Shubhayu Chatterjee, Harvard University › sites › default › files › PosterSession...Shubhayu...

2015 Summer School on Emergent Phenomena in Quantum Materials POSTER SESSION

Shubhayu Chatterjee, Harvard University

Probing excitations in insulators by injecting spin-currents Observation of fractional excitations in insulating spin-systems has been a long-sought goal in physics. In spite of promising evidence for observation of spin liquids, the exact nature of possible ground states, and in particular, the presence of a spin-gap is still unclear. Most experiments till this point have focused on thermodynamic measurements. We suggest a transport measurement as an alternate window into the nature of excitations of insulating spin systems. We couple a metal with a non-equilibrium spin-accumulation to an equilibrium insulating spin-system [1], and develop a general formalism to compute the spin-current. We use this to calculate the current into ordered antiferromagnets as well as spin liquids, and note salient features in the spin conductance. [1] Takei et. al. Phys. Rev. B 90, 094408 (2014)

Print Date 7/1/2015 1/31


Jing-Yuan Chen, University of Chicago

Kinetic Theory of Chiral Fermions Chiral kinetic theory is a useful tool for studying chiral fermion ensemble in far-from-equilibrium states. The realization of Lorentz invariance for chiral fermions in the classical limit is highly non-trivial; in particular, it requires the particles to undergo side jump during a collision. We find a non-local collision kernel, as well as a novel contribution to the particle number current, as results of the side jump. We also find an H-function that satisfies Boltzmann's H-theorem. As a demonstration of their use, we find the general equilibrium solution and the chiral vortical effect transport coefficients.

Print Date 7/1/2015 2/31


Xiao Chen, University of Illinois, Urbana Champaign

Many-Body localization phase transition in a Rokhsar-Kivelson type wavefunction We construct a one dimensional many-body wavefunction to study the many-body localization phase transition. The wavefunction is a Rokhsar-Kivelson type wavefunction, in which the weight for the con figurations is associated to a classical spin glass Hamilontian, with a random sign structure to represent a highly excited state. This wavefunction shows two distinct phases, a thermalized phase and a many-body localized phase, and they are characterized by different scaling behaviors of Renyi entropy. For the thermalized phase, there is a regime where the effective temperature is infinitely high and the Renyi entropies with di erent Renyi index all equal to the thermal entropy. Near the phase transition point in the thermal phase, the fluctuation of Renyi entropy is non-Gaussian. The Renyi entropies with different Renyi index have different scaling behavior, suggesting that the effective temperature drops down to a finite value. After entering into the many-body localization, the Renyi entropy saturates to a constant term despite of the random sign structure in the wavefunction.

Print Date 7/1/2015 3/31


Debanjan Chowdhury, Harvard University

Density wave instabilities of fractionalized Fermi liquids

Authors: Debanjan Chowdhury and Subir Sachdev

Recent experiments in the underdoped regime of the hole-doped cuprates have found evidence for an incommensurate charge-density wave (CDW) state. I'll present an analysis of the charge-ordering instabilities in a metal with strong antiferromagnetic fluctuations, where the electronic excitations are coupled to the fractionalized excitations of the fluctuating antiferromagnet. The resulting CDW emerging out of such a fractionalized Fermi-liquid (FL*) has wavevectors of the form (±δ, 0), (0, ±δ), with a predominantly d−wave form-factor, in agreement with experiments on a number of different families of the cuprates. This result sheds new light on the nature of the normal state out of which the CDW emerges.

Print Date 7/1/2015 4/31


Andreas Eberlein, Harvard University

Coexistence of incommensurate magnetism and superconductivity in the two-dimensional Hubbard model We analyze the competition of magnetism and superconductivity in the two-dimensional Hubbard model with a moderate interaction strength, including the possibility of incommensurate spiral magnetic order. Using an unbiased renormalization group approach, we compute magnetic and superconducting order parameters in the ground state. In addition to previously established regions of Neel order coexisting with d-wave superconductivity, the calculations reveal further coexistence regions where superconductivity is accompanied by incommensurate magnetic order.

