Entropy Flow Through Near-Critical Quantum Junctions

Transcription

1 Entropy Flow Through Near-Critical Quantum Junctions Daniel Friedan Journal of Statistical Physics ISSN Volume 67 Combined 3-4 J Stat Phys (207) 67: DOI 0.007/s

2 J Stat Phys (207) 67: DOI 0.007/s Entropy Flow Through Near-Critical Quantum Junctions Daniel Friedan,2 Received: 27 September 206 / Accepted: 2 February 207 / Published online: March 207 Springer Science+Business Media New York 207 Abstract This is the continuation of Friedan (J Stat Phys, 207. doi: 0.007/s ). Elementary formulas are derived for the flow of entropy through a circuit junction in a near-critical quantum circuit close to uilibrium, based on the structure of the energy momentum tensor at the junction. The entropic admittance of a near-critical junction in a bulk-critical circuit is expressed in terms of commutators of the chiral entropy currents. The entropic admittance at low fruency, divided by the fruency, gives the change of the junction entropy with temperature the entropic capacitance. As an example, and as a check on the formalism, the entropic admittance is calculated explicitly for junctions in bulk-critical quantum Ising circuits (free fermions, massless in the bulk), in terms of the reflection matrix of the junction. The half-bit of information capacity per end of critical Ising wire is re-derived by integrating the entropic capacitance with respect to temperature, from T = 0toT =. Keywords Quantum computers Quantum statistical mechanics Quantum transport Entropy transport Summary This paper continues the elementary investigation of entropy flow in near-critical quantum circuits close to uilibrium begun in 7]. The entropy current operator, which is just the In memory of Leo Kadanoff. This is the second part of a two-part work originally published in 2005 as 5]and6], here revised for clarity following helpful suggestions of the referee. The first part is 7] in this volume. B Daniel Friedan dfriedan@gmail.com New High Energy Theory Center, Rutgers University, Piscataway, NJ, USA 2 Natural Science Institute, University of Iceland, Reykjavík, Iceland

3 Entropy Flow Through Near-Critical Quantum Junctions 855 energy current divided by the temperature, j S (x, t) = kβtt x (x, t), is used to analyze the flow of entropy through the general junction in a near-critical quantum circuit, especially when the quantum wires are critical in the bulk so that the only departure from criticality is in the junction. Entropy flows in and out of the junction in response to changes in the entropic potentials in the wires, which are just the temperature drops in the wires. The linear response coefficient for the entropy current is the entropic admittance of the junction. The junction entropy, s(t ), is the charge in the junction. The entropic capacitance the temperature derivative, ds/dt is extracted from the low fruency limit of the entropic admittance. The information capacity of the junction, s( ) s(0) is found by integrating with respect to T. The junction entropy itself needs a global calculation, but changes in the junction entropy can be determined locally, by studying the entropy currents near the junction. A trivial example the general junction in a bulk-critical Ising circuit (i.e., free fermions, massless in the bulk) is done explicitly, to illustrate the formalism and check that it works. Definitions, notation and motivation are carried over from 7]. As explained there, the near-critical one-dimensional quantum systems under consideration are those that are described by + dimensional relativistic quantum field theories, and they are assumed to be close to uilibrium. A junction consists of N quantum wires joined at a single point. The wires are labelled by indices A, B =...N. Each wire is parametrized by a spatial coordinate x 0. The endpoints, x = 0, are all identified to form a single point, the junction. The entropy flow through the junction is analyzed in terms of the entropy currents in the N external wires attached to the junction. The junction is treated as a black box, from the point of view of the larger circuit containing it. The single point x = 0 might stand for a complicated sub-circuit, but the internal structure of the junction shows itself only by its effects on the flow of entropy in and out of the junction. Suppose that a small variation is made in the entropic potential in each wire, V S (t) B = e iωt V S (0) B, alternating at fruency ω. The change in the entropic potential is the local temperature drop, V S (t) B = T (t) B, on wire B outside the junction. The changing entropic potentials in the wires cause entropy currents to flow through the wires, in and out of the junction, given by a linear response formula: I S (t) A = Y S (ω) AB V S (t) B. () The entropic admittance matrix, Y S (ω) AB, describes the entropy flow characteristics of the junction. Entropic admittance has fundamental units k 2 / h. When the wires are bulk-critical, the entropy current is a sum of chiral currents, j S (x, t) A = j R (x, t) A j L (x, t) A,where j R (x, t) A = j R (x vt) A is the right-moving entropy current in wire A and j L (x, t) A = j L (x + vt) A is the left-moving entropy current, v being the speed of light in the relativistic quantum field theory of the one-dimensional bulk-critical system. These are just the chiral energy currents of the conformally invariant bulk-critical system, divided by the temperature, T. The Kubo formula for the entropic admittance is re-written in terms of the chiral entropy currents: Y S (ω) AB = iω dt e iω(t t 2 ) i jl (0, t ) B, j R (0, t 2 ) A j L (0, t 2 ) ] A. (2) h The time integral does not have the restriction t t 2 which expresses causality in the general Kubo formula: the fact that the response happens after the perturbation. Causality is automatically enforced in (2) by chirality, since a right-moving current cannot affect the

4 856 D. Friedan junction, and a perturbation at the junction cannot produce any left-moving current. Equation (2) exhibits Y S (ω) AB as a well-behaved analytic function of ω, free from the possibility of a short-time divergence, which can arise in the general Kubo formula in quantum field theory because of the restriction t t 2. The Kubo formula is also re-written, again for bulk-critical wires, in the form iω t2 dt e iω(t t 2 ) i js (x, t ) B, j S (x 2, t 2 ) ] A h = e iω(x +x 2 )/v Y S (ω) AB (e ) iω x x 2 /v e iω(x +x 2 )/v c k 2 ( ) ] 2 hβ δ AB + ω 2 2 h where x, x 2 > 0 are arbitrary points on the wires and c is the bulk conformal central charge. This formula gives a way to extract Y S (ω) AB from a measurement performed at a distance from the junction, by generating entropy current at an arbitrary point x > 0 in wire A,then detecting the induced entropy current at an arbitrary point x 2 > 0 in wire B. Thesetwo elementary formulas for the entropic admittance are the basic results of this paper. The net entropy current leaving the junction is N I S (t) A, so the change in the junction entropy, s(t), isgivenby t s(t) = iω s(t) = N I S (t) A,so s(t) = iω A, (3) Y S (ω) AB V S (t) B. (4) The change in the junction entropy with the junction temperature is ds dt = lim Y S (ω) AB (5) ω 0 iω A, because raising the temperature of the junction by T is uivalent to dropping the temperature on all of the wires outside the junction by the same amount, V S (t) B = T (t) B = T. Equation (5) also gives the specific heat of the junction, T ds/dt. Integrating ds/dt with respect to T gives the junction entropy, s(t ), asafunctionoftemperature,uptoan additive constant. Integrating ds/dt from T = 0 to T = gives the total information capacity of the junction, s( ) s(0). This does not give the junction entropy itself. That ruires a global calculation. But changes in s(t ) can be calculated locally, near the junction, using the entropy current. Y S (ω) AB is a response function, therefore analytic in the upper half-plane. Equation (2) exhibits Y S (ω) AB as the Fourier transform in time of the uilibrium expectation value of a commutator of local operators, so the only singularities in Y S (ω) AB at temperature T > 0 should be poles in the lower half-plane. The gap between the real axis and the nearest pole of Y S (ω) AB is the inverse of the characteristic response time of the junction in the channel AB. The zero temperature limit of these gaps can serve as analogs of the mass gap in a bulk quantum field theory. The conformal invariance of the bulk-critical wires implies a vanishing formula: Y S (i/ hβ) AB = 0. (6) In a separate paper 9], this vanishing formula is used to re-write the proof of the gradient formula for the junction beta-function in terms of the real time statistical mechanics of the

