In modern physics, spontaneous symmetry breaking is one of the most important and ubiquitous ideas. It is well known that the breakdown of global and continuous symmetry gives rise to gapless excitations called Nambu-Goldstone (NG) bosons. This is called NG theorem. Since the generators of broken symmetries are bosonic, the excitations are also bosons. On the other hand, excitations are thought of as fermionic when a generator of broken symmetry is fermionic. The most famous example of fermionic symmetry is supersymmetry (SUSY). SUSY is a symmetry that exchanges bosons and fermions. In high-energy physics, it is known that spontaneous SUSY breaking gives rise to massless modes called NG fermions or Nambu- Goldstinos. In non-relativistic systems, fewer examples of models with SUSY are studied. Therefore, spontaneous SUSY breaking in non-relativistic systems is less understood than that in nonrelativistic systems or spontaneous symmetry breaking. In this thesis, we introduce interacting Majorana models with supersymmetry (SUSY) and discuss the properties in terms of spontaneous SUSY breaking and NG modes.
In modern physics, spontaneous symmetry breaking is one of the most important and ubiquitous ideas. It is well known that the breakdown of global and continuous symmetry gives rise to gapless excitations called Nambu-Goldstone (NG) bosons. This is called NG theorem. Since the generators of broken symmetries are bosonic, the excitations are also bosons. On the other hand, excitations are thought of as fermionic when a generator of broken symmetry is fermionic. The most famous example of fermionic symmetry is supersymmetry (SUSY). SUSY is a symmetry that exchanges bosons and fermions. In high-energy physics, it is known that spontaneous SUSY breaking gives rise to massless modes called NG fermions or Nambu- Goldstinos. In non-relativistic systems, fewer examples of models with SUSY are studied. Therefore, spontaneous SUSY breaking in non-relativistic systems is less understood than that in nonrelativistic systems or spontaneous symmetry breaking. In this thesis, we introduce interacting Majorana models with supersymmetry (SUSY) and discuss the properties in terms of spontaneous SUSY breaking and NG modes.
We introduce and study a lattice fermion model in one dimension with explicit N=1 supersymmetry (SUSY). The Hamiltonian of the model is defined by the square of a supercharge built from Majorana fermion operators. The model describes interacting Majorana fermions, and its properties depend only on a single parameter g. When g=1, we find that SUSY is unbroken and the ground states are identical to those of the frustration-free Kitaev chains. We also find a parameter regime in which SUSY is restored in the infinite-volume limit. For sufficiently large g, we prove that SUSY is spontaneously broken and the low-lying excitations are gapless, which can be thought of as Nambu-Goldstone fermions. We then show numerically that these gapless modes have cubic dispersion at long wavelengths.
The subject of condensed matter physics is very rich — there are an infinite number of parameters producing a diversity of exciting phenomena. As a theorist, my goal is to distill general principles out of this complexity — to construct theories that can coherently explain many known examples altogether.This thesis is composed of several attempts to develop such theories in topics related to spontaneously symmetry breaking. A remarkable feature of many-body interacting systems is that although they are described by equations respecting various symmetries, they may spontaneously organize into a state that explicitly breaks symmetries. Examples are numerous: various types of crystalline and magnetic orders, Bose-Einstein condensates of cold atoms, superfluids of liquid helium, chiral symmetry in QCD, neutron stars, and cosmic inflation.These systems with spontaneously broken continuous symmetries have gapless excitations, so called Nambu-Goldstone bosons (NGBs). Although the properties of NGBs are well understood in Lorentz-invariant systems, surprisingly, some basic properties of NGBs such as their number and dispersion in nonrelativistic systems have not been discussed from a general perspective. In the first part of this thesis, we solve this issue by developing and analyzing an effective Lagrangian that coherently captures the low-energy, long-distance physics of many different symmetry-breaking states all at once.Next, we examine whether these NGBs originating from spontaneous symmetry breaking remain to be well-defined excitations inside a metal, where low-energy electrons near Fermi surface can collide with them. Our result is a one equation criterion that specifies whether the interactions between electrons and NGBs can be ignored, or whether it completely changes their character. In the latter case, unusual phases of matter such as non-Fermi liquids may arise; in that case, NGBs are overdamped and cannot form particle-like excitations in spite of the assumed symmetry breaking.In the last part of this thesis, we investigate the possibility of spontaneously breaking of Hamiltonian itself. The homogeneity of time is one of the most fundamental symmetries of nature, underlying the conservation of the energy. The question is whether this symmetry can be spontaneously broken, as suggested recently by Wilczek, in analogy with ordinary crystals. Contrary to his proposal that attracted a significant attention and stimulated many further studies, we prove a no-go theorem that rules out spontaneously breaking of time translation, in the ground state or in the canonical ensemble of a general Hamiltonian.
