furthermore, the partition function of the model exactly factorizes so that SðTÞ ¼ Sf ðTÞ þ SIsing ðTÞ;

rr0 2Lðfxr gÞ

where xr ¼ 0, 1 specifies whether we take the operator (1  cyr cr ) or cyr cr in the product and where Lðfxr gÞ is a set of links such that the links form closed paths that only terminate at points where xr ¼ 1. Roughly speaking, we choose an assignment of strings of z operators that end on sites with xr ¼ 1. We make an arbitrary choice for the set L for fixed fxr g; however, in the constrained subspace any two choices of L are actually equivalent because they differ only by closed loops. Since we have p ¼ 1 for all elementary plaquettes, such closed loop operators built from elementary loops do nothing to the state. To define U, it is also necessary to restrict to the subspace containing an even number of fermions; i.e., we must also restrict the sum P over fxr g so that r xr is even. The reason for considering U becomes apparent when we use the relation zrr0 ¼ xr xr0 . We find that all the strings of z operators dramatically simplify, and we are left with a factor of xr for every site with xr ¼ 1. We have Y XY ð1  cyr cr Þ1xr ðcyr cr Þxr ðxr Þxr : (12) U¼ fxr g r

With the use of this form, it follows that Ucr Uy ¼ xr cr ¼ fr



and that y

Uzr Uy ¼ ð1Þcr cr zr ¼ ~zr :


Hence, U converts from c and  to f and ~, and U decouples the Hamiltonian X X X XY UHUy ¼w cyr cr0  cyr cr gJ zrr0 J xrr0 : hrr0 i


where SðTÞ is the thermal entropy. As a technical subtlety, this entropy result only applies if we first send V ! 1 so that no states with p  1 are excited since otherwise the  variables are insufficient to describe the full Hilbert space. Indeed, since the Ising part at low temperatures is either gapped or at worst a 2 þ 1 dimensional conformal field theory with SIsing  T 2 it immediately follows that the thermal entropy is dominated by the f Fermi surface at low temperatures. Entanglement.—How we compute the entanglement entropy depends on what cut we choose to make. For example, if we regard c and  as the local degrees of freedom, then we cannot make a cut in the  variables since these are nonlocally related to the  variables. To begin, let us determine the ground state j0i of the system. We know that the Hamiltonian is decoupled in the f and ~ variables. To relate these degrees of freedom to the original c and  variables, we introduce a unitary transformation X Y Y U¼ ð1  cyr cr Þ1xr ðcyr cr Þxr zrr0 ; (11) fxr ¼0;1g r

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hrr0 i

r hr0 ri

(15) The ground state of H is Uy jcðw; ÞijðgÞi;


where jci and ji are the ground states in the decoupled model Eq. (15). The ground state jciji manifestly obeys the entanglement sum rule. Thus the important question is how does U modify the entanglement of region A? It is useful to decompose the fermion state into parity eigenstates by writing jci ¼ jc; 0i þ jc; 1i;


where 0, 1 ¼ NA mod 2. This decomposition is useful because when ð1ÞNA ¼ 1 we can choose the set L such that all z operators act within A and B with no boundary terms. Similarly, when ð1ÞNA ¼ 1, L may be chosen so that only a single specified link connecting A to B carries a z operator. Hence, in both cases U factorizes into UAi UBi with the unitaries UAi and UBi each acting just on A or B (we adopt the convention that the boundary links belong to A). Let us compare the reduced density matrix of A with and without the action of U. Without U, we have trB ððjc; 0i þ jc; 1iÞðhc; 0j þ hc; 1jÞ  jihjÞ;


which reduces to ð0Þ þ ð0Þ trB ððj0ih0j þ j1ih1jÞ  jihjÞ  A0 A1


since the cross terms cancel. With U, we have trB ðUA0 j0ih0j  jihjðUA0 Þy þ UA1 j1ih1j  jihjðUA1 Þy Þ 0 y 1 ð0Þ 1 y ¼ UA0 ð0Þ A0 ðUA Þ þ UA A1 ðUA Þ :


Note that once again the cross terms cancel since, for example, we can perform the trace in the number basis with no matrix elements between different parity sectors. However, we now see that since the state of A before U acted was diagonal in parity, it follows that the state after U acted can be related to the state before U acted by a unitary transformation. Define UA to be ! UA0 0 ; (21) 0 UA1


PRL 111, 100405 (2013)


where the two components refer to the parity sectors. Computing ð0Þ y UA ðð0Þ A0 þ A1 ÞUA


