Fix typos, citation readme

README.md

@@ -181,9 +181,21 @@ Publications which make use of MPSDynamics:
      * [10.5281/zenodo.4352728](https://doi.org/10.5281/zenodo.4352728)
 
 # Citation
-
-If you use this package in your research, please cite it:
-* Dunnett, A. (2021). angusdunnett/MPSDynamics: (v1.0). Zenodo. https://doi.org/10.5281/zenodo.5106435
+If you use the package in your research, please consider citing it.
+You can add the Zenodo record to your BibTex file:
+
+```tex
+@misc{mpsdynamics_zenodo2021,
+        title = {shareloqs/{MPSDynamics}},
+        shorttitle = {{MPSDynamics.jl}},
+        url = {https://zenodo.org/record/5106435},
+        abstract = {Tensor network simulations for finite temperature, open quantum system dynamics},
+        publisher = {Zenodo},
+        author = {Dunnett, Angus and Lacroix, Thibaut and Le Dé, Brieuc and Riva, Angela},
+        year = {2021},
+        doi = {10.5281/zenodo.5106435},
+}
+```
 
 # How to Contribute
 Contributions are welcome! Don't hesitate to contact us if you

docs/src/examples/anderson-model.md

@@ -406,7 +406,7 @@ plot(p2, p3, p4, p5, p1, layout = (3, 2), size = (1400, 1200))
 ```
 
 ________________
-### References
+## Bibliography
 
 [^khon_efficient_2021]: 
     > Kohn, L.; Santoro, G. E. Efficient mapping for anderson impurity problems with matrix product states. Phys. Rev. B 2021, 104 (1), 014303. https://doi.org/10.1103/PhysRevB.104.014303.

docs/src/examples/bath-observables.md

@@ -194,7 +194,7 @@ plot(p1, p2, p3, p4, layout = (2, 2), size = (1400, 1200))
 ```
 
 ___________________
-## References
+## Bibliography
 
 [^chin_exact_2010]:
     > Chin, A. W.; Rivas, Á.; Huelga, S. F.; Plenio, M. B. Exact Mapping between System-Reservoir Quantum Models and Semi-Infinite Discrete Chains Using Orthogonal Polynomials. Journal of Mathematical Physics 2010, 51 (9), 092109. https://doi.org/10.1063/1.3490188.

docs/src/examples/protontransfer.md

@@ -2,9 +2,9 @@
 
 ## Context
 
-The MPS formalism can also be used for physical chemistry problems [^chin_ESIPT_2024]. One development done with MPSDynamics is the introduction of a reaction coordinate tensor, allowing the system to be described in space. This description requires the tensor of the system to be linked to another tensor representing an harmonic oscillator. 
+The MPS formalism can also be used for physical chemistry problems [^chin_ESIPT_2024]. One development done with the `MPSDynamics.jl` package is the introduction of a reaction coordinate tensor, allowing the system to be described in space. This description requires the tensor of the system to be linked to another tensor representing an harmonic oscillator. 
 
-Here is an example of two electronic configurations : the enol named `|e\rangle` and the keto named `|k\rangle`. The reaction coordinate oscillator is expressed as RC.
+Here is an example of two electronic configurations undergoing a tautomerization : the enol named $|e\rangle$ and the keto named $|k\rangle$. The reaction coordinate oscillator is expressed as RC and represents the reaction path of the proton.
 ```math
 H_S + H_{RC} + H_{int}^{S-RC} = \omega^0_{e} |e\rangle \langle e| + \omega^0_{k} |k\rangle \langle k| + \Delta (|e\rangle \langle k| + |k\rangle \langle e|) + \omega_{RC} (d^{\dagger}d + \frac{1}{2}) + g_{e} |e\rangle \langle e|( d + d^{\dagger})+ g_{k} |k \rangle \langle k|( d + d^{\dagger})
 ```
@@ -17,7 +17,8 @@ H_B + H_{int}^{RC-B} = \int_{-∞}^{+∞} \mathrm{d}k \omega_k b_k^\dagger b_k -
 
 
 ## The code
-First we load the `MPSdynamics.jl` package to be able to perform the simulation, the `Plots.jl` one to plot the results, and the `LaTeXStrings.jl` one to be able to use ``\LaTeX`` in the plots. The function [`MPSDynamics.disp`](@ref) is also imported.
+
+First we load the `MPSdynamics.jl` package to be able to perform the simulation, the `Plots.jl` one to plot the results, and the `LaTeXStrings.jl` one to be able to use ``\LaTeX`` in the plots.
 
 ```julia
 using MPSDynamics, Plots, LaTeXStrings, ColorSchemes, PolyChaos, LinearAlgebra
@@ -320,7 +321,7 @@ end
 display(gif(anim, "gif_reducedrho.gif", fps = 2.5))
 ```
 ________________
-### References
+## Bibliography
 