Print Date 7/1/2015 5/31


Dominic Else, University of California, Santa Barbara

Classifying symmetry-protected topological phases through the anomalous action of the symmetry on the edge. A symmetry-protected topological (SPT) phase is a topological phase of matter which is distinguished from the trivial phase only in the presence of a certain symmetry. In (1+1)-D, it is well known that SPT phases can be identified from the projective action of the symmetry at an edge. The higher-dimensional version of this statement is that there is an obstruction to consistently implementing the edge symmetry in the presence of a boundary. We show that classifying such `anomalous' symmetries on a 1-dimensional edge recovers the known classification of (2+1)-D SPT phases by the third cohomology group H^3(G, U(1)). In spatial dimension greater than two we can also relate anomalous symmetries to higher cohomology groups, provided that they are of a certain special form. Our method also allows us to understand the proposed `supercohomology' classification of fermion systems in (2+1)-D.

Print Date 7/1/2015 6/31


Azar Eyvazov, Cornell University Supercooled Spin Liquid State in Pyrochlore Dy2Ti2O7

A “supercooled” liquid develops when a fluid does not crystallize upon cooling below its ordering temperature. Instead, the microscopic relaxation times diverge so rapidly that, upon further cooling, equilibration eventually becomes impossible and glass formation occurs. Classic supercooled liquids exhibit specific identifiers including microscopic relaxation times diverging on a Vogel–Tammann-Fulcher (VTF) trajectory, a Havriliak–Negami (HN) form for the dielectric function ɛ(ω, T), and a general Kohlrausch–Williams–Watts (KWW) form for time-domain relaxation. Recently, the pyrochlore Dy2Ti2O7 has become of interest because its frustrated magnetic interactions may, in theory, lead to highly exotic magnetic fluids. However, its true magnetic state at low temperatures has proven very difficult to identify unambiguously. We developed high precision, boundary-free magnetization transport techniques based upon toroidal geometries to gain an improved understanding of the time- and frequency-dependent magnetization dynamics of Dy2Ti2O7. A virtually universal HN form for the magnetic susceptibility χ(ω, T), a general KWW form for the real time magnetic relaxation, and a divergence of the microscopic magnetic relaxation rates with the VTF trajectory that we observe in this material all indicate that low-temperature Dy2Ti2O7 exhibits the characteristics of a supercooled magnetic liquid. One implication is that this translationally invariant lattice of strongly correlated spins may be

Print Date 7/1/2015 7/31


evolving toward an unprecedented magnetic glass state, perhaps due to many-body localization of spin. Xu-Gang He, SUNY Stony Brook U & BNL

Topological phase transition of insulators in the complex crystal momentum space The analytic properties of the band structure of the topological insulating system are considered. Specific attentions have been put on the non-analytic transition in the complex momentum space, which is required by a topological phase transition. In the complex space, the band structure has the doubly degenerate point--branch point, belonging to the two successive bands. We will show that the branch point contains essential information of the topological order. For some models, this fundamental relation can be exhibited explicitly by investigating the residue of the reduced Berry curvature around the branch points.

Print Date 7/1/2015 8/31


Biao Huang, The Ohio State University

Spinor BEC on a Cylindrical Surface in Synthetic Gauge Fields In the study of quantum matter, one usually deals with Euclidean space. However, many recent theoretical studies have indicated that fundamental properties of many systems can be revealed by changing their background manifold, like the Wen-Zee shift, the Hall viscosity, and topological orders. Here we discuss the experimental scheme to create a Bose-Einstein condensate (BEC) on a cylindrical surface with synthetic gauge field, and demonstrate the unique properties of this system. It is shown that both the isolated vortex and vortex arrays di er qualitatively from those in a planar surface. As the synthetic field strength changes, there is a sequence of transitions in the pattern of vortex array. These vortices will result in dramatic signatures in time-of-flight measurements and therefore can be revealed easily in experiments.

Print Date 7/1/2015 9/31


Yi-Ting Hsu, Cornell University

Recent observation of superconductivity in the n-doped transition metal dichalcogenides(TMDs) has generated much interest in these atomically thin semiconductors. Recent observation of superconductivity in the n-doped transition metal dichalcogenides(TMDs) has generated much interest in these atomically thin semiconductors. In turns out that the spin-orbit coupling in these materials lifts the spin degeneracy in a particle-hole asymmetric fashion leaving the conduction band nearly spin-degenerate but valence band spin-split. Hence the phase space of possible superconductivities for p-doped system is quite distinct from that for the existing n-doped systems. Specifically we show that p-doped monolayer TMDs can be a topological modulated superconductor whose Cooper pairs carry a finite center-of-mass momentum and have protected gapless edge excitations. For this we use a perturbative renormalization group(RG) treatment to predict the dominant pairing channel given microscopic interactions to be both the electron-phonon coupling and electronic repulsion. We then discuss strategies for materializing and detecting this new state of matter.

