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This content was downloaded from IP address 130.237.165.40 on 15/11/2018 at 18:58 JSTAT (2004) P06002 of a $30.00 A 27 ρ is large but in an infinite ξ ,were-derivetheresult c models, which are solvable 2 , 1 is the number of boundary points XXZ A Theory and Experiment Theory is a finite interval of length A ,where ξ when 6) log and John Cardy c/ 3 ( , et al 1 ∼A corresponding to the reduced density matrix A A S ρ of Holzhey log conformal field theory, integrable spin chains (vertex models), consists of an arbitrary number of disjoint intervals. For such a A We carry out a systematic study of entanglement entropy in . For the case of a 1+1-dimensional critical system, whose continuum ρ A A stacks.iop.org/JSTAT/2004/P06002 Tr 3) log [email protected] − c/ ( = . These results are verified for a free massive field theory, which is also used ∼ A Rudolf Peierls Centre for Theoretical Physics, 1All Keble Souls Road, College, Oxford, UK A A relativistic quantum field theory.S This is defined as the von Neumann entropy entanglement in extended quantum(theory), systems new (theory), applications quantum of phase statistical transitions mechanics Keywords: Abstract. 1 2 doi:10.1088/1742-5468/2004/06/P06002 Pasquale Calabrese Oxford OX1 3NP, UK E-mail: Received 24 May 2004 Accepted 29 May 2004 Published 11 June 2004 Online at by corner transfer matrixhigher methods. dimensions, and Finally used thethe to free entanglement motivate field entropy a results near scaling are a form extended quantum for phase to the transition. singular part of subsystem system, and extend itand to when many othersystem cases: away finite from systems,finite, its we finite show critical that temperatures, point, when the correlation length of to confirm a scalingintegrable ansatz lattice for models, the such as case the of Ising finite and size off-critical systems, and for limit is a conformal field theory with central charge S ournal of Statistical Mechanics: of Statistical ournal
An IOP and SISSA journal An IOP and 2004 IOP Publishing Ltd PII: S1742-5468(04)81343-8 1742-5468/04/P06002+ Author to whom any correspondence should be addressed.
c J 3
field theory Entanglement entropy and quantum JSTAT (2004) P06002 2 2 5 6 A 12 19 23 25 25 26 S =0. A S . The entanglement ρ B ], even in the partially 1 ) =Tr associated with this reduced A 2) degrees of freedom, with A , and an observer A measures ρ / | ...... 16 ρ Ψ . Suppose the whole system is in log A Ψ | Entanglement entropy and quantum field theory ρ A 2, which agrees with our intuitive idea = / Tr ρ entropy -sheeted Riemann geometry: infinite − =1 . For an unentangled product state, N = θ B ...... 12 2 S A S ...... 8 = A copies of the state available, by making only local ...... 19 S stacks.iop.org/JSTAT/2004/P06002 ...... 14 M -dimensional models 1 ] is entanglement + + + 1 1 , where A observes only the first spin and B the second, , with density matrix ...... 7 Ψ |↓↑ (2004) P06002 ( | ...... 9 θ of a complete set of commuting observables, while another observer B +sin should be a maximum for a maximally entangled state. A A S |↑↓ θ =cos For example, for a system with two binary (spin-1 One of these measures [ 3.1. Single interval 3.2. Finite system3.3. with a General boundary case 4.1. Massive field4.2. theory—general case Free bosonic4.3. field theory Integrable models and the corner transfer matrix 5.1. FSS in the Gaussian model Acknowledgments Appendix: the two-point function in an and finite volume results References Ψ a pure quantum state | J. Stat. Mech.: Theor. Exp. takes its maximum value of log 2 when cos Recently thereentanglement has and been applyingfreedom, considerable them such to as interest quantum extended spin in quantum chains. systems formulating with measures many of degrees of quantum 1. Introduction 2. von Neumann entropy and3. Riemann surfaces Entanglement entropy in 2D conformal field theory 5. Finite size effects 6. Higher dimensions: scaling of entropy and area law 1. Introduction Contents entropy is just the von Neumann entropy density matrix. It is easy to see that of maximal entanglement.entangled For case, this that system if it there has are been shown [ may measure the remainder. A’s reduced density matrix is only a subset 4. Entropy in non-critical Conversely, JSTAT (2004) P06002 3 : 1 (1.3) (1.2) (1.1) − ,and β increases, number of disjoint ]. is much larger than 5 , the field theory is 1 . ξ ] on a more systematic ]–[ − 7 3 a [ ) arbitrary | et al ], in the context of black hole k ], the entropy is most readily 7 ]. We also treat the case when , in an infinitely long chain. It [ 2 for large 9 , 2and1[ can be much larger than unity, so cannot field theory (CFT). 1 / with open boundaries, divided at is the conformal anomaly number | ], the concept has been applied to c A c 6 S 1 L is a single interval of length . et al + − 4 =1 Entanglement entropy and quantum field theory 1 . ]–[ . ξ g -sheeted Riemann surface, in the limit 1 c c 3 A A c n and its complement, we find S consists of an )) + )) + consists of a commuting set of components conformal ), where )) + )), and when the whole system has itself a A ,by A /a M π/L π/β such that 3.32 π/L k of length states which are maximally entangled. The optimal )sin( 3) log( A stacks.iop.org/JSTAT/2004/P06002 )sin( ) sinh( c/ ( L/πa
Entanglement entropy and quantum field theory which counts the number of boundary ξ 3) log( -sheeted Riemann surface. In section A c/ n ( , proportional to ∼ A Heisenberg spin chains. In section / A A S S is devoted to the discussion of higher dimensions. 6 XXZ stacks.iop.org/JSTAT/2004/P06002 is the same, but non-universal. . ) consists of a collection of disjoint intervals, each of length 1 c model, computing exactly at all values of the coupling the A is large. ξ/a ξ . This agrees with our continuum result ( S ], the result (the 1D analogue of surface area). XXZ 11 (2004) P06002 ( 6) log( B c/ ( . ) to be multiplied by a factor and ∼ 1.4 A A S = 1 the entropy is exactly calculable in the case of a free field theory. We verify universal A ], was obtained by the following argument: it was assumed that the entropy For a massive 1 + 1-dimensional relativistic QFT (which corresponds to an off-critical While all this work was being carried out, some other related papers have appeared For The analysis for the free theory is straightforward to extend to higher dimensions, at The layout of this paper is as follows. In the next section, we discuss the entropy 7 , we expect ( [ ξ of the system. For a lattice model in this geometry, with al quantum spin chain where the correlation length J. Stat. Mech.: Theor. Exp. is simply related to Baxter’sweights corner satisfy transfer a matrix, Yang–Baxter and relation,treat thus, all for explicitly its integrable the models eigenvalues case can whose spin be of chain, determined the and exactly. also Ising We the model and its anisotropic limit, the transverse Ising finite part of the entropy pointed out by Srednicki, theand coefficient is of therefore this non-universal. term,there for However, on should the be basis a of non-leading our calculations, piece we in propose that In all these cases, the constant in the literature. In [ we consider the 1to + 1-dimensional show conformal that case. theconstant, We partition by use function a the on calculablecomplex powerful the correlation methods plane. Riemann function of surface of CFT Similarallows of vertex results us operators interest apply to is in to given, a derive a up all system to the a withfrom special completely boundaries. cases general properties This describedthen of verified general above. for the result a stress Inlattice free tensor version section bosonic in of massive the this field.entropy relevant problem for In geometry. to the the the This case last corner is the of part transfer off-critical the of matrix, case Ising this and in and sectionof a compute we a finite the relate size free off-critical the system, massivesequence propose field of a theory. approximations. scaling We law, Section compute and verify the it relevant for scaling the function case in a systematic L the above formula, and exhibit the finite size crossover which occurs when should be conformally invariant and therefore some function infinite system divided atthat some the point entanglement into entropy is two semi-infinite finite, pieces. and derive In the this universal case formula we verify least in suitable geometries. This leads to the well-known law, first found by Srednicki [ when the correlation length that the entropy should be proportional to the surface area a singular waymoreover, on the couplings near a quantum phase transition, and whose form is, case of a massive 1 + 1-dimensional field theory. We derive the result in terms of the Euclidean path integral on an In the more general case when points between JSTAT (2004) P06002 5 )is and and
