1. Introduction
The Ising model is one of the simplest models in statistical physics [1, 2]. For the one-dimensional Ising model, the ground state is either a ferromagnetic order or an antiferromagnetic one [3]. At finite temperatures, due to thermal fluctuations, the long-range order disappears. When considering the transverse external field, we have a transverse Ising model [4–8]. Now, quantum phase transition caused by quantum fluctuations occurs between the long-range order and the spin-polarized state [9–12]. Near the quantum critical point, it manifests as a nonanalytical behavior in the ground state energy of the system. These greatly impact the understanding of the thermodynamic and dynamic properties of many condensed-matter systems.
At the same time, in recent years, the emergence of non-Hermitian physics has opened up a new direction for the study of physics. It refers to that the Dirac Hermitian conjugation of the Hamiltonian in the non-Hermitian systems is not itself [13]. This differs from the Hermitian system which has the conservation of probability and the real-valuedness of the expected value of energy versus a quantum state. In contrast, its probability is non-conservative due to the loss of energy and particles and information to external environments and the expected value of energy versus a quantum state is complex [14]. These characteristics lead to non-Hermitian systems exhibiting a great deal of astonishing physical phenomena, such as biorthogonality of eigenstates [15], exceptional points [16, 17], parity-time symmetry [18, 19], and pseudo-Hermiticity [20, 21]. As a result, significant progress in non-Hermitian quantum systems has been made in theoretical and experimental aspects [22–33], including non-Bloch bulk-boundary correspondence [34–40] and non-Hermitian skin effect [41–43]. However, the properties of non-Hermitian quantum systems at finite temperatures have not been fully investigated so far.
In the paper, we explore the physical properties of the one-dimensional non-Hermitian Ising model at finite temperature according to the quantum Liouvillian statistical theory. Based on the Liouvillian Hamiltonian of the system, we find that there exists a 'pseudo-phase transition' between the 'topological' phase and the 'non-topological' phase, which is accompanied by the Liouvillian energy gap closed rather than the usual energy gap closed. In addition, we point out that the one-dimensional non-Hermitian Ising model at finite temperature can be equivalent to an effective anisotropic XY model in the transverse field. For the case of a strong non-Hermitian strength limit, the system becomes an effective isotropic XY model in the transverse field. In the high-temperature limit, the system appears the non-thermalization, which is different from the Hermitian system. This work can provide a reference for the study of non-Hermitian systems at finite temperatures.
The outline of this paper is as follows: In section 2 , we show the quantum Liouvillian statistical theory for non-Hermitian quantum systems at finite temperatures. In section 3 , we study the one-dimensional non-Hermitian Ising model at finite temperature and give its analytical solutions. For its Liouvillian Hamiltonian, the 'pseudo-phase transition' is explored between the 'topological' phase and the 'non-topological' phase. Furthermore, we find the one-dimensional non-Hermitian Ising model at finite temperature can be equivalent to an effective anisotropic XY model in the transverse field. In this model, we discuss its thermodynamic properties in Hermitian and non-Hermitian cases, and in the zero-temperature and high-temperature limits, respectively. In section 4 , we draw the conclusion.
2. Quantum Liouvillian statistical theory
