Entanglement Entropy Via QFT

Entanglement entropy via QFT

Paul Rozdeba1 1Department of Physics, University of California at San Diego, La Jolla, CA 92093 The entanglement entropy is a useful measure of the “degree” of entanglement in a quantum system. Certain systems can eﬀectively be described by a corresponding QFT; in the case of a 1-dimensional system near a phase transition, the entropy can be calculated using a relativistic, massless CFT in 1+1 spacetime dimensions.

INTRODUCTION be finite or (semi-)infinite. At finite inverse temperature β, the density matrix is defined by Measuring the “amount” of entanglement in a system 1 −βHˆ is a matter of theoretical interest [1], but seems loosely ρij = h{φi(xi)}|e |{φj(xj)}i. (2) defined on its own. One metric of the entanglement is Z(β) something called the entanglement entropy; as we’ll see, Z(β) is the partition function, defined in the usual way this is a good metric to study, because in certain cases −βHˆ (namely, when the theory is reducible to a conformal field as Z(β) ≡ Tr e . Here, the trace becomes a path theory, or CFT), this is an exactly calculable quantity. integral over the collection of φ’s; explicitly Since we could easily be talking about systems which 1 Z Y aren’t in the realm of QFT, for example a chain of spins, ρ = [dφ(x, τ)] δ(φ(x, 0) − φi(xi)) Z you might ask why I’m bothering to mention field theory x R β Y − dτLE at all. It turns out to be useful for such systems near × δ(φ(x, β) − φj(xj))e 0 . (3) critical points in their phase diagrams, in which case cor- x relation lengths grow much larger than any microscopic “graininess” like a lattice spacing. If we want Z(β), it’s just the numerator of (2) but with I’ll start by defining the entanglement entropy, then the identification φi(x) = φj(x); in the path integral this talk about CFT a little bit before moving on to a spe- has the effect of “stitching” together the integration sur- cific example due to [4], in which case it is clear how face at τ = 0 and τ = β, so it now looks like a tube with the machinery of CFT is applicable and powerful in the circumference β. computation of the entropy. At some point we want to take a subset of our domain, which might be a collection of intervals (u1, v1) ... (uN , vN ); call it A. To calculate the reduced ENTANGLEMENT ENTROPY density matrix ρA, we should only stitch the edges of the integration surface in (3) at values of x which are not Imagine subdividing a quantum system into several contained in A. This leaves open cuts on our tube. If we parts [1]; for now, let’s say two, called A and B. Also join together n copies of this tube (bear with me, there imagine that the system A ∪ B is a pure quantum state is a goal in mind) at the cuts, i.e. in such a way that we k k+1 |Ψi, so that we can define the density matrix ρ = |ΨihΨ|. identify φi (x) = φj (x) for all x ∈ A, where 1 ≤ k ≤ n We can define the reduced density matrix for either sub- i.e. k labels the tube, this is like taking the trace of n n system by tracing out the degrees of freedom of its com- powers of ρ, i.e. Tr ρA. The figure below is a really nice plement àla pictorial representation of this surface, courtesy of [2].

ρA = Tr Bρ (1) and vice versa for ρB. The entanglement entropy is de- ﬁned for subsystem A, for example, in an identical man- ner to the von Neumann entropy as SA = −Tr ρA log ρA. To (heavily) paraphrase Holzhey, et. al. [4], this entropy describes correlations between A and A ∪ B by counting the number of states in B consistent with measurements restricted to A, given a priori knowledge that A ∪ B is a pure state. P Furthermore, it is true that Tr ρA = λ λ where the Let’s think about a speciﬁc system [1]; namely, a set λ’s are the eigenvalues of ρA, and λ ∈ [0, 1) since it’s ˆ n P n of local commuting observables {φ(xi)} living on a one- a density matrix; this means Tr ρA = λ λ converges dimensional lattice with sites labeled by the xi, with lat- and is thus an analytic function for n > 1. In addition, tice spacing a. The domain of x has length L, which may using the usual rule for derivatives of an exponential, we 2

