Entanglement entropy of the large Wilson-Fisher conformal field theory
We compute the entanglement entropy of the Wilson-Fisher conformal field theory (CFT) in 2+1 dimensions with O() symmetry in the limit of large for general entanglement geometries. We show that the leading large result can be obtained from the entanglement entropy of Gaussian scalar fields with their mass determined by the geometry. For a few geometries, the universal part of the entanglement entropy of the Wilson-Fisher CFT equals that of a CFT of massless scalar fields. However, in most cases, these CFTs have a distinct universal entanglement entropy even at . Notably, for a semi-infinite cylindrical region it scales as in the Wilson-Fisher theory, in stark contrast to the -linear result of the Gaussian fixed point.
The entanglement entropy (EE) has emerged as an important tool in characterizing strongly interacting quantum systems Calabrese and Cardy (2004); Ryu and Takayanagi (2006); Kitaev and Preskill (2006); Levin and Wen (2006); Fradkin and Moore (2006); Dong et al. (2008); Hsu et al. (2009); Metlitski et al. (2009); Casini and Huerta (2012); Zhang et al. (2012). In the context of relativistic theories in 2 spatial dimensions, the so-called theorem uses the EE on a circular disk to place constraints on allowed renormalization group flows Myers and Sinha (2010, 2011); Jafferis et al. (2011); Klebanov et al. (2011, 2012); Casini and Huerta (2012); Grover (2014). For quantum systems with holographic duals, the EE can be computed via the Ryu-Takayanagi formula Ryu and Takayanagi (2006), and this is a valuable tool in restricting possible holographic duals of strongly interacting theories Ogawa et al. (2012); Huijse et al. (2012).
Despite its importance, the list of results for the EE of strongly interacting gapless field theories in 2+1 dimensions is sparse. The most extensive results are for CFTs on a circular disk geometry in the vector large- and small- expansions Klebanov et al. (2011, 2012); Giombi and Klebanov (2015); Fei et al. (2015); Giombi et al. (2016); Fei et al. (2016); Tarnopolsky (2016). Some results have also been obtained Metlitski et al. (2009) in the infinite cylinder geometry in an expansion in , where is the spatial dimension, but the extrapolation of these results to is not straightforward.
In this paper we show how the vector large expansion can be used to obtain the EE in essentially all entanglement geometries, generalizing results that were only available so far in the circular disk geometry. The large expansion was also used in Ref. Metlitski et al. (2009) in the infinite cylinder geometry, but the results were limited to the universal deviation of the EE when the CFT is tuned away from the critical point by a relevant operator. For a region with a smooth boundary, the groundstate of a CFT has an EE which obeys
where is a short-distance UV length scale, is the area law coefficient depending on the regulator, is an infrared length scale associated with the entangling geometry, and is the universal part of the EE we are interested in. We will compute for the Wilson-Fisher CFT with O() symmetry on arbitrary smooth regions in the plane, and in the cylinder and torus geometries. Our methods generalize to other geometries, and also to other CFTs with a vector large limit. We also obtain universal entanglement entropies associated with geometries with sharp corners.
Our analysis relies on a general result which will be established in Section II. We consider the large limit of the Wilson-Fisher CFT on a general geometry using the replica method, which requires the determination of the partition function on a space which is a -sheeted Riemann surface. The large limit maps the CFT to a Gaussian field theory with a self-consistent, spatially dependent mass Metlitski et al. (2009). Determining this mass for general is a problem of great complexity, given the singular and non-translationally invariant -sheeted geometry; complete results for such a spatially dependent mass are not available. However, we shall show that a key simplification occurs in the limit required for the computation of the EE: the spatially dependent part of the mass does not influence the value of the EE. This simplification leads to the main results of our paper. We note here that the simplification does not extend to the Rényi entropies , so we shall not obtain any results for the Rényi entropies of the Wilson-Fisher CFT in the large limit.
