What are the differences between Classical Monte Carlo methods and Quantum Monte Carlo methods in condensed matter physics? If one want to study strongly correlated systems with Quantum Monte Carlo method, does he/she need to study Classical Monte Carlo method first? (just like if you want to study quantum mechanics you shall study classical mechanics first?)
[Physics] Difference between Monte Carlo and Quantum Monte Carlo methods
computational physicscondensed-matter
Related Solutions
The slave particle approach is based on the assumption of spin-charge separation in the strongly correlated electron systems (typically Mott insulators). It was proposed that the electrons can decay into spinons and chargons (holons/doublons). But to preserve the fermion statistics of the electrons, the spinon-chargon bound state must be fermionic, so the simplest way is to ascribe the fermion statistics to one of them: if the spinon is fermionic then the chargon should be bosonic (slave-boson), or if the chargon is fermionic then the spinon should be bosonic (slave-fermion). The differences are just a matter of which degrees of freedom (spin or charge) should the fermion statistics be ascribed to.
Within one kind of slave particle formalism, the supersymmetry is possible but not necessary. Whether or not the spinon and chargon are suppersymmetric to each other depends on their spectrums (which are the details). If the spinon and chargon have different spectrums (which is always the case), then the effective theory has no supersymmetry.
It seems that there is a certain kind of duality between the slave-boson and slave-fermion approaches, one may conjecture if the two approaches can be unified into a single theory. And it was indeed the case. Now we know that the two approaches are just two low-energy effective theories of the complete-fractionalization theory, which has two equivalent versions: the Chern-Simons version by Kou, Qi and Weng Phys. Rev. B 71, 235102 (2005), or the Majorana version by Xu and Sachdev Phys. Rev. Lett. 105, 057201 (2010).
In the complete-fractionalization theory, electron is fully fractionalized into bosonic spinon, bosonic chargon, and a mutual Chern-Simons gauge field (or Majorana fermion) to take care of the fermion statistics. Spinons and chargons are treated equally in this theory, both are bosonic degrees of freedom. In the mutual Chern-Simons theory, the spinons and chargons are coupled to a Chern-Simons theory of the K matrix $$K=\left(\begin{matrix}0&2\\2&0\end{matrix}\right),$$ meaning that the spinon and the chargon will mutually see each other as a $\pi$-vortex. This mutual $\pi$-vortex binding renders the spinon-chargon bound state a fermion, corresponding to the electron. If both the spinons and chargons are gapped, the remaining low-energy effective theory will be a $\mathbb{Z}_2$ gauge theory, which supports 3 kinds of topological excitations: electric charges, magnetic fluxes (visons) and fermions; corresponding to chargons, spinons and electrons respectively. Thus in this phase, the relation between the chargon and the spinon is just like that between the charge and the flux, i.e. the spin and charge degrees of freedom are electromagnetically dual to each other, and the duality is supported by the underlying $\mathbb{Z}_2$ topological order. This exotic topological phase was proposed to be the basic physics behind the pseudo-gap phase of the cuprates superconductors Phys. Rev. Lett. 106, 147002 (2011) (although more complexity should be added to fully explain the phenomena).
By tuning the relative density of spinons and chargons (presumably achievable by doping in real materials), the system can be driven into the ordered phase by condensing one of the fractionalized degrees of freedom (note that now both spinons and chargons are bosons and can condense). Condensing spinon results in a spin ordered state (i.e. the Neel antiferromagnet), while condensing chargon results in a charge superfluid state (i.e. d-wave superconductivity). But the mutual Chern-Simons theory forbidden both spinon and chargon to condense at the same time, which is consistent with the fact that we can never Bose-condense the electrons.
The Majorana theory is similar, but more comprehensive. In SU(2) operator matrix form, the decomposition reads $$C=B\Xi Z,$$ where $C$, $B$, $Z$ collect electron, chargon and spinon operators in matrices $$C=\left(\begin{matrix}c_\uparrow & c_\downarrow\\-c_\downarrow^\dagger & c_\uparrow^\dagger\end{matrix}\right),B=\left(\begin{matrix}b_d & b_h^*\\-b_h & b_d^*\end{matrix}\right),Z=\left(\begin{matrix}z_\uparrow & z_\downarrow\\-z_\downarrow^* & z_\uparrow^*\end{matrix}\right),$$ and $\Xi=\xi_0\sigma_0+i\xi_1\sigma_1+i\xi_2\sigma_2+i\xi_3\sigma_3$ contains the Majorana operators. Both $b$ and $z$ are bosonic, and the fermion statistics is carried by the Majorana fermions $\xi$. This is the most general scheme of spin-charge separation for electrons. The emergent gauge structure of this theory is $O(4)$ (representing the rotation among 4 Majorana fermions), which can be factorized into two $SU(2)$ gauge structures, as $O(4)\simeq SU(2)_B\times SU(2)_Z$, coupling to the chargons and spinons respectively. Because the $SU(2)$ fluctuation is confining in (2+1)D spacetime, without any topological order, the fractionalized particles will all be confined into electrons. But if we condense one of the bosonic degrees of freedom, say the chargons $B$, then $SU(2)_B$ can be Higgs out, and the remaining $SU(2)_Z$ gauge fluctuation would confine the Majorna fermion $\Xi$ with the spinon $Z$ into a composite particle $\Xi Z$, rendering the bosonic spinon to fermionic and reducing the Majorana theory to the slave-boson theory (which is exactly the same as Wen's $SU(2)$ theory Quantum Orders and Symmetric Spin Liquids). If we condense the spinon first, then the story will be reversed, ending up with the slave-fermion theory. Therefore the choice of which slave-particle approach depends entirely on the phase we wish to study (the degrees of freedom that we wish to condense). Different orders in the ground state would support different low-energy effective theories, which appear to us as different slave-particle approaches.
The mapping between the quantum and the classical system is formal, but as you say, we can usually interpret a quantum phase transition of a $d$ dimensional quantum system (that is, a phase transition at zero temperature) as a (classical) phase transition in a $d+1$ dimensional classical system. The temperature of the quantum system maps to the size of the $d+1$th dimension of the classical system. Furthermore, a quantum phase transition is at zero temperature, and therefore is driven by a non-thermal control parameter (an external pressure, a magnetic field, etc.). Usually, we can modelize the effect of this control parameter by the parameter $r_0$ in front of the quadratic term in the action, which changes sign at the transition: $r_0 \phi^2$.
Here enter the confusion: in classical systems, one also assume that the parameter $r_0$ drives the transition (changes sign), and one usually assumes that $r_0\propto (T-T_c)$, where $T$ is the temperature of the classical system, and $T_c$ is the (meanfield) critical temperature. But this has nothing to do with the quantum-classical mapping, and it is just specific to stat-mech.
Best Answer
That depends on what you want to achieve and what literature you have access to. Monte Carlo (MC) and Quantum MC are based on the same methods. Usually textbooks describe MC first and then build on this to explain QMC. One good resource to start is this book: Thijssen: Computational Physics. It does not cover everything in depth but gives a good overview.
Another good resource is this course website. It's free and includes many many code examples.
From the way I learned about those things, I would answer your question with: yes, you should at least take a look at MC before you start with Quantum MC.
EDIT: I just found this resource as well. A good intro to Quantum MC.