Extended quantum distance on thermo-field dynamics and its applications

Physica A 522 (2019) 1–8

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Physica A journal homepage: www.elsevier.com/locate/physa

Extended quantum distance on thermo-field dynamics and its applications ∗

Yoichiro Hashizume a , , Masuo Suzuki b , Takashi Nakajima a,c , Soichiro Okamura a a

Department of Applied Physics, Tokyo University of Science, 6-3-1 Niijuku, Katsushika, Tokyo, 125-8585, Japan Computational Astrophysics Laboratory, RIKEN, 2-1 Hirosawa, Wako, Saitama, 351-0198, Japan c PRESTO, Japan Science and Technology Agency, Kawaguchi, Saitama, 332-0012, Japan b

highlights • We show how to describe the difference between two thermodynamic states by thermo-field dynamics. • The difference between two thermodynamic states is well defined by thermodynamic state vectors. • The present method can be applied to non-equilibrium flux currents.

article

info

Article history: Received 12 July 2018 Received in revised form 29 October 2018 Available online 1 February 2019 Keywords: Thermo-field dynamics Difference between arbitrary two thermodynamic states Quantum distance/thermodynamic distance Phase transition Quantum–classical crossover Flux current

a b s t r a c t We investigated how to describe the difference between arbitrary two thermodynamic states by means of the thermo-field dynamics. Thermo-field dynamics plays a role of formulating to discuss statistical systems by focusing on the thermodynamic state itself. In this scheme, a thermodynamic state is expressed by a thermodynamic state vector. Thus, the difference between two thermodynamic states is well defined using the inner product of the relevant thermodynamic state vectors. In the present paper, we introduce the thermodynamic distance Dthermo to measure quantitatively the difference between the relevant two thermodynamic states, and we show some typical applications. As a result, we clarify that the concept of the thermodynamic distance includes a lot of information about statistical systems and it is helpful for us to obtain some characteristic parameters such as critical exponents or the quantum–classical crossover point. Moreover, we find that the present method can be applied to flux currents. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Thermo-field dynamics (TFD) [1–5] is a possible way to describe a thermodynamic state on a system by a ‘‘thermodynamic state vector’’. A thermodynamic state is a mixed state according to the Gibbs ensemble. This thermodynamic state is represented by a state vector defined in a ‘‘double Hilbert space’’ on the scheme of TFD [1–5]. As is briefly summarized in the succeeding section, the double Hilbert space is introduced by extending the original Hilbert space defined by an ordinal Hamiltonian. Using the thermodynamic state vector defined in the double Hilbert space, we can treat the thermodynamic state at finite temperatures as a state vector in the ‘‘extended’’ quantum mechanics. Since TFD enables us to focus on the ∗ Corresponding author. E-mail addresses: [email protected] (Y. Hashizume), [email protected] (M. Suzuki), [email protected] (T. Nakajima), [email protected] (S. Okamura). https://doi.org/10.1016/j.physa.2019.01.135 0378-4371/© 2019 Elsevier B.V. All rights reserved.

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Y. Hashizume, M. Suzuki, T. Nakajima et al. / Physica A 522 (2019) 1–8