 [^chin_ESIPT_2024]:
     > Le Dé, B.; Huppert, S.; Spezia, R.; Chin, A.W Extending Non-Perturbative Simulation Techniques for Open-Quantum Systems to Excited-State Proton Transfer and Ultrafast Non-Adiabatic Dynamics https://arxiv.org/abs/2405.08693

docs/src/theory.md

@@ -2,11 +2,11 @@
 
 ## Chain-Mapping of bosonic environments
 
-We consider, in the Schrödinger picture, a general Hamiltonian where a non-specified system interacts linearly with a bosonic environments
+We consider, in the Schrödinger picture, a general Hamiltonian where a non-specified system interacts linearly with a bosonic environments ($\hbar = 1$)
 
 ```math
 \begin{aligned}
-    \hat{H} =& \hat{H}_S + \int_0^{+\infty} \hbar\omega\hat{a}^\dagger_\omega\hat{a}_\omega\ \mathrm{d}\omega + \hat{A}_S\int_0^{+\infty}\sqrt{J(\omega)}\left(\hat{a}_\omega + \hat{a}^\dagger_\omega\right)\mathrm{d}\omega
+    \hat{H} =& \hat{H}_S + \int_0^{+\infty} \omega\hat{a}^\dagger_\omega\hat{a}_\omega\ \mathrm{d}\omega + \hat{A}_S\int_0^{+\infty}\sqrt{J(\omega)}\left(\hat{a}_\omega + \hat{a}^\dagger_\omega\right)\mathrm{d}\omega
 \end{aligned}
 ```
 
@@ -89,7 +89,7 @@ This modified bath has the same correlation function $S(t)$ and thus allows us t
 The Hamiltonian of the system interacting with this extended bath now includes temperature-dependent interactions:
 
 ```math
-\hat{H} = \hat{H}_S + \int_{-\infty}^{+\infty} d\omega \omega \hat{a}_\omega^\dagger \hat{a}_\omega + \hat{A}_S \otimes \int_{-\infty}^{+\infty} d\omega \sqrt{J(\omega,\beta)}\left(\hat{a}_\omega^\dagger+\hat{a}_\omega\right),
+\hat{H} = \hat{H}_S + \int_{-\infty}^{+\infty} \mathrm{d}\omega \omega \hat{a}_\omega^\dagger \hat{a}_\omega + \hat{A}_S \otimes \int_{-\infty}^{+\infty} \mathrm{d}\omega \sqrt{J(\omega,\beta)}\left(\hat{a}_\omega^\dagger+\hat{a}_\omega\right),
 ```
 
 This method simplifies the simulation of finite temperature effects by treating them within an effective zero-temperature framework, thereby keeping the computational advantages of using pure states. In conclusion: the dynamics of the system resulting from the interaction with the original bath, starting in a thermal state at finite temperature, is exactly the same as the one resulting from the interaction with the extended environment, starting in the vacuum state at zero temperature. Once computed the chain coefficients at a given inverse temperature $\beta$, the time evolution of the vacuum state interacting with the extended environment can be efficiently simulated using MPS time evolution methods.
@@ -264,6 +264,7 @@ There exist different versions of the TDVP algorithm. In `MPSDynamics.jl` three
 
 The main advantage of the one-site 1TDVP algorithm is that it preserves the unitarity of the MPS during the time evolution. Its main problem, conversely, is that the time evolution is constrained to happen on a manifold constituted by tensors of fixed bond dimension, a quantity closely related to the amount of entanglement in the MPS, and such a bond dimension has therefore to be fixed before the beginning of the time evolution. This strategy will necessarily be non optimal: the growth of the bond dimensions required to describe the quantum state should ideally mirror the entanglement growth induced by the time evolution. 2TDVP allows for such a dynamical growth of the bond dimensions, and therefore better describes the entanglement in the MPS. It suffers however of other drawbacks: first of all, a truncation error is introduced (by the means of an SVD decomposition), which entails a loss of unitarity of the time-evolved MPS. Moreover, 2TDVP has bad scaling properties with the size of the local dimensions of the MPS: this is a major issue when dealing with bosons. The DTDVP algorithm combines the best features of 1TDVP and 2TDVP: it preserves unitarity, it has the same scaling properties of 1TDVP, and it adapts the bond dimensions to the entanglement evolution at each site and at each time-step. DTDVP does not suffer from a truncation error, but introduces only a projection error.
 
+________________
 ## Bibliography
 [^chin_exact_2010]:
     > Chin, A. W.; Rivas, Á.; Huelga, S. F.; Plenio, M. B. Exact Mapping between System-Reservoir Quantum Models and Semi-Infinite Discrete Chains Using Orthogonal Polynomials. Journal of Mathematical Physics 2010, 51 (9), 092109. https://doi.org/10.1063/1.3490188.