Print Date 7/1/2015 10/31


Chung Koo Kim, Brookhaven National Lab

Imaging Dirac-Mass Disorder from Magnetic Dopant-Atoms in the Ferromagnetic Topological Insulator Crx(Bi0.1Sb0.9)2-xTe3

To achieve and utilize the most exotic electronic phenomena predicted for the surface states of 3D topological insulators (TI), it is necessary to open a “Dirac-mass gap” in their spectrum by breaking time-reversal symmetry. Use of magnetic dopant atoms to generate a ferromagnetic state is the most widely used approach. But it is unknown how the spatial arrangements of the magnetic dopant atoms influence the Dirac-mass gap at the atomic scale or, conversely, whether the ferromagnetic interactions between dopant atoms are influenced by the topological surface states. Here we image the locations of the magnetic (Cr) dopant atoms in the ferromagnetic TI Cr0.08(Bi0.1Sb0.9)1.92Te3. Simultaneous visualization of the Dirac-mass gap (r) Δ

reveals its intense disorder, which we demonstrate directly is related to fluctuations in n(r), the Cr atom areal density in the termination layer. We find the relationship of surface-state Fermi wavevectors to the anisotropic structure of (r) consistent with predictions for surface ferromagnetism Δ

mediated by those states. Moreover, despite the intense Dirac-mass disorder,

the anticipated relationship is confirmed throughout, and exhibits an electron-dopant interaction energy J*=145 meV•nm2. These observations reveal how magnetic dopant atoms actually generate the TI mass gap locally and that, to achieve the novel physics expected of

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time-reversal-symmetry breaking TI materials, control of the resulting Dirac-mass gap disorder will be essential.

Print Date 7/1/2015 12/31


Junhyun Lee, Harvard University

Wess-Zumino-Witten Terms in Graphene Landau Levels We consider the interplay between the antiferromagnetic and Kekulé valence bond solid orderings in the zero energy Landau levels of neutral monolayer and bilayer graphene. We establish the presence of Wess-Zumino-Witten terms between these orders and present two independent proofs. This topological field theory for the two order parameters implies that their quantum fluctuations are described by the deconfined critical theories of quantum spin systems. We also present implications for experiments, including the possible presence of excitonic superfluidity in bilayer graphene.

Print Date 7/1/2015 13/31


Kyungmin Lee, Cornell University

Reconciling the existence of cold spots with short range charge order Recent advances in experiments established short range orders associated with tendencies for spatial symmetry breaking as universal phenomena of underdoped cuprates. This brings the question of the relationship between these orders and superconductivity to the forefront of the study of high Tc superconductivity. Especially, the coexistence of such short range orders with highly homogeneous electronic structure close to the Fermi energy demands explanation. Here we study this issue paying special attention to the role form-factors play. Using both non-self-consistent and self-consistent Bogoliubov-de Gennes equations with real-space realization of short range order, we investigate how the short ranged order affect the electronic structure as well as superconducting tendencies. Typically an inhomogeneous potential due to short-ranged ordering patterns will act as a scatterer that is detrimental to unconventional superconductor which is not protected through Anderson's theorem. However we find that that form factor of the short ranged ordering form can make consequential differences in the way short-range order interact with superconductivity.

Print Date 7/1/2015 14/31


Shuo Mi, Leiden University

Fake Majorana zero-modes in a trivial superconducting quantum dot Abstract: The study of Majorana zero-modes in topological systems has become a hot spot in recently. Here we report a special kind of zero-mode in trivial superconducting quantum dots in Class D with broken spin-rotation symmetry: Y-shape Andreev resonances, which we called Fake Majorana zero-modes. They are the poles of the scattering matrix which should to be distinguished with X-shape resonances profiles, and featured with unquantized zero-bias peak in the differential conductance curve. Further, we studied the evoloution of the system with respect of an external parameter, say magnetic field or electrical gate voltage. And we find out the merging and spliting process of the Fake Majorana modes during this evolution. We argue that this new kind of zero-modes should be distinguish with the real Majorana zeor-modes. And in furure, we hope to realized Majorana emitter out of these Fake modes. Reference: JETP 119, 1018-1027 (2014)

