Sharp entanglement thresholds in the logarithmic negativity of disjoint blocks in the transverse-field Ising chain

Entanglement plays a central role in quantum many-body theory. Exotic quantum phases such as spin liquids [1, 2], topological [3, 4], or many-body localized systems [5–8] find their characterization in their entanglement properties. Moreover, quantum phase transitions are signaled by a universal entanglement contribution determined solely by the universality class of the transition [9–14]. This can be used to detect quantum phase transitions without prior knowledge on the nature of the transition [15], e.g., the order parameter, since entanglement is a general system-independent quantity. In the ongoing efforts to characterize quantum many-body systems via their entanglement properties, the entanglement entropy, measuring the entanglement between a subsystem and its remainder, is taking over a key role. However, a major limitation of the entanglement entropy is that it is a valid entanglement measure only for pure states. This is a particular challenge in view of experiments where thermal excitations or other imperfections leading to mixed states are generally unavoidable. Nevertheless, recent works on quantum simulators have demonstrated that entanglement in quantum many-body systems can be accessible in experiments. In systems of trapped ions, full-state tomography provides access to various entanglement measures [15–20]. In ultra-cold atoms it is possible to measure Renyi entropies [21] as also demonstrated in experiments [22, 23]. Recent theoretical works have outlined new approaches for measuring entanglement using unitary n-designs [24, 25] or machine learning techniques [26].

In this work, we map out the spatial entanglement structure of a low-dimensional quantum system, the transverse-field Ising chain, both in the ground state and in thermal states. For this purpose we use the logarithmic negativity [27–31], which shares many of the central features of the entanglement entropy in pure states, such as the area law for ground states of gapped Hamiltonians [32–34] or the aforementioned universal contribution appearing at quantum critical points [12, 35, 36]. In contrast to the entanglement entropy, however, the logarithmic negativity remains an entanglement measure also for mixed states [12, 36]. In order to obtain information about the spatial entanglement structure, we study the logarithmic negativity of two disjoint blocks of identical size ℓ as a function of their separation d, which can be viewed as the entanglement analog to a conventional quantum correlation function. For an illustration of our setup see figure 1(a). We find that for any fixed size ℓ of the two blocks there appears a sharp entanglement threshold ${d}^{* }$ beyond which the logarithmic negativity vanishes identically. For larger distance than ${d}^{* }$ the two blocks become unentangled, accordingly, as measured by the logarithmic negativity. In figure 1(b) we show the results for the entanglement threshold ${d}^{* }$ as a function of ℓ for different parameters of the transverse-field Ising chain, where one can see that the spatial extent of entanglement is restricted to rather short distances even when the system resides at the quantum phase transition where quantum correlations are long-ranged.

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While for the case where the two blocks consist of single qubits this result is well known [37–41], here we study systematically the crossover from the single-particle to the multi-particle case. We compute the logarithmic negativity numerically for large systems using the time-evolving block decimation (TEBD). In addition, we develop a simple effective model explaining our numerical observations.

This paper organized as follows: in section 2 we start with an introduction to the logarithmic negativity. In section 3 we introduce the transverse-field Ising chain for which we study the entanglement thresholds. The method, that we use for our numerical simulations, is described in detail in section 4. Afterwards we present our results in section 5. First, we consider the ground state properties in section 5.1 for an extensive range of system parameters and block sizes. Afterwards, we investigate how thermal fluctuations affect the entanglement in section 5.2. In section 6 we introduce a simple model of the reduced density matrix which captures the main features of the decay of the logarithmic negativity observed in section 5. We end our work with discussions and conclusions in the section 7.

The aim of this work is to study the spatial structure of entanglement in equilibrium states of the transverse-field Ising chain as depicted in figure 1. The entanglement entropy, which is the paradigmatic entanglement measure for the characterization of quantum many-body systems in ground states, cannot be used for that purpose since it can only access the entanglement between a subsystem and its remainder, but not the entanglement between two subsystems. The entanglement entropy of two distant blocks with the remainder has been studied already in the literature [42–44]. Here, we use the logarithmic negativity ${{ \mathcal E }}_{{ \mathcal N }}$ to quantify the entanglement between two distant blocks.

Let us denote by ρ the density matrix of the system, which can be either pure or mixed. To compute the logarithmic negativity, it is necessary to access the reduced density matrix ρ_A,B of two subsystems A and B which can be obtained from ρ by tracing out all the degrees of freedom not belonging to A or to B:

$$\begin{eqnarray}&&{\rho }_{A,B}={{\rm{Tr}}}_{\overline{{AB}}}\ \rho .\end{eqnarray} \tag{ 1 }$$

Here, ${{\rm{Tr}}}_{\overline{{AB}}}$ denotes the trace over the complement $\overline{{AB}}$ of A and B. The reduced density matrix ρ_A,B can be represented as

$$\begin{eqnarray}&&{\rho }_{A,B}=\displaystyle \sum _{\displaystyle \genfrac{}{}{0em}{}{\mu ,\nu }{m,n}}{C}_{m,n}^{\mu ,\nu }| \mu \rangle \langle \nu | \otimes | m\rangle \langle n| ,\end{eqnarray} \tag{ 2 }$$

where $| \mu \rangle$ and $| \nu \rangle$ label the basis states of the local Hilbert space ${{\mathbb{H}}}_{A}$ of subsystem A, and $| m\rangle$ and $| n\rangle$ of ${{\mathbb{H}}}_{B}$ accordingly. ${C}_{m,n}^{\mu ,\nu }$ are the coefficients given by ${C}_{m,n}^{\mu ,\nu }=\langle \mu ,m| {\rho }_{A,B}| \nu ,n\rangle$.

