New season

The condensed matter journal club kicked off the new season with a talk by Luke titled Electronic correlations in iron superconductors: insight from optical spectroscopy. Stay tuned for a summary of Luke's talk.

Igor has also started an email listserv which will be used to post updates. Subscribe to keep track of events.

An organization meeting was held last week where we came up with a list of topics and a schedule.
Join the listserv to get a link to the schedule. Feel free to pick a topic (or add one you're interested in) and choose a date for your talk!

Blogon 1: Schwinger Bosons

Opening talk by Tyler Dodds on June 9th, 2011.

Tyler presented a pedagogical talk on Schwinger Boson's with regards to Heisenberg models. Representing spin in terms of boson operators, together with a constraint, was shown to naturally lead to a gapped Z2 spin liquid, or to a magnetically ordered state if the bosons condense.

Last meeting of the Summer: Bosonization II

Journal club talk by Shunsuke Furukawa, continued from Part I.

Furukawa-san discussed 1d spin chains, in particular, the XXZ model. Using the Jordan-Wigner transformation, the spin model was mapped to a system of interacting fermions. The interacting fermions were, in turn, mapped onto a bosonic field theory. Wrapping up, Furukawa-san discussed his own work in mapping out the phase diagram of the XXZ model.

-----------------------------------------------------------------------

The journal club is adjourned until the summer of 2011, barring exceptionally interesting developments/papers that members might want to discuss.

Of mean-fields and convergences

Journal club talk by guest speaker Ioannis Anapolitanos
from the Department of Mathematics, UofT, on 26 Aug, 2010.

Reference: http://arxiv.org/abs/0904.4514

We had a lively session of condensed matter mathematics, with Ioannis sketching the status of mean-field theories in mathematical physics. He introduced Hartree mean-field theory for bosonic systems systematically, drawing an analogy to centre of mass motion in classical mechanics.

He then discussed the convergence of expectation values of general p-particle observables in Hartree mean field theory. At the end, he briefly outlined his own work which has improved the bound.

There were interesting discussions about fermionic systems and mixed density matrices.

Introduction to Bosonization

Journal club talk by Shunsuke Furukawa on August 19, 2010

References:
D. Senechal, cond-mat/9908262
F. D. M. Haldane, Phys. Rev. B 25, 4925 (1982).
K. Nomura and K. Okamoto, J. Phys. A 27, 5773 (1994)

Furukawa-san first introduced the idea of bosonization, broadly discussing notions of effective field theory and Luttinger parameter. He then demonstrated the correspondence between a free-fermion-model and a 1d bosonic field theory.
Coming up in Part II: Applications to interacting fermions and spin chains.

Nematicity in the Pseudogap Phase of the Cuprates

Journal club talk by Christoph M. Puetter on August 12, 2010

Reference:
M. Lawler et al., Nature 466, 347, (2010)
(http://www.nature.com/nature/journal/v466/n7304/full/nature09169.html)

Christoph chose this paper, written by the founder of this Journal Club, for an interesting discussion this week. There were broad discussions on the Cuprate phase diagram, nematicity and STM measurements. A lively debate ensued about the key finding of the paper that the nematic order tracks the pseudogap.

Mechanics of energy transfer in light-harvesting antenna proteins: scale-dependent descriptions

Journal club talk by guest speaker Hoda-Hossein Nejad from the Department of Chemistry, UofT, on August 5, 2010

Abstract:
Recent observations of persistent coherence in light-harvesting
antenna proteins pose some interesting questions regarding the role of
quantum effects in photosynthesis: Has nature chosen quantum mechanics as
an strategy for survival? How important are the quantum effects, and on
what scale are these effects significant?

It is plausible to argue that a fully quantum treatment captures the
dynamics of energy transfer on the scale of a few chromophores, but is
both unnecessary and unrealistic for descriptions of long-range energy
transfer between multiple proteins. In this talk,, I will present two
different formulations of the exciton transport problem as applied to the
photosynthetic light-harvesting complex PE545: 1) A fully quantum
description on the scale of a single protein, which takes into account the
interference between pathways leading to a given site. 2) A mixed quantum
classical treatment on the scale of multiple proteins, in which energy
transfer is regarded as a stochastic hop between delocalized eigenstates.