To study the properties of non-Hermitian systems at finite temperatures, the quantum Liouvillian statistical theory introduces the definition of the non-Hermitian thermal state. It indicates that a non-Hermitian system at finite temperature can evolve over time from any initial state to a unique state, which is called a non-Hermitian thermal state [44]. Such a non-Hermitian thermal state dominates the physical properties of the system.
For non-Hermitian systems in which the eigenvalues of its non-Hermitian Hamiltonian ${\hat{H}}_{\mathrm{NH}}$ are all real, people always call it a quasi-Hermitian system. For the quasi-Hermitian system, the density matrix in the non-Hermitian thermal state is [44]
$\begin{eqnarray}{\rho }_{\mathrm{NHTS}}^{S}=\hat{S}{\rho }_{0}{\hat{S}}^{\dagger },\end{eqnarray}$
where $\hat{S}$ is a similar transformation which can transform non-Hermitian Hamiltonian ${\hat{H}}_{\mathrm{NH}}$ into the Hermitian Hamiltonian ${\hat{H}}_{0}$ by $\begin{eqnarray*}{\hat{H}}_{0}={\hat{S}}^{-1}{\hat{H}}_{\mathrm{NH}}\hat{S}.\end{eqnarray*}$
The energy levels En of ${\hat{H}}_{\mathrm{NH}}$ (or ${\hat{H}}_{\mathrm{NH}}^{\dagger }$) are the same as those of ${\hat{H}}_{0}$. ${\rho }_{0}={{\rm{e}}}^{-{\beta }_{T}{\hat{H}}_{0}}$ is the density matrix of the Hermitian model ${\hat{H}}_{0}$ at the thermodynamic equilibrium, where βT is the inverse temperature. As a result of $\hat{S}{\rho }_{0}{\hat{S}}^{\dagger }={{\rm{e}}}^{-{\beta }_{T}{\hat{H}}_{L}}$, this type of non-Hermitian system at finite temperature can be expressed by the Liouvillian Hamiltonian ${\hat{H}}_{L}$ $\begin{eqnarray}{\hat{H}}_{L}=-\displaystyle \frac{1}{{\beta }_{T}}\mathrm{ln}[\hat{S}({{\rm{e}}}^{-{\beta }_{T}{\hat{H}}_{0}}){\hat{S}}^{\dagger }].\end{eqnarray}$
Its corresponding eigenvalue EnL and eigenstate are the Liouvillian energy level and Liouvillian state, respectively. It should be pointed out that ${\hat{H}}_{L}={\hat{H}}_{L}^{\dagger }$ and EnL must be real.In particular, there is non-thermalization at high temperature. Because when the temperature is high, βT → 0, the density matrix ${\rho }_{\mathrm{NHTS}}^{S}$ for a non-Hermitian system with all real spectra at the non-Hermitian thermal state is reduced to
$\begin{eqnarray}{\rho }_{\mathrm{NHTS}}^{S}\sim \hat{S}{\hat{S}}^{\dagger }\ne \hat{I}.\end{eqnarray}$
As a result, the weights of microscopic quantum states are not the same and the system is not thermalized. This is very different from that of the usual Hermitian model, in which the density matrix at the thermodynamic equilibrium in high temperature is ${{\rm{e}}}^{-{\beta }_{T}{\hat{H}}_{0}}\sim \hat{I}$. This means the weights of all microscopic quantum states of the Hermitian systems are the same and the system exists the thermalization effect.3. One-dimensional non-Hermitian Ising model in the transverse field
According to the above theory, we calculate the Liouvillian Hamiltonian of the one-dimensional non-Hermitian Ising model at finite temperature in the case of all real eigenvalues of the non-Hermitian Hamiltonian. Followed by it, we study its 'topological' properties and thermodynamic characteristics in detail.
3.1. The Jordan–Wigner transformation and the Liouvillian Hamiltonian
We consider a one-dimensional non-Hermitian Ising model, in which its Hamiltonian ${\hat{H}}_{\mathrm{NHIS}}$ can be written as [45]
$\begin{eqnarray}{\hat{H}}_{\mathrm{NHIS}}=-J\displaystyle \sum _{j}^{N}{\sigma }_{j}^{x,{\beta }_{\mathrm{NH}}}{\sigma }_{j+1}^{x,{\beta }_{\mathrm{NH}}},\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{l}{\sigma }_{j}^{x,{\beta }_{\mathrm{NH}}}=\hat{S}{\sigma }^{x}{\hat{S}}^{-1}=\cosh ({\beta }_{\mathrm{NH}})\cdot {\sigma }_{j}^{x}\\ \,+\,\mathrm{isinh}({\beta }_{\mathrm{NH}})\cdot {\sigma }_{j}^{y}.\end{array}\end{eqnarray}$