thermodynamic state itself, it helps us to investigate physics at finite temperatures. Actually, this feature of TFD was useful for extending the search algorithm of ground states such as the density matrix renormalization group method (DMRG) [6–8] or multiscale entanglement renormalization ansatz (MERA) [9] to finite temperatures. For another example, we have also clarified the thermodynamic properties of quantum entanglement [10,11]. In particular, as shown by one of the authors (MS) in 1986 [12], the above advantage of TFD (which focuses on the state itself directly) was used for the study on the resonated valence bond state (RVB state). This previous study [12] was helpful to explain experimental results [13,14]. As shown in the above discussions, TFD is suitable for describing the thermodynamic state at finite temperatures. On the other hand, the discussion of state transitions is still not studied well in the concept of TFD. However, understanding the state transitions recently becomes more and more important to clarify non-equilibrium statistical physics. Therefore, in the present research, we introduce a simple indicator, namely thermodynamic distance, in order to measure the difference between two arbitrary thermodynamic state vectors. To achieve this purpose, we refer to the quantum distance and fidelity which are well known measures to compare two quantum states at zero temperature. Here, we extend the quantum distance for finite-temperature systems using TFD, and we call it a ‘‘thermodynamic distance’’. Furthermore, we apply the thermodynamic distance to some typical cases. Here, we mention the definition of ‘‘thermodynamic state’’. The recent progress on thermalization processes of quantum system is highly developed. In their discussion, mainly on the bases of the eigenstate thermalization hypothesis (ETH) [15], the system is isolated from the heat bath, that is, the statistical ensemble is assumed as micro-canonical. On the other hand, the ‘‘thermodynamic state’’ described by thermo-field dynamics (TFD) is based on the thermal mixed state (the mixing structure is controlled by the tilde space), and its equilibrium state is assumed to be a canonical ensemble. In the present stage, it is still not clear how the TFD formalism corresponds to the micro-canonical thermalization process, though a similar form of thermodynamic pure state is obtained in the previous studies [16,17]. Thus, in the present discussion, the ‘‘thermalization process’’ from a quantum pure state is not considered. In the succeeding section, we give a brief summary of the TFD as well as the concept of the quantum distance, and that of an entropy operator to use the following sections. In Section 3, we introduce the thermodynamic distance in a general form on the TFD. The applications of the thermodynamic distance are shown in Section 4 in order to show that the thermodynamic distance includes a lot of information about state transitions along a quasi-equilibrium thermodynamic process in addition to a non-equilibrium current effect. Summary and discussions are given in Section 5. 2. Introduction to thermo-field dynamics and useful parameters to describe the difference between thermodynamic states Here, we make a brief summary about thermo-field dynamics in the present notation. Furthermore, two useful parameters to describe the difference between arbitrary two thermodynamic states, namely ‘‘quantum distance’’ and ‘‘entropy operator’’, are also introduced in the present formulation. 2.1. Thermo-field dynamics On thermo-field dynamics (TFD) formalisms [1–5], a thermodynamic state is described by a state vector |Ψ ⟩. This thermodynamic state vector |Ψ ⟩ is introduced in the double Hilbert space defined by the direct product of the original Hilbert space and its isomorphic space (tilde space). In particular, when the original Hilbert space is defined by a complete set of eigenstates {|n⟩} (n = 1, 2, . . . ) of the Hamiltonian H, the tilde space is also defined by the bases {|˜n⟩} (n = 1, 2, . . . ), whose space is isomorphic to the original space. The bases of the double Hilbert space are hence shown as {|n⟩ ⊗ |˜n⟩} ≡ {|n, n˜ ⟩} = {|ˆn⟩}. On the above double Hilbert space, the thermodynamic state vector |Ψ ⟩ is defined as

|Ψ ⟩ = ρ 1/2 |I ⟩,

(1)

where

|I ⟩ =

∑ |n, n˜ ⟩,

(2)

n

and the parameter ρ denotes the density matrix. For example, the density matrix ρ corresponds to the Gibbs factors. Thus, we have ρ = e−β H /Z in equilibrium with the partition function Z . Here and after, the parameter β denotes the Boltzmann’s inverse temperature 1/kB T at the temperature T using the Boltzmann constant kB . On the other hand, when we consider a non-equilibrium state, the density matrix ρ is obtained by the von Neumann equation [18] ih ¯

∂ ρ (t) = [H, ρ (t)], ∂t

(3)

or the dissipative von Neumann equation [19–22] ih ¯

∂ ρ (t) = [H, ρ (t)] + (dissipation terms). ∂t

(4)

On the bases of the above non-equilibrium density matrix, we can treat non-equilibrium systems on TFD [23–29]. Note that the density matrix ρ (as well as the original Hamiltonian H) operates just to the vector elements on the original Hilbert space.