Print Date 7/1/2015 15/31


Masaya Nakagawa, Kyoto University Laser-induced Kondo effect in ultracold alkaline-earth fermions

Kondo effect is a ubiquitous phenomenon in condensed matter physics. For example, the Kondo effect plays a central role in many heavy-fermion materials, leading to plethora of intriguing phenomena such as quantum criticality associated with the transition from the Kondo state to the RKKY magnetism. In this presentation, we propose a novel all-optical scheme to realize the Kondo effect in ultracold atoms. Motivated by recent development of manipulation techniques of ultracold alkaline-earth atoms [1-4], we consider a two-orbital model of ultracold SU(N) fermions coupled with external laser [5]. We show that optical transitions between the two orbitals induce effective hybridization between them, and result in the emergence of the Kondo effect under the laser field. The heavy-fermion liquid realized by the laser shows intriguing behavior which is different from the usual SU(N) Kondo effect, since the laser field strongly couples with the spin degrees of freedom. We demonstrate that the laser-induced Kondo state leads to spin-selective renormalization of effective masses, and also exhibits peculiar dependence on the polarization of the light. Finally, we discuss the competition between the laser-induced Kondo effect and other Kondo screening channels which come from the interatomic exchange interactions. [1] A. V. Gorshkov et al., Nat. Phys. 6, 289 (2010). [2] G. Cappellini et al., Phys. Rev. Lett. 113, 120402 (2014). [3] F. Scazza et al., Nat. Phys. 10, 779 (2014). [4] X. Zhang et al., Science 345, 1467 (2014).

[5] M. Nakagawa and N. Kawakami, arXiv:1506.02947

Print Date 7/1/2015 16/31


Laimei Nie, Stanford University Fluctuating orders and quenched randomness in the cuprates Authors: Laimei Nie, Lauren E. Hayward Sierens, Roger G. Melko, Subir Sachdev, Steven A. Kivelson

We study a quasi-2D classical Landau-Ginzburg-Wilson effective field theory in the presence of quenched disorder in which incommensurate charge-density wave and superconducting orders are intertwined. The disorder precludes long-range charge-density wave order, but not superconducting or nematic order. We select three representative sets of input parameters and compute the corresponding charge-density wave structure factors using both large-N techniques and classical Monte Carlo simulations. Where nematicity and superconductivity coexist at low temperature, the peak height of the charge-density wave structure factor decreases monotonically as a function of increasing temperature, unlike what is seen in X-ray experiments on YBa2Cu3O6+x. Conversely, where the thermal evolution of the charge-density wave structure factor qualitatively agrees with experiments, the nematic correlation length, computed to one-loop order, is shorter than the charge-density wave correlation length.

Print Date 7/1/2015 17/31


Aavishkar Patel, Harvard University

Hyperscaling violation at the onset of spin density wave order in two-dimensional metals

The hyperscaling property implies that spatially isotropic critical quantum states in d spatial dimensions have a specific heat which scales with temperature as Td/z, and an optical conductivity which scales with frequency as ω(d−2)/z for ω≫ T , where z is the dynamic critical exponent. We examine the spin-density- wave critical fixed point found by Sur and Lee, and find that it violates hyperscaling in d = 2, with a specific heat ∼T1/z, and an optical conductivity ∼ω−1/z, with subdominant logarithmic corrections. These hyperscaling violating scaling laws are the same as those found previously at the onset of Ising- nematic order in metals. However, unlike the Ising-nematic critical point, the spin density wave critical point also has a contribution to the DC conductivity ∼T−1/z (up to logarithms): this is a consequence of the particle-hole symmetric nature of the critical theory at “hot-spots” on the Fermi surface. The remaining “cold” Fermi surface yields an additive contribution to the DC conductivity, which can be described by memory function methods (Phys. Rev. B 90, 165146 (2014)).

Print Date 7/1/2015 18/31


Srinidhi Ramamurthy, University of Illinois, Urbana Champaign

Electromagnetic response of line-node semimetals.

Abstract: Topological semimetals are gapless states of matter which have robust surface states and interesting electromagnetic responses. We consider the electromagnetic response of a novel gapless phase in 3+1-dimensions which has a line-node and show that a two form $B_{\mu\nu}$ emerges in the parity odd part of its effective response. The two form gives us the polarization/magnetization of the sample. We also show that $\mathcal{TI}$ symmetry ensures the robustness of the phase.

