The study of quantum phase transitions has been a major focus of theoretical and experimental wor... more The study of quantum phase transitions has been a major focus of theoretical and experimental work in systems of correlated electrons and in correlated ultracold atoms in recent years. However, some of the best-characterized and understood examples of quantum phase ...
We describe the nature of charge transport at non-zero temperatures (T ) above the two-dimensiona... more We describe the nature of charge transport at non-zero temperatures (T ) above the two-dimensional (d) superfluid-insulator quantum critical point. We argue that the transport is characterized by inelastic collisions among thermally excited carriers at a rate of order k B T /h. This implies that the transport at frequencies ω k B T /h is in the hydrodynamic, collision-dominated (or 'incoherent') regime, while ω k B T /h is the collisionless (or 'phasecoherent') regime. The conductivity is argued to be e 2 /h times a non-trivial universal scaling function ofhω/k B T , and not independent ofhω/k B T , as has been previously claimed, or implicitly assumed. The experimentally measured d.c. conductivity is the hydrodynamichω/k B T → 0 limit of this function, and is a universal number times e 2 /h, even though the transport is incoherent. Previous work determined the conductivity by incorrectly assuming it was also equal to the collisionlesshω/k B T → ∞ limit of the scaling function, which actually describes phase-coherent transport with a conductivity given by a different universal number times e 2 /h. We provide the first computation of the universal d.c. conductivity in a disorder-free boson model, along with explicit crossover functions, using a quantum Boltzmann equation and an expansion in = 3 − d. The case of spin transport near quantum critical points in antiferromagnets is also discussed. Similar ideas should apply to the transitions in quantum Hall systems and to metal-insulator transitions. We suggest experimental tests of our picture and speculate on a new route to self-duality at two-dimensional quantum critical points.
We present the critical theory of a number of zero temperature phase transitions of quantum antif... more We present the critical theory of a number of zero temperature phase transitions of quantum antiferromagnets and interacting boson systems in two dimensions. The most important example is the transition of the S = 1/2 square lattice antiferromagnet between the Néel state (which breaks spin rotation invariance) and the paramagnetic valence bond solid (which preserves spin rotation invariance but breaks lattice symmetries). We show that these two states are separated by a secondorder quantum phase transition. This conflicts with Landau-Ginzburg-Wilson theory, which predicts that such states with distinct broken symmetries are generically separated either by a first-order transition, or by a phase with co-existing orders. The critical theory of the second-order transition is not expressed in terms of the order parameters characterizing either state, but involves fractionalized degrees of freedom and an emergent, topological, global conservation law. A closely related theory describes the superfluid-insulator transition of bosons at half-filling on a square lattice, in which the insulator has a bond density wave order. Similar considerations are shown to apply to transitions of antiferromagnets between the valence bond solid and the Z2 spin liquid: the critical theory has deconfined excitations interacting with an emergent U(1) gauge force. We comment on the broader implications of our results for the study of quantum criticality in correlated electron systems.
The theory of second order phase transitions is one of the foundations of modern statistical mech... more The theory of second order phase transitions is one of the foundations of modern statistical mechanics and condensed matter theory. A central concept is the observable `order parameter', whose non-zero average value characterizes one or more phases and usually breaks a symmetry of the Hamiltonian. At large distances and long times, fluctuations of the order parameter(s) are described by a continuum field theory, and these dominate the physics near such phase transitions. In this paper we show that near second order quantum phase transitions, subtle quantum interference effects can invalidate this paradigm. We present a theory of quantum critical points in a variety of experimentally relevant two-dimensional antiferromagnets. The critical points separate phases characterized by conventional `confining' order parameters. Nevertheless, the critical theory contains a new emergent gauge field, and `deconfined' degrees of freedom associated with fractionalization of the order parameters. We suggest that this new paradigm for quantum criticality may be the key to resolving a number of experimental puzzles in correlated electron systems.