5 Entropy Flow Through Near-Critical Quantum Junctions 857 entropy current. The gradient formula, s/ λ a = g ab (λ)β b, was originally proved in the less physical language of euclidean quantum field theory 8]. As an example, and as a check on the formalism, (2) is used to derive an explicit formula for Y S (ω) AB for the general junction in a bulk-critical quantum Ising circuit. The + dimensional quantum field theory of such a circuit is a theory of free Majorana fermions, massless in the bulk 2,4,0,]. An Ising junction is characterized by its reflection matrix, R(ω) B A,which gives the amplitude for left-moving fermions of energy hω to enter the junction through wire B, then leave the junction as right-moving fermions through wire A. Equation (2)isusedto express the entropic admittance in terms of the reflection matrix: Y S (ω) AB = k2 hβ 2 ] dω dω 2 δ(ω + ω 2 ω) R(ω ) B A ω R(ω 2) B A δ B A ( ) 8 (ω ω 2 ) 2 + e β hω 2 e β hω. (7) + Substituting in (5) gives the specific heat of the junction, T ds dt = k i dη Tr R R (η) ] (β hη)2 e β hη 2 ( e β hη + ) 2. (8) Integrating with respect to T gives s(t ) up to a constant. These formulas apply to the general Ising junction. R(ω) can depend on the temperature, as when the junction has internal structure. When R(ω) does not depend on the temperature, the integral over T can be performed explicitly, giving s(t ) s(0) = k i The information capacity is then s( ) s(0) = dη Tr R R (η) ] 2 k i β hη e β hη + + ln ( + e β hη)]. (9) dη Tr R R (η) ] ln 2. (0) 2 The information capacity consists of one half-bit for each pole of R(ω) B A in the lower halfplane (or each zero in the upper half-plane), when R(ω) is independent of temperature. An elementary Ising junction, without substructure, has a reflection matrix that is independent of temperature, of the form R(ω) = ω iλt λ ω + iλ T λ where the matrix λb A parametrizes the junction couplings (generalizing the boundary magnetic field in a -wire junction). Equation (0) thengivess( ) s(0) = (N/2)k ln 2. The information capacity of an elementary Ising junction is one half-bit per end of critical Ising wire. This reproduces a well-known result, at least for N =. For the case N =, the formula for s(t ) s(0), obtained from (9)and(), agrees with the formula for the boundary free energy and entropy of a single bulk-critical Ising wire, calculated directly 4,]. The case N = 2 is uivalent to a pair of free Majorana fermion fields on a single wire with boundary. When the reflection matrix is U()-invariant, this is uivalent to a single free massless Dirac fermion field on a wire with boundary, which is uivalent to the Toulouse limit of the spin-/2 Kondo model. In this case, Eq. (9) for the boundary entropy is identical to the formula for the boundary entropy in the spin-/2 Kondo model, specialized to the Toulouse limit, as discussed in 2], for example. Equation (8) for the junction specific ()

6 858 D. Friedan heat is somewhat more general, allowing for the reflection matrix R(ω) to be temperaturedependent. Still, the field theory of a bulk-critical Ising circuit is a theory of free fermions, so there is no difficulty to calculate the junction specific heat. It is done here as an illustration and check of the method of using the entropy current to describe the flow of entropy through the junction. The direct calculation of the junction entropy is global. First, the entropy of the whole system is calculated, then the bulk entropy of the wires is subtracted. What remains is the junction entropy. This is a subtle calculation, because boundary conditions are needed at the far ends of the wires, whose contributions to the total entropy must also be subtracted. Using the entropy current to calculate changes in the junction entropy is a local method that can be carried out in an arbitrarily small neighborhood of the junction. 2 Entropy Flow Through a Junction The energy momentum tensor of a quantum circuit consists of a bulk part, located in the wires, and a contribution located in the junction (see Appendix ). The bulk energy momentum tensor in wire A is Tν μ (x, t) A. The junction contributes only to the energy density. Its contribution is δ(x)tt t(t) junct, wherett t(t) junct is the junction energy. The junction energy is conventionally written Tt t(t) junct = θ(t). Energy and momentum are locally conserved in each bulk wire: 0 = μ Tν μ (x, t) A. (2) Energy is conserved at the junction: The hamiltonian is 0 = t θ(t) + H 0 = θ(t) + 0 T x t (0, t) A. (3) dx T t t (x, t) A. (4) The departure from criticality is expressed by the trace of the energy momentum tensor, written (x, t) = T μ μ (x, t). It consists of the bulk contribution in each wire, (x, t) A = T μ μ (x, t) A, and the contribution located in the junction, δ(x)θ(t). When the wires are bulkcritical, the (x, t) A all vanish. The departure from criticality is then entirely in the junction, expressed by θ(t) = β a (λ)φ a (t) (5) where the φ a (t) are the relevant and marginal operators located in the junction, and β a (λ) is the junction beta-function. The junction is critical when a change of scale has no effect, which is uivalent to θ(t) being a multiple of the identity, which is uivalent to 0 = t θ(t) = which is also the condition that the boundary energy be stationary. The entropy current operators in the wires are T x t (0, t) A (6) j S (x, t) A = kβt x t (x, t) A. (7)