(2.1) (2.2) B } . β A ) β ∪ x intervals ( A φ 5 ( { =2.These S copies of the . The density may be found − n N =0. ˆ H ) A = 1, and is found τ ρ B ρ , . This will have the , E ∩ A ] it was observed that S } − A ) 13 ( ; finally the limit )e x , S x , semi-infinite, or infinite. in the disjoint ( ) ) L φ 12 − [ ,bymaking x x |{ ) n ( β ˆ H φ B β ( −
− S e ), along the line ) j is }| Entanglement entropy and quantum field theory ,v β ) )+ j which are not in x, β u ( A x ( x φ ( ( S φ , and sewing them together cyclically along δ to form a cylinder of circumference ). n { ≡ ; finally it was assumed (with no justification x 1 β − ≤ ) ) ) A, B k β ( ) ] have considered the case of two intervals is valid for all values of ( →∞ 6 F x ≤ A, B Z ( x ( F φ stacks.iop.org/JSTAT/2004/P06002 , and the lattice sites are labelled by a discrete variable F =0and a as − , for any positive integer )= τ } n A x the Euclidean Lagrangian. (For a spin model this would with 1 0) ρ ) and integrating over these variables. This has the effect of E k x x, L ( } ( form for φ ) φ 3) log x ( x ( δ c/ }|{ is the partition function. φ (2004) P06002 ( x ) { , and their eigenvalues and corresponding eigenstates by ,with ˆ H can be finite, i.e. some interval of length ); by comparing with known case } τ β ). An expression for the reduced density matrix x ) = )] d 0, and is given by a universal logarithmic function of the cross-ratio ( x − N x } φ E ( be a subsystem consisting of the points ˆ { ,v φ L → ) x, τ π/β ( { ( N β x ρ A 0 ] based on this quantity a ( u φ ( respectively. For a bosonic lattice field theory, these will be the fundamental φ 14 [d { )=Tre = in a thermal state at inverse temperature ) sinh( ) by sewing together only those points β } ( ) E ρ 1 ,..., x S Z 2.2 ) ( − β/π 1 φ Very recently Casini and Huerta [ Now let This may be expressed in the standardZ way as a (Euclidean) path integral: The normalization factor of the partition function ensures that Tr We may then compute Tr ) should behave as ( ,v |{ x 1 =( , and argued that the quantity = ( x This restriction is not necessary, but the set-up is easier to picture in this case. . The domain of -theorem [ u ⊗ was taken. Infrom the first present principles paper, of we CFT. should stress, we have derived all these statements being given) that the log B Time is considered tobe be denoted continuous. by A complete set of local commuting observables will 5 x F Consider a lattice quantuminfinite theory line. in The one lattice spacing space is and one time dimension, initially on the 2. von Neumann entropy and Riemann surfaces x above, labelled by an integer ρ J. Stat. Mech.: Theor. Exp. be replaced by a coherent state path integral.) matrix bosonic fields of theThe theory; for dynamics a of spin model the some theory particular is component of described the by local spin. a time-evolution operator sewing together the edges along authors, however, assume thestress conformal that invariance in of the thetensor. present entropy, while, paper These once wec again, derive authors this we also from give fundamental properties a of very the nice stress alternative derivation of Zamolodchikov’s UV finite as of the four end points. This corresponds to, and agrees with, our case where where from ( effect of leaving open cuts, one for each interval ( by setting ( JSTAT (2004) P06002 . ) 6 j a u ( then (2.5) (2.3) (2.4) n ,and c x 1)), and ) should , 1+ of the 2.3 will emerge → a n ). However, as )isthesameas 2.3 -sheeted Riemann n 2.3 . Let us denote the A in a domain with total (which lie in [0 ∈ Z A ) x ρ terms proportional to log )onan is supposed to go over into E x, τ S . For the moment let us simply ( a φ ). Its dependence on ) for all 1 0, the right-hand side of ( is finite, the limit as ) Entanglement entropy and quantum field theory universal x A, n λ ( ). Then ( → ) gives an expression for the Tsallis ) has a unique analytic continuation to φ A Z a ( therefore also exists and is analytic in the log 2.3 2.3 behaves as n = , λ n Z n L λ ) ··· = 1 gives the required entropy: x . In this limit, ( j are the eigenvalues of n + φ v − 1 λ − = . ]therearealso a stacks.iop.org/JSTAT/2004/P06002 ) 8 and A L (and 0 keeping all other lengths fixed. The points 2 ρ A n j in the ratio of partition functions in ( ( f u Z n +1 ).) k → ,where + Z n ) n . 2 x a ) λ − − ( ∂ λ ∂n A a n φ cancel ( (1 1 Z A n / 1 = → (2004) P06002 ( 1. The derivative wrt -sheeted structure by Z lim f n 1) = k n ) = − = n A − x n> ( ρ n A n A = Z are the non-universal bulk and boundary free energies. Note, however, φ ρ ρ A 2 f S Tr log = 1, it follows that the left-hand side is absolutely convergent and therefore ](Tr and A 15 ρ 1 and with boundaries of total length 1 and that its first derivative at f A Now, since Tr We conclude that the right-hand side of ( So far, everything has been in the discrete space domain. We now discuss the We show that in this case the ratio of partition functions in ( n> region. Moreover, if the entropy where path integral on this analytic for all Re J. Stat. Mech.: Theor. Exp. the cuts so that first derivative converges to this value. Re assume real values, and the path integral is over fields surface, with branch points at continuum limit, in which These arise from pointscase, of these non-zero are curvatureis conical of precisely singularities the these at manifold logarithmicfinal and the terms result its branch for which boundary. the points. giveremark entropy rise In that, on In in to our the order the fact, short-distance to non-trivial cut-off achieve as a dependence we finite of limit shall the as show, it that these leading terms be multiplied by some renormalization constant initially consider the case of zero temperature. the Euclidean action forwhen this a is quantum Lorentz fieldbrought invariant, theory. to since bear. the We full shall TheIn power behaviour restrict two of of dimensions, attention relativistic the partition to field logarithm functions theory the of in can a case this then general limit be partition has function been well studied. area (Note that even before taking this limit, ( entropy [ since Tr was argued by Cardy and Peschel [ the correlation function arising from the insertion of primary scaling operators Φ Now specialize the discussionrelativistic of and the massless, previous i.e. section a to conformal the field case theory when (CFT), the with field central theory charge is 3. Entanglement entropy in 2D conformal field theory from the later analysis. JSTAT (2004) P06002 . ) 7 n . /n w C )is 1 ( ) (3.1) (3.2) (3.5) (3.6) (3.4) (3.3) α )) v 3.3 v ( + n − − with two w , w n ( )Φ = T / ∆ u 2 ) ( ) w . Now consider u n ¯ u Φ C − → . − .Equation( w . C v n w 2 ), into each of the ∂ (¯ ) ∆ ∂v ) 2 2 2 C v v − v . In particular, taking =(( − ( -plane n ) /n 2 n n z − 2 v 1 1 − ) maps the branch points ) − /n ( 2∆ ) v 2∆ 1 /z w ) u n w − ) ( − ζ u )Φ − | 2 2 − − u ) u n + = ( − )Φ z to the u w v n ∆ ( ( z − u v w n ∂ 2) / ( ( − 12)(1 ∂u v ) / 2 )Φ n | R ] of the correlator of ) )d¯ → u u c/ w Φ w (3 v ). This is related to the transformed ( ( copies of the given CFT. We expect w Entanglement entropy and quantum field theory ζ 16 = 0 by translational and rotational = 1 − − (¯ ) w T − − 2 =( ( . The effect of this is to insert a factor n C T ) C w w n ) , T w n z ) ) is a complex coordinate on a single sheet ( } w = R /n v + 2 z ( ] =( ( ) ( z ) 2 (1 n w α 24 n u ) ζ T 16 − z, w R v C ) − =2∆ ( { n -point correlation function is computable from − n E → )Φ n − R c i S ∆ (1 N ( in an infinitely long 1D quantum system, at zero 12 w u c − π 1 w ( ( X E w 2 S n ( )e ) which have the same complex scaling dimensions − = = Φ v )+ stacks.iop.org/JSTAT/2004/P06002 w − + ( z } ( ]e C n ( 2 w φ T − ) ) T ] u [d v 2 z, w )d φ n ( ) ): { n 2 w − -sheeted Riemann surface [d ∆ = 1 and no boundaries, that is the case considered by Holzhey w ) ( − c n 12 d w T ( N /n ) )Φ ] z/ )andΦ ≡ = u w (1 (2004) P06002 ( u ( ( n n 16 ( = n α R R − n C C )Φ ) ) )=(d ) w w w )by[ v i ( ( ( w ( z ( π 1 24)(1 T T T n ( 2 is the Schwartzian derivative ( T − T c/ } are normalized so that ), with scaling dimensions )Φ j =( u n v ( ( ± ). This is then uniformized by the mapping z, w n n n ] of a single interval of length { − 7 ∆ ∞ [ )Φ which acts identically on all the sheets of , In the language of string theory, the objects we consider are correlators of orbifold = w ( C n , which is now decoupled from the others. We have therefore shown that T et al where J. Stat. Mech.: Theor. Exp. We first consider the case 3.1. Single interval and Φ invariance, we find the expectation value of this, and using This is toprimary be operators compared Φ with the standard form [ equivalent to the conformal Ward identity [ the holomorphic component ofstress the tensor stress tensor Now consider the effect of an infinitesimal conformal transformation temperature. The conformal mapping ∆ In writing the above,C we are assuming that This maps the whole of the points in theories whose target space consists of the Ward identities of CFT. (disconnected) sheets. Moreover, this 2 where Φ of that some of our results may therefore have appeared in the literature of the subject. to (0 JSTAT (2004) P06002 , . . 8 n = x β n 2. (3.7) (3.8) (3.9) +i et al
] (3.10) ∆ τ L/ 16 = = = ), and the n . w ∞ , must be unity. behaves (apart ··· ) 1 n A . The insertion of 2 c , as obtained from ρ v ¯ with ∆ in a thermal mixed w , n Tr n A 2 . By arranging this to ρ v w and , which maps the whole maps each sheet in the ∝ u /n = 0. Once again, we work ) 1 n )Φ( w and 1 )) ¯ . The sheets are now sewn /Z w ), using the formula [ u ) β , z 1 ( )log A +i ) for its free energy. ( w w . π . It is maximal when n 1 w x<,τ 2 ( c Φ( Z = /β / − ≤ β/ ) Entanglement entropy and quantum field theory . n z 1 (corresponding to the renormalization 6)( i L c )) + 2∆ | a =( − → ) → j πc/ , z = 1, we recover the result of Holzhey w ) ( z -sheeted Riemann surface then consists of )) + ( w π/β n /n n encircles the points 1 w means that it is simple to compute in other − | =(( ∼− → C n n π/L z j ) as before, while, in the opposite limit ± 6)( w ) sinh( βF c/ ). The ( ] = /a ) determines all the properties under conformal − , corresponds to the entropy of a subsystem of length stacks.iop.org/JSTAT/2004/P06002 )sin( , ) 13 L , with periodic boundary conditions, in its ground state. β/πa 3.4 , .Thepowerof and setting /a ··· L n ) → . This leads to the result for the entropy ) 12 n u 3) log( L/πa 1 2 ∆ ¯ β z − n c/ − , 0, sewn together along 0 β ( 2 v 3) log(( z ≥ transforms under a general conformal transformation as a two- (( ∼ c/ . x n ( are not determined by this method. However (2004) P06002 ( 3) log(( n A n )Φ( c A 1 ρ ). Since this is to be inserted on each sheet, the right-hand side gets n c/ ¯ ∼ z S c = ( , ) 1 3.3 n A β z ). In particular, this means that ∼ ). In this limit, the von Neumann entropy is extensive, and its density ( ρ 2 ) A A Φ( S Tr S /β is the finite interval [0 /n we find ) has been inserted so as the make the final result dimensionless, as it should (1 A 3)( β Z th power of two-point function of a primary operator Φ − n πc/
( Since the Ward identity ( For example, the transformation )isgivenby( The fact that Tr Alternatively, we may orient the branch cut perpendicular to the axis of the cylinder, ∼ 24)(1 w -plane into an infinitely long cylinder of circumference ( in a finite 1D system of length A c/ transformations, we conclude that the renormalized constant be. The constants copies of the half-plane initially at zero temperature. It is convenient to use the complex variable J. Stat. Mech.: Theor. Exp. Next consider the case when the 1D system is a semi-infinite line, say [0 3.2. Finite system with a boundary multiplied by a factor T The uniformizing transformation is now into the path integral, where the contour Note that this expression is symmetric under S where the exponent is just 4 geometries, obtained by a conformal mapping point function of primary operators Φ ( state at finite temperature subsystem For agrees with that of the Gibbs entropy of an isolated system of length Differentiating with respect to from a possibleto overall constant) the under scale and conformal transformations identically together along a branch cut joining the images of the points w the standard CFT expression [ lie parallel to the axis of the cylinder, we get an expression for Tr This gives which, with the replacement JSTAT (2004) P06002 , 9 ) (i (3.11) (3.15) (3.16) (3.12) (3.13) (3.14) n =0(so )Φ i w α ( i T are related by , etc, to second T inthecaseofa z = 0 by rotational j u and i ) ,with z i ± T ( α ) T i ], with the normalization ) w ,or 17 j − v , w . , first discussed by Affleck and . j ( 1 1 g i u c c , ) has the same form as + i Entanglement entropy and quantum field theory g . w )) + ˜ = , we need to compute i , at zero temperature, divided into two 2 i 3.11 L B − z B z can be 3 2 )) + w π/β i 2 , but it is interesting to consider the more , − w i z 2 and ) /n i z , π/L 1 1. In this geometry, ) A + ± , this is meromorphic, has a double pole at each z /n +i 2 ) sinh(2 2 1 ) w ) c = − i .Equation( |≤ w )sin( 12 ∗ n ( z stacks.iop.org/JSTAT/2004/P06002 i w i ) | (2 2 . Hence it has the form ) α = n β/πa A w we find the same extensive entropy as before. However, 12)( − ( i } ∆ c/ L/πa β ( T w − ) ( as the boundary entropy , we similarly find →∞ . z, w
w 1 1 i ( { )= ) as before. Note that in the half-plane, c we find /a w c 6) log(( 2 − w c 1 ), we find − (2 12 − = (¯ c/ − n L = n ( (2004) P06002 ( 6) log((2 1 T ˜ n )+˜ c β )as 3.1 = c − R 2 c/ ∼ /z ∼ − ) ) /a ) } and . . As a function of n A w β n w w =( ρ ( ( ( ∆ /z 24)(1 A A z, w T T − , the algebra is more complicated, but the method is the same. The } Tr S S { ) c/ = 0. In order to determine N i 6) log(2 z, w =( B ]. { c/ i 9 ( =(2 n ,andisO( i ∼ ) w The analysis then proceeds in analogy with the previous case. We find For a completely finite 1D system, of length Once again, this result can be conformally transformed into a number of other cases. A (i S n = Φ w where ∆ Consider where J. Stat. Mech.: Theor. Exp. uniformizing transformation now has the form For general 3.3. General case Riemann surface to the unit disc Ludwig [ pieces of lengths At finite temperature general transformation, and the notation is simpler. Once again we have so there is no singularity at infinity). Here we can now identify ˜ By taking the limit when boundary. In our case, we have analytic continuation: which follows from the Ward identities of boundary CFT [ invariance, so, using ( JSTAT (2004) P06002 10 (3.19) (3.20) (3.27) (3.22) (3.24) (3.23) (3.25) (3.26) .
