Quantum entanglement in SU(3) lattice Yang-Mills theory at zero and finite temperatures
QQuantum entanglement in SU(3) lattice Yang-Millstheory at zero and finite temperatures
Y. Nakagawa ∗ Graduate School of Science and Technology, Niigata University, Niigata, 950-2181, JapanE-mail: [email protected]
A. Nakamura
Research Institute for Information Science and Education, Hiroshima University,Higashi-Hiroshima, Hiroshima, 739-8521, JapanE-mail: [email protected]
S. Motoki
Graduate School of Bio-Sphere Science, Hiroshima University, Higashi-Hiroshima, Hiroshima,739-8521, JapanE-mail: [email protected]
V.I. Zakharov
ITEP, B. Cheremushkinskaya 25, Moscow, 117218, RussiaMax-Planck-Institut für Physik, Föhringer Ring 6, 80805 Münich, GermanyE-mail: [email protected]
We examine the entanglement properties of the Yang-Mills theory by calculating α entanglemententropy with α = α entropy withrespect to the size l of the subregion, whose entanglement properties are interested in, scales as1 / l , and a clear discontinuity cannot be found within our statistical errors. The α entropy inthe deconfinement phase saturates at large l . The saturation value is comparable with the thermalentropy of the pure Yang-Mills theory, indicating that the α entropy obeys the volume law at large l in the deconfinement phase. The XXVIII International Symposium on Lattice Field TheoryJuly 14-19 2010Villasimius, Italy ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/ a r X i v : . [ h e p - l a t ] A p r ntanglement entropy Y. Nakagawa
1. Introduction
Various quantum systems show entanglement properties and they receive much attention inquantum information theory and condensed matter physics. Entanglement entropy is one of quan-tities measuring quantum entanglement. A typical example of the entangled state in quantum me-chanical systems is a two spin-1/2 system in spin singlet state, which is widely used to discuss theEPR paradox, one of major topics in quantum physics. Entanglement entropy can be defined notonly in quantum mechanical systems but also in quantum field theories.In quantum mechanical systems, fundamental degrees of freedom are particles and quantumentanglement measures how much two or more particles are quantum mechanically correlated witheach others. In quantum field theories, we focus on quantum entanglement of two or more sibre-gions. The entanglement entropy between two subregions, a subregion A of size l and its comple-ment B , measures how the spatial subregion in a total system is entangled quantum mechanicallywith its complement.Quantum entanglement of ground states has been widely discussed in condensed matter physics(for a review, see [1]). For example, the entanglement entropy in the Ising chain model shows adivergent behavior at the critical point while it saturates in the non-critical regime. It means thatthe entanglement entropy serves as an order parameter of quantum phase transitions. Therefore,the entanglement entropy is a useful quantity to investigate phase structures of quantum systems.The entanglement entropy of the pure Yang-Mills theory is particularly interesting. A schematicpicture of the pure Yang-Mills system is drawn in Fig. 1. The Yang-Mills theory is an asymptot-ically free theory and the high energy phenomena in QCD can well be described by gluon andquark degrees of freedom using the perturbation theory. At low energies, on the other hand, colordegrees of freedom are confined in hadrons and the Yang-Mills system is described by the colorlesshadrons. This may remind us of the deconfinement phase transition; the color degrees of freedomare released above the critical temperature, and gluons (and quarks in QCD) play a major role aseffective degrees of freedom while those in the confinement phase are glueballs (or hadrons). Thus,one might ask if there is a critical distance scale at which the effective