Nodal Line Entanglement Entropy: Generalized Widom Formula from Entanglement Hamiltonians

Nodal line entanglement entropy: Generalized Widom formula from entanglement Hamiltonians

Michael Pretko Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA (Dated: May 22, 2017) A system of fermions forming a Fermi surface exhibits a large degree of quantum entanglement, even in the absence of interactions. In particular, the usual case of a codimension one Fermi surface leads to a logarithmic violation of the area law for entanglement entropy, as dictated by the Widom formula. We here generalize this formula to the case of arbitrary codimension, which is of particular interest for nodal lines in three dimensions. We first rederive the standard Widom formula by calculating an entanglement Hamiltonian for Fermi surface systems, obtained by repurposing a trick commonly applied to relativistic theories. The entanglement Hamiltonian will take a local form in terms of a low-energy patch theory for the Fermi surface, though it is nonlocal with respect to the microscopic fermions. This entanglement Hamiltonian can then be used to derive the entanglement entropy, yielding a result in agreement with the Widom formula. The method is then generalized to arbitrary codimension. For nodal lines, the area law is obeyed, and the magnitude of the coefficient for a particular partition is non-universal. However, the coefficient has a universal dependence on the shape and orientation of the nodal line relative to the partitioning surface. By comparing the relative magnitude of the area law for different partitioning cuts, entanglement entropy can be used as a tool for diagnosing the presence and shape of a nodal line in a ground state wavefunction.

I. INTRODUCTION violations8, which may well be valid, but these results are less well understood.) This is quite a striking result, The concept of entanglement is a powerful tool for since it can occur even in a free system (though the result 9 understanding, characterizing, and classifying quantum also survives in an interacting Fermi liquid ). This is in phases of matter. For example, topologically ordered contrast to most other highly entangled phases, which systems, such as gapped spin liquids, can be character- require strong interactions between the microscopic de- ized by the inability of their spatial entanglement pattern grees of freedom. The high degree of entanglement in to be smoothly deformed to a direct product state1. Such these systems is facilitated by the large number of gapless highly-entangled phases of matter often have exotic exci- modes, a very distinct feature of Fermi surfaces. This tations, such as anyons, and have been the focus of much logarithmic violation of the area law then serves as an research in recent years. In order to characterize the en- important diagnostic tool for Fermi surfaces in a ground tanglement in a many-body state, a useful prescription state wavefunction. Obviously the result applies to free is to partition our system into two distinct spatial re- fermion systems, but it is arguably much more important gions. We can then form the reduced density matrix ρ in the study of strongly correlated systems, such as in 10 for a specific region by tracing out all degrees of freedom composite Fermi liquids and spin liquids with spinon 11 in its complement. We then define the entanglement en- Fermi surfaces , where the entanglement entropy can tropy of the partition as the von Neumann entropy as- provide compelling evidence for emergent Fermi surfaces. d−1 sociated with this density matrix, S = −T r[ρ log ρ]. For Furthermore, the coefficient of the L log L term in the ground states of local Hamiltonians in dimensions higher entanglement entropy is universal, in that it depends than one, the entanglement entropy typically obeys an only on the shape and size of the Fermi surface and the area law2, in that the entropy associated with a region of chosen partition, but does not depend on short-distance characteristic size L in d dimensions scales as the surface lattice physics. This is in contrast to a standard area area, S ∝ Ld−1. law, where the coefficient is highly sensitive to short- A notable exception to this rule, however, occurs in distance physics and can therefore be different in two Fermi surface systems. For a garden variety codimension systems with the same low-energy description. The en-

