Beyond the Landau-Ginzburg-Wilson paradigm: analytical studies of non-standard quantum criticality

Even a century after Heisenberg, Born, Jordan, and others laid the foundations of modern quantum mechanics, quantum phenomena continue to fascinate and puzzle us, with no end to their mysteries in sight. A particularly rich and active field, grounded in a variety of deeply rooted quantum-mechanical concepts, is that of quantum phase transitions, i.e., phase transitions at zero temperature controlled by nonthermal parameters. While a remarkable number of quantum phase transitions are successfully described within the Landau-Ginzburg-Wilson paradigm, particular attention is drawn to cases where a description within the paradigm fails. Prominent examples that evade the Landau-Ginzburg-Wilson paradigm are phase transitions beyond conventional symmetry breaking and phase transitions enriched by additional gapless modes. This dissertation seeks to further advance the theory of phase transitions beyond the Landau-Ginzburg-Wilson paradigm by proposing and analyzing several novel routes beyond the framework and relating them to established concepts in the field. First, we identify an exotic quasiuniversal behavior near the all-in-all-out Weyl quantum critical point in three-dimensional Luttinger semimetals, such as the pyrochlore iridates R₂Ir₂O₇, with R a rare-earth element. The quasiuniversal behavior is characterized by power laws with exponents that vary slowly over several orders of magnitude in energy or length. However, in contrast to the quasiuniversality discussed in the context of deconfined criticality, the present case is characterized by a genuinely universal ultra-low-temperature behavior. In this limit, the pertinent critical exponents can be computed exactly within a renormalization group analysis. This exotic behavior will manifest generically at any quantum critical fixed point possessing a marginally irrelevant operator. Experimental implications for the pyrochlore iridates are outlined and compared with recent experimental observations in Y₂Ir₂ $O₇ thin films. Second, we propose a novel mechanism for continuous order-to-order quantum phase transitions that, unlike deconfined criticality, operates independently of fractionalization. The proposed mechanism is based on the collision and annihilation of two renormalization group fixed points: a quantum critical fixed point and an infrared stable fixed point. The annihilation of these fixed points rearranges the flow topology, eliminating the disordered phase associated with the infrared stable fixed point and promoting a second critical fixed point, unaffected by the collision, to a quantum critical point between distinct orders. We argue that this mechanism is relevant to a broad spectrum of physical systems. In particular, it can manifest in Luttinger fermion systems in three spatial dimensions, leading to a continuous quantum phase transition between an antiferromagnetic Weyl semimetal state, which breaks time-reversal symmetry, and a nematic topological insulator, characterized by broken lattice rotational symmetry. This continuous antiferromagnetic-Weyl-to-nematic-insulator transition might be observed in rare-earth pyrochlore iridates R₂Ir₂O₇. Other possible realizations include kagome quantum magnets, quantum impurity models, and quantum chromodynamics with supplemental four-fermion interactions. This broad spectrum of potential realizations provides an ideal setting to explore the proposed fixed-point-annihilation mechanism and its interrelations with other concepts, such as fractionalization. Third, we present multiple strands of ongoing research, all ultimately working towards a putative Sachdev-Ye-Kitaev fixed point in three spatial dimensions. As a starting point, we investigate the large-$ N$ behavior of a partially bosonized Luttinger fermion theory at a large but finite number of quadratic band touching points N. Building on this, we analyze a random variant of the deterministic Luttinger fermion model, now in the limit of infinitely many fermionic and bosonic degrees of freedom, motivated by low-dimensional Yukawa-Sachdev-Ye-Kitaev theories. In both cases, complications, ultimately resulting from the absence of Lorentz symmetry, arise and motivate the study of a slightly simpler analogue, a relativistic Luttinger fermion theory. Not only does this third study improve our understanding of Luttinger fermion systems at large N in general, but it also reveals a novel fermionic conformal fixed point, which corresponds to an infrared stable fixed point in 8/3 < D < 4 space-time dimensions and to an ultraviolet stable fixed point in 4 < D < 6 space-time dimensions. While the former gives rise to a previously unknown conformal phase, the latter unveils