degrees of freedom changefrom colorful objects to colorless objects as the critical temperature of the deconfinement phasetransition.Recently, gauge/gravity correspondence has been extensively studied and it provides a power-ful tool to study non-perturbative infrared dynamics of confining gauge theories. Beginning withthe pioneering work by Ryu and Takayanagi [2], the holographic approach is applied to the calcu-lation of the entanglement entropy (for a review on the holographic calculation, see [3]). In thisapproach, the entanglement entropy of gauge theories is obtained by calculating geodesics (mini-mal surface bending down to the bulk space) in the gravity side, similar to the calculation of theWilson loop in the holographic approach. The boundary of geodesics coincides the boundary ofpartitioned subsystems in gauge theory side. The entanglement entropy has been studied for vari-ous confining backgrounds [4, 5]. It has been argued that the entanglement entropy could exhibit anon-analytic behavior with respect to the size l of the subregion; an O ( N c ) solution dominates atsmall l , and a l -independent O ( ) solution dominates above some critical length l ∗ (see Fig. 2). Thisindicates that the effective degrees of freedom change from colorful objects to colorless objects,and the critical length l ∗ plays a role of the inverse of the critical temperature of the deconfinement2 ntanglement entropy Y. Nakagawa effective d.o.f.= glueballs (colorless)effective d.o.f.= gluons (colorful)confinement phasedeconfinement phase T c ~ 300[MeV] ( in SU(3)) asymptotic freedomasymptotic slavery T SU( N c ) pure Yang-Mills theoryeffective d.o.f.= gluons (colorful) l -1 Figure 1:
Schematic picture of the pure Yang-Mill system. l ∂ S / ∂ l ~ N c2 / l ~ O( l c Figure 2:
A typical example of the holograhicprediction showing the discontinuity of the en-tanglement entropy S A . phase transition.The entanglement entropy in SU(2) lattice gauge theory has been studied by Velytsky [6] andBividovich and Polikarpov [7]. In Ref. [6], SU(N) lattice gauge theories are studied in Migdal-Kadanoff approximation, and in Ref. [7], SU(2) lattice gauge theory is numerically investigated,and there is an indication that the derivative of the entanglement entropy shows a discontinuouschange at some critical length scale l ∗ and it vanishes at large l .In this study, we investigate α entanglement entropy in SU(3) pure Yang-Mills theory usinglattice Monte Carlo simulations. Instead of directly calculating the entropy, we adopt numericaltechnique to evaluate the entanglement entropy, which has also been used in [7] (originally pro-posed in [8, 9] in order to calculate the pressure in the deconfined phase).
2. Entanglement entropy Figure 3:
The complementary regions A and B separated by an imaginaryboundary at x = l . Entanglement entropy of a pure state | Ψ (cid:105) is definedas follows. We divide the total system into subregion A and its complement B (see Fig. 3). Let l be the size ofthe system A in the x direction. The density matrix of thesystem is ρ = | Ψ (cid:105)(cid:104) Ψ | . At zero temperature, the groundstate is a pure state and the von Neumann entropy of thesystem is zero. The reduced density matrix obtained bytracing out the degrees of freedom in the region B , ρ A = Tr B ρ = Tr B | Ψ (cid:105)(cid:104) Ψ | , (2.1)describes the density matrix for an observer who can onlyaccess to the subregion A . Although we start off with apure state with vanishing von Neumann entropy, the state corresponding to the reduced densitymatrix is generally a mixed state. ρ A contains the information on the quantum degrees of freedomtraced out. The entanglement entropy is defined as the von Neumann entropy of the reduced density3 ntanglement entropy Y. Nakagawa matrix, S A = − Tr ρ A ln ρ A . (2.2)Some properties of the entanglement entropy can be found in [10].