The logarithmic negativity is an entanglement measure based on the positive partial transpose (PPT) criterion [45, 46], which provides a necessary condition for ρ_AB to be separable and therefore to contain no entanglement. Central to the PPT criterion is the partial transpose operation T_B performed on one of the two subsystems, B say:

$$\begin{eqnarray}&&{\rho }_{A,B}^{{T}_{B}}=[{I}_{A}\otimes {T}_{B}]{\rho }_{A,B}=\displaystyle \sum _{\displaystyle \genfrac{}{}{0em}{}{\mu ,\nu }{m,n}}{C}_{n,m}^{\mu ,\nu }| \mu \rangle \langle \nu | \otimes | m\rangle \langle n| ,\end{eqnarray} \tag{ 3 }$$

which leaves the basis states in A unchanged but performs a transpose on B. In the end, this operation is equivalent to ${C}_{m,n}^{\mu ,\nu }\to {C}_{n,m}^{\mu ,\nu }$ when comparing equation (2) with (3). While the eigenvalues of ρ_A,B are probabilities and therefore non-negative real numbers, this is not necessarily the case for the partially transposed ${\rho }_{A,B}^{{T}_{B}}$. When ρ_A,B is separable and therefore contains no entanglement, the eigenvalues λ of ${\rho }_{A,B}^{{T}_{B}}$ have to be non-negative, which is the aforementioned PPT criterion. In turn, this means that in case there exists a negative eigenvalue of ${\rho }_{A,B}^{{T}_{B}}$, the reduced density matrix ρ_A,B has to be entangled. The logarithmic negativity ${{ \mathcal E }}_{{ \mathcal N }}$ quantifies to which extent the partially transposed density matrix ${\rho }^{{T}_{B}}$ between two subsystems fails to be non-negative. More specifically, ${{ \mathcal E }}_{{ \mathcal N }}$ is defined as

$$\begin{eqnarray}&&{{ \mathcal E }}_{{ \mathcal N }}={\mathrm{log}}_{2}\parallel {\rho }^{{T}_{B}}{\parallel }_{1}={\mathrm{log}}_{2}\left[1+\displaystyle \sum _{\lambda }(| \lambda | -\lambda )\right],\end{eqnarray} \tag{ 4 }$$

where $\parallel .{\parallel }_{1}$ denotes the trace norm, and λ the eigenvalues of ${\rho }_{{AB}}^{{T}_{B}}$. In general, the PPT criterion is only a necessary but not a sufficient criterion for entanglement, i.e., there might be states that signal a vanishing logarithmic negativity that are, however, not separable. In this context it is important that ${{ \mathcal E }}_{{ \mathcal N }}$ constitutes an upper bound to the distillable entanglement [47]. A vanishing ${{ \mathcal E }}_{{ \mathcal N }}$ therefore means that such a Bell pair distillation is not possible.

For quantum many-body systems the logarithmic negativity has been studied extensively in the literature [27–29, 45, 48–53]. In particular, it has been found that ${{ \mathcal E }}_{{ \mathcal N }}$ displays the same universal contributions at quantum critical points [12, 35, 36, 54], as does the entanglement entropy [13, 33, 35]. In particular, the logarithmic negativity for two adjacent large blocks of size ℓ₁ and ℓ₂ becomes [35, 54]

$$\begin{eqnarray}&&{{ \mathcal E }}_{{ \mathcal N }}\sim \displaystyle \frac{c}{4}\mathrm{ln}\left[\displaystyle \frac{{{\ell }}_{1}{{\ell }}_{2}}{{{\ell }}_{1}+{{\ell }}_{2}}\right],\end{eqnarray} \tag{ 5 }$$

with c the central charge of the corresponding conformal field theory, which is a universal property of the underlying quantum phase transition. For ${{\ell }}_{2}\to \infty$, a situation which is equivalent to measuring the entanglement between a subsystem and its remainder, one obtains ${{ \mathcal E }}_{{ \mathcal N }}\sim (c/4)\mathrm{log}({{\ell }}_{1})$. The entanglement entropy has been intensively studied analytically [13, 55–57] and numerically [50, 58–61] for the ground state of the 1D transverse Ising model. On general grounds the entanglement entropy is characterized by an area law [33, 34, 62], although at the critical point a logarithmic dependence on the size ℓ₁ emerges leading to $S\sim (c/3)\mathrm{log}({{\ell }}_{1}),$ which has the same functional dependence as the logarithmic negativity.

In the case of disjoint blocks, the setup that we aim to address in this work, much less is known in general. Using conformal field theory it is possible to prove that the logarithmic negativity is a scale-invariant quantity at the critical point [35, 54, 63, 64]. Specifically, ${{ \mathcal E }}_{{ \mathcal N }}$ is a function only of the dimensionless quantity $y=\tfrac{({v}_{1}-{u}_{1})({v}_{2}-{u}_{2})}{({u}_{2}-{u}_{1})({v}_{2}-{v}_{1})}$, where u₁, v₁ are, respectively, the left and right edges of the first block, and u₂, v₂ of the second block.

One case that has been studied already extensively is when each of the two blocks contains a single spin [37–41]. Then, the entanglement between the two spins exactly vanishes beyond a distance of a few lattice sites, a phenomenon that has been termed 'entanglement sudden death' [65, 66]. How entanglement behaves for disjoint blocks larger than a single spin, is, however, not yet known.

In view of the sudden drop towards vanishing entanglement known for the single spin case, we introduce in the following the notion of the entanglement threshold ${d}^{* }$. We define ${d}^{* }$ to be the maximum distance d between two subsystems such that the two systems remain entangled. It is the main goal of this work to study this entanglement threshold in the transverse-field Ising chain⁵ .

The model we consider is the one-dimensional Ising model with a transverse field (TFIM) described by the following Hamiltonian:

$$\begin{eqnarray}&&H=-\displaystyle \frac{1}{2}\left(J\displaystyle \sum _{i=1}^{L-1}{\sigma }_{i}^{x}{\sigma }_{i+1}^{x}+h\displaystyle \sum _{i}^{L}{\sigma }_{i}^{z}\right),\end{eqnarray} \tag{ 6 }$$

where J denotes the spin–spin coupling, h the transverse field and ${\sigma }_{i}^{x(z)}$ the Pauli matrices acting on the ith lattice site. For convenience, we set the lattice spacing a = 1 and choose open boundary conditions. This model undergoes a quantum phase transition [9, 67] at zero temperature when J = h. For h < J, the system is in a ferromagnetic phase, while for h > J in a paramagnetic one. The order parameter of the transition is the magnetization ${m}_{x}={L}^{-1}{\sum }_{l}{\sigma }_{l}^{x}$ along the spin–spin coupling direction which is nonzero in the symmetry-broken phase and vanishes in the paramagnetic one. At nonzero temperature a symmetry-broken phase cannot exist for this one-dimensional system according to the Mermin–Wagner theorem [68–70].