Inverse Spin Galvanic Effect in the Interface between a Topological Insulator and a Ferromagnet

Journal club talk by William Witczak-Krempa on July 29, 2010

References:
I. Garate, M. Franz, Phys. Rev. Lett. 104, 146802 (2010)
http://prl.aps.org/abstract/PRL/v104/i14/e146802
arXiv: http://arxiv.org/abs/0911.0106

William went over a recent PRL paper by I. Garate and M. Franz [1], in
which they explore the behaviour of a thin ferromagnetic layer coated on the surface of a 3d topological insulator. They suggest the
possibility of dissipationless current-induced magnetization reversal in the ferromagnet, which might have applications in the field of spintronics.

All you wanted to know about photonic crystals but were afraid to ask

Journal club talk by Wah-Tung Lau on July 22, 2010

The slides can be found here.

References:
[1] S. John, Phys. Rev. Lett. 58 2486 (1987).
[2] S. John, "Localization of Light", Physics Today 44 32 (May 1991).
[3] S. Fan et. al., Phys. Rev. Lett. 78 3294 (1997).
[4] S. Johnson et. al., Optic Express 9 748 (2001).
[5] W. T. Lau et. al., Appl. Phys. Lett. 92 103106 (2008).
[6] Experiments: Prof. S. G. Johnson's slides at
http://ab-initio.mit.edu/photons/tutorial/L3-fab.pdf

Spin Liquid state of the S=1/2 Heisenberg model on the anisotropic triangular lattice

Journal Club talk by Dariush Heidarian on July 8, 2010
(click here for slides in pdf format)

Summary by Vijay Shankar Venkataraman

References:

Theoretical Papers:
1) D. Heidarian, S. Sorella, and F. Becca, Phys. Rev. B 80, 012404 (2009)
2) T. Pardini and R. R. P. Singh, Phys. Rev. B 77, 214433 (2008)
3) O. A. Starykh and L. Balents, Phys. Rev. Lett. 98, 077205 (2007)
4) S. Yunoki and S. Sorella, Phys. Rev. B 74, 014408 (2006)
5) Zheng Weihong, R. H. McKenzie, and R. P. Singh, Phys. Rev. B 59, 14367 (1999)
6) M. Bocquet, F. H. L. Essler, A. M. Tsvelik, and A. O. Gogolin, Phys. Rev. B 64, 094425 (2001).

Experimental Papers:
1) R. Coldea, D. A. Tennant, A. M. Tsvelik, and Z. Tylczynski, Phys. Rev. Lett. 86, 1335 (2001)
2) Y. Shimizu, K. Miyagawa, K. Kanoda, M. Maesato, and G. Saito, Phys. Rev. Lett. 91, 107001 (2003).

The existence of spin-liquids is still mired in controversy although there are various promising candidates. Prominent among these is a quasi-two-dimensional class of organic compounds comprised of dimers of an organic molecule. A single electron is localized on each dimer and the compound forms layers of triangular lattices.

Dariush began with a quick review of the experimental results of Shimizu et al where these materials show no sign of long range order down to very low temperatures. He then introduced the anisotropic Heisenberg model on the triangular lattice with nearest-neighbour interactions but with exchange anisotropy. This led to a vigorous discussion on the possible types (Ferromagnetic/Antiferromagnetic) and strengths of coupling constants that would cause frustration. After the audience reached a consensus, Dariush gave a brief survey of previous theoretical work on the model.

Dariush moved on to discussing his own work studying the Hamiltonian by using a variational wavefunction approach. After discussing different methods of doing a variational calculation, he showed us the different variational wavefunctions that he used in his work. Dariush then presented his results for the ground state energy and compared it with previous work.

Discussing results for the spin-spin correlation function as a function of anisotropy for different lattice sizes, he showed how the structure factor scales as a function of lattice size for different anisotropies. Dariush then showed us the phase diagram as a function of anisotropy which told us that the variational calculation points towards the existence of a stable spin-liquid state in the anisotropic triangular lattice.