7 Entropy Flow Through Near-Critical Quantum Junctions 859 The entropy density operators in the wires are ρ S (x, t) A = kβtt t (x, t) A kβtt t (x, t) A. (8) In addition, the junction makes a contribution, δ(x)q S (t), to the entropy density operator, where q S (t) = kβθ(t) + kβθ(t). (9) q S (t) is the junction entropy operator. The entropy density operator measures the variation of the entropy density away from its uilibrium value. q S (t) measures the variation of the junction entropy away from its uilibrium value: s(t) = q S (t). (20) Conservation of energy at the junction implies conservation of entropy: 0 = t q S (t) + j S (0, t) A. (2) Any change in the junction entropy is ual to the net flow of entropy into the junction. The entropy flow characteristics of the junction are its responses to small changes in the entropic potentials on the wires. First consider a -wire junction, which is simply the boundary of a wire. Put a small alternating entropic potential, V S (t), outside the junction, constant along the wire: S (x, t) = V S (t) = e iωt V S (0) for x > x (22) where x is some fixed point very close to 0. The hamiltonian is perturbed by H = 0 dx ρ S (x, t) S (x, t) = V S (t) x dx ρ S (x, t). (23) Entropy current will flow in response to the perturbation, in and out of the boundary, at fruency ω. In the linear response approximation, the induced current is I S (t) = j S (0, t) = Y S (ω) V S (t), (24) where the linear response coefficient, Y S (ω), is the entropic admittance of the boundary. Y S (0) = 0 for a -wire junction, because the entropy flowing into the boundary encounters the same entropic potential as the entropy flowing out, when ω = 0. A uniform small change in the entropic potential, constant over the whole system, has no effect, because the hamiltonian is perturbed by a multiple of itself. Therefore, raising the entropic potential by V S (t) everywhere outside the boundary is uivalent to lowering the potential on the boundary by V S (t). The perturbation can just as well be written H(t) = V S (t)q S (t). Now measure the entropy current at a point x 2 > 0 very near the boundary. The Kubo formula for the entropy current is j S (x 2, t 2 ) = t 2 dt i H(t ), j h S (x 2, t 2 ) ] = V S (0, t 2 ) t 2 dt e iω(t t 2 ) i qs (t h ), j S (x 2, t 2 ) ] (25)

8 860 D. Friedan so the entropic admittance is Y S (ω) = lim x 2 0 t2 dt e iω(t t 2 ) i qs (t ), j S (x 2, t 2 ) ] h Conservation of entropy at the junction, 0 = t q S (t) + j S (0, t), allows the Kubo formula to be written t2 Y S (ω) = lim dt e iω(t t 2 ) i js (0, t ), j S (x 2, t 2 ) ]. (27) x 2 0 iω h The change of the boundary entropy is then In the static limit, where (26) s(t) = q S (t) = (iω) j S (t) = (iω) Y S (ω) V S (t). (28) s = C S V S (29) C S = lim ω 0 (iω) Y S (ω). (30) Change of entropic charge divided by change of entropic potential might be called the entropic capacitance of the boundary (abusing the electrical analogy). The entropic potential of the boundary has been lowered by V S, which is uivalent to raising the temperature of the boundary by T = V S, so the entropic capacitance is C S = ds dt. (3) The specific heat of the boundary is T ds/dt = TC S. For an N-wire junction, impose small alternating potentials, V S (t) B, B =...N, on the wires outside the junction: S (x, t) B = V S (t) B = e iωt V S (0) B for x > x. (32) The induced entropy current flowing out of the junction through wire A is I S (t) A = lim x 2 0 j S (x 2, t) A = N Y S (ω) AB V S (t) B. (33) The linear response coefficients, Y S (ω) AB, form the entropic admittance matrix of the junction. The entropic potential in the wires perturbs the hamiltonian by H(t) = dx ρ S (x, t) B V S (t) B (34) x which is gauge uivalent to That is, the difference is H(t) = H(t) H(t) = t N j S (x, t) B iω V S(t) B. (35) ] dx ρ S (x, t) B x iω V S(t) B. (36)

9 Entropy Flow Through Near-Critical Quantum Junctions 86 The Kubo formula for the induced entropy current at a point, x 2, in wire A is j S (x 2, t 2 ) A = t2 dt = t 2 dt i h ] H(t ), j S (x 2, t 2 ) A ] i H(t ), j h S (x 2, t 2 ) A (37) x iω V S(t 2 ) B dx h ī ] ρs (x, t 2 ) B, j S (x 2, t 2 ) A. If x 2 < x, then the ual-time commutator in the second term is zero, because the operators are separated in space. On the other hand, if x 2 > x,then dx ī ρs (x, t 2 ) B, j S (x 2, t 2 ) ] A = kβ dx ī ] T t x h x h t (x, t 2 ) B, j S (x 2, t 2 ) A = kβδ AB t j S (x 2, t 2 ) A (38) which vanishes, since nothing changes with time when the system is in uilibrium. Since the second term on the rhs of (37) always vanishes, j S (x 2, t 2 ) A = N t2 and the entropic admittance matrix is Y S (ω) AB = dt i iω V S(t ) B js (x, t ) B, j S (x 2, t 2 ) ] A h lim x,x 2 0 The change in the junction entropy is iω t2 e iω(t t 2 ) i dt (39) js (x, t ) B, j S (x 2, t 2 ) ] A. (40) h so In the static limit, where t s(t) = t qs (t) = s(t) = iω s = js (0, t) A (4) Y S (ω) AB V S (t) B. (42) C SB = lim ω 0 iω C SB V B S (43) Y S (ω) AB. (44)

10 862 D. Friedan C SB is the entropic capacitance of the junction (still abusing the electrical analogy). Again, raising the temperature of the junction by T is uivalent to raising the entropic potential everywhere outside the junction by VS B = T,so ds dt = C S = C SB = lim ω 0 iω A, where C S is the total entropic capacitance of the junction. Y S (ω) AB (45) 3 The Continuity Equation at a Junction The change in the junction entropy is given by the time evolution uation t qs (t) = t q S (t) i + H(t), qs (t) ]. (46) h The ual-time commutator vanishes, because the entropic potential, S (x, t) A = V S (t) A, is imposed slightly outside the junction, in the region x > x. But the change in the junction entropy should properly include also the changes in entropy in the wires very near the junction: t N x2 0 dx ρ S (x, t) A. (47) This can be calculated using the continuity uation for entropy flow in wires derived in 7]: t ρs (x, t) + x js (x, t) = kβ x S (x, t) j S (x, t) kβ x S (x, t) j S (x, t) ] (48). The result is t q S (t) + N x2 0 dx ρ S (x, t ) A + N j S (x 2, t) A = kβ V S (t) A j S (x, t) A + j S (x 2, t) A. (49) Note that this is still a local calculation at the junction. The result depends only on the values of the entropic potential in the wires near the junction. Now take x, x 2 0, including in the change of junction entropy the changes of entropy in the wires near the junction: t qs (t) + N js (0, t) A = 2kβ V S (t) A j S (0, t) A. (50) This is the continuity uation for entropy at the junction. It is an exact uation. There is no linear response approximation, no assumption that V S (t) A is small, so it is better written t qs (t) N + js (0, t) A = 2kβ V S (t) A j S (0, t) A (5)

