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arXiv:1305.0006 · Physical Review B

Un-Fermi liquids: Unparticles in strongly correlated electron matter

Philip Phillips, Brandon W. Langley, Jimmy A. Hutasoit

Published Sep 17, 2013• 42 citations
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Abstract

Since any nontrivial infrared dynamics in strongly correlated electron matter must be controlled by a critical fixed point, we argue that the form of the single-particle propagator can be deduced simply by imposing scale invariance. As a consequence, the unparticle picture proposed by Georgi14 is the natural candidate to describe such dynamics. Unparticle stuff is scale-invariant matter with no particular mass. Scale invariance dictates that the propagator has an algebraic form which can admit zeros and hence is a candidate to explain the ubiquitous pseudogap state of the cuprates. We refer to the nonperturbative electronic state formed out of unparticles as an un-Fermi liquid. We show that the underlying action of the continuous mass formulation of unparticles can be recast as an action in anti--de Sitter space which serves as the generating functional for the propagator. We find that this mapping fixes the scaling dimension of the unparticle to be ${d}_{U}=d/2+\sqrt{{d}^{2}+4}/2$ and ensures that the corresponding propagator has zeros with $d$ the space-time dimension of the unparticle field. Should $d=2+1$, unparticles acquire the nontrivial phase $2\ensuremath{\pi}{d}_{U}$ upon interchange. Because ${d}_{U}$ is noninteger and in general not half integer, clockwise and counterclockwise interchange of unparticles do not lead to the same phase and time-reversal symmetry is broken spontaneously as reported in numerous experiments in the pseudogap phase of the cuprates. The possible relevance of this mechanism to such experiments is discussed. We then formulate the analogous BCS gap using unparticles and find that in contrast to the Fermi-liquid case, the transition temperature increases as the attractive interaction strength decreases, indicating that unparticles are highly susceptible to a superconducting instability.

DOI
10.1103/physrevb.88.115129

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K.N.P. Kumar, C.J. Prabhakara, S.K. Narasimha Murthy +1 more

Since any non-trivial infrared dynamics in strongly correlated electron matter must be controlled by a critical fixed point, we argue that the form of the single-particle propagator can be deduced simply by imposing scale invariance. As a consequence, the unparticle picture proposed by Georgi\cite {georgi} is the natural candidate to describe such dynamics. Unparticle stuff is scale-invariant matter with no particular mass. Scale invariance dictates that the propagator has an algebraic form which can admit zeros and hence is a candidate to explain the ubiquitous pseudogap state of the cuprates. The non-perturbative electronic state formed out of unparticles we refer to as an un-Fermi liquid. Philip W. Phillips, Brandon W. Langley, Jimmy A. Hutasoit show that the underlying action of the continuous mass formulation of unparticles can be recast exactly as an action in anti de Sitter space. They find that this mapping fixes the scaling dimension of the unparticle to be dU=d/2+d2+4 The full paper: http://www.iiste.org/PDFshare/APTA-PAGENO-354010-358233.pdf

Advances in physics theories and applications•Jan 1, 2013•0 citations

Beyond Particles: Unparticles in Strongly Correlated Electron Matter

Philip Phillips

I am concerned in these lectures with the breakdown of the particle concept in strongly correlated electron matter. I first show that the standard procedure for counting particles, namely Luttinger's theorem, breaks down anytime pole-like excitations are replaced by ones that have a divergent self-energy. Such a breakdown obtains in electronic systems whose pole-like excitations do not extend to the edge of the Brillouin zone, as in Fermi arcs in the cuprates. Since any non-trivial infrared dynamics in strongly correlated electron matter must be controlled by a critical fixed point, unparticles are the natural candidate to explain the presence of charged degrees of freedom that have no particle content. The continuous mass formulation of unparticles is recast as an action in anti de Sitter space. Such an action serves as the generating functional for the propagator. This mapping fixes the scaling dimension of the unparticle to be $d_U=d/2+\sqrt{d^2+4}/2$ and ensures that the corresponding propagator has zeros with $d$ the spacetime dimension of the unparticle field. The general {\it dynamical} mechanism by which bulk operators, such as the Pauli term, couple to the scaling dimension of the boundary operator and thereby lead to a vanishing of the spectral weight at zero energy is reviewed in the context of unparticles and zeros. The analogue of the BCS gap equations with unparticles indicates that the transition temperature increases as the attractive interaction strength decreases, indicating that unparticles are highly susceptible to a superconducting instability.