Time had run out by this point and an excursion into the technicalities of the variational Monte Carlo calculation was reserved for another journal club meeting.

Imaging the Fano lattice to hidden order transition in URu_2Si_2

Journal club talk by Fazel Fallah Tafti on July 15, 2010

Reference:
Nature, 465, 570

Quantum spin Hall effect: Dirac particle approach

Journal club talk by Bohm-Jung Yang on June 17, 2010
(Slides can be found here)

References:
(1) "The anomalous hall effect and magnetic monopoles in momentum
space", Z. Fang et al., Science, 302, 92 (2003)
(2) "Quantum spin hall effect and topological phase transition in HgTe
Quantum wells", B. Andrei Bernevig et al., Science, 314, 1757 (2006)
(3) "Topological quantization of the spin Hall effect in two-dimensional
paramagnetic semiconductors", X. -L. Qi et al, PRB, 74, 085308 (2006)
(4) "Topological insulators", M. Z. Hasan et al., Arxiv: 1002.3895

Charge Density Waves and Superconductivity in 2H-NbSe_2

Journal club talk by Igor Fridman on June 3, 2010
(click here for slides in pdf format)
Summary by William Witczak-Krempa

References

[1] Kiss et al., Nat. Phys. 3, 720 (2007)

http://dx.doi.org/10.1038/nphys699

[2] Borisenko et al., PRL 102, 166402 (2009)

http://dx.doi.org/10.1103/PhysRevLett.102.166402

[3] Johannes et al., PRB 73, 205102 (2006)

http://dx.doi.org/10.1103/PhysRevB.73.205102

The interplay between density wave and superconducting order has been a

subject of intense interest in systems ranging from the cuprates to the iron-pnictides. Along this line, Igor discussed the coexistence of charge density waves (CDWs) and superconductivity (SC) in the material 2H-NbSe_2. The main point was the comparison of two recent ARPES studies [1,2] that differ in their conclusions about the interplay of the two aforementioned orders. Although NbSe_2 has been long known to host both CDW and SC, the nature of the relationship between the condensates and the mechanism responsible for CDW remain under debate.

An introduction concerning 2H-NbSe_2 was first given. We learned that it is a layered material with a two-layer periodicity. It becomes superconducting below T_c = 7.2 K, whereas the CDW order appears around T_CDW ~ 33 K. A DFT study by Johannes et al. [3] demonstrated that the Fermi surface (FS) is composed of three bands: two of them two-dimensional and one three-dimensional. See the slides or [3] for the detailed structure. The superconductivity, of s-wave type, exists on on all bands but the SC gap is not isotropic as established for e.g. by [1,2]. The H_c2 anisotropy is 3 with H_c2^c = 5 T and H_c2^ab = 15 T.

To explain the CDW order, the simple Peierls mechanism was argued to be too naive. In the absence of a reliable microscopic model, a phenomenological approach is usually taken. We were told that there are two main candidates: some argue that the FS nesting leads to an enhancement of the charge susceptibility at the hot-spots. Others propose that saddle points at the Fermi energy can be unstable against CDW formation. (Saddle points are van Hove singularities and lead to an enhanced density of states.) The arena was ready for the fight of the ARPES groups.

The first group/contender, Kiss et al. [1], have conducted ARPES measurements across both the CDW and SC transitions. Their results favour the saddle-point explanation: they found that there is CDW-induced spectral-weight depletion at K-points, which evolves into the largest SC gaps. These gaps also exhibit the highest electron-phonon coupling and Fermi velocity. They concluded, against the prevailing view, that the charge order enhances SC in this system. Love, not war...

