# Entanglement Entropy and Quantum Field Theory

###### Abstract

We carry out a systematic study of entanglement entropy in relativistic quantum field theory. This is defined as the von Neumann entropy corresponding to the reduced density matrix of a subsystem . For the case of a 1+1-dimensional critical system, whose continuum limit is a conformal field theory with central charge , we re-derive the result of Holzhey et al. when is a finite interval of length in an infinite system, and extend it to many other cases: finite systems, finite temperatures, and when consists of an arbitrary number of disjoint intervals (See note added). For such a system away from its critical point, when the correlation length is large but finite, we show that , where is the number of boundary points of . These results are verified for a free massive field theory, which is also used to confirm a scaling ansatz for the case of finite-size off-critical systems, and for integrable lattice models, such as the Ising and XXZ models, which are solvable by corner transfer matrix methods. Finally the free-field results are extended to higher dimensions, and used to motivate a scaling form for the singular part of the entanglement entropy near a quantum phase transition.

## I Introduction.

Recently there has been considerable interest in formulating measures of quantum entanglement and applying them to extended quantum systems with many degrees of freedom, such as quantum spin chains.

One of these measuresBennett is entanglement entropy. Suppose the whole system is in a pure quantum state , with density matrix , and an observer A measures only a subset of a complete set of commuting observables, while another observer B may measure the remainder. A’s reduced density matrix is . The entanglement entropy is just the von Neumann entropy associated with this reduced density matrix. It is easy to see that . For an unentangled product state, . Conversely, should be a maximum for a maximally entangled state.

For example, for a system with
two binary (spin-) degrees of freedom, with
, where A observes only the first spin and B the second,
takes its maximum value of when , which
agrees with our intuitive idea of maximal entanglement.
For this system it has been shown,Bennett even in the partially
entangled case, that if there are
copies of the state available, by making only local operations A can
produce states which are maximally entangled. The optimal
conversion ratio is given, for large , by .^{1}^{1}1As
far as we are aware, this analysis has not been extended to systems with
many degrees of freedom such as we consider in this paper. Indeed, we
shall see that for such systems can be much larger than unity, so
cannot have such a simple interpretation.

Although there are other measures of entanglement,othermeasures the entropy is most readily suited to analytic investigation. In several papersVidal ; Korepin ; leb ; Casini , the concept has been applied to quantum spin chains. Typically, the subset consists of a commuting set of components of the spin degrees of freedom in some interval of length , in an infinitely long chain. It is found that the entanglement entropy generally tends to a finite value as increases, but that this value diverges as the system approaches a quantum critical point. At such a critical point, the entropy grows proportional to for large .

Close to a quantum critical point, where the correlation length is much larger than the lattice spacing , the low-lying excitations and the long-distance behaviour of the correlations in the ground state of a quantum spin chain are believed to be described by a quantum field theory in 1+1 dimensions. If the dispersion relation of the low-lying excitations is linear for wave numbers such that , the field theory is relativistic. We shall consider only those cases in this paper. At the critical point, where , the field theory is massless, and is a conformal field theory (CFT).

In this case, the von Neumann entropy of subsystem corresponding to an interval of length was calculated some time ago by Holzhey et al.Holzhey , in the context of black hole physics (although that connection has been questioned), where it was termed ‘geometric’ entropy. Using methods of conformal field theory, based in part on earlier work of Cardy and PeschelCardyPeschel , they found , where is the conformal anomaly number (sometimes called the central charge) of the corresponding CFT.

This result has been verified by analytic and numerical calculations on integrable quantum spin chains corresponding to CFTs with and . Vidal ; Korepin ; leb

In this paper, we first put the CFT arguments of Holzhey et al.Holzhey on a more systematic basis, and generalise their result in a number of ways. Our methods are based on a formula for the entropy in terms of the partition function in the path integral formulation of the quantum theory as a euclidean field theory on an -sheeted Riemann surface, in the limit .

For a 1+1-dimensional theory at a critical point, we derive analogous formulas for the entropy in the cases when the subsystem consists of an arbitrary number of disjoint intervals of the real line (see Eq. 41), and when the whole system has itself a finite length . For example, for the case when is a single interval of length , and periodic boundary conditions are imposed on the whole system, we find

(1) |

On the other hand, for a finite system of total length with open boundaries, divided at some interior point into an interval of length and its complement, we find

(2) |

where is the boundary entropy of Affleck and LudwigAffleckLudwig . We also treat the case when the system is infinitely long but is in a thermal mixed state at finite temperature :

(3) |

In all these cases, the constant is the same, but non-universal.