3. Replica method
B AB Z ( l , α ) = Z = A αββ Figure 4:
Schematic picturefor the system with α cutsin x − t plane. In the region A ( B ), the periodic boundarycondition is imposed withthe period αβ ( β ). In order to evaluate the entanglement entropy, we apply thereplica trick. The detail of the derivation is given in [11]. The pointis that the entanglement entropy defined in Eq. (2.2) can be repre-sented in the form, S A = − lim α → ∂ / ∂ α ln Tr A ρ α A . The trace of the α -th power of the reduced density matrix ρ A is given by the ratio ofthe partition functions, Tr ρ α A = Z ( l , α ) Z α . (3.1)Here Z ( l , α ) is the partition function of the system having specialtopology, the α -sheeted Riemann surface, and Z = Z ( α = ) . Thefield variables in the region A is periodically identified with the inter-val αβ ( β = / T is the lattice size in the temporal direction) whilein the region B the periodic boundary condition is imposed with theperiod β (see Fig. 4).The entanglement entropy is then given by S A ( l ) = − lim α → ∂∂ α ln (cid:18) Z ( l , α ) Z α (cid:19) . (3.2)The derivative of S A ( l ) with respect to l , which is free of the ultraviolet divergence, can be expressedas follows; ∂ S A ( l ) dl = ∂∂ l (cid:20) − lim α → ∂∂ α ln (cid:18) Z ( l , α ) Z α (cid:19)(cid:21) = lim α → ∂∂ l ∂∂ α F [ l , α ] . (3.3)That is, in order to calculate ∂ S A / ∂ l , we evaluate the free energy of the system having α cuts asis depicted in Fig. 4, take the derivative with respect to α and l , and then take the limit α → α cuts.
4. Lattice setup and observables
In the lattice simulations, the derivative in Eq. (3.3) is replaced by the finite difference, and weestimate the derivative bylim α → ∂∂ l ∂∂ α F [ A , α ] → ∂∂ l lim α → ( F [ l , α + ] − F [ l , α ]) → F [ l + a , α = ] − F [ l , α = ] a . (4.1)We note that ∂ F [ l , α = ] / ∂ l drops out since F [ l , α = ] does not depend on l . The differenceof the free energies can be evaluated numerically by introducing an ‘interpolating action’ whichinterpolates two actions corresponding to two free energies [8, 9], S int = ( − γ ) S l [ U ] + γ S l + a [ U ] .4 ntanglement entropy Y. Nakagawa S l and S l + a represent the actions corresponding to F [ l , α = ] and F [ l + a , α = ] in Eq. (4.1). It iseasy to show that F [ l + a , α = ] − F [ l , α = ] = − (cid:90) d γ ∂∂ γ ln Z ( l , γ ) = (cid:90) d γ (cid:104) S l + a [ U ] − S l [ U ] (cid:105) γ . (4.2)Here (cid:104)·(cid:105) γ refers to the Monte Carlo average with the interpolating action S int . Therefore, the α = α = γ ,and performing a numerical integration over γ . In order to evaluate the integral in Eq. (4.2), wecalculated the action differences from γ = β and lattice size is about3000 to 8000. The statistical errors are estimated by the jackknife method.
5. Simulation results α entanglement entropy at zero temperature The derivative of S A ( l ) with respect to l in the confinement phase is plotted in Fig. 5. ∂ S A ( l ) / ∂ l is normalized by the area of the common boundary of the two subregions, | ∂ A | . We observe thatdata on 12 and 16 agree within statistical errors. This implies that the derivative of the α = l region, the α = / l from thedimensional analysis. That is, ∂ S A / ∂ l behaves as 1 / l at small l . This behavior is exactly what theentanglement entropy in conformal field theory in (3+1)-dimensional spacetime shows. In order to l [fm] -10010203040506070 ( / | ∂ A | ) ∂ S / ∂ l [f m - ] , β =5.7016 , β =5.7016 , β =5.7516 , β =5.8016 , β =5.85 c / l d , c =0.149(48), d =3.06(20) l [fm] -50510 ( / | ∂ A | ) ∂ S / ∂ l [f m - ] , β =5.7016 , β =5.7016 , β =5.7516 , β =5.8016 , β =5.85 Figure 5: ( / | ∂ A | ) ∂ S A / ∂ l in the confinement phase. The dashed curve is the fit of the data by the function c / l d with the fitted values c = . ( ) , d = . ( ) . The right panel shows the zoom up of the left panelto make near-zero region more visible. ntanglement entropy Y. Nakagawa confirm this, we fitted data with the function ∂ S A / ∂ l = c ( / l ) d , and we obtain c = . ( ) , d = . ( ) , χ / nd f = . ∂ S A / ∂ l . Since the derivative