arXiv:1609.07502v2 [cond-mat.str-el] 8 Jun 2017 tanglement entropy (per spin species) in a Fermi surface one Fermi surface, the entanglement entropy is known 4 (both theoretically and numerically) to have a logarith- system is given by the well-known Widom formula : mic violation of the area law, S ∝ Ld−1 log L .3,4 The log L Z Z S = |nˆ · nˆ | (1) general intuition is that a Fermi surface is the result (2π)d−112 r k of sewing together many one-dimensional gapless theo- P.S. F.S. ries, which also violate the area law logarithmically5,6. The two integrals are over the partitioning surface and To date, this is the largest violation of the area law in the Fermi surface, respectively. (Note that the integral a well-established physical model. (The same sort of over the partition provides the factor of area, Ld−1.) logarithmic violation also occurs in systems with “Bose The integrand is the inner product of the unit normals surfaces”7, where bosonic excitations have vanishing en- of the two surfaces, Fermi and partitioning. This uni- ergy. There are proposals for states with more severe versal coefficient gives us important information about 2 both the size and shape of the Fermi surface. In par- the normal to the partitioning surface. The integrals are ticular, by choosing different partitions, we could deter- over the partitioning surface and the nodal line, respec- mine any cross-sectional area of the Fermi surface and tively. N is a non-universal numerical constant, and a thereby mostly reconstruct the original shape. (We note is the lattice cutoff, which we will address in a moment. that Bose surface systems obey a similar, but slightly For general codimension q, the entanglement entropy be- different Widom formula.13) comes: While the usual case of a codimension one Fermi sur- Z Z Nq q face is well-understood, comparatively little has been Sq = |Pknˆr| (3) aq−1 said about entanglement for Fermi surfaces of higher P.S. F.S. codimension, probably because there are far fewer exam- where Pknˆr is the projection of the partitioning surface ples that are not purely of academic interest. The sim- normaln ˆr into the normal subspace of the Fermi surface, plest case is a zero-dimensional (codimension d) Fermi which we will explain in more detail later. (This result surface, i.e. a Fermi point, such as a Dirac cone. In also applies to the case of Fermi points, where the Fermi this case, for d > 1, the entanglement entropy simply surface integral should be replaced by a discrete sum, has an area law with a non-universal coefficient, and it and the projection operator becomes the identity. The is difficult to extract anything useful. However, there above equation then gives the standard area law.) is another important case which is a bit more interest- Note that the presence of the short-distance cutoff in ing: a Fermi nodal line in three dimensions (codimension these formulas makes the overall magnitude of the area two). As compared to a normal Fermi surface, a nodal law coefficient non-universal. Nevertheless, we find an line is harder to realize. Naively, the simplest exam- interesting universal dependence on the shape of the ple would be a nodal line semimetal in a free fermion nodal line relative to the partitioning surface. While system. Indeed, it is possible to have a stable band- the overall magnitude for a specific partition is not par- touching along a line in momentum space, protected by ticularly meaningful, we can examine the ratio of the symmetries14–18. However, it is not possible for this line area law between different partitions, for which the non- to exist stably at the Fermi energy through any finite universal prefactors will cancel out. This will give us amount of symmetry protection14,19. (Nevertheless, a important information about the relative size of various Fermi surface which is only weakly deformed from such cross-sections of the line. The overall scale of the nodal a line will behave roughly like a nodal line semimetal in line cannot be determined by this procedure and must certain parameter regimes, and recent experiments seem be found through other means. But once a scale is fixed, to indicate that the free fermion nodal line is of direct this formula allows one to map out the shape of the nodal relevance for real materials20.) In actuality, the simplest line. We will also discuss some of the unique features of realization of a nodal line occurs in certain supercon- Equation 2 which can be used to distinguish nodal lines ductors, where there is no corresponding instability, and from other anisotropic entanglement patterns. there is experimental evidence that such nodal lines oc- In order to find this generalized Widom formula, we cur in certain heavy-fermion superconductors21,22. Per- must investigate the entanglement spectrum of Fermi haps most interestingly, it has recently been shown that surface systems. The entanglement spectrum of a par- there exist stable nodal line spin liquids, consisting of a tition is defined as the spectrum of the reduced density nodal line of fermionic spinons coupled to an emergent matrix ρ associated with a particular region. This in- U(1) gauge field23. Unlike their band-theoretic cousins, formation can be nicely encoded in an object called the the nodal lines of these spin liquid states can be pro- entanglement Hamiltonian H˜ (a.k.a. modular Hamilto- tected by symmetry to lie exactly at the Fermi energy. nian), defined via: In such a system, the entanglement entropy will have a ˜ large contribution from the many gapless modes along ρ = e−H (4) the nodal line (along with a separate gauge field contribution). Finding the entanglement entropy for nodal Since ρ is hermitian and has all non-negative eigenval- ues, it is always guaranteed that we can write ρ in this lines is therefore relevant for a number of different phys- ˜ ical systems. way. In this sense, H = − log ρ is simply a trivial rewrit- In this work, we will find a generalized Widom for- ing of the problem. However, the usefulness of this form mula which applies to Fermi surfaces of arbitrary codi- is the fact that, in many important cases, the entangle- mension, with emphasis on the nodal line. We will find ment Hamiltonian turns out to be local (i.e. a sum of that, unlike the codimension one case, higher codimen- local operators). For example, it is always local for any sion Fermi surfaces obey the area law. For nodal lines relativistic field theory partitioned into two half-spaces, in three dimensions (codimension two), we will find that as we will discuss in the next section. By repurposing the entanglement entropy is given by: this relativistic tool, we will find that the entanglement Hamiltonian for Fermi surface systems, of any codimen- Z Z sion, can also be written in local form in terms of a N ˆ 2 S = |t × nˆr| (2) suitably defined set of low-energy variables, specifically a P.S. N.L. a set of decoupled patch variables representing the dif- where tˆ is the unit tangent to the nodal line andn ˆr is ferent portions of the surface. However, the result will 3 still be nonlocal with respect to the original microscopic fermions, in agreement with earlier treatments of free fermion entanglement Hamiltonians24.