J > 0 is the coupling between the neighboring spins. This model can be transformed into the Hermitian Ising model $\begin{eqnarray}{\hat{H}}_{0}=-J\displaystyle \sum _{j}^{N}{\sigma }_{j}^{x}{\sigma }_{j+1}^{x},\end{eqnarray}$
under the similar transformation $\hat{S}={{\rm{e}}}^{{\beta }_{\mathrm{NH}}{\hat{H}}^{{\prime} }}$, where ${\hat{H}}^{{\prime} }=\tfrac{1}{2}{\sum }_{j}{\sigma }_{j}^{z}$ and βNH denotes the non-Hermitian strength. Due to the non-Hermitian similarity, the energy levels for ${\hat{H}}_{\mathrm{NHIS}}$ and ${\hat{H}}_{0}$ are the same. In this paper, we set J = 1.It is well known that the one-dimensional Hermitian Ising model can be mapped to the superconducting states by the Jordan–Wigner transformation to explore the corresponding many-body physics. For the Hermitian model ${\hat{H}}_{0}=-J{\sum }_{j}^{N}{\sigma }_{j}^{x}{\sigma }_{j+1}^{x}$, the Jordan–Wigner transformation is defined by the string-like creation and annihilation operators
$\begin{eqnarray}{c}_{j}^{\dagger }={\sigma }_{j}^{\dagger }\displaystyle \prod _{l\lt j}(-{\sigma }_{l}^{z}),\quad {c}_{j}=\displaystyle \prod _{l\lt j}(-{\sigma }_{l}^{z}){\sigma }_{j}^{-}.\end{eqnarray}$
So the fermion number operator becomes $\begin{eqnarray}{\hat{n}}_{j}={c}_{j}^{\dagger }{c}_{j}=\displaystyle \frac{1}{2}(1+{\sigma }_{j}^{z}).\end{eqnarray}$
Under Jordan–Wigner transformation, the one-dimensional Hermitian Ising model becomes a one-dimensional Kitaev model with an imbalanced p-wave superconductor paring [46–48] $\begin{eqnarray}{\hat{H}}_{0}=-J\displaystyle \sum _{j=1}^{N}\left({c}_{j}^{\dagger }{c}_{j+1}+{c}_{j+1}^{\dagger }{c}_{j}+{c}_{j}^{\dagger }{c}_{j+1}^{\dagger }+{c}_{j+1}{c}_{j}\right).\end{eqnarray}$
Based on the Fourier transform ${c}_{j}=\tfrac{1}{\sqrt{N}}{\sum }_{k}{c}_{k}{{\rm{e}}}^{{\rm{i}}{{kR}}_{j}}$ and ${c}_{j}^{\dagger }=\tfrac{1}{\sqrt{N}}{\sum }_{k}{c}_{k}^{\dagger }{{\rm{e}}}^{-{\rm{i}}{{kR}}_{j}}$, we can rewrite the fermion Hamiltonian ${\hat{H}}_{0}$ as $\begin{eqnarray}{\hat{H}}_{0}=\displaystyle \sum _{k\gt 0}{\psi }_{k}^{\dagger }{\hat{h}}_{0}(k){\psi }_{k},\end{eqnarray}$
in momentum space by introducing ${\psi }_{k}=\left(\begin{array}{c}{c}_{k}\\ {c}_{-k}^{\dagger }\end{array}\right)$ and ${\psi }_{k}^{\dagger }=\left(\begin{array}{cc}{c}_{k}^{\dagger } & {c}_{-k}\end{array}\right)$, where $\begin{eqnarray}{\hat{h}}_{0}(k)=2J\left(-\cos k\cdot {\tau }^{z}+\sin k\cdot {\tau }^{y}\right),\end{eqnarray}$
and τy,z are the Pauli matrices. Here, we assume that N is even. $k=0,\pm \tfrac{2}{N}\pi ,\cdots ,\pm \tfrac{N-2}{N}\pi ,\pi $ for cj+N = cj and $k=\pm \tfrac{1}{N}\pi ,\pm \tfrac{3}{N}\pi \cdots ,\pm \tfrac{N-1}{N}\pi $ for cj+N = − cj. The dispersion of the quasiparticle is given by $\begin{eqnarray}{E}_{0}(k)=\pm 2J.\end{eqnarray}$
Moreover, we have $\begin{eqnarray}{\hat{H}}^{{\prime} }=\displaystyle \frac{1}{2}\displaystyle \sum _{j}{\sigma }_{j}^{z}\to {\hat{H}}^{{\prime} }=\displaystyle \frac{1}{2}\displaystyle \sum _{k}{\psi }_{k}^{\dagger }{\tau }^{z}{\psi }_{k}.\end{eqnarray}$
For the non-Hermitian model, the Jordan–Wigner transformation turns into
$\begin{eqnarray}\begin{array}{rcl}{\tilde{c}}_{j}^{\dagger } & = & \hat{S}{\sigma }_{j}^{\dagger }\displaystyle \prod _{l\lt j}(-{\sigma }_{l}^{z}){\hat{S}}^{-1}={{\rm{e}}}^{{\beta }_{\mathrm{NH}}}{c}_{j}^{\dagger },\\ {\tilde{c}}_{j} & = & \hat{S}\displaystyle \prod _{l\lt j}(-{\sigma }_{l}^{z}){\sigma }_{j}^{-}{\hat{S}}^{-1}={{\rm{e}}}^{-{\beta }_{\mathrm{NH}}}{c}_{j}.\end{array}\end{eqnarray}$
Then one-dimensional non-Hermitian Ising model is transformed into the one-dimensional non-Hermitian Kitaev model $\begin{eqnarray}\begin{array}{rcl}{\hat{H}}_{\mathrm{NHK}} & = & -J\displaystyle \sum _{j=1}^{N}\left({c}_{j}^{\dagger }{c}_{j+1}+{c}_{j+1}^{\dagger }{c}_{j}\right.\\ & & \left.+{{\rm{e}}}^{2{\beta }_{\mathrm{NH}}}{c}_{j}^{\dagger }{c}_{j+1}^{\dagger }+{{\rm{e}}}^{-2{\beta }_{\mathrm{NH}}}{c}_{j+1}{c}_{j}\right).\end{array}\end{eqnarray}$