a new asymptotically safe quantum field theory.Even a century after Heisenberg, Born, Jordan, and others laid the foundations of modern quantum mechanics, quantum phenomena continue to fascinate and puzzle us, with no end to their mysteries in sight. A particularly rich and active field, grounded in a variety of deeply rooted quantum-mechanical concepts, is that of quantum phase transitions, i.e., phase transitions at zero temperature controlled by nonthermal parameters. While a remarkable number of quantum phase transitions are successfully described within the Landau-Ginzburg-Wilson paradigm, particular attention is drawn to cases where a description within the paradigm fails. Prominent examples that evade the Landau-Ginzburg-Wilson paradigm are phase transitions beyond conventional symmetry breaking and phase transitions enriched by additional gapless modes. This dissertation seeks to further advance the theory of phase transitions beyond the Landau-Ginzburg-Wilson paradigm by proposing and analyzing several novel routes beyond the framework and relating them to established concepts in the field. First, we identify an exotic quasiuniversal behavior near the all-in-all-out Weyl quantum critical point in three-dimensional Luttinger semimetals, such as the pyrochlore iridates R₂Ir₂O₇, with R a rare-earth element. The quasiuniversal behavior is characterized by power laws with exponents that vary slowly over several orders of magnitude in energy or length. However, in contrast to the quasiuniversality discussed in the context of deconfined criticality, the present case is characterized by a genuinely universal ultra-low-temperature behavior. In this limit, the pertinent critical exponents can be computed exactly within a renormalization group analysis. This exotic behavior will manifest generically at any quantum critical fixed point possessing a marginally irrelevant operator. Experimental implications for the pyrochlore iridates are outlined and compared with recent experimental observations in Y₂Ir₂$O₇ thin films. Second, we propose a novel mechanism for continuous order-to-order quantum phase transitions that, unlike deconfined criticality, operates independently of fractionalization. The proposed mechanism is based on the collision and annihilation of two renormalization group fixed points: a quantum critical fixed point and an infrared stable fixed point. The annihilation of these fixed points rearranges the flow topology, eliminating the disordered phase associated with the infrared stable fixed point and promoting a second critical fixed point, unaffected by the collision, to a quantum critical point between distinct orders. We argue that this mechanism is relevant to a broad spectrum of physical systems. In particular, it can manifest in Luttinger fermion systems in three spatial dimensions, leading to a continuous quantum phase transition between an antiferromagnetic Weyl semimetal state, which breaks time-reversal symmetry, and a nematic topological insulator, characterized by broken lattice rotational symmetry. This continuous antiferromagnetic-Weyl-to-nematic-insulator transition might be observed in rare-earth pyrochlore iridates R₂Ir₂O₇. Other possible realizations include kagome quantum magnets, quantum impurity models, and quantum chromodynamics with supplemental four-fermion interactions. This broad spectrum of potential realizations provides an ideal setting to explore the proposed fixed-point-annihilation mechanism and its interrelations with other concepts, such as fractionalization. Third, we present multiple strands of ongoing research, all ultimately working towards a putative Sachdev-Ye-Kitaev fixed point in three spatial dimensions. As a starting point, we investigate the large-$N$ behavior of a partially bosonized Luttinger fermion theory at a large but finite number of quadratic band touching points N. Building on this, we analyze a random variant of the deterministic Luttinger fermion model, now in the limit of infinitely many fermionic and bosonic degrees of freedom, motivated by low-dimensional Yukawa-Sachdev-Ye-Kitaev theories. In both cases, complications, ultimately resulting from the absence of Lorentz symmetry, arise and motivate the study of a slightly simpler analogue, a relativistic Luttinger fermion theory. Not only does this third study improve our understanding of Luttinger fermion systems at large N in general, but it also reveals a novel fermionic conformal fixed point, which corresponds to an infrared stable fixed point in 8/3 < D < 4 space-time dimensions and to an ultraviolet stable fixed point in 4 < D < 6 space-time dimensions. While the former gives rise to a previously unknown conformal phase, the latter unveils a new asymptotically safe quantum field theory.

