Linking Phase Transitions and Quantum Entanglement at Arbitrary Temperature
LLinking Phase Transitions and Quantum Entanglement at Arbitrary Temperature
Bo-Bo Wei ∗ School of Physics and Energy, Shenzhen University, 518060 Shenzhen, China
In this work, we establish a general theory of phase transitions and quantum entanglement in the equilibriumstate at arbitrary temperatures. First, we derived a set of universal functional relations between the matrixelements of two-body reduced density matrix of the canonical density matrix and the Helmholtz free energyof the equilibrium state, which implies that the Helmholtz free energy and its derivatives are directly relatedto entanglement measures because any entanglement measures are defined as a function of the reduced densitymatrix. Then we show that the first order phase transitions are signaled by the matrix elements of reduceddensity matrix while the second order phase transitions are witnessed by the first derivatives of the reduceddensity matrix elements. Near second order phase transition point, we show that the first derivative of thereduced density matrix elements present universal scaling behaviors. Finally we establish a theorem whichconnects the phase transitions and entanglement at arbitrary temperatures. Our general results are demonstratedin an experimentally relevant many-body spin model.
I. INTRODUCTION
Quantum phase transition is a transition between di ff erentquantum phases of a many-body system at zero temperature[1, 2]. It comes from diverging quantum fluctuations and maybe observed by varying the control parameter of the systemat zero temperature [1]. In recent years, a large amount ofe ff ort has been made in investigating phase transitions fromthe perspective of quantum information science [3], in partic-ular the quantum entanglement [4–7] and the quantum fidelity[8–10]. The advantage of investigating phase transitions fromquantum information science approach compared to the con-ventional approach is that one do not need to know the localorder parameter of the phase transitions and specific symme-tries of microscopic Hamiltonian [11–37].Previous investigations on the relations between phase tran-sitions and entanglement are based primarily on specificmany-body models [4–7]. Recently, Wu and his collaborators[38, 39] studied the relations between quantum phase transi-tions and quantum entanglement in a general settings and theirtheories are valid for a very broad class of many-body systems[38–41]. However, Wu’s results are valid only at zero temper-atures. Realistic experiments are performed at nonzero tem-peratures. It is thus highly desirable to investigate whether thegeneral relations between entanglement and phase transitionssurvive at nonzero temperature.Motivated by the works of Wu and his collaborators [38,39], in the present work, we study the general relations ofphase transitions and entanglement in the equilibrium state atarbitrary temperatures. We derived a set of universal func-tional relations between the matrix elements of two-body re-duced density matrix of the canonical equilibrium state andthe Helmholtz free energy of the equilibrium state. This re-veals that the Helmholtz free energy and its derivatives aredirectly related to entanglement measures since any entangle-ment measures are defined from the reduced density matrix.We show that the first order phase transitions are signaled by ∗ Electronic address: [email protected] the matrix elements of reduced density matrix while the sec-ond order phase transitions are witnessed by the first deriva-tives of the reduced density matrix elements. Close to secondorder phase transition point, we show that the first derivativesof the reduced density matrix elements present universal scal-ing behaviors. Finally we establish a theorem which connectsthe phase transitions and entanglement at arbitrary tempera-tures. We demonstrated our general conclusions in the Lipkin-Meshkov-Glick (LMG) model which presents both quantumphase transitions and thermal phase transitions.This paper is structured as follows. In Sec. II, we estab-lish the general framework and derived the relations betweenHelmholtz free energy and the reduced density matrix ele-ments. In Sec. III, we establish the relations between phasetransitions and reduced density matrix. Sec. IV is devotedto study the relations between phase transitions and entangle-ment. In Sec. V, we study the LMG model to demonstrateour general results. Finally Sec. VI is a brief summary anddiscussion.