Section II will compute the EE for the Wilson-Fisher CFT on arbitrary smooth regions in an infinite plane, and for regions containing a sharp corner, in which case (1) is modified. In both these cases, and for other entangling regions in the infinite plane, the EE is equal to that of a CFT of free scalar fields. Section III will consider the case of an entanglement cut on an infinite cylinder. A non-zero limit of as was obtained in Ref. Metlitski et al. (2009) for the free field case. We will show that a very different result applies to the Wilson-Fisher CFT, with . Section IV considers the case of a torus with two cuts: here is non-zero for both the free field and Wilson-Fisher cases, but the values are distinct from each other.
Ii Mapping to a Gaussian theory
In this section we consider the EE of the critical O() model at large-, and show that it can be mapped to the EE of a Gaussian scalar field.
ii.1 Replica method
We first recall how the EE can be computed in a quantum field theory using the replica method introduced in Refs. Holzhey et al. (1994); Calabrese and Cardy (2004). The EE associated with a region is given by
where is the reduced density matrix in . A closely related measure of the entanglement is the -th Rényi entropy, which is defined as
where is an integer. In the replica method, outlined below, the Rényi entropies are directly computed from a path integral construction. One can then analytically continue to non-integer values, and obtain the EE as a limit
Equivalently, one can consider expanding to leading order in , obtaining
Thus, the small behavior of is sufficient to compute the entropy .
The computation of proceeds as follows. We first consider the matrix element of the reduced density matrix between two field configurations on , and . This can be computed using the Euclidean path integral
where is the Euclidean action of the system. We then write the trace over in terms of these matrix elements
Here, is the partition function over the -sheeted Riemann surface obtained by performing the integrations in Eq. (7). In particular, we consider copies of our Euclidean field theory, but we glue the spatial region of the th copy to the spatial region of the th copy, repeating until we glue the th copy to the first copy. This construction introduces conical singularities at the boundary of .
ii.2 Entanglement entropy for the O() model at large
We now specialize to the critical O() model in -dimensions. We use a non-linear model formulation, writing the -sheeted action as
where runs from to and is summed over. Here, and denote the integration measure and the Laplacian on the -sheeted Riemann surface, respectively. The field is a Lagrange multiplier enforcing the local constraint . In the limit, the path integral is evaluated using the saddle point method:
In the last equality, the saddle point configuration of the field is determined by solving the gap equation
where is the Green’s function on the -sheeted surface:
and the critical coupling is determined by demanding that the gap vanishes for the infinite volume theory on the plane:
In the absence of the entangling cut, , we denote the saddle point value of as
We assume that the one-sheeted geometry is translation-invariant, so is independent of position. On the infinite plane we have , but we will also consider geometries where one or both dimensions are finite, in which case becomes a universal function of the geometry of the system determined by
On the -sheeted Riemann surface, is always a nontrivial function of position, and the exact form of this function depends on the shape of the entangling surface and the number of Riemann sheets . The problem of determining this function can be addressed numerically for fixed , but for the purposes of obtaining the EE, we only need its spatial dependence to first-order in . In particular, we assume that we can write
for some function of space-time . Then to leading order in and , we have
Then using the definition of and ,
implying that that last line of Eq. (17) vanishes, and does not contribute to the EE. After using , we can write
This final expression is equal to the quantity computed for free scalars with mass and the action
Therefore, the EE of the critical O() model at order is equal to the EE of free scalar fields, where the free fields have the same mass gap as the O() model on the physical, one-sheeted surface. Similar results will apply to other large- vector models. For instance, in Appendix B we follow very similar steps to show that the EE of the fermionic Gross-Neveu model maps to that of free Dirac fermions. The mass of the free fermions is determined self-consistently by the spatial geometry of the physical single-sheeted spacetime.
ii.3 Entanglement entropy on the infinite plane
We first consider the EE when the system is on the infinite plane. In this case, , and the EE associated with a region is equal to the EE of massless free scalars in the same region.