The theory of second order phase transitions is one of the foundations of modern statistical mech... more The theory of second order phase transitions is one of the foundations of modern statistical mechanics and condensed matter theory. A central concept is the observable `order parameter', whose non-zero average value characterizes one or more phases and usually breaks a symmetry of the Hamiltonian. At large distances and long times, fluctuations of the order parameter(s) are described by a continuum field theory, and these dominate the physics near such phase transitions. In this paper we show that near second order quantum phase transitions, subtle quantum interference effects can invalidate this paradigm. We present a theory of quantum critical points in a variety of experimentally relevant two-dimensional antiferromagnets. The critical points separate phases characterized by conventional `confining' order parameters. Nevertheless, the critical theory contains a new emergent gauge field, and `deconfined' degrees of freedom associated with fractionalization of the order parameters. We suggest that this new paradigm for quantum criticality may be the key to resolving a number of experimental puzzles in correlated electron systems.
We consider charge transport properties of 2+1 dimensional conformal field theories at non-zero t... more We consider charge transport properties of 2+1 dimensional conformal field theories at non-zero temperature. For theories with only Abelian U(1) charges, we describe the action of particle-vortex duality on the hydrodynamic-to-collisionless crossover function: this leads to powerful functional constraints for self-dual theories. For N =8 supersymmetric, SU(N ) Yang-Mills theory at the conformal fixed point, exact hydrodynamic-to-collisionless crossover functions of the SO(8) Rcurrents can be obtained in the large N limit by applying the AdS/CFT correspondence to Mtheory. In the gravity theory, fluctuating currents are mapped to fluctuating gauge fields in the background of a black hole in 3+1 dimensional anti-de Sitter space. The electromagnetic self-duality of the 3+1 dimensional theory implies that the correlators of the R-currents obey a functional constraint similar to that found from particle-vortex duality in 2+1 dimensional Abelian theories.
We study the finite temperature crossovers in the vicinity of a zero temperature quantum phase tr... more We study the finite temperature crossovers in the vicinity of a zero temperature quantum phase transition. The universal crossover functions are observables of a continuum quantum field theory. Particular attention is focussed on the high temperature limit of the continuum field theory, the so-called ``quantum-critical'' region. Basic features of crossovers are illustrated by a simple solvable model of dilute spinless fermions, and a partially solvable model of dilute bosons. The low frequency relaxational behavior of the quantum-critical region is displayed in the solution of the transverse-field Ising model. The insights from these simple models lead to a fairly complete understanding of the system of primary interest: the two-dimensional quantum rotor model, whose phase transition is expected to be in the same universality class as those in antiferromagnetic Heisenberg spin models. Recent work on the experimental implications of these results for the cuprate compounds is briefly reviewed.
Magnetic insulators have proved to be fertile ground for studying new types of quantum many body ... more Magnetic insulators have proved to be fertile ground for studying new types of quantum many body states, and I survey recent experimental and theoretical examples. The insights and methods transfer also to novel superconducting and metallic states. Of particular interest are critical quantum states, sometimes found at quantum phase transitions, which have gapless excitations with no particle-or wave-like interpretation, and control a significant portion of the finite temperature phase diagram. Remarkably, their theory is connected to holographic descriptions of Hawking radiation from black holes.
Motivated by recent neutron scattering experiments, we study the ordering of spins in the iron-ba... more Motivated by recent neutron scattering experiments, we study the ordering of spins in the iron-based superconductors La(O_{1-x}F_x)FeAs, assuming them in proximity to a Mott insulator in the phase diagram. The ground state of the parent system with x = 0 is a spin density wave with ordering wave vector Q = (0, \pi) or (\pi, 0). Upon raising the temperature, we find the system to restore SU(2) symmetry, while an Ising symmetry remains broken, explaining the experimentally observed lattice distortion to a monoclinic crystal structure. Upon further temperature increase, the spins finally disorder at a second transition. The phase transition driven by doping with charge carriers similarly splits into an O(3) transition, and an Ising transition with z = 3 at larger doping.