Y. Hashizume, M. Suzuki, T. Nakajima et al. / Physica A 522 (2019) 1–8

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In Eqs. (1) and (2), we use a complete set of eigenstates {|n⟩} (n = 1, 2, . . . ) of the Hamiltonian H. However, by the general representation theorem [30–32], we can choose any complete set of the original Hilbert space, that is, the thermodynamic state vector |Ψ ⟩ itself does not depend on the bases. Thus, we define the thermodynamic state vector |Ψ ⟩ as

|Ψ ⟩ = ρ 1/2

∑ |α, α⟩ ˜

(5)

α

using any complete set {|α⟩} of the original Hilbert space [30]. The thermodynamic expectation value (thermal average) ⟨Q ⟩ of a physical quantity Q is obtained as

⟨Q ⟩ = ⟨Ψ |Q |Ψ ⟩.

(6)

Here, we show a proof of the relation (6), which may be helpful for understanding the following sections. The thermal average ⟨Q ⟩ is defined in the statistical physics as ⟨Q ⟩ = Trρ Q . On the other hand, the right hand side of Eq. (6) is derived as

⟨Ψ |Q |Ψ ⟩ =

∑ ∑ 1/2 ⟨α|⟨α|ρ ˜ Q ρ 1/2 |α ′ ⟩|α˜ ′ ⟩ = ⟨α|ρ 1/2 Q ρ 1/2 |α ′ ⟩⟨α| ˜ α˜ ′ ⟩ α,α ′

α,α ′

∑ ∑ = ⟨α|ρ 1/2 Q ρ 1/2 |α ′ ⟩δα,α′ = ⟨α|ρ 1/2 Q ρ 1/2 |α⟩ α

α,α ′

= Trρ

1/2

Qρ

1/2

= Trρ Q

(7)

Thus, the relation (6) is proved [30]. Of course, substituting Q = 1, we can obtain ⟨Ψ |Ψ ⟩ = Trρ . As shown in the above derivation, operators defined in the original space do not operate to elements on the tilde space. This situation is more ˜ where 1˜ denotes the unit clearly shown by such mathematical descriptions of the operators Q and H as Q ⊗ 1˜ and H ⊗ 1, operator of the tilde space. 2.2. Quantum distance Quantum distance is introduced to clarify the difference of two quantum states at zero temperature [33]. In the scheme of quantum mechanics, a quantum state is described by a state vector |ψ⟩. This state vector depends on a mechanical parameter x, such as system size, potential scale, external fields, etc. Once the parameter changes as x → x + dx, the state vector |ψ (x)⟩ changes into |ψ (x + dx)⟩. Then, the difference between the two quantum states |ψ (x)⟩ and |ψ (x + dx)⟩ is characterized by the inner product F ≡ |⟨ψ (x)|ψ (x + dx)⟩|. This parameter F is called the ‘‘fidelity’’ [34,35]. In many cases, the quantum distance Dq is defined as Dq = 1 − |⟨ψ (x)|ψ (x + dx)⟩|2

(8)

using the fidelity. In this definition, when the state does not change, the distance takes the minimum value 0. On the other hand, it takes the maximum value 1 for two completely different states satisfying the condition F = 0. This quantum difference plays a role of a useful tool to compare two quantum states. In the present study, we extend the definition of the distance Dq between two thermodynamic states, and this extended distance enables us to compare the thermodynamic states as shown in Section 3. A similar distance has been given by Bures, Helstrom and Wootters (which is called as ‘‘Bures distance’’ or ‘‘Helstrom distance’’ in the area of quantum information geometry) [36–38] by using the quantum density matrix, whose definition is given as

(

DBures = 2 1 −

√

)

Trρ 1/2 (x)ρ (x + dx)ρ 1/2 (x) .