In the following we study the entanglement properties of the transverse-field Ising chain as a function of temperature. Therefore, in general, our system resides in a thermal mixed state given by the density matrix ρ of the canonical ensemble:

$$\begin{eqnarray}&&\rho =\displaystyle \frac{1}{Z}{{\rm{e}}}^{-\beta H},\end{eqnarray} \tag{ 7 }$$

with $\beta =\displaystyle \frac{1}{T}$ the inverse temperature, H the Hamiltonian and $Z=\mathrm{Tr}({{\rm{e}}}^{-\beta H})$ the partition function.

Although the TFIM is exactly solvable by mapping the problem to a free fermionic theory using a Jordan–Wigner transformation [9], the computation of the logarithmic negativity remains complicated. The main problems arise when performing the partial transpose operation, which in terms of the fermionic degrees of freedom does not have a solvable structure [30, 48, 71]. Therefore, numerical techniques are required and we use for that purpose the TEBD in the following [72, 73]. Since we aim to study both the ground as well as nonzero temperature states, we use both the pure state matrix product states (MPS) and finite temperature MPS formalism [74–76]. As depicted in figure 2, the quantum state of the system can in general be written in an MPS representation

$$\begin{eqnarray}&&| \psi \rangle =\displaystyle \sum _{\displaystyle \genfrac{}{}{0em}{}{{\sigma }_{1}\ldots {\sigma }_{L}}{{\alpha }_{1}\ldots {\alpha }_{L-1}}}{T}_{{\alpha }_{1}}^{{\sigma }_{1}}\ldots {T}_{{\alpha }_{i-1},{\alpha }_{i}}^{{\sigma }_{i}}\ldots {T}_{{\alpha }_{L-1}}^{{\sigma }_{L}}\ | {\sigma }_{1}\ldots {\sigma }_{i}\ldots {\sigma }_{L}\rangle ,\end{eqnarray} \tag{ 8 }$$

where each ${T}_{{\alpha }_{i-1},{\alpha }_{i}}^{{\sigma }_{i}}$ is a rank-3 tensor, which therefore depends on the local state $| {\sigma }_{i}\rangle$. Note that indices α_i−1 and α_i refer to the bond dimension on site i and sums over them run from 1 to its maximum value at each bond, χ_max with α₀ = α_L = 1.

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We use imaginary time-evolution to compute both ground state and thermal state

$$\begin{eqnarray}&&| {\psi }_{{\rm{GS}}}\rangle =\mathop{{\rm{lim}}}\limits_{\tau \to \infty }\displaystyle \frac{U(\tau )| \psi \rangle }{\parallel U(\tau )| \psi \rangle \parallel }=\mathop{{\rm{lim}}}\limits_{\tau \to \infty }\displaystyle \frac{{{\rm{e}}}^{-\tau H}| \psi \rangle }{\parallel {{\rm{e}}}^{-\tau H}| \psi \rangle \parallel },\end{eqnarray} \tag{ 9 }$$

with τ imaginary time. The TEBD algorithm relies on the Suzuki–Trotter decomposition [77] of the time-evolution operator U(τ). For this one first needs to decompose the U(τ) into N small time steps dτ, i.e., $U(\tau )={[U({\rm{d}}\tau =\tau /N)]}^{N}$ where N is a large enough that the time interval dτ = τ/N is small compared to any internal time scale of the system.

We employ the second order Suzuki–Trotter decomposition

$$\begin{eqnarray}&&U({\rm{d}}\tau )=\displaystyle \prod _{\displaystyle \genfrac{}{}{0em}{}{i}{{\rm{even}}}}{U}_{i}\left(\displaystyle \frac{{\rm{d}}\tau }{2}\right)\displaystyle \prod _{\displaystyle \genfrac{}{}{0em}{}{i}{{\rm{odd}}}}{U}_{i}({\rm{d}}\tau )\displaystyle \prod _{\displaystyle \genfrac{}{}{0em}{}{i}{{\rm{even}}}}{U}_{i}\left(\displaystyle \frac{{\rm{d}}\tau }{2}\right)+O({\rm{d}}{\tau }^{3}),\end{eqnarray} \tag{ 10 }$$

where ${U}_{i}(d\tau )=\exp (-{{\rm{i}}{h}}_{i,i+1}d\tau )$ are the time-evolution operators of bond i with ${h}_{i,i+1}=-\tfrac{1}{2}[J{\sigma }_{i}^{x}{\sigma }_{i+1}^{x}+h({\sigma }_{i}^{z}+{\sigma }_{i+1}^{z})]$ acting on the bonds i which can be even or odd.

For thermal states, we need to purify the thermal density matrix. This is done by introducing new degrees of freedom $\{{\bar{\sigma }}_{1}\ldots {\bar{\sigma }}_{i}\ldots {\bar{\sigma }}_{L}\}\in Q$ as an auxiliary spin-$\tfrac{1}{2}$ for each lattice site in the MPS representation [78]. For infinite temperature $T=\infty$ for each site i one can choose $| {\psi }_{\beta =0}^{i}\rangle =\tfrac{1}{\sqrt{2}}(| {\sigma }_{i}=\uparrow ,{\bar{\sigma }}_{i}=\downarrow \rangle -| {\sigma }_{i}=\downarrow ,{\bar{\sigma }}_{i}=\uparrow \rangle )$ which yields the full density matrix ${\rho }_{\beta =0}={2}^{-L}I$ after tracing out the auxiliary degrees of freedom. The thermal state $| {\psi }_{\beta }\rangle$ can be obtained from $| {\psi }_{\beta =0}\rangle$ with imaginary time-evolution, $| {\psi }_{\beta }\rangle ={{\rm{e}}}^{-\beta H/2}| {\psi }_{\beta =0}\rangle$. In this way one can compute the thermal density matrix by tracing out the auxiliary degrees of freedom as ${\rho }_{\beta }={{\rm{Tr}}}_{Q}| {\psi }_{\beta }\rangle \langle {\psi }_{\beta }|$.