II. FREE ENERGY AND REDUCED DENSITY MATRIX
Let us consider a general Hamiltonian up to two-body in-teractions, H = (cid:88) i ,α,β E i αβ | α i (cid:105)(cid:104) β i | + (cid:88) i , j ,α,β,γ,δ V i , j αβγδ | α i β j (cid:105)(cid:104) γ i δ j | . (1)Here {| α i (cid:105)} is a complete basis for the Hilbert space and α, β, γ, δ ∈ [0 , , · · · , d −
1] with d being the dimension ofthe Hilbert space and i , j are the indices labelling d -level sys-tems (qudits). This Hamiltonian is the same as that discussedin [38] where quantum phase transitions and reduced densitymatrix are discussed. In the present work, we generalize theconnections between phase transitions and entanglement atzero temperature to arbitrary temperatures. At non-zero tem-perature, the canonical density matrix of a many-body systemwith Hamiltonian H which is in thermal equilibrium with aheat bath at fixed temperature T is given by ρ = e − β H Z , (2) a r X i v : . [ qu a n t - ph ] J a n where β = / T is the inverse temperature of the bath (Weset the Boltzmann constant k B =
1) and Z = Tr[ e − β H ] is thecanonical partition function of the system. From the canonicaldensity matrix (2), the Helmholtz free energy is thus given by F = E − T S , (3) = Tr[ ρ H ] + T Tr[ ρ ln ρ ] , (4) = (cid:88) i ,α,β E i αβ Tr (cid:2) ρ | α i (cid:105)(cid:104) β i | (cid:3) + (cid:88) i , j ,α,β,γ,δ V i , j αβγδ Tr (cid:104) ρ | α i β j (cid:105)(cid:104) γ i δ j | (cid:105) + T Tr[ ρ ln ρ ] , (5) = (cid:88) i ,α,β E i αβ ρ i αβ + (cid:88) i , j ,α,β,γ,δ V i , j αβγδ ρ i j γδ,αβ + T Tr[ ρ ln ρ ] , (6) = (cid:88) i j Tr[ U i j ρ i j ] + T Tr[ ρ ln ρ ] . (7)Here in Equation (3), E is the internal energy and S is the en-tropy. In Equation (7), ρ i j is two-body reduced density matrixof the canonical density matrix ρ and U i j is defined by U i j αβ,γδ = E i αγ δ j βδ / N i + V i , j αβγδ . (8)Here N i is the number of qudits that the qudit i interact withand δ j βδ is the Kronecker delta function on the qudit j . Theinternal energy of the equilibrium state can be given by E = Tr[ ρ H ] = (cid:88) i j Tr[ U i j ρ i j ] . (9)The functional relations in Equation (7) and (9) tell us that theinternal energy of the equilibrium state is fully determined bythe two-body reduced density matrix of the canonical densitymatrix but the free energy do not. Equation (9) not only holdsfor the internal energy but also for other physical observable.For example the average value of an arbitrary two-body opera-tor M = (cid:80) i j M i j in the equilibrium state is fully characterizedby the two body reduced density matrix as (cid:104)M(cid:105) = (cid:88) i j Tr (cid:104) ρ i j M i j (cid:105) . (10)Note that Equation (9) and Equation (10) can be easily gen-eralized to Hamiltonian with n -body interactions, where theinternal energy and the average value of any physical observ-able are connected to n -body reduced density matrix. III. PHASE TRANSITIONS AND THE REDUCEDDENSITY MATRIX
In this section, we establish the relations between phasetransitions and the reduced density matrix. We assume themany-body Hamiltonian H in Equation (1) depends on con-trol parameter λ through E i αβ and V i , j αβγδ . Assuming that E i αβ and V i , j αβγδ are smooth functions of the control parameter ofthe system λ . From the definition of free energy, Equation (4) we have ∂ F ∂λ = Tr (cid:34) ∂ρ∂λ H + ρ ∂ H ∂λ (cid:35) + T Tr (cid:34) ∂ρ∂λ ln ρ + ρ ∂ ln ρ∂λ (cid:35) , (11) = Tr (cid:34) ∂ρ∂λ H + ρ ∂ H ∂λ (cid:35) + T Tr (cid:34) ∂ρ∂λ ln ρ (cid:35) , (12) = Tr (cid:34) ∂ρ∂λ H + ρ ∂ H ∂λ (cid:35) − T Tr (cid:34) ∂ρ∂λ ( β H + ln Z ) (cid:35) , (13) = Tr (cid:34) ρ ∂ H ∂λ (cid:35) , (14) = (cid:88) i j Tr (cid:34) ∂ U i j ∂λ ρ i j (cid:35) . (15)In the above derivation, the last term in Equation (11) whichreduces to Tr[ ∂ λ ρ ] = ρ ] =