Many condensed matter experiments explore the finite temperature dynamics of systems near quantum... more Many condensed matter experiments explore the finite temperature dynamics of systems near quantum critical points. Often, there are no well-defined quasiparticle excitations, and so quantum kinetic equations do not describe the transport properties completely. The theory shows that the transport co-efficients are not proportional to a mean free scattering time (as is the case in the Boltzmann theory of quasiparticles), but are completely determined by the absolute temperature and by equilibrium thermodynamic observables. Recently, explicit solutions of this quantum critical dynamics have become possible via the AdS/CFT duality discovered in string theory. This shows that the quantum critical theory provides a holographic description of the quantum theory of black holes in a negatively curved anti-de Sitter space, and relates its transport co-efficients to properties of the Hawking radiation from the black hole. We review how insights from this connection have led to new results for experimental systems: (i) the vicinity of the superfluid-insulator transition in the presence of an applied magnetic field, and its possible application to measurements of the Nernst effect in the cuprates, (ii) the magnetohydrodynamics of the plasma of Dirac electrons in graphene and the prediction of a hydrodynamic cyclotron resonance. this state, but there is a well-developed theory of the transport properties of the TL liquid. Historically, this theory evolved over several decades of research on quantum many body systems in one dimension. Key in the historical development were exact solutions of model Hamiltonians via the Bethe Ansatz. Insights gained from the structure of excitations in the Bethe Ansatz solutions led to a more general understanding of the low energy excitations of a generic Hamiltonian, and a universal low energy theory of the TL liquid. Thus while the exact solutions were restricted to artificial models, they played a key role in the development of the general theory. Of course, after the fact, with the general theory of the TL liquid before us, we can justify it in its own terms, and largely dispense with reference to the Bethe Ansatz solutions.
It is now widely accepted that the cuprate superconductors are characterized by the same long-ran... more It is now widely accepted that the cuprate superconductors are characterized by the same long-range order as that present in the Bardeen-Cooper-Schrieffer (BCS) theory: that associated with the condensation of Cooper pairs. We argue that many physical properties of the cuprates require interplay with additional order parameters associated with a proximate Mott insulator. We review a classification of Mott insulators in two dimensions, and contend that the experimental evidence so far shows that the class appropriate to the cuprates has collinear spin correlations, bond order, and confinement of neutral, spin S=1/2 excitations. Proximity to second-order quantum phase transitions associated with these orders, and with the pairing order of BCS, has led to systematic predictions for many physical properties. We use this context to review the results of recent neutron scattering, fluxoid detection, nuclear magnetic resonance, and scanning tunnelling microscopy experiments.
We show that strongly coupled field theories with holographic gravity duals at finite charge dens... more We show that strongly coupled field theories with holographic gravity duals at finite charge density and low temperatures can undergo de Haas - van Alphen quantum oscillations as a function of an external magnetic field. Exhibiting this effect requires computation of the one loop contribution of charged bulk fermions to the free energy. The one loop calculation is performed using a formula expressing determinants in black hole backgrounds as sums over quasinormal modes. At zero temperature, the periodic nonanalyticities in the magnetic susceptibility as a function of the inverse magnetic field depend on the low energy scaling behavior of fermionic operators in the field theory, and are found to be softer than in weakly coupled theories. We also obtain numerical and WKB results for the quasinormal modes of charged bosons in dyonic black hole backgrounds, finding evidence for nontrivial periodic behavior as a function of the magnetic field.
We present a general hydrodynamic theory of transport in the vicinity of superfluid-insulator tra... more We present a general hydrodynamic theory of transport in the vicinity of superfluid-insulator transitions in two spatial dimensions described by "Lorentz"-invariant quantum critical points. We allow for a weak impurity scattering rate, a magnetic field B, and a deviation in the density, ρ, from that of the insulator. We show that the frequency-dependent thermal and electric linear response functions, including the Nernst coefficient, are fully determined by a single transport coefficient (a universal electrical conductivity), the impurity scattering rate, and a few thermodynamic state variables. With reasonable estimates for the parameters, our results predict a magnetic field and temperature dependence of the Nernst signal which resembles measurements in the cuprates, including the overall magnitude. Our theory predicts a "hydrodynamic cyclotron mode" which could be observable in ultrapure samples. We also present exact results for the zero frequency transport co-efficients of a supersymmetric conformal field theory (CFT), which is solvable by the AdS/CFT correspondence. This correspondence maps the ρ and B perturbations of the 2+1 dimensional CFT to electric and magnetic charges of a black hole in the 3+1 dimensional anti-de Sitter space. These exact results are found to be in full agreement with the general predictions of our hydrodynamic analysis in the appropriate limiting regime. The mapping of the hydrodynamic and AdS/CFT results under particle-vortex duality is also described.