(9)

We can extend the Bures distance to finite temperature systems in the same way as shown in the following sections. However, their definition is very complicated to treat typical statistical systems. Then we mainly consider the definition (8) of Dq as the quantum distance. 2.3. Entropy operator An entropy operator is also useful for the present study. The entropy operator S is defined [22] using density matrix ρ as S = −kB log ρ.

(10)

In equilibrium cases, the density matrix ρ is described as ρ = exp[−β (H − F )], where F denotes the free energy. Thus, the entropy operator S (eq) for equilibrium systems is obtained [22] as H−F

. (11) T This notation includes the Hamiltonian H directly, and then this is still an operator, where F and T are c-number parameters. The thermal average of the entropy operator yields easily the entropy S as S (eq) =

S = ⟨S ⟩ = −kB ⟨log ρ⟩.

(12)

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3. Generalized difference between two thermodynamic states As shown in the previous section, the thermodynamic state vector |Ψ ⟩ describes the thermodynamic state directly even in a finite temperature state. On the bases of this feature, we try to describe the difference of states on a finite temperature in the scheme of TFD. For this purpose, we introduce the ‘‘thermodynamic distance’’ (which corresponds to the extended quantum distance) as Dthermo = 1 − |⟨Ψ (x)|Ψ (x + dx)⟩|2 .

(13)

Here, the parameter x denotes not only a dynamical parameter included in the Hamiltonian (such as external fields) but also a thermodynamic parameter such as temperature. When we apply the thermodynamic distance Dthermo to the zero temperature condition, it is equal to the quantum distance Dq , i.e., Dthermo (T = 0) = Dq . The thermodynamic state vector |Ψ (x)⟩ is obtained as

|Ψ (x)⟩ =

∑

ρ 1/2 (x)|α, α⟩ ˜ =

∑

α

e−S (x)/2kB |α, α⟩ ˜

(14)

α

using Eqs. (5) and (10). The parameter x is included in the density matrix ρ (x) or the entropy operator S (x), while the bases {|α, α⟩} ˜ do not depend on it. Then, when the parameter x changes into x + dx, the thermodynamic state vector changes as |Ψ (x)⟩ → |Ψ (x + dx)⟩. According to this scheme, we expand the |Ψ (x + dx)⟩ up to the second order with respect to dx as

|Ψ (x + dx)⟩ = |Ψ (x)⟩ +

1 ∂ 2 |Ψ (x)⟩ 2 ∂|Ψ (x)⟩ dx + dx . ∂x 2 ∂ x2

(15)

Thus, using the Eqs. (14) and (15), the thermodynamic distance is obtained as

(

2

])

Dthermo = −2Re ⟨Ψ (x)|Ψ ′ (x)⟩ dx − |⟨Ψ (x)|Ψ ′ (x)⟩| + Re ⟨Ψ (x)|Ψ ′′ (x)⟩

[

]

[

dx2 ,

(16)

where |Ψ ′ (x)⟩ = ∂|Ψ (x)⟩/∂ x and |Ψ ′′ (x)⟩ = ∂ 2 |Ψ (x)⟩/∂ x2 . Here Re[. . . ] denotes the real part of (. . . ). This general thermodynamic distance expresses the difference between the thermodynamic states |Ψ (x)⟩ and |Ψ (x + dx)⟩. When we treat quasi-equilibrium processes, the density matrix satisfies the normalization relation ⟨Ψ (x)|Ψ (x)⟩ = Trρ (x) = 1. Then, the differential of the normalization relation leads to the condition Re ⟨Ψ (x)|Ψ ′ (x)⟩ = 0.