The MPS representation of thermal states can be constructed as for pure states but with an extra index ${\bar{\sigma }}_{i}$ for auxiliary Hilbert space for each site.

$$\begin{eqnarray}| {\psi }_{\beta }\rangle & = & \displaystyle \sum _{\displaystyle \genfrac{}{}{0em}{}{\genfrac{}{}{0em}{}{{\sigma }_{1}\ldots {\sigma }_{L}}{{\bar{\sigma }}_{1}\ldots {\bar{\sigma }}_{L}}}{{\alpha }_{1}\ldots {\alpha }_{L-1}}}{T}_{{\alpha }_{1}}^{{\sigma }_{1}{\bar{\sigma }}_{1}}\ldots {T}_{{\alpha }_{i-1},{\alpha }_{i}}^{{\sigma }_{i}{\bar{\sigma }}_{i}}\ldots \\ & & \ldots {T}_{{\alpha }_{L-1}}^{{\sigma }_{L}{\bar{\sigma }}_{L}}| {\sigma }_{1}{\bar{\sigma }}_{1}\ldots {\sigma }_{i}{\bar{\sigma }}_{i}\ldots {\sigma }_{L}{\bar{\sigma }}_{L}\rangle .\end{eqnarray} \tag{ 11 }$$

To compute the logarithmic negativity for a generic state $| \psi \rangle$, we need access to the reduced density matrix and its partial transpose. Therefore one needs to compute $\rho =| \psi \rangle \langle \psi |$ and trace out those sites which are not included in the blocks A and B. Note that in the case of thermal states the auxiliary degrees of freedom must be traced out. The reduced density matrix and its partial transpose for both pure and thermal states will have the same form. Figure 2 shows the reduced density matrix and its partial transpose using MPS based diagrams.

$$\begin{eqnarray}{\rho }_{A,B} & = & {{\rm{Tr}}}_{\overline{A,B}}(| \psi \rangle \langle \psi | )\\ & = & \displaystyle \sum _{{\sigma }_{i},{\sigma }_{i}^{{\prime} }\in \{A,B\}}{C}_{{({\sigma }_{1}\ldots {\sigma }_{{\ell }})}_{B},{({\sigma }_{1}^{{\prime} }\ldots {\sigma }_{{\ell }}^{{\prime} })}_{B}}^{{({\sigma }_{1}\ldots {\sigma }_{{\ell }})}_{A},{({\sigma }_{1}^{{\prime} }\ldots {\sigma }_{{\ell }}^{{\prime} })}_{A}}\ | \ {({\sigma }_{1}\ldots {\sigma }_{i}\ldots {\sigma }_{{\ell }})}_{A},{({\sigma }_{1}\ldots {\sigma }_{i}\ldots {\sigma }_{{\ell }})}_{B}\rangle \\ & & \times \langle {({\sigma }_{1}^{{\prime} }\ldots {\sigma }_{i}^{{\prime} }\ldots {\sigma }_{{\ell }}^{{\prime} })}_{A},{({\sigma }_{1}^{{\prime} }\ldots {\sigma }_{i}^{{\prime} }\ldots {\sigma }_{{\ell }}^{{\prime} })}_{B}\ | ,\end{eqnarray} \tag{ 12 }$$

where the coefficient matrix C for pure states reads as

$$\begin{eqnarray}&&{C}_{{({\sigma }_{1}\ldots {\sigma }_{{\ell }})}_{B},{({\sigma }_{1}^{{\prime} }\ldots {\sigma }_{{\ell }}^{{\prime} })}_{B}}^{{({\sigma }_{1}\ldots {\sigma }_{{\ell }})}_{A},{({\sigma }_{1}^{{\prime} }\ldots {\sigma }_{{\ell }}^{{\prime} })}_{A}}=\displaystyle \sum _{\displaystyle \genfrac{}{}{0em}{}{\genfrac{}{}{0em}{}{{\sigma }_{i},{\sigma }_{i}^{{\prime} }\notin \{A,B\}}{{\alpha }_{1}\ldots {\alpha }_{L-1}}}{{\alpha }_{1}^{{\prime} }\ldots {\alpha }_{L-1}^{{\prime} }}}({T}_{{\alpha }_{1}}^{{\sigma }_{1}}{T}_{{\alpha }_{1}^{{\prime} }}^{{\sigma }_{1}^{{\prime} }\dagger })\ldots ({T}_{{\alpha }_{i-1},{\alpha }_{i}}^{{\sigma }_{i}}{T}_{{\alpha }_{i-1}^{{\prime} },{\alpha }_{i}^{{\prime} }}^{{\sigma }_{i}^{{\prime} }\dagger })\ldots ({T}_{{\alpha }_{L-1}}^{{\sigma }_{L}}{T}_{{\alpha }_{L-1}^{{\prime} }}^{{\sigma }_{L}^{{\prime} }\dagger }),\end{eqnarray} \tag{ 13 }$$