It is now widely accepted that the cuprate superconductors are characterized by the same longrang... more It is now widely accepted that the cuprate superconductors are characterized by the same longrange order as that present in the Bardeen-Cooper-Schrieffer (BCS) theory: that associated with the condensation of Cooper pairs. The author argues that many physical properties of the cuprates require interplay with additional order parameters associated with a proximate Mott insulator. A classification of Mott insulators in two dimensions is proposed. Experimental evidence so far shows that the class appropriate to the cuprates has collinear spin correlations, bond order, and confinement of neutral, spin S = 1/2 excitations. Proximity to second-order quantum phase transitions associated with these orders, and with the pairing order of BCS, has led to systematic predictions for many physical properties. In this context the author reviews the results of recent neutron scattering, fluxoid detection, nuclear magnetic resonance, and scanning tunnelling microscopy experiments.
This paper is concerned with the weak-moment magnetism in heavy-fermion materials and its relatio... more This paper is concerned with the weak-moment magnetism in heavy-fermion materials and its relation to the non-Fermi liquid physics observed near the transition to the Fermi liquid. We explore the hypothesis that the primary fluctuations responsible for the non-Fermi liquid physics are those associated with the destruction of the large Fermi surface of the Fermi liquid. Magnetism is suggested to be a low-energy instability of the resulting small Fermi surface state. A concrete realization of this picture is provided by a fractionalized Fermi liquid state which has a small Fermi surface of conduction electrons, but also has other exotic excitations with interactions described by a gauge theory in its deconfined phase. Of particular interest is a three-dimensional fractionalized Fermi liquid with a spinon Fermi surface and a U(1) gauge structure. A direct second-order transition from this state to the conventional Fermi liquid is possible and involves a jump in the electron Fermi surface volume. The critical point displays non-Fermi liquid behavior. A magnetic phase may develop from a spin density wave instability of the spinon Fermi surface. This exotic magnetic metal may have a weak ordered moment although the local moments do not participate in the Fermi surface. Experimental signatures of this phase and implications for heavy-fermion systems are discussed.
High temperature superconductivity emerges in the cuprate compounds upon changing the electron de... more High temperature superconductivity emerges in the cuprate compounds upon changing the electron density of an insulator in which the electron spins are antiferromagnetically ordered. A key characteristic of the superconductor is that electrons can be extracted from them at zero energy only if their momenta take one of four specific values (the `nodal points'). A central enigma has been the evolution of the zero energy electrons in the metallic state between the antiferromagnet and the superconductor, and recent experiments yield apparently contradictory results. The oscillation of the resistance in this metal as a function of magnetic field indicate that the zero energy electrons carry momenta which lie on elliptical `Fermi pockets', while ejection of electrons by high intensity light indicates that the zero energy electrons have momenta only along arc-like regions. We present a theory of new states of matter, which we call `algebraic charge liquids', which arise naturally between the antiferromagnet and the superconductor, and reconcile these observations. Our theory also explains a puzzling dependence of the density of superconducting electrons on the total electron density, and makes a number of unique predictions for future experiments.
We compute the direct current resistivity of a scale-invariant, d-dimensional strange metal with ... more We compute the direct current resistivity of a scale-invariant, d-dimensional strange metal with dynamic critical exponent z and hyperscaling-violating exponent θ, weakly perturbed by a scalar operator coupled to random-field disorder that locally breaks a Z 2 symmetry. Independent calculations via Einstein-Maxwell-Dilaton holography and memory matrix methods lead to the same results. We show that random field disorder has a strong effect on resistivity: charge carriers in the infrared are easily depleted, as the relaxation time for momentum is surprisingly small. In the course of our holographic calculation we use a non-trivial dilaton coupling to the disordered scalar, allowing us to study a strongly-coupled scale invariant theory with θ ̸ = 0. Using holography, we are also able to determine the disorder strength at which perturbation theory breaks down. Curiously, for locally critical theories this breakdown occurs when the resistivity is proportional to the entropy density, up to a possible logarithmic correction. *
The study of quantum phase transitions has been a major focus of theoretical and experimental wor... more The study of quantum phase transitions has been a major focus of theoretical and experimental work in systems of correlated electrons and in correlated ultracold atoms in recent years. However, some of the best-characterized and understood examples of quantum phase ...