arXiv:1412.1098•Jul 22, 2015•2 citations

Power-law liquid in cuprate superconductors from fermionic unparticles

Zhidong Leong, Chandan Setty, Kridsanaphong Limtragool +1 more

Recent photoemission spectroscopy measurements (T. J. Reber et al., arXiv:1509.01611) on cuprate superconductors have inferred that over a wide range of doping, the imaginary part of the electron self-energy scales as ${\mathrm{\ensuremath{\Sigma}}}^{\ensuremath{''}}\ensuremath{\sim}{({\ensuremath{\omega}}^{2}+{\ensuremath{\pi}}^{2}{T}^{2})}^{a}$ with $a=1$ in the overdoped Fermi-liquid state and $a<0.5$ in the optimal to underdoped regime. We show that this non-Fermi-liquid scaling behavior can naturally be explained by the presence of a scale-invariant state of matter known as unparticles. We evaluate analytically the electron self-energy due to interactions with fermionic unparticles. We find that, in agreement with experiments, the imaginary part of the self-energy scales with respect to temperature and energy as ${\mathrm{\ensuremath{\Sigma}}}^{\ensuremath{''}}\ensuremath{\sim}{T}^{2+2\ensuremath{\alpha}}$ and ${\ensuremath{\omega}}^{2+2\ensuremath{\alpha}}$, where $\ensuremath{\alpha}$ is the anomalous dimension of the unparticle propagator. In addition, the calculated occupancy and susceptibility of fermionic unparticles, unlike those of normal fermions, have significant spectral weights even at high energies. This unconventional behavior is attributed to the branch cut in the unparticle propagator which broadens the unparticle spectral function over a wide energy range and nontrivially alters the scattering phase space by enhancing (suppressing) the intrinsic susceptibility at low energies for negative (positive) $\ensuremath{\alpha}$. Our work presents evidence suggesting that unparticles might be important low-energy degrees of freedom in strongly coupled systems such as the cuprate superconductors.

arXiv:1705.07130•Physical review. B./Physical review. B•Nov 1, 2017•17 citations

Realizing infrared power-law liquids in the cuprates from unparticle interactions

Kridsanaphong Limtragool, Chandan Setty, Zhidong Leong +1 more

Recent photoemission experiments [1] reveal that the excitations along the nodal region in the strange metal phase of the cuprates, rather than corresponding to poles in the single-particle Green function, exhibit power-law scaling as a function of frequency and temperature. Because such power-law scaling is indicative of a scale-invariant sector, as a first step, we perturbatively evaluate the electron self-energy due to interactions with scale-invariant unparticles. We focus on a ${G}_{0}W$-type diagram with an interaction $W$ mediated by a bosonic scalar unparticle. We find that in the high-temperature limit, the imaginary part of the self-energy $\mathrm{Im}\mathrm{\ensuremath{\Sigma}}$ is linear in temperature. In the low-temperature limit, $\mathrm{Im}\mathrm{\ensuremath{\Sigma}}$ exhibits the same power law in both temperature and frequency, with an exponent that depends on the scaling dimension of an unparticle operator. Such behavior is qualitatively consistent with the experimental observations. We then expand the unparticle propagator into coherent and incoherent contributions and study how the incoherent part violates the density of states (DOS) and density-density correlation function sum rules (f-sum rule). Such violations, in principle, can be observed experimentally. Our work indicates that the physical mechanism for the origin of the power-law scaling is the incoherent background, which is generated from the Mott-scale physics.

