Linking phase transitions and quantum entanglement at arbitrary temperature

Bo-Bo Wei

Phys. Rev. A 97, 042115 – Published 19 April 2018

Abstract

In this work, we establish a general theory of phase transitions and quantum entanglement in the equilibrium state at arbitrary temperatures. First, we derive a set of universal functional relations between the matrix elements of a two-body reduced density matrix of the canonical density matrix and the Helmholtz free energy of the equilibrium state, which implies that the Helmholtz free energy and its derivatives are directly related to entanglement measures because any entanglement measures are defined as a function of the reduced density matrix. Then we show that the first-order phase transitions are signaled by the matrix elements of the reduced density matrix while the second-order phase transitions are witnessed by the first derivatives of the reduced density matrix elements. Near the second-order phase-transition point, we show that the first derivative of the reduced-density-matrix elements present universal scaling behaviors. Finally, we establish a theorem which connects the phase transitions and bipartite entanglement at arbitrary temperatures. Our general results are demonstrated in an experimentally relevant many-body spin model.

Received 31 January 2018

DOI:https://doi.org/10.1103/PhysRevA.97.042115

Physics Subject Headings (PhySH)

Critical phenomena Phase transitions Quantum entanglement Quantum phase transitions

Statistical Physics & ThermodynamicsQuantum Information, Science & Technology

Authors & Affiliations

Bo-Bo Wei^*

School of Physics and Energy, Shenzhen University, 518060 Shenzhen, China

^*bbwei@szu.edu.cn

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Vol. 97, Iss. 4 — April 2018

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Figure 1
Finite-size scaling of the first derivative of the two-body reduced-density-matrix element close to the quantum phase transition point in the LMG model at zero temperature $T = 0$ and $γ = 0.5$ . (a) The first derivative of the two-body reduced-density-matrix element with respect to control parameter $λ, \partial ρ_{11}^{i j} / \partial λ$ , in the LMG model as a function of control parameter $λ$ for different number of spins, $N$ . The solid black line is $N = 500$ , the dashed orange line is $N = 1000$ , the dotted magenta line is $N = 1500$ , and the dash-dotted blue line is $N = 2000$ . (b) Data collapse of the first derivative of the two-body reduced-density-matrix element in the LMG model shown in (a). According to scaling arguments, $\partial ρ_{11}^{i j} / \partial λ$ is a function of $N^{1 / ν} (λ - λ_{m})$ only with $λ_{m}$ being the position of the maximum of $\partial ρ_{11}^{i j} / \partial λ$ and $ν = 1.49$ being chosen so that data in (a) for different $N$ collapse perfectly. $\partial ρ_{11}^{i j} / \partial λ_{m}$ is a shorthand notation for $\partial ρ_{11}^{i j} {/ \partial λ |}_{λ = λ_{m}}$ .
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Figure 2
The first derivative of the two-body reduced-density-matrix element close to the quantum critical point in the LMG model at finite temperature. (a) The first derivative of the two-body reduced-density-matrix element with respect to control parameter $λ, \partial ρ_{11}^{i j} / \partial λ$ , in the LMG model at temperature $T = 0.2$ as a function of control parameter $λ$ for different number of spins, $N$ . The solid black line is $N = 150$ , the dashed orange line is $N = 200$ , the dotted magenta line is $N = 250$ , and the dash-dotted blue line is $N = 300$ . (b) The same as (a) but for a higher temperature, $T = 0.5$ .
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Figure 3
Scaling of the first derivative of the two-body reduced-density-matrix element in the quantum critical regime. (a) The first derivative of the two-body reduced-density-matrix element with respect to control parameter $λ$ at critical point $λ_{c} = 1, \partial ρ_{11}^{i j} / \partial λ_{c}$ , in the LMG model with $γ = 0.5$ as a function of temperature $T$ for different number of spins, $N$ . The solid black line is $N = 200$ , the dashed orange line is $N = 250$ , the dotted magenta line is $N = 300$ , and the dash-dotted blue line is $N = 400$ . (b) Data collapse of the first derivative of the two-body reduced-density-matrix element at low temperature in the LMG model. According to scaling arguments, $\partial ρ_{11}^{i j} / \partial λ_{c}$ is a function of $T N^{z}$ with $z$ being the dynamical critical exponent and $z = 0.32$ being chosen so that the data in (a) for different $N$ collapse perfectly at low temperature. $\partial ρ_{11}^{i j} / \partial λ_{c}$ is a shorthand notation for $\partial ρ_{11}^{i j} {/ \partial λ |}_{λ = λ_{c}}$ .
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Figure 4
Finite-size scaling of the first derivative of the two-body reduced density matrix close to the thermal phase-transition point in the LMG model with $λ = 0$ and $γ = 0.5$ . (a) The first derivative of the two-body reduced-density-matrix element with respect to temperature, $\partial ρ_{23}^{i j} / \partial T$ , in the LMG model as a function of temperature $T$ for different numbers of spins, $N$ . The solid black line is $N = 200$ , the dashed orange line is $N = 300$ , the dotted magenta line is $N = 400$ , and the dash-dotted blue line is $N = 500$ . Here we only show the real part of $ρ_{23}^{i j}$ because it is a complex number. (b) Data collapse of the first derivative of the two-body reduced-density-matrix element with respect to temperature in the LMG model shown in (a). According to scaling arguments, $\partial ρ_{23}^{i j} / \partial T$ is a function of $N^{1 / ν} (T - T_{m})$ only with $T_{m}$ being the position of the maximum of $\partial ρ_{23}^{i j} / \partial T$ and $ν = 2.01$ being chosen so that data in (a) for different $N$ collapse perfectly. $\partial ρ_{23}^{i j} / \partial T_{m}$ is a shorthand notation for $\partial ρ_{23}^{i j} {/ \partial T |}_{T = T_{m}}$ .
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