II. A RELATIVISTIC TRICK

In order to find the entanglement entropy for Fermi surface systems, we will repurpose a trick from the high energy playbook. Consider the ground state of any rela- FIG. 1. On scales small compared with L, the partitioning tivistic field theory, and let us choose our partition to be surface looks planar. into two half spaces, x1 < 0 and x1 > 0. Let the Hamil- tonian density of our theory be H, such that the Hamil- tonian is H = R ddxH. After tracing out the half-space On this scale, the curvature is insignificant and the sur- x1 < 0, the entanglement Hamiltonian of the ground face looks flat (see Figure 1), corresponding to the planar state in the remaining region takes the form25,26: surface considered earlier. Since the dominant contribution to entanglement is local (i.e. intrapatch, not inter- Z 2πx patch), we can evaluate the corresponding contribution H˜ = ddx 1 H (5) to the entanglement entropy from one patch, then add x1>0 c up the contributions from each such patch to get the This result is sometimes known as the Bisognano- total entanglement entropy. From this perspective, the Wichmann (BW) theorem. (Note that we will keep the area law seems inevitable, coming from integrating over speed of light c explicit throughout, for reasons which all patches which make up the partitioning surface. The will be apparent later. Other constants, like ~ and kB, entanglement entropy must contain a factor of area, and will always be taken to be unity.) A short version of the it is the deviations from the area law which seem more proof of this statement is given in Appendix A. The intu- difficult to explain. In order to see how these viola- ition is fairly simple and is closely related to the fact that tions arise, we have to develop a method for actually H˜ is the generator of boosts in the x1 direction. Notably, extracting the entanglement entropy from the entangle- the proof only relies on a Lorentz symmetry between the ment Hamiltonian. x0 (time) and x1 coordinates. The other spatial coordi- To do so, we must take a closer look at our reduced nates can simply be regarded as parameters and come density matrix: along for the ride. Z The result for a planar partitioning surface may seem 2πx1 ρ ∝ exp − ddx H (6) like a special case, but it is not. In fact, this example is x1>0 c sufficient to provide a general understanding of the area law for entanglement entropy26,27. For the ground state We note that this density matrix looks very much like of a local Hamiltonian, the largest part of the entangle- a thermal density matrix, e−βH , except that now the ment is generally between degrees of freedom separated temperature depends on position. We write our density by very short distances, since they are directly coupled. matrix as: This is manifest from the fact that the area law coef- Z ficient typically diverges as the lattice scale a is taken ρ ∝ exp − β(x)H(x) (7) to 0, indicating that most of the entanglement is coming from short-distance physics. Thus, in order to get the leading contribution to the entanglement entropy, we where our locally defined temperature is T (x) = only need to consider the region in the immediate vicin- β−1(x) = c . Essentially, the problem of ground state 2πx1 ity of the partitioning surface. In order to get important entanglement has mapped onto a thermodynamic prob- subleading corrections, such as topological entanglement lem, one where the partitioning surface is incredibly hot, entropy, one needs to be a bit more careful. But for the and the temperature falls off to zero in the bulk. This dominant term of the entanglement entropy, which is makes some intuitive sense, in that far away from the what we will concern ourselves with here, we only need surface the ground state is unaffected by the tracing out to worry about local properties of the partitioning sur- procedure, due to the primarily local nature of the en- face. tanglement. Using this intuition, instead of viewing the partition- With this interpretation in hand, we can then find the ing surface globally, we zoom in on a small patch. We dominant contribution to the entanglement entropy by choose the dimensions of the patch to be large compared summing up the local thermal entropy densities: to the lattice scale a, so that the continuum description is valid, but small compared to the size of the partition Z S = ddx s (T (x)) (8) L, so that the precise shape of the surface is irrelevant. ent therm x1>0 4

This local thermal framework is reminiscent of a have such a large low-temperature thermal entropy. If Thomas-Fermi sort of approximation and should be suffi- it is gapless, then it is likely described by a CFT, which cient for extracting the leading piece of the entanglement always has η ≥ 1. If it is gapped, then the thermal en- entropy. To demonstrate the power and the essential tropy is exponentially small at low temperatures. Thus, correctness of this thermodynamic method, let’s show logarithmic violations of the area law are likely the most that it reproduces the exact universal result for a one- severe type of violation possible in a relativistic system. dimensional conformal field theory, as outlined in28. To avoid confusion with the speed of light c, we will denote central charges by C. In a relativistic CFT, the thermal III. APPLICATION TO FERMI SURFACE entropy density is given by29: SYSTEMS πC stherm(T ) = T (9) For relativistic theories, we have seen that the entan- 3c glement Hamiltonian is relatively simple. In a condensed Let us take our partition to be a finite segment of length matter context, however, we are typically concerned with L. The integral over area then becomes a sum over the explicitly nonrelativistic systems, where exact results are two endpoints of the segment. Using our prescription of difficult to come by. One important result is that, for a integrating the thermal entropy over the half-space, we quadratic Hamiltonian, the entanglement Hamiltonian is obtain the entanglement entropy as: also quadratic24. For example, in a system of fermions freely hopping on a lattice, H = P t c†c , the entan- Z ij ij i j πC c glement Hamiltonian must take the similar form: S = 2 dx1 = (10) x1>0 3c 2πx1 X † C Z 1 C H˜ = hijc cj (12) dx = log(L/a) (11) i 1 ij 3 x1>0 x1 3 where we have taken a short-distance cutoff at the lat- where c† and c are the fermion creation and annihila- tice scale, a, and a large-distance cutoff at the size of tion operators and the sum runs over pairs of lattice the interval, L. Notice that the speed of light has com- sites. Formally, this looks just like another free fermion pletely canceled out. This is the exact result known for Hamiltonian. Unfortunately, the matrix elements hij are the entanglement entropy of finite segments in confor- in general neither translationally invariant nor even lo- mal field theories30. Furthermore, it readily generalizes cal. Except in certain special cases, there is no closed to the case where the region in question is a disjoint set form solution for the matrix elements, and they can typ- of intervals, instead of just a single interval. The fact ically only be determined numerically. While in prin- that these equations match exactly, not just in terms of ciple we know that such a quadratic result must hold, dependence on parameters but in numerical coefficient as in practice it is difficult for us to directly extract the well, is strong evidence that this method can be taken entropy from this form of the entanglement Hamilto- seriously. nian. (The quadratic result has been useful, however, Notice that the result for a one-dimensional CFT vio- in the study of entanglement in topological insulators lates the area law (which in one dimension would simply and superconductors33.) In Appendix B, we will show be S = constant) by a logarithmic factor. This logarith- how to make connection with this microscopic result, but mic divergence comes about from the large x1 behavior for now, we will take a different approach. of the integral in Equation 11, which corresponds to the In order to proceed, we shall now adapt the BW re- low-temperature thermal behavior s(T ) ∼ T in one di- sult from the previous section to obtain the entangle- mension. Such connections between entanglement and ment Hamiltonian for a Fermi surface system with a thermodynamics have been noted before31,32. In higher- spatial partition into two half-spaces. We will then ob- dimensional CFTs, we would have s(T ) ∼ T d. The re- tain the entanglement entropy for a generic partition by sulting thermal integral will then converge at large x1, integrating over the partitioning surface, as before. To resulting in an area law. More generally, we can say for accomplish these goals, we will work with an effective a relativistic system that whether or not the area law low-energy patch theory of the Fermi surface, which will is obeyed is a direct consequence of the low-temperature essentially be equivalent to a set of lower-dimensional thermal entropy behavior. Suppose the low-temperature relativistic theories. We will then apply the BW result η thermal entropy obeys a power-law, stherm(T ) ∼ T . (with subtle modifications) separately to each patch. To For η > 1, the thermal integral of Equation 8 will con- realize this explicitly, we shall break up a Fermi surface of verge at large x1, giving an area law. For η = 1, such as characteristic size kF into small patches of size Λ kF , in one-dimensional CFTs, the area law is violated log- as depicted in Figure 2. All modes outside these patches arithmically. (We will see that similar logic holds for have already been integrated out of our effective low- Fermi surfaces.) For η < 1, there should be power law energy theory. Each patch of the Fermi surface (labeled violations of the area law, S ∼ Ld−η. However, it seems by index p) will be described by its own independent P unlikely that one can engineer a relativistic system to Hamiltonian, H = p Hp. 5

