160 research outputs found
The RSB order parameter in finite-dimensional spin glasses: numerical computation at zero temperature
This thesis is focused on the computation of the overlap distribution which characterizes spin glasses with finite connectivity upon an RSB transition at zero temperature.
Two models are considered: the J± Bethe lattice spin glass and the Edwards-Anderson spin glass in three dimensions with random regular bond dilution (random dilution with the constraint of fixed connectivity z = 3). The approach is based on the study of the effects of a bulk perturbation on the energy landscape. In ultrametric spin glasses, the distribution of the excited states is known to be related to the order parameter through a universal formula. This formula is used for deriving the order parameter from the experimental distributions. In addition, the finite-size corrections to the ground state energy are computed for the two models
Unveiling ground state sign structures of frustrated quantum systems via non-glassy Ising models
Identification of phases in many-body quantum states is arguably among the
most important and challenging problems of computational quantum physics. The
non-trivial phase structure of geometrically frustrated or finite-density
electron systems is the main obstacle that severely limits the applicability of
the quantum Monte Carlo, variational, and machine learning methods in many
important cases. In this paper, we focus on studying real-valued signful
ground-state wave functions of several frustrated quantum spins systems. Under
the assumption that the tasks of finding wave function amplitudes and signs can
be separated, we show that the signs of the wave functions are easily
reconstructed with almost perfect accuracy by means of combinatorial
optimization. To this end, we map the problem of finding the wave function sign
structure onto an auxiliary classical Ising model which is defined on the
Hilbert space basis. Although the parental quantum system might be highly
frustrated, we demonstrate that the Ising model does not exhibit significant
frustrations and is solvable with standard optimization algorithms such as
Simulated Annealing. In particular, given the ground state amplitudes, we
reconstruct the signs of the wave functions of a fully-connected random
Heisenberg model and the antiferromagnetic Heisenberg model on the Kagome
lattice, thereby revealing the unelaborated hidden simplicity of many-body sign
structures.Comment: 7 pages, 4 figures in the main text; 5 pages, 2 figures in the
supplemental informatio
Nematic phase in the J-J square lattice Ising model in an external field
The J-J Ising model in the square lattice in the presence of an
external field is studied by two approaches: the Cluster Variation Method (CVM)
and Monte Carlo simulations. The use of the CVM in the square approximation
leads to the presence of a new equilibrium phase, not previously reported for
this model: an Ising-nematic phase, which shows orientational order but not
positional order, between the known stripes and disordered phases. Suitable
order parameters are defined and the phase diagram of the model is obtained.
Monte Carlo simulations are in qualitative agreement with the CVM results,
giving support to the presence of the new Ising-nematic phase. Phase diagrams
in the temperature-external field plane are obtained for selected values of the
parameter which measures the relative strength of the
competing interactions. From the CVM in the square approximation we obtain a
line of second order transitions between the disordered and nematic phases,
while the nematic-stripes phase transitions are found to be of first order. The
Monte Carlo results suggest a line of second order nematic-disordered phase
transitions in agreement with the CVM results. Regarding the stripes-nematic
transitions, the present Monte Carlo results are not precise enough to reach
definite conclusions about the nature of the transitions.Comment: 13 pages, 10 figure
Study of Condensed Matter Systems with Monte Carlo Simulation on Heterogeneous Computing Systems
We study the Edwards-Anderson model on a simple cubic lattice with a finite constant external field. We employ an indicator composed of a ratio of susceptibilities at finite momenta, which was recently proposed to avoid the difficulties of a zero momentum quantity, for capturing the spin glass phase transition. Unfortunately, this new indicator is fairly noisy, so a large pool of samples at low temperature and small external field are needed to generate results with a sufficiently small statistical error for analysis. We thus implement the Monte Carlo method using graphics processing units to drastically speed up the simulation. We confirm previous findings that conventional indicators for the spin glass transition, including the Binder ratio and the correlation length do not show any indication of a transition for rather low temperatures. However, the ratio of spin glass susceptibilities does show crossing behavior, albeit a systematic analysis is beyond the reach of the present data. This reveals the difficulty with current numerical methods and computing capability in studying this problem. One of the fundamental challenges of theoretical condensed matter physics is the accurate solution of quantum impurity models. By taking expansion in the hybridization about an exactly solved local limit, one can formulate a quantum impurity solver. We implement the hybridization expansion quantum impurity solver on Intel Xeon Phi accelerators, and aim to apply this approach on the Dynamic Hubbard Models
Dynamical properties of classical and quantum spin systems
The Kibble-Zurek mechanism (KZM) was originally proposed to describe the evolution and "freezing" of defects in the early universe, but later it was generalized to study other quantum and classical systems driven by a varying parameter. The basic idea behind the KZM is that, as long as the changing rate (velocity) of the parameter is below a certain critical velocity, _crit, the system will remain adiabatic (for isolated quantum systems) or quasi-static (for classical systems with a heat bath).