11 Entropy Flow Through Near-Critical Quantum Junctions 863 or t s(t) + I S (t) A = 2kβ V S (t) A I S (t) A (52) where I S (t) A = j S (0, t) A. 4 Junctions in Bulk-Critical Quantum Circuits Assume now that all the wires are critical in the bulk: (x, t) A = T μ μ (x, t) A = 0. (53) Any departure from criticality is in the junction. The entropy current in each bulk-critical wire separates into chiral entropy currents (see Appendix 2): where is the right-moving entropy current, and j S (x, t) A = j R (x, t) A j L (x, t) A (54) ρ S (x, t) A = v j R (x, t) A + v j L (x, t) A (55) j R (x, t) A = j R (x vt) A (56) j L (x, t) A = j L (x + vt) A (57) is the left-moving entropy current. The operators j R (x vt) A and j L (x + vt) A are defined on the entire real line, because, although x cannot be negative, the time t can range from to +. The rate of change of the junction entropy is the net rate of inflow, t q S = j L (0, t) A j R (0, t) A. (58) The uilibrium expectation values of commutators of the chiral currents vanish in regions of space-time where effects are forbidden by causality and chirality: jr (x, t ) B, j R (x 2, t 2 ) A ] = 0 if A = B or t 2 t = (x 2 x )/v (59) jl (x, t ) B, j L (x 2, t 2 ) A ] = 0 if A = B or t 2 t = (x x 2 )/v (60) jr (x, t ) B, j L (x 2, t 2 ) A ] = 0 if t t 2 <(x + x 2 )/v (6) jl (x, t ) B, j R (x 2, t 2 ) A ] = 0 if t 2 t <(x + x 2 )/v. (62) The first two identities express the fact that any commutator of two currents of the same chirality is an ual-time commutator. The last two identities express the fact that the leftmoving current at x cannot affect the right-moving current at x 2 before a signal has time to travel from x to the junction then back to x 2, and the fact that the right-moving current cannot affect the left-moving current at all. For the first identity, use time-translation invariance and chirality to write

12 864 D. Friedan jr (x, t ) B, j R (x 2, t 2 ) A ] = jr (x vt + vt 2 + vt, t 2 ) B, j R (x 2 + vt, t 2 ) A ] (63) for any t sufficiently large, such that x 2 + vt > 0andx vt + vt 2 + vt > 0. The commutator on the rhs is then an ual-time commutator, which vanishes if A = B or if x x 2 vt + vt 2 = 0. The second identity, (60), is derived in the same way. For the third identity, (6), write jr (x, t ) B, j L (x 2, t 2 ) A ] = jr (x vt + vt 2 vt, t 2 ) B, j L (x 2 + vt, t 2 ) A ] (64) whenever x vt + vt 2 vt > 0andx 2 + vt > 0, which is possible if and only if t t 2 <(x 2 + x )/v. The commutator on the rhs is then an ual-time commutator at nonzero spatial separation, so is zero. The fourth identity, (62), follows in the same way. For t < t 2,Eqs.(59), (60), and (6) imply that js (x, t ) B, j S (x 2, t 2 ) A ] ] jr (x, t ) B j L (x, t ) B, j R (x 2, t 2 ) A = ] jl (x, t ) B, j R (x 2, t 2 ) A j L (x 2, t 2 ) A if x < x 2 if x > x 2. (65) Then t2 dt e iω(t t 2 ) i js (x, t ) B, j S (x 2, t 2 ) ] A iω h = dt e iω(t t 2 ) iω i h ] jr (x, t ) B j L (x, t ) B, j R (x 2, t 2 ) A if x < x 2 ] (66) i h jl (x, t ) B, j R (x 2, t 2 ) A j L (x 2, t 2 ) A if x > x 2 where the restriction to t t 2 in the time integrals on the rhs has been dropped because the integrands vanish when t > t 2,by(59), (60), and (62). A possible ambiguity in Eq. (40)forY S (ω) AB is now apparent. Taking the limit x, x 2 0 with x < x 2 gives one expression for Y S (ω) AB : Y S (ω) 2 AB = iω i dt e iω(t t 2 ) Taking the limit with x > x 2 gives another expression: Y S (ω) 2 AB = iω i jr (0, t ) B j L (0, t ) B, j R (0, t 2 ) ] A. (67) h dt e iω(t t 2 ) jl (0, t ) B, j R (0, t 2 ) A j L (0, t 2 ) ] A. (68) h

13 Entropy Flow Through Near-Critical Quantum Junctions 865 The difference involves only the commutators of currents of the same chirality. These commutators are completely determined by bulk conformal invariance (see (7) and(72) of Appendix 2], with the result that Y S (ω) 2 AB Y S(ω) 2 AB = δ AB2kβ j S (0, t) A, (69) so the ambiguity only exists when there are nonzero entropy currents flowing in uilibrium. As explained in 7]at the end ofsect., persistent uilibrium currents are possible because of the isolation of the near-critical degrees of freedom from the environment. In the circuit laws for entropy, the choice between Y S (ω) 2 AB and Y S(ω) 2 AB is a matter of convention, as long as the convention is used consistently. For the local analysis of a junction connected to open wires, the appropriate choice depends on the situation. To describe the change in the entropy currents due to a change T (t) in the junction temperature, the appropriate choice is Y S (ω) 2 AB : I S (t) A = Y S (ω) 2 AB T (t). (70) The entropic potential on the junction is changed by T (t), then the resulting currents are observed outside the junction, i.e., x < x 2. On the other hand, to describe the change in the junction entropy due to changes, T (t) B = V S (t) B, in the temperatures of the wires, the appropriate choice is Y S (ω) 2 AB : s(t) = ( iω Y S (ω) 2 AB ) V S (t) B. (7) The entropic potential is changed in the wires outside the junction, then the net flow of entropy into the junction in measured at the junction, i.e., x > x 2. For either choice of convention, Y S (ω) AB is manifestly a well-defined analytic function of the fruency, given by (67) or(68). The original Kubo formula, (40), has the possibility of a short-time divergence, because of the restriction to t t 2 in the integral over time. There is no such possibility in (67) or(68), since Y S (ω) AB is simply the Fourier transform in time of a commutator of local operators. Define the Fourier modes of the chiral entropy currents by j R (x, t) A = dω e iω(t x/v) j R (ω) A (72) j L (x, t) A = dω e iω(t+x/v) j L (ω) A. (73) Equations (67)and(68) are uivalent to δ(ω + ω) hωy S (ω) 2 AB = jr (ω ) B j L (ω ) B, j R (ω) A ] (74) δ(ω + ω) hωy S (ω) 2 AB = jl (ω ) B, j R (ω) A j L (ω) A ]. (75) These formulas are used below to calculate Y S (ω) AB for junctions in bulk-critical Ising circuits. From now on, to simplify the discussion, assume that no uilibrium entropy currents are flowing through the junction. With this assumption, Y S (ω) AB = Y S (ω) 2 AB = Y S(ω) 2 AB. (76)