and for thermal states reads as

$$\begin{eqnarray}&&{C}_{{({\sigma }_{1}\ldots {\sigma }_{{\ell }})}_{B},{({\sigma }_{1}^{{\prime} }\ldots {\sigma }_{{\ell }}^{{\prime} })}_{B}}^{{({\sigma }_{1}\ldots {\sigma }_{{\ell }})}_{A},{({\sigma }_{1}^{{\prime} }\ldots {\sigma }_{{\ell }}^{{\prime} })}_{A}}=\displaystyle \sum _{\displaystyle \genfrac{}{}{0em}{}{\genfrac{}{}{0em}{}{{\sigma }_{i},{\sigma }_{i}^{{\prime} }\notin \{A,B\},}{{\bar{\sigma }}_{i},{\bar{\sigma }}_{i}^{{\prime} },{\alpha }_{1}\ldots {\alpha }_{L-1}}}{{\alpha }_{1}^{{\prime} }\ldots {\alpha }_{L-1}^{{\prime} }}}({T}_{{\alpha }_{1}}^{{\sigma }_{1}{\bar{\sigma }}_{1}}{T}_{{\alpha }_{1}^{{\prime} }}^{{\sigma }_{1}^{{\prime} }{\bar{\sigma }}_{1}^{{\prime} }\dagger })\ldots ({T}_{{\alpha }_{i-1},{\alpha }_{i}}^{{\sigma }_{i}{\bar{\sigma }}_{i}}{T}_{{\alpha }_{i-1}^{{\prime} },{\alpha }_{i}^{{\prime} }}^{{\sigma }_{i}^{{\prime} }{\bar{\sigma }}_{i}^{{\prime} }\dagger })\ldots ({T}_{{\alpha }_{L-1}}^{{\sigma }_{L}{\bar{\sigma }}_{L}}{T}_{{\alpha }_{L-1}^{{\prime} }}^{{\sigma }_{L}^{{\prime} }{\bar{\sigma }}_{L}^{{\prime} }\dagger }).\end{eqnarray} \tag{ 14 }$$

the partially transposed ${\rho }_{A,B}^{{T}_{B}}$ is given by

$$\begin{eqnarray}{\rho }_{A,B}^{{T}_{B}} & = & \displaystyle \sum _{{\sigma }_{i},{\sigma }_{i}^{{\prime} }\in \{A,B\}}{C}_{{({\sigma }_{1}^{{\prime} }\ldots {\sigma }_{{\ell }}^{{\prime} })}_{B},{({\sigma }_{1}\ldots {\sigma }_{{\ell }})}_{B}}^{{({\sigma }_{1}\ldots {\sigma }_{{\ell }})}_{A},{({\sigma }_{1}^{{\prime} }\ldots {\sigma }_{{\ell }}^{{\prime} })}_{A}}\ | \ {({\sigma }_{1}\ldots {\sigma }_{i}\ldots {\sigma }_{{\ell }})}_{A},{({\sigma }_{1}\ldots {\sigma }_{i}\ldots {\sigma }_{{\ell }})}_{B}\rangle \\ & & \times \langle {({\sigma }_{1}^{{\prime} }\ldots {\sigma }_{i}^{{\prime} }\ldots {\sigma }_{{\ell }}^{{\prime} })}_{A},{({\sigma }_{1}^{{\prime} }\ldots {\sigma }_{i}^{{\prime} }\ldots {\sigma }_{{\ell }}^{{\prime} })}_{B}\ | .\end{eqnarray} \tag{ 15 }$$

Note that in equation (15), the partial transpose operation is performed by acting on block B by exchanging the indices in the coefficient matrix C.

In our calculations we consider a chain of length L = 200, which is sufficiently large that boundary or finite size effects can be neglected for both ground state and nonzero temperature. The two blocks have the same size ℓ and are situated in the middle of the chain, i.e., the positions of the left edge of each block are, respectively, $\tfrac{L}{2}\pm \tfrac{d}{2}$, with d the distance between them. For the TEBD calculations we ensure that our results are converged with respect to the bond dimension χ_max. In particular, we find that in both the phases (h/J < 1 or h/J > 1), χ_max = 32 is sufficient to get a converged results for block sizes ℓ = 1 ... 5. For larger values of ℓ > 5 it is difficult to go to higher values of χ_max due to larger memory requirement, however, we have checked carefully that the ground states of the calculations are converged with respect to the chosen χ_max values for all h/J, see figure 3. For nonzero temperature, we employ a second order Suzuki– Trotter decomposition with an imaginary time step of δβ = 0.005/J, to cool the system from β = 0.0 down to the considered temperature $\beta =\tfrac{1}{T}$.

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After having presented our numerical techniques, we will now present our results. In section 5.1 we discuss the entanglement properties for the ground state, and afterwards in subsection 5.2, we consider the case of thermal states.

5.1. Logarithmic negativity in ground states

5.2. Logarithmic negativity at nonzero temperature

Switching from zero to finite temperature, thermal excitations start to play an important role. For example, the one-dimensional TFIM has a phase transition only at zero temperature [79]. This means that the correlation length stays finite through all values of the transverse field h/J.

Figure 4 shows the logarithmic negativity as a function of temperature T/J and the field h/J. We consider a chain of L = 200 lattice sites, and each of the two partitions contains ℓ = 4 spins. From figures 4(a) to (d) we increase the distance d between the two partitions from d = 0 to d = 3. Generally we notice that the higher the temperature, the more entanglement is suppressed. This observation is in agreement with the expectation that thermal fluctuations tend to suppress quantum coherence and consequently entanglement. In the opposite regime of low temperature, the logarithmic negativity shows a peak in the vicinity of the quantum phase transition which also survives at nonzero temperature.

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We will now study quantitatively how the logarithmic negativity decays by increasing the distance d between the two partitions at finite temperature. Figure 5 shows ${{ \mathcal E }}_{{ \mathcal N }}$ as a function of the distance d for different values of the inverse temperature βJ at a fixed h/J = 1.0. At very large temperature, here βJ = 5.0 in figure 5(a), the thermal fluctuations have a strong influence on the entanglement. For the partitions of size ℓ = 1 the logarithmic negativity drops to zero immediately, i.e., ${d}^{* }=0$ means two spins are entangled only when they are nearest-neighbors. For ℓ ≥ 2, the logarithmic negativity vanishes after the separation of one lattice site, i.e., ${d}^{* }=1$.

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By reducing the temperature to βJ = 15, thermal fluctuations remain sufficiently strong to restrict the entanglement threshold considerably. As shown in figure 5(b), for ℓ = 1, the logarithmic negativity between two blocks vanishes as soon as the distance between them is more than zero site. For ℓ = 2 and ℓ = 3 however two blocks remain entangled for a few more sites but of substantially shorter distance compared to the ground state. The thermal fluctuations show their dominant effect better for larger block size. One can see this by looking at the cases ℓ = 4 and ℓ = 5. Both drop to zero at approximately the same distance. Reducing the temperature further, the effect of thermal fluctuations becomes smaller as expected. For example in figure 5 for βJ = 25.0 and βJ = 35.0, the logarithmic negativity for ℓ = 4 and ℓ = 5 drops to zero at different threshold distances as a consequence of the less dominant effect of thermal fluctuations. The value of ${d}^{* }$ for l = 3 has converged for these βJ's but not for ℓ = 4, 5.