The nonequilibrium finite-size scaling (FSS) method based on KZM has been exploited systematically. Through applying the scaling hypothesis, we can extract the critical exponents and study the dynamic properties of the system.
In the first few chapters of this dissertation, we discuss the applications of KZM in several classical systems: first, we study the dynamics of 2D and 3D Ising model under a varying temperature as well as a varying magnetic field. Secondly, we examine the classical ℤ₂ gauge model, in which we show that KZM also works for topological phase transitions. Moreover, we also investigate the dynamics of other models with topological ordering only at T=0, where KZM cannot be applied. Lastly, we explore the 2D Ising spin glass with bimodal and gaussian couplings. With bimodal couplings, we find dual time scales associated with the order parameter and the energy correspondingly, while in the gaussian case one unique time scale is involved.
The systems mentioned above are all classical and the dynamics are approached through simulated annealing (SA), in which thermal fluctuations drives systems to explore the energy landscape in finding the ground state. In the last chapter, we explore the efficiency of Quantum Annealing (QA) on a fully-connected spin glass (or Sherington-Kirkpatrick model) with a transverse field. QA is the counterpart of SA, where quantum fluctuations drive the system toward the ground state when the quantum terms are reduced. QA is currently widely explored as a paradigm for quantum computing to solve optimization problems. Here we compare the scaling of the dynamics (with system size) of the fully-connected spin glass through QA versus SA
The Quantum Adiabatic Algorithm applied to random optimization problems: the quantum spin glass perspective
Among various algorithms designed to exploit the specific properties of
quantum computers with respect to classical ones, the quantum adiabatic
algorithm is a versatile proposition to find the minimal value of an arbitrary
cost function (ground state energy). Random optimization problems provide a
natural testbed to compare its efficiency with that of classical algorithms.
These problems correspond to mean field spin glasses that have been extensively
studied in the classical case. This paper reviews recent analytical works that
extended these studies to incorporate the effect of quantum fluctuations, and
presents also some original results in this direction.Comment: 151 pages, 21 figure
Understanding Disordered Systems Through Numerical Simulation and Algorithm Development
Disordered systems arise in many physical contexts. Not all matter is uni-
form, and impurities or heterogeneities can be modeled by fixed random disor-
der. Numerous complex networks also possess fixed disorder, leading to appli-
cations in transportation systems [1], telecommunications [2], social networks
[3, 4], and epidemic modeling [5], to name a few.
Due to their random nature and power law critical behavior, disordered
systems are difficult to study analytically. Numerical simulation can help
overcome this hurdle by allowing for the rapid computation of system states.
In order to get precise statistics and extrapolate to the thermodynamic limit,
large systems must be studied over many realizations. Thus, innovative al-
gorithm development is essential in order reduce memory or running time
requirements of simulations.