1. From Equa-tion (12) to Equation (13), we have made use of Equation (2).From Equation (13) to Equation (14), we have made use ofTr[ ∂ λ ρ ] = ∂ F ∂λ = (cid:88) i j Tr (cid:34) ∂ U i j ∂λ ρ i j (cid:35) . (16)Di ff erentiating both sides of Equation (16) with respect to λ ,we have ∂ F ∂λ = (cid:88) i j Tr (cid:34) ∂ U i j ∂λ ρ i j (cid:35) + (cid:88) i j Tr (cid:34) ∂ U i j ∂λ ∂ρ i j ∂λ (cid:35) . (17)From Equation (9), the first derivative of the free energy withrespect to the temperature satisfies that F − T ∂ F ∂ T = (cid:88) i j Tr[ U i j ρ i j ] . (18)Di ff erentiating both sides of Equation (18) with respect totemperature T , we get ∂ F ∂ T = − T (cid:88) i j Tr (cid:34) U i j ∂ρ i j ∂ T (cid:35) . (19)Equation (16),(17), (18) and (19) are the first central results ofthe paper. We now make several comments on their implica-tions to the phase transitions:1. Equation (16),(17), (18) and (19) connect the macroscopicquantities of a thermodynamic equilibrium state, the free en-ergy and its derivatives, to the microscopic state of the system,two-body reduced density matrix of the canonical density ma-trix. In addition, Equation(16) and (17) recovers the results in[38] at zero temperature.2. Implications for first order phase transitions: First orderphase transitions usually mean that the free energy is analyticfunction of the control parameters, such as λ, T , but the firstderivatives of the free energy with respect to λ, T are nonana-lytic functions of the control parameters (nonanalytic may beeither diverge or discontinuous). If we assume that E i αβ and V i , j αβγδ are smooth functions of the control parameters of thesystem λ . From Equation (16) and (18), the nonanalytic be-havior of the first derivative of the free energy with respect to λ, T must come from the nonanalytic behavior of matrix ele-ments of the two-body reduced density matrix of the canonicaldensity matrix, ρ i j .3. Implications for second order phase transitions: Second or-der phase transitions mean that the free energy and its firstderivatives are analytic functions of the control parameters λ, T but the second derivatives of the free energy with respectto the control parameters λ, T are nonanalytic (either divergeor discontinuous). We assume that E i αβ and V i , j αβγδ are smoothfunctions of the control parameters of the system λ . FromEquation (17) and (19), the nonanalytic behavior of the sec-ond derivatives of the free energy must come from the non-analytic behavior of matrix elements of the first derivatives ofthe two-body reduced density matrix with respect to λ, T , i.e. ∂ λ ρ i j and ∂ T ρ i j . Near thermal phase transitions, the singularpart of the free energy F s presents the scaling behavior [2], F s (cid:32) N , δ T (cid:33) = Ψ (cid:32) bN , b /ν δ T (cid:33) . (20)Here b is a scaling factor and δ T = T − T c with T c being thecritical temperature and δλ = λ − λ c with λ c being the criti-cal control parameter. ν is the correlation length critical expo-nent of the thermal phase transitions and Ψ ( x , y ) is a universalscaling function. Because Equation (19) tells us that the sin-gular part of the free energy must come from the singularity ofthe two-body reduced density matrix, we then expect that oneof the matrix elements of the two-body reduced density ma-trix of the canonical density matrix near critical point satisfiesthe following scaling relation, ∂ρ i j ∂ T ∝ Ψ (cid:32) bN , b /ν δ T (cid:33) . (21)Here Ψ ( x , y ) is a universal scaling function. In the quantumcritical region, the free energy density