For a massive 1+1-dimensional relativistic QFT (which corresponds to an off-critical quantum spin chain where the correlation length ) the simplest results are for an infinite system divided at some point into two semi-infinite pieces. In this case we verify that the entanglement entropy is finite, and derive the universal formula

(4) |

In the more general case when consists of a collection of disjoint intervals, each of length , we expect (4) to be multiplied by a factor which counts the number of boundary points between and (the 1d analogue of surface area.)

For the entropy is exactly calculable in the case of a free field theory. We verify the above formula, and exhibit the finite-size cross-over which occurs when is of the size of the system. For a lattice model in this geometry, with infinite, we point out that is simply related to Baxter’s corner transfer matrix, and thus, for integrable models whose weights satisfy a Yang-Baxter relation, all its eigenvalues can be determined exactly. We treat explicitly the case of the Ising model and its anisotropic limit, the transverse Ising spin chain, and also the XXZ model, computing exactly at all values of the coupling the finite part of the entropy . This agrees with our continuum result (4) for when the correlation length is large.

The analysis for the free theory is straightforward to extend to higher dimensions, at least in suitable geometries. This leads to the well-known law, first found by Srednickis-93 , that the entropy should be proportional to the surface area of the subsystem . As pointed out by Srednicki, the coefficient of this term, for , depends on the UV cut-off and is therefore non-universal. However, based on our calculations, we propose that there should be a non-leading piece in , proportional to , which depends in a singular way on the couplings near a quantum phase transition, and whose form is, moreover, universal.

The layout of this paper is as follows. In the next section, we discuss the entropy in terms of the euclidean path integral on an -sheeted Riemann surface. In Sec. III we consider the 1+1-dimensional conformal case. We use the powerful methods of CFT to show that the partition function on the Riemann surface of interest is given, up to a constant, by a calculable correlation function of vertex operators in a CFT in the complex plane. Similar results apply to a system with boundaries. This general result allows us to derive all the special cases described above. In Sec. IV we consider the case of a massive 1+1-dimensional field theory. We derive the result from completely general properties of the stress tensor in the relevant geometry. This is then verified for a free bosonic massive field. In the last part of this section we relate the lattice version of this problem to the corner transfer matrix, and compute the off-critical entropy for the case of the Ising and XXZ Heisenberg spin chains. In Sec. V we study the off-critical case in a finite-size system, propose a scaling law, and verify it for the case of a free massive field theory. We compute the relevant scaling function in a systematic sequence of approximations. Sec. VI is devoted to the discussion of higher dimensions.

While all this work was being carried out, some other related papers have appeared in the literature. In Ref. Korepin2 , the result , first found by Holzhey et al.Holzhey , was obtained by the following argument: it was assumed that the entropy should be conformally invariant and therefore some function of the variable ; by comparing with known case BCN ; Affleck it was observed that should behave as as ; finally it was assumed (with no justification being given) that the form for is valid for all values of ; finally the limit was taken. In the present paper, we should stress, we have derived all these statements from first principles of CFT.

Very recently Casini and HuertaCasini have considered the case of two intervals and , and argued that the quantity is UV finite as , and is given by a universal logarithmic function of the cross-ratio of the four end points. This corresponds to, and agrees with, our case . These authors, however, assume the conformal invariance of the entropy, while, once again, we stress that in the present paper we derive this from fundamental properties of the stress tensor. These authors also give a very nice alternative derivation of Zamolodchikov’s -theoremZam based on this quantity .

## Ii von Neumann Entropy and Riemann surfaces.

Consider a lattice quantum theory in one space and one time dimension, initially on the infinite line. The lattice spacing is , and the lattice sites are labelled by a discrete variable . The domain of can be finite, i.e. some interval of length , semi-infinite, or infinite. Time is considered to be continuous. A complete set of local commuting observables will be denoted by , and their eigenvalues and corresponding eigenstates by and respectively. For a bosonic lattice field theory, these will be the fundamental bosonic fields of the theory; for a spin model some particular component of the local spin. The dynamics of the theory is described by a time-evolution operator . The density matrix in a thermal state at inverse temperature is

(5) |

where is the partition function.

This may be expressed in the standard way as a (euclidean) path integral:

(6) |

where , with the euclidean lagrangian. (For a spin model this would be replaced by a coherent state path integral.)