Finally, under the periodic boundary condition, the non-Hermitian Hamiltonian of the one-dimensional non-Hermitian Ising model in momentum space is $\begin{eqnarray}{\hat{H}}_{\mathrm{NHK}}(k)=\displaystyle \sum _{k\gt 0}{\psi }_{k}^{\dagger }{\hat{h}}_{\mathrm{NHK}}(k){\psi }_{k},\end{eqnarray}$
where $\begin{eqnarray}{\hat{h}}_{\mathrm{NHK}}(k)=2J(\sin k\cdot {\tau }^{y,{\beta }_{\mathrm{NH}}}-\cos k\cdot {\tau }^{z}),\end{eqnarray}$
with $\begin{eqnarray}{\tau }_{j}^{y,{\beta }_{\mathrm{NH}}}=\cosh (2{\beta }_{\mathrm{NH}})\cdot {\tau }_{j}^{y}-\mathrm{isinh}(2{\beta }_{\mathrm{NH}})\cdot {\tau }_{j}^{x}.\end{eqnarray}$
It is obvious that the energy levels and the dispersion of the quasiparticle for ${\hat{H}}_{\mathrm{NHK}}$ and ${\hat{H}}_{0}$ are the same, i.e. $\begin{eqnarray}{E}_{\mathrm{NHK}}(k)\equiv \pm 2J.\end{eqnarray}$
This is a system with a flat band.Here, we only discuss the case of all real eigenvalues of the non-Hermitian Hamiltonian ${\hat{H}}_{\mathrm{NHK}}$. According to the quantum Liouvillian statistical theory, we obtain the Liouvillian Hamiltonian of the one-dimensional Ising model at finite temperature
$\begin{eqnarray}\begin{array}{rcl}{\hat{H}}_{L} & = & -\displaystyle \frac{1}{{\beta }_{T}}\mathrm{ln}[\hat{S}({{\rm{e}}}^{-{\beta }_{T}{\hat{H}}_{0}}){\hat{S}}^{\dagger }]\\ & = & \displaystyle \sum _{k\gt 0}{\psi }_{k}^{\dagger }\left[{\hat{h}}_{L}(k)\right]{\psi }_{k},\end{array}\end{eqnarray}$
where $\begin{eqnarray}{\hat{h}}_{L}(k)=\displaystyle \frac{{\varepsilon }_{k}}{\sqrt{{[{A}_{y}(k)]}^{2}+{[{A}_{z}(k)]}^{2}}}\left[{A}_{y}(k)\cdot {\tau }^{y}+{A}_{z}(k)\cdot {\tau }^{z}\right].\end{eqnarray}$
The Liouvillian energy levels are ${E}_{\pm }^{L}=\pm {\varepsilon }_{k}$, where $\begin{eqnarray}\begin{array}{rcl}{\varepsilon }_{k} & = & -\displaystyle \frac{1}{{\beta }_{T}}{\cosh }^{-1}[\cosh \left(2{\beta }_{\mathrm{NH}}\right)\cdot \cosh \left(2{\beta }_{T}J\right)\\ & & +\cos k\cdot \sinh \left(2{\beta }_{\mathrm{NH}}\right)\cdot \sinh \left(2{\beta }_{T}J\right)].\end{array}\end{eqnarray}$
Therefore, we can denote the energy bands of the system by a rotor${\vec{N}}_{L}(k)=\tfrac{1}{\sqrt{{\left.{\left[{A}_{x}(k)\right)}^{2}+{\left({A}_{y}(k)\right)}^{2}+({A}_{z}(k)\right]}^{2}}}({A}_{x}(k),{A}_{y}(k),{A}_{z}(k))$ with
$\begin{eqnarray}\begin{array}{rcl}{A}_{x}(k) & = & 0,\\ {A}_{y}(k) & = & \displaystyle \frac{-\sin k}{\cosh \left(2{\beta }_{\mathrm{NH}}\right)},\\ {A}_{z}(k) & = & \cos k+\tanh \left(2{\beta }_{\mathrm{NH}}\right)\cdot \coth \left(2{\beta }_{T}J\right).\end{array}\end{eqnarray}$
As a result, the non-Hermitian terms lead to unexpected physics consequences: On the one hand, they change the energy dispersion from ENHK(k) = ±2J to ${E}_{\pm }^{L}(k)=\pm {\varepsilon }_{k}$. See the illustration in figure 1; on the other hand, they rotate the rotor of energy bands from $(0,J\sin k,-J\cos k)$ to $\tfrac{1}{\sqrt{{[{A}_{y}(k)]}^{2}+{[{A}_{z}(k)]}^{2}}}(0,{A}_{y}(k),{A}_{z}(k))$. With increasing the non-Hermitian strength βNH, the original flat energy bands become dispersive. In addition, the non-Hermitian terms change the energy gap from ${\rm{\Delta }}=4\left|J\right|$ to ${\rm{\Delta }}=4\left|\tfrac{{\beta }_{\mathrm{NH}}}{{\beta }_{T}}\pm J\right|$.
Figure 1. The Liouvillian energy levels ${E}_{\pm }^{L}=\pm {\varepsilon }_{k}$ of the Liouvillian Hamiltonian ${\hat{H}}_{L}$. Here, J = 1. |
3.2. The 'topological phase diagram' and 'pseudo-phase transition'
We use the Jordan–Wigner transformation to map the non-Hermitian Ising model to a one-dimensional Kitaev model with an imbalanced p-wave superconductor. Therefore, by studying the topological properties of the one-dimensional Kitaev model with an imbalanced p-wave superconductor, we can know the properties of the non-Hermitian Ising model.