arXiv:1608.06637•Physical review. B./Physical review. B•Dec 8, 2016•14 citations

Fermi liquids beyond the forward-scattering limit: The role of nonforward scattering for scale invariance and instabilities

Han Ma, Sung-Sik Lee

Landau Fermi liquid theory is a fixed-point theory of metals that includes the forward-scattering amplitudes as exact marginal couplings. However, the fixed-point theory that only includes the strict forward scatterings is nonlocal in real space. In this paper, we revisit the Fermi liquid theory for charge-neutral fermions using the field-theoretic functional renormalization group formalism and show how the scale-invariant fixed point emerges as a local theory, which includes not only the forward scatterings but also nonforward scatterings with small but nonzero momentum transfers. In the low-energy limit, the near-forward scattering and pairing interactions take scale-invariant forms if the momentum transfer and the center-of-mass momentum of Cooper pairs, respectively, are comparable to the energy. The coupling functions fully capture the universal low-energy dynamics of the collective modes and instabilities of Fermi liquids. A runaway flow of the coupling function in the particle-hole channel beyond a critical interaction suggests an instability toward an ordered phase with a wave vector that depends on the interaction strength. At the critical interaction, the instability corresponds to the uniform Pomeranchuk or Stoner instability, but the momentum of the leading instability becomes nonzero for stronger attractive interaction. In the particle-particle channel, the coupling function reveals the dynamics of the unstable mode associated with the BCS instability. When an unstable normal metal evolves into the superconducting state, there exists a period in which a superconducting state with spatially nonuniform phase appears due to the presence of unstable Cooperon modes with nonzero momenta.

arXiv:2302.06828•Physical review. B./Physical review. B•Jan 23, 2024•7 citations

Non-perturbative approaches to strongly correlated electron systems

Ka Wai Lo

"Strong electron correlation phenomena are ubiquitous, such as unconventional superconductivity and the metal-insulator transitions. Having a proper understanding of strongly correlated electron systems requires a non-perturbative analysis because Fermi liquid theory inevitably breaks down due to the strong correlation. In this thesis, we will study two non-perturbative approaches to tackle strongly correlated electronic system: 1.) multi-dimensional bosonization and 2.) the AdS/CFT correspondence. Multidimensional bosonization is the generalization of bosonization to spatial dimensions larger than one. The bosonized theory is an effective field theory in terms of the bosonic particle-hole fluctuations in different patches of the Fermi surface. The bosonized theory is quadratic and hence can be solved non-perturbatively. The AdS/CFT correspondence is a conjecture that d-dimensional field theory is dual to a d+1 dimensional quantum gravitational theory. Problems in strongly coupled field theory then have an equivalent description using weakly interacting gravitational theory, allowing a non-perturbative analysis. Chapter 1 will serve as an introduction. We will review several experiments which show a breakdown of Fermi liquid theory. Formalisms for multidimensional bosonization and the AdS/CFT correspondence will then be reviewed. In chapter 2, the two-orbital Hubbard model with degenerate d{xz} and d{yz} orbitals are investigated. We apply multidimensional bosonization to solve this problem exactly and discover a z=3 overdamped collective modes that emerges at the orbital-ordering quantum critical point. These modes modify the single-particle density of states and lead to non-Fermi liquid behavior which can provide a possible explanation for the recently observed zero-bias enhancement in the point contact spectroscopy signal on iron pnictides. Chapters 3, 4 and 5 discuss the applications of the AdS/CFT correspondence to model various condensed matter systems. In chapter 3, We consider an interaction term between a bulk spinor field and a gauge field in the Reissner-Nordstr\""om AdS background. When the Pauli interaction term is large enough, a dynamical gap is generated and spectral weight transfer is observed in the spectral density, mimicking the behavior of the Hubbard model. We further consider the finite temperature case and discover that the ratio between the dynamical gap and the critical temperature has the same order as that of VO^2. The Pauli coupling is also studied in the superconducting background. Chapter 4 consider the propagation of a neutral scalar field in the geometry called the electron star. The electron star has Lifshitz scaling with finite dynamical critical exponent at the interior, which is suitable for the modeling of quantum criticality and quantum phase transition with a neutral order parameter, for example for antiferromagnetism. We find that the quantum phase transition has the Berezinski-Kosterlitz-Thouless characteristic and the dynamical critical exponent can change across the quantum critical point. In chapter 5, we construct a gravity dual of the nematic phase by studying the condensation of a spin-two field in the Schwarzschild-AdS background. The condensation of a spin-two field can distort the rotational symmetry of the Fermi surface as long as the spin-two field is coupled to a probe spinor field in the bulk gravity dual."