In the other corner, the ARPES collaboration of Borisenko et al. [2], reached different conclusions. First, they claim to have the first direct observation of the CDW gap, which opens in regions connected by CDW vectors, hence suggesting that the nesting mechanism is at work and not the saddle-point one. To invalidate the latter they pointed out that according to their data the CDW vectors are too short to connect the saddle-points along the \Gamma-K line, instead they connect Fermi “arcs” on the K-M line! They make the additional claim that there is a CDW pseudogap: in analogy with the cuprates the CDW gap persists in the normal state. The gap was claimed to increase with temperature, which caused some agitation in the crowd. Cookies were immediately distributed to reinstate order. It should be noted that the band structure obtained by [2] agrees very well with the first principles calculation of Johannes et al. [3]. Finally, and most importantly, the results of [2] lead to the conclusion that SC competes against the CDW order, instead of helping it.

Although the data of [2] seemed more reliable, the debate is not closed. Solid experimental proofs are needed as well as a better understanding of the microscopic mechanism on the theoretical side.The marathon continues.

Mean field theory for spin liquids: Large-N slave-fermion approach

Journal club talk by Jeffrey Rau on May 27, 2010
Summary to be posted soon.

References:

* L. Balents, Nature 464, 191 (2010)
-A nice survey of recent theoretical and experimental results for spin liquids

* Affleck and Marston, PRB 37,3774 (longer version in PRB 39,11538)
* Quantum Field Theory of Many-Body Systems, X.G. Wen, Chapter Nine
-Reasonably pedagogical sources

* Read and Sachdev, Nucl. Phys. B316, 609
-Not so pedagogical sources

Experimental Evidence for Spin Liquid States

Journal Club talk by Andrea Lupascu
on May 20, 2010 (click here for ppt file)
Summary by R. Ganesh


References
Yamashita et al.: Nature Physics, 4, 459 (2008)
Nakatsuji et al.: Science, 309, 1697 (2005)
Helton et al., PRL 98, 107204 (2007)

Andrea presented a crisp and topical journal club talk on Quantum Spin Liquids. After a broad defintion, she listed essential features of the QSL states. While there is no conclusive experimental signature, there are many experimental hints. She discussed candidate QSL states which all have antiferromagnetic correlations seen from negative Curie Weiss temperatures. All candidates are also Mott insulators (the candidates from the organic family are weakly Mott
insulating).

Taking two prominent examples - Herbertsmithite with a Kagome lattice structure and an organic salt with a triangular lattice structure, Andrea discussed experimental pointers to a spin liquid ground state. In particular, she showed susceptibility and heat capacity data as a function of temperature. These quantities show no sign of ordering and point to the existence of a Fermi surface, even though the systems are Mott insulators as seen from resistivity. This Fermi surface is conjectured to be associated with spinon excitations of a spin liquid ground state.

With this background, she presented her own results on an organic salt in which Mn ions form a 'star' lattice. Its large value of the \gamma coefficient makes it a promising candidate for a QSL. This led to copious and enjoyable discussion on issues such as the lattice structure, the phonon contribution to specific heat, other probes and possibility of ordering at lower temperatures.

Introduction to topological insulators

Journal Club Talk by Ting Pong Choy on 13th May 2010
Summary by Fazel Fallah Tafti

References:
Nature 452, 970-974 (2008)
Phys. Rev. Lett. 98, 106803 (2007)
Phys. Rev. B 74, 195312 (2006)
Phys. Rev. Lett. 95, 146802 (2005)


Ting Pong introduced two model topological insulators. The first model was a simple two band honey comb lattice with a Hamiltonian consisting of a NN hopping term and an "effective" spin-orbit NNN hopping term. This Hamiltonian can be decoupled in k space and the eigenvalues may be studied in two cases; (a) if the spin-orbit coupling is zero it simply describes the tight binding Graphene system with two Dirac points related through the inversion symmetry (b) if the spin-orbit coupling is non-zero a gap opens at both Dirac points with each band being doubly degenerate with spin up and spin down states. However due to very small spin-orbit coupling in C atoms, this is mostly a theoretical toy model with no experimental realization.

