Session W16: Quantum Criticality

Pairing of critical Fermi-surface states

States of matter with a sharp Fermi-surface but no well-defined Landau quasiparticles are expected to arise in a number of physical systems. Examples include i) quantum critical points associated with the onset of order in metals, ii) the spinon Fermi-surface (U(1) spin-liquid) state of a Mott insulator and iii) the Halperin-Lee-Read composite fermion charge liquid state of a half-filled Landau level. In this work, we use renormalization group techniques to investigate possible instabilities of such non-Fermi-liquids to pairing. We show that for a large class of phase transitions in metals, the attractive interaction mediated by order parameter fluctuations always leads to a superconducting instability, which preempts the non-Fermi-liquid effects. On the other hand, the spinon Fermi-surface and the Halperin-Lee-Read states are stable against pairing for a sufficiently weak attractive short-range interaction. However, once the strength of attraction exceeds a critical value, pairing sets in. We describe the ensuing quantum phase transition between i) the U(1) and the Z$_2$ spin-liquid states, and ii) the Halperin-Lee-Read and Moore-Read states.   

Fractionalized Fermi liquids as ground states of single band tJ models

Recently it has been argued that the normal state of underdoped cuprates might realize a fractionalized Fermi liquid (FL*), where electron-like quasiparticles couple to fractionalized excitations of a fluctuating antiferromagnetic background. The corresponding Fermi surface consists of pockets, the area of which is determined by the density of doped charge carriers alone as opposed to the total density of electrons, thereby violating Luttinger's theorem. Most previous studies of the FL* phase were based either on two-band models such as the Kondo lattice model, or on phenomenological models where Fermions are coupled to a fluctuating unit vector field representing the local Neel order. In this talk I will show that the FL* phase can indeed arise as the ground-state of a single band tJ model and discuss its implications.   

Doping the Kane-Mele-Hubbard model: A Slave-Boson Approach

We study the Kane-Mele-Hubbard model both at half-filling and away from half-filling using a slave-boson mean-field approach at zero temperature. We obtain a phase diagram at half-filling and discuss its connection to recent results from quantum Monte Carlo, cellular dynamical mean field, slave-rotor, and $Z_2$ mean-field studies. In particular, we find a small window in parameter space where a spin liquid phase with gapped spin and charge excitations reside. Upon doping, we show the spin liquid state becomes a superconducting state by explicitly calculating the singlet pairing order parameters. Interestingly, we find an ``optimal" doping for such superconductivity. Our work reveals some of the phenomenology associated with doping an interacting system with strong spin-orbit coupling and intermediate strength electron-electron interactions.   

Stripe melting and quantum criticality in correlated metals

In the last several years evidence for the occurrence of stripe and related orders has accumulated in many underdoped cuprates. With increasing doping the stripe ordering tendency disappears. This has given rise to the idea that a stripe melting quantum phase transition in the ``underlying normal state'' may play a role in some of the physics of the optimally doped strange metal. However there is currently no controlled understanding of a stripe-disordering phase transition in the presence of a Fermi-surface of electrons. We obtain a controlled critical theory of a continuous melting transition of charge stripes in a metal by proliferating pairs of dislocations in the stripe-order parameter, without proliferating single dislocations. At such a (deconfined) quantum critical point (QCP) the fluctuations of the stripe order parameter are strongly coupled, yet tractable. They also decouple dynamically from the Fermi-surface. We find that the full Fermi-surface and the associated Landau quasiparticles remain sharply defined at the QCP. On the stripe ordered side the reconstruction of the Fermi surface occurs at an energy scale that is parametrically different from that associated with the onset of stripe order.   

Quantum Criticality for Extended Nodes on the Bethe Lattice

Theoretical description of anisotropic systems, such as layered superconductors and coupled spin chains, is often a challenge due to the different natures of interactions along different directions. As a model of such a system, we present an analytical study (1) of $d$-dimensional ``nodes" arranged as the vertices of a Bethe lattice, where each node has nonzero spatial dimension and is described by an $O(N)$ quantum rotor model, and there is hopping between neighboring nodes. In the limit of large connectivity on the Bethe lattice, the hopping can be treated by constructing a self-consistent effective action for a single node. This procedure is akin to dynamical mean field theory, but generalized so that spatial as well as quantum fluctuations are taken into account on each node. The quantum phase transition is studied using this effective action for both infinite and finite $N$. The importance of the Perron-Frobenius uniform mode on the Bethe lattice is discussed, and its elimination via an ``infinite range hopping" term shifts the transition, leading to nontrivial critical behavior. We calculate critical exponents and find that the internode hopping reduces the upper and lower critical dimensions each by one. \\ (1) arXiv:1111.2011   