We describe the nature of charge transport at non-zero temperatures (T ) above the two-dimensiona... more We describe the nature of charge transport at non-zero temperatures (T ) above the two-dimensional (d) superfluid-insulator quantum critical point. We argue that the transport is characterized by inelastic collisions among thermally excited carriers at a rate of order k B T /h. This implies that the transport at frequencies ω k B T /h is in the hydrodynamic, collision-dominated (or 'incoherent') regime, while ω k B T /h is the collisionless (or 'phasecoherent') regime. The conductivity is argued to be e 2 /h times a non-trivial universal scaling function ofhω/k B T , and not independent ofhω/k B T , as has been previously claimed, or implicitly assumed. The experimentally measured d.c. conductivity is the hydrodynamichω/k B T → 0 limit of this function, and is a universal number times e 2 /h, even though the transport is incoherent. Previous work determined the conductivity by incorrectly assuming it was also equal to the collisionlesshω/k B T → ∞ limit of the scaling function, which actually describes phase-coherent transport with a conductivity given by a different universal number times e 2 /h. We provide the first computation of the universal d.c. conductivity in a disorder-free boson model, along with explicit crossover functions, using a quantum Boltzmann equation and an expansion in = 3 − d. The case of spin transport near quantum critical points in antiferromagnets is also discussed. Similar ideas should apply to the transitions in quantum Hall systems and to metal-insulator transitions. We suggest experimental tests of our picture and speculate on a new route to self-duality at two-dimensional quantum critical points.
We present the critical theory of a number of zero temperature phase transitions of quantum antif... more We present the critical theory of a number of zero temperature phase transitions of quantum antiferromagnets and interacting boson systems in two dimensions. The most important example is the transition of the S = 1/2 square lattice antiferromagnet between the Néel state (which breaks spin rotation invariance) and the paramagnetic valence bond solid (which preserves spin rotation invariance but breaks lattice symmetries). We show that these two states are separated by a secondorder quantum phase transition. This conflicts with Landau-Ginzburg-Wilson theory, which predicts that such states with distinct broken symmetries are generically separated either by a first-order transition, or by a phase with co-existing orders. The critical theory of the second-order transition is not expressed in terms of the order parameters characterizing either state, but involves fractionalized degrees of freedom and an emergent, topological, global conservation law. A closely related theory describes the superfluid-insulator transition of bosons at half-filling on a square lattice, in which the insulator has a bond density wave order. Similar considerations are shown to apply to transitions of antiferromagnets between the valence bond solid and the Z2 spin liquid: the critical theory has deconfined excitations interacting with an emergent U(1) gauge force. We comment on the broader implications of our results for the study of quantum criticality in correlated electron systems.
The theory of second order phase transitions is one of the foundations of modern statistical mech... more The theory of second order phase transitions is one of the foundations of modern statistical mechanics and condensed matter theory. A central concept is the observable `order parameter', whose non-zero average value characterizes one or more phases and usually breaks a symmetry of the Hamiltonian. At large distances and long times, fluctuations of the order parameter(s) are described by a continuum field theory, and these dominate the physics near such phase transitions. In this paper we show that near second order quantum phase transitions, subtle quantum interference effects can invalidate this paradigm. We present a theory of quantum critical points in a variety of experimentally relevant two-dimensional antiferromagnets. The critical points separate phases characterized by conventional `confining' order parameters. Nevertheless, the critical theory contains a new emergent gauge field, and `deconfined' degrees of freedom associated with fractionalization of the order parameters. We suggest that this new paradigm for quantum criticality may be the key to resolving a number of experimental puzzles in correlated electron systems.