The normalisation factor of the partition function ensures that , and is found by setting and integrating over these variables. This has the effect of sewing together the edges along and to form a cylinder of circumference .

Now let be a subsystem consisting of the points in the
disjoint^{2}^{2}2This restriction is not necessary, but the set-up is
easier to picture in this case.
intervals . An expression for the
the reduced density matrix
may be found from (6)
by sewing together only those points which are not in . This
will have the effect of leaving open cuts, one for each interval
, along the the line .

We may then compute , for any positive integer , by making copies of the above, labelled by an integer with , and sewing them together cyclically along the the cuts so that (and ) for all . Let us denote the path integral on this -sheeted structure by . Then

(7) |

Now, since , where are the eigenvalues of (which lie in ,) and since , it follows that the left hand side is absolutely convergent and therefore analytic for all . The derivative wrt therefore also exists and is analytic in the region. Moreover, if the entropy is finite, the limit as of the first derivative converges to this value.

We conclude that the right hand side of (7) has a unique analytic continuation to and that its first derivative at gives the required entropy:

(8) |

(Note that even before taking this limit, (7) gives an expression for the Tsallis entropyTsallis .)

So far, everything has been in the discrete space domain. We now discuss the continuum limit, in which keeping all other lengths fixed. The points then assume real values, and the path integral is over fields on an -sheeted Riemann surface, with branch points at and . In this limit, is supposed to go over into the euclidean action for a quantum field theory. We shall restrict attention to the case when this is Lorentz invariant, since the full power of relativistic field theory can then be brought to bear. The behaviour of partition functions in this limit has been well studied. In two dimensions, the logarithm of a general partition function in a domain with total area and with boundaries of total length behaves as

(9) |

where and are the non-universal bulk and boundary free energies. Note, however, that these leading terms cancel in the ratio of partition functions in (7). However, as was argued by Cardy and PeschelCardyPeschel there are also universal terms proportional to . These arise from points of non-zero curvature of the manifold and its boundary. In our case, these are conical singularities at the branch points. In fact, as we shall show, it is precisely these logarithmic terms which give rise to the non-trivial dependence of the final result for the entropy on the short-distance cut-off . For the moment let us simply remark that, in order to achieve a finite limit as , the right hand side of (7) should be multiplied by some renormalisation constant . Its dependence on will emerge from the later analysis.

## Iii Entanglement entropy in 2d conformal field theory.

Now specialise the discussion of the previous section to the case when the field theory is relativistic and massless, i.e. a conformal field theory (CFT), with central charge , and initially consider the case of zero temperature.

We show that in this case the ratio of partition functions in (7) is the same as the correlation function arising from the insertion of primary scaling operators and , with scaling dimensions , into each of the (disconnected) sheets. Moreover, this -point correlation function is computable from the Ward identities of CFT.

In the language of string theory, the objects we consider are correlators of orbifold points in theories whose target space consists of copies of the given CFT. We expect that some of our results may therefore have appeared in the literature of the subject.

### iii.1 Single interval

We first consider the case and no boundaries, that is the case considered by Holzhey et al.Holzhey of a single interval of length in an infinitely long 1d quantum system, at zero temperature. The conformal mapping maps the branch points to . This is then uniformised by the mapping . This maps the whole of the -sheeted Riemann surface to the -plane . Now consider the holomorphic component of the stress tensor . This is related to the transformed stress tensor byBPZ

(10) |

where is the Schwartzian derivative . In particular, taking the expectation value of this, and using by translational and rotational invariance, we find

(11) |

This is to be compared with the standard formBPZ of the correlator of with two primary operators and which have the same complex scaling dimensions :

(12) |

where are normalised so that . (12) is equivalent to the conformal Ward identityBPZ

(13) |

In writing the above, we are assuming that is a complex coordinate on a single sheet , which is now decoupled from the others. We have therefore shown that

(14) |

Now consider the effect of an infinitesimal conformal transformation of which act identically on all the sheets of . The effect of this is to insert a factor

(15) |

into the path integral, where the contour encircles the points and . The insertion of is given by (12). Since this is to be inserted on each sheet, the right hand side gets multiplied by a factor .

Since the Ward identity (13) determines all the properties under conformal transformations, we conclude that the renormalised behaves (apart from a possible overall constant) under scale and conformal transformations identically to the th power of two-point function of a primary operator with . In particular, this means that

(16) |

where the exponent is just . The power of (corresponding to the renormalisation constant ) has been inserted so as the make the final result dimensionless, as it should be. The constants are not determined by this method. However must be unity. Differentiating with respect to and setting , we recover the result of Holzhey et al.