One entangling region for which there are known results is the circular disk. According to the F-theorem Casini and Huerta (2012), the universal part of the EE for the disk is given by
Here, is the finite part of the Euclidean partition function on a three-sphere spacetime. This quantity was computed in Ref. Klebanov et al. (2011) for massless free scalar fields and for the large- O() model, and they were found to be equal at order in agreement with our general result given above. Explicitly,
where . Our results also apply to regions with sharp corners, in which case we can make non-trivial checks of our general result, as we now discuss.
Entanglement entropy of regions with corners
When region (embedded in the infinite plane) contains a sharp corner of opening angle , the EE of a CFT (1) is modified by a subleading logarithmic correction Casini and Huerta (2009); Hirata and Takayanagi (2007)
where the dimensionless coefficient is universal, and encodes non-trivial information about the quantum system. Since we work in the infinite plane, according to our analysis above, the large- value of will be the same as for free scalars, namely
The non-trivial function for a single free scalar was studied numerically and analytically by a number of authors for a wide range of angles Casini and Huerta (2009); Bueno et al. (2015); Elvang and Hadjiantonis (2015); Laflorencie et al. (2015); Bueno and Witczak-Krempa (2016); Helmes et al. (2016). Interestingly, we can make an analytical verification of the relation (24) in the nearly smooth limit, by virtue of the following identity that holds for any CFT Bueno et al. (2015); Bueno and Myers (2015); Faulkner et al. (2016)
Here, is a non-negative coefficient determining the groundstate two-point function of the stress tensor :
where is a dimensionless tensor fixed by conformal symmetry Osborn and Petkou (1994). Eq. (25) was conjectured Bueno et al. (2015) for general CFTs in two spatial dimensions, and subsequently proved using non-perturbative CFT methods Faulkner et al. (2016). Now, is the same at the Wilson-Fisher and Gaussian fixed points Sachdev (1993) at leading order in :
The knowledge of can be used to make a statement about away from the nearly smooth limit because the existence of the following lower bound Bueno and Witczak-Krempa (2016): , which follows from the strong subadditivity of the EE, and (25). We see that applying this bound to the large- Wilson-Fisher fixed point is consistent with our result (24).
Iii Infinite cylinder
We now compute the EE of the semi-infinite region obtained by tracing out half of an infinite cylinder. The relevant geometry is pictured in Fig. 1. We can take the position of the cut to be at by translation invariance. As for the disk, we can write the EE as
where is the universal part. The existence of a universal in critical theories was first established Hsu et al. (2009); Oshikawa (2010); Hsu and Fradkin (2010) for the quantum Lifshitz model using the methods of Ref. Fradkin and Moore (2006). In the context of CFTs, this geometry was considered in Ref. Metlitski et al. (2009), where the entropy was computed for massless free fields and for the Wilson-Fisher fixed point in the expansion (where the extra dimensions introduced in the -expansion are made periodic with circumference ).
We first review the calculation of the entropy for free massive fields, which will allow us to calculate the EE for the Wilson-Fisher fixed point for large . We allow for twisted boundary conditions along the finite direction
Here, . We note that unless , the fields are complex. In this case, we are considering complex fields, and the O() symmetry of the theory breaks down to U(1)SU().
This geometry allows a direct analytic computation of the -sheeted partition function for free fields by mapping to radial coordinates, . In these coordinates, the -sheeted surface is fully parametrized by giving the angular coordinate a periodicity of . In Refs. Calabrese and Cardy (2004); Metlitski et al. (2009), it was shown that the -sheeted partition function for free fields can be written in terms of the one-sheeted Green’s function:
Then using Eq. (5), the EE is given by
In Appendix A, we compute the Green’s function for a massive free field on the cylinder. Using Eq. (48), and making the cutoff dependence explicit, we find the regularized part of the EE to be (see also Ref. Chen et al. (2016))
For , this reduces to Eq. (5.12) of Ref. Metlitski et al. (2009), and indeed displays a divergence for a periodic boundary condition due to the zero mode. We note that the universal contribution to the EE of complex free scalar fields is of order , as one would expect from a free field theory with degrees of freedom.