The theory of second order phase transitions is one of the foundations of modern statistical mech... more The theory of second order phase transitions is one of the foundations of modern statistical mechanics and condensed matter theory. A central concept is the observable `order parameter', whose non-zero average value characterizes one or more phases and usually breaks a symmetry of the Hamiltonian. At large distances and long times, fluctuations of the order parameter(s) are described by a continuum field theory, and these dominate the physics near such phase transitions. In this paper we show that near second order quantum phase transitions, subtle quantum interference effects can invalidate this paradigm. We present a theory of quantum critical points in a variety of experimentally relevant two-dimensional antiferromagnets. The critical points separate phases characterized by conventional `confining' order parameters. Nevertheless, the critical theory contains a new emergent gauge field, and `deconfined' degrees of freedom associated with fractionalization of the order parameters. We suggest that this new paradigm for quantum criticality may be the key to resolving a number of experimental puzzles in correlated electron systems.
We consider charge transport properties of 2+1 dimensional conformal field theories at non-zero t... more We consider charge transport properties of 2+1 dimensional conformal field theories at non-zero temperature. For theories with only Abelian U(1) charges, we describe the action of particle-vortex duality on the hydrodynamic-to-collisionless crossover function: this leads to powerful functional constraints for self-dual theories. For N =8 supersymmetric, SU(N ) Yang-Mills theory at the conformal fixed point, exact hydrodynamic-to-collisionless crossover functions of the SO(8) Rcurrents can be obtained in the large N limit by applying the AdS/CFT correspondence to Mtheory. In the gravity theory, fluctuating currents are mapped to fluctuating gauge fields in the background of a black hole in 3+1 dimensional anti-de Sitter space. The electromagnetic self-duality of the 3+1 dimensional theory implies that the correlators of the R-currents obey a functional constraint similar to that found from particle-vortex duality in 2+1 dimensional Abelian theories.
We study the finite temperature crossovers in the vicinity of a zero temperature quantum phase tr... more We study the finite temperature crossovers in the vicinity of a zero temperature quantum phase transition. The universal crossover functions are observables of a continuum quantum field theory. Particular attention is focussed on the high temperature limit of the continuum field theory, the so-called ``quantum-critical'' region. Basic features of crossovers are illustrated by a simple solvable model of dilute spinless fermions, and a partially solvable model of dilute bosons. The low frequency relaxational behavior of the quantum-critical region is displayed in the solution of the transverse-field Ising model. The insights from these simple models lead to a fairly complete understanding of the system of primary interest: the two-dimensional quantum rotor model, whose phase transition is expected to be in the same universality class as those in antiferromagnetic Heisenberg spin models. Recent work on the experimental implications of these results for the cuprate compounds is briefly reviewed.
Magnetic insulators have proved to be fertile ground for studying new types of quantum many body ... more Magnetic insulators have proved to be fertile ground for studying new types of quantum many body states, and I survey recent experimental and theoretical examples. The insights and methods transfer also to novel superconducting and metallic states. Of particular interest are critical quantum states, sometimes found at quantum phase transitions, which have gapless excitations with no particle-or wave-like interpretation, and control a significant portion of the finite temperature phase diagram. Remarkably, their theory is connected to holographic descriptions of Hawking radiation from black holes.
Motivated by recent neutron scattering experiments, we study the ordering of spins in the iron-ba... more Motivated by recent neutron scattering experiments, we study the ordering of spins in the iron-based superconductors La(O_{1-x}F_x)FeAs, assuming them in proximity to a Mott insulator in the phase diagram. The ground state of the parent system with x = 0 is a spin density wave with ordering wave vector Q = (0, \pi) or (\pi, 0). Upon raising the temperature, we find the system to restore SU(2) symmetry, while an Ising symmetry remains broken, explaining the experimentally observed lattice distortion to a monoclinic crystal structure. Upon further temperature increase, the spins finally disorder at a second transition. The phase transition driven by doping with charge carriers similarly splits into an O(3) transition, and an Ising transition with z = 3 at larger doping.
