This study used quantum Monte Carlo simulations to find the critical temperatures and energy densities of the one-dimensional long-range transverse-field Ising model as a function of transverse field strength. They obtained binder cumulants and energy expectation values to identify the critical points. The critical temperatures and energies decreased with increasing transverse field strength due to greater disorder. These results help characterize the phase transition and can guide further study of eigenstate thermalization in this system.

1. Objective
What are the critical temperatures and critical energy
densities of the long-range transverse-field Ising model
in one-dimension?
Critical Temperatures and Energies of the
Long-Range Transverse-Field Ising Model in One-Dimension
Syrian Truong, Keith Fratus, Dr. Mark Srednicki
Department of Physics
University of California, Santa Barbara
Materials & Methods
• C++ based stochastic series expansion quantum Monte Carlo
simulations upon the one-dimensional long-range transverse-field
Ising model were executed.
• Binder cumulants and energies’ expectation values were obtained.
Results
Future Work
• Identified critical energy density locations
allow for more directed samples of points
well within the broken symmetry phase.
• Other important quantities of this model can
be calculated, such as specific heat, transverse
field, and transverse
field susceptibility.
Introduction
• Eigenstate thermalization links expectation
values in energy eigenstates and thermal
values at the same energy.
• Previous work has shown that spontaneous symmetry breaking is
compatible with eigenstate thermalization.
• There is doubt as to whether the observed critical energy locations
and expectation values agree with those found in larger systems.
• Comparisons of those results with those found in a thermodynamic
system are crucial.
References
• A. W. Sandvik, http://physics.bu.edu/~sandvik/programs/ssebasic/ssebasic.html.
• A. W. Sandvik, Physical Review E 68 (5), 056701 (2003).
• K. R. Fratus and M. Srednicki, Physical Review E 92 (4), 040103 (2015).
• M. Srednicki, Physical Review E 50 (2), 888-901 (1994).
• R. Mondaini, K. R. Fratus, M. Srednicki and M. Rigol, Physical Review E 93 (3), 032104 (2016).
Acknowledgements
I would like to thank the McNair Scholars Program, for funding and helping me
navigate the many tasks associated with undergraduate research; Dr. Mark
Srednicki, for your guidance, patience, and for opening the realm of theoretical
physics research to me; and Keith Fratus, for your guidance, patience, and helping
me get into McNair and physics research in the first place.
Conclusions
• Critical temperatures/energy densities lower
as transverse field strength increases.
• As transverse field strength increases, so does
the disorder in the system.
• These critical values identify regions well
within the broken symmetry phase, aiding the
study of eigenstate thermalization and
spontaneous symmetry breaking compatibility
Summary
• Found critical temperatures and critical
energy densities, as a function of transverse
field strength, of the long-range transverse-
field Ising model in one-dimension.
. . .. . .
FIGURE 2. Additional specifications.
Left shows ferromagnetism and right shows
periodic boundary conditions of this model.
FIGURE 1. Representation of
the one-dimensional long-range
transverse-field Ising model.
FIGURE 3. Binder cumulant (left) and energy expectation value (right).
Dashed lines at intersections show critical temperature/energy density.
T = Temperature T = Temperature
Binder
cumulant
quantity
Energy
expectation
value
Critical
temperature
Critical
energy
density
h = Transverse field strength
h = Transverse field strength
FIGURE 4. Critical temperature vs. transverse field strength.
Critical temperature as a function of transverse field strength (h), with respect
to the following Ising model Hamiltonian:
FIGURE 5. Critical energy density vs. transverse field strength.
Critical energy density as a function of transverse field strength (h), with
respect to the following Ising model Hamiltonian:
T = Temperature
FIGURE 6. Specific
Heat plot.
Specific heat as a function
of temperature of various
system sizes at h=1.5.
Binder cumulant intersections found critical temperatures
Critical temperatures identify critical energy densities
N=1024 Sites
N=512 Sites
N=256 Sites
N=128 Sites
N=64 Sites
N=32 Sites
N=16 Sites
Specific
Heat
N=512 Sites
N=256 Sites
N=512 Sites