the normal direction to the Fermi surface (which is different at each patch), and we assume that all modes with k⊥ > Λ have already been integrated out of the problem. The Fermi velocity depends on the patch, but is a constant within a specific patch. Letting x⊥ be the spatial coordinate corresponding to k⊥, we can perform a Fourier transform within each patch (cutting off all k integrals at Λ) and write the Hamiltonian in the following real-space form: Z X d H = d x vF Ψp(σx∂x⊥ )Ψp ≡ patches Z (14) X d d xHp patches FIG. 2. Though our Fermi surface has size of order kF , all low-energy degrees of freedom lie within order Λ of the Fermi Importantly, note that the real space field Ψp(x) is not surface. For our arguments, we look at individual patches, the same as the original microscopic fermion field, which such as the red box shown above, which is characterized by would have involved a global Fourier transform, as op- size Λ in all directions. posed to a patchwise Fourier transform. The corresponding Lagrangian is: The Bisognano-Wichmann result discussed above is X Z L = ddx Ψ (∂ − v σ ∂ )Ψ = only strictly applicable to theories with relativistic in- p t F x x⊥ p variance. However, the coordinates parallel to the parti- patches Z (15) tioning surface are irrelevant to the proof. We just need X d d x Ψp(σ · ∂)Ψp a spacetime symmetry between the one perpendicular patches spatial coordinate (x1) and time. Importantly, the effective temperature of the local thermal distribution is where σ ≡ {1, σx, 0, 0, ...}. We now have a set of in- determined by the appropriate “speed of light” of the dependent relativistic-looking theories for each patch of system. The correct velocity to use is technically the the Fermi surface. However, we must be careful in ap- velocity characterizing the spacetime symmetry between plying our relativistic results. As we have seen before, x0 and x1. For a strictly relativistic system, this point using the BW theorem amounts to a spacetime rota- would be irrelevant since the velocity is the same in all tional symmetry between the time direction and the x1 directions, but this distinction will be crucial in our ap- direction (partitioning surface normal), not the x⊥ di- plication to Fermi surfaces. rection (Fermi surface normal). In order to use the BW result, we must first rewrite the Lagrangian as:

X Z A. Codimension One L = ddx patches (16) As a warmup, we will find the entanglement Hamil- tonian for a codimension one Fermi surface, then use it Ψp(∂t − vF σx cos θ∂x1 − vF σx sin θ∂x2 )Ψp to rederive the Widom formula. We start with a bare bones model of a codimension one Fermi surface of spin- where cos θ = |nˆp · xˆ1| represents the angle between the 1/2 fermions characterized by a Fermi velocity vF (k), Fermi surface normaln ˆp (at the given patch) and the 34 which in general may vary over the Fermi surface. We partitioning surface normalx ˆ1. The coordinate x2 is then break the Fermi surface up into patches, as in Fig- an appropriately chosen orthogonal coordinate. We can ure 2, chosen small enough that vF is constant within now regard {x2, ..., xd} as parameters, viewing the above each patch and we can view the surface as locally flat. Lagrangian as a sum of one-dimensional theories with Each such patch on the surface essentially represents an velocity vF cos θ. We can therefore simply quote the independent chiral one-dimensional mode propagating in BW result for a given patch, provided we use the ve- the normal direction with linear dispersion. We can then locity vF cos θ as the “speed of light.” Our entanglement write a Hamiltonian for the system as follows35: Hamiltonian then becomes: Z Z 2πx X d ˜ X d 1 H = d k vF Ψp(k⊥σx)Ψp (13) H = d x Hp = x1>0 vF |nˆp · xˆ1| patches |k|<Λ patches Z (17) X 2πx1 ddx Ψ (σ ∂ )Ψ We have an independent Weyl fermion field Ψp for each p x x⊥ p |nˆp · xˆ1| patches x1>0 patch, labeled by index p. The variable k⊥ represents 6