, ) 2 . ] k
i w ··· ( w k ] 1 Φ − , )+ ··· z. i w k
j w w )+ j ··· − i j
w − i w α + w j k ( ) ∂ − j i α ; (3.18) − ∂w w i z C w )and . i
w k j w 2 i ( = + − w α
i i j α ) i i ··· = , 1 C k j − 2 α k w 2 − [ + w ) α + ) i / k ( w k 2 j i i } k = α Entanglement entropy and quantum field theory − j w 2 α α i w ] i + Φ 2)(1 j 1 ][ α 2 i α − 2 / − − 2 α k ) α ··· i ) i i ··· i i (1 w w w i
w ( ∂ +3 =(1 )+ i ∂w 2 k ∆ i ; (3.17) − i )+ i − = i log α z j w + α k = w
w i w 3 i α ( ) 2 − j ( − ) 2 i ∂ −
w i − 1 w i α ∂w j j w w i α − ( w + − w α − i ( − = j stacks.iop.org/JSTAT/2004/P06002 i − 3 i is w C = − w α i j C α = (1 2 i w i 2 α 3 1 2 w ( . After some algebra, we find − 1) ) ) w + = . i j
+2 i ) i
j i j ) 2 i i − α w i 2 − w w 2 i i . w i = w ) α k i ); (3.21) + w j i α j 2 i 3 2 i w − ( − α α 1) + 2 i i − − w α α 2 i ( α i i k + i i c − w w Φ α 12 = i − α i w i − ( w i − α j w − i ( (2004) P06002 ( α α α i − α − 1 i w = 2 ) α i 2 i − (1 ( ( 2 i = 1 w α ) i j
1 2 i C
) α
i α w α [ − 2) + 3 w ( ). This reduces to α = k = = ≡ 3 2 = − ( = 1 T i i i ∂ − T ∂w (1 i, k i z C B z − z
A − α 1)( − i α ( i α [ Let Then the coefficient of ( J. Stat. Mech.: Theor. Exp. order in their singularities at Thus we have shown that For these to be equivalent, we must have ∆ This is to be compared with the conformal Ward identity from which we find after a little more algebra that A necessary and sufficient condition for this is that { for each pair ( JSTAT (2004) P06002 a 11 )we (3.30) (3.28) (3.31) (3.33) (3.32) (3.29) . of the 1 3.31 η Nc + . corresponding k i ¯ 1. Interestingly σ ) i w σ − )¯ /a 2 , ) , we find ) α j can depend however E, j ], which corresponds v − v /n 6 that the result should 1 E by taking the intervals − − )+ or n n k 12)(1 k c j v ) c/ w 6)( . ( u , ) ) c/ k ( − k = /n i ¯ w k 1 w log(( i assumed ) ) w σ − i w 2 1 − n . . Differentiating with respect to − v σ i u n i j − i, k u w 1 according to whether ) B ( i j = j A B ) ) 1, and half the ( ( j Φ j n A ( j σ ,and ρ − n ∼ ∩ j Φ k S for each pair ( log /n n A Tr A v . i j k ( ρ η n ∂ α =1 ≡ S ∂w ,with log Tr ± i log(( α ) k behaves under conformal transformations in the same way as = ασ A ≤ . A similar calculation with ( i j i n A is independent of all the n = α ρ w S i E c 3 = 2. In fact, it may be generalized to the ratio of Tsallis entropies: from ( α Differentiating with respect to If these conditions are satisfied, A similar expression holds in the case of a boundary, with half of the Finally we comment on the recent result of Casini and Huerta [ = N gives the result of Casini and Huerta [ A on the ¯ that Tr J. Stat. Mech.: Theor. Exp. that is, four points. Noticedisappeared, that and in so have this the expression non-universal the numbers dependence on the ultraviolet cut-off S to be far apart from each other, insection: comparison to their lengths. Taking now n depend only on so The overall constant is fixed in terms of the previously defined enough, this is precisely the case we need, with where to the image points. have where to find JSTAT (2004) P06002 - c 12 (4.2) (4.5) (4.8) (4.7) (4.6) , while ξ | of a massive z | µν T . These are related ¯ zz T ) =4 -sheeted geometry, however, ¯ z z n T ] for the proof of his famous 14 Entanglement entropy and quantum field theory -function C ); (4.4) ¯ , they approach the CFT values (see previous z , . ξ z ) ( )) 2 / 2 ) R . ¯ R 0. ( z )wehave 0 with the correlation length (inverse mass) fixed. ξ ( n z . , and the trace Θ = 4 ( n ;. (4.3) ¯ G z stacks.iop.org/JSTAT/2004/P06002 G → n 4.1 ¯ z 2 2 1 4 is constant and non-vanishing: it measures the explicit → 1 4 G ¯ -dimensional models z n T G 1 4 both approach zero exponentially fast for /z / a G ) ) Θ = ≡ n )+ ¯ ¯ z z . )= +1 2 + ), 1 z z 2 G )= 2 ( ( T R 1+ n R − n n ( , Θ ( n F F G n F zz 1 4 ( and Θ=0;Θ=0 (4.1) C T − = = ¯ z z (2004) P06002 ( n ) + ∂ ∂ ≡ − 2 ≡ ) F 1 4 1 4 ) ) n ) ¯ ¯ ¯ z z z ∂ R 2 F ( T + + 24)(1 R z, z, z, ∂ )( ( ( ( ¯ T T z is much greater than 2 n c/ ¯ z z T T Θ( z ( ( C ∂ R ∂ L both vanish, but → n T F at 0 and infinity. However, for the non-critical case, the branch point at infinity n Consider the expectation values of these components. In the single-sheeted geometry, This means that if we define an effective Our argument parallels that of Zamolodchikov [ and is the negative real axis, so that the appropriate Riemann surface has branch points of T they all acquireorigin, a they have non-trivial the spatial form dependence. By rotational invariance about the breaking of scale invariance in the non-critical system. In the J. Stat. Mech.: Theor. Exp. In this section we considerlimit an where infinite non-critical the model lattice in 1+1 spacing dimensions, in the scaling 4. Entropy in non-critical 4.1. Massive field theory—general case This corresponds to a massiveA relativistic QFT. We firstorder consider the case when the subset is unimportant: wewhose should length arrive at the same expression by considering a finite system theorem. Let us consider the expectation value of the stress tensor From the conservation conditions ( Euclidean QFT on suchzero a components: Riemann surface. In complex coordinates, there are three non- section) in the opposite limit, on distance scales then Now we expect that by the conservation equations JSTAT (2004) P06002 , a 13 (4.9) should is the (4.12) (4.10) (4.11) (4.13) c k = .Theabove A n .Inanyquantum and its complement , we expect to see a n ξ . This suggests that, A Φ k = 1, we find the main ), where , we would have found 1 n n Φ ) ξ/a Θ is a union of disjoint intervals, ≤ A , 6) log( n ) 2 c/ should obey cluster decomposition: − Θ ( n Entanglement entropy and quantum field theory , , discussed earlier, to make the result ) ) ) i 2 − : w − ∼A Z 6 to a scale transformation, i.e. to a change ( ξ/a n , n ] A Z ± 6)(1 − S Z Φ = 1), and we have inserted a power of c/ -sheeted surface. Now this integral (multiplied log ( 0, that is = 1 of correlators of operators Φ 1 n =1 log 6) log( k i c , it should approach 6)(1 − n n ξ πn ≤ c/ and its complement. This would be the 1D version , c/ − ) ( − n A n /n − stacks.iop.org/JSTAT/2004/P06002 G = 1 )=( Z all ) dependence for the exponent of the Tsallis entropy is − R n | 2 )= j /n -theorem. However, we can still derive an integrated form ma 2 c 1 w 12)( )d R ) [log 1 c/ − ( − d( ) i n Θ ) 6) log( w 2 | . However, for lattice integrable models, we shall show how it is ma ∂/∂m (2004) P06002 ( 2 c/ R ( − a ( ( ( n R n n c m ) ]. When the interval lengths are of the order of G . Θ corresponding to the conventional normalization of the stress tensor) = π ∼− ξ IR 10 ∞ ( n c n (2 π A 0 – 2 Z , since this is the only dimensionful parameter of the renormalized theory. Z S − / UV m c is a constant (with however is the correlation length. We re-emphasize that this result was obtained only for n -theorem, using the boundary conditions c ξ c So far we have considered the simplest geometry in which the set If we were able to argue that This shows that the ( are semi-infinite intervals. The more general case, when We have assumed that theory is trivial in the infrared. If the RG flow is towards a non-trivial theory, where the scaling limit possible to obtain the full dependence without this restriction. J. Stat. Mech.: Theor. Exp. 6 corresponding to thedimensionless. renormalization constant a general property ofresult of the this continuum section: theory. Differentiating at where bereplacedby of the number of boundary points between of the area lawcomplicated [ but universal scaling form for the crossover. field theory a more general correlator is more difficultexpressed in in the terms massive of case. the derivative However, at it is still true that the entropy can be B where the integral is over the whole of the by a factor 1 that is, for separations in this limit, the entropy should behave as or equivalently giving finally calculation can be thought of in terms of the one-point function