The behavior of the entanglement threshold as function of temperature for different ℓ and transverse field is shown in figure 6. Away from criticality the entanglement threshold saturates quickly to a constant value for each ℓ, see figures 6(a) and (c). With reducing temperature, ${d}^{* }$ does not change and reaches to its final value at ground state which is an upper bound for ${d}^{* }$ at finite T. Let us point out that the yellow curve in figure 6(a) corresponding to ℓ = 3, seems to reach convergence already at βJ = 50.0. Nevertheless, this value differs by one lattice site from the result obtained for the ground state. This is due to how close we are to the ground state. The thermal activation of the lowest energy excitation is proportional to ${{\rm{e}}}^{-\beta s}$ with s the gap. For h/J = 1.0, we have s = 2π/L such that ${{\rm{e}}}^{-\beta s}\sim {10}^{-1}$ which means we need to be of much lower temperature to suppress thermal excitations.

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At criticality, the entanglement threshold ${d}^{* }$ increases with decreasing temperature, figure 6(b). For small ℓ ≤ 2 the value of ${d}^{* }$ converges to its value in the ground state at some temperature. The convergence to the entanglement threshold in the ground state becomes slower for large value of ℓ. In other words ${d}^{* }$ increases by increasing ℓ and reducing the temperature, see figure 6(b) for ℓ = 3–5.

In this section we want to shed some light on the sudden drop of the logarithmic negativity for two spins by providing a simple effective model. Of central importance in this analysis is the possibility of writing, on general grounds, any Hermitian operator of a L-spin system in terms of direct products of Pauli operators. In particular, we focus on the density matrix since it plays the main role in computing the logarithmic negativity:

$$\begin{eqnarray}&&\rho =\displaystyle \sum _{{n}_{1},\,\mathrm{...},\,{n}_{L}=1}^{4}{\rho }_{{n}_{1},\ldots ,{n}_{L}}\ {\sigma }_{{l}_{1}}^{{a}_{{n}_{1}}}\otimes ...\otimes {\sigma }_{{l}_{L}}^{{a}_{{n}_{L}}},\end{eqnarray} \tag{ 16 }$$

where a_m = 0, x, y, z, and ${\sigma }^{0}={{\mathbb{1}}}_{2}$ the 2 × 2 unit matrix. From equation (16), the density matrix is fully determined by the values of the correlation functions since ${\rho }_{{n}_{1},\ldots ,{n}_{L}}={\rm{Tr}}[\rho \ {\sigma }_{{l}_{1}}^{{a}_{1}}\otimes ...\otimes {\sigma }_{{l}_{L}}^{{a}_{L}}]$.

The case we study is a two-spin problem. We consider two spins and label the position of one of them at site 1 and the other at site 1 + d. This choice permits us to deal with a small number of correlation functions, leading to a simple and intuitive analytical condition for having non-vanishing logarithmic negativity. In particular we focus on the paramagnetic phase, where the structure of the reduced density matrix allows us to derive a condition for nonzero logarithmic negativity from an effective two-level system.

6.1. Reduced density matrix in the paramagnetic phase

In the paramagnetic phase, the 4 × 4 reduced density matrix ρ_A,B written in the basis $\{| \downarrow ,\downarrow \rangle ,| \downarrow ,\uparrow \rangle ,| \uparrow ,\downarrow \rangle ,| \uparrow ,\uparrow \rangle \}$, is characterized by having nonzero entries only on the diagonal and the anti-diagonal:

$$\begin{eqnarray}{\rho }_{A,B}=\ \left(\begin{array}{cccc}{\rho }_{A,B}(1,1) & 0 & 0 & {\rho }_{A,B}(1,4)\\ 0 & {\rho }_{A,B}(2,2) & {\rho }_{A,B}(2,3) & 0\\ 0 & {\rho }_{A,B}(3,2) & {\rho }_{A,B}(3,3) & 0\\ {\rho }_{A,B}(4,1) & 0 & 0 & {\rho }_{A,B}(4,4)\end{array}\right).\end{eqnarray} \tag{ 17 }$$

The reason for the vanishing of the other matrix elements is symmetries of the Hamiltonian, as one can directly see from writing those entries in terms of the respective two-point correlation functions. For example, let us consider ${\rho }_{A,B}(1,2)={\rho }_{0,x}-{\rho }_{z,x}+{\rm{i}}({\rho }_{0,y}-{\rho }_{z,y})$, in which ${\rho }_{{a}_{1},{a}_{1+d}}=\langle {\sigma }_{1}^{{a}_{1}}{\sigma }_{1+d}^{{a}_{1+d}}\rangle$. Since we are evaluating the correlation functions in the ground state and the system is symmetric under time-reversal, it follows that ρ_0,y = ρ_z,y = 0. Moreover, in the paramagnetic phase where the ground state does not break the ${{\mathbb{Z}}}_{2}$ symmetry, we also have ${\rho }_{0,x}={\rho }_{z,x}=0$. Taking into account all these considerations we conclude that ρ_A,B(1, 2) = 0 and similar argumentations hold for the other matrix elements ${\rho }_{A,B}(1,3),\ {\rho }_{A,B}(2,4)$ and ρ_A,B(3, 4).