•Jan 1, 2013•0 citations
Since any nontrivial infrared dynamics in strongly correlated electron matter must be controlled by a critical fixed point, we argue that the form of the single-particle propagator can be deduced simply by imposing scale invariance. As a consequence, the unparticle picture proposed by Georgi14 is the natural candidate to describe such dynamics. Unparticle stuff is scale-invariant matter with no particular mass. Scale invariance dictates that the propagator has an algebraic form which can admit zeros and hence is a candidate to explain the ubiquitous pseudogap state of the cuprates. We refer to the nonperturbative electronic state formed out of unparticles as an un-Fermi liquid. We show that the underlying action of the continuous mass formulation of unparticles can be recast as an action in anti--de Sitter space which serves as the generating functional for the propagator. We find that this mapping fixes the scaling dimension of the unparticle to be dU=d/2+d2+4/2{d}_{U}=d/2+\sqrt{{d}^{2}+4}/2dU​=d/2+d2+4​/2 and ensures that the corresponding propagator has zeros with ddd the space-time dimension of the unparticle field. Should d=2+1d=2+1d=2+1, unparticles acquire the nontrivial phase 2\ensuremathπdU2\ensuremath{\pi}{d}_{U}2\ensuremathπdU​ upon interchange. Because dU{d}_{U}dU​ is noninteger and in general not half integer, clockwise and counterclockwise interchange of unparticles do not lead to the same phase and time-reversal symmetry is broken spontaneously as reported in numerous experiments in the pseudogap phase of the cuprates. The possible relevance of this mechanism to such experiments is discussed. We then formulate the analogous BCS gap using unparticles and find that in contrast to the Fermi-liquid case, the transition temperature increases as the attractive interaction strength decreases, indicating that unparticles are highly susceptible to a superconducting instability.
Since any non-trivial infrared dynamics in strongly correlated electron matter must be controlled by a critical fixed point, we argue that the form of the single-particle propagator can be deduced simply by imposing scale invariance. As a consequence, the unparticle picture proposed by Georgi\cite {georgi} is the natural candidate to describe such dynamics. Unparticle stuff is scale-invariant matter with no particular mass. Scale invariance dictates that the propagator has an algebraic form which can admit zeros and hence is a candidate to explain the ubiquitous pseudogap state of the cuprates. The non-perturbative electronic state formed out of unparticles we refer to as an un-Fermi liquid. Philip W. Phillips, Brandon W. Langley, Jimmy A. Hutasoit show that the underlying action of the continuous mass formulation of unparticles can be recast exactly as an action in anti de Sitter space. They find that this mapping fixes the scaling dimension of the unparticle to be dU=d/2+d2+4 The full paper: http://www.iiste.org/PDFshare/APTA-PAGENO-354010-358233.pdf
I am concerned in these lectures with the breakdown of the particle concept in strongly correlated electron matter. I first show that the standard procedure for counting particles, namely Luttinger's theorem, breaks down anytime pole-like excitations are replaced by ones that have a divergent self-energy. Such a breakdown obtains in electronic systems whose pole-like excitations do not extend to the edge of the Brillouin zone, as in Fermi arcs in the cuprates. Since any non-trivial infrared dynamics in strongly correlated electron matter must be controlled by a critical fixed point, unparticles are the natural candidate to explain the presence of charged degrees of freedom that have no particle content. The continuous mass formulation of unparticles is recast as an action in anti de Sitter space. Such an action serves as the generating functional for the propagator. This mapping fixes the scaling dimension of the unparticle to be dU=d/2+d2+4/2d_U=d/2+\sqrt{d^2+4}/2dU​=d/2+d2+4​/2 and ensures that the corresponding propagator has zeros with ddd the spacetime dimension of the unparticle field. The general {\it dynamical} mechanism by which bulk operators, such as the Pauli term, couple to the scaling dimension of the boundary operator and thereby lead to a vanishing of the spectral weight at zero energy is reviewed in the context of unparticles and zeros. The analogue of the BCS gap equations with unparticles indicates that the transition temperature increases as the attractive interaction strength decreases, indicating that unparticles are highly susceptible to a superconducting instability.