Note the presence of two different non-orthogonal coor- Using the temperature found in Equation 18, we then dinates in the last formula: x1 represents the coordinate find that the total entanglement entropy for the planar along the normal to the entangling surface, and x⊥ rep- partitioning surface is given by: resents the (spatial) coordinate along the direction of the Z Z π2 v |nˆ · xˆ | normal to the Fermi surface. As in our earlier analysis, S = ddx F p 1 = −H˜ 3(2π)d v (2πx ) we can interpret the resulting density matrix, ρ = e , F.S. x1>0 F 1 (21) as a local thermal ensemble, with position dependent log L Z Z d−1 dx2...dxd (|nˆp · xˆ1|) temperature. The main difference now is that the tem- (2π) 12 F.S. perature depends not only on spatial location but also on the patch considered, so each patch of the Fermi surface where the leading integral is over the Fermi surface. The is at its own independent temperature given by: remaining spatial integrals represent an integral over the planar partitioning surface. For a more general parti- vF |nˆp · xˆ1| tioning surface, we simply break it up into small patches, Tp(x) = (18) 2πx1 which locally look flat. The same formula will apply, re- placingx ˆ1 by the appropriate normal direction. For a (Recall that vF in general can vary from one patch to general partitioning surface, the entanglement entropy another.) Patches of the Fermi surface which align with (per spin species) is then given by: the partitioning surface have the highest temperature and will contribute the most to the entanglement en- log L Z Z S = d−1 |nr · nk| (22) tropy. At the other extreme, patches orthogonal to the (2π) 12 F.S. P.S. partitioning surface have zero temperature and make no contribution to the entanglement entropy whatsoever. where the integrals are over the Fermi surface and the This temperature profile is illustrated in Figure 3. partitioning surface, which have normalsn ˆk andn ˆr respectively. This is precisely the Widom formula for the In order to obtain the entanglement entropy, we will 4 once again invoke our procedure of integrating the lo- entanglement entropy of a Fermi surface . It is interest- R ing to note that the Fermi velocity cancels out on each cal thermal density, Sent = stherm(T (x)). This x1>0 patch of the Fermi surface, so the entanglement entropy amounts to a fairly simple thermodynamics problem. We depends only on the geometry of the Fermi surface, not recall that the density of states (per real space volume on the details of the dispersion. and per spin species) contributed by a small patch of the Before moving on, we pause to put this equation into Fermi surface of area dA is given by: k a useful alternative form. We note that multiplying the dAk Fermi surface area element by the factor |nˆr ·nˆk| gives the dρ = d (19) area of the projection of that patch onto the partition- (2π) vF (k) ing surface. Taking a convex Fermi surface for simplicity, and its contribution to the low-temperature entropy den- it is not hard to see that integrating this projected area sity is given by: over the whole Fermi surface will give twice the extremal cross-section of the Fermi surface, as seen from then ˆ di- 2 2 r π π T rection (in other words, the size of its shadow). We shall ds = dρ T = d dAk (20) 3 3(2π) vF (k) denote this cross-section by σk(ˆnr). Then the Widom formula becomes: log L Z S = d−1 σk(ˆnr) (23) (2π) 6 P.S. The Widom formula thereby allows us to obtain direct useful information about the size and shape of the Fermi surface. In particular, choosing a planar partitioning surface in then ˆr direction will directly tell us the cross-section of the Fermi surface in that direction. The Widom formula therefore allows us to (almost) com- pletely map out the shape of the Fermi surface using only the ground state entanglement entropy.

B. General Codimension

FIG. 3. A map of the effective temperature on a spherical All of the above results have been derived for a stan- Fermi surface. The surface is hottest (red) along the poles dard codimension one Fermi surface, which is effectively determined by the direction ofn ˆr, cooling off to zero temper- a surface of quasi-one-dimensional modes, but the gen- ature (blue) along the equator. eral idea can be carried over to the general codimension 7

then:

vF |Pkxˆ1| Tp(x) = (25) 2πx1 The entanglement Hamiltonian is given by: Z ˜ X d 2πx1 H = d x Ψp(γ · ∂x⊥ )Ψp (26) vF |Pkxˆ1| patches x1>0