14 866 D. Friedan When there are no uilibrium entropy currents, bulk conformal invariance implies (see (90) of Appendix 2): jr (ω ] ) A, j R (ω) B = δ AB δ(ω + ω) c ( ) ] 2 hβ 2 k2 ω + ω 2 (77) jl (ω ] ) A, j L (ω) B = δ AB δ(ω + ω) c ( ) ] 2 hβ 2 k2 ω + ω 2 (78) This allows (66) to be re-written: t2 dt e iω(t t 2 ) i js (x, t ) B, j S (x 2, t 2 ) ] A iω h = e iω(x +x 2 )/v Y S (ω) AB (e ) iω x x 2 /v e iω(x +x 2 )/v c k 2 ( ) ] 2 hβ δ AB + ω 2 2 h for any x, x 2 0. The lhs of this uation describes an experiment in which the system is perturbed at some arbitrary point, x, away from the junction, then the response is detected at a second arbitrary point, x 2, also away from the junction. Equation (79) says that the entropic admittance of the junction can be extracted from this measurement. (79) 5 Properties of Y S (ω) AB in a Bulk-Critical Quantum Circuit The entropic admittance of a junction in a bulk-critical circuit satisfies: A. Y S (ω) AB is analytic in the upper half-plane. B. Y S (ω) AB = Y S ( ω) AB. C. Y S (ω) + Y S (ω) 0 for all real ω. D. N Y S (0) AB = N Y S (0) AB = 0. E. The only singularities of Y S (ω) AB are poles lying below the real axis. F. Y S (i/ hβ) AB = 0. All of these properties can be checked in the explicit example given by (42)below. Property A expresses causality. The reality condition, property B, follows directly from (74) and(75) and the fact that the currents are self-adjoint. For the positivity condition, property C, write (74)and(75) in terms of the uilibrium two-point functions: δ(ω + ω) hωy S(ω) 2 AB e β hω = j R (ω ) B j R (ω) A j L (ω ) B j R (ω) A (80) δ(ω + ω) hωy S(ω) 2 AB e β hω = j L (ω ) B j L (ω) A j L (ω ) B j R (ω) A. (8) Take the complex conjugate of (8), use the self-adjointness of the currents and the the reality condition, property B, and exchange A with B and ω with ω to get δ(ω + ω) hωy S(ω) 2 BA e β hω = j L (ω ) B j L (ω) A j R (ω ) B j L (ω) A (82)

15 Entropy Flow Through Near-Critical Quantum Junctions 867 Add (80) and(82) and use the self-adjointness of the currents to get property C (still assuming no uilibrium entropy currents, so Y S (ω) 2 AB = Y S(ω) 2 AB = Y S(ω) AB ). For property D, note that conservation at the junction, (58), implies In (80), set ω = 0toget 0 = ] jl (0) A j R (0) A. (83) ( ) N N δ(ω) Y S (0) 2 AB β = ] jr (0) B j L (0) B jr (ω) A so N Y S (0) 2 AB = 0. In (8), set ω = 0toget ( ) N δ(ω ) β Y S (0) 2 AB = j L (ω ) B N = 0 (84) jl (0) A j R (0) A] = 0 (85) so N Y S (0) 2 AB = 0. These results are exactly what is ruired in order that Eqs. (70) and 7 be physically sensible at ω = 0. Property E follows from (67) or(68), each of which displays Y S (ω) AB as the Fourier transform in time of the uilibrium expectation value of a commutator of local operators. The junction is a bounded system at nonzero temperature, so such an expectation value is a meromorphic function of ω. Causality implies that all its singularities lie below the real axis. Property F, Y S (i/ hβ) AB = 0, is derived from bulk conformal invariance, by Wick rotating to euclidean time τ = it. The euclidean space-time of each bulk wire is a semiinfinite cylinder: x > 0, τ τ + hβ. This euclidean space-time can be reinterpreted with x/v as the euclidean time coordinate and and vτ as the spatial coordinate. Space is then a circle of circumference hvβ. In this quantization, correlation functions on the semi-infinite cylinder are expectation values between the conformally invariant ground state at late time, at large x/v, and a boundary state at time x/v = 0. In this quantization, T R (x + ivτ) A = hv2 n= e n(x+ivτ)/ hvβ L n (86) where the L n are the Virsaro operators of the bulk conformal field theory in wire A (the constant of proportionality is found in Appendix ). The L n for n, annihilate the conformally invariant ground state when they act to the left, at large time x/v. In a correlation function containing T R (x + ivτ) A for sufficiently large x, the contributions of the operators L n, n, vanish, because they are acting on the conformally invariant ground state. The two-point function given by the Fourier transform of (80)is: (kβ) 2 T R (0) B T L (0) B ] T R (x vt) A = dω e iω(t x/v) hωy S (ω) AB e β hω. (87)

16 868 D. Friedan The product of operators on the lhs is time-ordered for t < 0 (in the original, physical quantization). The Wick rotation of this uation is sensible in the region hβ <τ<0, (kβ) 2 T R (0) B T L (0) B ] T R (x + ivτ) A = dω e ω(τ+ix/v) hωy S (ω) AB e β hω, (88) because the integrand decays exponentially when ω ±, as long as hβ < τ < 0. Deforming the integration contour into the upper half-plane yields a sum over the thermal poles: (kβ) 2 T R (0) B T L (0) B ] T R (x + ivτ) A = n= n β Y S (in/ hβ) AB e n(x+ivτ)/ hvβ The n = term must vanish because it is due to the conformal generator L acting on the conformally invariant ground state. So bulk conformal invariance implies Y S (i/ hβ) AB = 0. (89) 6 Mass Gaps for Junctions The real-time response functions dω e iω(t 2 t ) Y S (ω) AB (90) describe the entropy flowing out of the junction at time t 2 in wire A, in response to a perturbation in wire B at time t < t 2. The integral can be evaluated by deforming the contour of integration into the lower half-plane, since t 2 t > 0. The response decays as e t/t r,where /t r is the gap between the real ω-axis and the nearest singularity in the lower half-plane. The time t r is the the characteristic response time of the junction in the channel AB. At nonzero temperature, there will always be a gap. In the Ising example, (42) below, the characteristic response time of a -wire junction is t r = λ 2 + π/ hβ, whereλ is the boundary coupling constant (the boundary magnetic field). The interesting question is, what happens to these gaps in the limit T 0? The question is better posed in terms of the energy response functions ī T x h t (0, t ) B, Tt x ] (0, t 2 ) A (9) because of the extra factor /T in the entropy current. When the junction is non-critical, then some of the T = 0 response functions will decay exponentially in time. If the junction is critical, then the T = 0 response functions will decay as powers of the time, or else vanish identically. The gaps, /t r, in each channel might be interpreted as the mass gaps of the junction, analogous to the mass gap in a bulk system. 7 Example: Junctions in Bulk-Critical Ising Circuits As an example, consider the general junction in a bulk-critical Ising circuit. The wires are in the universality class of the + dimensional Ising model at its critical point. The circuit