alternative formulation of the in the mass Thus the left-hand side is equal to measures the response of the free energy JSTAT (2004) P06002 ) is < 14 x ( n k k. 7 Z J k (4.14) (4.16) (4.17) (4.15) (4.19) (4.18) ), and d θ (2004) P06002 ( G ) are the modified Bessel functions of the first and second kind ) Z m 2 G in ( θ 1 2 x kθ/n d ( ≡ + πn k k log ) 2 2 r r r, θ, r K 2 ( ( 2 1 < = n n ∂ ]. −∇ ∂m ) is the two-point correlation function in the ( S G G = )=cos( 18 r n θ, θ , )and Z r (0) = 1, and all its derivatives at the origin vanish: however, it goes to zero x θ, θ ( ≤ ( ( n F k k log I G C 2 The sum over To obtain the entanglement entropy we should know the ratio Letusfirstregularizeeachsumover ∂ and 0 ∂m This holds only for non-interacting theories: in the presence of interactions the sum of all the zero-point diagrams respectively [ − divergence which would occurorder of even the in sum the and plane. integration we However, find if we formally exchange the calculate such partition function. In the following, we find easier to use the identity is, as before, considered onfix a on the real negative axis. 7 4.2. Free bosonic field theory In this subsection we verify equation ( field theory (Gaussian model). The action has to be taken into account. J. Stat. Mech.: Theor. Exp. need the combination Its solution (see the appendix) may be expressed in polar coordinates as (here 0 are the Bessel functions of the first kind. At coincident points (i.e. where where the partition function in the after integrating over ∞ Interpreting the last sum as 2 derive more systematically. where so that JSTAT (2004) P06002 15 ] )the 18 (4.24) (4.25) (4.20) (4.23) (4.21) (4.26) (4.22) ,where n 4.21 , · 2 . a 1 2 − , ξ =Λ 2 m n , = ≥ log m , (0) 1 2 12 1) +1 can be scaled out by − j − j 1 nm (2 n , we should think of this = f 24 θ =2 ) j )! i , =1and =1 2 − ) j n c r B ( ) (2 Λ) /n +1 1 ]), we obtain j =2 ∞ 2 j − , k/n n 18 ( D ), with − Entanglement entropy and quantum field theory F )[ the factor of 24)( 1 n +1 ) k / j =0 for ( 2 (0) j k . In the last term in equation ( (1 4.13 0 − 2 2 k n ) mr . Thus we should choose Λ f 2 ( =0 ). K n B )! i a i k 2 j 1 δθ | . 2 (made following the recipe for the integration − 12 ∂ i k/n ∂k m 2 (2 2 Λ) m = − ( K − (2 1 1 m n m ) /∂k / k ≥ ]. Using standard identities of the Bessel function 2 j =0 ∂ k/n a )) 24 − ∂n k ( is conjugate to the angle 18 mr | r )= )d x ( ) =1 n ( − d k x F stacks.iop.org/JSTAT/2004/P06002 log x k ( ( ) r r ( k f k/n = k K k/n 2 = ∞ ) I K )d mr ∞ 0 ) k ( kI m x , we use the Euler–MacLaurin (EML) sum formula [ 1 n 0 =1 n ( ∂ x d k n ) is vanishing. k ( mr Z Z 24 k/n I ( k n ∞ ( 2 0 i ρ )+ K 0 = ) xI )= ∂ 4.21 d n k rK Tr r mr ( =log (2004) P06002 ( Z n x ( mr d ∞ f 2 ( 0 =0and 0 )= ∂ n k Z r ∂n ∞ x ρ d K 0 log ( k/n ∞ =0 − i n k 0 2 1 =0 | ∞ i 6 D i k ) kI = ∂ x ∂ d . Thus this potentially divergent term cancels in the required combination 1 ∂m ∂k 2 − ρ log Tr + ( k ∞ nk = 0 K . are the Bernoulli numbers [ log n k 1 . Having taken this combination, we may now remove the regulator to find the ρ → n 1 ∂ − Z 2 ) k B Tr nG δθ log )=2 ( − To perform the sum over Now the point is that in the integral over Thus, still with the regulator in place, r 2 − ( ∼ = n n ∂ ∂m The integration of the last expression wrt S and finally the entanglement entropy (namely J. Stat. Mech.: Theor. Exp. sufficiently fast at infinity. Since cut-off as being equivalent to a discretization where wherewehaveused G limits given in the previous section) gives main result wherewedefine Λ that agrees with the general formula we derive ( G letting order of the integral, derivative, and sum can be exchanged and each term in the sum is i.e. the sum in equation ( JSTAT (2004) P06002 ] ]) = 16 21 the 20 A ρ )isa (4.27) (4.29) (4.28) XXZ B 8 Z, direction, 4.29 )isatwo- x models it is ,with | 4 with respect +log Ψ 4.29 π/ Z XXZ Ψ | ∂9 log B ∂ 9 − ) =Tr = A ] identified. direction to be one; thus the ρ A ρ 21 B x Entanglement entropy and quantum field theory , and we normalize the Hamiltonian +logTrˆ n is also eigenstate of the transfer matrix A ρ A H ρ , = 1; thus the usual density matrix is log ˆ 1,sinceforthesemodelswecancomputethe Tr ˆ . The classical equivalent of ( +1 the scale giving the distance between the energy A . A z n ˆ λ ρ ρ 9 ] = 0. Therefore the reduced density matrix of a σ → z n CTM Tr ˆ n σ H stacks.iop.org/JSTAT/2004/P06002 1 − − H, T to 0, with the spins in = 0 the ground state of the Hamiltonian ( − =1 ) for the transverse Ising chain and the uniaxial ,with n L = , ˆ λ O ∞ λ A 9 when 4.13 ρ CTM − =Tre n ˆ = H ρ x n A − log σ ρ 1 A of ] the density matrix of the quantum chain is the partition function CTM ρ − =1 (2004) P06002 ( =e n L n H 4 19 Tr [ =Trˆ ˆ − A − Z exactly. = = ) is the partition function of two half-infinite strips, one extending from = is difficult. The difficulties arising in a direct calculation can be avoided I A A et al A . If the lattice is chosen in a clever way (i.e. rotated by ρ ρ ˆ S H ρ ˆ of the chain (defined in the introduction as ]) by imposing the transverse field in the is an operator with integer eigenvalues (for the Ising and A is an effective Hamiltonian, which, for the models under consideration, may ] ˆ 21 A O are the Pauli matrices at the site 20 . CTM n A ˆ σ H ρ to 0 and the other from + Although these systems are integrable and their ground state is known, a direct The Ising model in a transverse field can be described by the one-dimensional The method just outlined is very general. However, for the integrable chains under This partition function is the product of four Baxter corner transfer matrices Tr ˆ / of a classical system satisfying [ In the rest of this section, all logarithms are assumed taken to base e. A ˆ it is possible to write where subsystem calculation of mapping the quantum chains ontoout two-dimensional classical by spin Nishino systems.of As first a pointed two-dimensional stripquantum with chain a described cutT by perpendicular a to Hamiltonian it. In fact the ground state of a eigenvalues of 8 4.3. Integrable models and the cornerIn transfer this matrix subsection we verify ( Hamiltonian transition is driven by the parameter (following [ dimensional Ising model.quantum ‘paramagnet’ with For all the spins aligned with the magnetic field in the J. Stat. Mech.: Theor. Exp. consideration (and indeed for any model satisfying suitable Yang–Baxter equations [ be diagonalized by means of fermionization (see for more details about this equivalence [ and references therein). Note that Tr ˆ where wherewedefined Heisenberg model. These resultsof are also the an derivative independent check wrt of the uniform convergence to the cut), one ends up in the infinite length limit with [ complement of −∞ ρ levels, and (CTMs) [ expressed in terms of free fermions). Using this property the entropy is given by JSTAT (2004) P06002 2, 17 ]. 