The second model is a Cd(Hg)Te quantum well system composed of a thin HgTe slab sandwiched between two thick CdTe layers. The band structure of the two compounds are inverted with respect to each other. The band structure of the quantum well is similar to CdTe in the thin regime but once the thickness of the HgTe is raised above some critical value (6 nm) the bands are inverted and at d=dc the gap must close. This material has been experimentally tested by measuring the hall resistivity of the quantum well. Ting Pong showed that the Hamiltonian of this system is identical to the toy Graphene model mentioned above in small momentum limit.

In the second half of the discussion, a robust definition of the topological insulator was given based on the spin Chern number assuming spin being conserved. The spin Chern number is always an integer and it is calculated by integrating the curl of a Berry phase term in k-space. This number can be calculated for each occupied state and the sum over all the occupied states gives the total spin Chern number which is proved to always be an integer. This number is even for a conventional band insulator and odd for a topological band insulator. A model calculation was done to show how one can derive the Chern number using the model Hamiltonian which preserves the time reversal symmetry. Time reversal symmetry was introduced as iSy.K where K is the complex conjugate operator. The calculation was done through a mapping from kx,ky space into theta,phi space using a Jacobian. The "topology" comes from the way we define this mapping.

Dynamics of entanglement in two coupled qubits

by Vivek M. Aji, Joel E. Moore

Journal club talk by Asma Al-Qasimi
Summary by So Takei



Today's discussion was led by Asma Al-Qasimi from quantum
optics who has recently joined our journal club. She
presented an article by Vivek Aji and Joel Moore which
discusses the time-evolution of entanglement between two
interacting qubits each coupled to a dissipative bath
(arXiv:cond-mat/0312145). Asma began by identifying
entanglement as the defining difference between a quantum
and a classical computer. She then stressed the difficulty
in preserving entanglement and its extreme fragility to
various sources of decoherence. She emphasized the particular
importance of considering decoherence effects in solid state
qubits because they are always subject to interactions with
a multitude of quantum degrees of freedom. Indeed, solid state
qubits generally possess much shorter decoherence times relative
to qubits implemented in, for example, cavity quantum
electrodynamics and ion traps. The speaker then underlined the
main topic of the article: the time evolution of an entangled
state of two solid state qubits in the presence of dissipation.

Asma proceeded by introducing the solid state qubit considered
in the article called the flux qubit, or the persistent current
qubit. This qubit consists of a superconducting loop with typically
three Josephson junctions that encloses a flux supplied by an
external magnetic field. By appropriately tuning the three
Josephson coupling energies and the magnetic field, the effective
low-energy physics of the system is engineered to possess two
metastable states, |0> and |1> with opposite circulating persistent
current. Tunnel splitting between the two states is determined by the
offset from (1/2)Phi_0 of the flux, where Phi_0 is the superconducting
flux quantum.

The article mutually couples two flux qubits via a Heisenberg-type
coupling of strength J. Assuming that the source of dissipation
arises from the noise in the external flux, the article couples
the z-component of the pseudospin to a bath of oscillators, which
models the environment. The primary motivation of the work is
to investigate the two qubit system with entanglement of formation
as a diagnostic. At t=0, the qubits are prepared in an entangled
state |3> in Eq.2.

The speaker then presented three main results from the work.
In the first case with J=0, the work reveals an interesting
speed up in the decay rate of entanglement relative to the
decoherence rate of individual qubits. The speed up factor
is explicitly derived to be 2/log(2). The decoherence rate is
found to be linearly proportional to temperature and the
strength of dissipation. A question was asked whether the
decay rate of entanglement is generally faster than the
decoherence rate. The speaker supported the generality of the
result by alluding to her previous studies of entanglement.

Second, she discussed the time evolution of entanglement for
a fixed temperature T and various J's. The work finds that
for larger J entanglement is completely lost in finite time.
This result seemingly implies that J encourages loss of
entanglement. However, an intriguing revival of entanglement
for large J is observed for long times. A question was asked
whether coherence also behaves in a similar oscillatory fashion.
The answer was yes. However, it was noted during the discussion
that coherence is not necessarily completely lost even at
times where entanglement is zero.