presents the scaling be-havior [1] F s (cid:32) N , T , δλ (cid:33) = Ψ (cid:32) bN , b z T , b /ν δλ (cid:33) . (22)Here ν and z are respectively the correlation length criticalexponent and the dynamical critical exponent of the quantumphase transitions and Ψ ( x , y , z ) is a universal scaling function.Because Equation (17) tells us that the singular part of the freeenergy must come from the singularity of the first derivative ofthe two-body reduced density matrix, we then expect that oneof the matrix elements of the two-body reduced density matrixof the canonical density matrix near critical point satisfies thefollowing scaling relation, ∂ρ i j ∂λ ∝ Ψ (cid:32) bN , b z T , b /ν δλ (cid:33) . (23)Equation (21) and (23) tell us that the first derivative ofthe matrix elements of the two-body reduced density matrixpresent scaling behaviors both at quantum critical point andthermal critical point. IV. PHASE TRANSITIONS AND ENTANGLEMENT
In recently years, entanglement measures have been usedto diagnostic universal behaviors in quantum many-body sys-tems, in particular phase transitions [4, 6, 7]. The most usefulentanglement measures are the R´enyi entropies and the vonNeumann entropy [4, 6, 7]. If a quantum system is preparedin a state ρ and a bipartition of the system into a subsystem A and its complement B , the reduced density matrix of part A is ρ A = Tr B [ ρ ]. The R´enyi entropies S n of part A are defined as[7], S ( n ) A = − n ln Tr[ ρ nA ] . (24)When n →
1, the Renyi entropy becomes von Neumann en-tropy, lim n → S ( n ) A = S A = − Tr[ ρ A ln ρ A ].In the previous section, we have established the connectionsbetween phase transitions and the reduced density matrix. Be-cause quantum entanglement measures are defined from thereduced density matrix [4, 6, 7], it is thus conceivable that en-tanglement and phase transitions are directly connected. NowWe first state the central theorem about phase transitions andentanglement at arbitrary temperatures, which is a generaliza-tion of the work by Wu and his collaborators [38] to finitetemperatures. Then we make a proof of the theorem. Theorem:
If the following conditions (i), (ii), (iii) are sat-isfied, then nonanalytic behavior in the R´enyi entanglemententropy and in the first derivative of the R´enyi entanglemententropy are respectively necessary and su ffi cient conditions tosignal a first order phase transition and a second order phasetransitions.(i) The first order phase transition and second order phasetransitions are associated to nonanalytic behavior of thefirst order derivative of the free energy and second orderderivative of the free energy respectively. Furthermore,the nonanalytic behavior in the first order derivative ofthe free energy and in the second order derivative of thefree energy exclusively originated from the elements ofthe ρ i j and not from the summation itself.(ii) The nonanalytic matrix elements of ρ i j and its firstderivatives ( ∂ λ ρ i j , ∂ T ρ i j ) appear in the expression ofR´enyi entanglement entropy do not either all acciden-tally vanish or cancel each other;(iii) The nonanalytic matrix elements of ρ i j and its firstderivatives ( ∂ λ ρ i j , ∂ T ρ i j ) appear in the expression ofR´enyi entanglement entropy do not either all acciden-tally vanish or cancel other terms in the expression forthe four equations (16),(17), (18) and (19).Now let us prove the above theorem:Proof: The case for first order phase transitions:
If condi-tion (i) is satisfied, then the first order phase transitions mustcome from nonanalytic behavior of one matrix elements of ρ i j ,as given by Equations (16) and Equation (18). Taking the con-dition (ii) into account, the first order phase transitions will beassociated to nonanalytic behavior in the R´enyi entanglemententropy. So nonanalytic behavior in the R´enyi entanglemententropy is a necessary condition for first order phase transi-tions. Considering condition (iii), nonanalytic behavior in thethe R´enyi entanglement entropy must come from the nonan-alytic behavior of one or more of the matrix elements of thereduced density matrix ρ i j . Assuming condition (i), a first or-der phase transitions follows. Thus nonanalytic behavior inthe R´enyi entanglement entropy is also a su ffi cient conditionfor first order phase transitions. The case for second order phase transitions:
If condi-tion (i) is satisfied, then the second order phase transitionsmust come from nonanalytic behavior of one or more of thematrix elements of the first derivative of the reduced densitymatrix ( ∂ T ρ i j , ∂ λ ρ i j ), as given by Equations (17) and Equa-tion (19). Taking the condition (ii) into account, the secondorder phase transitions will be associated to nonanalytic be-havior in the first derivative of R´enyi entanglement entropy.So nonanalytic behavior in the first derivative of R´enyi en-tanglement entropy is a necessary condition for second orderphase transitions. Considering condition (iii), the nonanalyticbehavior in the first derivative of R´enyi entanglement entropymust come from the nonanalytic behavior of one or more ofthe matrix elements of the first derivative of the reduced den-sity matrix, ( ∂ T ρ i j , ∂ λ ρ i j ). Assuming condition (i), a secondorder phase transitions follows. Thus nonanalytic behavior inthe first derivative of the R´enyi entanglement entropy is also asu ffi cient condition for second order phase transitions. There-fore the theorem is proved. V. PHYSICAL MODEL DEMONSTRATION
To demonstrate the above ideas, we study a many-bodyspin model with both quantum phase transitions and finitetemperature phase transitions, namely the Lipkin-Meshkov-Glick (LMG) model [42–44] and the Hamiltonian of the LMGmodel is H = − JN (cid:88) i < j (cid:16) σ xi σ xj + γσ yi σ yj (cid:17) − λ (cid:88) j σ zj . (25)Here J is the ferromagnetic coupling strength between twopauli spins (cid:126)σ i and (cid:126)σ j at arbitrary two sites along the x and y directions, γ is the anisotropy of the ferromagnetic couplingin the y direction, λ is the magnetic field along z direction.The LMG model and its various extensions have been experi-mentally realized in trapped ion systems [45–47] and also maybe implemented in the nitrogen-vacancy centers system [48].Thus investigations in this work could be verified experimen-tally in near future.Let us now relate the derivative of free energy and the ma-trix elements of the two-body reduced density matrices of thecanonical density matrix:
1. Free energy and its derivatives with respect to the con-trol parameter λ : First, one can show that the diagonal ma-trix elements of the two-body reduced density matrix and theaverage value of physical quantity in the LMG model are re- λ ∂ ρ ij / ∂ λ - - ( λ - λ m ) N / ν ( ∂ ρ ij / ∂ λ m - ∂ ρ ij / ∂ λ ) / ∂ ρ ij / ∂ λ ( a )( b ) FIG. 1: (color online). Finite-size scaling of the first derivative ofthe two-body reduced density matrix element close to the quantumphase transition point in the LMG model. (a). The first derivative ofthe two-body reduced density matrix element with respect to controlparameter λ , ∂ρ ij /∂λ , in the LMG model as a function of controlparameter λ for di ff erent number of spins N . The black solid line is N = N = N = N = ∂ρ ij /∂λ is a function of N /ν ( λ − λ m ) only with λ m beingthe position of the maximum of ∂ρ ij /∂λ and ν = .