0in π/ 1 (with (4.32) (4.35) (4.36) (4.34) (4.30) (4.33) (4.31) − i → n 9 λ, λ 1, (0) = K → , =min[ ) λ 2 )=log2)andthe ) has two possible k k ∞ ∞ ) ( − k ( S = 1 ) = log 2, in agreement K λ and the resulting entropy √ ], and ∞ ( ) ) can be approximated by ( x 18 K S 9 1 π 2 4.34 . ) π 12 , = 1 1), 9 , characterized by a divergence at C = +1) λ j dependence (we use + Entanglement entropy and quantum field theory (2 . λ> ) )and( ξ ) − with λ , x j 2 2 (0) = 0 and − log 4.33 − = 0) has only one configuration available S +1) 1 j 12 λ (2 the magnetic field is negligible and the ground 1, 1, ]) log(1 + e − )= ∞ 18 k =0 ∞ λ< λ> log(1 + e j )[ = 1. The (second-order) transition between these two − 0 )is +log(1+e 1+e ) stacks.iop.org/JSTAT/2004/P06002 λ + =0 ± ∞ for for j x x 2 . = =0 ∞ 4.28 1 + − j x9 +1) log(1 . − j 2 | z n j =1.Notethat j = 9 +1 2 (2 1 1 σ ˆ n 12 1+e j j λ j ] 9 − 2 2 CTM λ +1) 21 x ˆ =0 1+e 1+e H ∞ − j d j ) + O((1 (2004) P06002 ( − j9 = 1. The exponent characterizing the divergence of the correlation 9 x 1 | (2 2 ∞ =0 =0 ∞ ∞ = 2 j j λ 0 ξ − 12 π 9 9 = =Tre = = CTM j ˆ S H 9 S S Z S 1, 2 log(1 / = 1, i.e. ) is the complete elliptic integral of the first kind [ 1 k ν = 0. In the opposite limit shows a plot of the entropy as a function of ( λ< − we mean in the critical region). The 1 z K n σ )= For The CTMof the Ising model may be diagonalized in terms of fermionic operators. Let us analyse in detail the behaviour at the critical point. For x ( state is ferromagnetic with and the entropy from equation ( (by K J. Stat. Mech.: Theor. Exp. and with the expectation that the pure ferromagnetic ground state ( Figure the quantum critical point The energy levels are Analogously, in the quantum ferromagnetic phase ( regimes happens at length is The CTMHamiltonian, written in terms of the fermion occupation number ˆ where accessible configurations with opposite sign of magnetization (i.e. eigenvalues 0 and 1), is [ both the phases; thus the sums in equations ( with all the spins aligned in the direction of the magnetic field pure quantum paramagnetic ground state ( is zero. the integral JSTAT (2004) P06002 18 .The (4.39) (4.38) (4.37) (4.40) XXZ λ ), with 1.6 4.13 )) ]: ]) ]). This model ). Close to the 4.35 21 20 20 4.34 ferromagnetic ground ) and approaching such ) XX XXX , ) +1 z n Entanglement entropy and quantum field theory σ 1 and (see equation ( z n λ σ − 0.8 1 1.2 1.4 1 and a planar ∆ is not universal. , i.e. log 2. +∆ 1 √ > 2 C has been used. This agrees with ( +1 √ →∞ y n 0.6 1 . σ λ − 1 y n | σ 9 k − 1 stacks.iop.org/JSTAT/2004/P06002 + for this model has been obtained in [ 0 describes the antiferromagnetic regime, in which we , − ∆ 1 1 < +1 √ , x n j − 2 1. The entropy is given by equation ( CTM σ ∝| 2 x ˆ n √ 2 ∆ H π j9n > σ ξ π 2 ( √ 12 2 n =0 Entanglement entropy for the 1D Ising chain as a function of ∞ √ j (2004) P06002 ( 2 9 1 = = 2 12 π 0 0.2 0.4 1(thecase∆ ξ a 1 0 CTM XXZ 0.8 0.6 0.4 0.2 < ˆ dashed line is the limit for Figure 1. S H S log H ∆ < 1 (whose classical equivalent is the Baxter six-vertex model [ = arccosh ∆, for ∆ > 2 for the Ising model. The constant 9 / The CTMHamiltonian Another model whose density matrix has been derived by using the CTMis the =1 isotropic point (∆ = 1) it holds that with because at ∆ = 1 a massless excitation (Goldstone mode) is present in the spectrum. J. Stat. Mech.: Theor. Exp. where in the last equality model c for ∆ has an Ising-like ferromagnetic state for ∆ a point from large ∆ values the correlation length diverges as (see e.g. [ are not interested). At ∆ = 1 the Hamiltonian is isotropic ( state at 0 JSTAT (2004) P06002 ). 19 (5.2) (5.3) (5.4) (5.1) A.2 )and (4.41) S model can in discs of finite n = log 2, since the S XXZ ) in a finite geometry. , , and finally integrate k 4.18 to 0. The Gaussian two- ) , in the infinite length limit. →∞ L ) 2 ξ − m r/L + 2 6) log k/n,i /L c/ Entanglement entropy and quantum field theory α ( =( 2 k/n,i 2 k/n , then sum over α J i , S ) )) , the propagator at coincident point is expansion: ) is the analogue of ( θ r x k/n,i ( 2 , L/ξ α ( G j ( ∞ j /L x s 2 ). +1 FSS 0 x s stacks.iop.org/JSTAT/2004/P06002 ( ≥ 2 k/n j ν = 1. Again, in the limit ∆ + J , J j c expansion: + L 2 -sheeted surface, but it now consists of =1 ∞ x x x n i j ,soastorecover s k x (log which forms e.g. the left half of a finite non-critical system of d 1 log , c 6 ≥ j ξ a − ), with A =0 ∞ th zero of k (2004) P06002 ( i and integrating over )= ) to write ∆ in terms of the correlation length, we have log )= )= 4.13 r x x )= 6 1 ( ( coefficients, and a large r L, ξ ,forlarge ( = ( 4.38 x n A r FSS FSS S s s S G , is first to perform the sum over log θ n should have the form denotes the also universal (in special situations logarithmic corrections could also be = (0) = 0 (this because we are referring to the scaling part; the constant term Z , sewn together along the negative real axis from L universal ν,i θ ∼− L ∞ j FSS α j ) ) should admit a small s s s x x As we showed in the previous section, the right order in which to proceed to get a The method used here for the transverse Ising model and the ( ( FSS FSS Setting point function in a finite geometry has been calculated in the appendix: equation ( sensitive radius ferromagnetic ground state is Ising-like. s s found before is not universal and it defines an overall additive normalization of J. Stat. Mech.: Theor. Exp. Once again we consider the 5.1. FSS in the Gaussian model Using equation ( So far, weor have studied away the froma entropy criticality either generalization in at ofentropy an the finite of critical infinite size a point, subsystem system. scaling for theory. a These This finite two subsystem, would regimes assert, may for be example, that linked the by 5. Finite size effects principle be appliedequations. to all those integrable models whose weights satisfy Yang–Baxter length 2 in agreement with ( with generated). In the next section we test this hypothesis for the free massive field theory. with with where JSTAT (2004) P06002 a 4) 20 / )is ,for 1 (5.5) (5.9) (5.7) (5.6) (5.8) x x ( − 2 , (1) FSS log )canbe ) s x ν/ , − ( ) 2 + FSS =1: i L 2of ( s 2 n )inthevariable c π (0) = m 2 ≥ mL, L/a ( ∼ L + 4.20 1 FSS x 2 s ,i F ν,i 2 0 m α ), with α ≡ ( ,i ) ) ) 5.1 0 ,i ,i 0 0 α ) α ) α ( ( ,i ,i . 0 1 0 0 J Y α α , and all the dependence upon ∞ ( ( (0) = 0 and ) 0 1 . 2 (the zeros of the Bessel function) is = 2 J Y 2 mL FSS Entanglement entropy and quantum field theory i /a − s L 2 m a i , L ) + + π 2 + 2 ] for the derivative of the Bessel functions − ,i 2 0 /L /L 18 α ,i =0and 0 )( mL, L/a α )) = 2 k/n,i 2 2 k/n,i (calculated as the sum of the first 1000 zeros of ( m α 1 L α 2 2 F , m ,L/a stacks.iop.org/JSTAT/2004/P06002 log ) ) ,i ,i =2 (0 + 0 0 i 1 ,i α α F 2 0 =1 ( ( k n 0 1 α d − ( J Y n ), truncating the EML formula equation ( ) ρ 2 π k i 5.5 Tr = 1 2 ) (2004) P06002 ( 2 ∂ =0 ∂n = m k isshowninfigure if the cut-off in the angular modes is properly chosen as in the infinite mL, L/a n − ) characterizes completely the crossover between the mass dominated ( − Z n k,i n 1 behaviour of the zeros of the Bessel functions is 2 = F ∂k i (1) FSS − 5.9 12 /Z ∂α s S a log n ( 2 Z ]), this sum diverges. 18 πL ) = 12( (it can be easily shown, since in this limit the sum can be replaced by an integral). = . This is really hard, requiring a sum of Bessel functions for generic argument, over x mL n r A first, qualitatively correct expression for the universal function However, the formal result is This gives the entropy as From this formula we may compare with the FSS ansatz ( Equation ( Aplotof ( Z at the first order in the derivative. This approximation gives a correct form of the ). From this figure we see that an optimum approximation for 0 (1) FSS J log volume case. The EML approximation at the first order is (we do not write the integral) J. Stat. Mech.: Theor. Exp. over the zeros of differentand Bessel integrations functions. andknow What try that one to this can understand operation do has what is to happens. invert be the done order From with of the care. the previous sums exercise we now finite. which reproduces the correct limits for (see e.g. [ Since the large that, as expected, isdisappears a after function the only subtraction. of This the agrees product with with respect to the order. The remaining sum over whichcanbederivedusingtheformulae[ result because it takesexpression into is account exact). the completein The infinite the integral volume ratio term result in (whose the truncated EML is divergent, but it cancels as before obtained from equation ( k whereweused The superscript (1)EML. is there to remind us that we made a first-order approximation in large (non-critical) and the geometry dominated (critical) regimes. s JSTAT (2004) P06002 21 (5.10) (5.11) (5.13) (5.12) 10 . ]: 5 2 − 22 10 × 2. 2. 