The Liouvillian Hamiltonian ${\hat{H}}_{L}={\sum }_{k}{\psi }_{k}^{\dagger }\left[{\hat{h}}_{L}(k)\right]{\psi }_{k}$ can be divided into three parts
$\begin{eqnarray}{\hat{H}}_{L}={\hat{H}}_{L,k\gt 0}+{\hat{H}}_{L,k=0}+{\hat{H}}_{L,k=\pi }.\end{eqnarray}$
After diagonalizing the fermion Hamiltonian ${\hat{H}}_{L}$ at the points k > 0, we have $\begin{eqnarray}{\hat{H}}_{L,k\gt 0}=\displaystyle \sum _{k\gt 0}\varepsilon (k){\alpha }_{k}^{\dagger }{\alpha }_{k}-\displaystyle \sum _{k\gt 0}\varepsilon (k){\alpha }_{-k}{\alpha }_{-k}^{\dagger },\end{eqnarray}$
where αk and ${\alpha }_{k}^{\dagger }$ are annihilation and creation operators of diagonalized quasiparticles and α±k annihilate the ground state ∣G⟩, i.e. α±k∣G⟩ = 0. Both α bands have positive energy at each point in momentum space k > 0. The Hamiltonian at k = 0 and k = π are diagonalized into $\begin{eqnarray}\begin{array}{rcl}{\hat{H}}_{L,k=0} & = & \varepsilon (k=0){\alpha }_{k=0}^{\dagger }{\alpha }_{k=0},\\ {\hat{H}}_{L,k=\pi } & = & \varepsilon (k=\pi ){\alpha }_{k=\pi }^{\dagger }{\alpha }_{k=\pi },\end{array}\end{eqnarray}$
where ϵ(k = 0) = $-2\left(\tfrac{{\beta }_{\mathrm{NH}}}{{\beta }_{T}}+J\right)$ and $\varepsilon (k=\pi )=-2\left(\tfrac{{\beta }_{\mathrm{NH}}}{{\beta }_{T}}-J\right)$.To describe the 'topological' properties of ${\hat{H}}_{L}$, we define two Z2 'topological' invariants—Pfaffians at high symmetry points k = 0 and k = π in momentum space
$\begin{eqnarray}\begin{array}{rcl}{\eta }_{{}_{k=0}} & = & \left\langle {\rm{\Psi }}\right|{c}_{k=0}^{\dagger }{c}_{k=0}\left|{\rm{\Psi }}\right\rangle ,\\ {\eta }_{{}_{k=\pi }} & = & \left\langle {\rm{\Psi }}\right|{c}_{k=\pi }^{\dagger }{c}_{k=\pi }\left|{\rm{\Psi }}\right\rangle .\end{array}\end{eqnarray}$
$\left|{\rm{\Psi }}\right\rangle $ denotes the 'ground state' of ${\hat{H}}_{L}$, and $\begin{eqnarray}\begin{array}{rcl}{\eta }_{{}_{k=0}} & = & \displaystyle \frac{\varepsilon [k=0]}{\left|\varepsilon [k=0]\right|}=\displaystyle \frac{-2\left(\tfrac{{\beta }_{\mathrm{NH}}}{{\beta }_{T}}+J\right)}{\left|-2\left(\tfrac{{\beta }_{\mathrm{NH}}}{{\beta }_{T}}+J\right)\right|},\\ {\eta }_{{}_{k=\pi }} & = & \displaystyle \frac{\varepsilon [k=\pi ]}{\left|\varepsilon [k=\pi ]\right|}=\displaystyle \frac{-2\left(\tfrac{{\beta }_{\mathrm{NH}}}{{\beta }_{T}}-J\right)}{\left|-2\left(\tfrac{{\beta }_{\mathrm{NH}}}{{\beta }_{T}}-J\right)\right|}.\end{array}\end{eqnarray}$
As a result, we have $\begin{eqnarray}\begin{array}{rcl}{\eta }_{{}_{k=0}} & = & \left\{\begin{array}{l}+1,\varepsilon \left(k=0\right)\gt 0\\ -1,\varepsilon \left(k=0\right)\lt 0\end{array},\right.\\ {\eta }_{{}_{k=0}} & = & \left\{\begin{array}{l}+1,\varepsilon \left(k=\pi \right)\gt 0\\ -1,\varepsilon \left(k=\pi \right)\lt 0\end{array}.\right.\end{array}\end{eqnarray}$
Table 1 shows the two Z2 'topological' invariants of four universal classes of 'topological' orders.
Table 1. The two Z2 'topological' variables of four universal classes of 'topological' orders. |
| ηk = 0 | −1 | −1 | 1 | 1 |
|---|---|---|---|---|
| ηk = π | −1 | 1 | −1 | 1 |
When ${\eta }_{{}_{k=0}}\cdot {\eta }_{{}_{k=\pi }}=-1$ ($({\eta }_{{}_{k=0}},{\eta }_{{}_{k=\pi }})=(-1,1)$, or (1, − 1)), we have a 'topological' phase; When ${\eta }_{{}_{k=0}}\cdot {\eta }_{{}_{k=\pi }}=1$ ($({\eta }_{{}_{k=0}},{\eta }_{{}_{k=\pi }})=(1,1)$, or (−1, −1)), we have a 'non-topological' phase. This is different from the case of the phase at zero temperature for the Hermitian Ising model, in which the phase of $({\eta }_{{}_{k=0}},{\eta }_{{}_{k=\pi }})=(-1,1)$ corresponds to ferromagnetic order; the phase of (1, − 1) corresponds to antiferromagnetic order; the phase of $({\eta }_{{}_{k=0}},{\eta }_{{}_{k=\pi }})=(1,1)$ corresponds to a spin-polarized phase denoted by ∣ → → → → ⟩; the phase of $({\eta }_{{}_{k=0}},{\eta }_{{}_{k=\pi }})=(-1,-1)$ corresponds to a spin-polarized phase denoted by ∣ ← ← ← ← ⟩.
We plot the 'global phase diagram' of the Liouvillian Hamiltonian ${\hat{H}}_{L}$ via the non-Hermitian strength βNH and the inverse temperature βT in figure 2, of which the two 'topological' invariants $({\eta }_{{}_{k=0}},{\eta }_{{}_{k=\pi }})$ are (−1, 1), (−1, −1), (1, 1) in region I, II, III, respectively. As a result, region I is the 'topological' phase; regions II and III are the 'non-topological' phases. The 'topological phase transition' occurs between these two phases, at which the Liouvillian energy gap is closed (${\rm{\Delta }}=4\left|\tfrac{{\beta }_{\mathrm{NH}}}{{\beta }_{T}}\pm J\right|=0$) that determines $\tfrac{{\beta }_{\mathrm{NH}}}{{\beta }_{T}}=\pm J$. Due to the nature of finite temperature, this 'topological phase transition' is a crossover. Thus, we call it 'pseudo-phase transition' and denote it by a dashed line in figure 2.