The fact that transforms under a general conformal transformation as a 2-point function of primary operators means that it is simple to compute in other geometries, obtained by a conformal mapping , using the formulaBPZ

(17) |

For example, the transformation maps each sheet in the -plane into an infinitely long cylinder of circumference . The sheets are now sewn together along a branch cut joining the images of the points and . By arranging this to lie parallel to the axis of the cylinder, we get an expression for in a thermal mixed state at finite temperature . This leads to the result for the entropy

(18) |

For we find as before, while, in the opposite limit , . In this limit, the von Neumann entropy is extensive, and its density agrees with that of the Gibbs entropy of an isolated system of length , as obtained from the standard CFT expressionBCN ; Affleck for its free energy.

Alternatively, we may orient the branch cut perpendicular to the axis of the cylinder, which, with the replacement , corresponds to the entropy of a subsystem of length in a finite 1d system of length , with periodic boundary conditions, in its ground state. This gives

(19) |

Note that this expression is symmetric under . It is maximal when .

### iii.2 Finite system with a boundary.

Next consider the case when the 1d system is a semi-infinite line, say , and the subsystem is the finite interval . The -sheeted Riemann surface then consists of copies of the half-plane , sewn together along . Once again, we work initially at zero temperature. It is convenient to use the complex variable . The uniformising transformation is now , which maps the whole Riemann surface to the unit disc . In this geometry, by rotational invariance, so that, using (10), we find

(20) |

where as before. Note that in the half-plane, and are related by analytic continuation: . (20) has the same form as , which follows from the Ward identities of boundary CFTJCbound , with the normalisation .

The analysis then proceeds in analogy with the previous case. We find

(21) |

so that .

Once again, this result can be conformally transformed into a number of other cases. At finite temperature we find

(22) |

By taking the limit when we find the same extensive entropy as before. However, we can now identify as the boundary entropy , first discussed by Affleck and LudwigAffleckLudwig .

For a completely finite 1d system, of length , at zero temperature, divided into two pieces of lengths and , we similarly find

(23) |

### iii.3 General case.

For general , the algebra is more complicated, but the method is the same. The uniformising transformation now has the form , with (so there is no singularity at infinity.) Here can be , , or in the case of a boundary. In our case, we have , but it is interesting to consider the more general transformation, and the notation is simpler. Once again we have

(24) |

Consider . As a function of , this is meromorphic, has a double pole at each , and is as . Hence it has the form

(25) |

where . In order to determine and , we need to compute , etc, to second order in their singularities at . After some algebra, we find

(26) | |||||

(27) | |||||

(28) |

Let

(29) |

Then the coefficient of is

from which we find after a little more algebra that

(30) | |||||

(31) |

Thus we have shown that

(32) |

This is to be compared with the conformal Ward identity

(33) |

For these to be equivalent, we must have and

(34) |

A necessary and sufficient condition for this is that

(35) |

for each pair . This reduces to

(36) |

that is, for each pair . Since the only way to satisfy this is to have , with , and half the and the remainder . Interestingly enough, this is precisely the case we need, with .

If these conditions are satisfied,

(37) |

so that

(38) |

where is independent of all the . In the case with no boundary, can depend however on the . A similar calculation with then gives a similar dependence. We conclude that behaves under conformal transformations in the same way as

(39) |

Taking now , and according as or , we find

(40) |

The overall constant is fixed in terms of the previously defined by taking the intervals to be far apart from each other, in comparison to their lengths.

Differentiating with respect to and setting , we find our main result of this section

(41) |

A similar expression holds in the case of a boundary, with half of the corresponding to the image points.

Finally we comment on the recent result of Casini and HuertaCasini , which corresponds to . In fact, it may be generalised to the ratio of Tsallis entropies: from (40) we find

(42) |

where , and the expression in parentheses is the cross-ratio of the four points. Notice that in this expression the dependence on the ultraviolet cut-off disappeared, so have the non-universal numbers . Differentiating with respect to gives the result of Casini and HuertaCasini , who however assumed that the result should depend only on .