49 being chosenso that data in (a) for di ff erent N collapse perfectly. ∂ρ ij /∂λ m is ashorthand notation for ∂ρ ij /∂λ | λ = λ m . lated by (See Appendix for detailed derivations) ρ i j = (cid:104) (cid:104) σ zi σ zj (cid:105) + (cid:104) σ zj (cid:105) + (cid:105) , (26) ρ i j = (cid:104) (cid:104) σ zi σ zj (cid:105) − (cid:104) σ zj (cid:105) + (cid:105) , (27) ρ i j = ρ i j = (cid:16) − (cid:104) σ zi σ zj (cid:105) (cid:17) . (28)Then the first derivative of the free energy can be calculatedas ∂ F ∂λ = (cid:42) ∂ H ∂λ (cid:43) , (29) = − N (cid:104) σ zj (cid:105) , (30) = − N (cid:104) ρ i j − ρ i j (cid:105) . (31)Here we have made use of the translation symmetry and Equa-tions (26) to (28). Di ff erentiating the above equation with re-spect to λ , we get ∂ F ∂λ = − N ∂ρ i j ∂λ − ∂ρ i j ∂λ . (32)We thus proved analytically Equations (16) and (17) in theLMG model. Because the LMG model presents a second or-der quantum phase transitions from a ferromagnetic phase toa paramagnetic phase at critical field λ c =
1, we thus expectthat ∂ρ ij ∂λ or ∂ρ ij ∂λ presents universal scaling behavior.In Figure 1, we show that the critical behavior of the firstderivative of the two-body reduced density matrix element ∂ρ /∂λ of the ground state as a function of the control param-eter λ . In Figure 1 (a), we plot ∂ρ /∂λ as a function of con-trol parameter λ for the system with di ff erent number of spins N = , , , ∂ρ /∂λ for systems with di ff erent number of spins crossat the quantum critical point λ c =
1. Second, ∂ρ /∂λ presentsa peak at λ m which is close to the critical point. As the sys-tem size increases, the position of control parameter λ m where ∂ρ /∂λ has a peak approaches the quantum critical point λ c .In Figure 1(b), we plot the ( ∂ρ /∂λ | λ m − ∂ρ /∂λ ) /∂ρ /∂λ asa function of scale parameter ( λ − λ m ) N /ν . We choose ν sothat the data in Figure 1(a) collapse perfectly and we foundthat the correlation length critical exponent ν = .
49, which isclose to the exact value ν = /
2. Free energy and its derivatives with respect to tem-perature T : The o ff -diagonal matrix elements of the two-body reduced density matrix and the average values of thephysical observable are related by (See Appendix for detailedderivations) (cid:60) ρ i j = (cid:104) (cid:104) σ xi σ xj (cid:105) + (cid:104) σ yi σ yj (cid:105) (cid:105) , (33) (cid:60) ρ i j = (cid:104) (cid:104) σ xi σ xj (cid:105) − (cid:104) σ yi σ yj (cid:105) (cid:105) . (34)The average value of the internal energy is (cid:104) E (cid:105) = (cid:104) H (cid:105) , (35) = − J ( N − (cid:104) (cid:104) σ xi σ xj (cid:105) + γ (cid:104) σ yi σ yj (cid:105) (cid:105) − λ N (cid:104) σ zj (cid:105) , (36) = − ( N − J (cid:104) (1 − γ ) (cid:60) ρ i j + (1 + γ ) (cid:60) ρ i j (cid:105) − λ N (cid:16) ρ i j − ρ i j (cid:17) , (37) = F − T ∂ F ∂ T . (38)In the above derivations, we have made use of the translationsymmetry and Equations (33) and (34). Di ff erentiating the T ∂ ρ ij / ∂ T - - ( T - T m ) N / ν ( ∂ ρ ij / ∂ T m - ∂ ρ ij / ∂ T ) / ∂ ρ ij / ∂ T ( a )( b ) FIG. 2: (color online). Finite-size scaling of the first derivative ofthe two-body reduced density matrix close to the thermal phase tran-sition point in the LMG model. (a). The first derivative of the two-body reduced density matrix element with respect to temperature, ∂ρ ij /∂ T , in the LMG model as a function of temperature T for dif-ferent number of spins N . The black solid line is N = N = N =
400 and theblue dash-dotted line is N = ρ ij because it is a complex number. (b). Data collapse of the firstderivative of the two-body reduced density matrix element with re-spect to temperature in the LMG model shown in (a). According toscaling arguments, ∂ρ ij /∂ T is a function of N /ν ( T − T m ) only with T m being the position of the maximum of ∂ρ ij /∂ T and ν = .