Print Date 7/1/2015 19/31


Rahul Sharma, Cornell University

Visualizing Topological Superconductivity Topological condensates have been known for almost half a century [1] but finding an electronic version of this state, the topological superconductor, has proven more elusive. A key signature of such a system would be a band of topologically protected surface states inside the superconducting energy gap [2]. Quasiparticle interference imaging (QPI) is a powerful technique that we have developed and used to reveal in-gap band structure of cuprate [3], pnictide [4], and heavy fermion superconductors [5]. In this poster, we discuss the potential for application of our QPI techniques for the study of topological superconductors. Predicted QPI signatures of different types of topological surface states like zero energy flat bands, arc surface states and helical Majorana modes will be discussed [6]. 1. D. D. Osheroff, R. C. Richardson & D. M. Lee Phys. Rev. Lett. 28, 885 (1972) 2. L.Fu and E. Berg, Phys. Rev. Lett. 105, 097001 ( 2010 ) 3. K. Fujita et al, Science 344, 612 (2014) 4. M. P. Allan et al, Science 336, 563 (2012) 5. M. P. Allan et al, Nature Physics 9, 468 (2013) 6. J.S. Hoffman, R. Queiroz and A Schnyder PRB 88 134505, (2013)

Print Date 7/1/2015 20/31


Ken Shiozaki, University of Illinois at

Urbana-Champaign

K-theory classification of topological insulators

I review the K-theory classification of topological crystalline insulators and superconductors, which is topological phases of free fermions with space group symmetry.

The twisted equivariant K-theory provides a calculable and unbiased way to classify topological phases for any symmetry classes including space group, time-reversal, and particle-hole symmetries.

Print Date 7/1/2015 21/31


Hiroaki Sumiyoshi, Kyoto University

Torsional response and quantum anomaly in Weyl semimetal with dislocation

Weyl semimetals are materials which are characterized by one or more pairs of Weyl cones protected by topological numbers. Recently, many efforts are devoted to investigate their physical properties associated with chiral anomaly, which is originally the concept of high-energy physics [1] and is considered to lead to exotic transport phenomena in condensed matter systems, for instance, anomalous Hall effect, chiral magnetic effect, and negative magnetoresistance [2]. Meanwhile, thanks to recent developments in microengineering, many both experimental and theoretical researches on the effect of lattice strain on the electronic property of other topological materials such as graphemes, weak topological insulators, and topological crystalline insulators, have been conducted and revealed many rich behaviors of these systems [3]. This study is focussing on Weyl semimetals with dislocation, and performed with both the quantum field theory with background vielbein and torsion in Cartan formalism, which is a useful tool to describe lattice strain and topological defects [4], and the numerical diagonalization method of tight-binding models. In this study, we obtained a new type of torsional response, named the chiral torsional magnetic effect, which is a current-torsion response. The physical consequence of this effect is the generation of ground-state current along screw or edge dislocation. We numerically confirmed that it exists even in lattice system, in contrast to the chiral magnetic effect in the narrow sense (i.e. ground state current) [5]. I will also discuss the physical reason. [1] For review: D. E. Kharzeev, Progress in Particle and Nuclear Physics 75, 133 (2014) [2] For review: P. Hosur and X. Qi, Comptes Rendus Physique 14, 857 (2013) [3] N. Levy et al., Science 329, 544 (2010) / Y. Ran et al., Nat. Phys. 5, 298 (2009) / E. Tang and L. Fu, Nat. Phys. 10, 964 (2014) [4] H. Kleinert, “Multivalued Fields: in Condensed Matter Electromagnetism, and Gravitation”, World Scientific (2008) [5] M.M. Vazifeh and M. Franz, Phys. Rev. Lett. 111, 027201 (2013)

Print Date 7/1/2015 22/31


Yu-Ting Tam, Brookhaven National Lab

Itinerancy enhanced quantum fluctuation of magnetic moments in iron-based superconductors We investigate the influence of itinerant carriers on dynamics and fluctuation of local moments in Fe-based superconductors, via linear spin-wave analysis of a spin-fermion model containing both itinerant and local degrees of freedom. Surprisingly against the common lore, instead of enhancing the (π,0) order, itinerant carriers with well nested Fermi surfaces is found to introduce significant amount of spatial and temporal quantum fluctuation that leads to the observed small ordered moment. Interestingly, the underlying mechanism is shown to be nesting-associated long-range coupling, rather than the previously believed ferromagnetic double-exchange effect. This challenges the validity of ferromagnetically compensated first-neighbor coupling reported from short-range fitting to the experimental dispersion, which turns out to result instead from the ferro-orbital order that is also found instrumental in stabilizing the magnetic order.