Finally, the presenter discussed the dynamics of entanglement
for a fixed J and various T's. Here, the dependence for the
second case is reproduced but for higher temperatures the long
time decay increases rapidly while the initial loss of
entanglement is hardly affected. A question was raised whether
qubits with Ising symmetry were considered. The answer was yes;
in the Ising case, no revival in entanglement is observed for
a fixed J and various T's.

Detection of geometric phases in superconducting nanocircuits

by Giuseppe Falci, Rosario Fazio, G. Massimo Palma, Jens Siewert and Vlatko Vedral

Journal club talk by Nakyeon Hwang
Summary by Patrick Morales



This week's talk, given by Nakyeon, was on a proposed experimental setup to detect geometrical phases in a superconducting device. The proposed device is an asymmetric SQUID where the thickness of the tunneling barrier is different in the two arms of the SQUID. The quantum interference due to the geometrical phases could be detected by measuring the charge state of the asymmetric SQUID as the Hamiltonian is adiabatically evolved by varying the offset charge and the flux through the SQUID in a cyclical fashion. Quantum interferometery based on these geometric phases can be used to develop a new design of gates for quantum computation using charge qubits.

When a quantum mechanical system, is evolved adiabatically such that the phase and the amplitude of the wave function describing the system is varied in a cyclical manner, the resulting wave function may differ from the original wave function by a phase factor. This geometrical phase, or Berry phase, results from the geometrical properties of the parameter space of the Hamiltonian. One example of this is the Aharonov-Bohm phase picked up from a charged particle encircling a magnetic field. The Aharonov-Bohm effect results in a periodic modulation of the critical current of a conventional SQUID as a function of magnetic field and provides evidence of macroscopic phase coherence of the superconducting condensate.

The Hamiltonian of an asymmetric SQUID operating in the charging regime, where the temperature is lower than the Josephson coupling energies of the junctions which in turn is much smaller than the charging energy of the SQUID contains two terms: one related to the charge of the SQUID and the other to the flux through it. The Hamiltonian can be swept through its phase space by applying a voltage across the SQUID, varying the charge of the SQUID, and by applying a perpendicular magnetic field to vary the flux through the SQUID. A non-trivial loop in phase space produces a Berry phase due to the asymmetry of the SQUID. The dynamical component of the phase can be subtracted by inverting the state of the SQUID and retracing the same path in phase space in the reverse direction. The resulting phase difference is due only to the geometrical component of the phase.

Ultimately, the geometric phase in an asymmetric SQUID could be used in the design of gates for quantum computation. Two capacitively coupled asymmetric SQUIDS could be used to create a universal two qubit gate. The gate voltages and the magnetic fluxes of each SQUID could be set independently. The effective charging of the target qubit would depend on the state of the control qubit, resulting in a controllable phase shift of the target qubit.

Quantum critical behaviour in the superfluid density of strongly underdoped ultrathin copper oxide films

by Iulian Hetel, Thomas R. Lemberger & Mohit Randeria

Journal club talk by Ganesh Ramachandran
Summary by Brandon Ramakko



This week's talk by Ganesh focused on experimental results for thin films of underdoped YBCO. To begin the discussion, he briefly explained the temperature-doping phase diagram for this HTSC. The transition in the overdoped regime of the superconducting dome can be explained by a mean-field transition but it is not clear what causes the transtion in the underdoped regime. This paper aims to explain the transition in this regime using their experimental results.

The speaker then discussed a plot of superfluid density versus temperature for Helium films. The superfluid density is constant and drops discontinuously to zero at Tc. The size of this drop increases linearly. This is characteristic of a 2D Kosterlitz-Thouless-Berezinski transition. So if the plot of superfluid density for YBCO has the same bahaviour, you could conclude that it is indeed a 2D Kosterlitz-Thouless-Berezinski transition.

The speaker went on to explain the two-coil mutual-inductance method used by the authors. The sample is placed between two coils and an AC current running through the first coil produces a flux which induces an EMF in the sample which drives a current. The flux due to the current in the sample goes through the second coil and by measuring the voltage in the second coil you can calculate the conductivity. Using London's equations you can get the superfluid density.