Figure 2. The 'global phase diagram' of the Liouvillian Hamiltonian ${\hat{H}}_{L}$ via the non-Hermitian strength βNH and the inverse temperature βT. The two Z2 'topological' invariants (ηk = 0, ηk = π) are (−1, 1), ( −1, −1), (1, 1) in region I, II, and III, respectively. The dashed line $\tfrac{{\beta }_{\mathrm{NH}}}{{\beta }_{T}}=\pm J$ denotes the 'pseudo-phase transition' between the 'topological' phase (region I) and the 'non-topological' phases (regions II and III). |
3.3. The thermodynamic properties
According to equations (20 )–(23 ) and using the Jordan–Wigner transformation and the Fourier transform, we can get
$\begin{eqnarray}\begin{array}{rcl}{\hat{H}}_{L} & = & \displaystyle \frac{{\varepsilon }_{k}}{\sqrt{{\left[{A}_{y}(k)\right]}^{2}+{\left[{A}_{z}(k)\right]}^{2}}}\displaystyle \sum _{j}\left[\displaystyle \frac{1+\tfrac{1}{\cosh \left(2{\beta }_{\mathrm{NH}}\right)}}{2}{\sigma }_{j}^{x}{c}_{j+1}^{x}\right.\\ & & +\displaystyle \frac{1-\tfrac{1}{\cosh \left(2{\beta }_{\mathrm{NH}}\right)}}{2}{\sigma }_{j}^{y}{c}_{j+1}^{y}\\ & & \left.+\tanh \left(2{\beta }_{\mathrm{NH}}\right)\cdot \coth \left(2{\beta }_{T}J\right){\sigma }_{j}^{z}\right].\end{array}\end{eqnarray}$
From this, we find the Liouvillian Hamiltonian ${\hat{H}}_{L}$ describes an effective anisotropic XY model in a transverse field [49–51], of which the effective transverse field is $\begin{eqnarray}h=\tanh (2{\beta }_{\mathrm{NH}})\cdot \coth (2J{\beta }_{T}),\end{eqnarray}$
and the anisotropy ratio is $\begin{eqnarray}\gamma =\displaystyle \frac{1}{\cosh (2{\beta }_{\mathrm{NH}})}.\end{eqnarray}$
Thus we use the effective transverse field h and the anisotropy ratio γ to show the 'global phase diagram' of the Liouvillian Hamiltonian ${\hat{H}}_{L}$ in figure 3, in which the white regions (I regions) are forbidden. Likewise, the 'pseudo-phase transition' denoted by a dashed line occurs between the 'topological' phase (II regions) and the 'non-topological' phase (III regions).
Figure 3. The 'Global phase diagram' of the Liouvillian Hamiltonian ${\hat{H}}_{L}$ via the effective transverse field $h=\tanh (2{\beta }_{\mathrm{NH}})\cdot \coth (2J{\beta }_{T})$ and the anisotropy ratio $\gamma =1/\cosh (2{\beta }_{\mathrm{NH}})$. The white regions (region I) are forbidden. The dashed line denotes the 'pseudo-phase transition' between the 'topological' phase (region II) and the 'non-topological' phase (region III). (a) The flow of the system with increasing non-Hermitian strength βNH at fixed temperature T. (b) The flow of the system with decreasing temperature T at fixed non-Hermitian strength βNH. |
3.3.1. The Hermitian case βNH = 0
For the Hermitian case βNH = 0, the Liouvillian Hamiltonian ${\hat{H}}_{L}$ is reduced to ${\hat{H}}_{0}$ and the results become trivial: there exists the correspondence between the ground states and the zero-temperature states, and the thermalization effect in the high-temperature limit. The system is on point 'c' in figure 3.