of S A rapidlydecreases, the signal-to-noise ratio becomes quite small at large l and it is very difficult to locatethe critical length of the entanglement entropy numerically, even if it exists. It can be safely statedthat our results exclude the possibility of the existence of the critical length at below 0.4 [fm]. α entanglement entropy below and above the critical temperature The left panel of Fig. 6 shows the derivative of the α entanglement entropy below the criticaltemperature. The fitted function at zero temperature, ( / | ∂ A | ) ∂ S A / ∂ l = . / l . , is drawn bythe dashed curve. We observe that the data agree with the fitted function of the zero temperature re-sult. This indicates that the α entanglement entropy does not show a clear temperature dependencebelow the critical temperature.The numerical result above the critical temperature is given in the right panel of Fig. 6. We seethat ∂ S A / ∂ l does not approach zero but saturates at large l . At zero temperature, the ground state isa pure state and the von Neumann entropy is zero. By contrast, the ground state at finite temperatureis a mixed state and the von Neumann entropy (thermal entropy) takes a finite value. This meansthat at finite temperature, the entanglement entropy measures not only the quantum mechanicalcorrelation between the two spatial subregions but also the thermal entropy of the subregion. Sincethe thermal entropy of the SU(3) Yang-Mills theory rapidly increases in the vicinity of the criticaltemperature, the saturation value of the entanglement entropy may be considered as the thermalentropy of the subregion A . We note that the asymptotic behavior of ( / | ∂ A | ) ∂ S A / ∂ l implies thatthe entanglement entropy obeys the volume law at large l above the critical temperature.We fitted the data with the function a / l + b , and we obtain a = . ( ) , b = . ( ) ( T / T c ∼ . ) a = . ( ) , b = . ( ) ( T / T c ∼ . ) . (5.1) l [fm] -1001020304050 ( / | ∂ A | ) dS / d l [f m - ] × β =5.70, T/T c ~ × β =5.82, T/T c ~0.9016 × β =5.86, T/T c ~0.98 l [fm] ( / | ∂ A | ) dS / d l [f m - ] × β =6.03, T/T c ~2.0220 × β =5.84, T/T c ~1.44 a / l + b, a =0.187(14), b =61.8(15) a / l + b, a =0.180(16), b =14.1(8) Figure 6:
Left panel: ( / | ∂ A | ) ∂ S A / ∂ l below the critical temperature. The dashed curve is the fitted functionat zero temperature. Right panel: ( / | ∂ A | ) ∂ S A / ∂ l above the critical temperature. The dotted curves showthe fits of the data by the function c / l + d . ntanglement entropy Y. Nakagawa
We note that the coefficient of the 1 / l term agrees with each other. In order to compare theasymptotic value of ∂ S A / ∂ l to the thermal entropy, we estimated the thermal entropy at T / T c ∼ .
44 and 2 .
02 by reading the values off the figure in Ref. [12]. A rough estimate gives s = ( T / T c ∼ . ) s = ( T / T c ∼ . ) . (5.2)These are comparable with the asymptotic values of ∂ S A / ∂ l .
6. Summary and conclusion
We studied the α entanglement entropy of the Yang-Mills vacuum with α = α = l is well fitted by the function c / l d with d=3.06(20). The exponent d is consistent withthat in the conformal field theory. A clear discontinuity in ∂ S A / ∂ l was not observed within the sta-tistical errors, which is arguded in the models of the gauge/gravity correspondence. Furthermore,we observe that the α entropy is almost temperature independent below the critical temperature.Above the critical temperature, α = l . Sincethe ground state of the finite temperature system is a mixed state, this implies that the entangle-ment entropy measures not only the correlation between the spatial subregions but also the thermalentropy of the subregion, which dominates at large l . Indeed, our fitted result of the asymptoticvalues is comparable with the thermal entropy of the pure SU(3) Yang-Mills theory. Acknowledgements
The simulation was performed on NEC SX-8R at RCNP, Osaka University and NEC SX-9 atCMC, Osaka University. The work is partially supported by Grant-in-Aid for Scientific Researchby Monbu-kagakusyo, No. 20340055.
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