Print Date 7/1/2015 23/31


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Min-Feng Tu, California Institute of Technology

Quantum transport simulation on Topological insulator Quantum transport simulation is a powerful tool to study spintronic system and spin-orbital coupling(SOC) in topological insulator. The interference of electrons contains the information of SOC and disorders in a semiconductor and can be observed by tuning magnetic field. We use this tool to study on engineered topological insulator, especially on graphene of which low energy physics cannot be described by Boltzmann transport theory. We hope to help experimentalists realize nature TI materials and detect its signature through simulation.

Print Date 7/1/2015 25/31


Yuxuan Wang, University of Wisconsin-Madison

Competing orders and symmetry breaking in underdoped cuprates

Recent experiments have provided strong evidence that there exist incommensurate static charge-density-wave (CDW) order with momenta (Q,0) and (0,Q) in underdoped cuprates. In the same doping range and at higher temperatures, there are evidence for broken rotational symmetry and broken time-reversal symmetry. In this talk we argue that magnetically-mediated interaction, which is known to give rise to d-wave superconductivity, can also lead to CDW order. We will then discuss the interplay between different charge order parameters and show that rotational symmetry and time-reversal symmetry are both broken in the ground state. Going beyond mean-field analysis we show that these discrete symmetries indeed get broken at higher temperatures than the CDW onset temperature. In the second part, we show an SU(2) particle-hole symmetry of the model leads to the coexistence of CDW order and a pair-density-wave (PDW) order, the latter defined as a superconducting order with a finite total Cooper pair momentum. The PDW order has been argued to exist in the pseudogap

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region, and we show the coexistence of CDW and PDW explains ARPES data. We make specific predictions for experiments.

Ran Wei, Cornell University

Anomalous topological charge pumping in a one-dimensional optical superlattice Abstract: We model atomic motion in a sliding superlattice potential to explore ``topological charge pumping" and to find optimal parameters for experimental observation of this phenomenon. We analytically study the band-structure, finding how the Wannier states evolve as two sinusoidal lattices are moved relative to one-another, and relate this evolution to the center of mass motion of an atomic cloud. We pay particular attention to counterintuitive or anomalous regimes, such as when the atomic motion is opposite to that of the lattice. We propose a practical cold-atom experiment to detect this anomalous behavior. Through numerical simulations, we find that a negative adiabatic current and a non-trivial Chern number ${\mathcal C}=-1$ is readily measured.

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Jhih-Shif You, University of California, San Diego

Unconventional Bose-Einstein Condensations of Two-component Bosons in the p-orbital Bands in Optical lattice We investigate the unconventional Bose-Einstein condensations (BECs) of two-species mixture with the p-wave symmetry in the second band of a bipartite optical lattice. A new modified imaginary-time propagation method is developed to numerically solve the Gross-Pitaevskii (GP) equation by truncating states in the lowest bands, and can be applicable to even higher orbital bands. Different from single-species case, the two-species boson mixture exhibits two non-equivalent complex BECs: One breaks time-reversal symmetry but one does not, in the dominant intra-species interaction regime. When the inter-species interaction is turned stronger, both states undergo a quantum phase transition at the SU(2) invariant point toward a real-valued checkerboard state with a staggered spin density structure. We also discuss the lattice asymmetry, strong interaction effect and experimental implication.

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Yizhi You, University of Illinois

Geometry theory in Chiral topological phases

We present the theory of geometry in a variety of Chiral topological phases including FQHE, Half-filled Landau Level and Weyl semimetals. By coupling the electron theory with the geometry metric, the effective field theory exhibits exotic coupling between the vielbein and the gauge degree of freedom.

In gapless system such as Half-filled Landau Level, the emergent gauge fluctuation would also contribute to the Odd viscosity response and Wen-Zee coupling.

In addition, by functional bosonization technique, we can dual our fermion theory into a Hydrodynamical gauge theory. We would show that the Hall viscosity response in the fermion theory is related with EM response in the dual gauge theory.

Print Date 7/1/2015 31/31

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