Quantum Phase Transitions Computational Challenges

They will be discussed in more detail later in this chapter. However, quantum Monte Carlo methods for fermions suffer from the notorious sign problem that originates in the antisymmetry of the many-fermion wave function and hampers the simulation severely. Techniques developed for dealing with the sign problem often reintroduce biases into the method, via, for instance. 

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In this section we discuss briefly—without any pretense of completeness— further computational approaches to quantum phase transitions. The conceptually simplest method for solving a quantum many-particle problem is (numerically) exact diagonalization. However, as already discussed in the section on Quantum Phase Transitions Computational Challenges, the exponential increase of the Hilbert space dimension with the number of degrees of freedom severely limits the possible system sizes. One can rarely simulate more than a few dozen particles even for simple lattice systems. Systems of this size are too small to study quantum phase transitions (which are a property of the thermodynamic limit of infinite system size) with the exception of, perhaps, certain simple one-dimensional systems. Even in one dimension, however, more powerful methods have largely superceded exact diagonalization. 

The purpose of this chapter is twofold In the following two sections. Phase Transitions and Critical Behavior and Quantum vs. Classical Phase Transitions, we give a concise introduction into the theory of quantum phase transitions, emphasizing similarities with and differences from classical thermal transitions. After that, we point out the computational challenges posed by quantum phase transitions, and we discuss a number of successful computational approaches together with prototypical examples. However, this chapter is not meant to be comprehensive in scope. We rather want to help scientists who are taking their first steps in this field to get off on the right foot. Moreover, we want to provide experimentalists and traditional theorists with an idea of what simulations can achieve in this area (and what they cannot do,. .. yet). Those readers who want to learn more details about quantum phase transitions and their applications should consult one of the recent review articles or the excellent textbook on quantum phase transitions by Sachdev. ... 

Perhaps the most important aspect of energetic salts that needs to be understood for their energetic applications is the mechanisms of thermal decomposition. The immediate challenge is to use computations, since experimental measurements are in many cases not feasible, to determine the initial chemical reactions for various conditions, i.e., phase, temperature, and pressure. This is critical for understanding both combustion and detonation. Quantum chemistry methods can be used to compute bond-dissociation energies and transition-state... 

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