The partial transpose of the density matrix is therefore determined by two uncoupled effective two-level systems:

$$\begin{eqnarray}{\rho }_{A,B}^{T}=\ \left(\begin{array}{cccc}{\rho }_{A,B}(1,1) & {\rho }_{A,B}(2,3) & 0 & 0\\ {\rho }_{A,B}(3,2) & {\rho }_{A,B}(4,4) & 0 & 0\\ 0 & 0 & {\rho }_{A,B}(2,2) & {\rho }_{A,B}(1,4)\\ 0 & 0 & {\rho }_{A,B}(4,1) & {\rho }_{A,B}(3,3)\end{array}\right).\end{eqnarray} \tag{ 18 }$$

For the sake of simplicity, let us focus only on one two-level system, since both of them have the same features:

$$\begin{eqnarray}{\rho }_{1}^{{\rm{TLS}}}=\ \left(\begin{array}{cc}{\rho }_{A,B}(1,1) & {\rho }_{A,B}(2,3)\\ {\rho }_{A,B}(3,2) & {\rho }_{A,B}(4,4)\end{array}\right).\end{eqnarray} \tag{ 19 }$$

Let us denote with δ = ρ_A,B(2, 3) the coupling between the two levels, with ${E}_{\pm }^{* }$ the eigenvalues of the matrix (19) and with ${E}_{-}={\rho }_{A,B}(1,1),{E}_{+}={\rho }_{A,B}(4,4)$ the unperturbed ones. For an illustration see figure 7.

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The picture of the two-level system in equation (18) gives a simple physical explanation for the spatial behavior of the logarithmic negativity. Although the reduced density matrix ρ_A,B always has positive eigenvalues since it is a semi-positively defined operator, the partially transposed matrix ${\rho }_{A,B}^{T}$ can have negative ones, when at least one of the two two-level systems has negative eigenvalues. This can lead to a non-vanishing logarithmic negativity.

6.1.1. Condition for non-vanishing logarithmic negativity

With increasing δ the splitting between E₊ and E₋ increases, which for sufficiently large δ turns one of the eigenvalues negative. In order to obtain a more quantitative description of the behavior of the logarithmic negativity, we solve the eigenvalue problem of the matrix (19), and similarly for the other two-level system, searching for the conditions which lead to a negative eigenvalue and therefore to a non-vanishing logarithmic negativity. As a result we obtain the following inequalities:

$$\begin{eqnarray}&&{\rho }_{A,B}^{2}(2,3)\gt {\rho }_{A,B}(1,1){\rho }_{A,B}(4,4),\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&{\rho }_{A,B}^{2}(1,4)\gt {\rho }_{A,B}(2,2){\rho }_{A,B}(3,3).\end{eqnarray} \tag{ 21 }$$

Equations (20) and (21) give a quantitative statement concerning how strong the couplings ρ_A,B(2, 3), ρ_A,B(1, 4) must be to lower the eigenvalue below zero.

To achieve a better physical intuition for the behavior of the logarithmic negativity as a function of distance, we express the conditions (20) and (21) in terms of the two-point correlation functions using the prescription in equation (16). These functions, in some particular limiting cases, are described by universal behaviors allowing a simple analysis of the conditions (20) and (21) and consequently it is possible to have a clear idea on the spatial structure of the logarithmic negativity for two spins. For simplicity we consider only equation (20), but similar observations hold for equation (21). Since

$$\begin{eqnarray}&&{\rho }_{A,B}(2,3)={\rho }_{x,x}+{\rho }_{y,y}\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&{\rho }_{A,B}(1,1)=1+{\rho }_{z,z}-{\rho }_{0,z}-{\rho }_{z,0}\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&{\rho }_{A,B}(4,4)=1+{\rho }_{z,z}+{\rho }_{0,z}+{\rho }_{z,0}.\end{eqnarray} \tag{ 24 }$$

Equation (20) reads

$$\begin{eqnarray}&&{(1-{\rho }_{z,z})}^{2}-{({\rho }_{z,0}-{\rho }_{0,z})}^{2}\\ &&\quad \lt {({\rho }_{x,x}-{\rho }_{y,y})}^{2}+{({\rho }_{x,y}+{\rho }_{y,x})}^{2}.\end{eqnarray} \tag{ 25 }$$

Equation (25) can be further simplified noting that the translational invariance of the system implies ρ_0,z = ρ_z,0. Moreover, the terms ${\rho }_{x,y}$ and ρ_y,x vanish because the entries of the reduced density matrix ρ_A,B have to be real due to time-reversal symmetry. Using all this information, equation (25) simplifies to

$$\begin{eqnarray}&&{(1-{\rho }_{z,z})}^{2}\lt {({\rho }_{x,x}-{\rho }_{y,y})}^{2}.\end{eqnarray} \tag{ 26 }$$

In other words, using the definition of the coefficients: ${\rho }_{x,x}=\langle {\sigma }_{1}^{x}{\sigma }_{d+1}^{x}\rangle ,{\rho }_{y,y}=\langle {\sigma }_{1}^{y}{\sigma }_{d+1}^{y}\rangle$ and ${\rho }_{z,z}=\langle {\sigma }_{1}^{z}{\sigma }_{d+1}^{z}\rangle$, we can rewrite equation (26) as following

$$\begin{eqnarray}&&{(1-\langle {\sigma }_{1}^{z}{\sigma }_{d+1}^{z}\rangle )}^{2}\lt {(\langle {\sigma }_{1}^{x}{\sigma }_{d+1}^{x}\rangle -\langle {\sigma }_{1}^{y}{\sigma }_{d+1}^{y}\rangle )}^{2}.\end{eqnarray} \tag{ 27 }$$

6.1.2. Vanishing logarithmic negativity at large distance

From equation (26) one can directly see the vanishing logarithmic negativity when the two spins are very far apart. In this regime, the correlation functions follow a generic behavior:

$$\begin{eqnarray}&&{\rho }_{x,x}=\langle {\sigma }_{1}^{x}{\sigma }_{d+1}^{x}\rangle \sim {{\rm{e}}}^{-d/{\xi }_{x}}\mathop{\longrightarrow }\limits_{d\to \infty }0,\end{eqnarray} \tag{ 28 }$$

$$\begin{eqnarray}&&{\rho }_{y,y}=\langle {\sigma }_{1}^{y}{\sigma }_{d+1}^{y}\rangle \sim {{\rm{e}}}^{-d/{\xi }_{y}}\mathop{\longrightarrow }\limits_{d\to \infty }0,\end{eqnarray} \tag{ 29 }$$

$$\begin{eqnarray}&&{\rho }_{z,z}=\langle {\sigma }_{1}^{z}{\sigma }_{d+1}^{z}\rangle \sim \langle {\sigma }_{1}^{z}\rangle \langle {\sigma }_{d+1}^{z}\rangle \ne 0.\end{eqnarray} \tag{ 30 }$$