[

]

(17)

On the other hand, using the representation Eq. (14), |Ψ (x)⟩ can be described as ′

|Ψ ′ (x)⟩ = −

1

∫ 1 ∑ 2kB

dλe(λ−1)S (x)/2kB

0

α

∂ S (x) −λS (x)/2kB e |α, α⟩. ˜ ∂x

(18)

Then, the term ⟨Ψ (x)|Ψ ′ (x)⟩ yields

⟨Ψ (x)|Ψ (x)⟩ = − ′

⟨

1 2kB

⟩ ∂ S (x) . ∂x

(19)

As is clearly shown in Eq. (19), the term ⟨Ψ (x)|Ψ ′ (x)⟩ is a real number under quasi-equilibrium conditions. Thus, from Eqs. (17) and (19), we can conclude that ⟨Ψ (x)|Ψ ′ (x)⟩ = 0, that is, the thermodynamic distance is obtained as (eq)

Dthermo = ⟨Ψ ′ (x)|Ψ ′ (x)⟩dx2 ,

(20)

because one more differential of Eq. (17) yields Re ⟨Ψ (x)|Ψ ′′ (x)⟩ = −⟨Ψ ′ (x)|Ψ ′ (x)⟩.

[

]

(21) (eq)

Using the entropy operator, the thermodynamic distance Dthermo is also given by the canonical correlation of S ′ ≡ ∂ S (x)/∂ x as (eq)

Dthermo =

1 4kB

Trρ (x)

1

β

β

∫ 0

dλeλH S ′ e−λH S ′ dx2 ≡

1 ⟨ 4kB

′ 2 S ′ (−ih ¯ λ); S dx .

⟩

(22)

These representations of thermodynamic distance in Eqs. (16) and (22) can be applied to some typical systems, and they include a lot of information of the state transition |Ψ (x)⟩ → |Ψ (x + dx)⟩. Some applications of the above formulation are shown in the following section. 4. Typical applications In the present section, we show some typical applications of the thermodynamic distance Dthermo to phase transitions, quantum–classical crossover, and non-equilibrium flux currents.

Y. Hashizume, M. Suzuki, T. Nakajima et al. / Physica A 522 (2019) 1–8

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4.1. Phase transitions In this subsection, we treat a second order phase transition. Here we study the critical exponent α of the specific heat C (T ), i.e., C (T ) ∝ (T − Tc )−α , where T and Tc denote temperature and the critical temperature, respectively. We compare the two thermodynamic states of the present system at the temperatures T and T + ∆T for small ∆T . The thermodynamic distance of these two states are defined as Dthermo (T , ∆T ) = 1 − |⟨Ψ (T )|Ψ (T + ∆T )⟩|2 .

(23)

Then, we obtain 1 ⟨

(eq)

Dthermo (T , ∆T ) = Dthermo (T , ∆T ) =

4k2B

2 ′ S ′ (−ih ¯ λ); S ∆T ,

⟩

(24)

using the representation (22), where S ′ = ∂ S /∂ T . Furthermore, the entropy operator in the equilibrium satisfies Eq. (11). Finally, the thermodynamic distance Dthermo (T , ∆T ) of this system is derived as Dthermo (T , ∆T ) =

= =

∆T 2

1

∫

4k2B

dλTrρ eβλH

(

H − ⟨H ⟩

T2

0

∆T

2

4k2B T 4 2

Trρ (H − ⟨H⟩)2 =

∆T C (T ) 4kB T 2

)

e−βλH

(

H − ⟨H ⟩

)

T2

∆T ⟨(H − ⟨H⟩) ⟩ 2

2

4k2B T 4

,

(25)

where we have used the relations C (T ) = ⟨(H − ⟨H⟩)2 ⟩/kB T 2 and [exp(βλH), H] = 0. Thus, near the critical temperature T = Tc + δ T , the thermodynamic distance is reduced to Dthermo (T , ∆T , δ T ) =

∆T 2 C (Tc + δ T ) 4kB Tc2 (1 + δ T /Tc )2

.