17 Entropy Flow Through Near-Critical Quantum Junctions 869 is described by the quantum field theory of a free Majorana fermion, massless in the bulk. There are two real, chiral, free fermion fields in each wire: ψ R (x, t) A = ψ R (x vt) A (92) ψ L (x, t) A = ψ L (x + vt) A. (93) The operators ψ R (x vt) A and ψ L (x + vt) A are defined on the entire real line, because, although x cannot be negative, the time t can range from to +. The fermion fields satisfy canonical ual-time anti-commutation relations in the bulk. For x, x > 0: ψr (x, t) A,ψ R (x, t) B] + = δ AB hvδ(x x) (94) ψl (x, t) A,ψ L (x, t) B] + = δ AB hvδ(x x) (95) ψr (x, t) A,ψ L (x, t) B] = 0. (96) + Their Fourier transforms are ψ R (x vt) A = dω e iω(t x/v) ψ R (ω) A (97) ψ L (x + vt) A = dω e iω(t+x/v) ψ L (ω) A (98) The hamiltonian, H 0, acts by H0, ψ R (ω) A] = hω ψ R (ω) A (99) H0, ψ L (ω) A] = hω ψ L (ω) A. (00) The junction dynamics is encoded in the reflection matrix, R(ω) A B : or, suppressing indices, ψ R (ω) A = R(ω) B A ψ L (ω) B (0) ψ R (ω) = R(ω) ψ L (ω). (02) The Fourier modes satisfy the anti-commutation relations ψ R (ω ) A, ψ R (ω) B] + = δ AB h δ(ω + ω) (03) ψ L (ω ) A, ψ L (ω) B] + = δ AB h δ(ω + ω) (04) ψ L (ω ) A, ψ R (ω) B] N + = δ AB R(ω) B B h δ(ω + ω). (05) B = The matrix R(ω) is analytic in the upper half-plane, by (96), and satisfies the reality and unitarity conditions R( ω) = R(ω) (06) R( ω) T R(ω) =. (07)

18 870 D. Friedan The anti-commutation relations and the action of the hamiltonian determine the uilibrium two-point functions: ψ R (ω ) A ψ R (ω) B = ( + eβ hω ) δ AB hδ(ω + ω) (08) ψ L (ω ) A ψ L (ω) B = ( + eβ hω ) δ AB hδ(ω + ω) (09) ψ L (ω ) A ψ R (ω) B = ( + N eβ hω ) δ AB R(ω) B B hδ(ω + ω). (0) The chiral energy momentum currents are B = T R (x vt) A = 2 vt t t (x, t) A + 2 T x t (x, t) A = i 2 :ψ R(x, t) A t ψ R (x, t) A : () T L (x + vt) A = 2 vt t t (x, t) A 2 T x t (x, t) A = i 2 :ψ L(x, t) A t ψ L (x, t) A : (2) and their Fourier transforms are T R (x vt) A = T L (x + vt) A = T R (ω) A = T L (ω) A = Their commutation relations with the fermion fields are dω e iω(t x/v) T R (ω) A (3) dω e iω(t+x/v) T L (ω) A (4) dω dω 2 δ(ω + ω 2 ω) ω 2 2 : ψ R (ω ) A ψ R (ω 2 ) A : (5) dω dω 2 δ(ω + ω 2 ω) ω 2 2 : ψ L (ω ) A ψ L (ω 2 ) A :. (6) T R (ω ) A, ψ R (ω) B] = δ B A h( 2 ω + ω) ψ R (ω + ω) B (7) T L (ω ) A, ψ L (ω) B] = δ B A h( 2 ω + ω) ψ L (ω + ω) B. (8) The hamiltonian is H 0 = T R (0) A = T L (0) A. (9) Equation (75)forY S (ω) AB can now be evaluated, using the entropy currents j R (ω) A = kβ T R (ω) A (20) j L (ω) A = kβ T L (ω) A (2) and Eqs. (5 6), (03 05), and (08 0). The result is Y S (ω) AB = k2 hβ 2 ] dω dω 2 δ(ω + ω 2 ω) R(ω ) B A ω R(ω 2) B A δ B A ( ) 8 (ω ω 2 ) 2 + e β hω 2 e β hω. (22) +

19 Entropy Flow Through Near-Critical Quantum Junctions 87 The entropic capacitance is The total entropic capacitance is C S = ds dt = C SB = lim N ω 0 iω Y S (ω) AB = T k i dη R R (η) ] B B N C SB = k T i (β hη) 2 e β hη 2(e β hη +) 2. dη Tr R R (η) ] () (β hη)2 e β hη 2 ( e β hη + ) 2. (24) Integrating with respect to T gives the junction entropy, up to a constant. When R(ω) is independent of temperature, the integral can be done explicitly, giving s(t ) s(0) = k dη Tr R R (η) ] β hη ( i 2 e β hη + + ln + e β hη)]. (25) The information capacity of the junction is then s( ) s(0) = k ln 2 i dη 2 Tr R R (η) ]. (26) The information capacity of the junction thus consists of a half-bit for each pole of R(ω) B A in the lower half-plane (or for each zero in the upper half-plane). An elementary Ising junction is a junction without substructure. It has a fermionic degree of freedom, χ(t) A, at the end of each wire, satisfying t χ(t) = λ(ψ R + ψ L )(0, t) (27) λ T χ(t) = (ψ R ψ L )(0, t) (28) where λ is a real N N matrix. The matrix elements λb A are the junction coupling constants. Eliminating χ(t) gives boundary conditions on the fermion fields: or, uivalently, ( t + λ T λ)ψ R (0, t) = ( t λ T λ)ψ L (0, t) (29) R(ω) = ω iλt λ ω + iλ T λ. (30) The information capacity of the elementary Ising junction is given immediately by (26): s( ) s(0) = N k ln 2 (3) 2 Equation () for the entropic capacitance is evaluated by analytic continuation into the upper half-plane, picking up the residues at the thermal poles. The result is C SB = ds dt = k2 β F (σ ) B B (32) where σ = β hλt λ (33) F (σ ) = σ σ 2 ψ ( + σ) 2 (34) ψ (z) = (ln Ɣ) (z) = (n + z) 2. (35) n=0

