Small corrections in the English manual in the section of Algorithm

docs/sphinx/en/algorithm/algorithms.rst

@@ -183,7 +183,7 @@ Real-time evolution using iTPS (and other tensor network states) differs signifi
 
 One major difference is the size of the quantum entanglement of the target quantum state. In imaginary time evolution, as the evolution progresses towards the ground state, the quantum entanglement of the state does not become excessively large. Thus, the description by iTPS works well. However, in real-time evolution, typically (unless the initial state's iTPS is an eigenstate of the Hamiltonian), quantum entanglement can increase over time. To maintain the approximation accuracy of iTPS, it is necessary to increase the bond dimension of iTPS as the time gets longer. Naturally, increasing the bond dimension also increases computational costs, so with realistic computational resources, accurately approximating real-time evolution using iTPS is limited to short times. The applicable time range depends on the model, but for example, in spin models, the limit is often around a time :math:`t = O(1/J)` with respect to the typical interaction strength :math:`J`.
 
-Another difference is the characteristics of the physical phenomenon to be reproduced. When using imaginary time evolution to calculate the ground state, it is sufficient to reach the ground state after a sufficiently long evolution, so minor deviations from the correct path of imaginary time evolution are not a significant issue. On the other hand, in real-time evolution, there is often interest not only in the final state but also in the time evolution of the quantum state itself. To accurately approximate the path of time evolution, it is necessary to not only increase the bond dimension of iTPS but also to make the time increment :math:`\delta t` of the Suzuki-Trotter decomposition sufficiently small. Depending on the situation, it may be more efficient to use higher-order Suzuki-Trotter decompositions. In TeNeS, it is possible to handle higher-order Suzuki-Trotter decompositions by editing the ``evolution`` section of the input file that is ultimately entered into tenes.
+Another difference is the characteristics of the physical phenomenon to be reproduced. When using imaginary time evolution to calculate the ground state, it is sufficient to reach the ground state after a sufficiently long evolution, so minor deviations from the correct path of imaginary time evolution are not a significant issue. On the other hand, in real-time evolution, there is often interest not only in the final state but also in the time evolution of the quantum state itself. To accurately approximate the path of time evolution, it is necessary to not only increase the bond dimension of iTPS but also to make the time increment :math:`\delta t` of the Suzuki-Trotter decomposition sufficiently small. Depending on the situation, it may be more efficient to use higher-order Suzuki-Trotter decompositions. In TeNeS, it is possible to handle higher-order Suzuki-Trotter decompositions by editing the ``evolution`` section of the input file that is ultimately entered into TeNeS.
 
 Finite temperature simulation
 ===========================
@@ -262,7 +262,7 @@ Similarly to pure states, once the converged corner transfer matrices and edge t
 .. image:: ../../img/trace_Sz.*
    :align: center
 
-, the local magnetization :math:`\mathrm{Tr} (\rho S_i^z)` is calculated using the same diagram as :math:`\langle \Psi|S_i^z|\Psi\rangle`.
+the local magnetization :math:`\mathrm{Tr} (\rho S_i^z)` is calculated using the same diagram as :math:`\langle \Psi|S_i^z|\Psi\rangle`.
 
 Lastly, it is important to mention the drawbacks of approximation by iTPO. The density matrix of a mixed state is Hermitian and positive semidefinite, with non-negative eigenvalues. However, when approximating the density matrix with iTPO, this positive semidefiniteness is not guaranteed, and physical quantities calculated from the iTPO approximation might exhibit unphysical behavior, such as energies lower than the ground state energy. This is a problem of iTPO representation, and cannot be avoided just by improving the accuracy of CTMRG in expectation value calculation by increasing the bond dimension :math:`\chi`. To recover physical behavior, it is necessary to increase the bond dimension :math:`D` of iTPO to improve the approximation accuracy of the density matrix.