To put these results to use, we need to know the entropy density contributed by each patch of the Fermi surface. This is a fairly straightforward thermodynamics problem. The thermal energy density per patch area at temperature T is given by: FIG. 4. A visualization of the problem at hand. The plane Z 1 vF |k| represents the partitioning surface, with normalx ˆ. In this e(T ) = dqk = (2π)d evF |k|/T + 1 case, we show a nodal line in three dimensions, which has (27) q = 2. At each point on this line, we find the appropriate 2πq/2(2q − 1) Γ(q + 1) ζ(q + 1)T q+1 BW velocity using the projection Pkxˆ of the unit vectorx ˆ (2π)d(2v )q Γ(q/2) onto the normal plane of the line at that point. F q+1 Making use of de = T ds, we will have s(T ) = q (e/T ), case with little trouble. For a Fermi surface of codimen- or with some simplification: sion q, the effective theory will be described by a set πq/2(2q − 1) Γ(q + 2) T q of quasi-q-dimensional relativistic theories. Suppose we s(T ) = ζ(q + 1) (2π)d2q Γ( q + 1) v have a specified codimension q Fermi surface (i.e. a sur- 2 F (28) face of dimension d − q embedded in d dimensions). The T q dispersion is generically linear around this surface. In ≡ A(q) vF close analogy with the codimension one case, each patch on this surface has its own independent q-dimensional where this equation defines the shorthand A(q). Follow- gapless field theory, which we take to be isotropic, so that ing the same procedure that we did in the codimension each patch is characterized by a single Fermi velocity.36 one case, the entanglement entropy for a generic parti- We do, however, allow the velocity to vary over the sur- tion then becomes: face. Unlike the codimension one case, we don’t just have Z Z Z q one normal direction to the Fermi surface, but rather a |Pknˆr| S = A(q) dx1 = q-dimensional normal surface within which we have lin- F.S. P.S. x1>0 2πx1 ear dispersion (see Figure 4). Within each patch of the A(q) Z Z q (29) Fermi surface, we can write an effective theory in terms q q−1 |Pknˆr| ≡ (2π) (q − 1)a F.S. P.S. of an independent Dirac field, yielding a Hamiltonian of Z Z the form: N(q) q q−1 |Pknˆr| Z a F.S. P.S. X d H = d x vF Ψp(γ · ∂x⊥ )Ψp (24) patches where the integrals are over the (d−q)-dimensional Fermi surface and the (d − 1)-dimensional partitioning surface, for an appropriate set of γ matrices. The symbol x⊥ and a represents a short-distance cutoff. The numerical represents the q spatial coordinates corresponding to the coefficient (which will not be terribly important) is given normal directions at each patch. Now choose a basis for by: x⊥ which includes the direction of the projection ofx ˆ1 into the normal surface (see Figure 4). It is now not hard πq/2(2q − 1) Γ(q + 2) N(q) = q ζ(q + 1) (30) to see that the appropriate BW velocity relating x0 and d+q q (2π) 2 (q − 1) Γ( 2 + 1) x1 will once again have a cos θ factor, where now the angle is that betweenx ˆ1 and its projection onto the normal Equation 29 is the generalization of the Widom formula surface. Sincex ˆ1 is a unit vector, this factor can be writ- to the general codimension case, q > 1. The factor of q ten more succinctly as |Pkxˆ1|, i.e. the magnitude of the |Pknˆr| is the natural generalization of the “flux factor” projection. (Note that the projector is different for each |nˆr · nˆk| found in the codimension one case. patch of the Fermi surface, so it has k dependence.) The There are a few notable features to this formula. First appropriate velocity to use in the Bisognano-Wichmann of all, assuming q > 1, there is no large distance di- theorem will then be vF |Pkxˆ1|. The appropriate tem- vergence to the integrals, and the area law is strictly perature profile for each patch of the Fermi surface is obeyed.37 Also, the short-distance cutoff a appears in the 8 prefactor, making the overall magnitude non-universal. Rather, it is only the relative magnitude of different choices of partitioning surfaces which is meaningful, as we will discuss below. In the q = 1 case, the dimensions of the Fermi surface, represented by kF , provided all the necessary dimensionful factors, with the prefactor of the d−1 logarithm scaling as (LkF ) . For general q, the scaling d−1 d−q q−1 goes as L kF /a . Another notable aspect is the q generalized flux factor, |Pknˆr| . Since |Pknˆr| < 1, this means that there is much less contribution to the entanglement entropy from patches of the Fermi surface that don’t “align” with the partitioning surface, in the sense ofn ˆr lying in the normal surface. And since q appears as an exponent, we see that the higher codimension case FIG. 5. We consider a circular nodal line and a planar par- is even more sensitive to these misalignments than the titioning surface which is misaligned with it by angle θ. The usual Widom formula. entanglement entropy is greatest when the two normal vectors line up, θ = 0.

C. Nodal Line Entanglement Entropy resulting entanglement entropy is then: We now specialize the results of the previous section to NA Z S = |tˆ× nˆ|2 (33) the case of Fermi nodal lines in three spatial dimensions, a N.L. which have q = 2. Let the nodal line have a unit tangent The integral is similar to, but not identical to, the length vector tˆ. The projection P nˆ can then be written as k r of the projection of the line onto the partitioning plane, nˆ −(tˆ·nˆ )ˆn . The entanglement entropy then takes the r r r which would be R |tˆ×nˆ|. While the present integral is not form: directly equal to the projective size, like the codimension Z Z one case, it is a proxy for it. N ˆ 2 S = |nˆr − (t · nˆr)ˆnr| = As a concrete example, let us take our nodal line to be a P.S. N.L. Z Z (31) circular in shape, lying in a plane, as illustrated in Figure N 2 (1 − (tˆ· nˆr) ) 5. (This is not extraordinarily unrealistic. Some form of a P.S. N.L. symmetry protection may cause the nodal line to exist in a specific plane. Also, the presence of interactions may where the inner integral is over the nodal line. This can smooth out the line under the renormalization group, equivalently be written in the form: resulting in a shape of high symmetry at low energies.) N Z Z Let the nodal circle lie in the xy plane. Without loss S = |tˆ× nˆ |2 (32) of generality, we then letn ˆ lie in the xz plane,n ˆ = a r P.S. N.L. (sin θ, 0, cos θ), with θ representing the angle betweenn ˆ Either of these equivalent forms can be regarded as the and the normal to the circle. Taking φ as the azimuthal ˆ Widom formula for nodal lines. Unlike the case of a angle in the plane, we have t = (− sin φ, cos φ, 0). The codimension one Fermi surface, this result will be pro- entanglement entropy is then: portional to the size of the partitioning surface, without NA Z 2π S(θ) = dφ(cos2 θ + sin2 θ cos2 φ) = any logarithmic factors, so the area law for entangle- a ment entropy will be obeyed. Also, we note the explicit 0 (34) πNA presence of the lattice scale a, making the coefficient (1 + cos2 θ) non-universal. But do not despair! While the overall a magnitude of the entanglement entropy for a given par- We therefore see that S is maximized when θ = 0, tition is not particularly meaningful, there is still impor- i.e. when the nodal line lies in the same plane as the tant information contained in its shape dependence. We partitioning surface, and is minimized when it is per- can examine the relative magnitude for different orien- pendicular to the partitioning surface. In particular, tations and shapes of the partitioning surface. The non- we can make the prediction that the entanglement en- universal prefactors will cancel out of such ratios. While tropy will be precisely half as large in the latter case, the entanglement entropy does not tell us the overall size S(π/2)/S(0) = 1/2. This is similar to the projective of the nodal line, it does allow us to map out the shape length `(θ) = R |tˆ× nˆ|, which would obey `(π/2)/`(0) = of the line. For example, suppose we choose a planar 1/π. We see that the planar entanglement entropy is partitioning surface, breaking the system up into two highly anisotropic, as might be expected. Moreover, a half-spaces. Let the area of the plane (dictated by the given shape will have a very specific form. This pro- system size) be A, and let it have fixed normaln ˆ. The vides a useful tool for mapping out the shape of a nodal 9 line, purely based on ground state entanglement properties. By examining the angular dependence of planar partitions, and comparing against the results for specific nodal line shapes, one can thereby get a general picture of the overall shape of the nodal line. However, there are likely other systems which also have such anisotropic entanglement. It would be nice to have a direct test which can verify that the anisotropy is indeed coming from a nodal line, as opposed to some other source. Luckily, Equation 33 for the planar entanglement entropy obeys a nontrivial relation which will not be obeyed by a generic anisotropic entropy source. Let us look at three separate planar partitions, corresponding to three mutually orthogonal normal vectors nˆ1,n ˆ2, andn ˆ3 (letting all planes have the same area A). The sum of the entanglement entropies for these three partitions will obey: FIG. 6. The wavefunction can be understood as a transition amplitude from φ0 to φany in imaginary time. Z NA X ijk i`m S1 + S2 + S3 = tˆjnˆk tˆ`nˆm = a N.L. nˆ1,nˆ2,nˆ3 NA Z 2NA 2 = ` a N.L. a (35) where ` is the length of the nodal line, and we have taken advantage of the completeness of the three vectors as a basis. This relation says that, for an arbitrary orthogonal basis {nˆ1, nˆ2, nˆ3}, the sum of the three entropies is always equal to the same constant, dictated by the length of the line. This result is not satisfied by a generic anisotropic entropy source, and is therefore a unique dis- tinguishing feature of nodal line anisotropy. By verifying this relation, joined with an analysis of the precise form of the anisotropy, the entanglement entropy thereby provides us with a useful tool for determining the existence FIG. 7. Equivalently, it can be understood as a transition and shape of a Fermi nodal line. amplitude from φA to φB in an angular time variable.