20 872 D. Friedan The total entropic capacitance is C S = ds dt = N C SB = k 2 β Tr F (σ ). (36) Integrating with respect to T gives the junction entropy, up to a constant: s(t ) s(0) = ktr ( F 2 (σ ) σ F 2 (σ )] = k β ) TrF 2 (σ ) (37) β where ( ) F 2 (σ ) = σ ln σ σ ln Ɣ 2 + σ + ln. (38) Therefore the boundary free energy, up to a constant, is ln z(t ) ln z(0) = TrF 2 (σ ). (39) This agrees, in the case N =, with the direct calculation of the boundary partition function and the boundary entropy of a bulk-critical Ising wire in a boundary magnetic field 4,]. The junction coupling constant, λ, is the boundary magnetic field. To have an explicit example of the entropic admittance, for illustration, calculate Y S (ω) AB for the case N =. The wire labels, A, B =, are suppressed. Equation (30) forthe reflection matrix is R(ω) = ω iλ2 ω + iλ 2 (40) In (22), use the identity ( ) ω + ω 2 R(ω )R(ω 2 ) = R(ω ) R(ω 2 )] (4) ω 2 ω then evaluate the integral by deforming the contour, to obtain the entropic admittance of the boundary of a bulk-critical quantum Ising wire: Y S (ω) = k 2 h β2 λ 2 iω ( ) 2 h 2 k2 β 3 λ 2 ω(ω + 2iλ 2 ) ( )( n=0 n β hλ2 n β hλ2 ( = k 2 h β2 λ 2 iω + k 2 h β2 λ 2 λ 2 iω ) 2 ( ψ 2 + β h ) (λ2 iω) iβ hω ( ψ 2 + β h )] λ2. (42) Acknowledgements I thank A. Konechny for many discussions. I thank the members of an informal Rutgers seminar S. Ashok, A. Ayyer, D. Belov, E. Dell Aquila, B. Doyon, and R. Essig for listening to a preliminary version of this work, and for their helpful comments and questions. I thank S. Lukyanov for pointing towards some of the condensed matter literature, leading in particular to Ref. 2]. This work was supported by the Rutgers New High Energy Theory Center. )

21 Entropy Flow Through Near-Critical Quantum Junctions 873 Appendix : The Energy Momentum Tensor at a Junction Consider N wires connected at a junction. Label the wires A =,...,N. Each wire is parametrized by a spatial coordinate, x 0. All the end-points, x = 0, are identified to a single point, which is the junction. The time coordinate is t; the euclidean time is τ = it. The euclidean space-time of each wire is parametrized by the complex coordinate z = x + ivτ = x vt, z = x ivτ = x + vt. The space-time metric on wire A,inthebulk, is g μν (x) A dx μ dx ν = v 2 (dt) 2 + (dx) 2 = dz 2. (43) At the junction, there is only the metric on time, g tt = v 2. Varying the space-time metric gives the correlation functions of the energy momentum tensor: δ = i ] dt δg tt (t)t tt (t) junct + dx δg μν (x, t) A T μν (x, t) A. 2 h 0 (44) An infinitesimal localized reparametrization of space-time is a collection of vector fields, v ν (x, t) A, one on each wire, all supported within a bounded region of space-time. They must agree at the junction, and must leave the junction in place, v ν (0, t) A = v ν (t) junct, (45) v x (0, t) A = v x (t) junct = 0. (46) Such a reparametrization of space-time is uivalent to changing the space-time metric by δg μν (x, t) A = μ v ν (x, t) A + ν v μ (x, t) A (47) δg tt (t) junct = 2 t v t (t) junct. (48) The physics does not depend on the parametrization of space-time, so ] 0 = dt t v t (t) junct Tt t (t) junct + dx μ v ν (x, t) A Tν μ (x, t) A 0 (49) for all localized reparametrizations of space-time. Integrating by parts gives the local conservation of energy and momentum: 0 = μ Tν μ (x, t) A (50) 0 = t Tt t (t) junct + Tt x (0, t) A. (5) Taking v μ (x, t) to be infinitesimal time translation identifies the hamiltonian: H 0 = T t t (t) junct + 0 dx T t t (x, t) A. (52) T t t (x, t) A is the bulk energy density in wire A. T t t (t) junct is the energy of the junction.

22 874 D. Friedan The departure from scale invariance is the trace of the energy momentum tensor, (x, t) = T μ μ (x, t). The junction contribution is conventionally written θ(t). Thatis, the junction energy is written and conservation of energy at the junction is written T t t (t) junct = θ(t) (53) 0 = t θ(t) + T x t (0, t) A. (54) Appendix 2: The Chiral Energy and Entropy Currents When the quantum wires are bulk-critical, the bulk energy density and current are linear combinations of chiral energy currents: T R (x, t) A = T R (x vt) A = 2 vt t t (x, t) A + 2 T x t (x, t) A (55) T L (x, t) A = T L (x + vt) A = 2 vt t t (x, t) A 2 T x t (x, t) A. (56) The bulk entropy current and density operators are linear combinations of chiral entropy currents: where j S (x, t) A = j R (x, t) A j L (x, t) A (57) ρ S (x, t) A = v j R(x, t) A + v j L(x, t) A (58) j R (x, t) A = j R (x vt) A = kβt R (x, t) A 2 kβ T R (x, t) A + T L (x, t) A (59) j L (x, t) A = j L (x + vt) A = kβt L (x, t) A 2 kβ T R (x, t) A + T L (x, t) A. (60) j R (x, t) A flows to the right j L (x, t) A flows to the left, both at the speed of light, v: j R (x, t + δt) A = j R (x vδt, t) A, j L (x, t + δt) A = j L (x + vδt, t) A. The uilibrium expectation values in (59)and(60) are subtracted so that ρ S (x, t) A = 0. Then jr (x, t) A = j L (x, t) A = js (x, t) A 2. (6) The goal is to calculate the commutators j R (x, t ] ) A, j R (x, t) B and jl (x, t ] ) A, j L (x, t) B. Because of the chirality of the currents, these commutators of two currents of the same chirality are completely determined by the ual-time commutators. The ual-time commutators vanish if A = B, so the commutators need only be calculated for A = B. The labels A, B are dropped during the calculation, to simplify the notation, then restored at the end. In + dimensional conformal field theory, the chiral components of the energy momentum tensor are usually written T (z), T ( z), where T (z) = hv 2 T R(x vt) (62) T ( z) = hv 2 T L(x + vt). (63)