Quantum Criticality on Graphs

One of the tenets of our understanding of strongly correlated systems is that their macroscopic critical behavior is often universal and independent of any microscopic details. Continuous phase transitions driven by either thermal or quantum fluctuations can be placed in universality classes defined by the spatial dimension and the symmetry of a Landau-like order parameter. In an attempt to further understand the role of the spatial dimension in universality, we imagine a system whose symmetries and connectivities are controlled by the low-lying spectrum of degrees of freedom on a graph. As an elucidating example, we study a quantum phase transition on large random regular graphs, with coupled vertices described by $d$-dimensional $\mathrm{O}(N)$ quantum rotor models.   

Entropic Signatures of Quantum Criticality in Itinerant Ferromagnets and Metamagnetic Systems

We investigate the thermodynamic properties of itinerant ferromagnets near quantum critical points described by a quantum Landau-Ginzburg $\phi^4$ theory. We show that the quartic coupling in this theory, which is dangerously irrelevant, has a singular contribution to the renormalized Gaussian free energy. We trace this singularity to some ultraviolet contributions, thereby demonstrating its unphysical nature. We introduce a procedure to regularize this singularity, and apply the prescription to calculate thermodynamic quantities across ferromagnetic quantum critical points in both two and three dimensions. Our calculation illustrates various thermodynamic signatures of quantum criticality, including the entropy accumulation at the quantum critical point which was first proposed on scaling grounds [1]. We systematically compare our theoretical results with the experimental data on entropy and specific heat as a function of magnetic field in Sr3Ru2O7 [2]. We demonstrate that the thermodynamic data are compatible with a quantum critical scenario, but the critical behavior does not agree well with the conventional itinerant ferromagnetic quantum criticality picture. [1] L. Zhu et al PRL 91, 066404 (2003). [2] A.W. Rost et al, Science 325, 1360 (2009).   

Quantum criticality in a dissipative (2+1)-dimensional $XY$ model of circulating currents in high-$T_{\rm c}$ cuprates

We present large-scale Monte Carlo results for the dynamical critical exponent $z$ and the spatio-temporal two-point correlation function of a (2+1)-dimensional quantum $XY$ model with bond dissipation, proposed to describe a quantum critical point in high-$T_c$ cuprates near optimal doping. The phase variables of the model, originating with a parametrization of circulating currents within the CuO$_2$ unit cells in cuprates, are compact, $\{ \theta_{vvr,\tau} \}$ $\in [-\pi,\pi \rangle$. The dynamical critical exponent is found to be $z \approx 1$, and the spatio-temporal correlation functions are explicitly demonstrated to be isotropic in space-imaginary time. The model thus has a fluctuation spectrum where momentum and frequency enter on equal footing, rather than having the essentially momentum-independent marginal Fermi liquid-like fluctuation spectrum previously reported for the same model.   

Strong Electron Correlation by Virtual Phonon Exchange in Jahn-Teller Crystals

In Jahn-Teller crystals - crystals with at least one sublattice of ions with orbitally degenerate electronic states - virtual phonon exchange is a major source of strong electron correlation. This type of electron correlation leads to different structural and magnetic transitions the interplay of which is especially interesting in case of the triple degeneracy of the electronic ground states. The interest to these systems lately has increased as it is related to some unusual situations in perovskite and spinel structure compounds that are of big practical interest (multiferroicity, piezomagnetism, and others). Mutual influence of the antiferrodistortive XY-type and ferroelastic ZZ-type orderings mediated by magnetic external or internal interactions in such a type of crystals is under discussion. It is found that even in case of stronger interactions leading to the ferroelastic ordering the presence of magnetic interaction causes new type of structural phase transition from the ZZ- type of ordering to the XY-type. These structural transformations are accompanied by specific anomalies in the temperature and external magnetic field dependences. The thermodynamics of the systems with strong electron correlation leading to such a phase transitions is analyzed.   