We now turn to the Wilson-Fisher fixed point. In a finite geometry, the Wilson-Fisher fixed point will acquire a mass gap which is proportional to and depends only on . This is computed by solving , which is done in Appendix A. The result is
Then from the arguments of Section II,
It happens that for the saddle point value of the mass, the universal part of the EE vanishes for all values of the twist . The leading contribution to will be of , a drastic reduction from Gaussian fixed point which is of order .
This result can be seen more directly from Eq. (31). The gap equation implies that , so without even solving for , the EE can immediately be written
However, the critical coupling is completely non-universal and independent of . Using a hard UV momentum cutoff ,
and the EE is pure area law, .
In fact, this result can be extended to other geometries. The result for the large- Wilson-Fisher fixed point occurred because the entropy is proportional to . However, the results of Refs. Calabrese and Cardy (2004); Metlitski et al. (2009) imply that the expression for the free-field entropy given in Eq. (31) holds for any system where the entangling cut is perpendicular to an infinite, translationally-invariant direction. Thus, if we consider the large- Wilson-Fisher CFT on any -dimensional spatial geometry with at least one infinite dimension, the universal part of the EE obtained by tracing out over half of that dimension is . This argument only holds in dimensions where the Wilson-Fisher CFT exists, so for . In particular, this result agrees with the large- limit of the -expansion calculation in Ref. Metlitski et al. (2009), which considered the Wilson-Fisher CFT on the -dimensional spatial region , where is the -dimensional torus. This constitutes a non-trivial consistency check on both calculations.
Finally, we note that similar results apply to other large models. As shown in Appendix B, the EE for the Gross-Neveu CFT maps to that of free Dirac fermions, where the mass of the fermions is determined by the spatial geometry of the one-sheeted spacetime, . Here, the critical coupling is again a non-universal quantity which cannot depend on the spatial geometry of the system, and is proportional to the UV cutoff. Then using the results of Ref. Calabrese and Cardy (2004), it can be shown that for free fermions on the spatial geometries discussed in the previous paragraph, and therefore for the large- Gross-Neveu CFT on the infinite cylinder.
where we have defined the ratio
and is the modular parameter, , for the rectangular torus we work with. is a universal term that we shall study at the large- Wilson-Fisher fixed point.
As discussed in Section II.2, the EE at leading order in is given by that of free complex scalars with a mass determined by the geometry. is thus the self-consistent mass on the torus for a single copy of the theory, which was recently computed at large in Ref. Whitsitt and Sachdev (2016). It obeys the scaling relation , where is the aspect ratio of the torus, and is a non-trivial dimensionless function given in Appendix C. depends on both twists along the - and -cycles of the torus, . Since for a massive free boson is not known, we will numerically study the -dependence of by working on the lattice.
However, before doing so, we describe two limits in which we can make statements about . First, we consider the so-called thin torus limit for which , while remain finite, i.e. and is fixed. For generic boundary conditions, we have that the torus EE approaches twice the semi-infinite cylinder value Chen et al. (2015); Witczak-Krempa et al. (2016) discussed above, . This holds in the absence of zero modes, which is the generic case. Our result Eq. (34) implies that in that limit. However, this cannot hold at all values of . Indeed, for any fixed let us consider the “thin slice” limit . There, reduces to the universal contribution of a thin strip of width in the infinite plane Chen et al. (2015); Witczak-Krempa et al. (2016), , where the universal constant can be computed in the infinite plane. is thus independent of the boundary conditions along . Applying our mapping to free fields, this means that at leading order in
where for a free scalar Casini and Huerta (2009). By continuity, we thus expect that for general and , will scale linearly with in the large- limit. We now verify this statement via a direct calculation.