23 Entropy Flow Through Near-Critical Quantum Junctions 875 The constant of proportionality, / hv 2, will be confirmed shortly. In euclidean space-time, T (z) and T ( z) satisfy operator product expansions T (z ) T (z) (z z) 4 c 2 + (z z) 2 2T (z) + (z z) T (z) (64) T ( z ) T ( z) ( z z) 4 c 2 + ( z z) 2 2 T ( z) + ( z z) T ( z). (65) Write the ual-time commutators as τ-ordered operator products, T (x ), T (x) ] = τ{t (x + iɛ) T (x)} τ{t (x iɛ) T (x)} (66) T (x ), T (x) ] = τ{ T (x iɛ) T (x)} τ{ T (x + iɛ) T (x)}, (67) and evaluate them using the operator product expansions, getting T (x ), T (x) ] = c i 2 δ(3) (x x) + ( x x ) δ(x x) T (x) ] (68) T (x ), T (x) ] = c i 2 δ(3) (x x) + ( x x ) δ(x x) T (x) ]. (69) Integrate over x to identify the energy density as T t t hv (x, t) = hv T (x vt) T (x + vt), (70) confirming the constant of proportionality in (62 63), between T (z) and T R (x vt) and between T ( z) and T L (x + vt). These ual-time commutation relations could have been derived ually well by combining conformal invariance with the general ual-time commutation relations derived in Appendix of 7]. The ual-time commutation relations, (68)and(69), can be written i TR (0, t ), T R (0, t) ] = h c h i h TL (0, t ), T L (0, t) ] = h The uilibrium expectation values of the commutators are i = h 2 3 t δ(t t ) + ( t t ) δ(t t )T R (0, t) ] (7) c 2 3 t δ(t t ) + ( t t ) δ(t t )T L (0, t) ]. (72) c 2 3 t δ(t t ) + 2 t δ(t t ) T R (0, t) (73) TR (0, t ), T R (0, t) ] h i TL (0, t ), T L (0, t) ] = h c h 2 3 t δ(t t ) + 2 t δ(t t ) T L (0, t). (74) Now restore the wire labels, A, B. Define the Fourier modes of the chiral energy momentum currents by T R (x, t) A = T R (x vt) A = dω e iω(t x/v) T R (ω) A (75) T L (x, t) A = T L (x + vt) A = dω e iω(t+x/v) T L (ω) A (76) The commutation relations, (7)and(72), are uivalent to T R (ω ] ) B, T R (ω) A = δab c ] 2 h2 ω 3 δ(ω + ω) + h(ω ω) T R (ω + ω) A (77) T L (ω ] ) B, T L (ω) A = δab c ] 2 h2 ω 3 δ(ω + ω) + h(ω ω) T L (ω + ω) A. (78)

24 876 D. Friedan Conservation of energy at the junction, (3), implies that T R (0) A T L (0) A = 0. (79) A A The hamiltonian acts by ] H0, T R (ω) A = hω T R (ω) A (80) ] H0, T L (ω) A = hω T L (ω) A (8) so the hamiltonian is H 0 = A T R (0) A = A T L (0) A. (82) A straightforward calculation shows these expressions for the hamiltonian to be ual to the spatial integral of the energy density. The uilibrium expectation values of the commutators are T R (ω ] ) B, T R (ω) A = δ AB c 2 T L (ω ] ) B, T L (ω) A = δ AB c 2 h ω3 2ω T R (0, t) A h ω3 2ω T L (0, t) A ] hδ(ω + ω) (83) ] hδ(ω + ω). (84) The uilibrium one-point functions, TR (x, t) A = T R (x vt) A (85) TL (x, t) A = T L (x + vt) A, (86) are independent of t, so are constant in x. They can be evaluated far from the junction, where the system is conformally invariant. When no uilibrium currents are flowing, when T x t (0, t) A = 0, the uilibrium bulk energy density is,3] Equivalently, T t t (x, t) A = c 2 hvβ 2. (87) TR (x, t) A = T L (x, t) A = c 24 hβ 2. (88) The uilibrium expectation values of the commutators are then T R (ω ] ) B, T R (ω) A = δ ABδ(ω + ω) c ( ) 2 2 h2 ω 3 + ω] (89) hβ T L (ω ] ) B, T L (ω) A = δ ABδ(ω + ω) c ( ) 2 2 h2 ω 3 + ω]. (90) hβ Bulk conformal invariance ruires that this commutator vanish at ω = i/ hβ. This condition gives another way to derive the uilibrium energy density.

25 Entropy Flow Through Near-Critical Quantum Junctions 877 References. Affleck, I.: Universal term in the free energy at a critical point and the conformal anomaly. Phys. Rev. Lett. 56, (986) 2. Affleck, I., Ludwig, A.W.: Universal noninteger ground state degeneracy in critical quantum systems. Phys. Rev. Lett. 67, 6 (99) 3. Blöte, H., Cardy, J., Nightingale, M.: Conformal invariance, the central charge, and universal finite-size amplitudes at criticality. Phys. Rev. Lett. 56, (986) 4. Chatterjee, R.: Exact partition function and boundary state of critical ising model with boundary magnetic field. Mod. Phys. Lett. A 0, (995) 5. Friedan, D.: Entropy flow in near-critical quantum circuits (2005). arxiv:cond-mat/ Friedan, D.: Entropy flow through near-critical circuit junctions (2005). arxiv:cond-mat/ Friedan, D.: Entropy flow in near-critical quantum circuits. J. Stat. Phys. (207). doi:0.007/s Friedan, D., Konechny, A.: Boundary entropy of one-dimensional quantum systems at low temperature. Phys. Rev. Lett. 93, 030,402 (2004) 9. Friedan, D., Konechny, A.: Infrared properties of boundaries in -d quantum systems. J. Stat. Phys. 0603, P04 (2006) 0. Ghoshal, S., Zamolodchikov, A.: Boundary s-matrix and boundary state in two-dimensional integrable quantum field theory. Int. J. Mod. Phys. A9, (994). Erratum ibid.a9, 4353 (994). Konechny, A.: Ising model with a boundary magnetic field: An example of a boundary flow. JHEP 2, 058 (2004) 2. Tsvelick, A., Wiegmann, P.: Exact results in the theory of magnetic alloys. Adv. Phys. 32(4), (983)