(26)

Generally, the temperature difference ∆T can be assumed as ∆T = δ T ≃ 0. Then the distance Dthermo is reduced as Dthermo (T , ∆T , ∆T ) ∝ (∆T )2−α .

(27)

Of course, in the limit ∆T → 0, the thermodynamic distance vanishes, because the state |Ψ (Tc + ∆T )⟩ approaches |Ψ (Tc )⟩. This vanishing process depends on the critical exponent α , as shown in the above formula (27). 4.2. Quantum–classical crossover As the second example, we apply the thermodynamic distance to detect the quantum–classical crossover point. In 1986, one of the authors (MS) investigated the resonated valence bond (RVB) state in a finite size frustrated system [12]. The Hamiltonian is assumed as H = −J(σ 1 · σ 2 + σ 2 · σ 3 + σ 3 · σ 1 )

(28)

with Heisenberg spins σ 1 , σ 2 , and σ 3 whose spins are 1/2. In the present subsection, the spin state (up/down) of σ j is denoted as |σj ⟩ = {|+⟩j or |−⟩j }. The interaction J is assumed to be negative for the anti-ferro magnetic model. From the previous study, the thermodynamic state vector at the temperature T is obtained by TFD [12] as 1

[

(

3|K |

)

]

|RVB⟩ ,

(29)

˜ 1 |+, +⟩ ˜ 2 |+, +⟩ ˜ 3 + |−, −⟩ ˜ 1 |+, +⟩ ˜ 2 |+, +⟩ ˜ 3 + ··· |I ⟩ = |+, +⟩ ∑ ∑ ∑ = |σ1 , σ˜ 1 ⟩|σ2 , σ˜ 2 ⟩|σ3 , σ˜ 3 ⟩,

(30)

|Ψ (T )⟩ = √

2 2 cosh 3K

−3|K |/2

e

|I ⟩ + 4 sinh

2

where K , |I ⟩, and |RVB⟩ are defined as K = −β|J |,

σ1 =±1 σ2 =±1 σ3 =±1

and

|RVB⟩ =

1( 3

) |ˆs⟩12 |I ⟩3 + |ˆs⟩23 |I ⟩1 + |ˆs⟩31 |I ⟩2 ,

(31)

respectively. In Eq. (31), |ˆs⟩ij denotes the singlet pair between σi and σj in the double Hilbert space as 1

|ˆs⟩ij ≡ |s⟩ij |˜s⟩ij and |s⟩ij = √ (|+, −⟩ij − |−, +⟩ij ), 2

(32)

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Y. Hashizume, M. Suzuki, T. Nakajima et al. / Physica A 522 (2019) 1–8

while |I ⟩k (k ̸ = i, j) denotes a not-entangled spin state as

˜ k + |−, −⟩ ˜ k. |I ⟩k = |+, +⟩

(33)

The state |RVB⟩ shows that the singlet pairs are resonated at the ground state, that is, the state vector |Ψ (T )⟩ → |RVB⟩ in the limit T → 0. This state |RVB⟩ is essentially a quantum state, because it is constructed by entangled states. On the other hand, for higher temperatures, spins takes random states. We classify this higher temperature state as a ‘‘classical state’’. So, there is a quantum–classical crossover point at a finite temperature. As is easily predicted from the above discussions, the state behavior drastically changes around the crossover point, because the |RVB⟩ character is dominant in the lower temperature regions while the classical state is dominant in higher temperature regions. Thus, the difference between |Ψ (T )⟩ and |Ψ (T + ∆T )⟩ will become a maximum value near the crossover point for the small constant ∆T . For these motivations, let us consider the thermodynamic distance described by Eq. (20) for the state vector (29), and let us detect the crossover point using the thermodynamic distance. As a result, we can easily obtain the thermodynamic distance as (eq)

Dthermo (T , ∆T ) = Dthermo (T , ∆T ) =

9k2B

K4

4J 2

cosh2 3K

∆T 2 .