Jan 1, 20260 citations

Three studies in quantum criticality

R. R. Ramazashvili

Jan 1, 19980 citations

Effective critical exponents in the large n Landau Ginzburg Wilson model

Jyoti Sharma

Jan 1, 20070 citations

Non Equilibrium Quantum Criticality: an intuitive approach

Emanuele G. Dalla Torre, Eugene Demler, Thierry Giamarchi +1 more

Bulletin of the American Physical SocietyFeb 28, 20120 citations

Mott Fermionic "Quantum" Criticality Beyond Ginzburg-Landau-Wilson

Zaira Nazario, David Santiago

The Mott critical point between a metal and a correlated insulator has usually been studied via density or spin density bosonic mode fluctuations according to the standard Ginzburg-Landau-Wilson phase transition paradigm. A moment's reflection leads to increasing doubts that such an approach should work as the transition is nonmagnetic, voiding the relevance of spin density modes. Charge density modes are irrelevelant since the long range Coulomb interaction leads to a large plasmon gap and their incompressibility. In solidarity with these doubts, recent measurements of the Mott critical point in low dimensional organic materials yield critical exponents in violent diasagreement with the bosonic mode criticality lore. We propose that fermionic fluctuations control the behavior of the Mott transition. The transition thus has an intrinsic quantum aspect despite being a finite temperature phase transition. We develop this hitherto unexplored physics, obtain experimental predictions and find agreement with one of the novel unexplained experimental exponents. We conclude that this Mott transition corresponds to a new universality class of finite temperature critical points that contains quantum effects and cannot be accounted for by conventional Ginzburg-Landau-Wilson wisdom.The Mott critical point between a metal and a correlated insulator has usually been studied via density or spin density bosonic mode fluctuations according to the standard Ginzburg-Landau-Wilson phase transition paradigm. A moment's reflection leads to increasing doubts that such an approach should work as the transition is nonmagnetic, voiding the relevance of spin density modes. Charge density modes are irrelevelant since the long range Coulomb interaction leads to a large plasmon gap and their incompressibility. In solidarity with these doubts, recent measurements of the Mott critical point in low dimensional organic materials yield critical exponents in violent diasagreement with the bosonic mode criticality lore. We propose that fermionic fluctuations control the behavior of the Mott transition. The transition thus has an intrinsic quantum aspect despite being a finite temperature phase transition. We develop this hitherto unexplored physics, obtain experimental predictions and find agreement with one of the novel unexplained experimental exponents. We conclude that this Mott transition corresponds to a new universality class of finite temperature critical points that contains quantum effects and cannot be accounted for by conventional Ginzburg-Landau-Wilson wisdom.

arXiv:0705.3653May 24, 20070 citations

O₇ thin films. Second, we propose a novel mechanism for continuous order-to-order quantum phase transitions that, unlike deconfined criticality, operates independently of fractionalization. The proposed mechanism is based on the collision and annihilation of two renormalization group fixed points: a quantum critical fixed point and an infrared stable fixed point. The annihilation of these fixed points rearranges the flow topology, eliminating the disordered phase associated with the infrared stable fixed point and promoting a second critical fixed point, unaffected by the collision, to a quantum critical point between distinct orders. We argue that this mechanism is relevant to a broad spectrum of physical systems. In particular, it can manifest in Luttinger fermion systems in three spatial dimensions, leading to a continuous quantum phase transition between an antiferromagnetic Weyl semimetal state, which breaks time-reversal symmetry, and a nematic topological insulator, characterized by broken lattice rotational symmetry. This continuous antiferromagnetic-Weyl-to-nematic-insulator transition might be observed in rare-earth pyrochlore iridates R₂Ir₂O₇. Other possible realizations include kagome quantum magnets, quantum impurity models, and quantum chromodynamics with supplemental four-fermion interactions. This broad spectrum of potential realizations provides an ideal setting to explore the proposed fixed-point-annihilation mechanism and its interrelations with other concepts, such as fractionalization. Third, we present multiple strands of ongoing research, all ultimately working towards a putative Sachdev-Ye-Kitaev fixed point in three spatial dimensions. As a starting point, we investigate the large-

N$ behavior of a partially bosonized Luttinger fermion theory at a large but finite number of quadratic band touching points N. Building on this, we analyze a random variant of the deterministic Luttinger fermion model, now in the limit of infinitely many fermionic and bosonic degrees of freedom, motivated by low-dimensional Yukawa-Sachdev-Ye-Kitaev theories. In both cases, complications, ultimately resulting from the absence of Lorentz symmetry, arise and motivate the study of a slightly simpler analogue, a relativistic Luttinger fermion theory. Not only does this third study improve our understanding of Luttinger fermion systems at large N in general, but it also reveals a novel fermionic conformal fixed point, which corresponds to an infrared stable fixed point in 8/3 < D < 4 space-time dimensions and to an ultraviolet stable fixed point in 4 < D < 6 space-time dimensions. While the former gives rise to a previously unknown conformal phase, the latter unveils a new asymptotically safe quantum field theory.