Many condensed matter experiments explore the finite temperature dynamics of systems near quantum... more Many condensed matter experiments explore the finite temperature dynamics of systems near quantum critical points. Often, there are no well-defined quasiparticle excitations, and so quantum kinetic equations do not describe the transport properties completely. The theory shows that the transport co-efficients are not proportional to a mean free scattering time (as is the case in the Boltzmann theory of quasiparticles), but are completely determined by the absolute temperature and by equilibrium thermodynamic observables. Recently, explicit solutions of this quantum critical dynamics have become possible via the AdS/CFT duality discovered in string theory. This shows that the quantum critical theory provides a holographic description of the quantum theory of black holes in a negatively curved anti-de Sitter space, and relates its transport co-efficients to properties of the Hawking radiation from the black hole. We review how insights from this connection have led to new results for experimental systems: (i) the vicinity of the superfluid-insulator transition in the presence of an applied magnetic field, and its possible application to measurements of the Nernst effect in the cuprates, (ii) the magnetohydrodynamics of the plasma of Dirac electrons in graphene and the prediction of a hydrodynamic cyclotron resonance. this state, but there is a well-developed theory of the transport properties of the TL liquid. Historically, this theory evolved over several decades of research on quantum many body systems in one dimension. Key in the historical development were exact solutions of model Hamiltonians via the Bethe Ansatz. Insights gained from the structure of excitations in the Bethe Ansatz solutions led to a more general understanding of the low energy excitations of a generic Hamiltonian, and a universal low energy theory of the TL liquid. Thus while the exact solutions were restricted to artificial models, they played a key role in the development of the general theory. Of course, after the fact, with the general theory of the TL liquid before us, we can justify it in its own terms, and largely dispense with reference to the Bethe Ansatz solutions.
It is now widely accepted that the cuprate superconductors are characterized by the same long-ran... more It is now widely accepted that the cuprate superconductors are characterized by the same long-range order as that present in the Bardeen-Cooper-Schrieffer (BCS) theory: that associated with the condensation of Cooper pairs. We argue that many physical properties of the cuprates require interplay with additional order parameters associated with a proximate Mott insulator. We review a classification of Mott insulators in two dimensions, and contend that the experimental evidence so far shows that the class appropriate to the cuprates has collinear spin correlations, bond order, and confinement of neutral, spin S=1/2 excitations. Proximity to second-order quantum phase transitions associated with these orders, and with the pairing order of BCS, has led to systematic predictions for many physical properties. We use this context to review the results of recent neutron scattering, fluxoid detection, nuclear magnetic resonance, and scanning tunnelling microscopy experiments.
We show that strongly coupled field theories with holographic gravity duals at finite charge dens... more We show that strongly coupled field theories with holographic gravity duals at finite charge density and low temperatures can undergo de Haas - van Alphen quantum oscillations as a function of an external magnetic field. Exhibiting this effect requires computation of the one loop contribution of charged bulk fermions to the free energy. The one loop calculation is performed using a formula expressing determinants in black hole backgrounds as sums over quasinormal modes. At zero temperature, the periodic nonanalyticities in the magnetic susceptibility as a function of the inverse magnetic field depend on the low energy scaling behavior of fermionic operators in the field theory, and are found to be softer than in weakly coupled theories. We also obtain numerical and WKB results for the quasinormal modes of charged bosons in dyonic black hole backgrounds, finding evidence for nontrivial periodic behavior as a function of the magnetic field.
We present a general hydrodynamic theory of transport in the vicinity of superfluid-insulator tra... more We present a general hydrodynamic theory of transport in the vicinity of superfluid-insulator transitions in two spatial dimensions described by "Lorentz"-invariant quantum critical points. We allow for a weak impurity scattering rate, a magnetic field B, and a deviation in the density, ρ, from that of the insulator. We show that the frequency-dependent thermal and electric linear response functions, including the Nernst coefficient, are fully determined by a single transport coefficient (a universal electrical conductivity), the impurity scattering rate, and a few thermodynamic state variables. With reasonable estimates for the parameters, our results predict a magnetic field and temperature dependence of the Nernst signal which resembles measurements in the cuprates, including the overall magnitude. Our theory predicts a "hydrodynamic cyclotron mode" which could be observable in ultrapure samples. We also present exact results for the zero frequency transport co-efficients of a supersymmetric conformal field theory (CFT), which is solvable by the AdS/CFT correspondence. This correspondence maps the ρ and B perturbations of the 2+1 dimensional CFT to electric and magnetic charges of a black hole in the 3+1 dimensional anti-de Sitter space. These exact results are found to be in full agreement with the general predictions of our hydrodynamic analysis in the appropriate limiting regime. The mapping of the hydrodynamic and AdS/CFT results under particle-vortex duality is also described.