(34)

Thus, the difference of thermodynamic state, Dthermo (T , ∆T ), for fixed small ∆T becomes maximum at T ≡ Tq/c ≃ 1.5|J |/kB which should be the crossover point. This estimated crossover point Tq/c is relevant to the previous studies [12–14]. As shown in the above two examples (phase transition case and quantum–classical crossover case), the thermodynamic distance yields a lot of information of the thermodynamic state, and it is useful to compare the states along a parameter even in the case of finite temperatures. Furthermore, because of this feature of thermodynamic distance, we can detect such a characteristic condition or parameter of an equilibrium system as the critical exponent or the crossover point. 4.3. Non-dissipative flux current In the present subsection, we discuss a system with the non-dissipative flux current J(x). When the non-dissipative flux current exists in a system, the phase factors {θα (x)} play an important role on the thermodynamic state. The corresponding thermodynamic state vector is described as

|Ψ (x)⟩ = e−S (x)/2kB

∑

eiθα (x) |α, α⟩. ˜

(35)

α

Here we assume that the entropy operator and the phase factor depend on the controllable mechanical parameter x such as the system size, an external field, etc. This kind of phase induced current appears typically in discussing non-equilibrium spin systems [39,40]. When we change the parameter x into x + dx, the effective non-dissipative current J(x) appears between the states |Ψ (x)⟩ and |Ψ (x + dx)⟩, owing to the change of phases {θα (x)}. Another simple example is given by setting the parameter x as a position. When x denotes the position variable, the thermodynamic distance Dthermo corresponds to the difference between the two thermodynamic states at the positions x and x + dx. Using the above thermodynamic state (35), the flux current J(x) is actually obtained as J(x) =

1 (

) ⟨Ψ (x)|Ψ ′ (x)⟩ − ⟨Ψ ′ (x)|Ψ (x)⟩ = ⟨θα′ (x)⟩,

2i

(36)

where ⟨θα′ (x)⟩ is defined as

⟨θα′ (x)⟩ ≡

∑

⟨α|ρ (x)

α

∂θα (x) |α⟩. ∂x

(37)

Thus, the non-dissipative current J(x) is actually given by the phase factors. In the present case, the relations ⟨Ψ (x)|Ψ (x)⟩ = 1, Re[⟨Ψ (x)|Ψ ′ (x)⟩] = 0, and Re[⟨Ψ (x)|Ψ ′′ (x)⟩] = −⟨Ψ ′ (x)|Ψ ′ (x)⟩ still hold. However, ⟨Ψ (x)|Ψ ′ (x)⟩ does not vanish, that is, ⟨Ψ (x)|Ψ ′ (x)⟩ = iJ(x) (̸ = 0). Considering these relations, we obtain the thermodynamic distance using Eq. (16), as

{ Dthermo =

1

⟨S (λ); S ⟩ + ∆J ′

4k2B

′

2

}

dx2 ,

(38)

where ⟨S ′ (λ); S ′ ⟩ and ∆J denote

⟨S ′ (λ); S ′ ⟩ =

1

∫

dλρ (x)eλS (x)/kB S ′ (x)e−λS (x)/kB S ′ (x),

(39)

0

and

∆J = 2

∑ α

∂θα (x) ⟨α|ρ (x) ∂x

(

≡ ⟨θα′ (x)2 ⟩ − ⟨θα′ (x)⟩2 ,

)2

(

∑ ∂θα (x) |α⟩ − ⟨α|ρ (x) |α⟩ ∂x α

)2

(40)