Recent photoemission spectroscopy measurements (T. J. Reber et al., arXiv:1509.01611) on cuprate superconductors have inferred that over a wide range of doping, the imaginary part of the electron self-energy scales as ${\mathrm{\ensuremath{\Sigma}}}^{\ensuremath{''}}\ensuremath{\sim}{({\ensuremath{\omega}}^{2}+{\ensuremath{\pi}}^{2}{T}^{2})}^{a}$ with a=1a=1a=1 in the overdoped Fermi-liquid state and a<0.5a<0.5a<0.5 in the optimal to underdoped regime. We show that this non-Fermi-liquid scaling behavior can naturally be explained by the presence of a scale-invariant state of matter known as unparticles. We evaluate analytically the electron self-energy due to interactions with fermionic unparticles. We find that, in agreement with experiments, the imaginary part of the self-energy scales with respect to temperature and energy as ${\mathrm{\ensuremath{\Sigma}}}^{\ensuremath{''}}\ensuremath{\sim}{T}^{2+2\ensuremath{\alpha}}$ and ${\ensuremath{\omega}}^{2+2\ensuremath{\alpha}}$, where $\ensuremath{\alpha}$ is the anomalous dimension of the unparticle propagator. In addition, the calculated occupancy and susceptibility of fermionic unparticles, unlike those of normal fermions, have significant spectral weights even at high energies. This unconventional behavior is attributed to the branch cut in the unparticle propagator which broadens the unparticle spectral function over a wide energy range and nontrivially alters the scattering phase space by enhancing (suppressing) the intrinsic susceptibility at low energies for negative (positive) $\ensuremath{\alpha}$. Our work presents evidence suggesting that unparticles might be important low-energy degrees of freedom in strongly coupled systems such as the cuprate superconductors.
Recent photoemission experiments [1] reveal that the excitations along the nodal region in the strange metal phase of the cuprates, rather than corresponding to poles in the single-particle Green function, exhibit power-law scaling as a function of frequency and temperature. Because such power-law scaling is indicative of a scale-invariant sector, as a first step, we perturbatively evaluate the electron self-energy due to interactions with scale-invariant unparticles. We focus on a G0W{G}_{0}WG0​W-type diagram with an interaction WWW mediated by a bosonic scalar unparticle. We find that in the high-temperature limit, the imaginary part of the self-energy $\mathrm{Im}\mathrm{\ensuremath{\Sigma}}$ is linear in temperature. In the low-temperature limit, $\mathrm{Im}\mathrm{\ensuremath{\Sigma}}$ exhibits the same power law in both temperature and frequency, with an exponent that depends on the scaling dimension of an unparticle operator. Such behavior is qualitatively consistent with the experimental observations. We then expand the unparticle propagator into coherent and incoherent contributions and study how the incoherent part violates the density of states (DOS) and density-density correlation function sum rules (f-sum rule). Such violations, in principle, can be observed experimentally. Our work indicates that the physical mechanism for the origin of the power-law scaling is the incoherent background, which is generated from the Mott-scale physics.
Landau Fermi liquid theory is a fixed-point theory of metals that includes the forward-scattering amplitudes as exact marginal couplings. However, the fixed-point theory that only includes the strict forward scatterings is nonlocal in real space. In this paper, we revisit the Fermi liquid theory for charge-neutral fermions using the field-theoretic functional renormalization group formalism and show how the scale-invariant fixed point emerges as a local theory, which includes not only the forward scatterings but also nonforward scatterings with small but nonzero momentum transfers. In the low-energy limit, the near-forward scattering and pairing interactions take scale-invariant forms if the momentum transfer and the center-of-mass momentum of Cooper pairs, respectively, are comparable to the energy. The