iv.1 Lattice numerics
The lattice Hamiltonian for a free boson of mass can be taken to be
where the theory is defined on a square lattice with unit spacing, is the operator canonically conjugate to , and . The index runs over the lattice sites in the -direction. Crystal momentum along the -direction remains a good quantum number in the presence of the entanglement cut, and is quantized as follows
where the integer runs from to , and is the twist along the -direction. We note that the Hamiltonian (40) corresponds to decoupled 1d massive boson chains: , each with an effective mass . This means that the EE is the sum over the corresponding 1d EEs: . For each 1d chain, the EE for the interval of length is obtained from the correlation functions and , where we have suppressed the label. The prescription for the EE is then Casini and Huerta (2009)
where are eigenvalues of the matrix , with the meaning that and are restricted to region . This method was previously used to study the EE of free fields on the torus Chen et al. (2015); Witczak-Krempa et al. (2016); Chojnacki et al. (2016); Chen et al. (2016).
To obtain the universal part of the entropy we first need to numerically determine the area law coefficient (37), which we find is . We can then isolate the universal part, , by subtracting the area law contribution. The result for a square torus, i.e. , is shown in Fig. 3, where we compare the Wilson-Fisher fixed point with the Gaussian fixed point. Only is shown because the other half is redundant by virtue of the identity , true for pure states. We set and (since fully periodic boundary conditions yield a divergent ), which leads to a purely imaginary mass for the Wilson-Fisher theory, while is naturally zero at the Gaussian fixed point. The imaginary mass does not cause a problem since in the presence of the twist, (42). From Fig. 3, we see that scales linearly with as was anticipated above. However, contrary to the case of the infinite plane, the EE of the Wilson-Fisher fixed point is reduced compared to the Gaussian fixed point, for all values of . The difference between the EE of the two theories decreases in the thin slice limit , where we have the divergence , with the same constant for both theories, Eq. (39). This constant has been calculated in a different context Casini and Huerta (2009), , and fits our data very well. Another consistency check is that should be convex decreasing Witczak-Krempa et al. (2016) for , which is indeed the case in Fig. 3.
The large limit of the Wilson-Fisher theory yields the simplest tractable strongly-interacting CFT in 2+1 dimensions. In this paper, we have succeeded in computing the entanglement entropy of this theory using a method which can be applied to essentially any entanglement geometry. In particular, for any region in the infinite plane, the EE of the large- Wilson-Fisher fixed point is the same as that of free massless bosons. In contrast, when space is compactified into a cylinder or a torus, the results will differ in general as we have illustrated using cylindrical entangling regions. Our results naturally extend to other large- vector theories, like the fermionic Gross-Neveu CFT (Appendix B).
In the case of the EE of the semi-infinite cylinder, , Ref. Metlitski et al. (2009) has compared its value at the UV Gaussian fixed point and the IR Wilson-Fisher one using the expansion. For these two specific fixed points, it was found that , suggestive of the potential of to “count” degrees of freedom. Our results at large , however, show that the opposite can occur for certain RG flows. Namely, is possible. Indeed, let us consider the flow from the Wilson-Fisher critical point to the stable fixed point describing the phase where the O symmetry is spontaneously broken Sachdev (2011). In the UV, we see that does not grow with , while at the IR fixed point due to the Goldstone bosons Metlitski and Grover (2011). This holds for generic boundary conditions .
It is also of interest to obtain the Rényi entropies of such an interacting CFT, notably for comparison with
large-scale quantum Monte Carlo simulations which can usually only
yield the second Rényi entropy Hastings et al. (2010); Inglis and Melko (2013).
Unfortunately, this is a much more
challenging problem, because the full -dependence of the saddle-point in (12) is needed on a -sheeted
Riemann surface. Numerical analysis will surely be required, supplemented by analytic
results in the limit of large and small .