Features of Fermi Systems near $\ell$=0 Pomeranchuk Instabilities: A Crossing Symmetric Approach

In Fermi systems, interactions can cause symmetry-breaking deformations of the Fermi surface, called Pomeranchuk instabilities. In Fermi liquid (FL) language, this occurs when one of the Landau harmonics F$_{\ell}^{a,s}$ $\rightarrow$ -(2$\ell$ + 1); e.g. F$_{0}^{a,s}$ = -1 are related to ferromagnetic (a), and density instabilities (s) resepctively. The corresponding point in parameter space may be viewed as a quantum critical point (QCP). Using graphical and numerical methods to solve coupled non-linear integral equations of a crossing symmetric equation (TSCE) scheme, we study the behavior of spin/density excitations; effective mass; ferromagnetic, spin density wave, phase separation, and pairing transitions near $\ell$=0 Pomeranchuk instabilities in a 3D Fermi system. Considering momentum dependence of the renormalized FL interactions, we find a number of results for repulsive and attractive couplings of arbitrary strengths; viz. attraction in both singlet and triplet pairing amplitudes (though singlet pairing is primarily favored); possibility of a second ferromagnetic transition due to spin waves, and possibility of phase separation with and without ferromagnetic transition. Some of our results may apply to ferromagnetic superconductors, such as UGe$_{2}$ and UIr.   

Phase reconstruction near to the two-dimensional ferromagnetic quantum critical point

We study the formation of new phases in two dimensions near to the putative quantum critical point of the itinerant ferromagnet to paramagnet phase transition. In addition to the first order and helimagnetic behaviour found in non-analytic extensions to Hertz-Millis theory [1] and in the quantum order-by-disorder approach [2], we find a small region of spin nematic order. Our approach also admits a concurrent formation of superconducting order. We further study the effect of small deformations from quadratic electron dispersion -- as previously found in three dimensions, these enlarge the region of spin nematic order at the expense of spiral order.\\[4pt] [1] D. Belitz, T.R. Kirkpatrick and T. Vojta, Rev. Mod. Phys. \textbf{77}, 579 (2005),. V. Efremov, J.J. Betouras, A.V. Chubukov Phys. Rev. B\textbf{ 77}, 220401(R), (2008)\\[0pt] [2] G. J. Conduit Phys. Rev. A \textbf{82}, 043604 (2010)   

Divergence of the effective mass in a strongly-interacting 2D electron system

The diffusion thermopower in a low-disorder, strongly-interacting 2D electron system in silicon increases with decreasing electron density and tends to infinity at a finite density $n_t$. Comparison with earlier data for a high-disorder silicon system indicates that the critical density $n_t$ does not depend on the degree of disorder. The increase of the thermopower is associated with a diverging electron mass in the close vicinity of an interaction-induced phase transition.   

A New View of the Mott-Hubbard Transition: Renormalization of the Fermi-Surface Topology

We present the renormalization of the (underlying) Fermi-surface topology in the Hubbard model on a square lattice with frustrating hopping, that is relevant for the physics of high-temperature superconductors. With the help of novel high precision variational tools, including Jastrow factors and backflow correlations, we show that the Fermi surface renormalizes to perfect nesting at the interaction-driven Mott-Hubbard transition and in the large interaction limit. Moreover, we present new results for the density-driven Mott-Hubbard transition, investigating the Fermi-surface renormalization flow as a function of doping, where the renormalization occurs only when the half-filled case is insulating. We associate the flow to the appearance of a van Hove singularity at the Fermi level at small doping, that is interpreted as an instability to magnetic order. Finally, we show also that Fermi surface renormalization is associated to a strong crossover at finite doping for the critical $U$ corresponding to the Mott-Hubbard transition.   

Detecting Quantum Phase Transitions using Classical Noises

We theoretically propose that the classical noise spectra provide an efficient and straightforward way to detect the quantum phase transition points in low-dimensional quantum spin systems. By using Ornstein-Uhlenbeck noise, we employ both a quadratic response theory and time-dependent density matrix renormalization group method to study the quantum system. In the non-Markovian region, the time evolutions of physical observables exhibit distinct behaviors for different quantum phases. In addition, we have the freedom to choose various noises to detect peculiar quantum phases. This method can be used to measure the three body correlation function directly. We demonstrate that the method can determine faithfully the quantum transition points of the transverse Ising model as well as spin-1 bilinear-biquadratic Heisenberrg model. The possible experimental realizations of noise detection are discussed.   

Topological non-Fermi liquids and continuous transitions to Fermi liquids in fractional Chern states

We develop a slave-particle formulation of the Halperin-Lee-Read (HLR) non-Fermi liquid state and extend it to situations without an external magnetic field. We use this formulation to develop a theory of a continuous transition to a Fermi liquid, producing an example in which we can understand how a Fermi surface is continuously destroyed to obtain a fractionalized Fermi liquid. We discuss senses in which the HLR state should be viewed as a topological non-Fermi liquid, and finally we discuss experimental possibilities for inducing such transitions by tuning the bandwidth of a topologically non-trivial bandstructure.