coupling functions fully capture the universal low-energy dynamics of the collective modes and instabilities of Fermi liquids. A runaway flow of the coupling function in the particle-hole channel beyond a critical interaction suggests an instability toward an ordered phase with a wave vector that depends on the interaction strength. At the critical interaction, the instability corresponds to the uniform Pomeranchuk or Stoner instability, but the momentum of the leading instability becomes nonzero for stronger attractive interaction. In the particle-particle channel, the coupling function reveals the dynamics of the unstable mode associated with the BCS instability. When an unstable normal metal evolves into the superconducting state, there exists a period in which a superconducting state with spatially nonuniform phase appears due to the presence of unstable Cooperon modes with nonzero momenta.
"Strong electron correlation phenomena are ubiquitous, such as unconventional superconductivity and the metal-insulator transitions. Having a proper understanding of strongly correlated electron systems requires a non-perturbative analysis because Fermi liquid theory inevitably breaks down due to the strong correlation. In this thesis, we will study two non-perturbative approaches to tackle strongly correlated electronic system: 1.) multi-dimensional bosonization and 2.) the AdS/CFT correspondence. Multidimensional bosonization is the generalization of bosonization to spatial dimensions larger than one. The bosonized theory is an effective field theory in terms of the bosonic particle-hole fluctuations in different patches of the Fermi surface. The bosonized theory is quadratic and hence can be solved non-perturbatively. The AdS/CFT correspondence is a conjecture that d-dimensional field theory is dual to a d+1 dimensional quantum gravitational theory. Problems in strongly coupled field theory then have an equivalent description using weakly interacting gravitational theory, allowing a non-perturbative analysis. Chapter 1 will serve as an introduction. We will review several experiments which show a breakdown of Fermi liquid theory. Formalisms for multidimensional bosonization and the AdS/CFT correspondence will then be reviewed. In chapter 2, the two-orbital Hubbard model with degenerate d{xz} and d{yz} orbitals are investigated. We apply multidimensional bosonization to solve this problem exactly and discover a z=3 overdamped collective modes that emerges at the orbital-ordering quantum critical point. These modes modify the single-particle density of states and lead to non-Fermi liquid behavior which can provide a possible explanation for the recently observed zero-bias enhancement in the point contact spectroscopy signal on iron pnictides. Chapters 3, 4 and 5 discuss the applications of the AdS/CFT correspondence to model various condensed matter systems. In chapter 3, We consider an interaction term between a bulk spinor field and a gauge field in the Reissner-Nordstr\""om AdS background. When the Pauli interaction term is large enough, a dynamical gap is generated and spectral weight transfer is observed in the spectral density, mimicking the behavior of the Hubbard model. We further consider the finite temperature case and discover that the ratio between the dynamical gap and the critical temperature has the same order as that of VO^2. The Pauli coupling is also studied in the superconducting background. Chapter 4 consider the propagation of a neutral scalar field in the geometry called the electron star. The electron star has Lifshitz scaling with finite dynamical critical exponent at the interior, which is suitable for the modeling of quantum criticality and quantum phase transition with a neutral order parameter, for example for antiferromagnetism. We find that the quantum phase transition has the Berezinski-Kosterlitz-Thouless characteristic and the dynamical critical exponent can change across the quantum critical point. In chapter 5, we construct a gravity dual of the nematic phase by studying the condensation of a spin-two field in the Schwarzschild-AdS background. The condensation of a spin-two field can distort the rotational symmetry of the Fermi surface as long as the spin-two field is coupled to a probe spinor field in the bulk gravity dual."