It is now widely accepted that the cuprate superconductors are characterized by the same longrang... more It is now widely accepted that the cuprate superconductors are characterized by the same longrange order as that present in the Bardeen-Cooper-Schrieffer (BCS) theory: that associated with the condensation of Cooper pairs. The author argues that many physical properties of the cuprates require interplay with additional order parameters associated with a proximate Mott insulator. A classification of Mott insulators in two dimensions is proposed. Experimental evidence so far shows that the class appropriate to the cuprates has collinear spin correlations, bond order, and confinement of neutral, spin S = 1/2 excitations. Proximity to second-order quantum phase transitions associated with these orders, and with the pairing order of BCS, has led to systematic predictions for many physical properties. In this context the author reviews the results of recent neutron scattering, fluxoid detection, nuclear magnetic resonance, and scanning tunnelling microscopy experiments.
This paper is concerned with the weak-moment magnetism in heavy-fermion materials and its relatio... more This paper is concerned with the weak-moment magnetism in heavy-fermion materials and its relation to the non-Fermi liquid physics observed near the transition to the Fermi liquid. We explore the hypothesis that the primary fluctuations responsible for the non-Fermi liquid physics are those associated with the destruction of the large Fermi surface of the Fermi liquid. Magnetism is suggested to be a low-energy instability of the resulting small Fermi surface state. A concrete realization of this picture is provided by a fractionalized Fermi liquid state which has a small Fermi surface of conduction electrons, but also has other exotic excitations with interactions described by a gauge theory in its deconfined phase. Of particular interest is a three-dimensional fractionalized Fermi liquid with a spinon Fermi surface and a U(1) gauge structure. A direct second-order transition from this state to the conventional Fermi liquid is possible and involves a jump in the electron Fermi surface volume. The critical point displays non-Fermi liquid behavior. A magnetic phase may develop from a spin density wave instability of the spinon Fermi surface. This exotic magnetic metal may have a weak ordered moment although the local moments do not participate in the Fermi surface. Experimental signatures of this phase and implications for heavy-fermion systems are discussed.
High temperature superconductivity emerges in the cuprate compounds upon changing the electron de... more High temperature superconductivity emerges in the cuprate compounds upon changing the electron density of an insulator in which the electron spins are antiferromagnetically ordered. A key characteristic of the superconductor is that electrons can be extracted from them at zero energy only if their momenta take one of four specific values (the `nodal points'). A central enigma has been the evolution of the zero energy electrons in the metallic state between the antiferromagnet and the superconductor, and recent experiments yield apparently contradictory results. The oscillation of the resistance in this metal as a function of magnetic field indicate that the zero energy electrons carry momenta which lie on elliptical `Fermi pockets', while ejection of electrons by high intensity light indicates that the zero energy electrons have momenta only along arc-like regions. We present a theory of new states of matter, which we call `algebraic charge liquids', which arise naturally between the antiferromagnet and the superconductor, and reconcile these observations. Our theory also explains a puzzling dependence of the density of superconducting electrons on the total electron density, and makes a number of unique predictions for future experiments.
We compute the direct current resistivity of a scale-invariant, d-dimensional strange metal with ... more We compute the direct current resistivity of a scale-invariant, d-dimensional strange metal with dynamic critical exponent z and hyperscaling-violating exponent θ, weakly perturbed by a scalar operator coupled to random-field disorder that locally breaks a Z 2 symmetry. Independent calculations via Einstein-Maxwell-Dilaton holography and memory matrix methods lead to the same results. We show that random field disorder has a strong effect on resistivity: charge carriers in the infrared are easily depleted, as the relaxation time for momentum is surprisingly small. In the course of our holographic calculation we use a non-trivial dilaton coupling to the disordered scalar, allowing us to study a strongly-coupled scale invariant theory with θ ̸ = 0. Using holography, we are also able to determine the disorder strength at which perturbation theory breaks down. Curiously, for locally critical theories this breakdown occurs when the resistivity is proportional to the entropy density, up to a possible logarithmic correction. *
Uploads
Papers by Subir Sachdev