Thus, in the limit $d\to \infty$, both ρ_x,x and ρ_y,y go to zero and therefore the inequality (26) cannot be satisfied leading to a vanishing logarithmic negativity. In addition, the two-level system description is able to predict that the logarithmic negativity is zero not only in the singular point $d=\infty$, but in an interval of nonzero extent $d\lt \infty$. For a general field h in the paramagnetic phase, $h\gt {h}_{c}$, both the magnetization and correlation along z are finite but smaller than one. Consequently, the diagonal elements in the matrix (19) are strictly larger than zero, as one can realize from equations (23) and (24). In order to argue the existence of a finite interval of distances where the logarithmic negativity vanishes, let us first begin from the case where the two spins are infinitely far apart from each other, meaning that the matrix (19) is diagonal because of the exponential suppression of the off-diagonal elements announced by equations (22), (28) and (29). As the distance d decreases, the off-diagonal element ρ_A(2, 3) = δ starts to have a nonzero value, affecting perturbatively the eigenvalue of the matrix (19). In particular, using perturbation theory in δ, the shift of the eigenvalues is proportional to the square of the coupling of the two-level system δ

$$\begin{eqnarray}&&{E}_{\pm }^{* }={E}_{\pm }\pm \displaystyle \frac{{\delta }^{2}}{{E}_{+}-{E}_{-}},\end{eqnarray} \tag{ 31 }$$

supposing that E_± are nondegenerate. Let us point out that the unperturbed eigenvalues E_±, appearing in equation (31), cannot assume negative values since they correspond to the diagonal elements of ρ_A,B which are probabilities. As a consequence, δ must be sufficiently large to make at least one eigenvalue negative. This can occur only when the distance between the two spins is less than a certain threshold, $d\lt \tilde{d}$, since the strength of δ is exponentially suppressed with d as suggested by equations (22), (28) and (29).

While the condition for nonzero logarithmic negativity in equation (26) holds also for small distances d, the exponential structures of the correlation functions in equations (28), (29) are no longer valid since they describe the asymptotic behavior in the limit $d\to \infty$. Nevertheless, the strength of δ decreases with d, as we observe from the nonzero entanglement between the two spins in the paramagnetic phase at short-distance, see figure 3 panels (d) and (e). Moreover, nonzero entanglement between two spins at short distances was already shown in a variety of works [37–41].

6.2. Reduced density matrix at the critical point

In this work we have studied the spatial entanglement structure of the transverse-field Ising chain at zero and nonzero temperatures. Specifically, we have investigated the logarithmic negativity between two disjoint blocks of equal size ℓ as a function of their separation, which is an entanglement analog to a quantum correlation function. While the cases of ℓ = 1 and of large blocks has been considered already in the literature [37–41], we studied the crossover regime between those limits.

We have found that for any fixed size ℓ of the blocks there exists an entanglement threshold at a distance ${d}^{* }$ beyond which the logarithmic negativity vanishes identically. This holds across the whole phase diagram of the system including also the quantum critical point where the system exhibits long-ranged quantum correlations. The influence of temperature onto the spatial entanglement structure as measured by the logarithmic negativity depends crucially on the size ℓ of the blocks. The larger ${d}^{* }$ (for increasing ℓ) the more important the influence of temperature, cutting off long-range entanglement.

For small blocks ℓ the entanglement threshold ${d}^{* }$ appears on short distances on the order of a few lattice spacings even at the quantum critical point. In this case the precise value of ${d}^{* }$ is determined by nonuniversal short-distance properties that depend on the microscopic details of the model. However, using a simple effective model we have found for the case ℓ = 1 that the existence of the threshold ${d}^{* }$ can be derived solely from the universal long-distance properties.

A vanishing logarithmic negativity for blocks of size ℓ = 1 implies that the two corresponding qubits are unentangled, because the PPT criterion (whose violation is measured by the negativity) for the separability of a quantum state is not only necessary but also sufficient. For larger blocks ℓ > 1 the PPT criterion is not sufficient anymore, such that a vanishing logarithmic negativity at distances larger than ${d}^{* }$ does not necessarily imply that the two blocks are completely unentangled. Thus, we cannot exclude that there exist other measures signaling nonzero entanglement. However, it is important to note that the logarithmic negativity gives a bound on the distillable entanglement, such that a vanishing logarithmic negativity implies that no Bell pairs can be extracted from the state.

At first sight the already known result of a finite entanglement threshold ${d}^{* }\lt \infty$ for ℓ = 1 at the critical point might not comply with expectations originating from strong quantum correlations or the well-established violation of the area law for the entanglement entropy. The results of our work provide a quantitative description of the crossover from ℓ = 1 to ℓ ≫ 1 upon increasing ℓ.

We have studied the spatial entanglement structure for the transverse-field Ising chain so that it is a natural question to which extent our results extend to a broader class of systems. The effective model for the reduced density matrix at ℓ = 1, which we used to argue about the existence of an entanglement threshold, can be straightforwardly applied to other models as well, independent of the dimension provided the blocks consist of spin-1/2 degrees of freedom and the system resides in a paramagnetic phase. Our conclusions also hold for the critical point whenever the quantum correlations are long-ranged along one particular direction. This might change, for example, in the case the transition is associated with a broken U(1) instead of ${{\mathbb{Z}}}_{2}$ symmetry. For larger block sizes ℓ > 1 the situation is much less clear on general grounds and deserves a further investigation.

We acknowledge Roderich Mössner for useful discussions. SB acknowledges support from DST, India, through Ramanujan Fellowship Grant No. SB/S2/RJN-128/2016. JHB was supported by the ERC Starting Grant No. 679722 and the Knut and Alice Wallenberg Foundation 2013-0093. MH acknowledges the support of the Deutsche Forschungsgemeinschaft via the Gottfried Wilhelm Leibniz Prize program.