IV. CONCLUSION useful tool for diagnosing these phases of matter.

In this work, we have generalized the Widom formula for Fermi surface entanglement entropy to the case of ACKNOWLEDGMENTS arbitrary codimension, by means of repurposing a trick for relativistic theories. Along the way, we found an en- I would like to thank Ari Turner, Brian Swingle, Yahui tanglement Hamiltonian for Fermi surface systems and Zhang, and Senthil Todadri for useful discussions. In a new derivation of the standard Widom formula. Most particular, I thank Ari Turner for sharing his ideas on importantly, we have found a Widom formula for nodal Fermi surface entanglement Hamiltonians, Brian Swingle lines, which depends sensitively on the shape of the for encouragement during the early stages of this project, nodal line and its orientation relative to the chosen par- and Yahui Zhang and Senthil Todadri for sharing and titioning surface. While the overall magnitude of the discussing their work on nodal line spin liquids. This entanglement entropy is non-universal, the generalized work was supported by NSF DMR-1305741. Widom formula displays a characteristic angular dependence. By comparing the entanglement entropy for different partitions, one can map out the shape of the nodal APPENDIX A: THE BISOGNANO-WICHMANN line, allowing for the detection of a nodal line purely THEOREM based on ground state entanglement properties. There is compelling evidence that nodal lines can exist in real- We review a quick argument for the Bisognano- istic systems, and entanglement entropy will serve as a Wichmann theorem for entanglement Hamiltonians in 10 relativistic systems. Let Ψ0[φ] = hφ|Ψ0i be the ground does not imply that it is local with respect to the orig- state of our system, as a function of φ, schematically inal microscopic fermions. In this section, we will make representing all fields in the theory. We take advantage connection with the microscopic fermions by employing of the fact that |Ψ0i can be obtained starting with a an alternate technique for calculating the entanglement state corresponding to an arbitrary field configuration, Hamiltonian. We will find that indeed it is nonlocal with −Hτ |φanyi, then evolving it in imaginary time, e |φanyi. respect to microscopic variables, but in a fairly simple As τ → ∞, the true ground state should be recovered way. The treatment here has been heavily influenced by (provided |φanyi has at least some nonzero overlap with a discussion with Ari Turner, who put forward the main the ground state, as a generic state does). Thus, we can logic of this method. write the ground state as: The important insight is to regard the higher- −Hτ hφ0(x)|Ψ0i ∝ lim hφ0(x)|e |φanyi (36) dimensional Fermi surface as a collection of one- τ→∞ dimensional Fermi surfaces. To illustrate the principle, The above formula represents the amplitude for the state we will work through a two-dimensional model. (Qual- |φ0(x)i at imaginary time τ = 0 to transition to the itatively similar results will hold in higher dimensions.) state |φanyi at imaginary time τ = ∞. In path integral We take a Hamiltonian of the form: language, this amplitude can be rewritten as:

Z R ∞ 0 Z 0 dτLE [φ ] 2 2 2 hφ0(x)|Ψ0i ∝ Dφ e 0 (37) H = dxdyΨ(−(∂x + ∂y ) − kF )Ψ (42) where LE is the Lagrangian expressed in imaginary time, 0 and the path integral is over all configurations φ (x, τ) modeling fermions at chemical potential µ = k2 just 0 F subject to the boundary conditions φ (x, 0) = φ0(x) and above the bottom of a band, where the dispersion is ω = 0 φ (x, ∞) = φany(x). Then we go to polar coordinates in k2 (setting the band mass to 1/2 for simplicity). Suppose the x1τ upper half-plane instead of Cartesian ones. We x is the normal direction to our partitioning cut. We can equally well regard this amplitude as that for the then Fourier transform in the transverse (y) direction: propagation from the configuration φA for x1 > 0 to the configuration φB for x1 < 0, with the angular coordinate taking the role of the time variable. (See Figures 6 and Z dk Z H = y dxΨ (−∂2 − (k2 − k2))Ψ (43) 7). Since the theory is relativistic, we can simply write 2π ky x F y ky this amplitude as:

−πHR hφ0(x)|Ψ0i ∝ hφA|e |φBi (38) We have written ky as a subscript on the field Ψky to em- 1 R d phasize that we can now treat it as a parameter. For each where HR = d x x1H is the generator of boosts c x1>0 fixed k , we have a theory of one-dimensional fermions in the x direction (x τ rotations). (If the theory were y 1 1 with dispersion ω = k2 and chemical potential given by not relativistic, then H would not be a conserved quan- x R µ(k ) = k2 −k2. For k > k (i.e. for the region outside tity along the trajectory, necessitating a sort of “time- y F y y F the original Fermi surface), the one-dimensional system ordering.”) The ground state density matrix for the sys- is (trivially) gapped and will not contribute significantly tem is then given by: to entanglement. We can safely ignore this region with- 0 −πHR 0 −πHR 0 ρ0(φ, φ ) ∝ hφA|e |φBihφB|e |φAi (39) out losing any interesting physics. We therefore need to find the entanglement Hamiltonian for ky < kF , where and the reduced density matrix for region A (x1 > 0) is we have a one-dimensional Fermi surface, consisting of given by: two points.

0 −2πHR 0 ρA(φA, φA) ∝ hφA|e |φAi (40) We note the following important fact about these one- dimensional theories: while the full theory does not have Thus, up to an additive constant, we may write: any relativistic invariance, the low-energy description Z 2πx1 does, with velocity given by the eﬀective Fermi velocity, H˜ = − log ρ = ddx H (41) q c 2 2 x1>0 vF (ky) = 2 kF − ky. When a theory with Hamiltonian which is what we set out to show. density H has relativistic invariance at speed v, we can write its entanglement Hamiltonian (for the half-space) 2π R d as H˜ = d x x1H. Since our theory does not v x1>0 APPENDIX B: CONNECTING WITH have full relativistic invariance, such a replacement will MICROSCOPIC VARIABLES not give the exact entanglement Hamiltonian for the theory. It will, however, give the correct contribution from As we have seen in the main text, the entanglement low energy modes, which is the dominant contribution. Hamiltonian takes a local form with respect to suit- We can therefore make the rough approximation of using ably chosen low-energy patch variables. However, this the BW result on each one-dimensional theory, resulting 11 in the following entanglement Hamiltonian: We note the power law decay and also the oscillations at wavevector kF , reminiscent of Friedel oscillations. Just Z kF Z ˜ dky π as in Friedel oscillations, the long-range physics here is H = q dx x 2π 2 2 due to the sharp change in occupation at the Fermi level. −kF k − k F y (44) We are essentially taking the Fourier transform of a function with a kink at k . One can verify that the decay Ψ (−∂2 − (k2 − k2))Ψ F ky x F y ky exponents 1/2 and 3/2 come directly from the behavior of the integral near kF . The large separation behavior We can now explicitly perform the inverse Fourier trans- of the entanglement Hamiltonian, and thus its degree of formation. Consider the ﬁrst term: nonlocality, is dictated entirely by physics near the Fermi Z Z surface. dx x dydy0

kF Z dk π 0 y iky (y −y) 0 2 q e Ψ(x, y )(−∂x)Ψ(x, y) = The ﬁeld Ψ represents our microscopic fermionic vari- −k 2π 2 2 F kF − ky able, and the nonlocality of H˜ can be directly interpreted Z Z as the nonlocality of the hopping elements hij of the 0 π 0 0 2 dx x dydy J0(kF (y − y)) Ψ(x, y )(−∂ )Ψ(x, y) quadratic form of the entanglement Hamiltonian. As 2 x mentioned earlier, for any free fermion system on a lat- (45) tice the entanglement Hamiltonian must take a quadratic form: where J0 is a Bessel function. A similar analysis on the other term yields the full form of the entanglement Hamiltonian as: π Z H˜ = dxdydy0 x 2 0 0 2 J0(kF (y − y))Ψ(x, y )(−∂x)Ψ(x, y)− (46) X † kF 0 0 ρ ∝ exp(− hijci cj) (48) J1(kF (y − y))Ψ(x, y )Ψ(x, y) (y0 − y) ij

When the separation y0 − y is large, we can use asymp- totic expressions for the Bessel functions. In this limit, the quantity in parentheses above becomes: 0 cos(kF (y − y)) s The result obtained in Equation 46 corresponds to the 2 0 2 continuum limit of this expression, and we can directly 0 Ψ(x, y )(−∂x)Ψ(x, y) πkF (y − y) see the nonlocality of the hopping elements. The hopping element between sites will decay as cos(k (y0 − p2k /π √ F − F Ψ(x, y0)Ψ(x, y) y))/ y0 − y at large separation in the y direction. The 0 3/2 (y − y) interaction in the x direction is still local, so hopping (47) should still be mostly nearest neighbor in that direction.

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