3.3.2. The zero temperature βT → ∞
For the case of βNH ≠ 0, the system becomes non-Hermitian. Near zero temperature (or βT → ∞ ), due to $\coth (2J{\beta }_{T})\to 1$, we have
$\begin{eqnarray}{h}^{2}+{\gamma }^{2}\equiv 1.\end{eqnarray}$
As a result, ${\hat{H}}_{L}$ becomes an anisotropic XY model near zero temperature and the system is on the quarter circle (red curve in figure 3). Now, with increasing non-Hermitian strength $\left|{\beta }_{\mathrm{NH}}\right|$, the system shifts from h = 0, γ = 1 (denoted by point 'c') to point h = 1, γ = 0 (denoted by point 'p') in figure 3. The existence of the quarter circle in figure 3 indicates the collapse of the correspondence between the ground states and the zero-temperature states. At a fixed finite temperature, with increasing non-Hermitian strength $\left|{\beta }_{\mathrm{NH}}\right|$, the quarter circle becomes the quarter ellipse, as seen in the purple curve in figure 3(a).To better reflect the characteristics of the system at zero temperature, we calculate the expected value of spin correlation which is defined as
$\begin{eqnarray}\left\langle {\sigma }_{i}^{\mu }{\sigma }_{i+L}^{\mu }\right\rangle =\displaystyle \frac{\mathrm{Tr}\left({\sigma }_{i}^{\mu }{\sigma }_{i+L}^{\mu }\cdot {\rho }_{\mathrm{NHTS}}^{{S}}\right)}{\mathrm{Tr}\left({\rho }_{\mathrm{NHTS}}^{{S}}\right)}.\end{eqnarray}$
Because ${\rho }_{\mathrm{NHTS}}^{{S}}$ is not involved any term about σy, we have $\left\langle {\sigma }_{i}^{y}{\sigma }_{i+L}^{y}\right\rangle =0$. The calculation of $\left\langle {\sigma }_{i}^{z}{\sigma }_{i+L}^{z}\right\rangle $ is obtained as $\left\langle {\sigma }_{i}^{z}{\sigma }_{i+L}^{z}\right\rangle ={\tanh }^{2}({\beta }_{\mathrm{NH}})$. The spin correlation $\left\langle {\sigma }_{i}^{x}{\sigma }_{i+L}^{x}\right\rangle $ can be mapped to the Green function of fermions [52], i.e. $\begin{eqnarray}\left\langle {\sigma }_{i}^{x}{\sigma }_{i+L}^{x}\right\rangle =\left|\begin{array}{cccc}{G}_{i,i+1} & {G}_{i,i+2} & ... & {G}_{i,i+L}\\ \vdots & ... & ... & \vdots \\ {G}_{i+L-1,i+1} & ... & ... & {G}_{i+L-1,i+L}\end{array}\right|,\end{eqnarray}$
where Gi,j is defined by ${G}_{i,j}=\left\langle ({c}_{i}^{\dagger }-{c}_{i})({c}_{j}^{\dagger }+{c}_{j})\right\rangle $. For the case of N → ∞ and i − j = r, we have $\begin{eqnarray}\begin{array}{rcl}{G}_{r} & = & \displaystyle \frac{1+1/\cosh (2{\beta }_{\mathrm{NH}})}{2}{L}_{r+1}+\displaystyle \frac{1-1/\cosh (2{\beta }_{\mathrm{NH}})}{2}{L}_{r-1}\\ & & +\tanh (2{\beta }_{\mathrm{NH}})\coth (2J{\beta }_{T}){L}_{r},\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{rcl}{L}_{r} & = & \displaystyle \frac{1}{\pi }{\displaystyle \int }_{0}^{\pi }\left(\displaystyle \frac{\cos ({rk}){\rm{d}}k}{\sqrt{1-{\sin }^{2}k\cdot {\tanh }^{2}(2{\beta }_{\mathrm{NH}})+2\cos k\cdot \tanh (2{\beta }_{\mathrm{NH}})\cdot \coth (2J{\beta }_{T})+{\tanh }^{2}(2{\beta }_{\mathrm{NH}})\cdot {\coth }^{2}(2J{\beta }_{T})}}\right.\\ & & \left.\cdot \sqrt{1-\displaystyle \frac{2}{\cosh \left(2{\beta }_{\mathrm{NH}}\right)\cdot \cosh \left(2J{\beta }_{T}\right)+\cos k\cdot \sinh \left(2{\beta }_{\mathrm{NH}}\right)\cdot \sinh \left(2J{\beta }_{T}\right)+1}}\right).\end{array}\end{eqnarray}$
We show the spin correlation SL at the low temperature of T = 0.02J in figure 4. With the increase of the non-Hermitian strength, the spin correlation function in the z-direction becomes larger, while that in the x-direction becomes smaller, indicating that all the spins of the system are shifted to the z-direction.
Figure 4. The spin correlations SLxx, SLyy and SLzz at low temperature T = 0.02J. Here, L = 10. |
3.3.3. The strongly non-Hermitian case βNH → ± ∞
For the case of a strong non-Hermitian strength limit, βNH → ± ∞ , the original non-Hermitian Hamiltonian ${\hat{H}}_{\mathrm{NHIS}}$ turns into
$\begin{eqnarray}-{{J}{\rm{e}}}^{\left|{\beta }_{\mathrm{NH}}\right|}\displaystyle \sum _{j}{\sigma }_{j}^{+}{\sigma }_{j+1}^{+},\end{eqnarray}$
or $\begin{eqnarray}-{{J}{\rm{e}}}^{\left|{\beta }_{\mathrm{NH}}\right|}\displaystyle \sum _{j}{\sigma }_{j}^{-}{\sigma }_{j+1}^{-}.\end{eqnarray}$
Due to $\gamma =1/\cosh (2{\beta }_{\mathrm{NH}})\to 0$, the Liouvillian Hamiltonian ${\hat{H}}_{L}$ is reduced into an effective isotropic XY model in the transverse field. Now, at finite temperature (βT $\nrightarrow $ ∞), the Liouvillian energy levels are obtained as $\begin{eqnarray}{E}_{\pm }^{L}\simeq \pm 2\displaystyle \frac{{\beta }_{\mathrm{NH}}}{{\beta }_{T}}\pm \displaystyle \frac{1}{{\beta }_{T}}\mathrm{ln}\left[\cosh (2J{\beta }_{T})+\cos k\cdot \sinh (2J{\beta }_{T})\right],\end{eqnarray}$