The speaker then discusses the experiental plot of superfluid density versus temperature. There is no discontinuous drop as expected in a Kosterlitz-Thouless-Berezinski transition. The drop is probably smoothed out due to complexities such as the measurement being taken at a frequency of 50 kHz, inhomogeneities, vortex pinning and possible new physics. A line was added that seems to go through each curve where the density starts to decrease quickly. This seemed to be consistent over a range of doping values. The Tc values were determined from this fit.

It was asked how thick the layers used were and the speaker replied that the thinnest sample was 2 layers thick. It was also asked why the behaviour of the density for T less than Tc was quadratic when it should be linear. This difference in behaviour might be due to disorder or the frequency of 50 kHz (intead of 0) being used.

The speaker then discussed a plot of Tc versus superfluid density(T=0). The measurements show a linear relationship consistent with a 2D Kosterlitz-Thouless-Berezinski transition for small doping or low temperature. It was asked at what temperature it stops behaving linearly. The speaker aswered that at 10K, Tc behaves like the square root of the density. Josephson scaling near a quantum critical point (QCP) implies that the critical point in the underdoped regime is a 2D QCP. Since the transition is mediated by vortex-antivortex pairs, if these could be suppressed you could have a room temperature superconductor.

Patrick Morales commented on the difficulty of the experiment. Having attemped similar experiments he said that the split coil experiment on a thin film is difficult because you have weak signals with large noise. You need a large homogeneous sample. He mentioned it is extremely difficult to get a nice uniform thin sample because YBCO does not grow well in the underdoped regime and it is very hard to remove oxygen in a homogeneous manner.

Dynamics of a Quantum Phase Transition in a Ferromagnetic Bose-Einstein Condensate

by Bogdan Damski and Wojciech H. Zurek

Journal club talk by Edward Taylor
Summary by Ganesh Ramachandran



Ed chose an interesting paper studying dynamics through a quantum phase transition in a spinor condensate. The discussion brought out some general ideas/features of the dynamics of quantum phase transitions.

Ed gave a lightning introduction to spinor condensates and wrote down the energy functional given in the paper. To put it in context, he described previous experimental study of the ferromagnetic-polar transition by Sadler et al.The phase diagram is known to have ferromagnetic, polar and normal(non-condensed) phases. Experiments have probed the dynamics as well as the detailed domain structure formed. The phase diagram is known to have ferromagnetic, polar and normal(non-condensed) phases.

The Hamiltonian possesses three Bogoliubov modes. In the polar phase, one is gapless, corresponding to the broken U(1) symmetry of the BEC. In the ferromagnet, broken U(1) and rotational symmetries give 2 gapless modes. The only energy scale then, is $\Delta$, the excitation gap of the third mode. As we move away from the critical point where even this mode is gapless(?), $\Delta$ rises from zero and eventually saturates. This gives us two time scales which should determine the dynamics - the relaxation time $1/\Delta$ and the transition time taken until saturation.

Tuning the rate of increase of magnetic field, we can explore an impulse regime and an adiabatic regime depending on which time scale dominates. The crossover between regimes occurs where the time scales are equal, which should scale as the one-third power of the 'quench time'. The numerical calculations in the paper do give this precise scaling.

Ed drew a typical plot of the z-magnetization as a function of changing magnetic field or time. The plot showed that the order parameter moved away from zero toward the expected equilibrium value, only after a delay. This was identified as the crossover between impulse and adiabatic regimes. The delay thus read off, showed the expected 1/3 scaling except for values close to zero.

Ed wound up with a neat quick summary, only to make way for a brisk discussion. Igor's question prompted a discussion on domain formation. Drawing a parallel to the early universe where the size of structures was limited by the speed of light, Ed brought out that the size of domains was given by the speed of sound. With two soundlike modes, for some reason, it is the slower mode velocity that plays a role. Kibble-Zurek theory predicts a 1/3 scaling for domain size, which has been observed in this paper.

There was a question from Michael which brought out that the above considerations only hold for intermediate time scales, where the transition is non-adiabatic, but slow enough. Michael also pointed out that it would be interesting to explore the normal region just above the critical point.