0 y 1 ð0Þ 1 y UA0 ð0Þ A0 ðUA Þ þ UA A1 ðUA Þ ;


we obtain

since ð0Þ Ax has a definite parity of x. To summarize, we have shown that the action of U on the state jciji reduces, after the trace over B, to a unitary transformation of the density matrix of A. Hence, all spectral data of A are preserved by the application of U despite the nonlocal transformation between  and . Our final result is Sn ðAÞ ¼ Sn ðA; cÞ þ Sn ðA; Þ since the state jciji is factorized. Note, however, that the trace is still over the  variables (not the  variables). In words, we find that the total Renyi entropy is simply the sum of the Renyi entropy of fermions with band structure specified by w,  (with no coupling to ) and the Renyi entropy of the  model (with no coupling to c). We emphasize again that while this means the fermionic component of the entropy may be computed directly from the state jfi, we must still convert from the nonlocal  variables to the local  variables to compute the  contribution to the entropy. Application to Fermi liquids.—The Ising part of the entanglement entropy never scales faster than L (L is the linear size of A), and thus the entanglement entropy has an L lnðLÞ term determined by the f Fermi surface for all g. The entropy of the system at g ¼ 1 is simply given by the f Fermi surface whereas the entropy at g ¼ 0 is given by S ¼ Sf þ ðj@Aj  1Þ ln2 in accord with the results of Ref. [9,16] (as usual, we do the splitting over  variables as in Ref. [14]). Our argument works at the level of a unitary transformation, and since the parity structure we used in our proof is also a property of excited states, we see that the full entanglement-thermal crossover function Sn ðA; TÞ is also exactly additive [31]. However, we must first send V ! 1 for the thermal results to hold in this simple form. Within the Fermi liquid phase, we have thus partially confirmed the universality claimed in Refs. [3,37] since we have exhibited a Fermi liquid state with variable quasiparticle residue in which the entropy is exactly the free fermion result for the L lnðLÞ term. Furthermore, because the  variables are gapped within the Fermi liquid phase, all the mutual information calculations of Ref. [38] also go through in the Fermi liquid state (the generalization of our results above to multiple regions is trivial if the  variables are gapped). Recently, an interesting numerical calculation [39] of the Renyi entropy in Fermi liquids has also largely confirmed the predictions of Refs. [3,37,40] concerning the universality of the Widom formula with small deviations observed only at very strong interactions. The precise origin of these discrepancies is not yet understood. Discussion.—In this Letter, we have derived an entanglement entropy sum rule for a wide class of models. There

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are many simple extensions of our arguments here: we may modify the band structure of the f fermions, add pairing terms for the f fermions, couple bosons instead of fermions to the Ising spins, generalize to Zn gauge fields or clock models [41], and work in higher dimensions. All these generalizations obey an entanglement sum rule as discussed in the Supplemental Material [35]. Thus, our results apply to a wide range of models where they provide a simple way to compute entropies as well as a demonstration that supposed universal terms are indeed independent of some microscopic parameters. For the models we considered, we have proven exact additivity for the full entropy. More generally, our results imply that the universal terms will add in any phase smoothly connected to one describable by our models. These phases include the Fermi liquid, the orthogonal metal [34], Z2 spin liquids [32], and much more. Moreover, we have the added ability to follow the entropy through various quantum critical points. We also give an example of a broader application of our ideas. It was noted in Ref. [34] that Luttinger’s theorem relating the Fermi surface volume as defined by the electron spectral function to the density of electrons is not satisfied in the orthogonal metal; i.e., the electron spectral function sees an overabundance of critical surfaces relative to the f Fermi surface. Luttinger’s theorem only applies in its original form to Fermi liquids, so nothing is surprising about this observation. However, the Luttinger count may be upheld, provided we use the entanglement entropy to measure the f Fermi surface. This suggestion makes contact with recent attempts to measure the Fermi surfaces of gauge charged particles in compressible states using entanglement entropy in a holographic context [42,43]. We have shown that the entropy does capture the ‘‘hidden’’ (hidden, because f is not a gauge invariant operator) f Fermi surface in the compressible orthogonal metal. Finally, let us conclude by noting that many recent experiment measurements on the organics [44,45] and other two-dimensional magnetic systems have revealed evidence of a spin liquid ground state, possibly one including a Fermi surface of spinons. However, the identification of the precise spin liquid state has remained elusive with various experiments pointing in different directions. It may be that the detailed information on entanglement presented here when combined with numerical work can help elucidate the precise nature of these exciting new phases. B. G. S. is supported by a Simons Fellowship through Harvard University and acknowledges helpful conversations on some of these topics with S. Sachdev, L. Huijse, and T. Senthil.

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Entanglement sum rules.

We compute the entanglement entropy of a wide class of models that may be characterized as describing matter coupled to gauge fields. Our principle re...
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