ﬁnd that conditions:

∂ n [gµν + (d − 2)∂µ∂ν ] ∂ · ε = 0 (9) SA = −Tr ρA log ρA = − lim Tr ρA (4) n→1+ ∂n where d is the number of dimensions (these conditions n holds specifically if we take the flat-space case g = δ ). which is well-defined because of the analyticity of Tr ρA. µν µν Finally, if we define the partition function on the n-sheet Is d = 2 somehow special? Answer: yes, it is! Eqs. (9) (comprised of the n stitched-up tubes) to be Zn(A), then reveal themselves to be the Cauchy-Riemann equations. we arrive at the result The conformal transformations in d = 2, therefore, can be written as holomorphic functions on z andz ¯, under the n Zn(A) substitutions z = x1 + ix2 andz ¯ = x1 − ix2. Soon, we’ll Tr ρA = (5) Zn use z andz ¯ with the Euclidean space-time coordinates; ∂ Zn(A) it turns out this is a useful construct. ⇒SA = − lim (6) n→1+ ∂n Zn where the first equality holds by definition. This is a AN EXAMPLE: RELATIVISTIC, MASSLESS nice result, because it lets us calculate SA without ever 1+1-DIM THEORY having to calculate a density matrix (which, in general, may be very difficult or impossible to compute). This example is originally due to [4], however this dis- We can make this look like quantum field theory by cussion follows [1] which is much clearer and is largely a taking the limit a → 0. As we’ll see, this leads to terms “redo” of the original treatment. More specifically than in S which are proportional to log a. This looks like a the section title would imply, this is a case in which the problem, but in fact this dependence on a can be shown system in question is restricted to a single interval on the to drop out of the final expression. In the section af- lattice with length `. Before we jump into this specific ter next, we’ll focus on a relativistic and massless 1+1 example, however, we should discuss the nature of the dimension field theory, which is writable as a so-called stress tensor in CFT. conformal field theory, or CFT. After analytic continuation [5] from Euclidean spacetime to z, z¯, we can write the stress tensor as a function WHAT IS A CFT? of these complex variables. Additionally, we can expand the insertion of T at two points z and w, the form of which is general and what is term-by-term allowed by Before moving on, it’s worth noting what a CFT ac- conformal symmetry: tually is. A CFT exhibits conformal symmetry in the c/2 2T (w) ∂T (w) sense that the action of the theory is invariant under con- T (z)T (w) ∼ + + , (10) formal transformations. The conformal group includes (z − w)4 (z − w)2 (z − w) the translation, rotation, and dilatation (scaling) trans- which is the operator product expansion (OPE) for formations (remember that a Lorentz transformation is T (z)T (w). The first term is called the conformal like a rotation after we analytically continue to Euclidean anomaly, characterized by the central charge c. This time!). As you may have learned long ago, these are the term is anomalous because, classically, the expectation transformations which preserve angle measures between value of T should vanish. In fact, c = 0 classicaly, so the intersecting lines. first term is purely a quantum effect. As an example [3], consider the two-point function for a The central charge is actually a pretty remarkable scalar field, hφ(x)φ(y)i. If the theory exhibits conformal thing. One way to think about [3] it is that it’s related to symmetry, then under a scaling of the coordinates x → a “soft” conformal symmetry breaking which arises from λx, the two-point function should change at most up to the introduction of a macroscopic scale into the system, a scale factor: which could be a boundary condition, for example. Even more interestingly, c lumps theories together into large hφ(x)φ(y)i = λ2∆ hφ(λx)φ(λy)i (7) universality classes; for example, c = 1 for the free bo- where ∆ is the scaling dimension for φ. In fact, conformal son, c = 1/2 for the free fermion, etc. In some sense, symmetry imposes translational and rotational ivariance it’s an extensive measure of the degrees of freedom in the as well, so that two-point functions must be of the form system. We’ll see how this manifests in the calculation of the entanglement entropy. hφ(x)φ(y)i = |x − y|−2∆ (8) Eq. (10) can be written in terms of the generator of the transformation w → z, then integrated to give the (up to the normalization of φ). transformation of T in the form Consider [5] a coordinate transformation of the form 2 µ µ µ dz c x → x + ε (x). It turns out that if this is a confor- T (z) → T (w) = T (z) + S[z; w] (11) mal transformation, εµ(x) is restricted by the following dw 12 3