This thesis presents a review of disordered systems, as well as a thorough
study of two particular systems through numerical simulation, algorithm de-
velopment and optimization, and careful statistical analysis of scaling proper-
ties.
Chapter 1 provides a thorough overview of disordered systems, the his-
tory of their study in the physics community, and the development of tech-
niques used to study them. Topics of quenched disorder, phase transitions, the
renormalization group, criticality, and scale invariance are discussed. Several
prominent models of disordered systems are also explained. Lastly, analysis
techniques used in studying disordered systems are covered.
In Chapter 2, minimal spanning trees on critical percolation clusters are
studied, motivated in part by an analytic perturbation expansion by Jackson
and Read [6] that I check against numerical calculations. This system has a
direct mapping to the ground state of the strongly disordered spin glass [7].
We compute the path length fractal dimension of these trees in dimensions
d = {2, 3, 4, 5} and find our results to be compatible with the analytic results
suggested by Jackson and Read.
In Chapter 3, the random bond Ising ferromagnet is studied, which is es-
pecially useful since it serves as a prototype for more complicated disordered
systems such as the random field Ising model and spin glasses. We investigate
the effect that changing boundary spins has on the locations of domain walls
in the interior of the random ferromagnet system. We provide an analytic
proof that ground state domain walls in the two dimensional system are de-
composable, and we map these domain walls to a shortest paths problem. By
implementing a multiple-source shortest paths algorithm developed by Philip
Klein [8], we are able to efficiently probe domain wall locations for all possible
configurations of boundary spins. We consider lattices with uncorrelated dis-
order, as well as disorder that is spatially correlated according to a power law.
We present numerical results for the scaling exponent governing the probabil-
ity that a domain wall can be induced that passes through a particular location
in the system’s interior, and we compare these results to previous results on
the directed polymer problem
Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond
We review recent developments in the physics of ultracold atomic and
molecular gases in optical lattices. Such systems are nearly perfect
realisations of various kinds of Hubbard models, and as such may very well
serve to mimic condensed matter phenomena. We show how these systems may be
employed as quantum simulators to answer some challenging open questions of
condensed matter, and even high energy physics. After a short presentation of
the models and the methods of treatment of such systems, we discuss in detail,
which challenges of condensed matter physics can be addressed with (i)
disordered ultracold lattice gases, (ii) frustrated ultracold gases, (iii)
spinor lattice gases, (iv) lattice gases in "artificial" magnetic fields, and,
last but not least, (v) quantum information processing in lattice gases. For
completeness, also some recent progress related to the above topics with
trapped cold gases will be discussed.Comment: Review article. v2: published version, 135 pages, 34 figure
Random Strain Induced Correlations in Materials with Intertwined Nematic and Magnetic Orders
Electronic nematicity is rarely observed as an isolated instability of a correlated electron system. Instead, in iron pnictides and in certain cuprates and heavy-fermion materials, nematicity is intertwined with an underlying spin-stripe or charge-stripe state. As a result, random strain, ubiquitous in any real crystal, creates both random-field disorder for the nematic degrees of freedom and random-bond disorder for the spin or charge ones. Here, we put forward an Ashkin-Teller model with random Baxter fields to capture the dual role of random strain in nematic systems for which nematicity is a composite order arising from a stripe state. Using Monte Carlo to simulate this random Baxter-field model, we find not only the expected breakup of the system into nematic domains, but also the emergence of nontrivial disorder-promoted magnetic correlations. Such correlations enhance and tie up the fluctuations associated with the two degenerate magnetic stripe states from which nematicity arises, leaving characteristic signatures in the spatial profile of the magnetic domains, in the configurational space of the spin variables, and in the magnetic noise spectrum. We discuss possible experimental manifestations of these effects in iron-pnictide superconductors. Our work establishes the random Baxter-field model as a more complete alternative to the random-field Ising model to describe complex electronic nematic phenomena in the presence of disorder
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