1.5 42 . ≤ 8 3 x x< expansion [ 1 x is a fit). We can also (1) 6 ∞ 0 ) let us summarize what mL ) s x ( 0.5 (which is exact and does not FSS ,s s 6 4 x − 0 10 log 0 approximants for Entanglement entropy and quantum field theory ), compared with quadratic, quartic, -0.2 -0.4 × ∼ x X x ( , 98 . (1) FSS , 1 23 s , 4 3456 1 ) approximations. Even the use of quadratic and − − 5 − ∞ 0 128 = = s behaviour 120 912 (this value of . (1) 5 = ,i ,i + 7 3 0 0 x 1 1 ,i x α α 5 0 stacks.iop.org/JSTAT/2004/P06002 1 =0 ) ) ) ) α ,i ,i ,i ,i ) ) 0 0 0 0 (log ∞ 0 ,s ,i ,i α α α α s 3 0 0 x ( ( ( ( − ): the exact expression obtained as numerical sums over the first 0 0 α 1 1 α ( ( x J J Y Y ( 0 1 10 Quartic approximation Sextic approximation Large x approximation Exact Quadratic approximation J Y × =1 =1 ∞ ∞ (1) FSS i i ,with s =1 ∞ (2004) P06002 ( i π π 15 ∞ 0 approximation may reproduce the right formula over the whole range. . s 1 − π − x + 024 = = = 0 -1 x -2 -1.5 -0.5 (1) 4 (1) 1 (1) 2 (1) 3 without any approximation. s Figure 2. S 1000 zeros of thesextic) Bessel and functions large compared with small (quadratic, quartic, and large Inset: the comparison of several small s s s log j s − (see also the inset) shows a plot of 2 )= x Before starting the full calculation of the function ( asy FSS J. Stat. Mech.: Theor. Exp. s and sextic approximations. The agreement is excellent in the region calculate analytically all the universal coefficients of the small Figure we can learnapproximation from of the the full first-order functionargument approximation may expansion in be obtained the with by EML matching the only expansion: the large first-order a small rather good depend upon thecoefficients approximation). Thus in the following we show how to calculate the JSTAT (2004) P06002 x 22 (5.19) (5.15) (5.14) (5.18) (5.17) (5.16) (5.20) , 2 k,i 1 =2)must α , 2 − i , ··· k +1) . In fact, applying . ∂ k n ∂k ( 2+ Z k k 1 6 1 d 2 d x 2 k/n,i − + k and α 2 2 ) 6 π 3 Z 2 k − 1+ d +1) = 3 4 k ( − , log ∞ k,i = 0 Entanglement entropy and quantum field theory − ∂k 2 ∂α j 2 1 2 k/n,i .... k k,i 2 = 0 part, obtaining the universal function α 2 k,i 1 2 x α +1) 3 4 ), but this time we will not use EML sum α . In particular we can write an FSS ansatz m − k − Z ( 5.5 j k 1+ 2 k,i n = 1+ d 217 401 α log . − . This formula has to be intended in a formal 2 0 n 4 (in agreement with what was previously found) k =1 / − n i 2 k,i 1 log | − 1 1 4 3 stacks.iop.org/JSTAT/2004/P06002 +1) ) α = n n − k k − , Z ( d i = /∂n k 3 = i ) d k k with coefficients x − ), one has +1) (1) 1 1 ( d k k s j 2 k 3 2 d 6 π +1) k ∂f x k 4( 5.18 +1 ( 3 4 j 2 k k j f j − (2004) P06002 ( − = 4( 1) 3 4 =3 = − 2 k,i ( ∂ 1 − )= ∂k α =1 x )=3 k ( )= n = = x d x ] ( j wrong 1 1 1 =1 ∞ ( i f f s s f FSS k f 22 s ∂ ∂n 3 ) − − mL = = The interesting object is log Our starting point is again equation ( = x ( wrong 1 wrong 1 s that has a first-order term formula. For this reason we have to be careful since the sum is divergent. J. Stat. Mech.: Theor. Exp. f also for this ‘free energy’ and so subtract the way: in fact itrecipe is the for difference the of cut-offcomputational two explained diverging point sums. in of the To make viewabout previous this it section the difference has is cut-off sensible to the simpler andzeros be to to of used. adjust make the Anyway, the the Bessel fromexpansion result calculation functions the of only without of at being generic the order careful end. cannot be The done numerical as sum before, over the but the small can be calculated analytically. For example but the integralexpressions is with not the vanishing. wrong cut-off. This The is value a of finite the integral difference (2 between the two divergent form which We use the superscriptin ‘wrong’ this because way, since it we is are implicitly not using strictly the correct same to cut-off make for the calculation s leads to which using [ be properly subtracted to give the right result: the EML to the sum in ( JSTAT (2004) P06002 . 23 A )a / x (6.3) (6.1) (6.2) A ( S ± f does not = s a -dimensional d ,usingthemore j s + 1 field theory. We d is the scaling dimension h y ) . g is proportional to the surface A S the external magnetic field, Entanglement entropy and quantum field theory , ) h z − dependence on the microscopic cut-off. (the parameter driving the transition) is 4, signalling that such approximation not d / g − 1 ,Tξ a h − y is not the total entropy: we also expect to find − 2 the entropy and , hξ sing ξ ) ( . s h ) is not the total free energy density: there is another d> ± y = 1. To calculate the entropy of a z s stacks.iop.org/JSTAT/2004/P06002 − = 1), and all other neglected operators are supposed refers to the two phases, and 1+ 1) z 1 dimensions are translationally invariant in a domain sing z − − hξ f d ± d ( ( ( − ], for ± − − d f ξ 10 ]. d Tξ − . Thus we should discuss the entropy per unit area 22 ξ is the correlation length, )= ∼ A =0and ν (2004) P06002 ( should be a universal function apart from the normalizations of )= − T t ∂S ∂T ± t, h g,h,T T ∝ s = ( ( ξ = h , sing sing c C f s . As for the free energy, ν /T 0 describes the relevant effect of the field conjugated to the order parameter, | − -dependent pieces which are, however, analytic in c | a c > T g h of the subsystem − y − A g T 0 the relevance of the temperature close to the quantum phase transition (in all the From the scaling of the entropy all the scaling laws can be obtained; e.g. the specific As argued by Srednicki [ This scaling can be explicitly checked in the case of the Gaussian model in all In fact, from the scaling of the singular part of the free energy density (here | | ), h = = system close to a quantum critical point, one has to consider a consider the geometry where J. Stat. Mech.: Theor. Exp. This is close to the first-order EML result ξ its arguments. The relation between non-universal piece which has an explicit examples considered up to now, z> area heat goes like explicitly However, this term is analytic in the thermodynamic variables. dimensions for Similar identities can be derived for other observables. universal function, the subscript t enter explicitly into thermodynamic relationsthermodynamic near the variables critical are point,trivial as suitably long scaling as normalized dimensions, the related and various to allowed the to various universal scale critical with exponents. their non- The scalingtransitions. hypothesis Crudely plays speaking, a it fundamental asserts that role the in microscopic length understanding scale classical phase complicated expressions for the sumreported of in higher the negative literature powers [ of zeros of Bessel functions 6. Higher dimensions: scaling of entropy and area law only reproduces theof qualitative 10%). physics but In is the also same quantitatively manner reliable one (at can the calculate all level the coefficients the critical behaviour oflaws all may the be thermodynamic derived. observables Note and that in particular the scaling of In analogy with theof classical the case, entropy we per may unit conjecture a area scaling near form a for quantum the phase singular transition: part to be irrelevant. where JSTAT (2004) P06002 24 → ,in 2 (6.6) (6.8) (6.7) (6.4) (6.5) (6.9) x must m 1 − , a ) in agreement θ, θ for large ( ) are in general 1 k d − C d /x ) 1 . m 2 ⊥ . λr k ∝ ( . 5 ) 1 n ) + 1-dimensional subspace. T k/n x 2 z ( − J − 2 directions and coincides ) ) d m n FSS ). − λr h, L + s )is ( h d ) 2 y , 2 ma λ ) k/n m ,L 4.25 J is a non-universal constant fixing | c log( λ + s g L/ξ d Entanglement entropy and quantum field theory ( 24 A 2 ⊥ ∝ and a cut-off proportional to − λ k FSS g , A ∞ | s . 