0 where S[z; w] is the Schwartzian derivative, where c1 is some constant which is not universal (unlike the central charge!). Notice that a length scale has been 3 3 2 2 (∂wz)(∂wz) − 2 (∂wz) introduced into the expression. As we take a → 0, we S[z; w] = 2 (12) (∂wz) might think of this as taking the lattice spacing to zero. The expression diverges logartihmically; in a crude sense, where you should remember that z = z(w). it’s like we’re trying to add up an infinite number of de- We want to know about the stress tensor in the first grees of freedom. This issue can be eliminated in some place because (5) is just the vacuum expectation value cases with the proper choice of renormalization factor [4]. h0|0iRn on the stitched-up Riemann n-surface; but this In fact, this amounts to redefining the entropy with re- is just hT (w)iRn . We can do even better by finding a spect to the ground state, since differences in entropies transformation that sends us to some z in which we know 0 are much more well-behaved. In this case, even c1 will the expectation value of T (z) to be zero, eliminating it disappear. from the expectation value of (11)! Namely, we can send the Riemann n-sheet to the plane , in which case we C We can redo the previous calculation, but at a finite know hT (z)iC = 0 by rotational invariance. The mapping that does the trick [1] is inverse temperature β, and obtain instead the result

w − u1/n w → z = (13) w − v c β π` S (β) ∼ log sinh + c0 (18) where u and v are the branch points on the Riemann sur- A 3 πa β 1 face at the edges of the interval in consideration (under z they get sent to 0 and ∞, respectively). Using the conformal Ward identities, one can also calculate a correlator like hT (w)φn(u)φ−n(v)i , where C in which case we recover the previous result in the low- φ±n are two operators with the same (complex) scal- temperature limit ` β. In the high-temperature limit ing dimension ∆n = ∆¯ n, and are normalized so that −4∆n ` β, the entropy goes to ∼ (πc/3)(`/β), which is an hφn(u)φ−n(v)i = |v − u| , with C extensive quantity, comfortingly in agreement with our " # c 1 2 classical expectations! We have thus recovered a result ∆ = 1 − . (14) n 24 n which, in some sense, is a connection between the classical and quantum regimes. The details of the intervening calculation is not as im- portant as the result, which is that Acknowledgements Zn(A) hT (w)φn(u)φ−n(v)iC n = hT (w)iRn = . (15) Z hφn(u)φ−n(v)i C Thanks for prof. J. McGreevy for an illuminating con- This is good, because as it turns out, the RHS is actually versation about the nature of the stress tensor and the an expression that is totally in terms of w, the branch central charge. points u, v and the scaling dimension ∆n! This gives us the LHS of (15) up to a constant cn:

−(c/6)(n−1/n) n v − u Tr ρA = cn (16) [1] P. Calabrese and J. L. Cardy, J. Stat. Mech. 0406, P06002 a (2004) [hep-th/0405152]. [2] P. Calabrese and J. L. Cardy, Int. J. Quant. Inf. 4, 429 where a has been inserted for dimensional reasons. The (2006) [quant-ph/0505193]. cn’s aren’t determined, but for n = 1 this constant should [3] P. Di Francesco, P. Mathieu and D. Senechal, New York, be 1 (since ρA is normalized). The result, after using the USA: Springer (1997) 890 p derivative trick, is that [4] C. Holzhey, F. Larsen and F. Wilczek, Nucl. Phys. B 424, 443 (1994) [hep-th/9403108]. c ` [5] S. V. Ketov, Singapore, Singapore: World Scientiﬁc (1995) S = log + c0 (17) A 3 a 1 486 p