## Iv Entropy in non-critical 1+1-dimensional models

### iv.1 Massive field theory - general case

In this section we consider an infinite non-critical model in 1+1-dimensions, in the scaling limit where the lattice spacing with the correlation length (inverse mass) fixed. This corresponds to a massive relativistic QFT. We first consider the case when the subset is the negative real axis, so that the appropriate Riemann surface has branch points of order at 0 and infinity. However, for the non-critical case, the branch point at infinity is unimportant: we should arrive at the same expression by considering a finite system whose length is much greater than .

Our argument parallels that of ZamolodchikovZam for the proof of his famous -theorem. Let us consider the expectation value of the stress tensor of a massive euclidean QFT on such a Riemann surface. In complex coordinates, there are three non-zero components: , , and the trace . These are related by the conservation equations

(43) | |||||

(44) |

Consider the expectation values of these components. In the single-sheeted geometry, and both vanish, but is constant and non-vanishing: it measures the explicit breaking of scale invariance in the non-critical system. In the -sheeted geometry, however, they all acquire a non-trivial spatial dependence. By rotational invariance about the origin, they have the form

(45) | |||||

(46) | |||||

(47) |

From the conservation conditions (43) we have

(48) |

Now we expect that and both approach zero exponentially fast for , while in the opposite limit, on distance scales , they approach the CFT values (see previous section) , .

This means that if we define an effective -function

(49) |

then

(50) |

If we were able to argue that , that is
, we would have
found alternative formulation of the -theorem.
However, we can still derive an integrated form of the -theorem,
using the boundary conditions:^{3}^{3}3We have assumed that theory is
trivial in the infrared. If the RG flow is towards a non-trivial theory,
should be replaced by .

(51) |

or equivalently

(52) |

where the integral is over the whole of the the -sheeted surface. Now this integral (multiplied by a factor corresponding to the conventional normalisation of the stress tensor) measures the response of the free energy to a scale transformation, i.e. to a change in the mass , since this is the only dimensionful parameter of the renormalised theory. Thus the left hand side is equal to

(53) |

giving finally

(54) |

where is a constant (with however ), and we have inserted a power of , corresponding to the renormalisation constant discussed earlier, to make the result dimensionless.

This shows that the dependence for the exponent of the Tsallis entropy is a general property of the continuum theory. Differentiating at , we find the main result of this section

(55) |

where is the correlation length. We re-emphasise that this result was obtained only for the scaling limit . However, for lattice integrable models, we shall show how it is possible to obtain the full dependence without this restriction.

So far we have considered the simplest geometry in the which set and its complement are semi-infinite intervals. The more general case, when is a union of disjoint intervals, is more difficult in the massive case. However it is still true that the entropy can be expressed in terms of the derivative at of correlators of operators . The above calculation can be thought of in terms of the one-point function . In any quantum field theory a more general correlator should obey cluster decomposition: that is, for separations all , it should approach . This suggests that, in this limit, the entropy should behave as , where is the number of boundary points between and its complement. This would be the 1d version of the area laws-93 . When the interval lengths are of the order of , we expect to see a complicated but universal scaling form for the cross-over.

### iv.2 Free bosonic field theory

In this subsection we verify Eq. (55) by an explicit calculation for a massive free field theory (Gaussian model.) The action

(56) |

is, as before, considered on a -sheeted Riemann surface with one cut, which we arbitrarily fix on the real negative axis.

To obtain the entanglement entropy we should know the ratio
, where is the partition function
in the -sheeted geometry. There are several equivalent ways to calculate
such partition function. In the following, we find easier to use the
identity^{4}^{4}4This holds only for non-interacting theories: in the
presence of interactions the sum of all the zero-point diagrams has to be
taken into account.

(57) |

where is the two-point correlation function in the -sheeted geometry. Thus we need the combination . obeys

(58) |

Its solution (see the Appendix) may be expressed in polar coordinates as (here and )

(59) |

where , , , and are the Bessel functions of the first kind. At coincident points (i.e. , ), and after integrating over and we have

(60) |

where and are the modified Bessel functions of the first and second kind respectively as .

The sum over in (60) is UV divergent. This reflects the usual short-distance divergence which would occur even in the plane. However, if we formally exchange the order of the sum and integration we find

(61) |

Interpreting the last sum as , we obtain the correct result, which we now derive more systematically.

Let us first regularise each sum over by inserting a function : is chosen so that , and all its derivatives at the origin vanish: however it goes to zero sufficiently fast at infinity. Since is conjugate to the angle , we should think of this cut-off as being equivalent to a discretisation . Thus we should choose , where .