Reduced density matrix and entanglement of interacting quantum field theories with Hamiltonian truncation

Entanglement is the fundamental difference between classical and quantum systems and has become one of the guiding principles in the exploration of high- and low-energy physics. The calculation of entanglement entropies in interacting quantum field theories, however, remains challenging. Here, we present the first method for the explicit computation of reduced density matrices of interacting quantum field theories using truncated Hamiltonian methods. The method is based on constructing an isomorphism between the Hilbert space of the full system and the tensor product of Hilbert spaces of subintervals. This naturally enables the computation of the von Neumann and arbitrary Rényi entanglement entropies as well as mutual information, logarithmic negativity, and other measures of entanglement. Our method is applicable to equilibrium states and nonequilibrium evolution in real time. It is model independent and can be applied to any Hamiltonian truncation method that uses a free basis expansion. We benchmark the method on the free Klein-Gordon theory finding excellent agreement with the analytic results. We further demonstrate its potential on the interacting sine-Gordon model, studying the scaling of von Neumann entropy in ground states and real-time dynamics following quenches of the model.

Figure 1

Schematic drawing of the algorithm. (Left) We split the system on the top by representing the field $\varphi$ living on the full interval $[0,L]$ with an equivalent setting: a pair of fields ${\varphi }_L$ and ${\varphi }_R$ living on subintervals $[0,\ell ]$ and $[\ell ,L]$ with an additional boundary condition at $\ell$. The figure depicts the case of Neumann boundary conditions at the cut. The boundary conditions at the edges are chosen to be Dirichlet. (Right) Quantization of the fields gives rise to two isomorphic Hilbert spaces $\mathcal{H}$ and ${\mathcal{H}}_{LR}={\mathcal{H}}_L\otimes {\mathcal{H}}_R$ and a unitary map between them. This maps density matrices to a form suitable for taking partial traces. The cones in the figure represent the exponentially growing number of states with the energy above the ground state. $\mathcal{H}$ and ${\mathcal{H}}_{LR}$ are generated on top of different vacua (horizontal line below the cone).

Figure 2

Illustration of the $M$ matrix for different cutoff schemes. The amount of coefficients for the left and the right partition in the matrices $u$ and $v$ differ depending on the cutoff scheme. The coefficients belonging to the left (right) side are displayed in red (blue).

Figure 3

Correlations $\langle \varphi \left(x\right)\varphi (L-x)\rangle$ of the Klein-Gordon model with mass $m=1.0$. The plot shows the correlations of the system computed with the original (full) mode decomposition of the fields as a reference. The system is cut at $\ell /L=1/3$.

Figure 4

Spatially resolved von Neumann entropy for the Klein-Gordon model at different masses $m$ (displayed in different colors). Different methods are encoded in the linestyle. The massless case is compared to the CFT result while the massive cases are compared to covariance matrices computations.

Figure 5

von Neumann entropy of thermal states of the Klein-Gordon system with $m=5$. The dots are the results of HT and the dashed lines are covariance matrix computations. Entropies at finite temperature are computed from a Boltzmann distribution at temperature $T$. The curve at zero temperature uses the ground state of the system.

Figure 6

Spatially resolved von Neumann entropy of the sine-Gordon model for Neumann boundary conditions at the cut. The curves are plotted for the interaction parameter $\lambda =7$ and soliton mass $M=25$ and for $\lambda =17$ for two different mass values, $M=25$ at $M=60.29$. At $M=60.29$ the gap (mass of the first breather) agrees with that of the $\lambda =7$ case and is $m_1=11.13L$. Thus, the correlation length is less than one tenth of the system size. The curves display a logarithmic scaling of the entanglement entropy.

Figure 7

Time evolution of the von Neumann entropy after quenches in the Klein-Gordon model. The system is split at $\ell /L=0.5$. Different panels show quenches from and to different masses (as indicated in the insets). Solid lines are results from HT and dashed lines represent the covariance matrix results. The inset in panel (d) details the behavior of the quench from $m_0=50$ to $m=200$ at short times.

Figure 8

Time evolution of the von Neumann entropy after quenches. The top panel shows the Klein-Gordon evolution of the entropy at $\ell /L=0.5$ for a quench from boson mass $m_0=7$ to $m=12$. The bottom panel depicts a quench of the sine-Gordon model for $\lambda =7$ from a soliton mass of $M_0=15.73$ to $M=26.96$. The masses are chosen such that the first breather masses $m_1$ of the sG model agree with the KG boson masses.

Figure 9

Frequency spectrum of the von Neumann entropy $S_N$ time evolution after a sine-Gordon mass quench from soliton mass $M_0=15$ to $M=20$ at $\lambda =7$. The spectrum is obtained using a discrete Fourier transform. The amplitude at frequencies $\omega$ is compared against energy levels of (moving) breathers. The breather energies are computed using reflection factors [21, 57, 58, 59], the expressions are listed in Appendix pp6. Due to the charge conjugation symmetry, only $C$-even breather state frequencies appear. The inset shows the original time evolution of the quench.

Figure 10

Check of the symplectic properties of the transformation from full modes to partial modes, Eq. (A1), for $s_F=10$. The fixed cutoff scheme does not faithfully reproduce the bosonic commutation relations except for the case of $x=0.5$. In this case, the two schemes coincide.

Figure 11

Effect of rounding schemes for the number of modes $s_L$ and $s_R$ in the left and right partition for Dirichlet boundary conditions at the cut. The system is a Klein-Gordon model with mass $m=1$. The transformation yields the correct bosonic commutation on commensurate cuts (green) with distance $1/s_F$. If we do not cut at commensurate splits, the $s_L$ can be either floored to the next integer (blue) or rounded (orange).

Figure 12

Tree to build all tuples of configurations of $\vec{n}=(2,2,2)$. The expression in parentheses are the ${\mathcal{J}}_i$ that still have to be distributed. The tuple that is added to the configuration at every step is noted in brackets. The final configuration of every path can be assembled by following the arrows and collecting the entries of all brackets. The missing leaf in the last layer on the right indicates a configuration that cannot be build due to the restrictions on the tuples.