01 beingchosen so that data in (a) for di ff erent N collapse perfectly. ∂ρ ij /∂ T m is a shorthand notation for ∂ρ ij /∂ T | T = T m . above equation with respect to temperature, we get ∂ F ∂ T = ( N − JT (1 − γ ) ∂ (cid:60) ρ i j ∂ T + (1 + γ ) ∂ (cid:60) ρ i j ∂ T + λ NT ∂ρ i j ∂ T − ∂ρ i j ∂ T . (39)We thus proved analytically Equations (17) and (19) in theLMG model.In Figure 2, we show that the critical behavior of the firstderivative of the two-body reduced density matrix element ∂ρ /∂ T of the equilibrium state as a function of tempera-ture T . In Figure 2 (a), we plot ∂ρ i j /∂ T as a function oftemperature T for the system with di ff erent number of spins N = , , ,
500 respectively. First, one can see that ∂ρ i j /∂ T for systems with di ff erent number of spins cross atthe thermal critical point T c =
1. Second, ∂ρ i j /∂ T presents apeak at T m which is close to the thermal critical point. As thesystem size increases, the position of temperature T m where ∂ρ i j /∂ T has a peak approaches the thermal critical point T c .In Figure 2(b), we plot the ( ∂ρ i j /∂ T | T m − ∂ρ i j /∂ T ) /∂ρ i j /∂ T as a function of scale variable ( T − T m ) N /ν . We choose ν sothat the data in Figure 2(a) collapse perfectly and we foundthat the correlation length critical exponent ν = .
01, which isclose to the exact value ν = VI. SUMMARY
In summary, we have established a general theory of phasetransitions and quantum entanglement in the equilibrium stateat arbitrary temperature. We derived a set of universal func-tional relations between the matrix elements of two-body re-duced density matrix of the canonical equilibrium state andthe Helmholtz free energy of the equilibrium state. These rela-tions imply that the free energy and its derivatives are directlyrelated to quantum entanglement in the canonical equilibriumstate. Furthermore, we showed that the first order phase tran-sitions are signaled by the matrix elements of reduced densitymatrix while the second order phase transitions are witnessedby the first derivatives of the reduced density matrix elements.Close to second order phase transitions, we showed that thefirst derivatives of the reduced density matrix elements presentuniversal scaling behaviors. We finally established a theoremwhich connects the phase transitions and entanglement at arbi-trary temperature. Our general results are demonstrated in theLMG model and could be verified experimentally in trappedion settings.
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (Grant Number 11604220).
Appendix: Derivation of the Relations between the MatrixElements of Two-body Reduced Density Matrix and the AverageValue of Physical Quantity in LMG Model
In this appendix we derive the relations between the matrixelements of two-body reduced density matrix and the averagevalue of physical quantity in LMG model. First the average of a Pauli spin along z direction can be calculated as (cid:104) σ zj (cid:105) = Tr[ ρσ zj ] , (A1) = Tr i j [ ρ i j σ zj ] , (A2) = (cid:88) α,β (cid:104) α i β j | ρ i j σ zj | α i β j (cid:105) , (A3) = ρ i j − ρ i j + ρ i j − ρ i j . (A4)Similarly, one gets (cid:104) σ zi (cid:105) = Tr[ ρσ zi ] , (A5) = ρ i j + ρ i j − ρ i j − ρ i j . (A6)Translation symmetry implies that (cid:104) σ zi (cid:105) = (cid:104) σ zj (cid:105) , which leadsto ρ i j = ρ i j . (A7)Thus (cid:104) σ zj (cid:105) = ρ i j − ρ i j . (A8)Besides, Tr[ ρ ] = ρ i j + ρ i j + ρ i j = . (A9)The correlation function of two Pauli spins along z directionis (cid:104) σ zi σ zj (cid:105) = Tr[ ρσ zi σ zj ] , (A10) = ρ i j − ρ i j − ρ i j + ρ i j , (A11) = ρ i j − ρ i j + ρ i j , (A12) = ρ i j + ρ i j ] − . (A13)Therefore, we have ρ i j = (cid:104) (cid:104) σ zi σ zj (cid:105) + (cid:104) σ zj (cid:105) + (cid:105) , (A14) ρ i j = (cid:104) (cid:104) σ zi σ zj (cid:105) − (cid:104) σ zj (cid:105) + (cid:105) , (A15) ρ i j = ρ i j = (cid:16) − (cid:104) σ zi σ zj (cid:105) (cid:17) . (A16)Thus Equations (26), (27) and Equation (28) in the main textare derived.One can see that the Hamiltonian of LMG model is invari-ant under a global rotation along z axis by an angle π . Thisleads to (cid:104) σ xi (cid:105) = , (A17) (cid:104) σ yi (cid:105) = . (A18)The average value of σ xi and σ yi can be given by the matrixelements of the two-body reduced density matrix, (cid:104) σ xj (cid:105) = Tr[ ρσ xj ] , (A19) = Tr i j [ ρ i j σ xj ] , (A20) = (cid:88) α,β (cid:104) α i β j | ρ i j σ xj | α i β j (cid:105) , (A21) = (cid:88) α,β,γ,δ (cid:104) α i β j | ρ i j | γ i δ j (cid:105)(cid:104) γ i δ j | σ xj | α i β j (cid:105) , (A22) = ρ i j + ρ i j + ρ i j + ρ i j , (A23) = (cid:104) (cid:60) ρ i j + (cid:60) ρ i j (cid:105) . (A24) (cid:104) σ yj (cid:105) = Tr[ ρσ yj ] , (A25) = Tr i j [ ρ i j σ yj ] , (A26) = (cid:88) α,β (cid:104) α i β j | ρ i j σ yj | α i β j (cid:105) , (A27) = (cid:88) α,β,γ,δ (cid:104) α i β j | ρ i j | γ i δ j (cid:105)(cid:104) γ i δ j | σ yj | α i β j (cid:105) , (A28) = − i ρ i j + i ρ i j − i ρ i j + i ρ i j , (A29) = (cid:104) (cid:61) ρ i j + (cid:61) ρ i j (cid:105) . (A30)Thus we have ρ i j = − ρ i j . (A31)The average value of two-body operators can be calculated as (cid:104) σ xi σ xj (cid:105) = Tr[ ρσ xi σ xj ] , (A32) = Tr i j [ ρ i j σ xi σ xj ] , (A33) = (cid:88) α,β (cid:104) α i β j | ρ i j σ xi σ xj | α i β j (cid:105) , (A34) = (cid:88) α,β,γ,δ (cid:104) α i β j | ρ i j | γ i δ j (cid:105)(cid:104) γ i δ j | σ xi σ xj | α i β j (cid:105) , (A35) = ρ i j + ρ i j + ρ i j + ρ i j , (A36) = (cid:104) (cid:60) ρ i j + (cid:60) ρ i j (cid:105) . (A37) In the above, we have made use of the Hermitian property ofthe two-body reduced density matrix. Moreover, (cid:104) σ yi σ yj (cid:105) = Tr[ ρσ yi σ yj ] , (A38) = Tr i j [ ρ i j σ yi σ yj ] , (A39) = (cid:88) α,β (cid:104) α i β j | ρ i j σ yi σ yj | α i β j (cid:105) , (A40) = (cid:88) α,β,γ,δ (cid:104) α i β j | ρ i j | γ i δ j (cid:105)(cid:104) γ i δ j | σ yi σ yj | α i β j (cid:105) , (A41) = − ρ i j + ρ i j + ρ i j − ρ i j , (A42) = (cid:104) (cid:60) ρ i j − (cid:60) ρ i j (cid:105) . (A43)We thus obtain (cid:60) ρ i j = (cid:104) (cid:104) σ xi σ xj (cid:105) + (cid:104) σ yi σ yj (cid:105) (cid:105) , (A44) (cid:60) ρ i j = (cid:104) (cid:104) σ xi σ xj (cid:105) − (cid:104) σ yi σ yj (cid:105) (cid:105) . (A45)Thus Equations (33) and Equation (34) in the main text arederived. [1] S. Sachdev, Quantum Phase Transitions (Cambridge UniversityPress, Cambridge, 2011).[2] J. L. Cardy,
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