When global continuous symmetries are spontaneously broken, there appear gapless collective excitations called Nambu-Goldstone modes (NGMs) that govern the low-energy property of the system. The application of this famous theorem ranges from high-energy, particle physics to condensed matter and atomic physics. When a symmetry breaking occurs in systems that lack the Lorentz invariance to start with, as is usually the case in condensed matter systems, the number of resulting NGMs can be fewer than that of broken symmetry generators, and the dispersion of NGMs is not necessarily linear. In this article, we review recently established formulas for NGMs associated with broken internal symmetries that work equally for relativistic and nonrelativistic systems. We also discuss complexities of NGMs originating from space-time symmetry breaking. In the process we cover many illuminating examples from various context. We also present a complementary point of view from the Lieb-Schultz-Mattis theorem.
The subject of condensed matter physics is very rich — there are an infinite number of parameters producing a diversity of exciting phenomena. As a theorist, my goal is to distill general principles out of this complexity — to construct theories that can coherently explain many known examples altogether.This thesis is composed of several attempts to develop such theories in topics related to spontaneously symmetry breaking. A remarkable feature of many-body interacting systems is that although they are described by equations respecting various symmetries, they may spontaneously organize into a state that explicitly breaks symmetries. Examples are numerous: various types of crystalline and magnetic orders, Bose-Einstein condensates of cold atoms, superfluids of liquid helium, chiral symmetry in QCD, neutron stars, and cosmic inflation.These systems with spontaneously broken continuous symmetries have gapless excitations, so called Nambu-Goldstone bosons (NGBs). Although the properties of NGBs are well understood in Lorentz-invariant systems, surprisingly, some basic properties of NGBs such as their number and dispersion in nonrelativistic systems have not been discussed from a general perspective. In the first part of this thesis, we solve this issue by developing and analyzing an effective Lagrangian that coherently captures the low-energy, long-distance physics of many different symmetry-breaking states all at once.Next, we examine whether these NGBs originating from spontaneous symmetry breaking remain to be well-defined excitations inside a metal, where low-energy electrons near Fermi surface can collide with them. Our result is a one equation criterion that specifies whether the interactions between electrons and NGBs can be ignored, or whether it completely changes their character. In the latter case, unusual phases of matter such as non-Fermi liquids may arise; in that case, NGBs are overdamped and cannot form particle-like excitations in spite of the assumed symmetry breaking.In the last part of this thesis, we investigate the possibility of spontaneously breaking of Hamiltonian itself. The homogeneity of time is one of the most fundamental symmetries of nature, underlying the conservation of the energy. The question is whether this symmetry can be spontaneously broken, as suggested recently by Wilczek, in analogy with ordinary crystals. Contrary to his proposal that attracted a significant attention and stimulated many further studies, we prove a no-go theorem that rules out spontaneously breaking of time translation, in the ground state or in the canonical ensemble of a general Hamiltonian.
We study a model of interacting spinless fermions in a one dimensional lattice with supersymmetry (SUSY). The Hamiltonian is given by the anticommutator of two supercharges Q and ${Q}^{\ifmmode\dagger\else\textdagger\fi{}}$, each of which is comprised solely of fermion operators and possesses one adjustable parameter g. When the parameter g vanishes, the model is identical to the one studied by Nicolai [H. Nicolai, J. Phys. A 9, 1497 (1976)], where the zero-energy ground state is exponentially degenerate. On the other hand, in the large-g limit the model reduces to the free-fermion chain with a four-fold degenerate ground state. We show that for finite chains SUSY is spontaneously broken when g>0. We also rigorously prove that for sufficiently large g the ground-state energy density is nonvanishing in the infinite-volume limit. We further analyze the nature of the low-energy excitations by employing various techniques such as rigorous inequalities, exact numerical diagonalization, and renormalization group method with bosonization. The analysis reveals that the low-energy excitations are described by massless Dirac fermions (or Thirring fermions more generally), which can be thought of as Nambu-Goldstone fermions from the spontaneous SUSY breaking.
We introduce a lattice fermion model in one spatial dimension with supersymmetry (SUSY) but without particle number conservation. The Hamiltonian is defined as the anticommutator of two nilpotent supercharges Q and ${Q}^{\ifmmode\dagger\else\textdagger\fi{}}$. Each supercharge is built solely from spinless fermion operators and depends on a parameter g. The system is strongly interacting for small g, and in the extreme limit g=0, the number of zero-energy ground states grows exponentially with the system size. By contrast, in the large-g limit, the system is noninteracting and SUSY is broken spontaneously. We study the model for modest values of g and show that under certain conditions spontaneous SUSY breaking occurs in both finite and infinite chains. We analyze the low-energy excitations both analytically and numerically. Our analysis suggests that the Nambu-Goldstone fermions accompanying the spontaneous SUSY breaking have cubic dispersion at low energies.