and the rotor of energy bands is fixed to be $\begin{eqnarray}\vec{N}(k)=(0,0,1).\end{eqnarray}$
The system is on the blue line in figure 3. With decreasing the temperature T, the system approaches the point h = 1, γ = 0 (denoted by point 'p' in figure 3) from h → ∞ , γ = 0. Near zero temperature with βT → ∞ , the system trends closer to the singular point at 'p' in figure 3. Now, Liouvillian energy dispersions are reduced into $\begin{eqnarray}{E}_{\pm }^{L}\simeq \pm \displaystyle \frac{2}{{\beta }_{T}}({\beta }_{\mathrm{NH}}+J{\beta }_{T}),\end{eqnarray}$
for k ∼ π or $\begin{eqnarray}{E}_{\pm }^{L}\simeq \pm \displaystyle \frac{2}{{\beta }_{T}}({\beta }_{\mathrm{NH}}-J{\beta }_{T}),\end{eqnarray}$
for k ∼ π. We have an effective 'Liouvillian Hamiltonian' ${\hat{H}}_{{\rm{L}}}^{{\prime} }\sim -2{\hat{H}}^{{\prime} }$ for k ∼ π at the effective 'temperature' ${T}_{\mathrm{eff}}=\tfrac{1}{{k}_{{\rm{B}}}({\beta }_{\mathrm{NH}}+J{\beta }_{T})}$ or ${\hat{H}}_{L}^{{\prime} }\sim -2{\hat{H}}^{{\prime} }$ for k ∼ π at effective 'temperature' ${T}_{\mathrm{eff}}=\tfrac{1}{{k}_{{\rm{B}}}({\beta }_{\mathrm{NH}}-J{\beta }_{T})}$. Nevertheless, at a given finite non-Hermitian strength, with decreasing the temperature T, the system is at the dodger blue line in figure 3(b).3.3.4. The high-temperature case βT → 0
In the high-temperature limit of βT → 0, we have a temperature-dependent Hamiltonian ${\hat{H}}_{{\rm{L}}}\sim -\tfrac{2}{{\beta }_{T}}{\hat{H}}^{{\prime} }$ at temperature T or a pseudo-temperature–independent pseudo-Hamiltonian ${\hat{H}}_{\mathrm{eff}}^{\mathrm{pseudo}}=-2{\hat{H}}^{{\prime} }$ at pseudo-temperature Tpseudo = 1/βNH. The situation is very different from that of the usual Hermitian model, in which there exists the thermalization effect in the high-temperature limit (T → ∞ ) due to ${{\rm{e}}}^{-{\beta }_{T}{\hat{H}}_{0}}\sim 1$ and the correspondence between the ground states and the zero-temperature states. In addition, as βT → 0 and $\left|{\beta }_{\mathrm{NH}}\right|\to \infty $, the 'ground state' of ${\hat{H}}_{\mathrm{eff}}^{\mathrm{pseudo}}$ describes a 'non-topological' state (regions II, III in figure 2).
To better see this phenomenon, we show the expected values of local magnetizations
$\begin{eqnarray}{n}^{\mu }=\displaystyle \frac{1}{N}\displaystyle \sum _{i}\left\langle {\sigma }_{i}^{\mu }\right\rangle ,\end{eqnarray}$
where the expected values of the spin operators ${\sigma }_{i}^{\mu }$ (μ = x, y, z) are $\begin{eqnarray}\left\langle {\sigma }_{i}^{\mu }\right\rangle =\displaystyle \frac{\mathrm{Tr}\left({\sigma }_{i}^{\mu }\cdot {\rho }_{\mathrm{NHTS}}^{{S}}\right)}{\mathrm{Tr}\left({\rho }_{\mathrm{NHTS}}^{{S}}\right)},\end{eqnarray}$
and N is the number of lattice sites in figure 5. After calculation, we have $\begin{eqnarray}\begin{array}{rcl}{n}^{x} & = & 0,\\ {n}^{y} & = & 0,\\ {n}^{z} & = & \tanh ({\beta }_{\mathrm{NH}}),\end{array}\end{eqnarray}$
for the system at the high temperature T = 20J. This result (nz ≠ 0 if βNH ≠ 0) clearly indicates the non-thermalization effect in the high-temperature limit.
Figure 5. The expected values of local magnetizations nμ (μ = x, y, z) at high temperature T = 20J. Here, L = 10. |
4. Conclusions
In the paper, we study the one-dimensional non-Hermitian Ising model at finite temperature with the help of the quantum Liouvillian statistical theory. A 'pseudo-phase transition' is discovered between the 'topological' phase and the 'non-topological' phase. Compared with the usual energy gap closed in the Hermitian system, the 'pseudo-phase transition' is accompanied by the Liouvillian energy gap closed. Moreover, we show that its Liouvillian Hamiltonian ${\hat{H}}_{{\rm{L}}}$ can be reduced to an effective anisotropic XY model in the transverse field. In the Hermitian case, the system shows the thermalization in the high-temperature limit and the correspondence between the ground states and the zero-temperature states. Near the zero temperature, we find that all spins of the system are shifted to one direction (z-direction). For the case of a strong non-Hermitian strength limit, the system becomes an effective isotropic XY model in the transverse field. In the high-temperature limit, the system appears the non-thermalization. This work will promote the study of the physical properties of non-Hermitian systems at finite temperatures.