0 (0) = 1 and )) /ν 1) log( 2 = 2 1 − 1 − k . However, the coefficient of this divergence ⊥ d ⊥ L m a − ( FSS d ( r k 1) d L/ξ s − 1 1 ) ( . + − + to stress that it is different from that appearing − − =0 L d ∞ π ⊥ FSS ( d d s k 2 ⊥ 2 ⊥ k s FSS − ⊥ d d k (2 k A 1 this singular term combines with the non-singular s r , and on the remaining two-dimensional plane there a 1) in front of all the equations. The space integration is in an arbitrary direction. The choice of a different 1 πn − A stacks.iop.org/JSTAT/2004/P06002 = 0, since the two-point function at coincident points + d 2 → ⊥ ( log ⊥ k ⊥ ⊥ ∞ − L 1 d r r r ⊥ − L 1 ⊥ k d k =0)= − 1 i ) d (we use e (log − π d T d 1 )= θ ⊥ c 6 against a constant term coming from the regular part, with d 1. This gives the anticipated form − (2 k d d = 1 ) r − → π d = d sing )= (2004) P06002 ( d s r d (2 n A 12 ⊥ ρ ) on the cut plane, i.e. r L, ξ L, g, h L, g, h, T − 1 behaviour may be seen as resulting from a cancellation between the ( ( ( )as − 1 d = L to give the previously found result )thatinsteadis 4.17 sing − sing sing )= ) a function which satisfies log Tr S S s s . There is, however, a finite piece which behaves as 1) 1) 6.4 − m ,θ d − = 0 this reads ( L/ξ ( − is the vector between the two points considered in the d ). Note however that as T behaviour of a , and the integration over ⊥ 2 FSS ⊥ r r, θ, r = 6.2 ∝ 1) s k ; − The two-point function is translationally invariant in The calculation proceeds as in the one-dimensional case with the substitution In this way the entropy is Just as for the classical free energy, one may also conjecture a finite size scaling form h ⊥ d =O(( + ( r 2 ( s − is a branchgeometry cut is going not from expected 0 to to change the main results. J. Stat. Mech.: Theor. Exp. delimited by a hypersurface of area with equation ( G has to be considered). Thus the analogue of equation ( in equation ( with obviously over d for the entropy of the form order to reproduce the infinite volume limit, and the normalization ofsense. the entropy. The In log one dimension, this relation again does not make L be understood for the integration over analytic in (and other sub-leading divergences which can occur for sufficiently large A piece with ( In general, the integral diverges like that has been explicitly checked for the Gaussian model in section For m where To have a finitesystem result, must both live in the a integrations finite are box in with a finite region. This means that the JSTAT (2004) P06002 . k 25 πn N k/n (A.1) ). )=0, ,with 2 = r ν,i ( m ν f α ) + 2 = 2 r m , i.e. L i + −∇ λ /n ]) 2 r 18 −∇ ). A complete set of θ , sin ) , νν ) δ ) r i θ, r ν,i λ ( α ν ( cos J r ) +1 2 ν νθ Entanglement entropy and quantum field theory J , the eigenvalues are 2 L 2 L )=( = is an integer multiple of 1 =0. . x, y r -sheeted Riemann geometry: infinite =sin(i ) ν )= r r N b ν,i . =( , r/L − ) ,i nm r in the sense of distributions. Imposing the 2 r r . δ ( ν ( ) d k α δ = ( φ stacks.iop.org/JSTAT/2004/P06002 ,φ ) r ν ) r k/n,i d 2 J r ( →∞ i )= α ) k ( λ )d r φ /L L ( , k 2 r +1 ν r r/L ( J N ( n ) 2 ) of the Helmholtz differential equation ( k/n ν,i φ G k J ) α r ) νθ ( 2 r , ν ( r k (2004) P06002 ( -sheeted Riemann surface that we are interested in, the eigenvalue m ( πn d m n 2 rJ G φ )= + d r 2 r = r =cos(i , r L ), eigenfunctions of the Helmholtz differential operator ( i,k m ( a ν,i 0 r −∇ ( G N φ N ( k φ ), that are also eigenfunctions of the same differential operator, do not enter in ) Bessel functions of the first kind. Note that the Bessel functions of the second th zero of the Bessel function. x x i ( ( ν ν ) admits an eigenfunction expansion J Y r Let us consider first the solution in a finite geometry. The infinite volume limit is Using the orthogonality relation of the Bessel functions (see e.g. [ In the case of an the , r ( ν,i periodicity boundary condition we have that Constraining the eigenfunctions to vanish at recovered by taking the limit J. Stat. Mech.: Theor. Exp. is a normalization constantof that the should eigenfunctions: be derived from the orthonormality requirement The Green function Appendix: the two-pointand function finite in volume an results JC thanks D Huse for firstin bringing part this problem by to his thewhile attention. EPSRC JC This under work was was Grantof supported a GR/R83712/01. Mathematics member and Thesupported the of initial by School the phase the of Institute was BellFunds Natural Fund, carried for for Sciences the out Natural James Advanced for Sciences. D Study. their Wolfensohn hospitality. Fund, and He This a thanks stay grant was in the aid School from the Acknowledgments the expansion since we require regularity at with kind G problem is solved in polar coordinates eigenfunctions is α we get from the orthonormality requirement in terms of with a specific boundary condition, is the solution of JSTAT (2004) P06002 , , , , ). 66 26 A JETP , 2004 , 1988 , 1994 (A.2) 096402 4.17 in the . ) 92 Phys. Rev. Phys. Rev. Quant. Inf. νν , 1986 δ . θ, θ ( π k Phys. Rev. , 2003 , 1986 C = hep-th/0405111 , 2004 ) 479 x ]. 2 /L 52 , 2002 23 r ,andthe sin λ Preprint x Phys.Rev.Lett. 2 k/n,i d ] → α m π ( ) 2 0 , 2004 /L + k/n , 2004 2 ν,i J. Stat. Phys. J = ) α /L x 2 r/L Concentrating partial entanglement by local ] , 1988 2 k/n,i hep-th/9303048 [ cos Entanglement entropy and quantum field theory α ] x k/n,i Infinite conformal symmetry in two-dimensional d 666 α ). The limit of the normalization factor ( π 565 71 λ 2 ] ] 0 742 k/n Diverging entanglement length in gapped quantum spin 43 − A subsystem-independent generalization of entanglement J 56 ) λ 333 ( Fine-grained entanglement loss along renormalization group and δ quant-ph/0311087 Scaling of entanglement close to a quantum phase transitions [ 241 Entanglement in quantum critical phenomena k/n,i π 2 ] Conformal invariance, the central charge, and universal finite-size α quant-ph/9511030 B becomes continuous, [ ( ] /L stacks.iop.org/JSTAT/2004/P06002 i =2 2 Geometric and renormalized entropy in conformal field theory +1 Entanglement and the phase transition in single mode super-radiance ] Phys. Rev. Lett. 087201 x 2046 quant-ph/0305023 quant-ph/0309027 Ground state entanglement in quantum spin chains [ [ 2 k/n 92 J 1d 53 Phys. Rev. Lett. Entanglement in a simple quantum phase transition π Universal non-integer ‘ground-state degeneracy’ in critical quantum systems A , 1993 2 0 Nucl. Phys. =1 ∞ i 161 107902 073602 ] Quantum spin chain, Toeplitz determinants and Fisher–Hartwig conjecture k Pis. Zh. Eksp. Teor. Fiz. , 1986 , the index quant-ph/0304108 d A finite entanglement entropy and the c-theorem 67 92 92 hep-th/9403108 (2004) P06002 ( Finite-size dependence of the free energy in two-dimensional critical systems quant-ph/0404120 , 1984 =0 ), which leads to the two-point function reported in the text ( ] ∞ 377 44 [ Irreversibility of the flux of the renormalization group in a 2D field theory 79 [ k 608 Phys. Rev. quant-ph/0304098 →∞ πn 300 424 Phys. Rev. Lett. 116 1 πn L (2 Universality of entropy scaling in 1D gap-less models 416 B B 2 Preprint 048 [ Entropy and area comes from , 1996 λ/ 4 k 731 (translation) 227902 746 k Possible generalization of Boltzmann–Gibbs statistics , 2004 Universal term in the free energy at a critical point and the conformal anomaly d quant-ph/0202162 d )= [ 43 90 56 Phys.Rev.Lett. Phys.Rev.Lett. Nature Phys.Rev.Lett. , 2004 ,θ )= cond-mat/0311056 λ 1991 Nucl. Phys. [ Lett. quantum field theory 2004 Lett. Nucl. Phys. Lett. 32110 Comput. J. Stat. Phys. amplitudes at criticality flows systems operations 2002 2004 ote H W J, Cardy J and Nightingale M P, ( In the limit k Latorre J I, Rico E and Vidal G, Zamolodchikov A B, Barnum H, Knill E, Ortiz G, Somma R and Viola L, Latorre J I, Lutken C A, Rico E and Vidal G, Verstraete F, Martin-Delgado M A and Cirac J I, Osborne T J and Nielsen M A, r, θ, r N ( [9] Affleck I and Ludwig A W W, [8] Cardy J and Peschel I, [3] Vidal G, Latorre J I, Rico E and Kitaev A, [2] Osterloh A, Amico L, Falci G and Fazio R, [1] Bennett C H, Bernstein H J, Popescu S and Schumacher B, [7] Holzhey C, Larsen F and Wilczek F, [4] Jin B-Q and Korepin V E, [5] Lambert N, Emary C and Brandes T, [6] Casini H and Huerta M, [10] Srednicki M, [11] Korepin V E, J. Stat. Mech.: Theor. Exp. 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