Note: While finishing this work, we became aware of a related forthcoming paper Hung et al. (2016) that studies the area law term in a large- supersymmetric version of the O model.
We thank Pablo Bueno, Shubhayu Chatterjee, Xiao Chen, Eduardo Fradkin, Janet Hung, Max Metlitski, and Alex Thomson for useful discussions. This research was supported by the NSF under Grant DMR-1360789 and the MURI grant W911NF-14-1-0003 from ARO. WWK was also supported by a postdoctoral fellowship and a Discovery Grant from NSERC, and by a Canada Research Chair (tier 2). WWK further acknowledges the hospitality of the Aspen Center for Physics, where part of this work was done, and which is supported by National Science Foundation grant PHY-1066293. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. SS also acknowledges support from Cenovus Energy at Perimeter Institute.
Appendix A Green’s function and large mass gap on the cylinder
In Section III, we used the Green’s function for a massive scalar on the infinite cylinder. This is given by
where we allow a twist in the finite direction
We evaluate this expression using Zeta function regularization. We first introduce an extra parameter , and consider the expression
This expression is convergent for . We evaluate this expression where it is convergent, and then analytically continue it to . After evaluating the integrals, we obtain
The remaining sum needs to be evaluated as a function of , which requires the use of generalized Zeta functions. General formulae for sums of this type can be found in Reference Elizalde (2012), and after evaluating this sum and taking , we find
We note that the original integral, Eq. (44), has a linear UV divergence which has been set to zero by our cutoff method. In other regularization methods, one generically expects an extra term proportional to the UV cutoff, , which contributes to the area law in Eq. (31).
We also find the mass gap for the Wilson-Fisher fixed point at large-. The gap equation is
However, in Zeta regularization we have
Then using Eq. (48), we find the energy gap on the cylinder
as quoted in the main text.
Appendix B Entanglement entropy of the Gross-Neveu model at large
We discuss the Gross-Neveu model Zinn-Justin (2002) in the large limit. The calculation of the entanglement entropy is very similar to the critical O() model, and we find a mapping to the free fermion entanglement analogous to the mapping derived in Section II.
The critical model is defined by the Euclidean Lagrangian
where the repeated index is summed over, running from 1 to . Here, is a Hubbard-Stratonovich field used to decouple the quartic interaction term . We now follow the steps in Eq. (10) to obtain the partition function using the saddle point method.
The saddle point configuration of is determined by the Gross-Neveu gap equation
Here, the trace is over spinor indices and we have left the identity matrix in spinor space implicit. The critical coupling is
Following our procedure for the O() model, we write the saddle point configuration as
to leading order in , for an unknown function . Then by a similar reasoning to the calculations in Section II, we find
This is the -sheeted partition function for free Dirac fermions with mass , where is the mass gap of the Gross-Neveu model on the one-sheeted physical spacetime, .
Just as for the O Wilson-Fisher fixed point, we can verify our result for the special case where region is a disk embedded in the infinite plane. The disk’s universal entanglement entropy in the Gross-Neveu CFT was found to be that of free massless Dirac fermions Klebanov et al. (2011), . This is exactly our result since for this geometry.
Appendix C Mass gap on the torus
In Section IV, we used the self-consistent mass of the large- Wilson-Fisher fixed point on the torus. This mass takes the form
where is a universal function of them modular parameter of the torus, , and the twists and . The calculation of was done in Ref. Whitsitt and Sachdev (2016). Here, we outline the results needed for the current work. Unlike Ref. Whitsitt and Sachdev (2016), we specialize to the rectangular torus, .
The Green’s function on the torus is
for integers and . As in Appendix A, we evaluate using Zeta regularization; the technical details of this calculation can be found in Ref. Whitsitt and Sachdev (2016). In this regularization, the gap equation is
After regularizing, we can write the Green’s function as
where we use the Jacobi Theta function
For given values of , , and , we compute the value of by numerically inverting the gap equation using the Eq. (62).
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