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respectively. According to Eq. (38), we find that the two thermodynamic states |Ψ (x)⟩ and |Ψ (x + dx)⟩ are different form each other, depending on the fluctuation of entropy (the first term of Eq. (38)) and the fluctuation of the flux current (the second term of Eq. (38)). These results seem to be reasonable intuitively. Furthermore, on the bases of the thermodynamic difference as shown in Eq. (38), it is suggested that the original effects (such as fluctuations of the entropy and the current) of the difference between two thermodynamic states will be clarified by the thermodynamic distance as shown in the above discussions. So it will be useful to study the thermodynamic states and non-equilibrium systems, using the present scheme. 5. Summary and discussions In the present study, we have extended the quantum distance so as to apply it to finite temperature states on the bases of the thermo-field dynamics. The extended quantum distance which we call ‘‘thermodynamic distance’’ has been applied to some typical systems. Thus, we have clarified that the characteristic parameters of such systems at finite temperatures can be obtained using the thermodynamic distance. Especially, we have shown that the thermodynamic distance gives such typical properties as critical exponents and quantum–classical crossover point in equilibrium systems. Furthermore, the present method can also be applied to non-equilibrium systems with non-dissipative currents. One of the new important results is that the thermodynamic distance is separable for the fluctuation of the entropy and that of currents. When the system includes dissipative currents, we predict the cross term of these fluctuations (fluctuations of entropy and current) will appear in the thermodynamic distance. The present method to investigate the difference of states using the thermodynamic distance Dthermo can be applied to many kinds of non-equilibrium systems. Of course, we can compare two different states by thermodynamic observables ⟨Q ⟩ of the parameter Q such as internal energy, entropy, etc. However, if we apply such observables to the measure of states, it depends on the inevitable features of the observable Q . For example, the entropy becomes larger, and the free energy becomes smaller near the equilibrium. In the present method focusing on the TFD state vector |Ψ ⟩ itself, we can treat the thermodynamic distance systematically. This point of view may be helpful for further discussions to investigate non-equilibrium systems or more complicated equilibrium systems. By the way, we discuss here the relation between the present method and the thermodynamic information geometry [41– 45]. The present thermodynamic distance can be extended to higher-dimensional parameter space. In higher-dimensional parameter space, we set the relevant parameters as {xµ } (µ = 1, 2, . . . ) and the corresponding thermodynamic state as |Ψ ({xµ })⟩. Expanding the extended thermodynamic state up to the second order with respect to x1 , x2 , x3 , . . . , we define the thermodynamic distance as Dthermo = 1 − |⟨Ψ ({xµ })|Ψ ({xµ + dxµ })⟩|2 . Then, for quasi-equilibrium processes described by the condition ⟨Ψ ({xµ })|∂µ Ψ ({xµ + dxµ })⟩ = 0, we can obtain the relation Dthermo =

1 4kB

⟨∂µ ∂ν S ⟩dxµ dxν ≡

1 4

gµν dxµ dxν ;

gµν =

1 kB

⟨∂µ ∂ν S ⟩

(41)

using Einstein’s summation convention in which repeated indices are implicitly summed over. As is well known, the information geometry [41] introduced by Amari et al. can be applied to the thermodynamics based on the above metric tensor gµν [42–44]. Thus, present research also shows that the TFD concept derives the information geometry applied to equilibrium thermodynamics. Especially, as shown in Eq. (41), our present thermodynamic distance corresponds to the one-dimensional case of information geometry applied to the thermodynamics. However, through the present study, it is suggested that the thermodynamic distance based on the thermodynamic state |Ψ ⟩ can be more widely used even for non-equilibrium systems, because the TFD method can be applied to non-equilibrium systems directly as discussed in Section 2. The above point of view strongly supports the recent study by Dimov et al. [45]. They treated an information geometry defined by an extended entanglement entropy on boson systems using TFD. Furthermore, they clarified time-dependent behaviors of the extended entanglement entropy on boson systems. Their study gives an example to show the relation between TFD and information geometry on non-equilibrium systems as discussed above. Acknowledgment One of the authors (Y.H.) is partially supported by JSPS, Japan KAKENHI Grant Number 17K14357 and 26800205. References [1] [2] [3] [4] [5] [6] [7]

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